3.3=20 功率谱分析及其应用</t= itle> <o:SmartTagType namespaceuri=3D"urn:schemas-microsoft-com:office:smarttags" name=3D"chsdate" downloadurl=3D""/>  <style>  </style>  </head> <body lang=3DZH-CN link=3Dblack vlink=3Dpurple style=3D'tab-interval:21.0pt= ;text-justify-trim: punctuation'> <div class=3DSection1 style=3D'layout-grid:15.2pt 0pt;mso-layout-grid-char-= alt: 0'> <h2>3.3 </= span>功率&#= 35889;分析及其应用</h2> 在第2章中讨论了周期= 信号和瞬态信号的时= 2495;波形与频域的幅值ค= 9;及相位谱之间的对应&= #20851;系，并了解到频域= 551;述可反映信号频率结= ;构组成。然而对于随&#= 26426;信号，由于其样本ć= 54;线的波形具有随机性= ，而且是时域无限信= 1495;，不满足傅里叶变ৰ= 2;条件，因而从理论上&= #35762;，随机信号不能直= 509;进行傅里叶变换作幅= ;值谱和相位谱分析，&#= 32780;是应用具有统计特ô= 49;的功率谱密度函数在= 频域内对随机信号作= 9057;谱分析，它是研究ॱ= 9;稳随机过程的重要方&= #27861;。功率谱密度函数= 998;自谱和互谱两种形式= ;。 <h3 style=3D'line-height:16.5pt'><a name=3D"_Toc139040641"></a><st1:chsdate Year=3D"1899" Month=3D"12" Day=3D"30" IsLunarDate=3D"False" IsROCDate=3D"Fa= lse" w:st=3D"on">3.3.1</span= ></st1:chsdate> 巴塞伐尔(Paseval)定理</= span></h3> 巴塞伐尔定理中= ，在时域中计算的信= 1495;总能量等于在频域ߑ= 3;计算的信号总能量，&= #21363; <![if !vml]><img width=3D157 height=3D31 src=3D"ch033.files/image002.gif" v:shapes=3D"_x0000_i1025"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; (3= -35) 该定理可以用傅= 里叶变换的卷积来证= 6126;。设有傅里叶变换ल= 5; <![if !vml]><img width=3D89 height=3D21 src=3D"ch033.files/image004.gif" v:shapes=3D"_x0000_i1026"><![endif]>= ，<= span lang=3DEN-US><![if !vml]><img width=3D92 height=3D21 src=3D"ch033.files/image006.gif" v:shapes=3D"_x0000_i1027"><![endif]>=  根据频域&#= 21367;积定理有 <![if !vml]><img width=3D263 height=3D31 src=3D"ch033.files/image008.gif" v:shapes=3D"_x0000_i1028"><![endif]>=  令f<s= pan lang=3DEN-US>0=3D0，x<s= pan lang=3DEN-US>1(t)=3Dx2(t)=3Dx(t)，则= <![if !vml]><img width=3D183 height=3D31 src=3D"ch033.files/image010.gif" v:shapes=3D"_x0000_i1029"><![endif]>=  式中，x(t)是实函数，则X(-</spa= n>jf)=3DX*(jf)，所以 <![if !vml]><img width=3D288 height=3D31 src=3D"ch033.files/image012.gif" v:shapes=3D"_x0000_i1030"><![endif]>=  式中，<sp= an lang=3DEN-US><![if !vml]><img width=3D53 height=3D23 src=3D"ch033.files/image014.gif" v:shapes=3D"_x0000_i1031"><![endif]>= 称为= 能谱，是沿频率轴的= 3021;量分布密度。 <span lang=3DEN-US style=3D'font-size:10.5pt;mso-bidi-font-size:10.0pt;font= -family: Arial;mso-fareast-font-family:SimHei;mso-bidi-font-family:"Times New Roman"; mso-ansi-language:EN-US;mso-fareast-language:ZH-CN;mso-bidi-language:AR-SA'= > <h3 style=3D'line-height:16.5pt'><a name=3D"_Toc139040642"></a><st1:chsdate Year=3D"1899" Month=3D"12" Day=3D"30" IsLunarDate=3D"False" IsROCDate=3D"Fa= lse" w:st=3D"on">3.3.2</span= ></st1:chsdate> 功率谱分析及其应用= </h3> <h4 style=3D'text-indent:21.0pt;line-height:16.5pt'>1. <= /span>= 功率谱密度函数的定= 0041;</h4> 对于平稳随机信号</spa= n>x(t)，若其均值为零= 且不含周期成分，则= 0854;自相关函数Rx</span= >(<![if !vml]><img width=3D12 height=3D13 src=3D"ch033.files/image016.gif" v:shapes=3D"_x0000_i1032"><![endif]>= →<s= pan lang=3DEN-US>∞)=3D0, 满足= 傅里叶变换条件 <![if !vml]><img width=3D105 height=3D31 src=3D"ch033.files/image018.gif" v:shapes=3D"_x0000_i1033"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; &nbs= p; (3-36) 于是存在&#= 22914;下关于Rx</span= >(<![if !vml]><img width=3D12 height=3D13 src=3D"ch033.files/image016.gif" v:shapes=3D"_x0000_i1034"><![endif]>= )的傅里叶变换对ᦂ= 6; <![if !vml]><img width=3D152 height=3D31 src=3D"ch033.files/image020.gif" v:shapes=3D"_x0000_i1035"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; &nbs= p; (3-37) <![if !vml]><img width=3D143 height=3D31 src=3D"ch033.files/image022.gif" v:shapes=3D"_x0000_i1036"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; &nbs= p; (3-38) 定义Sx</span= >(jf)为随机信号x(t)的自功率谱密度= 函数，简称自谱或自= 1151;率谱。Rx</span= >(<![if !vml]><img width=3D12 height=3D13 src=3D"ch033.files/image016.gif" v:shapes=3D"_x0000_i1037"><![endif]>= )是对信号x(t)的时延域分析，= Sx</span= >(jf)是在频域分析，= 它们所包含的信息是= 3436;全相同的。 而对于平稳随机信Ö= 95;x(t)、<= span lang=3DEN-US>y(t)，在满足傅里叶= 变换条件下存在如下= 0851;于Rxy(<![if !vml]><img width=3D12 height=3D13 src=3D"ch033.files/image016.gif" v:shapes=3D"_x0000_i1038"><![endif]>= ) 的傅里叶变换对ᦂ= 6; <![if !vml]><img width=3D159 height=3D31 src=3D"ch033.files/image024.gif" v:shapes=3D"_x0000_i1039"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; (3-39)<= /p> <![if !vml]><img width=3D152 height=3D31 src=3D"ch033.files/image026.gif" v:shapes=3D"_x0000_i1040"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; (3-40)</= span> 定义Sxy</spa= n>(jf)为随机信号x(t)、y(t)的互= 谱密度函数，简称互= 5889;或互功率谱。Sxy</spa= n>(jf)保留了R<s= ub>xy(<![if !vml]><img width=3D12 height=3D13 src=3D"ch033.files/image016.gif" v:shapes=3D"_x0000_i1041"><![endif]>= )的全部信息。 Rx(</sp= an><![if !vml]><img width=3D12 height=3D13 src=3D"ch033.files/image016.gif" v:shapes=3D"_x0000_i1042"><![endif]>= )为实偶函数，故</span= >Sx</span= >(jf)也为实偶函数。= 互相关函数Rxy(<![if !vml]><img width=3D12 height=3D13 src=3D"ch033.files/image016.gif" v:shapes=3D"_x0000_i1043"><![endif]>= )为非奇非偶函数ᦁ= 2;因此Sxy</spa= n>(jf)具有虚、实两部= 分。Sx</span= >(jf)是(<![if !vml]><img width=3D23 height=3D12 src=3D"ch033.files/image028.gif" v:shapes=3D"_x0000_i1044"><![endif]>= ，<= span lang=3DEN-US><![if !vml]><img width=3D15 height=3D12 src=3D"ch033.files/image030.gif" v:shapes=3D"_x0000_i1045"><![endif]>= )频率= 范围内的自功率谱，= 5152;以称为双边自谱。ஸ= 1;于Sx</span= >(jf)为实偶函数，而= 在实际应用中频率不= 3021;为负值，因此，用ࢷ= 2;(0，= <![if !vml]><img width=3D15 height=3D12 src=3D"ch033.files/image030.gif" v:shapes=3D"_x0000_i1046"><![endif]>= )频率= 范围内的单边自谱Gx</span= >(jf)表示信号的全部= 功率谱(见图3.28)，即= <![if !vml]><img width=3D100 height=3D21 src=3D"ch033.files/image033.gif" v:shapes=3D"_x0000_i1047"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; &nbs= p; (3-41) <![if !vml]><img width=3D351 height=3D171 src=3D"ch033.files/image035.gif" v:shapes=3D"_x0000_i1048"><![endif]></span= > <h5>图= 3.28 单边&#= 33258;谱和双边自谱</h5> <h4 style=3D'text-indent:21.0pt'>2. = 功率谱密度函数的物= 9702;意义</h4> 当<![if !vml]><img width=3D12 height=3D13 src=3D"ch033.files/image016.gif" v:shapes=3D"_x0000_i1049"><![endif]>= =3D0时，= 根据式(3-38)，有= <![if !vml]><img width=3D139 height=3D31 src=3D"ch033.files/image037.gif" v:shapes=3D"_x0000_i1050"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; </s= pan>(3-42) 而根据自相关函= 数的定义式(3-18)，则= <![if !vml]><img width=3D195 height=3D76 src=3D"ch033.files/image039.gif" v:shapes=3D"_x0000_i1051"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; (3-43) 比较上述两式，= 则 <![if !vml]><img width=3D165 height=3D39 src=3D"ch033.files/image041.gif" v:shapes=3D"_x0000_i1052"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; (3-44) 在机械系= 2479;中，如果x</sp= an>(t= )是Ë= 01;移-时间历程，<sp= an lang=3DEN-US style=3D'mso-bidi-font-size:10.5pt;letter-spacing:-.1pt'>x<sup= >2(t= )就Ö= 53;映蓄积在弹性体上的= 势能；而如果x(t)是速= 度-时间历程，x2(t)就反= 映系统运动的动能。= 2240;此，x2(t)可以= 看作信号的能量，<= span lang=3DEN-US>x2(t)/T</span= >表示信号x(t)的功= 率，而<![if !vml]><img width=3D84 height=3D39 src=3D"ch033.files/image043.gif" v:shapes=3D"_x0000_i1053"><![endif]>= 则为= 信号x(t)的总功率。由式= (3-43)可知= ，Sx</span= >(jf)曲线下的总面积= 与x2(t)/T</span= >曲线下的总面积= 相等。故Sx</span= >(jf )<= span style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han= si-font-family: "Times New Roman"'>曲线下的总面积= 就是信号的总功率。= 3427;是由无数不同频率ߍ= 8;的功率元Sx</span= >(jf )df组成，Sx</span= >(jf )<= span style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han= si-font-family: "Times New Roman"'>的大小表示总功= 率在不同频率处的功= 9575;分布。因此，Sx</span= >(jf )<= span style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han= si-font-family: "Times New Roman"'>表示信号的功率= 密度沿频率轴的分布= 5292;故又称Sx</span= >(jf)为功率谱密度函= 数，如图3.29所示= 。用同样的方法，可= 0197;解释互谱密度函数</span= >Sxy</spa= n>(jf)。 <![if !vml]><img width=3D485 height=3D169 src=3D"ch033.files/image045.gif" v:shapes=3D"_x0000_i1054"><![endif]></span= > <h5>图= 3.29 自功&#= 29575;谱的图形解释</h5> <h4 style=3D'text-indent:21.0pt'>3. = 自功率谱密度函数Sx</span= >(jf)= 和幅值谱X(jf)= 的关系</h4> 由巴塞伐尔定理，Õ= 63;式(3-35)，信= 号的平均功率表示为</s= pan> <!-= -[if gte vml 1]><v:shape id=3D"_x0000_i1055" type=3D"#_x0000_t75" style=3D'width:3in;height:27.75pt= ' o:ole=3D""> <v:imagedata src=3D"ch033.files/image046.wmz" o:title=3D""/> </v:shape><![endif]--><![if !vml]><img width=3D288 height=3D37 src=3D"ch033.files/image047.gif" v:shapes=3D"_x0000_i1055"><![endif]>=  = (3-45) 由式(3-43)<= span style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han= si-font-family: "Times New Roman"'>和式(3-44)<= span style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han= si-font-family: "Times New Roman"'>，式(3-45)<= span style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han= si-font-family: "Times New Roman"'>变为 <![if !vml]><img width=3D215 height=3D31 src=3D"ch033.files/image049.gif" v:shapes=3D"_x0000_i1056"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; (3-46) 因此，自谱(双边、单&#= 36793;)Sx(jf)、Gx(jf)和幅值谱X(jf)的关= 系为 <![if !vml]><img width=3D139 height=3D37 src=3D"ch033.files/image051.gif" v:shapes=3D"_x0000_i1057"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; &nbs= p; (3-47) <![if !vml]><img width=3D140 height=3D37 src=3D"ch033.files/image053.gif" v:shapes=3D"_x0000_i1058"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; </s= pan>(3-48) 利用这一关系，通ñ= 20;就可以对时域信号直= 接作傅里叶变换来计= 1639;其功率谱。 <span lang=3DEN-US style=3D'font-size:10.5pt;font-family:"Times New Roman"; mso-fareast-font-family:KaiTi_GB2312;mso-font-kerning:1.0pt;mso-ansi-langua= ge: EN-US;mso-fareast-language:ZH-CN;mso-bidi-language:AR-SA'> <h4 style=3D'text-indent:21.0pt'>4. = 功率谱的估计</h4> 在实际测试中，观ė= 79;只能在有限的时间区= 域[T1,T2]内，因而所得到= 的平均功率只是近似= 0540;。根据功率谱密度࠲= 9;数的定义，信号的自&= #35889;估计应当先根据原= 987;信号计算出其相关函= ;数，然后对自相关函&#= 25968;作傅里叶变换。在ê= 54;际自谱估计时，往往= 采用更为方便可行的= 6041;法。 在用模拟分析方法Ë= 16;自谱估计时，通常采= 用窄带滤波器和适当= 0340;模拟电路来实现。ஷ= 2;中心频率为ƒ、带= 宽为B的带通滤波器对= 时域信号进行滤波，= 1487;得中心频率为ƒ处信= 号的平均功率为 <![if !vml]><img width=3D187 height=3D37 src=3D"ch033.files/image055.gif" v:shapes=3D"_x0000_i1059"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; (3-49) 显然，它是ƒ处带= 宽B的函数。由于自= 谱表示信号的功率密= 4230;沿频率轴的分布，ࡘ= 3;单位频率上的平均功&= #29575;。因此 <!-= -[if gte vml 1]><v:shape id=3D"_x0000_i1060" type=3D"#_x0000_t75" style=3D'width:201.75pt;height:35= .25pt' o:ole=3D""> <v:imagedata src=3D"ch033.files/image056.wmz" o:title=3D""/> </v:shape><![endif]--><![if !vml]><img width=3D269 height=3D47 src=3D"ch033.files/image057.gif" v:shapes=3D"_x0000_i1060"><![endif]>=  = (3-50) 由此，自谱的估= 计为 <sub= ><![if !vml]><img width=3D163 height=3D37 src=3D"ch033.files/image059.gif" v:shapes=3D"_x0000_i1061"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; (3-51) 在模拟分析过程中ʌ= 92;自动或手动调节可调= 中心频率的带通滤波= 2120;的中心频率ƒ，对= 信号依次进行扫频、= 8388;波、平方、积分和༊= 0;法运算，最后由记录&= #20202;得到<![if !vml]><img width=3D44 height=3D24 src=3D"ch033.files/image061.gif" v:shapes=3D"_x0000_i1062"><![endif]>= -<= /span>ƒ图，= 其原理框图如图3.30所示= 。 <![if !vml]><img width=3D416 height=3D99 src=3D"ch033.files/image063.gif" v:shapes=3D"_x0000_i1063"><![endif]></span= > <h5>图= 3.30 自谱&#= 30340;模拟分析原理框图</spa= n></h5> 类似地，模拟信号į= 40;互谱估计为 <![if !vml]><img width=3D148 height=3D37 src=3D"ch033.files/image065.gif" v:shapes=3D"_x0000_i1064"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; (3-52)</= span> 上述功率谱估计的Ą= 41;法都是基于模拟分析= 的方法，它受模拟分= 6512;仪记录时间T、滤波器带宽等= 参数的影响，是一种= 6817;似估计。由于模拟࠳= 8;析仪电路复杂、价格&= #26114;贵，目前较实用、= 377;效的方法是基于数字= ;信号处理技术，通过</= span>FFT来进= 行功率谱估计。 根据式(3-48)，离= 散随机序列x(n)的自功率谱密度= 为 </spa= n><![if !vml]><img width=3D123 height=3D37 src=3D"ch033.files/image067.gif" v:shapes=3D"_x0000_i1065"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; </s= pan>(3-53) 因此，自谱的估= 计为 = <![if !vml]><img width=3D101 height=3D37 src=3D"ch033.files/image069.gif" v:shapes=3D"_x0000_i1066"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; &nbs= p; (3-54) <span lang=3DEN-US style=3D'font-size:10.5pt;mso-bidi-font-size:10.0pt;font= -family: "Times New Roman";mso-fareast-font-family:SimSun;mso-ansi-language:EN-US; mso-fareast-language:ZH-CN;mso-bidi-language:AR-SA'> 而由式(3-52)= ，互谱的估计为= <![if !vml]><img width=3D132 height=3D37 src=3D"ch033.files/image071.gif" v:shapes=3D"_x0000_i1067"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; (3-55) 离散序列{x(n)}的傅里叶变换<= span lang=3DEN-US> X(K)具有周期函数的性ũ= 36;，因而这种功率谱估= 计的方法称为周期图= 7861;。它是一种最简单ӌ= 9;常用的功率谱估计算&= #27861;。 <h4 style=3D'text-indent:21.0pt'>5. = 功率谱的应用</h4> 1) </s= pan>获取系统的频率= 结构特性 与幅值谱X(f)相似= ，自谱Sx</span= >(jf)也反映信号的频= 率结构。由于自谱Sx</span= >(jf)反映的是信号幅= 值的平方，因而其频= 9575;结构特性更为明显ᦁ= 2;如图3.31所示= 。 <![if !vml]><img width=3D400 height=3D444 src=3D"ch033.files/image073.gif" v:shapes=3D"_x0000_i1068"><![endif]></span= > <h5>图= 3.31 幅值&#= 35889;和自谱图</h5> 若有一线性系统= 如图 3.32 所示= ，其输入为x(t)，输出为y(t)，系= 统的频率响应函数为</s= pan>H(jf)，则Y(jf)= =3DH(jf)<![if !vml]><img width=3D8 height=3D11 src=3D"ch033.files/image075.gif" v:shapes=3D"_x0000_i1069"><![endif]>= X(jf </sp= an>)。<o:p></o:p> <![if !vml]><img width=3D176 height=3D43 src=3D"ch033.files/image077.gif" v:shapes=3D"_x0000_i1070"><![endif]></span= > <h5 style=3D'margin-bottom:0cm;margin-bottom:.0001pt'>图3.32 = 理想单输入/输出系统<o:= p></o:p></h5> <span lang=3DEN-US style=3D'font-size:10.5pt;mso-bidi-font-size:10.0pt;font= -family: "Times New Roman";mso-fareast-font-family:SimSun;mso-ansi-language:EN-US; mso-fareast-language:ZH-CN;mso-bidi-language:AR-SA'> 根据自谱和幅值谱į= 40;关系可以证明 </spa= n>Sy(jf)=3D|H(jf) |2 <!= --[if gte vml 1]><v:shape id=3D"_x0000_i1071" type=3D"#_x0000_t75" style=3D'width:6pt;height:8.25pt'= o:ole=3D""> <v:imagedata src=3D"ch033.files/image074.wmz" o:title=3D""/> </v:shape><![endif]--><![if !vml]><img width=3D8 height=3D11 src=3D"ch033.files/image075.gif" v:shapes=3D"_x0000_i1071"><![endif]>= Sx(jf) = ; &n= bsp; = ; = (3-56) </spa= n>Gy(jf)=3D|H(jf) |2 <!= --[if gte vml 1]><v:shape id=3D"_x0000_i1072" type=3D"#_x0000_t75" style=3D'width:6pt;height:8.25pt'= o:ole=3D""> <v:imagedata src=3D"ch033.files/image074.wmz" o:title=3D""/> </v:shape><![endif]--><![if !vml]><img width=3D8 height=3D11 src=3D"ch033.files/image075.gif" v:shapes=3D"_x0000_i1072"><![endif]>= Gx</sub= >(jf) &= nbsp; &nbs= p; &= nbsp; </s= pan>(3-57) Sxy (jf)=3DH(jf) <![if !vml]><img width=3D8 height=3D11 src=3D"ch033.files/image075.gif" v:shapes=3D"_x0000_i1073"><![endif]>= Sx</sub= >(jf) &= nbsp; &nbs= p; &= nbsp; </s= pan>(3-58) 式(3-56)和式(3-57)<= span style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han= si-font-family: "Times New Roman"'>表明，通过输入= /输出的自&#= 35889;分析，就能得出系ń= 79;的幅频特性。但由于= 自谱是自相关函数的= 0613;里叶变换，而自相ࠥ= 1;函数丢失了相位信息&= #65292;因而自谱分析同样= 002;掉了相位信息，利用= ;自谱分析仅仅能获得&#= 31995;统的幅频特性，而Ç= 81;能得到系统的相频特= 性。 由式(3-58)<= span style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han= si-font-family: "Times New Roman"'>可知，从输入的= 自谱和输入/输出的互&#= 35889;可以得到系统的频ħ= 75;响应函数，该式与式= (3-56)和式= (3-57)不同= 的是，所得到的H(jf)不仅含有幅频特= 性而且含有相频特性= 5292;这是因为互相关函ਤ= 8;中包含着相位信息。<= /span> 【例3.11】 应用互功率谱从ࡢ= 3;外界干扰的信息中获&= #21462;测试系统频率响应= 989;数。 如图3.33所示= 的测试系统受外界干= 5200;，n<s= pan lang=3DEN-US>1(t)为输入噪声，n<s= pan lang=3DEN-US>2(t)为加在系统中间= 环节的噪声，n<s= pan lang=3DEN-US>3(t)为加在输出端的= 噪声。该系统的输出</s= pan>y(t)为 <![if !vml]><img width=3D184 height=3D21 src=3D"ch033.files/image079.gif" v:shapes=3D"_x0000_i1074"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; (3-59) 式中，x'(t)、<= span lang=3DEN-US>n'1(t)、n'<= span lang=3DEN-US>2(t)分别为系统对x(t)、<= span lang=3DEN-US>n1(t)、n<s= pan lang=3DEN-US>2(t)的响应。 输入x(t)和输出y</= span>(t)的互= 相关函数为 <![if !vml]><img width=3D244 height=3D24 src=3D"ch033.files/image081.gif" v:shapes=3D"_x0000_i1075"><![endif]>=  &= nbsp; &nbs= p; (3-60) 由于输入</= span>x(t)和噪声n1</su= b>(t)、<= span lang=3DEN-US>n2(t)和n<s= pan lang=3DEN-US>3(t)是独立无关的，= 故互相关函数<![if !vml]><img width=3D43 height=3D24 src=3D"ch033.files/image083.gif" v:shapes=3D"_x0000_i1076"><![endif]>= 、<= span lang=3DEN-US><![if !vml]><img width=3D44 height=3D24 src=3D"ch033.files/image085.gif" v:shapes=3D"_x0000_i1077"><![endif]>= 和<= span lang=3DEN-US><![if !vml]><img width=3D43 height=3D24 src=3D"ch033.files/image087.gif" v:shapes=3D"_x0000_i1078"><![endif]>= 均为= 零，所以 = <![if !vml]><img width=3D91 height=3D23 src=3D"ch033.files/image089.gif" v:shapes=3D"_x0000_i1079"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; (3-61) <a name=3D"OLE_LINK1"> </spa= n><![if !vml]><img width=3D184 height=3D23 src=3D"ch033.files/image091.gif" v:shapes=3D"_x0000_i1080"><![endif]>=  &= nbsp; &nbs= p; (3-62)</spa= n></a> 式中，= H(jf)=3D H<sp= an style=3D'mso-bidi-font-style:italic'>1(jf) ·= ; H2(jf)为系统的频率响= 应函数。 <![if !vml]><img width=3D304 height=3D97 src=3D"ch033.files/image093.gif" v:shapes=3D"_x0000_i1081"><![endif]></span= > <h5>图= 3.33 受外&#= 30028;干扰的系统</h5> 可见，利用互相关Ñ= 89;数分析可排除噪声的= 影响，这是互相关函= 5968;分析方法的突出优୅= 7;。然而应当注意到，&= #21033;用式(3-58)求线= 性系统的频率响应函= 5968;H(jf)时，尽管其中的= 互谱可以不受噪声的= 4433;响，但是输入信号௚= 0;自谱仍然无法排除输&= #20837;端测量噪声的影响A= 292;从而形成测量误差。= ; 2) </s= pan>测定系统的滞后= 时间 互谱分析可用来测ê= 50;滞后时间。一个系统= 输入x(t)和输出y</= span>(t)的互= 谱中的幅角<![if !vml]><img width=3D47 height=3D23 src=3D"ch033.files/image095.gif" v:shapes=3D"_x0000_i1082"><![endif]>= 表示= 了系统输出与输入在= 9057;率ƒ处的= 相位差，因此，在任= 9968;频率ƒ上通= 过系统的滞后时间为</s= pan> <![if !vml]><img width=3D69 height=3D43 src=3D"ch033.files/image097.gif" v:shapes=3D"_x0000_i1083"><![endif]>=  3) </s= pan>利用功率谱分析= 对设备进行故障诊断</s= pan> 【例3.12】 图3.34是由= 汽车变速箱上测取的= 5391;动加速度信号经功஍= 5;谱分析处理后所得的&= #21151;率谱图。一般地，= 491;常运行的机器其功率= ;谱是稳定的，而且各&#= 35889;线对应零件不同运ů= 16;状态的振源。在机器= 运行不正常时，例如= 5292;转系的动不平衡、๘= 4;承的局部损伤、齿轮&= #30340;不正常等，都会引$= 215;谱线的变动。图(b)中，= 在9.2Hz和<= span lang=3DEN-US>18.4Hz两处= 出现额外峰谱，这显= 1034;了机器的某些不正६= 0;，而且指示了异常功&= #29575;消耗所在的频率。$= 825;就为寻找与此频率相= ;对应的故障部位提供&#= 20102;依据。 <![if !vml]><img width=3D529 height=3D123 src=3D"ch033.files/image099.gif" v:shapes=3D"_x0000_i1084"><![endif]></span= > = (a) 变速箱正&#= 24120;工作时的谱图 = (b) 变速箱不&#= 27491;常工作时的谱图 <h5>图= 3.34 汽车&#= 21464;速箱的振动功率谱Þ= 70;</h5> <h2 style=3D'text-align:justify;text-justify:inter-ideograph'><o:p> </o:p></h2> </div> </body> </html> ------=_NextPart_01C7AF42.A50F8AB0 Content-Location: file:///C:/D064C693/ch033.files/image001.wmz Content-Transfer-Encoding: base64 Content-Type: image/x-wmz H4sIAAAAAAACC51UTWgTQRR+bzYbm02bv1pBKLUNWtJKpBY8FpKmwXqohqbFHoSQNhvdkjTSpLaF QkuE4sl6EVFRvCwoePBS8CCheNCD2qsnpT30LmF7qjS+md2NtVZCHXbmffPmzfvmfTPs1y8fnoBo G54thi6O+kYRCEgvGQCDbbEqU2eIFkJWq9UEuoCnLZ+b2XHNbItteFoJdTu90A41HkzNx1kIr1Nf 8QA8pB1uK6oZRtKlW2OLt1WAFZHFxfMxjk6yK/iYtRPaY5GaedoHKAwD35iWV4udV9X5ztFCPj0D lcnd+Y/UzYA+cNAYpolCtl8xNxHq33StoJl9C89KZvY3+/+XHYX9Qc62OguXMtSTWQpN9yxlSMUi 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MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_NextPart_01C7AF42.A50F8AB0" 此文档为“单个文件网页”，也称为“Web 档案”文件。如果您看到此消息，但是您的浏览器或编辑器不支持“Web 档案”文件。请下载支持“Web 档案”的浏览器，如 Microsoft Internet Explorer。 ------=_NextPart_01C7AF42.A50F8AB0 Content-Location: file:///C:/D064C693/ch033.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" 3.3=20 功率谱分析及其应用</t= itle> <o:SmartTagType namespaceuri=3D"urn:schemas-microsoft-com:office:smarttags" name=3D"chsdate" downloadurl=3D""/>  <style>  </style>  </head> <body lang=3DZH-CN link=3Dblack vlink=3Dpurple style=3D'tab-interval:21.0pt= ;text-justify-trim: punctuation'> <div class=3DSection1 style=3D'layout-grid:15.2pt 0pt;mso-layout-grid-char-= alt: 0'> <h2>3.3 </= span>功率&#= 35889;分析及其应用</h2> 在第2章中讨论了周期= 信号和瞬态信号的时= 2495;波形与频域的幅值ค= 9;及相位谱之间的对应&= #20851;系，并了解到频域= 551;述可反映信号频率结= ;构组成。然而对于随&#= 26426;信号，由于其样本ć= 54;线的波形具有随机性= ，而且是时域无限信= 1495;，不满足傅里叶变ৰ= 2;条件，因而从理论上&= #35762;，随机信号不能直= 509;进行傅里叶变换作幅= ;值谱和相位谱分析，&#= 32780;是应用具有统计特ô= 49;的功率谱密度函数在= 频域内对随机信号作= 9057;谱分析，它是研究ॱ= 9;稳随机过程的重要方&= #27861;。功率谱密度函数= 998;自谱和互谱两种形式= ;。 <h3 style=3D'line-height:16.5pt'><a name=3D"_Toc139040641"></a><st1:chsdate Year=3D"1899" Month=3D"12" Day=3D"30" IsLunarDate=3D"False" IsROCDate=3D"Fa= lse" w:st=3D"on">3.3.1</span= ></st1:chsdate> 巴塞伐尔(Paseval)定理</= span></h3> 巴塞伐尔定理中= ，在时域中计算的信= 1495;总能量等于在频域ߑ= 3;计算的信号总能量，&= #21363; <![if !vml]><img width=3D157 height=3D31 src=3D"ch033.files/image002.gif" v:shapes=3D"_x0000_i1025"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; (3= -35) 该定理可以用傅= 里叶变换的卷积来证= 6126;。设有傅里叶变换ल= 5; <![if !vml]><img width=3D89 height=3D21 src=3D"ch033.files/image004.gif" v:shapes=3D"_x0000_i1026"><![endif]>= ，<= span lang=3DEN-US><![if !vml]><img width=3D92 height=3D21 src=3D"ch033.files/image006.gif" v:shapes=3D"_x0000_i1027"><![endif]>=  根据频域&#= 21367;积定理有 <![if !vml]><img width=3D263 height=3D31 src=3D"ch033.files/image008.gif" v:shapes=3D"_x0000_i1028"><![endif]>=  令f<s= pan lang=3DEN-US>0=3D0，x<s= pan lang=3DEN-US>1(t)=3Dx2(t)=3Dx(t)，则= <![if !vml]><img width=3D183 height=3D31 src=3D"ch033.files/image010.gif" v:shapes=3D"_x0000_i1029"><![endif]>=  式中，x(t)是实函数，则X(-</spa= n>jf)=3DX*(jf)，所以 <![if !vml]><img width=3D288 height=3D31 src=3D"ch033.files/image012.gif" v:shapes=3D"_x0000_i1030"><![endif]>=  式中，<sp= an lang=3DEN-US><![if !vml]><img width=3D53 height=3D23 src=3D"ch033.files/image014.gif" v:shapes=3D"_x0000_i1031"><![endif]>= 称为= 能谱，是沿频率轴的= 3021;量分布密度。 <span lang=3DEN-US style=3D'font-size:10.5pt;mso-bidi-font-size:10.0pt;font= -family: Arial;mso-fareast-font-family:SimHei;mso-bidi-font-family:"Times New Roman"; mso-ansi-language:EN-US;mso-fareast-language:ZH-CN;mso-bidi-language:AR-SA'= > <h3 style=3D'line-height:16.5pt'><a name=3D"_Toc139040642"></a><st1:chsdate Year=3D"1899" Month=3D"12" Day=3D"30" IsLunarDate=3D"False" IsROCDate=3D"Fa= lse" w:st=3D"on">3.3.2</span= ></st1:chsdate> 功率谱分析及其应用= </h3> <h4 style=3D'text-indent:21.0pt;line-height:16.5pt'>1. <= /span>= 功率谱密度函数的定= 0041;</h4> 对于平稳随机信号</spa= n>x(t)，若其均值为零= 且不含周期成分，则= 0854;自相关函数Rx</span= >(<![if !vml]><img width=3D12 height=3D13 src=3D"ch033.files/image016.gif" v:shapes=3D"_x0000_i1032"><![endif]>= →<s= pan lang=3DEN-US>∞)=3D0, 满足= 傅里叶变换条件 <![if !vml]><img width=3D105 height=3D31 src=3D"ch033.files/image018.gif" v:shapes=3D"_x0000_i1033"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; &nbs= p; (3-36) 于是存在&#= 22914;下关于Rx</span= >(<![if !vml]><img width=3D12 height=3D13 src=3D"ch033.files/image016.gif" v:shapes=3D"_x0000_i1034"><![endif]>= )的傅里叶变换对ᦂ= 6; <![if !vml]><img width=3D152 height=3D31 src=3D"ch033.files/image020.gif" v:shapes=3D"_x0000_i1035"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; &nbs= p; (3-37) <![if !vml]><img width=3D143 height=3D31 src=3D"ch033.files/image022.gif" v:shapes=3D"_x0000_i1036"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; &nbs= p; (3-38) 定义Sx</span= >(jf)为随机信号x(t)的自功率谱密度= 函数，简称自谱或自= 1151;率谱。Rx</span= >(<![if !vml]><img width=3D12 height=3D13 src=3D"ch033.files/image016.gif" v:shapes=3D"_x0000_i1037"><![endif]>= )是对信号x(t)的时延域分析，= Sx</span= >(jf)是在频域分析，= 它们所包含的信息是= 3436;全相同的。 而对于平稳随机信Ö= 95;x(t)、<= span lang=3DEN-US>y(t)，在满足傅里叶= 变换条件下存在如下= 0851;于Rxy(<![if !vml]><img width=3D12 height=3D13 src=3D"ch033.files/image016.gif" v:shapes=3D"_x0000_i1038"><![endif]>= ) 的傅里叶变换对ᦂ= 6; <![if !vml]><img width=3D159 height=3D31 src=3D"ch033.files/image024.gif" v:shapes=3D"_x0000_i1039"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; (3-39)<= /p> <![if !vml]><img width=3D152 height=3D31 src=3D"ch033.files/image026.gif" v:shapes=3D"_x0000_i1040"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; (3-40)</= span> 定义Sxy</spa= n>(jf)为随机信号x(t)、y(t)的互= 谱密度函数，简称互= 5889;或互功率谱。Sxy</spa= n>(jf)保留了R<s= ub>xy(<![if !vml]><img width=3D12 height=3D13 src=3D"ch033.files/image016.gif" v:shapes=3D"_x0000_i1041"><![endif]>= )的全部信息。 Rx(</sp= an><![if !vml]><img width=3D12 height=3D13 src=3D"ch033.files/image016.gif" v:shapes=3D"_x0000_i1042"><![endif]>= )为实偶函数，故</span= >Sx</span= >(jf)也为实偶函数。= 互相关函数Rxy(<![if !vml]><img width=3D12 height=3D13 src=3D"ch033.files/image016.gif" v:shapes=3D"_x0000_i1043"><![endif]>= )为非奇非偶函数ᦁ= 2;因此Sxy</spa= n>(jf)具有虚、实两部= 分。Sx</span= >(jf)是(<![if !vml]><img width=3D23 height=3D12 src=3D"ch033.files/image028.gif" v:shapes=3D"_x0000_i1044"><![endif]>= ，<= span lang=3DEN-US><![if !vml]><img width=3D15 height=3D12 src=3D"ch033.files/image030.gif" v:shapes=3D"_x0000_i1045"><![endif]>= )频率= 范围内的自功率谱，= 5152;以称为双边自谱。ஸ= 1;于Sx</span= >(jf)为实偶函数，而= 在实际应用中频率不= 3021;为负值，因此，用ࢷ= 2;(0，= <![if !vml]><img width=3D15 height=3D12 src=3D"ch033.files/image030.gif" v:shapes=3D"_x0000_i1046"><![endif]>= )频率= 范围内的单边自谱Gx</span= >(jf)表示信号的全部= 功率谱(见图3.28)，即= <![if !vml]><img width=3D100 height=3D21 src=3D"ch033.files/image033.gif" v:shapes=3D"_x0000_i1047"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; &nbs= p; (3-41) <![if !vml]><img width=3D351 height=3D171 src=3D"ch033.files/image035.gif" v:shapes=3D"_x0000_i1048"><![endif]></span= > <h5>图= 3.28 单边&#= 33258;谱和双边自谱</h5> <h4 style=3D'text-indent:21.0pt'>2. = 功率谱密度函数的物= 9702;意义</h4> 当<![if !vml]><img width=3D12 height=3D13 src=3D"ch033.files/image016.gif" v:shapes=3D"_x0000_i1049"><![endif]>= =3D0时，= 根据式(3-38)，有= <![if !vml]><img width=3D139 height=3D31 src=3D"ch033.files/image037.gif" v:shapes=3D"_x0000_i1050"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; </s= pan>(3-42) 而根据自相关函= 数的定义式(3-18)，则= <![if !vml]><img width=3D195 height=3D76 src=3D"ch033.files/image039.gif" v:shapes=3D"_x0000_i1051"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; (3-43) 比较上述两式，= 则 <![if !vml]><img width=3D165 height=3D39 src=3D"ch033.files/image041.gif" v:shapes=3D"_x0000_i1052"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; (3-44) 在机械系= 2479;中，如果x</sp= an>(t= )是Ë= 01;移-时间历程，<sp= an lang=3DEN-US style=3D'mso-bidi-font-size:10.5pt;letter-spacing:-.1pt'>x<sup= >2(t= )就Ö= 53;映蓄积在弹性体上的= 势能；而如果x(t)是速= 度-时间历程，x2(t)就反= 映系统运动的动能。= 2240;此，x2(t)可以= 看作信号的能量，<= span lang=3DEN-US>x2(t)/T</span= >表示信号x(t)的功= 率，而<![if !vml]><img width=3D84 height=3D39 src=3D"ch033.files/image043.gif" v:shapes=3D"_x0000_i1053"><![endif]>= 则为= 信号x(t)的总功率。由式= (3-43)可知= ，Sx</span= >(jf)曲线下的总面积= 与x2(t)/T</span= >曲线下的总面积= 相等。故Sx</span= >(jf )<= span style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han= si-font-family: "Times New Roman"'>曲线下的总面积= 就是信号的总功率。= 3427;是由无数不同频率ߍ= 8;的功率元Sx</span= >(jf )df组成，Sx</span= >(jf )<= span style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han= si-font-family: "Times New Roman"'>的大小表示总功= 率在不同频率处的功= 9575;分布。因此，Sx</span= >(jf )<= span style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han= si-font-family: "Times New Roman"'>表示信号的功率= 密度沿频率轴的分布= 5292;故又称Sx</span= >(jf)为功率谱密度函= 数，如图3.29所示= 。用同样的方法，可= 0197;解释互谱密度函数</span= >Sxy</spa= n>(jf)。 <![if !vml]><img width=3D485 height=3D169 src=3D"ch033.files/image045.gif" v:shapes=3D"_x0000_i1054"><![endif]></span= > <h5>图= 3.29 自功&#= 29575;谱的图形解释</h5> <h4 style=3D'text-indent:21.0pt'>3. = 自功率谱密度函数Sx</span= >(jf)= 和幅值谱X(jf)= 的关系</h4> 由巴塞伐尔定理，Õ= 63;式(3-35)，信= 号的平均功率表示为</s= pan> <!-= -[if gte vml 1]><v:shape id=3D"_x0000_i1055" type=3D"#_x0000_t75" style=3D'width:3in;height:27.75pt= ' o:ole=3D""> <v:imagedata src=3D"ch033.files/image046.wmz" o:title=3D""/> </v:shape><![endif]--><![if !vml]><img width=3D288 height=3D37 src=3D"ch033.files/image047.gif" v:shapes=3D"_x0000_i1055"><![endif]>=  = (3-45) 由式(3-43)<= span style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han= si-font-family: "Times New Roman"'>和式(3-44)<= span style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han= si-font-family: "Times New Roman"'>，式(3-45)<= span style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han= si-font-family: "Times New Roman"'>变为 <![if !vml]><img width=3D215 height=3D31 src=3D"ch033.files/image049.gif" v:shapes=3D"_x0000_i1056"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; (3-46) 因此，自谱(双边、单&#= 36793;)Sx(jf)、Gx(jf)和幅值谱X(jf)的关= 系为 <![if !vml]><img width=3D139 height=3D37 src=3D"ch033.files/image051.gif" v:shapes=3D"_x0000_i1057"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; &nbs= p; (3-47) <![if !vml]><img width=3D140 height=3D37 src=3D"ch033.files/image053.gif" v:shapes=3D"_x0000_i1058"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; </s= pan>(3-48) 利用这一关系，通ñ= 20;就可以对时域信号直= 接作傅里叶变换来计= 1639;其功率谱。 <span lang=3DEN-US style=3D'font-size:10.5pt;font-family:"Times New Roman"; mso-fareast-font-family:KaiTi_GB2312;mso-font-kerning:1.0pt;mso-ansi-langua= ge: EN-US;mso-fareast-language:ZH-CN;mso-bidi-language:AR-SA'> <h4 style=3D'text-indent:21.0pt'>4. = 功率谱的估计</h4> 在实际测试中，观ė= 79;只能在有限的时间区= 域[T1,T2]内，因而所得到= 的平均功率只是近似= 0540;。根据功率谱密度࠲= 9;数的定义，信号的自&= #35889;估计应当先根据原= 987;信号计算出其相关函= ;数，然后对自相关函&#= 25968;作傅里叶变换。在ê= 54;际自谱估计时，往往= 采用更为方便可行的= 6041;法。 在用模拟分析方法Ë= 16;自谱估计时，通常采= 用窄带滤波器和适当= 0340;模拟电路来实现。ஷ= 2;中心频率为ƒ、带= 宽为B的带通滤波器对= 时域信号进行滤波，= 1487;得中心频率为ƒ处信= 号的平均功率为 <![if !vml]><img width=3D187 height=3D37 src=3D"ch033.files/image055.gif" v:shapes=3D"_x0000_i1059"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; (3-49) 显然，它是ƒ处带= 宽B的函数。由于自= 谱表示信号的功率密= 4230;沿频率轴的分布，ࡘ= 3;单位频率上的平均功&= #29575;。因此 <!-= -[if gte vml 1]><v:shape id=3D"_x0000_i1060" type=3D"#_x0000_t75" style=3D'width:201.75pt;height:35= .25pt' o:ole=3D""> <v:imagedata src=3D"ch033.files/image056.wmz" o:title=3D""/> </v:shape><![endif]--><![if !vml]><img width=3D269 height=3D47 src=3D"ch033.files/image057.gif" v:shapes=3D"_x0000_i1060"><![endif]>=  = (3-50) 由此，自谱的估= 计为 <sub= ><![if !vml]><img width=3D163 height=3D37 src=3D"ch033.files/image059.gif" v:shapes=3D"_x0000_i1061"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; (3-51) 在模拟分析过程中ʌ= 92;自动或手动调节可调= 中心频率的带通滤波= 2120;的中心频率ƒ，对= 信号依次进行扫频、= 8388;波、平方、积分和༊= 0;法运算，最后由记录&= #20202;得到<![if !vml]><img width=3D44 height=3D24 src=3D"ch033.files/image061.gif" v:shapes=3D"_x0000_i1062"><![endif]>= -<= /span>ƒ图，= 其原理框图如图3.30所示= 。 <![if !vml]><img width=3D416 height=3D99 src=3D"ch033.files/image063.gif" v:shapes=3D"_x0000_i1063"><![endif]></span= > <h5>图= 3.30 自谱&#= 30340;模拟分析原理框图</spa= n></h5> 类似地，模拟信号į= 40;互谱估计为 <![if !vml]><img width=3D148 height=3D37 src=3D"ch033.files/image065.gif" v:shapes=3D"_x0000_i1064"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; (3-52)</= span> 上述功率谱估计的Ą= 41;法都是基于模拟分析= 的方法，它受模拟分= 6512;仪记录时间T、滤波器带宽等= 参数的影响，是一种= 6817;似估计。由于模拟࠳= 8;析仪电路复杂、价格&= #26114;贵，目前较实用、= 377;效的方法是基于数字= ;信号处理技术，通过</= span>FFT来进= 行功率谱估计。 根据式(3-48)，离= 散随机序列x(n)的自功率谱密度= 为 </spa= n><![if !vml]><img width=3D123 height=3D37 src=3D"ch033.files/image067.gif" v:shapes=3D"_x0000_i1065"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; </s= pan>(3-53) 因此，自谱的估= 计为 = <![if !vml]><img width=3D101 height=3D37 src=3D"ch033.files/image069.gif" v:shapes=3D"_x0000_i1066"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; &nbs= p; (3-54) <span lang=3DEN-US style=3D'font-size:10.5pt;mso-bidi-font-size:10.0pt;font= -family: "Times New Roman";mso-fareast-font-family:SimSun;mso-ansi-language:EN-US; mso-fareast-language:ZH-CN;mso-bidi-language:AR-SA'> 而由式(3-52)= ，互谱的估计为= <![if !vml]><img width=3D132 height=3D37 src=3D"ch033.files/image071.gif" v:shapes=3D"_x0000_i1067"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; (3-55) 离散序列{x(n)}的傅里叶变换<= span lang=3DEN-US> X(K)具有周期函数的性ũ= 36;，因而这种功率谱估= 计的方法称为周期图= 7861;。它是一种最简单ӌ= 9;常用的功率谱估计算&= #27861;。 <h4 style=3D'text-indent:21.0pt'>5. = 功率谱的应用</h4> 1) </s= pan>获取系统的频率= 结构特性 与幅值谱X(f)相似= ，自谱Sx</span= >(jf)也反映信号的频= 率结构。由于自谱Sx</span= >(jf)反映的是信号幅= 值的平方，因而其频= 9575;结构特性更为明显ᦁ= 2;如图3.31所示= 。 <![if !vml]><img width=3D400 height=3D444 src=3D"ch033.files/image073.gif" v:shapes=3D"_x0000_i1068"><![endif]></span= > <h5>图= 3.31 幅值&#= 35889;和自谱图</h5> 若有一线性系统= 如图 3.32 所示= ，其输入为x(t)，输出为y(t)，系= 统的频率响应函数为</s= pan>H(jf)，则Y(jf)= =3DH(jf)<![if !vml]><img width=3D8 height=3D11 src=3D"ch033.files/image075.gif" v:shapes=3D"_x0000_i1069"><![endif]>= X(jf </sp= an>)。<o:p></o:p> <![if !vml]><img width=3D176 height=3D43 src=3D"ch033.files/image077.gif" v:shapes=3D"_x0000_i1070"><![endif]></span= > <h5 style=3D'margin-bottom:0cm;margin-bottom:.0001pt'>图3.32 = 理想单输入/输出系统<o:= p></o:p></h5> <span lang=3DEN-US style=3D'font-size:10.5pt;mso-bidi-font-size:10.0pt;font= -family: "Times New Roman";mso-fareast-font-family:SimSun;mso-ansi-language:EN-US; mso-fareast-language:ZH-CN;mso-bidi-language:AR-SA'> 根据自谱和幅值谱į= 40;关系可以证明 </spa= n>Sy(jf)=3D|H(jf) |2 <!= --[if gte vml 1]><v:shape id=3D"_x0000_i1071" type=3D"#_x0000_t75" style=3D'width:6pt;height:8.25pt'= o:ole=3D""> <v:imagedata src=3D"ch033.files/image074.wmz" o:title=3D""/> </v:shape><![endif]--><![if !vml]><img width=3D8 height=3D11 src=3D"ch033.files/image075.gif" v:shapes=3D"_x0000_i1071"><![endif]>= Sx(jf) = ; &n= bsp; = ; = (3-56) </spa= n>Gy(jf)=3D|H(jf) |2 <!= --[if gte vml 1]><v:shape id=3D"_x0000_i1072" type=3D"#_x0000_t75" style=3D'width:6pt;height:8.25pt'= o:ole=3D""> <v:imagedata src=3D"ch033.files/image074.wmz" o:title=3D""/> </v:shape><![endif]--><![if !vml]><img width=3D8 height=3D11 src=3D"ch033.files/image075.gif" v:shapes=3D"_x0000_i1072"><![endif]>= Gx</sub= >(jf) &= nbsp; &nbs= p; &= nbsp; </s= pan>(3-57) Sxy (jf)=3DH(jf) <![if !vml]><img width=3D8 height=3D11 src=3D"ch033.files/image075.gif" v:shapes=3D"_x0000_i1073"><![endif]>= Sx</sub= >(jf) &= nbsp; &nbs= p; &= nbsp; </s= pan>(3-58) 式(3-56)和式(3-57)<= span style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han= si-font-family: "Times New Roman"'>表明，通过输入= /输出的自&#= 35889;分析，就能得出系ń= 79;的幅频特性。但由于= 自谱是自相关函数的= 0613;里叶变换，而自相ࠥ= 1;函数丢失了相位信息&= #65292;因而自谱分析同样= 002;掉了相位信息，利用= ;自谱分析仅仅能获得&#= 31995;统的幅频特性，而Ç= 81;能得到系统的相频特= 性。 由式(3-58)<= span style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han= si-font-family: "Times New Roman"'>可知，从输入的= 自谱和输入/输出的互&#= 35889;可以得到系统的频ħ= 75;响应函数，该式与式= (3-56)和式= (3-57)不同= 的是，所得到的H(jf)不仅含有幅频特= 性而且含有相频特性= 5292;这是因为互相关函ਤ= 8;中包含着相位信息。<= /span> 【例3.11】 应用互功率谱从ࡢ= 3;外界干扰的信息中获&= #21462;测试系统频率响应= 989;数。 如图3.33所示= 的测试系统受外界干= 5200;，n<s= pan lang=3DEN-US>1(t)为输入噪声，n<s= pan lang=3DEN-US>2(t)为加在系统中间= 环节的噪声，n<s= pan lang=3DEN-US>3(t)为加在输出端的= 噪声。该系统的输出</s= pan>y(t)为 <![if !vml]><img width=3D184 height=3D21 src=3D"ch033.files/image079.gif" v:shapes=3D"_x0000_i1074"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; (3-59) 式中，x'(t)、<= span lang=3DEN-US>n'1(t)、n'<= span lang=3DEN-US>2(t)分别为系统对x(t)、<= span lang=3DEN-US>n1(t)、n<s= pan lang=3DEN-US>2(t)的响应。 输入x(t)和输出y</= span>(t)的互= 相关函数为 <![if !vml]><img width=3D244 height=3D24 src=3D"ch033.files/image081.gif" v:shapes=3D"_x0000_i1075"><![endif]>=  &= nbsp; &nbs= p; (3-60) 由于输入</= span>x(t)和噪声n1</su= b>(t)、<= span lang=3DEN-US>n2(t)和n<s= pan lang=3DEN-US>3(t)是独立无关的，= 故互相关函数<![if !vml]><img width=3D43 height=3D24 src=3D"ch033.files/image083.gif" v:shapes=3D"_x0000_i1076"><![endif]>= 、<= span lang=3DEN-US><![if !vml]><img width=3D44 height=3D24 src=3D"ch033.files/image085.gif" v:shapes=3D"_x0000_i1077"><![endif]>= 和<= span lang=3DEN-US><![if !vml]><img width=3D43 height=3D24 src=3D"ch033.files/image087.gif" v:shapes=3D"_x0000_i1078"><![endif]>= 均为= 零，所以 = <![if !vml]><img width=3D91 height=3D23 src=3D"ch033.files/image089.gif" v:shapes=3D"_x0000_i1079"><![endif]>=  &= nbsp; &nbs= p; &= nbsp; (3-61) <a name=3D"OLE_LINK1"> </spa= n><![if !vml]><img width=3D184 height=3D23 src=3D"ch033.files/image091.gif" v:shapes=3D"_x0000_i1080"><![endif]>=  &= nbsp; &nbs= p; (3-62)</spa= n></a> 式中，= H(jf)=3D H<sp= an style=3D'mso-bidi-font-style:italic'>1(jf) ·= ; H2(jf)为系统的频率响= 应函数。 <![if !vml]><img width=3D304 height=3D97 src=3D"ch033.files/image093.gif" v:shapes=3D"_x0000_i1081"><![endif]></span= > <h5>图= 3.33 受外&#= 30028;干扰的系统</h5> 可见，利用互相关Ñ= 89;数分析可排除噪声的= 影响，这是互相关函= 5968;分析方法的突出优୅= 7;。然而应当注意到，&= #21033;用式(3-58)求线= 性系统的频率响应函= 5968;H(jf)时，尽管其中的= 互谱可以不受噪声的= 4433;响，但是输入信号௚= 0;自谱仍然无法排除输&= #20837;端测量噪声的影响A= 292;从而形成测量误差。= ; 2) </s= pan>测定系统的滞后= 时间 互谱分析可用来测ê= 50;滞后时间。一个系统= 输入x(t)和输出y</= span>(t)的互= 谱中的幅角<![if !vml]><img width=3D47 height=3D23 src=3D"ch033.files/image095.gif" v:shapes=3D"_x0000_i1082"><![endif]>= 表示= 了系统输出与输入在= 9057;率ƒ处的= 相位差，因此，在任= 9968;频率ƒ上通= 过系统的滞后时间为</s= pan> <![if !vml]><img width=3D69 height=3D43 src=3D"ch033.files/image097.gif" v:shapes=3D"_x0000_i1083"><![endif]>=  3) </s= pan>利用功率谱分析= 对设备进行故障诊断</s= pan> 【例3.12】 图3.34是由= 汽车变速箱上测取的= 5391;动加速度信号经功஍= 5;谱分析处理后所得的&= #21151;率谱图。一般地，= 491;常运行的机器其功率= ;谱是稳定的，而且各&#= 35889;线对应零件不同运ů= 16;状态的振源。在机器= 运行不正常时，例如= 5292;转系的动不平衡、๘= 4;承的局部损伤、齿轮&= #30340;不正常等，都会引$= 215;谱线的变动。图(b)中，= 在9.2Hz和<= span lang=3DEN-US>18.4Hz两处= 出现额外峰谱，这显= 1034;了机器的某些不正६= 0;，而且指示了异常功&= #29575;消耗所在的频率。$= 825;就为寻找与此频率相= ;对应的故障部位提供&#= 20102;依据。 <![if !vml]><img width=3D529 height=3D123 src=3D"ch033.files/image099.gif" v:shapes=3D"_x0000_i1084"><![endif]></span= > = (a) 变速箱正&#= 24120;工作时的谱图 = (b) 变速箱不&#= 27491;常工作时的谱图 <h5>图= 3.34 汽车&#= 21464;速箱的振动功率谱Þ= 70;</h5> <h2 style=3D'text-align:justify;text-justify:inter-ideograph'><o:p> </o:p></h2> </div> </body> </html> ------=_NextPart_01C7AF42.A50F8AB0 Content-Location: 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