【文章內(nèi)容簡(jiǎn)介】 The distribution of power density along frequency axis .功率密度沿頻率軸的分布 Auto power spectrum density function is an even funtion,its frequency range is (∞， +∞） .It is also called two sided auto power spectrum density funtion. Function value over frequency range (∞， 0） is a symmetric 對(duì)稱(chēng) mapping映射 of those over frequency range (0， +∞） .Hence:自功率譜密度函數(shù)是偶函數(shù) ,它的頻率范圍是 (∞，+∞）又稱(chēng)雙邊自功率譜密度函數(shù)。它在頻率范圍 (∞， 0）的函數(shù)值是其在 (0， +∞）頻率范圍函數(shù)值的對(duì)稱(chēng)映射，因此， )(2)( fSfG xx ?One sided spectrum and two sided spectrum單邊譜和雙邊譜 Signal Analysis in Frequency Domain信號(hào)的頻域分析 0 f )(),( fGfS xx )( fGx )( fSx? Applications of power spectrum 功率譜的應(yīng)用 1） Relations among Sx(f),|X(f)| and H(f).自 功率譜密度 Sx(f)與幅值譜 |X(f)|及系統(tǒng)頻率響應(yīng)函數(shù) H(f)的關(guān)系 )()()(fXfYfH ?Signal Analysis in Frequency Domain信號(hào)的頻域分析 )()(fHth)()(fYty)()(fXtx)()()()()()()()()(fGfGfSfSfXfXfXfYfHxxxyxxxy ??????)().()().()()()()()()()()()(.).()()()(*239。239。239。)(239。39。22239。39。fXfYfXfYfd t YetxdtetydtetxttdetydttxtdetydttxdedttytxdeRfSftjftjftjttfjfjfjfjxyxy???????????????? ?? ?? ?? ?????????????????????????????????????????????????????????????????t令)()(2s i n)(2cos)()(2s i n)()(2cos)()()(2s i n)(2cos)()()(*)(22fXfXf t d ttxjf t d ttxt d tftxjt d tftxdtetxfXf t d ttxjf t d ttxdtetxfXtfjftj??????????????? ?? ??? ?????????????????????????????????????????????)()()()()()()()()(fGfGfSfSfXfXfXfYfHxxxyxxxy ?????? )()()( fjXfXfXIR ?? )()()( fjXfXfX IR ???If 若 222 )()()()()( fXfXfXfXfXIR ???? 2* )()()()()()()()()( fHfSfSfXfYfXfYfHfHxy ??????)()()(fSfSfHxy?Signal Analysis in Frequency Domain信號(hào)的頻域分析 )(|)(|)( 2 fSfHfS xy ? )(|)(|)( 2 fGfHfGxy ??Relations among input auto power spectrum density function, output auto power spectrum density function and system frequency response function .輸入 /輸出自功率譜密度函數(shù)與系統(tǒng)頻率響應(yīng)函數(shù)關(guān)系 Amplitude frequency characteritics of system can be obtained by analysis of input and output auto power spectrum density function. But phase information has been lost and phase frequency characteristic of system cann’t be obtained by this spetral analysis 、輸出自譜的分析，就能得出系統(tǒng)的幅頻特性。但這樣的譜分析丟失了相位信息，不能得出系統(tǒng)的相頻特性。 ?Ideal linear system with single input and single output 單輸入、單輸出的理想線性系統(tǒng) )()()( fSfHfS xxy ?Signal Analysis in Frequency Domain信號(hào)的頻域分析 2） noise influence can be eliminated by crossing )()()()()( 321 39。 tntntntxty ?????? )()()()()(321 ????? xnnxnxxxxy RRRRR ???? ???Because input and noise are independent .so:由于輸入和噪聲是獨(dú)立無(wú)關(guān)的 )()( ?? xxxy RR ?? )()()()( fSfHfSfS xxxxy ?? ?Signal Analysis in Frequency Domain信號(hào)的頻域分析 )()(11fHth)()(fYty)()(fXtx+ )()(22fHth+ + + + + + + + )()(11 fN tn)((22 fN tn )( )(33 fN tn3） Applications of power spectrum in equipments diagnosis功率譜在設(shè)備診斷中的應(yīng)用 Power spectrum figure of acceleration signal at gearbox of a 車(chē)變速箱上加速度信號(hào)的功率譜圖 Normal 正常 Abnormal 異常 故障頻率 Signal Analysis in Frequency Domain信號(hào)的頻域分析 4） water fall figure 瀑布圖 Signal Analysis in Frequency Domain信號(hào)的頻域分析 Frequency spectrum figure at different rotating speed不同轉(zhuǎn)速下的頻譜圖 5） Canbel figure 坎貝爾（ Canbel）圖 Signal Analysis in Frequency Domain信號(hào)的頻域分析 Auto spectrum of oscillation at that rotating speed 該轉(zhuǎn)速點(diǎn)上振動(dòng)的自譜。 coherence function γxy2(f) 相干函數(shù) )1)(0()()(|)(|)( 222 ??? ffSfSfSf xyyxxyxy ???γxy2(f)=0,it isn’t coherent between output and －輸出信號(hào)與輸入信號(hào)不相干 ?γxy2(f)=1,it is entirely coherent between output and 1－輸出與輸入信號(hào)完全相干 ?γxy2(f)=0~1,there are three possibilities:(1)there is disturbance of outside noise while testing。(2)output is a synthetic output of input and other inputs.(3) The system is 0~1之間－有如下三種可能： (1）測(cè)試中有外界噪聲干擾； (2）輸出是輸入和其它輸入的綜合輸出； (3）系統(tǒng)是非線性的。 1)()( )()()()( )()()()( )()()()( |)(|)(22222 ?????fSfSfSfSfSfSfSfHfSfSfSfHfSfSfSfyxxyyxxyxxyxxyxy??For a linear system:對(duì)于線性系統(tǒng) Signal Analysis in Frequency Domain信號(hào)的頻域分析 Sx(f) is Real number 實(shí)數(shù) Coherence analysis between pressure pulsation and oil tube oscillation 油壓脈動(dòng)與油管振動(dòng)的相干分析 auto spectrum of signal x(t) b)auto spectrum of signal y(t) c) coherence function a)信號(hào) x(t)的自譜 b)信號(hào) y(t)的自譜 c)相干函數(shù) Base frequenccy of pressure pulsation of oil tube 壓油管壓力脈動(dòng)的基頻 Rotating speed of lube pump : n=781rpm, number of gear teeth of pump gear : z=14 . 潤(rùn)滑油泵轉(zhuǎn)速為 n=781rpm,油泵齒輪的齒數(shù)為 z=14 Signal Analysis in Frequency Domain信號(hào)的頻域分析 Cepstrum analysis 倒譜分析 ? Cepreum analysis is also called the second frequency spectrum analysis. 倒頻譜分析亦稱(chēng)為二次頻譜分析 ? Detect periodic structure of frequency spectrum of plex signals ,separate and draw periodic ponents among signals in dense and universal frequency spectrum檢測(cè)復(fù)雜信號(hào)頻譜上的周期結(jié)構(gòu)，分離和提取在密集泛頻譜信號(hào)中周期成分 Cepstrum function倒頻譜函數(shù) ))(( ceps tr umpo werqCpC q C q F S fp x0() () |{log ( )}|? ??? ??? ?fSFqC yy log1???? ? ?? ?R F S fy? ? ?1Comparison:比較 Signal Analysis in Frequency Domain信號(hào)的頻域分析 It may be defined as :也可定義為 : Applications of cepstrum倒頻譜的應(yīng)用 1） separate the influence of routeway to In state inspection and fault diagnosis of the machine ,signals measured usually are the response of fault sources which have passed through system routeway. To get the source signal, the influence of routeway to signal must be eliminated. 在機(jī)械狀態(tài)監(jiān)測(cè)和故障診斷中，所測(cè)得的信號(hào)，往往是由故障源經(jīng)系統(tǒng)路徑的傳輸而得到的響應(yīng)，如欲得到該源信號(hào)，必須消除傳遞通道的影響。 2|)(|)()( fHfSfS xy ??Signal Analysis in Frequency Domain信號(hào)的頻域分析 )()(fHth)()(fYty)()(fXtx}|)(|{log)]([log)}({log 2fHFfSFfSF xy ???? ??