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no-repeat 6px;background-position:center;margin-top:10px;min-width:55px;display:none;} @media (min-width:768px){.pzSfk .floats-partial-section .floats-partial-footer .floats-partial-caption .expand_btn{display:none;}} .pzSfk .floats-partial-section .floats-partial-footer .floats-partial-caption .caption-text{-webkit-flex:9;-ms-flex:9;flex:9;text-align:left;padding:21px 37px 10px 0px;font-size:16px;font-weight:700;} @media (min-width:768px){.pzSfk .floats-partial-section .floats-partial-footer .floats-partial-caption .caption-text{padding:19px 18px 10px 18px;}} @media (min-width:992px){.pzSfk .floats-partial-section .floats-partial-footer .floats-partial-caption .caption-text{padding:19px 21px 10px 21px;}} .pzSfk .floats-partial-section .floats-partial-footer .floats-partial-caption 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no-repeat 6px;background-position:center;padding:16px 0 16px 0;min-width:62px;max-width:62px;} @media (min-width:768px){.pzSfk .floats-partial-section .floats-partial-footer .floats-partial-caption .caption-partial-url{padding:16px 0 17px 0;min-width:52px;max-width:52px;}} @media (min-width:992px){.pzSfk .floats-partial-section .floats-partial-footer .floats-partial-caption .caption-partial-url{padding:15px 0 17px 0;min-width:60px;max-width:60px;}} .pzSfk .floats-partial-section .floats-partial-footer .floats-partial-caption .caption-partial-url:hover{-webkit-text-decoration:none;text-decoration:none;} .pzSfk .floats-partial-section .floats-partial-footer .floats-partial-caption-text{font-size:14px;padding:0px 18px 17px 22px;} @media (min-width:768px){.pzSfk .floats-partial-section .floats-partial-footer .floats-partial-caption-text{padding:0px 15px 15px 18px;}} @media (min-width:992px){.pzSfk .floats-partial-section .floats-partial-footer .floats-partial-caption-text{padding:0px 18px 19px 21px;}} .pzSfk .floats-partial-section .floats-partial-footer .floats-partial-caption-text figcaption{margin:0;padding:0;} .pzSfk .floats-partial-section .floats-partial-footer .floats-partial-caption-text svg{max-width:100%;} .pzSfk ol{font-size:16px;padding-left:32px;font-family:STIXGeneral-Regular;line-height:24px;margin:15px 0;} .pzSfk ol li{margin:10px 0;padding-left:10px;word-break:normal;} .pzSfk ol li a{word-break:break-all;} .pzSfk ol li .reflinks{display:block;} .pzSfk ol li .reflinks a{color:#4D8A17;-webkit-text-decoration:none;text-decoration:none;cursor:pointer;} .pzSfk ol li .reflinks a:hover{-webkit-text-decoration:underline;text-decoration:underline;} .pzSfk ol li .reflinks .sep{color:#000000;} .pzSfk .table-wrapper{width:auto;overflow-x:auto;} .pzSfk .table-wrapper::-webkit-scrollbar{-webkit-appearance:none;width:5px;height:5px;} .pzSfk .table-wrapper::-webkit-scrollbar-thumb{border-radius:5px;background-color:rgba(0,0,0,.5);-webkit-box-shadow:0 0 1px rgba(255,255,255,.5);} @media (min-width:768px){.pzSfk .table-wrapper{overflow-x:auto;width:auto;-webkit-break-word:word-break;break-word:word-break;margin:0 auto;}} @media (min-width:992px){.pzSfk .table-wrapper{width:auto;}} @media (min-width:1324px){.pzSfk .table-wrapper{width:auto;}} @media (min-width:320px){.pzSfk{margin-bottom:33px;padding:0;}} @media (min-width:992px){.pzSfk{padding:0;margin-bottom:100px;}} @media (min-width:1325px){.pzSfk{margin-bottom:100px;}} .pzSfk .owl-stage{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;} .pzSfk .floats-partial-middle--thumbs{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;} .pzSfk .floats-partial-middle--thumbs .partial-carousel-dot svg{fill:#4D8A17;} .pzSfk .floats-partial-middle--thumbs .partial-carousel-dot strong{color:#4D8A17;font-weight:500;} .pzSfk .floats-partial-middle--thumbs .partial-carousel-dot.active strong{color:#007B8B;} .pzSfk .floats-partial-middle--thumbs 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height="24"><span class="text">提交</年代pan></span><span class="sc-htoDjs kNoiZu sc-dqBHgY cQgXqt fileArrow icon-colored undefined icon-themeable" title="" font-size="16"></span></a> <div class="sc-krDsej ewCAZX hide__submenu journalSubmenuCheck" id="journalnav__submenu"> <button class="sc-itybZL kdvWPs journalSubmenuCheck"><span class="sc-htoDjs gvAGTV sc-fzsDOv hAFFGk journalSubmenuCheck icon-colored _1CwL9R82TPl2HxDleeYGDR icon-themeable" title="" font-size="9"></span></button> </div> <div class="sc-dTdPqK lkWWFY hide__submenu journal__nav__desktop" id="journalnav__submenudesktop"> <div order="0" class="sc-kUaPvJ ehYbwJ"> <div class="sc-gPWkxV legory"></div> <div class="sc-gPWkxV legory"></div> </div> <div class="sc-jVODtj epgILe"></div> <span class="sc-fONwsr kCfRMJ"></span> </div> </nav> </div> <div class="sc-nrwXf byflLi breadcrumb__wrapper journal_article__BreadCrumbStyle-sc-12fk6zy-2 bjwAQR article"> <a class="sc-htpNat bUhGXt link sc-eitiEO jXeALb breadCrumb" href="//www.newsama.com/journals/jps/" aria-label="Journal of Probability and Statistics">概率统计杂志</A.>/<A.ClA.年代年代="年代C-htpNat bUhGXt link sc-eitiEO jXeALb breadCrumb" href="//www.newsama.com/journals/jps/contents/year/2012/" aria-label="2012">2012</A.>/<年代pan class="sc-bhlBdH UVThf">文章</年代pan> </div> <div class="threeColumn__ThreeColumnWrapper-sc-1bf4078-0 dDyxFF threeColumnWrapper"> <div class="threeColumn__LeftWrapper-sc-1bf4078-1 fkVFlF leftWrapper"> <button type="button" class="sc-eKZiaR dzZrgS section__button " id="sectionButton"><span class="sc-htoDjs diutvq icon__dots dots__detector icon-colored _3n-7Ho7DlfjMlZbrRr9Q4a icon-themeable" title="" font-size="22"></span>文章章节</button> <div class="sc-bEjcJn jLUVhb static_contents__section_links links__hidden "> <div id="leftSectionMenu" class="sc-cbkKFq prcY sc-bGbJRg dVCLgr left_section_menu"> <p>在这一页上</p> <span class="sc-likbZx bBmZJZ grey" property="background-color" color="#D8D8D8"><a class="sc-htpNat bUhGXt link sc-ePZHVD ewHMLz" href="#abstract" aria-label="">摘要</A.></年代pan> <span class="sc-likbZx bBmZJZ grey" property="background-color" color="#D8D8D8"><a class="sc-htpNat bUhGXt link sc-ePZHVD ewHMLz" href="#introduction-and-background" aria-label="">导言和背景</A.></年代pan> <span class="sc-likbZx bBmZJZ grey" property="background-color" color="#D8D8D8"><a class="sc-htpNat bUhGXt link sc-ePZHVD ewHMLz" href="#references" aria-label="">参考文献</A.></年代pan> <span class="sc-likbZx bBmZJZ grey" property="background-color" color="#D8D8D8"><a class="sc-htpNat bUhGXt link sc-ePZHVD ewHMLz" href="#copyright" aria-label="">版权</A.></年代pan> </div> </div> </div> <div class="threeColumn__ContentWrapper-sc-1bf4078-3 bbzREB contentWrapper"> <div class="threeColumn__Content-sc-1bf4078-4 jknsZc content"> <div class="articleContent__ArticleContentsWrapper-sc-2yl9jy-0 eSvjPB article_contents"> <div class="articleContent__ArticleContentsSectionsWrapper-sc-2yl9jy-1 jntdRu article_contents__sections_wrapper"> <div class="articleContent__ArticleWrapper-sc-2yl9jy-2 hAzjHP"> <div class="sc-chbbiW mBGSe article_header_class"> <p class="sc-kZmsYB bkiEfx article_type">研究文章|开放存取</p> <div class="sc-RcBXQ ztvYg article_citation"> 体积 2012 <span class="sc-fZwumE cUsFHx">|</年代pan>物品ID 738636 <span class="sc-fZwumE cUsFHx">|</年代pan> <a class="sc-htpNat bUhGXt link sc-fQejPQ duuwge" href="https://doi.org/10.1155/2012/738636" aria-label="Doi-link">https://doi.org/10.1155/2012/738636</A.></D我v> <div class="sc-LKuAh lcOiDF article_full_citation"> <div class="sc-hzNEM gsmOCv isHide"> <p>梦元徐，克里斯塔B. Cohlmia，Wayne A. Woodward，Henry L. Gray, "<年代pan class="adjust-article-svg-size">G-滤波非平稳时间序列</年代pan>",<i>概率统计杂志</我>, 卷。2012, 物品ID738636, 15   页面, 2012． https://doi.org/10.1155/2012/738636</p> </div> <button class="sc-hSdWYo etauus sc-iBEsjs kPshFw" color="">引证</button> </div> <h1 class="sc-iRbamj jZOxOO sc-iSDuPN gaKwvw article_title adjust-article-svg-size heading-title">G-滤波非平稳时间序列</H1.></D我v> <div class="sc-jrIrqw eHMFfo article_authors"> <hr class="sc-cEvuZC XkrFe hr__author"> <span class="">梦元徐,<年代up>1.</年代up></span> <span class="">克丽斯塔·B·科米亚,<年代up>2.</年代up></span> <span class=""><b>韦恩·a·伍德沃德</b><a class="sc-htpNat bUhGXt link" href="mailto:" aria-label="Mail Option"><span class="sc-htoDjs kNoiZu sc-hjRWVT irNWJI icon-colored _2zt-1j0LmaF8UjTQAM413G icon-themeable" title="" font-size="16"></span></a>,<年代up>3.</年代up> 和</年代pan> <span class="">亨利·l·格雷<年代up>3.</年代up></span> <div class="sc-fHxwqH dMuGGA isHide"> <p><sup>1.</年代up><span>美国北卡罗来纳州三角研究园NIH/NIEHS（国家卫生研究院/国家环境卫生科学研究院）生物统计学分支，美国北卡罗来纳州27709</年代pan></p> <p><sup>2.</年代up><span>美国德克萨斯州奥德萨市奥德萨学院数学系，邮编：79764</年代pan></p> <p><sup>3.</年代up><span>南卫理公会大学统计科学系，德州达拉斯75205</年代pan></p> </div> <button class="sc-hSdWYo etauus sc-iQtOjA kfDwZY" color="">显示更多</button> </div> <div class="articleContent__EditorWrapper-sc-2yl9jy-3 evMkvU"> <div class="articleContent__TextWrapper-sc-2yl9jy-8 fYrLtk"> <b>学术编辑器:</b>周善忠</D我v> <div class="articleContent__PublicationTimeLineWrapper-sc-2yl9jy-4 fjviok"> <div class="articleContent__PublcationCol-sc-2yl9jy-5 hgKieC notPublished"> <span class="articleContent__PublicationField-sc-2yl9jy-6 cIribm">收到了</年代pan> <span class="articleContent__PublicationValue-sc-2yl9jy-7 khHUcu">2011年8月16日</年代pan> </div> <div class="articleContent__PublcationCol-sc-2yl9jy-5 hgKieC notPublished"> <span class="articleContent__PublicationField-sc-2yl9jy-6 cIribm">修改</年代pan> <span class="articleContent__PublicationValue-sc-2yl9jy-7 khHUcu">2011年11月15日</年代pan> </div> <div class="articleContent__PublcationCol-sc-2yl9jy-5 hgKieC notPublished"> <span class="articleContent__PublicationField-sc-2yl9jy-6 cIribm">认可的</年代pan> <span class="articleContent__PublicationValue-sc-2yl9jy-7 khHUcu">2011年11月15日</年代pan> </div> <div class="articleContent__PublcationCol-sc-2yl9jy-5 hgKieC isPublished"> <span class="articleContent__PublicationField-sc-2yl9jy-6 cIribm">发表</年代pan> <span class="articleContent__PublicationValue-sc-2yl9jy-7 khHUcu">2012年3月04</年代pan> </div> </div> </div> <article class="sc-jtRlXQ pzSfk article_body "> <div> <div class="xml-content"> <h4 class="header" id="abstract">摘要</H4.><p>经典线性滤波器可以成功地滤除频率内容不随时间变化的时间序列中的分量，以及具有时变频率（TVF）的非平稳时间序列中的分量不重叠的分量。但是，对于许多类型的非平稳时间序列，TVF分量往往在时间上重叠。在这种情况下，经典的线性滤波方法无法从原始过程中提取分量。在本文中，我们介绍并从理论上发展了基于时间变形技术的G滤波器。仿真实例和实际蝙蝠回波定位实例表明，G滤波器可以成功地滤除TVF分量随时间重叠的G平稳过程。</p> <span class="end-abs"></span> <h4 id="introduction-and-background">1.导言和背景</H4.><p>本文针对具有时变频率(TVF)行为的非平稳过程G-stationary process开发了滤波器。我们从线性滤波器的简单讨论开始，因为我们在这里发展的滤波器是线性滤波器的推广。传统的线性滤波器定义为<年代vg height="20.4375" id="M1" style="vertical-align:-4.47675pt;width:244.3625px;" version="1.1" viewbox="0 0 244.3625 20.4375" width="244.3625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,20.4375)"> <g transform="translate(72,-55.65)"> <text transform="matrix(1,0,0,-1,-71.95,60.17)"> <tspan style="font-size: 12.50px; " x="90.440002" y="0.029999999"> ∫</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.689105" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.852604" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.766043" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="22.392874" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="34.433262" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="47.0863" y="0"> ∗</t年代pan> <tspan style="font-size: 12.50px; " x="56.401031" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="62.67754" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="66.841042" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="70.754478" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="78.393806" y="0"> =</t年代pan> </text> <text transform="matrix(1,0,0,-1,24.19,67.16)"> <tspan style="font-size: 8.75px; " x="0" y="0"> +</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> ∞</t年代pan> <tspan style="font-size: 8.75px; " x="-2.25" y="11.46"> −</t年代pan> <tspan style="font-size: 8.75px; " x="3.74512" y="11.46"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,40.87,60.17)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8773699" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="14.040869" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.729975" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="32.070194" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="37.996616" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="42.160114" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="48.436623" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="52.600121" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="58.526543" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="62.690041" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="69.341637" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="75.268059" y="0"> ,</t年代pan> </text> </g> </g> </svg>哪里<年代vg height="13.45" id="M2" style="vertical-align:-2.21957pt;width:86.1875px;" version="1.1" viewbox="0 0 86.1875 13.45" width="86.1875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-67.78,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="23.580658" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="32.145214" y="0"> ∞</t年代pan> <tspan style="font-size: 12.50px; " x="43.722992" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="48.936741" y="0"> ∞</t年代pan> <tspan style="font-size: 12.50px; " x="60.514519" y="0"> )</t年代pan> </text> </g> </g> </svg>．卖<年代vg height="14.6" id="M3" style="vertical-align:-3.13504pt;width:43.625px;" version="1.1" viewbox="0 0 43.625 14.6" width="43.625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.52,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-56.59,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.102664" y="0"> )</t年代pan> </text> </g> </g> </svg>和<年代vg height="14.6" id="M4" style="vertical-align:-3.13504pt;width:37.150002px;" version="1.1" viewbox="0 0 37.150002 14.6" width="37.150002" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.52,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.59,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.102664" y="0"> )</t年代pan> </text> </g> </g> </svg>表示平稳输入和输出过程的功率谱<年代vg height="13.45" id="M5" style="vertical-align:-2.21957pt;width:32.987499px;" version="1.1" viewbox="0 0 32.987499 13.45" width="32.987499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8773699" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="14.040869" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.954308" y="0"> )</t年代pan> </text> </g> </g> </svg>和<年代vg height="13.45" id="M6" style="vertical-align:-2.21957pt;width:23.7875px;" version="1.1" viewbox="0 0 23.7875 13.45" width="23.7875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.689105" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.852604" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.766043" y="0"> )</t年代pan> </text> </g> </g> </svg>基本滤波定理分别指出：<年代vg height="17.887501" id="M7" style="vertical-align:-3.13504pt;width:137.46249px;" version="1.1" viewbox="0 0 137.46249 17.887501" width="137.46249" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.8875)"> <g transform="translate(72,-57.69)"> <text transform="matrix(1,0,0,-1,-71.95,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.52,57.74)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.59,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.102664" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="18.729494" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="30.769882" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="33.270481" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="42.235134" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="46.398632" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="53.337799" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="57.501297" y="0"> |</t年代pan> </text> <text transform="matrix(1,0,0,-1,2.42,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2.</t年代pan> </text> <text transform="matrix(1,0,0,-1,7.29,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,14.72,57.74)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,22.66,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.102664" y="0"> )</t年代pan> </text> </g> </g> </svg>哪里<年代vg height="19.35" id="M8" style="vertical-align:-4.47675pt;width:156.625px;" version="1.1" viewbox="0 0 156.625 19.35" width="156.625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.35)"> <g transform="translate(72,-56.52)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="39.740002" y="0.02"> ∫</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.9646511" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="13.12815" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.067314" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="27.694145" y="0"> =</t年代pan> </text> <text transform="matrix(1,0,0,-1,-26.51,68.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> <tspan style="font-size: 8.75px; " x="-2.25" y="11.46"> −</t年代pan> <tspan style="font-size: 8.75px; " x="3.74512" y="11.46"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-12.08,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.2765059" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.440005" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.353444" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="18.516943" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,12.01,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 2.</t年代pan> <tspan style="font-size: 8.75px; " x="10.37112" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="15.683584" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="18.405457" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="23.262815" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,38.51,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.6515961" y="0"> </t年代pan> </text> </g> </g> </svg>为频响函数[<A.Href="#B14">1.</A.>]．此结果的重要性是它提供了有关过滤器的信息，如被平方频率响应函数所示，<年代vg height="16.9" id="M9" style="vertical-align:-2.34499pt;width:42.762501px;" version="1.1" viewbox="0 0 42.762501 16.9" width="42.762501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.9)"> <g transform="translate(72,-58.48)"> <text transform="matrix(1,0,0,-1,-71.95,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5006001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.465251" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.62875" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.567915" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.731415" y="0"> |</t年代pan> </text> <text transform="matrix(1,0,0,-1,-42.72,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2.</t年代pan> </text> </g> </g> </svg>，影响输出信号的频率行为。这在设计具有特定属性的过滤器方面非常有用。线性滤波器通常用于“滤除”某些频率从时间序列实现。在本讨论中，我们将考虑低通和高通滤波器，分别旨在消除高频行为和低频。我们将使用流行的Butterworth滤波器（见[<A.Href="#B3">2.</A.>])。的频率响应函数的平方<年代vg height="10.325" id="M10" style="vertical-align:-0.0pt;width:14.8375px;" version="1.1" viewbox="0 0 14.8375 10.325" width="14.8375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.325)"> <g transform="translate(72,-63.74)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>th阶低通巴特沃斯滤波器由下式给出<年代pan class="equation" id="EEq1"> <svg height="41.974998" id="M11" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 41.974998" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,41.975)"> <g transform="translate(72,-38.42)"> <text transform="matrix(1,0,0,-1,145.46,57.83)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1600001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="30.32" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="30.32" y="3.1600001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5" y="1.58"> </t年代pan> <tspan style="font-size: 12.50px; " x="13.540149" y="1.58"> (</t年代pan> <tspan style="font-size: 12.50px; " x="17.703648" y="1.58"> </t年代pan> <tspan style="font-size: 12.50px; " x="26.143173" y="1.58"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,178.28,63)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2.</t年代pan> </text> <text transform="matrix(1,0,0,-1,186.63,56.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="43.66" y="-7.25"> 1.</t年代pan> </text> <g> <path d="m 199.86,59.37 67.11,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,199.86,43.03)"> <tspan style="font-size: 12.50px; " x="20.370001" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="9.0271664" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="26.219999" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="33.159164" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="36.634998" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,243.44,39.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,248.01,40.52)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,253.86,50.57)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2.</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,268.16,56.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="196.59" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="200.75349" y="0"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="207.005" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="210.13075" y="0"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="216.38225" y="0"> )</t年代pan> </text> </g> </g> </svg></span>哪里<年代vg height="14.5875" id="M12" style="vertical-align:-3.2316pt;width:14.5125px;" version="1.1" viewbox="0 0 14.5125 14.5875" width="14.5125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.5875)"> <g transform="translate(72,-60.33)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.01,60.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>是截止频率，并且<年代vg height="10.325" id="M13" style="vertical-align:-0.0pt;width:14.8375px;" version="1.1" viewbox="0 0 14.8375 10.325" width="14.8375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.325)"> <g transform="translate(72,-63.74)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>是滤波器的阶数。高通巴特沃斯滤波器的频率响应函数具有类似的形式。传统的线性滤波器（如巴特沃斯滤波器）可以成功地从平稳过程中提取成分，其中数据的频率行为不随时间变化。然而，对于许多非平稳时间序列对于时变频率（TVF）行为，传统滤波器的时不变假设常常导致无法从此类过程中提取TVF分量<A.Href="#ex1">1．1</A.>检验巴特沃斯滤波器在包含时变频率的数据上的应用。</p> <p><span class="statement" id="ex1"><span class="statement-title"><i class="statement-title">示例1.1（应用于TVF数据的线性滤波器）。</我></年代pan>数字<A.Href="//www.newsama.com/journals/jps/2012/738636/fig1/" target="_blank">1（a）</A.>显示长度的实现<年代vg height="10.9125" id="M14" style="vertical-align:-0.17555pt;width:50.724998px;" version="1.1" viewbox="0 0 50.724998 10.9125" width="50.724998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.9125)"> <g transform="translate(72,-63.27)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.6773224" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.71771" y="0"> 4.</t年代pan> <tspan style="font-size: 12.50px; " x="27.969212" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="34.220711" y="0"> 0</t年代pan> </text> </g> </g> </svg>从模型中<年代pan class="equation" id="EEq2"> <svg height="19.35" id="M15" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 19.35" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.35)"> <g transform="translate(72,-56.52)"> <text transform="matrix(1,0,0,-1,33.88,61.13)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="61.290001" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.71" y="-0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.873499" y="-0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.786938" y="-0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="28.413769" y="-0.0099999998"> =</t年代pan> <tspan style="font-size: 12.50px; " x="40.454159" y="-0.0099999998"> C</t年代pan> <tspan style="font-size: 12.50px; " x="46.005489" y="-0.0099999998"> o</t年代pan> <tspan style="font-size: 12.50px; " x="52.256989" y="-0.0099999998"> 年代</t年代pan> <tspan style="font-size: 12.50px; " x="66.080002" y="7.1054274e-015"> 3.</t年代pan> <tspan style="font-size: 12.50px; " x="72.331497" y="7.1054274e-015"> 6.</t年代pan> <tspan style="font-size: 12.50px; " x="78.583" y="7.1054274e-015"> </t年代pan> <tspan style="font-size: 12.50px; " x="88.260323" y="7.1054274e-015"> l</t年代pan> <tspan style="font-size: 12.50px; " x="91.736153" y="7.1054274e-015"> n</t年代pan> <tspan style="font-size: 12.50px; " x="102.17" y="-0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="106.3335" y="-0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="113.0101" y="-0.0099999998"> +</t年代pan> <tspan style="font-size: 12.50px; " x="124.36282" y="-0.0099999998"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="130.61432" y="-0.0099999998"> 7.</t年代pan> <tspan style="font-size: 12.50px; " x="136.86583" y="-0.0099999998"> 5.</t年代pan> <tspan style="font-size: 12.50px; " x="143.11732" y="-0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="150.05649" y="-0.0099999998"> +</t年代pan> <tspan style="font-size: 12.50px; " x="161.39671" y="-0.0099999998"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,203.55,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1.</t年代pan> </text> <text transform="matrix(1,0,0,-1,208.42,58.62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="51.209999" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.5700002" y="-2.51"> +</t年代pan> <tspan style="font-size: 12.50px; " x="18.910221" y="-2.51"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="22.035971" y="-2.51"> 5.</t年代pan> <tspan style="font-size: 12.50px; " x="30.362968" y="-2.51"> C</t年代pan> <tspan style="font-size: 12.50px; " x="35.914303" y="-2.51"> o</t年代pan> <tspan style="font-size: 12.50px; " x="42.165802" y="-2.51"> 年代</t年代pan> <tspan style="font-size: 12.50px; " x="56" y="-2.51"> 8.</t年代pan> <tspan style="font-size: 12.50px; " x="62.251499" y="-2.51"> 5.</t年代pan> <tspan style="font-size: 12.50px; " x="68.502998" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="78.180321" y="-2.51"> l</t年代pan> <tspan style="font-size: 12.50px; " x="81.656158" y="-2.51"> n</t年代pan> <tspan style="font-size: 12.50px; " x="92.080002" y="-2.52"> (</t年代pan> <tspan style="font-size: 12.50px; " x="96.2435" y="-2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="102.9326" y="-2.52"> +</t年代pan> <tspan style="font-size: 12.50px; " x="114.27283" y="-2.52"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="120.52432" y="-2.52"> 7.</t年代pan> <tspan style="font-size: 12.50px; " x="126.77583" y="-2.52"> 5.</t年代pan> <tspan style="font-size: 12.50px; " x="133.02733" y="-2.52"> )</t年代pan> <tspan style="font-size: 12.50px; " x="139.96649" y="-2.52"> +</t年代pan> <tspan style="font-size: 12.50px; " x="151.30672" y="-2.52"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,368,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2.</t年代pan> </text> <text transform="matrix(1,0,0,-1,372.88,58.62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.8699999" y="-2.51"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="91.870003" y="-2.51"> (</t年代pan> <tspan style="font-size: 12.50px; " x="96.033501" y="-2.51"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="102.285" y="-2.51"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="105.41075" y="-2.51"> 2.</t年代pan> <tspan style="font-size: 12.50px; " x="111.66225" y="-2.51"> )</t年代pan> </text> </g> </g> </svg></span>哪里<年代vg height="14.6" id="M16" style="vertical-align:-3.13504pt;width:43.75px;" version="1.1" viewbox="0 0 43.75 14.6" width="43.75" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-63.69,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1.</t年代pan> </text> <text transform="matrix(1,0,0,-1,-55.34,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 1.</t年代pan> </text> </g> </g> </svg>和<年代vg height="14.7875" id="M17" style="vertical-align:-3.13504pt;width:63.287498px;" version="1.1" viewbox="0 0 63.287498 14.7875" width="63.287498" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.7875)"> <g transform="translate(72,-60.17)"> <text transform="matrix(1,0,0,-1,-71.95,63.35)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-63.69,60.22)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2.</t年代pan> </text> <text transform="matrix(1,0,0,-1,-55.34,63.35)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="18.279387" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="21.405136" y="0"> 5.</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> 9</t年代pan> </text> </g> </g> </svg>．在图中的数据中，TVF行为都很清楚<A.Href="//www.newsama.com/journals/jps/2012/738636/fig1/" target="_blank">1（a）</A.>以及图中所示的两个组件<A.Href="//www.newsama.com/journals/jps/2012/738636/fig1/" target="_blank">1（b）</A.>和<A.Href="//www.newsama.com/journals/jps/2012/738636/fig1/" target="_blank">1（c）</A.>。但是，应注意的是，“低频”成分开头的频率内容（图<A.Href="//www.newsama.com/journals/jps/2012/738636/fig1/" target="_blank">1（b）</A.>)与图中所示“高频”分量的末端大致相同<A.Href="//www.newsama.com/journals/jps/2012/738636/fig1/" target="_blank">1（c）</A.>。因此，巴特沃斯型过滤器无法完全分离部件。如图所示<A.Href="//www.newsama.com/journals/jps/2012/738636/fig1/" target="_blank">1 (d)</A.>和<A.Href="//www.newsama.com/journals/jps/2012/738636/fig1/" target="_blank">1 (e)</A.>我们展示了应用于图中数据的3RD订单低通巴特滤波器的结果<A.Href="//www.newsama.com/journals/jps/2012/738636/fig1/" target="_blank">1（a）</A.>与<年代vg height="14.9" id="M18" style="vertical-align:-3.2316pt;width:53.4375px;" version="1.1" viewbox="0 0 53.4375 14.9" width="53.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.9)"> <g transform="translate(72,-60.08)"> <text transform="matrix(1,0,0,-1,-71.95,63.35)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.01,60.22)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-56.97,63.35)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="15.153636" y="0"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="21.405136" y="0"> 5.</t年代pan> </text> </g> </g> </svg>和<年代vg height="11.0375" id="M19" style="vertical-align:-0.17555pt;width:19.6625px;" version="1.1" viewbox="0 0 19.6625 11.0375" width="19.6625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0375)"> <g transform="translate(72,-63.17)"> <text transform="matrix(1,0,0,-1,-71.95,63.4)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="3.1257501" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="9.3772497" y="0"> 6.</t年代pan> </text> </g> </g> </svg>,分别。在图中<A.Href="//www.newsama.com/journals/jps/2012/738636/fig1/" target="_blank">1 (d)</A.>截止频率，<年代vg height="14.9" id="M20" style="vertical-align:-3.2316pt;width:62.549999px;" version="1.1" viewbox="0 0 62.549999 14.9" width="62.549999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.9)"> <g transform="translate(72,-60.08)"> <text transform="matrix(1,0,0,-1,-71.95,63.35)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.01,60.22)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-56.97,63.35)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="15.153636" y="0"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="21.405136" y="0"> 5.</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> ,</t年代pan> </text> </g> </g> </svg>提取时间的低频行为是否有很好的工作<年代vg height="12.3" id="M21" style="vertical-align:-1.29163pt;width:47.849998px;" version="1.1" viewbox="0 0 47.849998 12.3" width="47.849998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.3)"> <g transform="translate(72,-62.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="25.668659" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="31.920158" y="0"> 0</t年代pan> </text> </g> </g> </svg>，但在接近信号末尾时，这两个分量都会通过滤波器。这是因为“高频”分量中的频率行为（随时间而降低）已降低到其通过滤波器的程度<年代vg height="14.9" id="M22" style="vertical-align:-3.2316pt;width:53.4375px;" version="1.1" viewbox="0 0 53.4375 14.9" width="53.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.9)"> <g transform="translate(72,-60.08)"> <text transform="matrix(1,0,0,-1,-71.95,63.35)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.01,60.22)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-56.97,63.35)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="15.153636" y="0"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="21.405136" y="0"> 5.</t年代pan> </text> </g> </g> </svg>. 图形<A.Href="//www.newsama.com/journals/jps/2012/738636/fig1/" target="_blank">1 (e)</A.>显示降低截止频率的效果<年代vg height="14.8375" id="M23" style="vertical-align:-3.2316pt;width:53.4375px;" version="1.1" viewbox="0 0 53.4375 14.8375" width="53.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.8375)"> <g transform="translate(72,-60.13)"> <text transform="matrix(1,0,0,-1,-71.95,63.4)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.01,60.27)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-56.97,63.4)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="15.153636" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="21.405136" y="0"> 6.</t年代pan> </text> </g> </g> </svg>. 对于<年代vg height="12.4875" id="M24" style="vertical-align:-1.29163pt;width:47.849998px;" version="1.1" viewbox="0 0 47.849998 12.4875" width="47.849998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.4875)"> <g transform="translate(72,-62.01)"> <text transform="matrix(1,0,0,-1,-71.95,63.35)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> ≥</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> 2.</t年代pan> <tspan style="font-size: 12.50px; " x="25.668659" y="0"> 5.</t年代pan> <tspan style="font-size: 12.50px; " x="31.920158" y="0"> 0</t年代pan> </text> </g> </g> </svg>滤波器执行仅提取低频分量的合理作业。然而，降低截止频率的效果是通过滤波器朝向数据的开头来既不高于低频分量。实际上，由于在图中信号的第一部分之间的频率内容之间的重叠<A.Href="//www.newsama.com/journals/jps/2012/738636/fig1/" target="_blank">1（b）</A.>以及图中信号的后半部分<A.Href="//www.newsama.com/journals/jps/2012/738636/fig1/" target="_blank">1（c）</A.>，没有截止频率将能够分离这两个成分使用滤波器，如巴特沃斯。<br>示例中的数据集<A.Href="#ex1">1．1</A.>是m -平稳过程的一种实现，m -平稳过程是g -平稳过程的一种特殊情况，引入g -平稳过程是为了推广平稳的概念[<A.Href="#B11">3.</A.>]．如上所述，为了从G-stationary过程中过滤数据，必须调整当前的时不变线性滤波。Baraniuk和Jones [<A.Href="#B1">4.</A.>]利用酉等价系统理论设计了基于交替坐标系的信号分析和处理系统的一般类。利用酉等价系统的原理，滤波过程是:(1)对数据进行预处理，(2)采用传统的时不变滤波方法，(3)将结果转换到原始的时间尺度。为了解决g -平稳过程的滤波问题，本文定义了g -滤波器，并说明了它在g -平稳过程信号中的应用。<br>本文的组织结构如下<A.Href="#sec2">2.</A.>我们讨论G-平稳模型，重点讨论拟合策略<年代vg height="13.6875" id="M25" style="vertical-align:-2.21957pt;width:31.6px;" version="1.1" viewbox="0 0 31.6 13.6875" width="31.6" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6875)"> <g transform="translate(72,-61.05)"> <text transform="matrix(1,0,0,-1,-71.95,63.31)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.727334" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="13.890833" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="21.017544" y="0"> )</t年代pan> </text> </g> </g> </svg>- Jiang等的平稳模型[<A.Href="#B11">3.</A.>]．节<A.Href="#sec3">3.</A.>我们提供了有关用于从G-静止模型的数据进行过滤的G-滤波器的结果，以及部分<A.Href="#sec4">4.</A.>我们介绍了一种直接实现的G滤波器，用于提取如图所示的组件<A.Href="//www.newsama.com/journals/jps/2012/738636/fig1/" target="_blank">1（b）</A.>和<A.Href="//www.newsama.com/journals/jps/2012/738636/fig1/" target="_blank">1（c）</A.>．</年代pan></p> <div class="floats-partial-section partial-carousel-wrapper" data-section="fig1"> <div class="floats-partial-content fig-content-fig1"> <div class="partial-carousel--single-image"> <img alt="（a） TVF信号"nA.me="738636.fig.001a" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.001a.jpg"> <br> <strong>（a） TVF信号</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="（b）低频分量"nA.me="738636.fig.001b" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.001b.jpg"> <br> <strong>（b）低频分量</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="(c)高频组件"nA.me="738636.fig.001c" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.001c.jpg"> <br> <strong>(c)高频组件</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="(d)低通滤波器输出，c ut o off =。１５"nA.me="738636.fig.001d" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.001d.jpg"> <br> <strong>(d)低通滤波器输出，c ut o off =。１５</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="（e）来自低通滤波器的输出，c u f=.0 6"nA.me="738636.fig.001e" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.001e.jpg"> <br> <strong>(e)低通滤波器输出，c ut o off =。0 6</年代trong> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs fig-thumbs-fig1"> <button class="partial-carousel-dot" aria-label="(a) TVF signal"><img alt="（a） TVF信号"nA.me="738636.fig.001a" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.001a.jpg"><br><strong>（a） TVF信号</年代trong></button> <button class="partial-carousel-dot" aria-label="(b) Low-frequency component"><img alt="（b）低频分量"nA.me="738636.fig.001b" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.001b.jpg"><br><strong>（b）低频分量</年代trong></button> <button class="partial-carousel-dot" aria-label="(c) High-frequency component"><img alt="(c)高频组件"nA.me="738636.fig.001c" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.001c.jpg"><br><strong>(c)高频组件</年代trong></button> <button class="partial-carousel-dot" aria-label="(d) Output from low-pass filter, c u t o ﬀ = . 1 5"><img alt="(d)低通滤波器输出，c ut o off =。１５"nA.me="738636.fig.001d" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.001d.jpg"><br><strong>(d)低通滤波器输出，c ut o off =。１５</年代trong></button> <button class="partial-carousel-dot" aria-label="(e) Output from low-pass filter, c u t o ﬀ = . 0 6"><img alt="（e）来自低通滤波器的输出，c u f=.0 6"nA.me="738636.fig.001e" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.001e.jpg"><br><strong>(e)低通滤波器输出，c ut o off =。0 6</年代trong></button> </div> <div class="partial-carousel-next-prev fig-nav-fig1"> <button class="carousel-prev" aria-label="Previous"></button> <button class="carousel-next" aria-label="Next"></button> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图1</年代pan> <a aria-label="full view" class="caption-partial-url" href="//www.newsama.com/journals/jps/2012/738636/fig1/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> <a href="#EEq2">１．２</A.><年代vg height="14.9" id="M26" style="vertical-align:-3.2316pt" version="1.1" viewbox="0 0 53.4375 14.9" width="53.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.9)"> <g transform="translate(72,-60.08)"> <text transform="matrix(1,0,0,-1,-71.95,63.35)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.01,60.22)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-56.97,63.35)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="15.153636" y="0"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="21.405136" y="0"> 5.</t年代pan> </text> </g> </g> </svg> <svg height="14.8375" id="M27" style="vertical-align:-3.2316pt" version="1.1" viewbox="0 0 53.4375 14.8375" width="53.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.8375)"> <g transform="translate(72,-60.13)"> <text transform="matrix(1,0,0,-1,-71.95,63.4)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.01,60.27)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-56.97,63.4)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="15.153636" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="21.405136" y="0"> 6.</t年代pan> </text> </g> </g> </svg> <table> (a)由信号输入()，（b）真低频分量，（c）真高频分量，（d）带，（e）从低通滤波器输出．</tA.ble> </div> </div> </div> <h4 id="time-deformation-method-and-the-g-stationary-process">2.时间变形法与g -平稳过程</H4.><p>平稳过程是分布不随时间变化的随机过程。对于平稳时间序列，已经发展了大量的统计理论和方法。然而，许多过程是非平稳的，其分布随时间而变化。已经提出了几种利用时间变形分析这些非平稳过程的技术。例如，Girardin和Senoussi [<A.Href="#B7">5.</A.>]提出了一个基于调和分析的半群框架，定义了一个半群平稳过程，包括一个局部平稳过程和一个时间扭曲过程。克莱克和马拉特[<A.Href="#B4">6.</A.>通过证明变形自协方差在小尺度上满足迁移方程，导出了变形的估计量。在不同类型的非平稳过程中，通过变形平稳过程的指标集得到的扭曲非平稳过程受到了特别的关注。例如，自相似过程是在平稳过程的时间尺度上应用Lamperti算子得到的[<A.Href="#B6">7.</A.>]．Hlawatsch和Tauböck [<A.Href="#B9">8.</A.>]和Hlawatsch等人[<A.Href="#B10">9</A.>]讨论使用时频位移算子在时频平面上位移信号。另见Sampson和Guttorp[<A.Href="#B17">10</A.>]和Perrin和Senoussi [<A.Href="#B15">11</A.>]．</p> <p>灰色和张[<A.Href="#B20">12</A.>在时间轴的对数变换上的基准时间变形。他们将在对数尺度上是平稳的过程称为m平稳过程，并表明得到的光谱表示是基于梅林变换的。Flandrin等[<A.Href="#B6">7.</A.>]指出Gray和Zhang的对数变换[<A.Href="#B20">12</A.>是兰珀蒂变换的一种特殊情况。Gray等人[<A.Href="#B8">13</A.>扩展m平稳模型，以分析在离散时间点收集的数据。Jiang et al. [<A.Href="#B11">3.</A.>]定义了一个更一般的非平稳过程族，称为G平稳过程，其频率随时间单调变化。另见Woodward等人[<A.Href="#B18">14</A.>，第13章]。在本文中，我们根据G-静止模型研究了时间变形的使用来过滤TVF数据。我们将生成的过滤器称为G-Filters。</p> <h5 id="sec2.1">2．1．G-Stationary流程</H5.><p><span class="statement" id="deff1"><span class="statement-title"><i class="statement-title">定义2.1。</我></年代pan>允许<年代vg height="13.575" id="M30" style="vertical-align:-2.26974pt;width:93.887497px;" version="1.1" viewbox="0 0 93.887497 13.575" width="93.887497" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.575)"> <g transform="translate(72,-61.14)"> <text transform="matrix(1,0,0,-1,-71.95,63.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ｛</t年代pan> <tspan style="font-size: 12.50px; " x="6.00144" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="15.87881" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="20.042309" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.955748" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="31.582579" y="0"> ∶</t年代pan> <tspan style="font-size: 12.50px; " x="41.447445" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="48.83672" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="60.877106" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="69.004059" y="0"> ｝</t年代pan> </text> </g> </g> </svg>是一个随机过程<年代vg height="10.95" id="M31" style="vertical-align:-0.3135pt;width:41.349998px;" version="1.1" viewbox="0 0 41.349998 10.95" width="41.349998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.95)"> <g transform="translate(72,-63.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.590281" y="0"> ⊂</t年代pan> <tspan style="font-size: 12.50px; " x="23.630671" y="0"> ℝ</t年代pan> </text> </g> </g> </svg>允许<年代vg height="13.6125" id="M32" style="vertical-align:-2.34499pt;width:49.912498px;" version="1.1" viewbox="0 0 49.912498 13.6125" width="49.912498" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.3897533" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.430141" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.581617" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="31.745117" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.658558" y="0"> )</t年代pan> </text> </g> </g> </svg>映射到一个集合上<年代vg height="16.424999" id="M33" style="vertical-align:-4.77652pt;width:49.200001px;" version="1.1" viewbox="0 0 49.200001 16.424999" width="49.200001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.425)"> <g transform="translate(72,-58.86)"> <text transform="matrix(1,0,0,-1,-71.95,63.67)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.7,60.55)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.07,63.67)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ⊂</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> ℝ</t年代pan> </text> </g> </g> </svg>，并让<年代vg height="16.700001" id="M34" style="vertical-align:-2.34499pt;width:22.025px;" version="1.1" viewbox="0 0 22.025 16.700001" width="22.025" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.7)"> <g transform="translate(72,-58.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.3,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 1.</t年代pan> </text> </g> </g> </svg>表示一个倒数，然后<年代vg height="13.45" id="M35" style="vertical-align:-2.21957pt;width:27.775px;" version="1.1" viewbox="0 0 27.775 13.45" width="27.775" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8773699" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="14.040869" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.954308" y="0"> )</t年代pan> </text> </g> </g> </svg>据说是一个<我>G-平稳过程</我>如果<年代pan class="list"><span class="list-item"><span class="list-label">（一世）</年代pan><span class="list-content"> <svg height="13.55" id="M36" style="vertical-align:-2.29482pt;width:78.25px;" version="1.1" viewbox="0 0 78.25 13.55" width="78.25" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.55)"> <g transform="translate(72,-61.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.9271421" y="0"> [</t年代pan> <tspan style="font-size: 12.50px; " x="13.090641" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.968012" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="27.13151" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.044949" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="35.208447" y="0"> ]</t年代pan> <tspan style="font-size: 12.50px; " x="42.835278" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="54.875668" y="0"> </t年代pan> </text> </g> </g> </svg>,</年代pan></span><span class="list-item"><span class="list-label">(ii)</年代pan><span class="list-content"> <svg height="16.5375" id="M37" style="vertical-align:-2.21957pt;width:125.8625px;" version="1.1" viewbox="0 0 125.8625 16.5375" width="125.8625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.5375)"> <g transform="translate(72,-58.77)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> v</t年代pan> <tspan style="font-size: 12.50px; " x="9.0271664" y="0"> A.</t年代pan> <tspan style="font-size: 12.50px; " x="14.578498" y="0"> r</t年代pan> <tspan style="font-size: 12.50px; " x="23.655676" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="33.533047" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="37.696545" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="41.609985" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="49.249317" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="61.277203" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-3.32,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2.</t年代pan> </text> <text transform="matrix(1,0,0,-1,5.03,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> ∞</t年代pan> </text> </g> </g> </svg>,</年代pan></span><span class="list-item"><span class="list-label">(iii)</年代pan><span class="list-content"> <svg height="17.674999" id="M38" style="vertical-align:-3.13504pt;width:310.79999px;" version="1.1" viewbox="0 0 310.79999 17.674999" width="310.79999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.675)"> <g transform="translate(72,-57.86)"> <text transform="matrix(1,0,0,-1,-67.78,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.9271421" y="0"> [</t年代pan> <tspan style="font-size: 12.50px; " x="13.090641" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="17.254141" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.13151" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="31.29501" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.208447" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="42.147614" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="53.487835" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="61.102161" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="65.265663" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="69.429161" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="79.306526" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="83.470032" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,22.35,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 1.</t年代pan> </text> <text transform="matrix(1,0,0,-1,33.21,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.314975" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="14.478474" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.391912" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="25.331078" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="36.671299" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="42.822777" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="46.986275" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="53.38781" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="57.551308" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="61.714809" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="68.666473" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="80.006699" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="87.621025" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="91.784523" y="0"> ]</t年代pan> <tspan style="font-size: 12.50px; " x="99.423859" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="111.45174" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,153.93,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,161.86,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.565035" y="0"> )</t年代pan> </text> </g> </g> </svg>．</年代pan></span></span></span></p> <p>虽然没有立即显而易见，但这个定义基本上如下。允许<年代vg height="9.9375" id="M39" style="vertical-align:-2.34499pt;width:8.4375px;" version="1.1" viewbox="0 0 8.4375 9.9375" width="8.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.9375)"> <g transform="translate(72,-64.05)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>是时间轴的转变，让<年代vg height="13.6125" id="M40" style="vertical-align:-2.34499pt;width:55.125px;" version="1.1" viewbox="0 0 55.125 13.6125" width="55.125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.1514759" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.314975" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.228414" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="21.855244" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="33.895634" y="0"> </t年代pan> </text> </g> </g> </svg>和<年代vg height="16.700001" id="M41" style="vertical-align:-2.34499pt;width:64.125px;" version="1.1" viewbox="0 0 64.125 16.700001" width="64.125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.7)"> <g transform="translate(72,-58.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-45.87,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 1.</t年代pan> </text> <text transform="matrix(1,0,0,-1,-35,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.089921" y="0"> )</t年代pan> </text> </g> </g> </svg>．最后，<年代vg height="16.700001" id="M42" style="vertical-align:-2.34499pt;width:155.2px;" version="1.1" viewbox="0 0 155.2 16.700001" width="155.2" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.7)"> <g transform="translate(72,-58.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8773699" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="14.040869" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.954308" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="25.581139" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="37.621529" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="47.498898" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="51.662395" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-13.63,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 1.</t年代pan> </text> <text transform="matrix(1,0,0,-1,-2.76,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.089921" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="14.25342" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="21.880251" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="33.920639" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="40.609745" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="44.773243" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="50.699665" y="0"> )</t年代pan> </text> </g> </g> </svg>哪里<年代vg height="16.700001" id="M43" style="vertical-align:-2.34499pt;width:67.337502px;" version="1.1" viewbox="0 0 67.337502 16.700001" width="67.337502" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.7)"> <g transform="translate(72,-58.64)"> <text transform="matrix(1,0,0,-1,-67.78,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.152436" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.192825" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="32.070194" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-29.05,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 1.</t年代pan> </text> </g> </g> </svg>．卖<年代vg height="13.6125" id="M44" style="vertical-align:-2.34499pt;width:53.162498px;" version="1.1" viewbox="0 0 53.162498 13.6125" width="53.162498" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.5022802" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.542669" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.694145" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="31.857643" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="38.259178" y="0"> )</t年代pan> </text> </g> </g> </svg>,然后<年代pan class="equation" id="eq1"> <svg height="19.475" id="M45" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 19.475" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.475)"> <g transform="translate(72,-56.42)"> <text transform="matrix(1,0,0,-1,72.88,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.71" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.559999" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,98.1,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 1.</t年代pan> </text> <text transform="matrix(1,0,0,-1,111.05,61.04)"> <tspan style="font-size: 12.50px; " x="66.889999" y="2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.890593" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="17.054092" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.967531" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="27.906696" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="39.259418" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="47.986515" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="52.150013" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="58.551548" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="62.71505" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="76.209999" y="0.0099999998"> =</t年代pan> <tspan style="font-size: 12.50px; " x="88.237885" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="99.515594" y="0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="103.67909" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="112.41869" y="0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="116.58218" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="120.49563" y="0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="127.43479" y="0.0099999998"> +</t年代pan> <tspan style="font-size: 12.50px; " x="138.77501" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="147.5146" y="0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="151.6781" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="158.07965" y="0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="162.24315" y="0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="169.88248" y="0.0099999998"> =</t年代pan> <tspan style="font-size: 12.50px; " x="181.91035" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="193.18806" y="0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="197.35156" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="206.05365" y="0.0099999998"> +</t年代pan> <tspan style="font-size: 12.50px; " x="217.40637" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="223.44533" y="0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="229.68433" y="0.0099999998"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="353.70001" y="0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="357.86349" y="0.0099999998"> 2.</t年代pan> <tspan style="font-size: 12.50px; " x="364.11499" y="0.0099999998"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="367.24075" y="0.0099999998"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="373.49225" y="0.0099999998"> )</t年代pan> </text> </g> </g> </svg></span>所以，它是这样的<年代pan class="equation" id="EEq3"> <svg height="19.475" id="M46" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 19.475" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.475)"> <g transform="translate(72,-56.42)"> <text transform="matrix(1,0,0,-1,-14.89,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-5.64,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,4.38,61.04)"> <tspan style="font-size: 12.50px; " x="42" y="2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="103.89" y="2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="109.74" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="122.45" y="2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="128.31" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.565035" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="18.204369" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="30.232254" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="46.790001" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="50.953499" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="63.656548" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="67.820045" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="71.733482" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="78.68515" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="90.025375" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="97.639702" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,139.34,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 1.</t年代pan> </text> <text transform="matrix(1,0,0,-1,152.29,61.04)"> <tspan style="font-size: 12.50px; " x="66.889999" y="2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="132.39" y="0.050000001"> [</t年代pan> <tspan style="font-size: 12.50px; " x="269.98001" y="0.050000001"> ]</t年代pan> <tspan style="font-size: 12.50px; " x="276.23001" y="0.0099999998"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.890593" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="17.054092" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.967531" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="27.906696" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="39.259418" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="47.986515" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="52.150013" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="58.551548" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="62.71505" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="75.519997" y="0.0099999998"> −</t年代pan> <tspan style="font-size: 12.50px; " x="86.860222" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="94.480003" y="2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="100.33141" y="2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="108.59" y="0.0099999998"> =</t年代pan> <tspan style="font-size: 12.50px; " x="120.61789" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="136.55" y="7.1054274e-015"> (</t年代pan> <tspan style="font-size: 12.50px; " x="140.7135" y="7.1054274e-015"> </t年代pan> <tspan style="font-size: 12.50px; " x="151.9787" y="7.1054274e-015"> (</t年代pan> <tspan style="font-size: 12.50px; " x="156.1422" y="7.1054274e-015"> </t年代pan> <tspan style="font-size: 12.50px; " x="162.06862" y="7.1054274e-015"> )</t年代pan> <tspan style="font-size: 12.50px; " x="169.02029" y="7.1054274e-015"> −</t年代pan> <tspan style="font-size: 12.50px; " x="180.36052" y="7.1054274e-015"> </t年代pan> <tspan style="font-size: 12.50px; " x="187.97484" y="7.1054274e-015"> )</t年代pan> <tspan style="font-size: 12.50px; " x="194.22633" y="7.1054274e-015"> (</t年代pan> <tspan style="font-size: 12.50px; " x="198.38983" y="7.1054274e-015"> </t年代pan> <tspan style="font-size: 12.50px; " x="209.65504" y="7.1054274e-015"> (</t年代pan> <tspan style="font-size: 12.50px; " x="213.81854" y="7.1054274e-015"> </t年代pan> <tspan style="font-size: 12.50px; " x="222.52063" y="7.1054274e-015"> +</t年代pan> <tspan style="font-size: 12.50px; " x="233.87335" y="7.1054274e-015"> </t年代pan> <tspan style="font-size: 12.50px; " x="239.91231" y="7.1054274e-015"> )</t年代pan> <tspan style="font-size: 12.50px; " x="246.85147" y="7.1054274e-015"> −</t年代pan> <tspan style="font-size: 12.50px; " x="258.19168" y="7.1054274e-015"> </t年代pan> <tspan style="font-size: 12.50px; " x="265.806" y="7.1054274e-015"> )</t年代pan> <tspan style="font-size: 12.50px; " x="312.45999" y="-0.039999999"> (</t年代pan> <tspan style="font-size: 12.50px; " x="316.6235" y="-0.039999999"> 2.</t年代pan> <tspan style="font-size: 12.50px; " x="322.875" y="-0.039999999"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="326.00076" y="-0.039999999"> 2.</t年代pan> <tspan style="font-size: 12.50px; " x="332.25226" y="-0.039999999"> )</t年代pan> </text> </g> </g> </svg></span>也就是说，定义<A.Href="#deff1">2．1</A.>给予条件<年代vg height="13.45" id="M47" style="vertical-align:-2.21957pt;width:27.775px;" version="1.1" viewbox="0 0 27.775 13.45" width="27.775" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8773699" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="14.040869" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.954308" y="0"> )</t年代pan> </text> </g> </g> </svg>和<年代vg height="13.6125" id="M48" style="vertical-align:-2.34499pt;width:23.112499px;" version="1.1" viewbox="0 0 23.112499 13.6125" width="23.112499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.1514759" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.314975" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.228414" y="0"> )</t年代pan> </text> </g> </g> </svg>这对于双重过程来说是必要和充分的，<年代vg height="13.45" id="M49" style="vertical-align:-2.21957pt;width:26.299999px;" version="1.1" viewbox="0 0 26.299999 13.45" width="26.299999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.689105" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.852604" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.779026" y="0"> )</t年代pan> </text> </g> </g> </svg>，弱静止。含义是，而且<年代vg height="13.45" id="M50" style="vertical-align:-2.21957pt;width:27.775px;" version="1.1" viewbox="0 0 27.775 13.45" width="27.775" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8773699" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="14.040869" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.954308" y="0"> )</t年代pan> </text> </g> </g> </svg>可能不稳定在原始指数集上(on<年代vg height="9.125" id="M51" style="vertical-align:-0.11285pt;width:5.0124998px;" version="1.1" viewbox="0 0 5.0124998 9.125" width="5.0124998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.125)"> <g transform="translate(72,-64.7)"> <text transform="matrix(1,0,0,-1,-71.95,64.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>)，它可以被映射到一个新的(变形的)时间尺度上，在这个时间尺度上它是静止的。我们会提到<年代vg height="13.45" id="M52" style="vertical-align:-2.21957pt;width:31.512501px;" version="1.1" viewbox="0 0 31.512501 13.45" width="31.512501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.689105" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.852604" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.779026" y="0"> )</t年代pan> </text> </g> </g> </svg>作为<年代vg height="13.45" id="M53" style="vertical-align:-2.21957pt;width:36.887501px;" version="1.1" viewbox="0 0 36.887501 13.45" width="36.887501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8773699" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="14.040869" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.954308" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="22.117807" y="0"> ,</t年代pan> </text> </g> </g> </svg>我们让我们<年代vg height="14.6" id="M54" style="vertical-align:-3.13504pt;width:230.89999px;" version="1.1" viewbox="0 0 230.89999 14.6" width="230.89999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-63.56,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-56.63,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.202448" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="17.829279" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="29.869667" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="38.79681" y="0"> [</t年代pan> <tspan style="font-size: 12.50px; " x="42.960308" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="47.123806" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="53.812912" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="57.97641" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="63.902832" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="70.841995" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="82.194725" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="89.809052" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="93.972549" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="98.136047" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="104.82515" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="108.98865" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="117.69074" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="129.03096" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="135.06992" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="142.00908" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="153.34929" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="160.96362" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="165.12712" y="0"> ]</t年代pan> </text> </g> </g> </svg>表示对偶过程的自协方差。<年代vg height="14.6" id="M55" style="vertical-align:-3.13504pt;width:40.025002px;" version="1.1" viewbox="0 0 40.025002 14.6" width="40.025002" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.7,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.76,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.552532" y="0"> )</t年代pan> </text> </g> </g> </svg>被称为G-平稳自协方差<年代vg height="14.6" id="M56" style="vertical-align:-3.13504pt;width:96.525002px;" version="1.1" viewbox="0 0 96.525002 14.6" width="96.525002" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-63.56,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-56.63,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.202448" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="17.829279" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="29.869667" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-17.5,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-9.56,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.552532" y="0"> )</t年代pan> </text> </g> </g> </svg>．g谱的定义在定义中<A.Href="#deff2">2．2</A.>．</p> <p><span class="statement" id="deff2"><span class="statement-title"><i class="statement-title">定义2.2。</我></年代pan>允许<年代vg height="13.575" id="M57" style="vertical-align:-2.26974pt;width:105.675px;" version="1.1" viewbox="0 0 105.675 13.575" width="105.675" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.575)"> <g transform="translate(72,-61.14)"> <text transform="matrix(1,0,0,-1,-71.95,63.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ｛</t年代pan> <tspan style="font-size: 12.50px; " x="6.00144" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="15.87881" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="20.042309" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.955748" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="28.119247" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="33.320496" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="40.709766" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="52.750156" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="56.913654" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="63.190163" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="68.391411" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="74.267822" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="78.43132" y="0"> ｝</t年代pan> </text> </g> </g> </svg>是一个g-instary过程，让<年代vg height="13.575" id="M58" style="vertical-align:-2.26974pt;width:131.1875px;" version="1.1" viewbox="0 0 131.1875 13.575" width="131.1875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.575)"> <g transform="translate(72,-61.14)"> <text transform="matrix(1,0,0,-1,-71.95,63.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ｛</t年代pan> <tspan style="font-size: 12.50px; " x="6.00144" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.690545" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.854044" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.780466" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.943966" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="32.145214" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="41.54747" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="53.587856" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="57.751358" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="66.31591" y="0"> ∞</t年代pan> <tspan style="font-size: 12.50px; " x="77.893692" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="83.09494" y="0"> ∞</t年代pan> <tspan style="font-size: 12.50px; " x="94.672714" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="98.836212" y="0"> ｝</t年代pan> </text> </g> </g> </svg>是与时间变形函数对应的对偶，<年代vg height="13.6125" id="M59" style="vertical-align:-2.34499pt;width:23.112499px;" version="1.1" viewbox="0 0 23.112499 13.6125" width="23.112499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.1514759" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.314975" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.228414" y="0"> )</t年代pan> </text> </g> </g> </svg>．G-stationary过程的g -谱，<年代vg height="13.45" id="M60" style="vertical-align:-2.21957pt;width:32.987499px;" version="1.1" viewbox="0 0 32.987499 13.45" width="32.987499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-67.78,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8773699" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="14.040869" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.954308" y="0"> )</t年代pan> </text> </g> </g> </svg>，定义为对偶过程的谱，即，<年代pan class="equation" id="eq2"> <svg height="36.737499" id="M61" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 36.737499" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,36.7375)"> <g transform="translate(72,-42.61)"> <text transform="matrix(1,0,0,-1,122.98,54.16)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,132.15,51.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,142.17,54.16)"> <tspan style="font-size: 12.50px; " x="42.990002" y="7.6999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.102664" y="0"> ;</t年代pan> <tspan style="font-size: 12.50px; " x="16.666498" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.318094" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="30.944925" y="0"> =</t年代pan> </text> <text transform="matrix(1,0,0,-1,192.17,68.19)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> <tspan style="font-size: 8.75px; " x="-1.26" y="25.530001"> −</t年代pan> <tspan style="font-size: 8.75px; " x="4.7351198" y="25.530001"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,207.6,54.16)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,213.16,59.16)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 2.</t年代pan> <tspan style="font-size: 8.75px; " x="10.37112" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="15.692336" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="18.414207" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="23.271568" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,243.24,54.16)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,252.49,51.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,261.51,54.16)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.189945" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="16.441444" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.09304" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="29.13199" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="203.24001" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="207.4035" y="0"> 2.</t年代pan> <tspan style="font-size: 12.50px; " x="213.655" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="216.78075" y="0"> 3.</t年代pan> <tspan style="font-size: 12.50px; " x="223.03224" y="0"> )</t年代pan> </text> </g> </g> </svg></span>如果映射,<年代vg height="13.6125" id="M62" style="vertical-align:-2.34499pt;width:23.112499px;" version="1.1" viewbox="0 0 23.112499 13.6125" width="23.112499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.1514759" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.314975" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.228414" y="0"> )</t年代pan> </text> </g> </g> </svg>，来自原始空间<年代vg height="13.45" id="M63" style="vertical-align:-2.21957pt;width:56.512501px;" version="1.1" viewbox="0 0 56.512501 13.45" width="56.512501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="23.580658" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="29.857164" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="35.070915" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="40.947327" y="0"> )</t年代pan> </text> </g> </g> </svg>到对偶空间<年代vg height="13.45" id="M64" style="vertical-align:-2.21957pt;width:88.699997px;" version="1.1" viewbox="0 0 88.699997 13.45" width="88.699997" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.3897533" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="21.430141" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="25.593641" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="34.158195" y="0"> ∞</t年代pan> <tspan style="font-size: 12.50px; " x="45.735973" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="50.949726" y="0"> ∞</t年代pan> <tspan style="font-size: 12.50px; " x="62.527504" y="0"> )</t年代pan> </text> </g> </g> </svg>那么，这是什么呢<年代pan class="equation" id="eq3"> <svg height="135.08749" id="M65" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 135.08749" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,135.0875)"> <g transform="translate(72,36.07)"> <text transform="matrix(1,0,0,-1,94.79,54.16)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,103.97,51.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,113.99,54.16)"> <tspan style="font-size: 12.50px; " x="42.990002" y="7.6999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.102664" y="0"> ;</t年代pan> <tspan style="font-size: 12.50px; " x="16.653996" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.305592" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="30.944925" y="0"> =</t年代pan> </text> <text transform="matrix(1,0,0,-1,163.98,68.19)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> <tspan style="font-size: 8.75px; " x="-1.25" y="25.530001"> −</t年代pan> <tspan style="font-size: 8.75px; " x="4.74512" y="25.530001"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,179.41,54.16)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,184.97,59.16)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 2.</t年代pan> <tspan style="font-size: 8.75px; " x="10.37112" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="15.692336" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="18.414207" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="23.271568" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,215.05,54.16)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,224.31,51.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,233.32,54.16)"> <tspan style="font-size: 12.50px; " x="-88.379997" y="38.259998"> =</t年代pan> <tspan style="font-size: 12.50px; " x="-76.339996" y="45.950001"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.202448" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="16.441444" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.09304" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,163.98,29.93)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> <tspan style="font-size: 8.75px; " x="-1.25" y="25.52"> −</t年代pan> <tspan style="font-size: 8.75px; " x="4.74512" y="25.52"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,179.41,15.9)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,184.97,20.9)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 2.</t年代pan> <tspan style="font-size: 8.75px; " x="10.37112" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="15.692336" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="18.414207" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="23.271568" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,215.05,15.9)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,223.44,12.78)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,233.46,13.39)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8600001" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,245.97,20.9)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 1.</t年代pan> </text> <text transform="matrix(1,0,0,-1,258.92,15.91)"> <tspan style="font-size: 12.50px; " x="28.66" y="2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="-113.98" y="40.34"> =</t年代pan> <tspan style="font-size: 12.50px; " x="-101.94" y="48.040001"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.202448" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="16.441444" y="0"> ;</t年代pan> <tspan style="font-size: 12.50px; " x="22.00528" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="36.599998" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="43.251595" y="0.0099999998"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,163.98,-10.41)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="-1.25" y="25.52"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,170.67,-24.43)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,176.24,-19.43)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 2.</t年代pan> <tspan style="font-size: 8.75px; " x="10.37112" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="15.683584" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="18.405457" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="23.262815" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="27.5688" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="30.483215" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="34.964241" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,216.08,-24.43)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,224.47,-27.56)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,234.49,-24.43)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.565035" y="0"> ;</t年代pan> <tspan style="font-size: 12.50px; " x="16.116367" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.767963" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="29.019463" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,270.17,-19.43)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,275.73,-24.43)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.552532" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="16.804031" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="26.29381" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="30.457308" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="36.858845" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="43.097839" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="189.02" y="-39.27"> (</t年代pan> <tspan style="font-size: 12.50px; " x="193.1835" y="-39.27"> 2.</t年代pan> <tspan style="font-size: 12.50px; " x="199.435" y="-39.27"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="202.56075" y="-39.27"> 4.</t年代pan> <tspan style="font-size: 12.50px; " x="208.81226" y="-39.27"> )</t年代pan> </text> </g> </g> </svg></span><br>当<年代vg height="13.6125" id="M66" style="vertical-align:-2.34499pt;width:85.449997px;" version="1.1" viewbox="0 0 85.449997 13.6125" width="85.449997" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-67.78,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.1514759" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.314975" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.228414" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="21.855244" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="33.895634" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="40.172138" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="46.861244" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="58.201466" y="0"> </t年代pan> </text> </g> </g> </svg>,<年代vg height="13.45" id="M67" style="vertical-align:-2.21957pt;width:86.1875px;" version="1.1" viewbox="0 0 86.1875 13.45" width="86.1875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-67.78,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="23.580658" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="32.145214" y="0"> ∞</t年代pan> <tspan style="font-size: 12.50px; " x="43.722992" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="48.936741" y="0"> ∞</t年代pan> <tspan style="font-size: 12.50px; " x="60.514519" y="0"> )</t年代pan> </text> </g> </g> </svg>， G-stationary process，<年代vg height="13.45" id="M68" style="vertical-align:-2.21957pt;width:27.775px;" version="1.1" viewbox="0 0 27.775 13.45" width="27.775" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8773699" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="14.040869" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.954308" y="0"> )</t年代pan> </text> </g> </g> </svg>，只是传统的弱平稳过程，当<年代vg height="13.725" id="M69" style="vertical-align:-2.34499pt;width:80.387497px;" version="1.1" viewbox="0 0 80.387497 13.725" width="80.387497" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.725)"> <g transform="translate(72,-61.02)"> <text transform="matrix(1,0,0,-1,-67.78,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.1514759" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.314975" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.228414" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="21.855244" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="33.895634" y="0"> l</t年代pan> <tspan style="font-size: 12.50px; " x="37.371468" y="0"> n</t年代pan> <tspan style="font-size: 12.50px; " x="47.786465" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="51.949966" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="55.863403" y="0"> )</t年代pan> </text> </g> </g> </svg>,<年代vg height="13.45" id="M70" style="vertical-align:-2.21957pt;width:63.612499px;" version="1.1" viewbox="0 0 63.612499 13.45" width="63.612499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="23.580658" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="29.832157" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="35.04591" y="0"> ∞</t年代pan> <tspan style="font-size: 12.50px; " x="46.623688" y="0"> )</t年代pan> </text> </g> </g> </svg>,<年代vg height="13.45" id="M71" style="vertical-align:-2.21957pt;width:32.987499px;" version="1.1" viewbox="0 0 32.987499 13.45" width="32.987499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8773699" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="14.040869" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.954308" y="0"> )</t年代pan> </text> </g> </g> </svg>被称为m-instary过程[<A.Href="#B8">13</A.>]．当<年代vg height="13.6125" id="M72" style="vertical-align:-2.34499pt;width:23.112499px;" version="1.1" viewbox="0 0 23.112499 13.6125" width="23.112499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.1514759" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.314975" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.228414" y="0"> )</t年代pan> </text> </g> </g> </svg>是Box-Cox变换，<年代pan class="equation" id="EEq4"> <svg height="33.9375" id="M73" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 33.9375" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,33.9375)"> <g transform="translate(72,-44.85)"> <text transform="matrix(1,0,0,-1,174.18,53.85)"> <tspan style="font-size: 12.50px; " x="37.68" y="-7.25"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.7270937" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="12.890593" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.804031" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="24.443365" y="0"> =</t年代pan> </text> <text transform="matrix(1,0,0,-1,215.77,66.1)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,223.78,61.1)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> 1.</t年代pan> </text> <g> <path d="m 211.86,56.98 29.52,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,223.24,45.1)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="241.50999" y="-8.75"> (</t年代pan> <tspan style="font-size: 12.50px; " x="245.67349" y="-8.75"> 2.</t年代pan> <tspan style="font-size: 12.50px; " x="251.925" y="-8.75"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="255.05075" y="-8.75"> 5.</t年代pan> <tspan style="font-size: 12.50px; " x="261.30225" y="-8.75"> )</t年代pan> </text> </g> </g> </svg></span> <svg height="13.45" id="M74" style="vertical-align:-2.21957pt;width:63.612499px;" version="1.1" viewbox="0 0 63.612499 13.45" width="63.612499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="23.580658" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="29.832157" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="35.04591" y="0"> ∞</t年代pan> <tspan style="font-size: 12.50px; " x="46.623688" y="0"> )</t年代pan> </text> </g> </g> </svg>,然后<年代vg height="13.45" id="M75" style="vertical-align:-2.21957pt;width:32.987499px;" version="1.1" viewbox="0 0 32.987499 13.45" width="32.987499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8773699" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="14.040869" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.954308" y="0"> )</t年代pan> </text> </g> </g> </svg>被称为<年代vg height="13.45" id="M76" style="vertical-align:-2.21957pt;width:30.275px;" version="1.1" viewbox="0 0 30.275 13.45" width="30.275" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> G</t年代pan> <tspan style="font-size: 12.50px; " x="9.0271664" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="13.190665" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.954788" y="0"> )</t年代pan> </text> </g> </g> </svg>静止的过程(<A.Href="#B11">3.</A.>]．当<年代vg height="16.700001" id="M77" style="vertical-align:-2.34499pt;width:91.224998px;" version="1.1" viewbox="0 0 91.224998 16.700001" width="91.224998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.7)"> <g transform="translate(72,-58.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.1514759" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.314975" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.228414" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="21.855244" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="33.895634" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="40.172138" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-27.86,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2.</t年代pan> </text> <text transform="matrix(1,0,0,-1,-20.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="11.327718" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.204128" y="0"> </t年代pan> </text> </g> </g> </svg>,<年代vg height="13.45" id="M78" style="vertical-align:-2.21957pt;width:68.824997px;" version="1.1" viewbox="0 0 68.824997 13.45" width="68.824997" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="23.580658" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="29.832157" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="35.04591" y="0"> ∞</t年代pan> <tspan style="font-size: 12.50px; " x="46.623688" y="0"> )</t年代pan> </text> </g> </g> </svg>与<年代vg height="12.4125" id="M79" style="vertical-align:-1.29163pt;width:107.9125px;" version="1.1" viewbox="0 0 107.9125 12.4125" width="107.9125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.4125)"> <g transform="translate(72,-62.07)"> <text transform="matrix(1,0,0,-1,-67.78,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7398367" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="21.780226" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="32.195225" y="0"> A.</t年代pan> <tspan style="font-size: 12.50px; " x="37.746555" y="0"> n</t年代pan> <tspan style="font-size: 12.50px; " x="43.998058" y="0"> D</t年代pan> <tspan style="font-size: 12.50px; " x="54.42556" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="63.765301" y="0"> ≥</t年代pan> <tspan style="font-size: 12.50px; " x="75.805687" y="0"> 0</t年代pan> </text> </g> </g> </svg>,然后<年代vg height="13.45" id="M80" style="vertical-align:-2.21957pt;width:32.987499px;" version="1.1" viewbox="0 0 32.987499 13.45" width="32.987499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8773699" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="14.040869" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.954308" y="0"> )</t年代pan> </text> </g> </g> </svg>称为线性啁啾平稳过程。看到刘(<A.Href="#B12">15</A.>，以及Robertson等人[<A.Href="#B16">16</A.>]．</年代pan></p> <h5 id="sec2.2">2.2.分析G-平稳过程的一般策略</H5.><p>我们给出了以下分析G平稳过程的概要。<年代pan class="list"><span class="list-item"><span class="list-label">(1)</年代pan><span class="list-content">转换时间轴以获得一个平稳的双实现(我们将在下面讨论这个)。</年代pan></span><span class="list-item"><span class="list-label">(2)</年代pan><span class="list-content">使用平稳时间序列的方法分析变换（双重）实现。例如，有时这是通过装配AR来实现的(<年代vg height="9.875" id="M81" style="vertical-align:-2.29482pt;width:7.7624998px;" version="1.1" viewbox="0 0 7.7624998 9.875" width="7.7624998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.875)"> <g transform="translate(72,-64.1)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>）或ARMA（<年代vg height="9.875" id="M82" style="vertical-align:-2.29482pt;width:22.0625px;" version="1.1" viewbox="0 0 22.0625 9.875" width="22.0625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.875)"> <g transform="translate(72,-64.1)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.1139669" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="11.315215" y="0"> </t年代pan> </text> </g> </g> </svg>）模型到双数据，然后根据手中的问题计算预测，光谱估计等（在当前设置中我们将过滤双数据）。</年代pan></span><span class="list-item"><span class="list-label">（3）</年代pan><span class="list-content">根据需要转换回原始时间刻度。</年代pan></span></span></p> <p><span class="statement" id="step1"><span class="statement-title"><i class="statement-title">步骤1（变换时间轴以获得平稳的双重实现）。</我></年代pan>寻找时间转换，<年代vg height="13.6125" id="M83" style="vertical-align:-2.34499pt;width:49.912498px;" version="1.1" viewbox="0 0 49.912498 13.6125" width="49.912498" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.3897533" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.430141" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.581617" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="31.745117" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.658558" y="0"> )</t年代pan> </text> </g> </g> </svg>，相当于确定抽样方案，以便<年代vg height="17.799999" id="M84" style="vertical-align:-3.2316pt;width:157.03751px;" version="1.1" viewbox="0 0 157.03751 17.799999" width="157.03751" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.8)"> <g transform="translate(72,-57.76)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8773699" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="14.040869" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-54,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-48.75,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="7.6268301" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="19.667219" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="29.544588" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="33.708088" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-8.39,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 1.</t年代pan> </text> <text transform="matrix(1,0,0,-1,2.48,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,12.57,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,17.81,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="11.802832" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="23.830717" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,48.34,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>哪里<年代vg height="14.3625" id="M85" style="vertical-align:-3.2316pt;width:15.0375px;" version="1.1" viewbox="0 0 15.0375 14.3625" width="15.0375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.3625)"> <g transform="translate(72,-60.51)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.26,60.66)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>是静止的。即对于g -平稳过程，通过抽样理论得到平稳对偶过程<年代vg height="13.45" id="M86" style="vertical-align:-2.21957pt;width:27.775px;" version="1.1" viewbox="0 0 27.775 13.45" width="27.775" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8773699" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="14.040869" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.954308" y="0"> )</t年代pan> </text> </g> </g> </svg>在点<年代vg height="17.799999" id="M87" style="vertical-align:-3.2316pt;width:101.5125px;" version="1.1" viewbox="0 0 101.5125 17.799999" width="101.5125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.8)"> <g transform="translate(72,-57.76)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-48.61,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-39.9,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-21.21,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 1.</t年代pan> </text> <text transform="matrix(1,0,0,-1,-10.34,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-0.25,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,4.99,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>.然而，一般而言，观测数据将在等间距点处获得，在这些点处不可用<年代vg height="13.0125" id="M88" style="vertical-align:-3.2316pt;width:11.5625px;" version="1.1" viewbox="0 0 11.5625 13.0125" width="11.5625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.0125)"> <g transform="translate(72,-61.59)"> <text transform="matrix(1,0,0,-1,-71.95,64.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-68.04,61.74)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>.插值主要用于处理此问题。参见Gray等人[<A.Href="#B8">13</A.>]蒋等[<A.Href="#B11">3.</A.>，和Woodward等人[<A.Href="#B18">14</A.>]。有关基于卡尔曼滤波的方法，请参见Wang等人[<A.Href="#B19">17</A.>]．<br>在下面的讨论中，我们将讨论安装<年代vg height="13.45" id="M89" style="vertical-align:-2.21957pt;width:30.275px;" version="1.1" viewbox="0 0 30.275 13.45" width="30.275" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> G</t年代pan> <tspan style="font-size: 12.50px; " x="9.0271664" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="13.190665" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.954788" y="0"> )</t年代pan> </text> </g> </g> </svg>-观测到的TVF数据的平稳模型<年代vg height="13.45" id="M90" style="vertical-align:-2.21957pt;width:30.275px;" version="1.1" viewbox="0 0 30.275 13.45" width="30.275" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> G</t年代pan> <tspan style="font-size: 12.50px; " x="9.0271664" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="13.190665" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.954788" y="0"> )</t年代pan> </text> </g> </g> </svg>模型具有足够的通用性，可以包含具有增加或减少频率行为的TVF数据。的值,<年代vg height="13.0125" id="M91" style="vertical-align:-3.2316pt;width:11.5625px;" version="1.1" viewbox="0 0 11.5625 13.0125" width="11.5625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.0125)"> <g transform="translate(72,-61.59)"> <text transform="matrix(1,0,0,-1,-71.95,64.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-68.04,61.74)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>，需要基于离散Box-Cox时间变换获得平稳对偶，<年代vg height="19.1625" id="M92" style="vertical-align:-4.3951pt;width:141.0625px;" version="1.1" viewbox="0 0 141.0625 19.1625" width="141.0625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.1625)"> <g transform="translate(72,-56.67)"> <text transform="matrix(1,0,0,-1,-71.95,61.1)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.239957" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.280346" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="26.443846" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-41.59,66.1)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.2799997"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-33.57,61.1)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="17.591721" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="21.75522" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="25.231054" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="29.394552" y="0"> Δ</t年代pan> </text> <text transform="matrix(1,0,0,-1,4.86,57.98)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,9.71,61.1)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.764123" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="13.703288" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="25.043509" y="0"> </t年代pan> </text> </g> </g> </svg>是<年代vg height="18.012501" id="M93" style="vertical-align:-3.2316pt;width:151.7625px;" version="1.1" viewbox="0 0 151.7625 18.012501" width="151.7625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,18.0125)"> <g transform="translate(72,-57.59)"> <text transform="matrix(1,0,0,-1,-71.95,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-68.04,57.74)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-59.32,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="20.354883" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="29.907175" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="41.259899" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="47.298847" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="51.462349" y="0"> Δ</t年代pan> </text> <text transform="matrix(1,0,0,-1,1.17,57.74)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,6.02,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.5397892" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="20.880011" y="0"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="27.13151" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,37.32,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1.</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> </t年代pan> </text> </g> </g> </svg>哪里<年代vg height="17.975" id="M94" style="vertical-align:-3.20526pt;width:84.400002px;" version="1.1" viewbox="0 0 84.400002 17.975" width="84.400002" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.975)"> <g transform="translate(72,-57.62)"> <text transform="matrix(1,0,0,-1,-71.95,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.189945" y="0"> Δ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-52.72,57.74)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-47.87,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.5397892" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="20.880011" y="0"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="27.13151" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,-16.57,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1.</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> </t年代pan> </text> </g> </g> </svg>被称为<我>实现偏移量</我>和<年代vg height="14.65" id="M95" style="vertical-align:-3.20526pt;width:17.4625px;" version="1.1" viewbox="0 0 17.4625 14.65" width="17.4625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.65)"> <g transform="translate(72,-60.28)"> <text transform="matrix(1,0,0,-1,-71.95,63.52)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Δ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.92,60.4)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>是采样增量[<A.Href="#B11">3.</A.>使用一个搜索例程来确定的值<年代vg height="10.8125" id="M96" style="vertical-align:-0.20064pt;width:13.7875px;" version="1.1" viewbox="0 0 13.7875 10.8125" width="13.7875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.8125)"> <g transform="translate(72,-63.35)"> <text transform="matrix(1,0,0,-1,-67.78,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>产生“最稳定”对偶的偏移量。对于每一组<年代vg height="10.8125" id="M97" style="vertical-align:-0.20064pt;width:13.7875px;" version="1.1" viewbox="0 0 13.7875 10.8125" width="13.7875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.8125)"> <g transform="translate(72,-63.35)"> <text transform="matrix(1,0,0,-1,-67.78,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>以及搜索中考虑的偏移值、数据、，<年代vg height="14.7125" id="M98" style="vertical-align:-3.2316pt;width:34.325001px;" version="1.1" viewbox="0 0 34.325001 14.7125" width="34.325001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.7125)"> <g transform="translate(72,-60.23)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8773699" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="14.040869" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-54,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-48.75,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>,<年代vg height="12.8875" id="M99" style="vertical-align:-1.76814pt;width:80.074997px;" version="1.1" viewbox="0 0 80.074997 12.8875" width="80.074997" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.8875)"> <g transform="translate(72,-61.69)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.239957" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.280346" y="0"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="28.531845" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="37.909096" y="0"> 2.</t年代pan> <tspan style="font-size: 12.50px; " x="44.160595" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="49.374348" y="0"> …</t年代pan> </text> </g> </g> </svg>的近似值为<年代vg height="14.7125" id="M100" style="vertical-align:-3.2316pt;width:24.362499px;" version="1.1" viewbox="0 0 24.362499 14.7125" width="24.362499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.7125)"> <g transform="translate(72,-60.23)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-68.04,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.79,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> '</t年代pan> <tspan style="font-size: 12.50px; " x="5.3637872" y="0"> 年代</t年代pan> </text> </g> </g> </svg>使用插值。然后索引<年代vg height="10.7375" id="M101" style="vertical-align:-0.13794pt;width:8.6000004px;" version="1.1" viewbox="0 0 8.6000004 10.7375" width="8.6000004" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.7375)"> <g transform="translate(72,-63.41)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>，与给定的双重实现<年代vg height="10.8125" id="M102" style="vertical-align:-0.20064pt;width:8.5749998px;" version="1.1" viewbox="0 0 8.5749998 10.8125" width="8.5749998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.8125)"> <g transform="translate(72,-63.35)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>并且获得偏移量，并且对于<年代vg height="10.8125" id="M103" style="vertical-align:-0.20064pt;width:8.5749998px;" version="1.1" viewbox="0 0 8.5749998 10.8125" width="8.5749998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.8125)"> <g transform="translate(72,-63.35)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>和偏移量，检查得到的双重实现的平稳性。江等[<A.Href="#B11">3.</A.>]建议通过检查转换后数据的前半部分和后半部分的特征(例如，样本自相关性)来测量平稳性。他们使用q统计量来测量两半的样本自相关性之间的差异。这种度量是基于这样一个事实，即在实现过程中，平稳性下的相关结构保持不变。软件，GWS，写在<年代vg height="11.325" id="M104" style="vertical-align:-0.51414pt;width:19.525px;" version="1.1" viewbox="0 0 19.525 11.325" width="19.525" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.325)"> <g transform="translate(72,-62.94)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> s</t年代pan> <tspan style="font-size: 12.50px; " x="6.9516678" y="0"> +</t年代pan> </text> </g> </g> </svg>，可用于执行此搜索过程，可从网站上的作者处获取<A.Href="http://www.texasoft.com/atsa" target="_blank">http://www.texasoft.com/atsa</A.>．该软件可用于适应一个<年代vg height="13.45" id="M105" style="vertical-align:-2.21957pt;width:30.275px;" version="1.1" viewbox="0 0 30.275 13.45" width="30.275" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> G</t年代pan> <tspan style="font-size: 12.50px; " x="9.0271664" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="13.190665" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.954788" y="0"> )</t年代pan> </text> </g> </g> </svg>对一组数据进行建模，并提供光谱分析、预测等方法。在这里涉及TVF数据分析的示例中，我们使用GWS软件包。</年代pan></p> <p><span class="statement" id="ex2"><span class="statement-title"><i class="statement-title">示例2.3（<年代vg height="13.45" id="M106" style="vertical-align:-2.21957pt;width:30.275px;" version="1.1" viewbox="0 0 30.275 13.45" width="30.275" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> G</t年代pan> <tspan style="font-size: 12.50px; " x="9.0271664" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="13.190665" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.954788" y="0"> )</t年代pan> </text> </g> </g> </svg>-分析示例中的TVF数据<A.Href="#ex1">1．1</A.>)．</我></年代pan>在本例中，我们执行<年代vg height="13.45" id="M107" style="vertical-align:-2.21957pt;width:30.275px;" version="1.1" viewbox="0 0 30.275 13.45" width="30.275" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> G</t年代pan> <tspan style="font-size: 12.50px; " x="9.0271664" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="13.190665" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.954788" y="0"> )</t年代pan> </text> </g> </g> </svg>在例子中讨论的时间序列的分析<A.Href="#ex1">1．1</A.>。我们还使用Wigner-Ville图，通过计算加窗版本的自协方差函数的傅里叶逆变换来显示数据中的时频行为（参见[<A.Href="#B18">14</A.>,<A.Href="#B13">18</A.>,<A.Href="#B2">19</A.>])．如前所述，<年代vg height="13.45" id="M108" style="vertical-align:-2.21957pt;width:27.775px;" version="1.1" viewbox="0 0 27.775 13.45" width="27.775" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8773699" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="14.040869" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.954308" y="0"> )</t年代pan> </text> </g> </g> </svg>在(<A.Href="#EEq2">１．２</A.>)是一个m平稳过程。数字<A.Href="//www.newsama.com/journals/jps/2012/738636/fig2/" target="_blank">2(一个)</A.>显示先前在图中显示的数据<A.Href="//www.newsama.com/journals/jps/2012/738636/fig1/" target="_blank">1（a）</A.>，图中<A.Href="//www.newsama.com/journals/jps/2012/738636/fig2/" target="_blank">2 (c)</A.>我们展示了相关的帕森光谱密度<A.Href="//www.newsama.com/journals/jps/2012/738636/fig2/" target="_blank">2 (c)</A.>所谓的“扩频”是指在大约<年代vg height="13.7375" id="M109" style="vertical-align:-2.34499pt;width:47.724998px;" version="1.1" viewbox="0 0 47.724998 13.7375" width="47.724998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.7375)"> <g transform="translate(72,-61.01)"> <text transform="matrix(1,0,0,-1,-71.95,63.4)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.402496" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="25.568636" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="31.820135" y="0"> 6.</t年代pan> </text> </g> </g> </svg>到<年代vg height="13.6125" id="M110" style="vertical-align:-2.34499pt;width:47.724998px;" version="1.1" viewbox="0 0 47.724998 13.6125" width="47.724998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.402496" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="25.568636" y="0"> 2.</t年代pan> <tspan style="font-size: 12.50px; " x="31.820135" y="0"> 2.</t年代pan> </text> </g> </g> </svg>这是由示例中记录的数据中的频率变化引起的<A.Href="#ex1">1．1</A.>．频率的变化可以在图中Wigner-Ville图中看到<A.Href="//www.newsama.com/journals/jps/2012/738636/fig2/" target="_blank">2 (e)</A.>。在绘图中，较暗的阴影对应于频率行为较强的位置。例如，图<A.Href="//www.newsama.com/journals/jps/2012/738636/fig2/" target="_blank">2(一个)</A.>显示一个非常明显的较低的TVF组件，它被朝大约的“条带”所示<年代vg height="13.6125" id="M111" style="vertical-align:-2.34499pt;width:39.912498px;" version="1.1" viewbox="0 0 39.912498 13.6125" width="39.912498" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.402496" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="25.568636" y="0"> 1.</t年代pan> </text> </g> </g> </svg>在数据的开头，而<年代vg height="10.9125" id="M112" style="vertical-align:-0.17555pt;width:47.849998px;" version="1.1" viewbox="0 0 47.849998 10.9125" width="47.849998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.9125)"> <g transform="translate(72,-63.27)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> 3.</t年代pan> <tspan style="font-size: 12.50px; " x="25.668659" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="31.920158" y="0"> 0</t年代pan> </text> </g> </g> </svg>与较低TVF分量相关的周期长度约为20<年代vg height="13.8" id="M113" style="vertical-align:-2.34499pt;width:58.137501px;" version="1.1" viewbox="0 0 58.137501 13.8" width="58.137501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.8)"> <g transform="translate(72,-60.96)"> <text transform="matrix(1,0,0,-1,-71.95,63.35)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.565995" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="26.606384" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="29.732134" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="35.983635" y="0"> 5.</t年代pan> <tspan style="font-size: 12.50px; " x="42.235134" y="0"> )</t年代pan> </text> </g> </g> </svg>它在视觉上与图中所示的组件一致<A.Href="//www.newsama.com/journals/jps/2012/738636/fig1/" target="_blank">1（b）</A.>。较高的TVF分量（由较轻的条带表示）表示频率约为<年代vg height="13.6125" id="M114" style="vertical-align:-2.34499pt;width:47.724998px;" version="1.1" viewbox="0 0 47.724998 13.6125" width="47.724998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.402496" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="25.568636" y="0"> 2.</t年代pan> <tspan style="font-size: 12.50px; " x="31.820135" y="0"> 2.</t年代pan> </text> </g> </g> </svg>（大约5个）在数据减少到约<年代vg height="13.6125" id="M115" style="vertical-align:-2.34499pt;width:39.912498px;" version="1.1" viewbox="0 0 39.912498 13.6125" width="39.912498" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.402496" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="25.568636" y="0"> 1.</t年代pan> </text> </g> </g> </svg>(周期约10)由约<年代vg height="10.9125" id="M116" style="vertical-align:-0.17555pt;width:47.849998px;" version="1.1" viewbox="0 0 47.849998 10.9125" width="47.849998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.9125)"> <g transform="translate(72,-63.27)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> 3.</t年代pan> <tspan style="font-size: 12.50px; " x="25.668659" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="31.920158" y="0"> 0</t年代pan> </text> </g> </g> </svg>.同样，这与图中所示一致<A.Href="//www.newsama.com/journals/jps/2012/738636/fig1/" target="_blank">1（c）</A.>．这两个条带之间的方格图案是干扰，并不表示强频率行为。Wigner-Ville的情节直观地说明了Example中提到的事实<A.Href="#ex1">1．1</A.>，较低频率的行为（底部条带）在数据开始时的开头处具有较高的频率，而不是数据集的末尾的较高频率分量（上部条带）。也就是说，不能绘制水平线，其完全在两条条内保持。使用GWS软件，看来一个<年代vg height="13.45" id="M117" style="vertical-align:-2.21957pt;width:30.275px;" version="1.1" viewbox="0 0 30.275 13.45" width="30.275" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> G</t年代pan> <tspan style="font-size: 12.50px; " x="9.0271664" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="13.190665" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.954788" y="0"> )</t年代pan> </text> </g> </g> </svg>转化为<年代vg height="10.9375" id="M118" style="vertical-align:-0.20064pt;width:35.787498px;" version="1.1" viewbox="0 0 35.787498 10.9375" width="35.787498" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.9375)"> <g transform="translate(72,-63.25)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.227454" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.267843" y="0"> 0</t年代pan> </text> </g> </g> </svg>（静止）和偏移175是生产静止双重的合理选择。基于该转换的双数据在图中给出<A.Href="//www.newsama.com/journals/jps/2012/738636/fig2/" target="_blank">2 (b)</A.>。高频和低频行为与图中所示类似<A.Href="//www.newsama.com/journals/jps/2012/738636/fig2/" target="_blank">2(一个)</A.>但这两种频率都没有表现出时变行为。图中对应的谱密度<A.Href="//www.newsama.com/journals/jps/2012/738636/fig2/" target="_blank">2 (d)</A.>显示两个不同的峰值（一个接近.07和另一个近.13）。也就是说，变换数据（即，G光谱密度）的光谱密度清楚地表明数据中存在两个主要的“频率”组件。图中的Wigner Ville情节<A.Href="//www.newsama.com/journals/jps/2012/738636/fig2/" target="_blank">2 (f)</A.>表示两条平行线，一条在<年代vg height="13.6125" id="M119" style="vertical-align:-2.34499pt;width:47.724998px;" version="1.1" viewbox="0 0 47.724998 13.6125" width="47.724998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.402496" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="25.568636" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="31.820135" y="0"> 7.</t年代pan> </text> </g> </g> </svg>还有一个大约<年代vg height="13.6125" id="M120" style="vertical-align:-2.34499pt;width:47.724998px;" version="1.1" viewbox="0 0 47.724998 13.6125" width="47.724998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.402496" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="25.568636" y="0"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="31.820135" y="0"> 3.</t年代pan> </text> </g> </g> </svg>.此图表明频率行为不随时间变化，这与平稳数据一致。</年代pan></p> <div class="floats-partial-section partial-carousel-wrapper" data-section="fig2"> <div class="floats-partial-content fig-content-fig2"> <div class="partial-carousel--single-image"> <img alt="(一)数据"nA.me="738636.fig.002a" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.002a.jpg"> <br> <strong>(一)数据</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="（b）双重数据"nA.me="738636.fig.002b" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.002b.jpg"> <br> <strong>（b）双重数据</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="（c）（a）的帕森谱密度"nA.me="738636.fig.002c" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.002c.jpg"> <br> <strong>（c）（a）的帕森谱密度</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="（d）（b）的帕森谱密度"nA.me="738636.fig.002d" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.002d.jpg"> <br> <strong>（d）（b）的帕森谱密度</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="（e）Wigner-Ville Plot for（a）"nA.me="738636.fig.002e" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.002e.jpg"> <br> <strong>（e）Wigner-Ville Plot for（a）</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="（f） Wigner Ville地块（b）"nA.me="738636.fig.002f" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.002f.jpg"> <br> <strong>（f） Wigner Ville地块（b）</年代trong> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs fig-thumbs-fig2"> <button class="partial-carousel-dot" aria-label="(a) Data"><img alt="(一)数据"nA.me="738636.fig.002a" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.002a.jpg"><br><strong>(一)数据</年代trong></button> <button class="partial-carousel-dot" aria-label="(b) Dual data"><img alt="（b）双重数据"nA.me="738636.fig.002b" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.002b.jpg"><br><strong>（b）双重数据</年代trong></button> <button class="partial-carousel-dot" aria-label="(c) Parzen spectral density for (a)"><img alt="（c）（a）的帕森谱密度"nA.me="738636.fig.002c" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.002c.jpg"><br><strong>（c）（a）的帕森谱密度</年代trong></button> <button class="partial-carousel-dot" aria-label="(d) Parzen spectral density for (b)"><img alt="（d）（b）的帕森谱密度"nA.me="738636.fig.002d" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.002d.jpg"><br><strong>（d）（b）的帕森谱密度</年代trong></button> <button class="partial-carousel-dot" aria-label="(e) Wigner-Ville plot for (a)"><img alt="（e）Wigner-Ville Plot for（a）"nA.me="738636.fig.002e" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.002e.jpg"><br><strong>（e）Wigner-Ville Plot for（a）</年代trong></button> <button class="partial-carousel-dot" aria-label="(f) Wigner-Ville plot for (b)"><img alt="（f） Wigner Ville地块（b）"nA.me="738636.fig.002f" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.002f.jpg"><br><strong>（f） Wigner Ville地块（b）</年代trong></button> </div> <div class="partial-carousel-next-prev fig-nav-fig2"> <button class="carousel-prev" aria-label="Previous"></button> <button class="carousel-next" aria-label="Next"></button> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图2</年代pan> <a aria-label="full view" class="caption-partial-url" href="//www.newsama.com/journals/jps/2012/738636/fig2/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> <a href="#EEq2">１．２</A.><年代vg height="10.9375" id="M121" style="vertical-align:-0.20064pt" version="1.1" viewbox="0 0 35.787498 10.9375" width="35.787498" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.9375)"> <g transform="translate(72,-63.25)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.227454" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.267843" y="0"> 0</t年代pan> </text> </g> </g> </svg> <table> (a)由信号输入()，（b）基于(C)（a）数据的Parzen谱密度，(d) (b)双数据的Parzen谱密度，(e) (a)数据的Wigner-Ville曲线图，(f) (b)数据的Wigner-Ville曲线图。</tA.ble> </div> </div> </div> <h4 id="the-g-filter">3.G-过滤器</H4.><p>在本节中，我们定义了用于过滤G平稳数据的G滤波器。G滤波器可被视为滤波G平稳信号的酉等效系统的具体示例（参见[<A.Href="#B1">4.</A.>])。</p> <p><span class="statement" id="deff3"><span class="statement-title"><i class="statement-title">定义3.1。</我></年代pan>给定一个随机过程<年代vg height="13.575" id="M122" style="vertical-align:-2.26974pt;width:116.1px;" version="1.1" viewbox="0 0 116.1 13.575" width="116.1" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.575)"> <g transform="translate(72,-61.14)"> <text transform="matrix(1,0,0,-1,-67.78,63.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ｛</t年代pan> <tspan style="font-size: 12.50px; " x="6.00144" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="15.87881" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="20.042309" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.955748" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="28.119247" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="37.483994" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="44.873268" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="56.913654" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="61.077156" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="67.353661" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="72.567413" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="78.443825" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="82.607323" y="0"> ｝</t年代pan> </text> </g> </g> </svg>和一个脉冲响应函数，<年代vg height="13.5625" id="M123" style="vertical-align:-2.21957pt;width:24.4375px;" version="1.1" viewbox="0 0 24.4375 13.5625" width="24.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.5625)"> <g transform="translate(72,-61.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ℎ</t年代pan> <tspan style="font-size: 12.50px; " x="7.214231" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="11.37773" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="15.291169" y="0"> )</t年代pan> </text> </g> </g> </svg>,<年代vg height="13.45" id="M124" style="vertical-align:-2.21957pt;width:56.512501px;" version="1.1" viewbox="0 0 56.512501 13.45" width="56.512501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="23.580658" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="29.857164" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="35.070915" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="40.947327" y="0"> )</t年代pan> </text> </g> </g> </svg>，表示G-滤波器或G-卷积<年代vg height="13.5625" id="M125" style="vertical-align:-2.21957pt;width:56.900002px;" version="1.1" viewbox="0 0 56.900002 13.5625" width="56.900002" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.5625)"> <g transform="translate(72,-61.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.653036" y="0"> ⊗</t年代pan> <tspan style="font-size: 12.50px; " x="25.956228" y="0"> ℎ</t年代pan> <tspan style="font-size: 12.50px; " x="33.17046" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="37.333958" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="41.247398" y="0"> )</t年代pan> </text> </g> </g> </svg>，被定义为<年代pan class="equation" id="eq4"> <svg height="15.75" id="M126" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 15.75" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.75)"> <g transform="translate(72,-59.4)"> <text transform="matrix(1,0,0,-1,118.79,63.2)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.265203" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.428702" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.34214" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.981474" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="39.009361" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="51.6749" y="0"> ⊗</t年代pan> <tspan style="font-size: 12.50px; " x="64.978088" y="0"> ℎ</t年代pan> <tspan style="font-size: 12.50px; " x="74.280319" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="78.443825" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="82.357262" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="89.984093" y="0"> ≡</t年代pan> <tspan style="font-size: 12.50px; " x="102.02448" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,230.7,60.07)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,239.16,63.2)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∗</t年代pan> <tspan style="font-size: 12.50px; " x="9.3147354" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,255.69,60.07)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,263.45,63.2)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.890593" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="17.054092" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.967531" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="25.131029" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="31.38253" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="201.3" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="205.4635" y="0"> 3.</t年代pan> <tspan style="font-size: 12.50px; " x="211.715" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="214.84074" y="0"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="221.09225" y="0"> )</t年代pan> </text> </g> </g> </svg></span>哪里<年代vg height="17.8125" id="M127" style="vertical-align:-3.24037pt;width:119.25px;" version="1.1" viewbox="0 0 119.25 17.8125" width="119.25" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.8125)"> <g transform="translate(72,-57.75)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.07,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-56.39,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.089921" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="17.716751" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="29.757139" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="39.63451" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="43.798008" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-5.94,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 1.</t年代pan> </text> <text transform="matrix(1,0,0,-1,4.93,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.089921" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="14.25342" y="0"> )</t年代pan> </text> </g> </g> </svg>和<年代vg height="17.8125" id="M128" style="vertical-align:-3.24037pt;width:112.5875px;" version="1.1" viewbox="0 0 112.5875 17.8125" width="112.5875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.8125)"> <g transform="translate(72,-57.75)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.74,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-59.06,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.089921" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="17.729254" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="29.757139" y="0"> ℎ</t年代pan> <tspan style="font-size: 12.50px; " x="36.971371" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="41.134869" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-11.26,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 1.</t年代pan> </text> <text transform="matrix(1,0,0,-1,-0.39,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.089921" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="14.25342" y="0"> )</t年代pan> </text> </g> </g> </svg>,<年代vg height="13.45" id="M129" style="vertical-align:-2.21957pt;width:83.487503px;" version="1.1" viewbox="0 0 83.487503 13.45" width="83.487503" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.3897533" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="21.430141" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="25.593641" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="34.158195" y="0"> ∞</t年代pan> <tspan style="font-size: 12.50px; " x="45.735973" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="50.949726" y="0"> ∞</t年代pan> <tspan style="font-size: 12.50px; " x="62.527504" y="0"> )</t年代pan> </text> </g> </g> </svg>是的对偶吗<年代vg height="13.45" id="M130" style="vertical-align:-2.21957pt;width:27.775px;" version="1.1" viewbox="0 0 27.775 13.45" width="27.775" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8773699" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="14.040869" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.954308" y="0"> )</t年代pan> </text> </g> </g> </svg>和<年代vg height="13.5625" id="M131" style="vertical-align:-2.21957pt;width:29.65px;" version="1.1" viewbox="0 0 29.65 13.5625" width="29.65" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.5625)"> <g transform="translate(72,-61.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ℎ</t年代pan> <tspan style="font-size: 12.50px; " x="7.214231" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="11.37773" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="15.291169" y="0"> )</t年代pan> </text> </g> </g> </svg>关于时间变形函数<年代vg height="13.6125" id="M132" style="vertical-align:-2.34499pt;width:28.325001px;" version="1.1" viewbox="0 0 28.325001 13.6125" width="28.325001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-67.78,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.1514759" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.314975" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.228414" y="0"> )</t年代pan> </text> </g> </g> </svg>,<年代vg height="14.8375" id="M133" style="vertical-align:-3.24037pt;width:66.112503px;" version="1.1" viewbox="0 0 66.112503 14.8375" width="66.112503" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.8375)"> <g transform="translate(72,-60.13)"> <text transform="matrix(1,0,0,-1,-71.95,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.07,60.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-53.61,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∗</t年代pan> <tspan style="font-size: 12.50px; " x="9.3022318" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-37.08,60.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-31.4,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.0769377" y="0"> )</t年代pan> </text> </g> </g> </svg>是通常的卷积，<年代vg height="18" id="M134" style="vertical-align:-4.03784pt;width:206.47501px;" version="1.1" viewbox="0 0 206.47501 18" width="206.47501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,18)"> <g transform="translate(72,-57.6)"> <text transform="matrix(1,0,0,-1,-71.95,61.67)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.07,58.55)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-53.61,61.67)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∗</t年代pan> <tspan style="font-size: 12.50px; " x="9.3022318" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-37.08,58.55)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-31.4,61.67)"> <tspan style="font-size: 12.50px; " x="27.75" y="0.02"> ∫</t年代pan> <tspan style="font-size: 12.50px; " x="35.529999" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.0769377" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="15.703768" y="0"> =</t年代pan> </text> <text transform="matrix(1,0,0,-1,14.01,58.55)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,19.69,61.67)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.552532" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="14.716031" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,41.63,58.55)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,47.31,61.67)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.852604" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="22.192825" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="28.59436" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="32.757858" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="39.409454" y="0"> </t年代pan> </text> </g> </g> </svg>．</年代pan></p> <p><span class="statement" id="thm1"><span class="statement-title">定理3.2。</年代pan><i>如果映射<年代vg height="13.6125" id="M135" style="vertical-align:-2.34499pt;width:23.112499px;" version="1.1" viewbox="0 0 23.112499 13.6125" width="23.112499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.1514759" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.314975" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.228414" y="0"> )</t年代pan> </text> </g> </g> </svg>来自原始空间<年代vg height="13.45" id="M136" style="vertical-align:-2.21957pt;width:56.512501px;" version="1.1" viewbox="0 0 56.512501 13.45" width="56.512501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="23.580658" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="29.857164" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="35.070915" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="40.947327" y="0"> )</t年代pan> </text> </g> </g> </svg>到对偶空间<年代vg height="13.45" id="M137" style="vertical-align:-2.21957pt;width:88.699997px;" version="1.1" viewbox="0 0 88.699997 13.45" width="88.699997" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.3897533" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="21.430141" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="25.593641" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="34.158195" y="0"> ∞</t年代pan> <tspan style="font-size: 12.50px; " x="45.735973" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="50.949726" y="0"> ∞</t年代pan> <tspan style="font-size: 12.50px; " x="62.527504" y="0"> )</t年代pan> </text> </g> </g> </svg>那么，这是什么呢</我><年代pan class="equation" id="eq5"> <svg height="39.450001" id="M138" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 39.450001" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,39.45)"> <g transform="translate(72,-40.44)"> <text transform="matrix(1,0,0,-1,56.42,52.08)"> <tspan style="font-size: 12.50px; " x="102.04" y="7.6999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.265203" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.428702" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.34214" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.981474" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="39.021862" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="51.6749" y="0"> ⊗</t年代pan> <tspan style="font-size: 12.50px; " x="64.978088" y="0"> ℎ</t年代pan> <tspan style="font-size: 12.50px; " x="74.280319" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="78.443825" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="82.357262" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="89.996597" y="0"> =</t年代pan> </text> <text transform="matrix(1,0,0,-1,165.46,66.1)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="-1.25" y="25.52"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,172.16,52.08)"> <tspan style="font-size: 12.50px; " x="38.740002" y="2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="44.599998" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.703048" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.866547" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.193066" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="29.444565" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,223.41,57.08)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 1.</t年代pan> </text> <text transform="matrix(1,0,0,-1,236.36,52.08)"> <tspan style="font-size: 12.50px; " x="66.809998" y="2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="74.75" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.890593" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="17.054092" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.967531" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="27.906696" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="39.259418" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="47.986515" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="52.150013" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="58.476532" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="62.64003" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,317.76,57.08)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,323.32,52.08)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.490017" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="16.729013" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.38061" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="29.707129" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="141.42999" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="145.59351" y="0"> 3.</t年代pan> <tspan style="font-size: 12.50px; " x="151.845" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="154.97075" y="0"> 2.</t年代pan> <tspan style="font-size: 12.50px; " x="161.22224" y="0"> )</t年代pan> </text> </g> </g> </svg></span><i></i></span></p> <p><span class="statement" id="proof1"><span class="statement-title"><i class="statement-title">证明。</我></年代pan>考虑以下:<年代pan class="equation" id="eq6"> <svg height="215.16251" id="M139" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 215.16251" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,215.1625)"> <g transform="translate(72,100.13)"> <text transform="matrix(1,0,0,-1,90.01,63.2)"> <tspan style="font-size: 12.50px; " x="26.99" y="30.51"> =</t年代pan> <tspan style="font-size: 12.50px; " x="39.029999" y="38.209999"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.265203" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.428702" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.34214" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.981474" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="39.021862" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="51.6749" y="0"> ⊗</t年代pan> <tspan style="font-size: 12.50px; " x="64.978088" y="0"> ℎ</t年代pan> <tspan style="font-size: 12.50px; " x="74.280319" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="78.443825" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="82.357262" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,136.04,46.72)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> <tspan style="font-size: 8.75px; " x="-1.25" y="25.530001"> −</t年代pan> <tspan style="font-size: 8.75px; " x="4.74512" y="25.530001"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,151.47,32.69)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,161.35,29.56)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,169.11,32.7)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1700001" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.096422" y="0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="16.335419" y="0.0099999998"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,192.66,29.56)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,200.43,32.7)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="-83.43" y="40.349998"> =</t年代pan> <tspan style="font-size: 12.50px; " x="-71.389999" y="48.040001"> </t年代pan> <tspan style="font-size: 12.50px; " x="4.1599998" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.887094" y="0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="17.050592" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.964031" y="0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="27.903196" y="0.0099999998"> −</t年代pan> <tspan style="font-size: 12.50px; " x="39.25592" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="45.182343" y="0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="51.421341" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="58.072937" y="0.0099999998"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,136.04,6.38)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="-1.25" y="25.530001"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,142.73,-7.65)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,152.61,-10.77)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,160.38,-7.64)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.089921" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="16.328918" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,183.93,-10.77)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,191.69,-7.64)"> <tspan style="font-size: 12.50px; " x="-74.690002" y="40.43"> =</t年代pan> <tspan style="font-size: 12.50px; " x="-62.650002" y="48.130001"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.890593" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="17.054092" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.967531" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="27.906696" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="39.259418" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="47.986515" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="52.150013" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="58.476532" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="62.64003" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="68.891533" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="75.543129" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="84.282722" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="88.44622" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="94.772743" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,136.04,-34.05)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="-1.25" y="25.52"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,142.74,-48.07)"> <tspan style="font-size: 12.50px; " x="38.740002" y="2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="44.59" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.703048" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.866547" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.193066" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="29.432062" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,193.99,-43.07)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 1.</t年代pan> </text> <text transform="matrix(1,0,0,-1,206.94,-48.07)"> <tspan style="font-size: 12.50px; " x="66.809998" y="2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="-89.940002" y="40.43"> =</t年代pan> <tspan style="font-size: 12.50px; " x="-77.900002" y="48.130001"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.890593" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="17.054092" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.967531" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="27.906696" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="39.259418" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="47.986515" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="52.150013" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="58.476532" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="62.64003" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="74.75" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="81.401596" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="90.128693" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="94.292191" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="100.61871" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,136.04,-74.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="-1.25" y="25.530001"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,142.74,-88.5)"> <tspan style="font-size: 12.50px; " x="38.740002" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="44.59" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.703048" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.866547" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.193066" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="29.432062" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,193.99,-83.5)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 1.</t年代pan> </text> <text transform="matrix(1,0,0,-1,206.94,-88.49)"> <tspan style="font-size: 12.50px; " x="66.809998" y="2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="74.75" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.890593" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="17.054092" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.967531" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="27.906696" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="39.259418" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="47.986515" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="52.150013" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="58.476532" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="62.64003" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,288.34,-83.5)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,293.9,-88.49)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.490017" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="16.729013" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.38061" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="29.707129" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="170.85001" y="-71.300003"> (</t年代pan> <tspan style="font-size: 12.50px; " x="175.0135" y="-71.300003"> 3.</t年代pan> <tspan style="font-size: 12.50px; " x="181.265" y="-71.300003"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="184.39075" y="-71.300003"> 3.</t年代pan> <tspan style="font-size: 12.50px; " x="190.64224" y="-71.300003"> )</t年代pan> </text> </g> </g> </svg></span></span></p> <p><span class="statement" id="coro1"><span class="statement-title">推论3.3。</年代pan><i>自从<年代vg height="14.8375" id="M140" style="vertical-align:-3.24037pt;width:156.5125px;" version="1.1" viewbox="0 0 156.5125 14.8375" width="156.5125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.8375)"> <g transform="translate(72,-60.13)"> <text transform="matrix(1,0,0,-1,-71.95,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.07,60.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-53.61,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∗</t年代pan> <tspan style="font-size: 12.50px; " x="9.3022318" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-37.08,60.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-31.4,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.089921" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="17.716751" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="29.757139" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,5.58,60.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,14.03,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∗</t年代pan> <tspan style="font-size: 12.50px; " x="9.3147354" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,33.23,60.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,38.91,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.089921" y="0"> )</t年代pan> </text> </g> </g> </svg>哪里<年代vg height="13.6125" id="M141" style="vertical-align:-2.34499pt;width:49.912498px;" version="1.1" viewbox="0 0 49.912498 13.6125" width="49.912498" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.3897533" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.430141" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.581617" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="31.745117" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.658558" y="0"> )</t年代pan> </text> </g> </g> </svg>紧接着<年代vg height="13.5625" id="M142" style="vertical-align:-2.21957pt;width:133.05px;" version="1.1" viewbox="0 0 133.05 13.5625" width="133.05" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.5625)"> <g transform="translate(72,-61.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.653036" y="0"> ⊗</t年代pan> <tspan style="font-size: 12.50px; " x="25.956228" y="0"> ℎ</t年代pan> <tspan style="font-size: 12.50px; " x="33.17046" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="37.333958" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="41.247398" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="48.88673" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="60.914616" y="0"> ℎ</t年代pan> <tspan style="font-size: 12.50px; " x="70.917015" y="0"> ⊗</t年代pan> <tspan style="font-size: 12.50px; " x="84.220207" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="94.09758" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="98.261078" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="102.17451" y="0"> )</t年代pan> </text> </g> </g> </svg>．</我></年代pan></p> <p><span class="statement" id="thm2"><span class="statement-title">定理3.4。</年代pan><i>允许<年代vg height="13.575" id="M143" style="vertical-align:-2.26974pt;width:110.8875px;" version="1.1" viewbox="0 0 110.8875 13.575" width="110.8875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.575)"> <g transform="translate(72,-61.14)"> <text transform="matrix(1,0,0,-1,-71.95,63.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ｛</t年代pan> <tspan style="font-size: 12.50px; " x="6.00144" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="15.87881" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="20.042309" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.955748" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="28.119247" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="37.483994" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="44.873268" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="56.913654" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="61.077156" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="67.353661" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="72.567413" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="78.443825" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="82.607323" y="0"> ｝</t年代pan> </text> </g> </g> </svg>为具有时间-变形函数的g -平稳输入过程<年代vg height="13.6125" id="M144" style="vertical-align:-2.34499pt;width:28.325001px;" version="1.1" viewbox="0 0 28.325001 13.6125" width="28.325001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-67.78,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.1514759" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.314975" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.228414" y="0"> )</t年代pan> </text> </g> </g> </svg>,然后</我><年代pan class="list"><span class="list-item"><span class="list-label">（a）</年代pan><span class="list-content"><i>输出的过程,<年代vg height="13.5625" id="M145" style="vertical-align:-2.21957pt;width:99.949997px;" version="1.1" viewbox="0 0 99.949997 13.5625" width="99.949997" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.5625)"> <g transform="translate(72,-61.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.689105" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.852604" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.766043" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="22.392874" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="34.433262" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="47.0863" y="0"> ⊗</t年代pan> <tspan style="font-size: 12.50px; " x="60.389488" y="0"> ℎ</t年代pan> <tspan style="font-size: 12.50px; " x="67.603722" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="71.76722" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="75.680656" y="0"> )</t年代pan> </text> </g> </g> </svg>,<年代vg height="13.45" id="M146" style="vertical-align:-2.21957pt;width:56.512501px;" version="1.1" viewbox="0 0 56.512501 13.45" width="56.512501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="23.580658" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="29.857164" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="35.070915" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="40.947327" y="0"> )</t年代pan> </text> </g> </g> </svg>哪里<年代vg height="13.5625" id="M147" style="vertical-align:-2.21957pt;width:29.1px;" version="1.1" viewbox="0 0 29.1 13.5625" width="29.1" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.5625)"> <g transform="translate(72,-61.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ℎ</t年代pan> <tspan style="font-size: 12.50px; " x="7.214231" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="11.37773" y="0"> t</t年代pan> <tspan style="font-size: 12.50px; " x="14.853564" y="0"> )</t年代pan> </text> </g> </g> </svg>是脉冲响应函数，也是G-insivalary，</我></年代pan></span><span class="list-item"><span class="list-label">（b）</年代pan><span class="list-content"><i>分子的G-谱<年代vg height="13.45" id="M148" style="vertical-align:-2.21957pt;width:32.987499px;" version="1.1" viewbox="0 0 32.987499 13.45" width="32.987499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8773699" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="14.040869" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.954308" y="0"> )</t年代pan> </text> </g> </g> </svg>和<年代vg height="13.45" id="M149" style="vertical-align:-2.21957pt;width:29px;" version="1.1" viewbox="0 0 29 13.45" width="29" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.689105" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.852604" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.766043" y="0"> )</t年代pan> </text> </g> </g> </svg>满足<年代pan class="equation" id="eq7"> <svg height="20.700001" id="M150" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 20.700001" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,20.7)"> <g transform="translate(72,-55.44)"> <text transform="matrix(1,0,0,-1,123.62,59.28)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,132.8,56.15)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,141.82,59.28)"> <tspan style="font-size: 12.50px; " x="42.98" y="-1.58"> |</t年代pan> <tspan style="font-size: 12.50px; " x="42.98" y="1.58"> |</t年代pan> <tspan style="font-size: 12.50px; " x="45.48" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.102664" y="0"> ;</t年代pan> <tspan style="font-size: 12.50px; " x="16.653996" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.305592" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="30.944925" y="0"> =</t年代pan> </text> <text transform="matrix(1,0,0,-1,198.76,56.15)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,206.39,59.28)"> <tspan style="font-size: 12.50px; " x="27.48" y="-1.58"> |</t年代pan> <tspan style="font-size: 12.50px; " x="27.48" y="1.58"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.102664" y="0"> ;</t年代pan> <tspan style="font-size: 12.50px; " x="16.653996" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.305592" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,236.37,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2.</t年代pan> </text> <text transform="matrix(1,0,0,-1,241.24,59.28)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,250.42,56.15)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,260.44,59.28)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.102664" y="0"> ;</t年代pan> <tspan style="font-size: 12.50px; " x="16.653996" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.305592" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="29.557093" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="204.31" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="208.4735" y="0"> 3.</t年代pan> <tspan style="font-size: 12.50px; " x="214.72501" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="217.85075" y="0"> 4.</t年代pan> <tspan style="font-size: 12.50px; " x="224.10225" y="0"> )</t年代pan> </text> </g> </g> </svg></span>哪里<年代vg height="20.512501" id="M151" style="vertical-align:-5.4048pt;width:254.85001px;" version="1.1" viewbox="0 0 254.85001 20.512501" width="254.85001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,20.5125)"> <g transform="translate(72,-55.59)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-60.5,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.102664" y="0"> ;</t年代pan> <tspan style="font-size: 12.50px; " x="16.653996" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.805471" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="30.444805" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="42.485195" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-3.5,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,1.55,55.72)"> <tspan style="font-size: 6.25px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,6.25,61.03)"> <tspan style="font-size: 12.50px; " x="30.780001" y="0.02"> ∫</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.102664" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="18.729494" y="0"> =</t年代pan> </text> <text transform="matrix(1,0,0,-1,42.73,68.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> <tspan style="font-size: 8.75px; " x="-2.25" y="11.46"> −</t年代pan> <tspan style="font-size: 8.75px; " x="3.74512" y="11.46"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,57.16,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,62.72,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 2.</t年代pan> <tspan style="font-size: 8.75px; " x="10.37112" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="15.692336" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="18.414207" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="23.271568" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,92.1,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,99.32,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,105,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.089921" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="14.25342" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.905016" y="0"> </t年代pan> </text> </g> </g> </svg>的频响函数是<年代vg height="14.8375" id="M152" style="vertical-align:-3.24037pt;width:34.0625px;" version="1.1" viewbox="0 0 34.0625 14.8375" width="34.0625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.8375)"> <g transform="translate(72,-60.13)"> <text transform="matrix(1,0,0,-1,-71.95,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.74,60.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-59.06,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.089921" y="0"> )</t年代pan> </text> </g> </g> </svg>．</我></年代pan></span></span><i></i></span></p> <p><span class="statement" id="proof2"><span class="statement-title"><i class="statement-title">证明。</我></年代pan>（a）让<年代vg height="14.7875" id="M153" style="vertical-align:-3.24037pt;width:147.47501px;" version="1.1" viewbox="0 0 147.47501 14.7875" width="147.47501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.7875)"> <g transform="translate(72,-60.17)"> <text transform="matrix(1,0,0,-1,-71.95,63.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ｛</t年代pan> <tspan style="font-size: 12.50px; " x="6.00144" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-56.07,60.32)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-50.39,63.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.089921" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="14.25342" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="23.618168" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="33.020424" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="45.060814" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="49.224312" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="57.788864" y="0"> ∞</t年代pan> <tspan style="font-size: 12.50px; " x="69.366646" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="74.580399" y="0"> ∞</t年代pan> <tspan style="font-size: 12.50px; " x="86.158173" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="90.321671" y="0"> ｝</t年代pan> </text> </g> </g> </svg>是静止的对偶过程<年代vg height="13.45" id="M154" style="vertical-align:-2.21957pt;width:27.775px;" version="1.1" viewbox="0 0 27.775 13.45" width="27.775" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8773699" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="14.040869" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.954308" y="0"> )</t年代pan> </text> </g> </g> </svg>和<年代vg height="14.8375" id="M155" style="vertical-align:-3.24037pt;width:141.55px;" version="1.1" viewbox="0 0 141.55 14.8375" width="141.55" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.8375)"> <g transform="translate(72,-60.13)"> <text transform="matrix(1,0,0,-1,-71.95,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ｛</t年代pan> <tspan style="font-size: 12.50px; " x="6.00144" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-58.73,60.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-53.05,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.089921" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="14.25342" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="21.530167" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="30.932423" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="42.972813" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="47.136311" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="55.700867" y="0"> ∞</t年代pan> <tspan style="font-size: 12.50px; " x="67.278641" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="72.492393" y="0"> ∞</t年代pan> <tspan style="font-size: 12.50px; " x="84.070175" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="88.233673" y="0"> ｝</t年代pan> </text> </g> </g> </svg>双重身份<年代vg height="13.5625" id="M156" style="vertical-align:-2.21957pt;width:23.9px;" version="1.1" viewbox="0 0 23.9 13.5625" width="23.9" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.5625)"> <g transform="translate(72,-61.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ℎ</t年代pan> <tspan style="font-size: 12.50px; " x="7.214231" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="11.37773" y="0"> t</t年代pan> <tspan style="font-size: 12.50px; " x="14.853564" y="0"> )</t年代pan> </text> </g> </g> </svg>. 自从<年代vg height="14.725" id="M157" style="vertical-align:-3.24037pt;width:37.387501px;" version="1.1" viewbox="0 0 37.387501 14.725" width="37.387501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.725)"> <g transform="translate(72,-60.22)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.07,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-56.39,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.089921" y="0"> )</t年代pan> </text> </g> </g> </svg>是静止的,然后<年代vg height="14.8375" id="M158" style="vertical-align:-3.24037pt;width:121.2875px;" version="1.1" viewbox="0 0 121.2875 14.8375" width="121.2875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.8375)"> <g transform="translate(72,-60.13)"> <text transform="matrix(1,0,0,-1,-71.95,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.26,60.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-59.58,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.089921" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="17.716751" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="29.757139" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-19.94,60.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-11.48,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∗</t年代pan> <tspan style="font-size: 12.50px; " x="9.3147354" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,5.05,60.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,10.73,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.089921" y="0"> )</t年代pan> </text> </g> </g> </svg>是静止的。因此<年代pan class="equation" id="eq8"> <svg height="15.75" id="M159" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 15.75" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.75)"> <g transform="translate(72,-59.4)"> <text transform="matrix(1,0,0,-1,59.53,63.2)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.265203" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.428702" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.34214" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.981474" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="39.021862" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="51.6749" y="0"> ⊗</t年代pan> <tspan style="font-size: 12.50px; " x="64.978088" y="0"> ℎ</t年代pan> <tspan style="font-size: 12.50px; " x="74.280319" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="78.443825" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="82.357262" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="89.996597" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="102.02448" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,171.44,60.07)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,179.9,63.2)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∗</t年代pan> <tspan style="font-size: 12.50px; " x="9.3147354" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,196.43,60.07)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,204.2,63.2)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.890593" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="17.054092" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.967531" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="25.131029" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="32.770363" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="44.810753" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,255.7,60.07)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,263.46,63.2)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.890593" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="17.054092" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.967531" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="25.131029" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="31.38253" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="50.72467" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="58.113945" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="72.229828" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="76.393333" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="82.669838" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="87.883591" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="93.759995" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="201.28999" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="205.45351" y="0"> 3.</t年代pan> <tspan style="font-size: 12.50px; " x="211.705" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="214.83075" y="0"> 5.</t年代pan> <tspan style="font-size: 12.50px; " x="221.08224" y="0"> )</t年代pan> </text> </g> </g> </svg></span>是G-平稳的。<br>(b)让<年代vg height="17.424999" id="M160" style="vertical-align:-5.4048pt;width:43px;" version="1.1" viewbox="0 0 43 17.424999" width="43" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.425)"> <g transform="translate(72,-58.06)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.52,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.61,58.18)"> <tspan style="font-size: 6.25px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-52.91,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.102664" y="0"> )</t年代pan> </text> </g> </g> </svg>和<年代vg height="17.424999" id="M161" style="vertical-align:-5.4048pt;width:47.775002px;" version="1.1" viewbox="0 0 47.775002 17.424999" width="47.775002" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.425)"> <g transform="translate(72,-58.06)"> <text transform="matrix(1,0,0,-1,-67.78,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-58.48,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-53.8,58.18)"> <tspan style="font-size: 6.25px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-49.1,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.102664" y="0"> )</t年代pan> </text> </g> </g> </svg>是过程的平稳对偶的谱<年代vg height="14.725" id="M162" style="vertical-align:-3.24037pt;width:42.599998px;" version="1.1" viewbox="0 0 42.599998 14.725" width="42.599998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.725)"> <g transform="translate(72,-60.22)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.07,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-56.39,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.089921" y="0"> )</t年代pan> </text> </g> </g> </svg>和<年代vg height="14.725" id="M163" style="vertical-align:-3.24037pt;width:38.612499px;" version="1.1" viewbox="0 0 38.612499 14.725" width="38.612499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.725)"> <g transform="translate(72,-60.22)"> <text transform="matrix(1,0,0,-1,-67.78,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-61.09,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-55.41,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.089921" y="0"> )</t年代pan> </text> </g> </g> </svg>的g谱<年代vg height="13.45" id="M164" style="vertical-align:-2.21957pt;width:27.775px;" version="1.1" viewbox="0 0 27.775 13.45" width="27.775" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8773699" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="14.040869" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.954308" y="0"> )</t年代pan> </text> </g> </g> </svg>和<年代vg height="13.45" id="M165" style="vertical-align:-2.21957pt;width:23.7875px;" version="1.1" viewbox="0 0 23.7875 13.45" width="23.7875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.689105" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.852604" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.766043" y="0"> )</t年代pan> </text> </g> </g> </svg>. 自从<年代vg height="17.424999" id="M166" style="vertical-align:-5.4048pt;width:42.599998px;" version="1.1" viewbox="0 0 42.599998 17.424999" width="42.599998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.425)"> <g transform="translate(72,-58.06)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.99,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.94,58.18)"> <tspan style="font-size: 6.25px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-53.24,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.102664" y="0"> )</t年代pan> </text> </g> </g> </svg>的频响函数是<年代vg height="14.8375" id="M167" style="vertical-align:-3.24037pt;width:39.262501px;" version="1.1" viewbox="0 0 39.262501 14.8375" width="39.262501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.8375)"> <g transform="translate(72,-60.13)"> <text transform="matrix(1,0,0,-1,-67.78,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-60.57,60.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.89,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.089921" y="0"> )</t年代pan> </text> </g> </g> </svg>然后根据基本的线性滤波定理，得出<年代pan class="equation" id="eq9"> <svg height="23.65" id="M168" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 23.65" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,23.65)"> <g transform="translate(72,-53.08)"> <text transform="matrix(1,0,0,-1,28.11,58.52)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,35.54,55.39)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,40.22,53.21)"> <tspan style="font-size: 6.25px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,47,58.53)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="32.279999" y="-2.3299999"> |</t年代pan> <tspan style="font-size: 12.50px; " x="32.279999" y="2.3399999"> |</t年代pan> <tspan style="font-size: 12.50px; " x="34.779999" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="4.1599998" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.599525" y="0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="20.238857" y="0.0099999998"> =</t年代pan> </text> <text transform="matrix(1,0,0,-1,90.74,55.39)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,95.79,53.21)"> <tspan style="font-size: 6.25px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,102.57,58.53)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.77" y="-2.3299999"> |</t年代pan> <tspan style="font-size: 12.50px; " x="16.77" y="2.3399999"> |</t年代pan> <tspan style="font-size: 12.50px; " x="4.1700001" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.597022" y="0.0099999998"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,121.84,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2.</t年代pan> </text> <text transform="matrix(1,0,0,-1,126.71,58.52)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,134.14,55.39)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,141.06,53.21)"> <tspan style="font-size: 6.25px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,147.84,58.53)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1599998" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.599525" y="0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="18.838522" y="0.0099999998"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="32.391773" y="0.0099999998"> t</t年代pan> <tspan style="font-size: 12.50px; " x="35.967632" y="0.0099999998"> H</t年代pan> <tspan style="font-size: 12.50px; " x="42.219131" y="0.0099999998"> A.</t年代pan> <tspan style="font-size: 12.50px; " x="47.770466" y="0.0099999998"> t</t年代pan> <tspan style="font-size: 12.50px; " x="55.409798" y="0.0099999998"> 我</t年代pan> <tspan style="font-size: 12.50px; " x="58.885632" y="0.0099999998"> 年代</t年代pan> <tspan style="font-size: 12.50px; " x="63.749298" y="0.0099999998"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="71.038544" y="0.0099999998"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,228.32,55.39)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,237.33,58.53)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="42.990002" y="-1.58"> |</t年代pan> <tspan style="font-size: 12.50px; " x="42.990002" y="1.59"> |</t年代pan> <tspan style="font-size: 12.50px; " x="45.490002" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="4.1599998" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.099165" y="0.0099999998"> ;</t年代pan> <tspan style="font-size: 12.50px; " x="16.650497" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.302094" y="0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="30.941425" y="0.0099999998"> =</t年代pan> </text> <text transform="matrix(1,0,0,-1,294.27,55.39)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,301.9,58.53)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="27.48" y="-1.58"> |</t年代pan> <tspan style="font-size: 12.50px; " x="27.48" y="1.59"> |</t年代pan> <tspan style="font-size: 12.50px; " x="4.1700001" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.109165" y="0.0099999998"> ;</t年代pan> <tspan style="font-size: 12.50px; " x="16.660498" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.312094" y="0.0099999998"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,331.88,65.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2.</t年代pan> </text> <text transform="matrix(1,0,0,-1,336.76,58.52)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,345.93,55.39)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,355.95,58.53)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1700001" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.109165" y="0.0099999998"> ;</t年代pan> <tspan style="font-size: 12.50px; " x="16.660498" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.312094" y="0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="29.563593" y="0.0099999998"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="108.8" y="0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="112.9635" y="0.0099999998"> 3.</t年代pan> <tspan style="font-size: 12.50px; " x="119.215" y="0.0099999998"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="122.34075" y="0.0099999998"> 6.</t年代pan> <tspan style="font-size: 12.50px; " x="128.59225" y="0.0099999998"> )</t年代pan> </text> </g> </g> </svg></span>如果映射<年代vg height="13.6125" id="M169" style="vertical-align:-2.34499pt;width:23.112499px;" version="1.1" viewbox="0 0 23.112499 13.6125" width="23.112499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.1514759" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.314975" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.228414" y="0"> )</t年代pan> </text> </g> </g> </svg>来自原始空间<年代vg height="13.45" id="M170" style="vertical-align:-2.21957pt;width:56.512501px;" version="1.1" viewbox="0 0 56.512501 13.45" width="56.512501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="23.580658" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="29.857164" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="35.070915" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="40.947327" y="0"> )</t年代pan> </text> </g> </g> </svg>到对偶空间<年代vg height="13.45" id="M171" style="vertical-align:-2.21957pt;width:83.487503px;" version="1.1" viewbox="0 0 83.487503 13.45" width="83.487503" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.3897533" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="21.430141" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="25.593641" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="34.158195" y="0"> ∞</t年代pan> <tspan style="font-size: 12.50px; " x="45.735973" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="50.949726" y="0"> ∞</t年代pan> <tspan style="font-size: 12.50px; " x="62.527504" y="0"> )</t年代pan> </text> </g> </g> </svg>那么，这是什么呢<年代vg height="22.9125" id="M172" style="vertical-align:-5.4048pt;width:422.52499px;" version="1.1" viewbox="0 0 422.52499 22.9125" width="422.52499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,22.9125)"> <g transform="translate(72,-53.67)"> <text transform="matrix(1,0,0,-1,-71.95,59.11)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-60.5,55.98)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.95,59.11)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.102664" y="0"> ;</t年代pan> <tspan style="font-size: 12.50px; " x="16.653996" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.805471" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="30.444805" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="42.485195" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-3.5,55.98)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,1.55,53.79)"> <tspan style="font-size: 6.25px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,6.25,59.11)"> <tspan style="font-size: 12.50px; " x="30.780001" y="0.029999999"> ∫</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.102664" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="18.729494" y="0"> =</t年代pan> </text> <text transform="matrix(1,0,0,-1,42.73,66.1)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> <tspan style="font-size: 8.75px; " x="-2.25" y="11.46"> −</t年代pan> <tspan style="font-size: 8.75px; " x="3.74512" y="11.46"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,57.16,59.11)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,62.72,64.11)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 2.</t年代pan> <tspan style="font-size: 8.75px; " x="10.37112" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="15.692336" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="18.414207" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="23.271568" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,92.73,59.11)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,99.94,55.98)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,105.62,59.11)"> <tspan style="font-size: 12.50px; " x="42.34" y="0.029999999"> ∫</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.089921" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="14.25342" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.905016" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="30.294769" y="0"> =</t年代pan> </text> <text transform="matrix(1,0,0,-1,153.66,66.1)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="-2.25" y="11.46"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,160.36,59.11)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,165.92,64.11)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 2.</t年代pan> <tspan style="font-size: 8.75px; " x="10.37112" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="15.692336" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="18.414207" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="23.271568" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="27.577553" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="30.491968" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="34.920479" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,206.34,59.11)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ℎ</t年代pan> <tspan style="font-size: 12.50px; " x="7.214231" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="11.37773" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.704248" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="21.867746" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,234.86,64.11)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,238.33,59.11)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.490017" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="14.653516" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="21.305113" y="0"> </t年代pan> </text> </g> </g> </svg>．</年代pan></p> <h5 id="sec3.1">３．１．的M-Filter</H5.><p>灰色和张[<A.Href="#B20">12</A.>]引入了m卷积或m滤波器对m -平稳过程进行滤波。当<年代vg height="13.725" id="M173" style="vertical-align:-2.34499pt;width:75.175003px;" version="1.1" viewbox="0 0 75.175003 13.725" width="75.175003" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.725)"> <g transform="translate(72,-61.02)"> <text transform="matrix(1,0,0,-1,-71.95,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.1514759" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.314975" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.228414" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="21.855244" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="33.895634" y="0"> l</t年代pan> <tspan style="font-size: 12.50px; " x="37.371468" y="0"> n</t年代pan> <tspan style="font-size: 12.50px; " x="47.786465" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="51.949966" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="55.863403" y="0"> )</t年代pan> </text> </g> </g> </svg>,<年代vg height="13.45" id="M174" style="vertical-align:-2.21957pt;width:63.612499px;" version="1.1" viewbox="0 0 63.612499 13.45" width="63.612499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="23.580658" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="29.832157" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="35.04591" y="0"> ∞</t年代pan> <tspan style="font-size: 12.50px; " x="46.623688" y="0"> )</t年代pan> </text> </g> </g> </svg>，然后就这样了<年代pan class="equation" id="eq10"> <svg height="36.887501" id="M175" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 36.887501" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,36.8875)"> <g transform="translate(72,-42.49)"> <text transform="matrix(1,0,0,-1,-4,54.16)"> <tspan style="font-size: 12.50px; " x="39.02" y="7.6999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.265203" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.428702" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.34214" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.981474" y="0"> =</t年代pan> </text> <text transform="matrix(1,0,0,-1,42.02,68.19)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> <tspan style="font-size: 8.75px; " x="-1.25" y="25.530001"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,52.71,54.16)"> <tspan style="font-size: 12.50px; " x="38.75" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="44.599998" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.703048" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.866547" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.193066" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="29.444565" y="0"> ℎ</t年代pan> </text> <text transform="matrix(1,0,0,-1,103.96,59.16)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 1.</t年代pan> </text> <text transform="matrix(1,0,0,-1,116.91,54.16)"> <tspan style="font-size: 12.50px; " x="66.82" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="120.3" y="7.6999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.890593" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="17.054092" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.967531" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="27.919199" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="39.259418" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="47.986515" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="52.150013" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="58.476532" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="62.64003" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="74.75" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="81.401596" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="90.128693" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="94.292191" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="100.61871" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="108.25804" y="0"> =</t年代pan> </text> <text transform="matrix(1,0,0,-1,244.21,68.19)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> <tspan style="font-size: 8.75px; " x="-1.25" y="25.530001"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,254.9,54.16)"> <tspan style="font-size: 12.50px; " x="38.75" y="4.9200001"> </t年代pan> <tspan style="font-size: 12.50px; " x="48.509998" y="-7.25"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.703048" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.866547" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.193066" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="29.444565" y="0"> ℎ</t年代pan> </text> <g> <path d="m 302.21,57.28 6.32,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,302.21,45.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.52" y="-3.8299999"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.209999" y="-8.75"> </t年代pan> <tspan style="font-size: 12.50px; " x="26.687275" y="-8.75"> l</t年代pan> <tspan style="font-size: 12.50px; " x="30.163109" y="-8.75"> n</t年代pan> <tspan style="font-size: 12.50px; " x="40.578106" y="-8.75"> (</t年代pan> <tspan style="font-size: 12.50px; " x="44.741608" y="-8.75"> </t年代pan> <tspan style="font-size: 12.50px; " x="51.068123" y="-8.75"> )</t年代pan> <tspan style="font-size: 12.50px; " x="58.707458" y="-8.75"> =</t年代pan> <tspan style="font-size: 12.50px; " x="70.747849" y="-8.75"> </t年代pan> <tspan style="font-size: 12.50px; " x="83.450897" y="-8.75"> #</t年代pan> <tspan style="font-size: 12.50px; " x="91.790398" y="-8.75"> ℎ</t年代pan> <tspan style="font-size: 12.50px; " x="101.09263" y="-8.75"> (</t年代pan> <tspan style="font-size: 12.50px; " x="105.25613" y="-8.75"> </t年代pan> <tspan style="font-size: 12.50px; " x="109.16956" y="-8.75"> )</t年代pan> <tspan style="font-size: 12.50px; " x="115.40856" y="-8.75"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="162.53999" y="-8.75"> (</t年代pan> <tspan style="font-size: 12.50px; " x="166.70351" y="-8.75"> 3.</t年代pan> <tspan style="font-size: 12.50px; " x="172.955" y="-8.75"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="176.08075" y="-8.75"> 7.</t年代pan> <tspan style="font-size: 12.50px; " x="182.33224" y="-8.75"> )</t年代pan> </text> </g> </g> </svg></span>即Gray和Zhang(1988)定义的m卷积。</p> <h4 id="filtering-data-using-the-g-filter">4.使用G过滤器过滤数据</H4.><p>释义<A.Href="#deff3">3．1</A.>结果表明，从原始空间看，G-滤波器不是线性滤波器，但从对偶空间看，G-滤波器是线性滤波器。因此，根据第节中有关G-滤波器的定义和结果<A.Href="#sec3">3.</A.>，我们使用Baraniuk和Jones提出的策略[<A.Href="#B1">4.</A.>]:<年代pan class="list"><span class="list-item"><span class="list-label">(1)</年代pan><span class="list-content">做适当的时间变形，<年代vg height="13.6125" id="M176" style="vertical-align:-2.34499pt;width:49.912498px;" version="1.1" viewbox="0 0 49.912498 13.6125" width="49.912498" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.3897533" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.430141" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.581617" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="31.745117" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.658558" y="0"> )</t年代pan> </text> </g> </g> </svg>，得到一个平稳的对偶过程。</年代pan></span><span class="list-item"><span class="list-label">(2)</年代pan><span class="list-content">在对偶空间上采用传统的线性滤波方法提取分量。</年代pan></span><span class="list-item"><span class="list-label">（3）</年代pan><span class="list-content">将过滤的双组件转换回原始时间空间。</年代pan></span></span></p> <p>在以下示例中，我们使用上述步骤说明了G过滤器的实现。</p> <p><span class="statement" id="ex3"><span class="statement-title"><i class="statement-title">示例4.1（示例<A.Href="#ex1">1．1</A.>和<A.Href="#ex2">2．3</A.>再次访问）。</我></年代pan>在本例中，我们将重新访问图中所示的TVF数据集<A.Href="//www.newsama.com/journals/jps/2012/738636/fig1/" target="_blank">1（a）</A.>在例子中讨论了这一点<A.Href="#ex1">1．1</A.>和<A.Href="#ex2">2．3</A.>。我们将对数据进行G筛选（基于G(<年代vg height="10.8125" id="M177" style="vertical-align:-0.20064pt;width:8.5749998px;" version="1.1" viewbox="0 0 8.5749998 10.8125" width="8.5749998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.8125)"> <g transform="translate(72,-63.35)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>)模型适合于数据)按照上面概述的步骤。<br><span class="list"><span class="list-item"><span class="list-label">(1)</年代pan><span class="list-content">第一步是在例子中完成的<A.Href="#ex2">2．3</A.>产生图中的双重数据<A.Href="//www.newsama.com/journals/jps/2012/738636/fig2/" target="_blank">2 (b)</A.>．</年代pan></span><span class="list-item"><span class="list-label">(2)</年代pan><span class="list-content">基于示例中的讨论<A.Href="#ex2">2．3</A.>图中的维格纳维尔情节<A.Href="//www.newsama.com/journals/jps/2012/738636/fig2/" target="_blank">2 (f)</A.>似乎可以用一个大约的截止频率来分离双数据中的低频和高频成分<年代vg height="14.7125" id="M178" style="vertical-align:-3.2316pt;width:45.612499px;" version="1.1" viewbox="0 0 45.612499 14.7125" width="45.612499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.7125)"> <g transform="translate(72,-60.23)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.01,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-56.97,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="15.153636" y="0"> 1.</t年代pan> </text> </g> </g> </svg>.数字<A.Href="//www.newsama.com/journals/jps/2012/738636/fig3/" target="_blank">3(一个)</A.>和<A.Href="//www.newsama.com/journals/jps/2012/738636/fig3/" target="_blank">3 (b)</A.>显示低通和高通巴特沃斯滤波器的结果，分别应用于图中的双重数据<A.Href="//www.newsama.com/journals/jps/2012/738636/fig2/" target="_blank">2 (b)</A.>。可以看出，这些滤波器很好地分离了双重数据中的低频和高频分量。</年代pan></span><span class="list-item"><span class="list-label">（3）</年代pan><span class="list-content">数据<A.Href="//www.newsama.com/journals/jps/2012/738636/fig3/" target="_blank">3 (c)</A.>和<A.Href="//www.newsama.com/journals/jps/2012/738636/fig3/" target="_blank">3 (d)</A.>显示数据<A.Href="//www.newsama.com/journals/jps/2012/738636/fig3/" target="_blank">3(一个)</A.>和<A.Href="//www.newsama.com/journals/jps/2012/738636/fig3/" target="_blank">3 (b)</A.>，分别绘制在原始时间刻度上（使用线性插值）。比较数字<A.Href="//www.newsama.com/journals/jps/2012/738636/fig3/" target="_blank">3 (c)</A.>和<A.Href="//www.newsama.com/journals/jps/2012/738636/fig3/" target="_blank">3 (d)</A.>有数字<A.Href="//www.newsama.com/journals/jps/2012/738636/fig1/" target="_blank">1（b）</A.>和<A.Href="//www.newsama.com/journals/jps/2012/738636/fig1/" target="_blank">1（c）</A.>，分别表明该程序在分离两个TVF频率方面做得很好，而我们以前在示例中无法分离这两个TVF频率<A.Href="#ex1">1．1</A.>使用标准的方法。峰高的某种参差不齐的行为是由于内插。目前正在研究线性插值的替代方法。</年代pan></span></span></span></p> <div class="floats-partial-section partial-carousel-wrapper" data-section="fig3"> <div class="floats-partial-content fig-content-fig3"> <div class="partial-carousel--single-image"> <img alt="（a）低通滤波双通道"nA.me="738636.fig.003a" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.003a.jpg"> <br> <strong>（a）低通滤波双通道</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="(b)高通滤波双通道"nA.me="738636.fig.003b" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.003b.jpg"> <br> <strong>(b)高通滤波双通道</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="（c）低通滤波数据"nA.me="738636.fig.003c" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.003c.jpg"> <br> <strong>（c）低通滤波数据</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="（d）高通滤波数据"nA.me="738636.fig.003d" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.003d.jpg"> <br> <strong>（d）高通滤波数据</年代trong> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs fig-thumbs-fig3"> <button class="partial-carousel-dot" aria-label="(a) Low pass-filtered dual"><img alt="（a）低通滤波双通道"nA.me="738636.fig.003a" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.003a.jpg"><br><strong>（a）低通滤波双通道</年代trong></button> <button class="partial-carousel-dot" aria-label="(b) High pass-filtered dual"><img alt="(b)高通滤波双通道"nA.me="738636.fig.003b" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.003b.jpg"><br><strong>(b)高通滤波双通道</年代trong></button> <button class="partial-carousel-dot" aria-label="(c) Low pass-filtered data"><img alt="（c）低通滤波数据"nA.me="738636.fig.003c" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.003c.jpg"><br><strong>（c）低通滤波数据</年代trong></button> <button class="partial-carousel-dot" aria-label="(d) High pass-filtered data"><img alt="（d）高通滤波数据"nA.me="738636.fig.003d" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.003d.jpg"><br><strong>（d）高通滤波数据</年代trong></button> </div> <div class="partial-carousel-next-prev fig-nav-fig3"> <button class="carousel-prev" aria-label="Previous"></button> <button class="carousel-next" aria-label="Next"></button> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图3</年代pan> <a aria-label="full view" class="caption-partial-url" href="//www.newsama.com/journals/jps/2012/738636/fig3/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> <svg height="14.7125" id="M179" style="vertical-align:-3.2316pt" version="1.1" viewbox="0 0 45.612499 14.7125" width="45.612499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.7125)"> <g transform="translate(72,-60.23)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.01,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-56.97,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="15.153636" y="0"> 1.</t年代pan> </text> </g> </g> </svg> <a href="//www.newsama.com/journals/jps/2012/738636/fig2/">2 (b)</A.><tA.ble> （a，b）低通和高通滤波器的输出应用于图中的双重数据;（c，d）在转换回原始时间尺度后是过滤的双组分。</tA.ble> </div> </div> </div> <p><span class="statement" id="ex4"><span class="statement-title"><i class="statement-title">示例4.2（过滤示例）。</我></年代pan>在这个例子中，我们认为<年代vg height="13.45" id="M180" style="vertical-align:-2.21957pt;width:30.275px;" version="1.1" viewbox="0 0 30.275 13.45" width="30.275" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> G</t年代pan> <tspan style="font-size: 12.50px; " x="9.0271664" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="13.190665" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.954788" y="0"> )</t年代pan> </text> </g> </g> </svg>固定的模型<年代pan class="equation" id="EEq5"> <svg height="39.587502" id="M181" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 39.587502" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,39.5875)"> <g transform="translate(72,-40.33)"> <text transform="matrix(1,0,0,-1,-15.73,50.12)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8600001" y="-2.52"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="14.874663" y="-2.52"> −</t年代pan> <tspan style="font-size: 12.50px; " x="26.227386" y="-2.52"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="32.478886" y="-2.52"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="35.604637" y="-2.52"> 6.</t年代pan> <tspan style="font-size: 12.50px; " x="41.856136" y="-2.52"> 2.</t年代pan> <tspan style="font-size: 12.50px; " x="48.107635" y="-2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="59.585392" y="-2.52"> +</t年代pan> <tspan style="font-size: 12.50px; " x="70.925613" y="-2.52"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="77.177109" y="-2.52"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="80.302864" y="-2.52"> 9</t年代pan> <tspan style="font-size: 12.50px; " x="86.554359" y="-2.52"> 9</t年代pan> <tspan style="font-size: 12.50px; " x="92.805862" y="-2.52"> 5.</t年代pan> <tspan style="font-size: 12.50px; " x="99.057365" y="-2.52"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,92.54,57.64)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2.</t年代pan> </text> <text transform="matrix(1,0,0,-1,97.41,50.12)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.9269018" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="13.79" y="-2.52"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="22.817165" y="-2.52"> −</t年代pan> <tspan style="font-size: 12.50px; " x="34.157387" y="-2.52"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="40.408886" y="-2.52"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="43.534637" y="-2.52"> 9</t年代pan> <tspan style="font-size: 12.50px; " x="49.786137" y="-2.52"> 6.</t年代pan> <tspan style="font-size: 12.50px; " x="56.037636" y="-2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="67.515388" y="-2.52"> +</t年代pan> <tspan style="font-size: 12.50px; " x="78.855614" y="-2.52"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="85.107109" y="-2.52"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="88.232864" y="-2.52"> 9</t年代pan> <tspan style="font-size: 12.50px; " x="94.48436" y="-2.52"> 9</t年代pan> <tspan style="font-size: 12.50px; " x="100.73586" y="-2.52"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,207.36,57.64)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2.</t年代pan> </text> <text transform="matrix(1,0,0,-1,212.23,50.12)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.9400001" y="-2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.65" y="4.8099999"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.219999" y="-10.74"> (</t年代pan> <tspan style="font-size: 12.50px; " x="35.383499" y="-10.74"> </t年代pan> <tspan style="font-size: 12.50px; " x="42.072605" y="-10.74"> +</t年代pan> <tspan style="font-size: 12.50px; " x="53.412827" y="-10.74"> 3.</t年代pan> <tspan style="font-size: 12.50px; " x="59.664326" y="-10.74"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="65.915825" y="-10.74"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,282.32,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 3.</t年代pan> </text> <text transform="matrix(1,0,0,-1,289.97,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> 1.</t年代pan> </text> <g> <path d="m 243.45,55.76 64.11,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,268.78,43.88)"> <tspan style="font-size: 12.50px; " x="39.98" y="-1.4299999"> </t年代pan> <tspan style="font-size: 12.50px; " x="73.230003" y="-1.4299999"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> 3.</t年代pan> <tspan style="font-size: 12.50px; " x="6.2515001" y="0"> ℎ</t年代pan> <tspan style="font-size: 12.50px; " x="52.830002" y="-8.7600002"> =</t年代pan> <tspan style="font-size: 12.50px; " x="64.857887" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="83.800003" y="-16.98"> (</t年代pan> <tspan style="font-size: 12.50px; " x="87.963501" y="-16.98"> </t年代pan> <tspan style="font-size: 12.50px; " x="94.652603" y="-16.98"> +</t年代pan> <tspan style="font-size: 12.50px; " x="105.99283" y="-16.98"> 3.</t年代pan> <tspan style="font-size: 12.50px; " x="112.24432" y="-16.98"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="118.49583" y="-16.98"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,391.45,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 3.</t年代pan> </text> <text transform="matrix(1,0,0,-1,399.1,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> 1.</t年代pan> </text> <g> <path d="m 352.58,55.76 64.11,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,377.91,43.88)"> <tspan style="font-size: 12.50px; " x="39.98" y="-1.4299999"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> 3.</t年代pan> <tspan style="font-size: 12.50px; " x="6.2515001" y="0"> ℎ</t年代pan> <tspan style="font-size: 12.50px; " x="51.439999" y="-8.7600002"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="86.823486" y="-8.7600002"> (</t年代pan> <tspan style="font-size: 12.50px; " x="90.986992" y="-8.7600002"> 4.</t年代pan> <tspan style="font-size: 12.50px; " x="97.238487" y="-8.7600002"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="100.36424" y="-8.7600002"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="106.61574" y="-8.7600002"> )</t年代pan> </text> </g> </g> </svg></span>[<年代vg height="13.45" id="M182" style="vertical-align:-2.21957pt;width:27.775px;" version="1.1" viewbox="0 0 27.775 13.45" width="27.775" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8773699" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="14.040869" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.954308" y="0"> )</t年代pan> </text> </g> </g> </svg>是由Jiang等人提出的[<A.Href="#B11">3.</A.>)作为一个<年代vg height="13.45" id="M183" style="vertical-align:-2.21957pt;width:44.400002px;" version="1.1" viewbox="0 0 44.400002 13.45" width="44.400002" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> G</t年代pan> <tspan style="font-size: 12.50px; " x="9.0271664" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="13.190665" y="0"> 4.</t年代pan> <tspan style="font-size: 12.50px; " x="19.442165" y="0"> ;</t年代pan> <tspan style="font-size: 12.50px; " x="24.993498" y="0"> 3.</t年代pan> <tspan style="font-size: 12.50px; " x="31.244997" y="0"> )</t年代pan> </text> </g> </g> </svg>显然，在时间转换之后<年代vg height="16.9" id="M184" style="vertical-align:-2.34499pt;width:191.4375px;" version="1.1" viewbox="0 0 191.4375 16.9" width="191.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.9)"> <g transform="translate(72,-58.48)"> <text transform="matrix(1,0,0,-1,-71.95,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.1514759" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.314975" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.228414" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="21.855244" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="33.895634" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="38.059132" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="42.22263" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="48.911736" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="60.251957" y="0"> 3.</t年代pan> <tspan style="font-size: 12.50px; " x="66.503456" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="72.754959" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,4.98,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 3.</t年代pan> </text> <text transform="matrix(1,0,0,-1,12.63,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="17.591721" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="21.75522" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="25.231054" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="29.394552" y="0"> 3.</t年代pan> <tspan style="font-size: 12.50px; " x="35.646053" y="0"> ℎ</t年代pan> <tspan style="font-size: 12.50px; " x="42.860283" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="50.499619" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="62.540005" y="0"> </t年代pan> </text> </g> </g> </svg>的双重模型<年代vg height="13.6125" id="M185" style="vertical-align:-2.34499pt;width:91.4375px;" version="1.1" viewbox="0 0 91.4375 13.6125" width="91.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.689105" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.852604" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.779026" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="24.405855" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="36.446243" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="46.323616" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="50.487114" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="56.638592" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="60.80209" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="64.71553" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="68.879028" y="0"> )</t年代pan> </text> </g> </g> </svg>是静态AR（4）模型<年代pan class="equation" id="EEq6"> <svg height="19.475" id="M186" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 19.475" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.475)"> <g transform="translate(72,-56.42)"> <text transform="matrix(1,0,0,-1,71.39,58.52)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8499999" y="-2.51"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="14.877166" y="-2.51"> −</t年代pan> <tspan style="font-size: 12.50px; " x="26.217386" y="-2.51"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="32.468887" y="-2.51"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="35.594639" y="-2.51"> 6.</t年代pan> <tspan style="font-size: 12.50px; " x="41.846138" y="-2.51"> 2.</t年代pan> <tspan style="font-size: 12.50px; " x="48.097637" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="59.57539" y="-2.51"> +</t年代pan> <tspan style="font-size: 12.50px; " x="70.928116" y="-2.51"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="77.179611" y="-2.51"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="80.305367" y="-2.51"> 9</t年代pan> <tspan style="font-size: 12.50px; " x="86.556862" y="-2.51"> 9</t年代pan> <tspan style="font-size: 12.50px; " x="92.808365" y="-2.51"> 5.</t年代pan> <tspan style="font-size: 12.50px; " x="99.059868" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,179.65,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2.</t年代pan> </text> <text transform="matrix(1,0,0,-1,184.53,58.52)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.9269018" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="13.78" y="-2.51"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="22.807165" y="-2.51"> −</t年代pan> <tspan style="font-size: 12.50px; " x="34.147388" y="-2.51"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="40.398888" y="-2.51"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="43.524635" y="-2.51"> 9</t年代pan> <tspan style="font-size: 12.50px; " x="49.776138" y="-2.51"> 6.</t年代pan> <tspan style="font-size: 12.50px; " x="56.027637" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="67.517891" y="-2.51"> +</t年代pan> <tspan style="font-size: 12.50px; " x="78.858116" y="-2.51"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="85.109612" y="-2.51"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="88.235367" y="-2.51"> 9</t年代pan> <tspan style="font-size: 12.50px; " x="94.486862" y="-2.51"> 9</t年代pan> <tspan style="font-size: 12.50px; " x="100.73837" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,294.47,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2.</t年代pan> </text> <text transform="matrix(1,0,0,-1,299.35,58.52)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.9299998" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,313.97,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,320.68,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.040389" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,339,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,342.24,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="122.51" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="126.6735" y="0"> 4.</t年代pan> <tspan style="font-size: 12.50px; " x="132.925" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="136.05075" y="0"> 2.</t年代pan> <tspan style="font-size: 12.50px; " x="142.30225" y="0"> )</t年代pan> </text> </g> </g> </svg></span>哪个有特征方程<年代vg height="16.5375" id="M187" style="vertical-align:-2.21957pt;width:279.35001px;" version="1.1" viewbox="0 0 279.35001 16.5375" width="279.35001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.5375)"> <g transform="translate(72,-58.77)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="13.190665" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="24.530886" y="0"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="30.782387" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="33.908134" y="0"> 6.</t年代pan> <tspan style="font-size: 12.50px; " x="40.159637" y="0"> 2.</t年代pan> <tspan style="font-size: 12.50px; " x="46.411137" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="55.4258" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="66.766022" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="69.891769" y="0"> 9</t年代pan> <tspan style="font-size: 12.50px; " x="76.143272" y="0"> 9</t年代pan> <tspan style="font-size: 12.50px; " x="82.394768" y="0"> 5.</t年代pan> <tspan style="font-size: 12.50px; " x="88.646271" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,22.95,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2.</t年代pan> </text> <text transform="matrix(1,0,0,-1,27.82,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="8.3269978" y="0"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="17.354164" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="28.694386" y="0"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="34.945885" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="38.071636" y="0"> 9</t年代pan> <tspan style="font-size: 12.50px; " x="44.323135" y="0"> 6.</t年代pan> <tspan style="font-size: 12.50px; " x="50.574635" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="59.589298" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="70.942024" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="74.067772" y="0"> 9</t年代pan> <tspan style="font-size: 12.50px; " x="80.319275" y="0"> 9</t年代pan> <tspan style="font-size: 12.50px; " x="86.57077" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,120.63,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2.</t年代pan> </text> <text transform="matrix(1,0,0,-1,125.5,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="7.6393328" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="19.667219" y="0"> 0</t年代pan> </text> </g> </g> </svg>它有两对复共轭根，非常接近单位圆。与每个分量相关的是一个系统频率。该模型的实现将以两个系统频率的邻域中的频率行为为特征，因此谱估计将具有接近这两个频率值的峰值。（见[<A.Href="#B18">14</A.>,<A.Href="#B19">17</A.>])。数字<A.Href="//www.newsama.com/journals/jps/2012/738636/fig4/" target="_blank">4(一)</A.>显示长度的实现<年代vg height="10.9125" id="M188" style="vertical-align:-0.17555pt;width:50.724998px;" version="1.1" viewbox="0 0 50.724998 10.9125" width="50.724998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.9125)"> <g transform="translate(72,-63.27)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.6773224" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.71771" y="0"> 2.</t年代pan> <tspan style="font-size: 12.50px; " x="27.969212" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="34.220711" y="0"> 0</t年代pan> </text> </g> </g> </svg>从(<A.Href="#EEq5">4．1</A.>)，可以看到频率随时间急剧增加（即，周期长度减少）。也有两个TVF的迹象。使用GWS代码，时间转换<年代vg height="11.0375" id="M189" style="vertical-align:-0.27588pt;width:47.5px;" version="1.1" viewbox="0 0 47.5 11.0375" width="47.5" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0375)"> <g transform="translate(72,-63.17)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.227454" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.267843" y="0"> 2.</t年代pan> <tspan style="font-size: 12.50px; " x="28.519342" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="31.645094" y="0"> 9</t年代pan> </text> </g> </g> </svg>选择offset 14。基于此时间变换，得到如图所示的对偶数据<A.Href="//www.newsama.com/journals/jps/2012/738636/fig4/" target="_blank">4 (b)</A.>这里的数据看起来是静止的，同样有两个基本频率。原始数据集中频率的极端增加在图中Wigner-Ville图中得到了说明<A.Href="//www.newsama.com/journals/jps/2012/738636/fig4/" target="_blank">4 (c)</A.>。对于较小的<年代vg height="9.125" id="M190" style="vertical-align:-0.11285pt;width:5.0124998px;" version="1.1" viewbox="0 0 5.0124998 9.125" width="5.0124998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.125)"> <g transform="translate(72,-64.7)"> <text transform="matrix(1,0,0,-1,-71.95,64.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>大约<年代vg height="13.6125" id="M191" style="vertical-align:-2.34499pt;width:39.912498px;" version="1.1" viewbox="0 0 39.912498 13.6125" width="39.912498" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.402496" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="25.568636" y="0"> 1.</t年代pan> </text> </g> </g> </svg>在<年代vg height="10.9125" id="M192" style="vertical-align:-0.17555pt;width:53.0625px;" version="1.1" viewbox="0 0 53.0625 10.9125" width="53.0625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.9125)"> <g transform="translate(72,-63.27)"> <text transform="matrix(1,0,0,-1,-67.78,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> 2.</t年代pan> <tspan style="font-size: 12.50px; " x="25.668659" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="31.920158" y="0"> 0</t年代pan> </text> </g> </g> </svg>.上部（不太可见）带与一个更高频率的TVF分量相关，该分量也从接近零的频率开始，并增加到约<年代vg height="13.8" id="M193" style="vertical-align:-2.34499pt;width:47.724998px;" version="1.1" viewbox="0 0 47.724998 13.8" width="47.724998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.8)"> <g transform="translate(72,-60.96)"> <text transform="matrix(1,0,0,-1,-71.95,63.35)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.402496" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="25.568636" y="0"> 2.</t年代pan> <tspan style="font-size: 12.50px; " x="31.820135" y="0"> 5.</t年代pan> </text> </g> </g> </svg>朝向<年代vg height="10.9125" id="M194" style="vertical-align:-0.17555pt;width:47.849998px;" version="1.1" viewbox="0 0 47.849998 10.9125" width="47.849998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.9125)"> <g transform="translate(72,-63.27)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> 2.</t年代pan> <tspan style="font-size: 12.50px; " x="25.668659" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="31.920158" y="0"> 0</t年代pan> </text> </g> </g> </svg>. 双重数据的Wigner-Ville图表明，这两个频率已经稳定，进一步支持双重数据的平稳性。双模式由两个频率控制，一个较低的频率约为<年代vg height="13.6125" id="M195" style="vertical-align:-2.34499pt;width:47.724998px;" version="1.1" viewbox="0 0 47.724998 13.6125" width="47.724998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.402496" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="25.568636" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="31.820135" y="0"> 3.</t年代pan> </text> </g> </g> </svg>频率上限（同样不太明显）约为<年代vg height="13.6125" id="M196" style="vertical-align:-2.34499pt;width:39.912498px;" version="1.1" viewbox="0 0 39.912498 13.6125" width="39.912498" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.402496" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="25.568636" y="0"> 1.</t年代pan> </text> </g> </g> </svg>.应用具有截止频率的三阶低通和高通巴特沃斯滤波器的结果<年代vg height="14.9" id="M197" style="vertical-align:-3.2316pt;width:68.337502px;" version="1.1" viewbox="0 0 68.337502 14.9" width="68.337502" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.9)"> <g transform="translate(72,-60.08)"> <text transform="matrix(1,0,0,-1,-67.78,63.35)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-59.34,60.22)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-51.3,63.35)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="15.153636" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="21.405136" y="0"> 6.</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> 5.</t年代pan> </text> </g> </g> </svg>如图所示<A.Href="//www.newsama.com/journals/jps/2012/738636/fig4/" target="_blank">4 (e)</A.>和<A.Href="//www.newsama.com/journals/jps/2012/738636/fig4/" target="_blank">4（f）</A.>,分别。数据<A.Href="//www.newsama.com/journals/jps/2012/738636/fig4/" target="_blank">4（g）</A.>和<A.Href="//www.newsama.com/journals/jps/2012/738636/fig4/" target="_blank">4（h）</A.>显示在原始时间轴上绘制的过滤数据集。<br>经过筛选的两个组件的行为与原始数据中的行为一致，如图所示<A.Href="//www.newsama.com/journals/jps/2012/738636/fig5/" target="_blank">5(一个)</A.>和<A.Href="//www.newsama.com/journals/jps/2012/738636/fig5/" target="_blank">5 (b)</A.>．此外，通过对比图可以看出，TVF组件与原始数据中的组件非常接近<A.Href="//www.newsama.com/journals/jps/2012/738636/fig5/" target="_blank">5（c）</A.>和<A.Href="//www.newsama.com/journals/jps/2012/738636/fig5/" target="_blank">5（d）</A.>有型<A.Href="//www.newsama.com/journals/jps/2012/738636/fig4/" target="_blank">4 (c)</A.>．</年代pan></p> <div class="floats-partial-section partial-carousel-wrapper" data-section="fig4"> <div class="floats-partial-content fig-content-fig4"> <div class="partial-carousel--single-image"> <img alt="（a） TVF数据"nA.me="738636.fig.004a" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.004a.jpg"> <br> <strong>（a） TVF数据</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="（b）双重数据"nA.me="738636.fig.004b" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.004b.jpg"> <br> <strong>（b）双重数据</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="（c）（a）的Wigner-Ville Plot"nA.me="738636.fig.004c" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.004c.jpg"> <br> <strong>（c）（a）的Wigner-Ville Plot</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="(d) (b) Wigner-Ville plot"nA.me="738636.fig.004d" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.004d.jpg"> <br> <strong>(d) (b) Wigner-Ville plot</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="(e)低频双分量"nA.me="738636.fig.004e" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.004e.jpg"> <br> <strong>(e)低频双分量</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="(f)高频双分量"nA.me="738636.fig.004f" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.004f.jpg"> <br> <strong>(f)高频双分量</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="(g)恢复的低频TVF分量"nA.me="738636.fig.004g" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.004g.jpg"> <br> <strong>(g)恢复的低频TVF分量</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="（h）恢复高频TVF分量"nA.me="738636.fig.004h" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.004h.jpg"> <br> <strong>（h）恢复高频TVF分量</年代trong> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs fig-thumbs-fig4"> <button class="partial-carousel-dot" aria-label="(a) TVF data"><img alt="（a） TVF数据"nA.me="738636.fig.004a" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.004a.jpg"><br><strong>（a） TVF数据</年代trong></button> <button class="partial-carousel-dot" aria-label="(b) Dual data"><img alt="（b）双重数据"nA.me="738636.fig.004b" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.004b.jpg"><br><strong>（b）双重数据</年代trong></button> <button class="partial-carousel-dot" aria-label="(c) Wigner-Ville plot for (a)"><img alt="（c）（a）的Wigner-Ville Plot"nA.me="738636.fig.004c" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.004c.jpg"><br><strong>（c）（a）的Wigner-Ville Plot</年代trong></button> <button class="partial-carousel-dot" aria-label="(d) Wigner-Ville plot for (b)"><img alt="(d) (b) Wigner-Ville plot"nA.me="738636.fig.004d" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.004d.jpg"><br><strong>(d) (b) Wigner-Ville plot</年代trong></button> <button class="partial-carousel-dot" aria-label="(e) Low-frequency dual component"><img alt="(e)低频双分量"nA.me="738636.fig.004e" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.004e.jpg"><br><strong>(e)低频双分量</年代trong></button> <button class="partial-carousel-dot" aria-label="(f) High-frequency dual Component"><img alt="(f)高频双分量"nA.me="738636.fig.004f" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.004f.jpg"><br><strong>(f)高频双分量</年代trong></button> <button class="partial-carousel-dot" aria-label="(g) Recovered Low-frequency TVF component"><img alt="(g)恢复的低频TVF分量"nA.me="738636.fig.004g" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.004g.jpg"><br><strong>(g)恢复的低频TVF分量</年代trong></button> <button class="partial-carousel-dot" aria-label="(h) Recovered high-frequency TVF component"><img alt="（h）恢复高频TVF分量"nA.me="738636.fig.004h" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.004h.jpg"><br><strong>（h）恢复高频TVF分量</年代trong></button> </div> <div class="partial-carousel-next-prev fig-nav-fig4"> <button class="carousel-prev" aria-label="Previous"></button> <button class="carousel-next" aria-label="Next"></button> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图4</年代pan> <a aria-label="full view" class="caption-partial-url" href="//www.newsama.com/journals/jps/2012/738636/fig4/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> <table> （a）（a）（a），（c）双数据，（c）（a），（d）Wigner-Ville图（b），（e）和（f）低通和高的结果在原始时间尺度上绘制的双数据（G，H）上的滤波器（G，H）过滤器滤波器组件。</tA.ble> </div> </div> </div> <div class="floats-partial-section partial-carousel-wrapper" data-section="fig5"> <div class="floats-partial-content fig-content-fig5"> <div class="partial-carousel--single-image"> <img alt="(a)滤波后的低频分量对信号"nA.me="738636.fig.005a" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.005a.jpg"> <br> <strong>(a)滤波后的低频分量对信号</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="(b)滤波高频分量对信号"nA.me="738636.fig.005b" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.005b.jpg"> <br> <strong>(b)滤波高频分量对信号</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="（c）低TVF分量的Wigner-Ville图"nA.me="738636.fig.005c" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.005c.jpg"> <br> <strong>（c）低TVF分量的Wigner-Ville图</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="（d）高TVF分量的Wigner-Ville图"nA.me="738636.fig.005d" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.005d.jpg"> <br> <strong>（d）高TVF分量的Wigner-Ville图</年代trong> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs fig-thumbs-fig5"> <button class="partial-carousel-dot" aria-label="(a) Filtered low-frequency component versus signal"><img alt="(a)滤波后的低频分量对信号"nA.me="738636.fig.005a" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.005a.jpg"><br><strong>(a)滤波后的低频分量对信号</年代trong></button> <button class="partial-carousel-dot" aria-label="(b) Filtered high-frequency component versus signal"><img alt="(b)滤波高频分量对信号"nA.me="738636.fig.005b" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.005b.jpg"><br><strong>(b)滤波高频分量对信号</年代trong></button> <button class="partial-carousel-dot" aria-label="(c) Wigner-Ville plot for low TVF component"><img alt="（c）低TVF分量的Wigner-Ville图"nA.me="738636.fig.005c" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.005c.jpg"><br><strong>（c）低TVF分量的Wigner-Ville图</年代trong></button> <button class="partial-carousel-dot" aria-label="(d) Wigner-Ville plot for high TVF component"><img alt="（d）高TVF分量的Wigner-Ville图"nA.me="738636.fig.005d" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.005d.jpg"><br><strong>（d）高TVF分量的Wigner-Ville图</年代trong></button> </div> <div class="partial-carousel-next-prev fig-nav-fig5"> <button class="carousel-prev" aria-label="Previous"></button> <button class="carousel-next" aria-label="Next"></button> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图5</年代pan> <a aria-label="full view" class="caption-partial-url" href="//www.newsama.com/journals/jps/2012/738636/fig5/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> <table> （a，b）两个过滤组件（实线），具有原始信号（虚线），（c，d）两个提取的组分的Wigner-Ville图。</tA.ble> </div> </div> </div> <p><span class="statement" id="ex5"><span class="statement-title"><i class="statement-title">示例4.3（蝙蝠回声定位数据）。</我></年代pan>在这个例子中，我们考虑了一个大型棕色蝙蝠的回声定位数据。这些数据是由伊利诺伊大学贝克曼研究所的Al Feng提供的。<A.Href="//www.newsama.com/journals/jps/2012/738636/fig6/" target="_blank">6（a）</A.>包括以7微秒间隔所拍摄的381个数据点，总持续时间<年代vg height="11.0375" id="M198" style="vertical-align:-0.17555pt;width:43.099998px;" version="1.1" viewbox="0 0 43.099998 11.0375" width="43.099998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0375)"> <g transform="translate(72,-63.17)"> <text transform="matrix(1,0,0,-1,-71.95,63.4)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="3.1257501" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="9.3772497" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="15.62875" y="0"> 2.</t年代pan> <tspan style="font-size: 12.50px; " x="21.880251" y="0"> 6.</t年代pan> <tspan style="font-size: 12.50px; " x="28.13175" y="0"> 6.</t年代pan> </text> </g> </g> </svg>秒。该信号相当复杂，可能由两个不同的信号组成。图中的Wigner-Ville情节<A.Href="//www.newsama.com/journals/jps/2012/738636/fig6/" target="_blank">6（c）</A.>表明数据包含两个TVF分量，并暗示可能有另一个TVF与高于两个主要分量的频率相关。格雷等人[<A.Href="#B8">13</A.>]分析了与偏移量203的M-静止模型的数据集。它们的分析表明，有三个主导的TVF组件，并且最高组件在时间序列开始时足够高的频率，以至于采样率太快了检测到直到<年代vg height="10.9125" id="M199" style="vertical-align:-0.17555pt;width:47.849998px;" version="1.1" viewbox="0 0 47.849998 10.9125" width="47.849998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.9125)"> <g transform="translate(72,-63.27)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="25.668659" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="31.920158" y="0"> 0</t年代pan> </text> </g> </g> </svg>．在我们的分析中，我们将使用Gray等人提出的时间变换[<A.Href="#B8">13</A.>]计算对偶数据。使用此时间转换生成图中的双数据<A.Href="//www.newsama.com/journals/jps/2012/738636/fig6/" target="_blank">6（b）</A.>．图中是Wigner-Ville的情节<A.Href="//www.newsama.com/journals/jps/2012/738636/fig6/" target="_blank">6（c）</A.>显示TVF条带之间的重叠，例如，与附近较低的TVF条带相关的频率<年代vg height="10.9125" id="M200" style="vertical-align:-0.17555pt;width:32.224998px;" version="1.1" viewbox="0 0 32.224998 10.9125" width="32.224998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.9125)"> <g transform="translate(72,-63.27)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> 0</t年代pan> </text> </g> </g> </svg>和上面的带子差不多吗<年代vg height="11.1" id="M201" style="vertical-align:-0.17555pt;width:47.849998px;" version="1.1" viewbox="0 0 47.849998 11.1" width="47.849998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.1)"> <g transform="translate(72,-63.12)"> <text transform="matrix(1,0,0,-1,-71.95,63.35)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> 2.</t年代pan> <tspan style="font-size: 12.50px; " x="25.668659" y="0"> 5.</t年代pan> <tspan style="font-size: 12.50px; " x="31.920158" y="0"> 0</t年代pan> </text> </g> </g> </svg>双数据的Wigner-Ville图如图所示<A.Href="//www.newsama.com/journals/jps/2012/738636/fig6/" target="_blank">6（d）</A.>从图中可以看出，两个主频已稳定下来，两个双频主频分离良好，在约<年代vg height="13.6125" id="M202" style="vertical-align:-2.34499pt;width:47.724998px;" version="1.1" viewbox="0 0 47.724998 13.6125" width="47.724998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.402496" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="25.568636" y="0"> 1.</t年代pan> <tspan style="font-size: 12.50px; " x="31.820135" y="0"> 8.</t年代pan> </text> </g> </g> </svg>和<年代vg height="13.6125" id="M203" style="vertical-align:-2.34499pt;width:39.912498px;" version="1.1" viewbox="0 0 39.912498 13.6125" width="39.912498" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.402496" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ．</t年代pan> <tspan style="font-size: 12.50px; " x="25.568636" y="0"> 3.</t年代pan> </text> </g> </g> </svg>.数字<A.Href="//www.newsama.com/journals/jps/2012/738636/fig6/" target="_blank">6（e）</A.>和<A.Href="//www.newsama.com/journals/jps/2012/738636/fig6/" target="_blank">6（f）</A.>分别显示绘制在原始时间轴上的低通和高通滤波器结果。</年代pan></p> <div class="floats-partial-section partial-carousel-wrapper" data-section="fig6"> <div class="floats-partial-content fig-content-fig6"> <div class="partial-carousel--single-image"> <img alt="(一)蝙蝠数据"nA.me="738636.fig.006a" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.006a.jpg"> <br> <strong>(一)蝙蝠数据</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="(b) Bat双重数据"nA.me="738636.fig.006b" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.006b.jpg"> <br> <strong>(b) Bat双重数据</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="（c）（a）的Wigner-Ville Plot"nA.me="738636.fig.006c" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.006c.jpg"> <br> <strong>（c）（a）的Wigner-Ville Plot</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="(d) (b) Wigner-Ville plot"nA.me="738636.fig.006d" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.006d.jpg"> <br> <strong>(d) (b) Wigner-Ville plot</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="（e）低频TVF分量"nA.me="738636.fig.006e" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.006e.jpg"> <br> <strong>（e）低频TVF分量</年代trong> </div> <div class="partial-carousel--single-image"> <img alt="（f）高频TVF分量"nA.me="738636.fig.006f" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.006f.jpg"> <br> <strong>（f）高频TVF分量</年代trong> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs fig-thumbs-fig6"> <button class="partial-carousel-dot" aria-label="(a) Bat data"><img alt="(一)蝙蝠数据"nA.me="738636.fig.006a" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.006a.jpg"><br><strong>(一)蝙蝠数据</年代trong></button> <button class="partial-carousel-dot" aria-label="(b) Bat dual data"><img alt="(b) Bat双重数据"nA.me="738636.fig.006b" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.006b.jpg"><br><strong>(b) Bat双重数据</年代trong></button> <button class="partial-carousel-dot" aria-label="(c) Wigner-Ville plot for (a)"><img alt="（c）（a）的Wigner-Ville Plot"nA.me="738636.fig.006c" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.006c.jpg"><br><strong>（c）（a）的Wigner-Ville Plot</年代trong></button> <button class="partial-carousel-dot" aria-label="(d) Wigner-Ville plot for (b)"><img alt="(d) (b) Wigner-Ville plot"nA.me="738636.fig.006d" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.006d.jpg"><br><strong>(d) (b) Wigner-Ville plot</年代trong></button> <button class="partial-carousel-dot" aria-label="(e) Low-frequency TVF component"><img alt="（e）低频TVF分量"nA.me="738636.fig.006e" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.006e.jpg"><br><strong>（e）低频TVF分量</年代trong></button> <button class="partial-carousel-dot" aria-label="(f) High-frequency TVF component"><img alt="（f）高频TVF分量"nA.me="738636.fig.006f" src="https://static-01.hindawi.com/articles/jps/volume-2012/738636/figures/738636.fig.006f.jpg"><br><strong>（f）高频TVF分量</年代trong></button> </div> <div class="partial-carousel-next-prev fig-nav-fig6"> <button class="carousel-prev" aria-label="Previous"></button> <button class="carousel-next" aria-label="Next"></button> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图6</年代pan> <a aria-label="full view" class="caption-partial-url" href="//www.newsama.com/journals/jps/2012/738636/fig6/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> <table> （a）棕色蝙蝠回声定位数据，（b）（a），（c）Wigner-Ville图（a），（d）Wigner-Ville图（b），（e）低通滤波数据，（f）高通滤波数据。</tA.ble> </div> </div> </div> <h4 id="concluding-remarks">5.结束语</H4.><p>在本文中，我们证明了经典的线性滤波方法不能直接应用于TVF分量随时间重叠的TVF数据。我们讨论了一系列模型，G-平稳模型，它已被证明是通常平稳模型的有用扩展，并可用于对一定范围的非平稳时间进行建模然后，我们介绍了一种从G-平稳模型中过滤数据的简单而直接的方法。这种G-过滤方法扩展了标准线性滤波器，并提供了过滤时间重叠的时变频率数据的技术。仿真实例和实际数据示例表明，该方法的有效性G-过滤器在过滤来自G-平稳模型或可由G-平稳模型近似的数据时的活性。</p> </div> <div class="xml-content"> <h4 class="references" id="references">参考文献</H4.><ol><l我我D="B14">a . 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height="24"><span class="sc-dRFtgE jrsDZP">下载其他格式</年代pan><span class="sc-gkFcWv gWTCXQ">更多</年代pan></span> <span class="sc-htoDjs bMdQcY sc-gbOuXE jGPvPL icon-colored _1CwL9R82TPl2HxDleeYGDR icon-themeable" title="" font-size="11"></span> </div> <div class="sc-fnwBNb etzuBr"> <a class="sc-htpNat bUhGXt link sc-iNhVCk gWgCNb open_panel" href="//www.newsama.com/downloads/journals/jps/2012/738636.epub" aria-label="ePub" property="background-color" color="#D8D8D8"> <div class="sc-eAKXzc WHkfb"> ePub</D我v><img alt="" class="sc-EHOje jOLhQl sc-bfYoXt cGuwDn" title="" role="presentation" src="data:image/svg+xml;base64,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" height="24"></a> <a class="sc-htpNat bUhGXt link sc-iNhVCk gWgCNb open_panel" href="//www.newsama.com/downloads/journals/jps/2012/738636.xml" aria-label="XML" property="background-color" color="#D8D8D8"> <div class="sc-eAKXzc WHkfb"> XML</D我v><img alt="" class="sc-EHOje jOLhQl sc-bfYoXt cGuwDn" title="" role="presentation" src="data:image/svg+xml;base64,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" 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height="24"><span class="sc-dRFtgE jrsDZP">订购印刷品</年代pan><span class="sc-gkFcWv gWTCXQ">顺序</年代pan></span><span class="sc-htoDjs kNoiZu sc-gbOuXE jGPvPL right__icon icon-colored _32KP5uexH-Bgs7Kj8Gk1Tz icon-themeable" title="" font-size="16"></span></a> <div class="sc-hBbWxd kdRRST"> <div class="sc-dyGzUR cHatnS"> <div> 意见<年代pan>1643</年代pan> </div> <div> 下载<年代pan>1385</年代pan> </div> </div> <div> <div class="sc-drKuOJ kxdGAZ"> <span class="__dimensions_badge_embed__" data-doi="10.1155/2012/738636" data-style="small_circle"></span>引用</D我v> </div> </div> <nav class="sc-bRBYWo fXBKFa sc-hUfwpO jHPgUP" id="toolbar__iconNav"> <h3 class="sc-Rmtcm hNPrIr">共享职位</H3.><ul class="sc-hzDkRC bnbfTS"> <li class="iconBorder"><a class="sc-htpNat bUhGXt link" href="https://twitter.com/share?url=https://doi.org/10.1155/2012/738636&via=Hindawi" aria-label="Sharing on Twitter" rel="noopener noreferrer" target="_blank"><img alt="" class="sc-EHOje jOLhQl socialTwitter" title="" role="presentation" 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