Non-Linear Time-Frequency Distributions (Mertins - Signal Analysis (Revised Edition)), страница 4

In order to illustratethis,theWignerVille spectrum will be discussed for various processes in connection with thestandard characterizations.Stationary Processes. For stationary processes the autocorrelation function only depends on r, and the Wigner-Ville spectrum becomes the powerspectral density:Ww,,(t,w)if z ( t ) is stationary.= S,,(w) =(9.128)2939.4.

The Wigner-ViUe SpectrumProcesses with Finite Energy. If we assume that the process z ( t ) hasfinite energy, an average energy density spectrum can be derived from theWigner-Ville spectrum asrcaFor the mean energy we then haveE, = E{lm1SWw z z ( t , w ) dw dt.lz(t)I2 d t } =27r--m(9.131)-mNon-Stationary Processes with Infinite Energy.

For non-stationaryprocesses with infinite energy the power spectral density is not defined.However, a mean power density is given by(9.132)Cyclo-Stationary Processes. For cyclo-stationary processes it is sufficientto integrate over one period T in order to derive the mean power density:(9.133)Example. As a simple example of a cyclo-stationary process, we considerthe signalcWz(t) =d ( i ) g ( t - iT).(9.134)a=-ccHere, g ( t ) is the impulse response of a filter that is excited with statisticallyindependent data d ( i ) , i E Z.The process d ( i ) is assumed to be zero-meanand stationary.

The signal z(t) can be viewed as the complex envelope of areal bandpass signal.Nowweconsider the autocorrelation function of the process z(t). We294Chapter 9. Non-Linear Time-Ekequency Distributionsobtainr,,(t+7,t)=E { x * ( t ) x ( t+ T ) }cccW=WE { d * ( i ) d ( j ) } g*(t - iT) g ( t - j T+T)i = - ~j=-WWg*(t - iT) g ( t - iT= c$+T).2=-W(9.135)As (9.135) shows, the autocorrelation function depends on t and T , and ingeneral the process ~ ( tis)not stationary. Nevertheless, it is cyclo-stationary,because the statistical properties repeatperiodically:rzz(t+T,t)=rzz(t+T+eT,t+eT),r:eEZ.(9.136)Typically, one chooses the filter g ( t ) such that its autocorrelationfunctionsatisfies the first Nyquist condition:(T)=1 for m = 0,otherwise.{0(9.137)Commonly used filters are the so-called raised cosine filters, which aredesigned as follows.

For the energy density S,”,(w) t)r:(t) we takes,”,(w)=11:[Ifor IwTl/n+ c o s [ g ( w ~ / -r (1 - r ) ) ] ]0for 1 - r5 1-r,5 l w ~ ~5/ 1r + r ,for IwTl/n 2 1+r.(9.138)Here, r is known as the roll-off factor, which can be chosen in the region0 5 r 5 1. For r = 0 we get the ideal lowpass. For P > 0 the energy densitydecreases in cosine form.From (9.138) we derive(9.139)As we see, for r > 0, r g ( t ) is a windowed version of the impulse response ofthe ideal lowpass. Because of the equidistantzeros of the si-function, condition(9.137) is satisfied for arbitrary roll-off factors.2959.4. The Wigner-ViUe SpectrumWith(9.140)the required impulse response g ( t ) can be derived from (9.138) by means ofan inverse Fourier transform:(4rtlT) cos(nt(1 r)/T) sin(nt(1- r)/T)(9.141)g(t) =nt [l - ( 4 ~ t / T ) ~ ]++whereg(0)=-Tl (1 +P(-=4 - 1), ,(9.142)Figure 9.9 shows three examples of autocorrelation functions with periodT and the corresponding Wigner-Ville spectra.

We observe that for largeroll-off factors there are considerable fluctuations in power in the course of aperiod. When stating the mean power density in the classical way accordingto (9.133) these effects are not visible (cf. Figure 9-10).As can be seen in Figure 9.9, the fluctuations of power decrease withvanishing roll-off factor. In the limit, the ideal lowpass is approached (T = 0),and the process ~ ( tbecomes)wide-sense stationary.

In orderto show this, theautocorrelation functionT,, (t+.r,t ) is written as theinverse Fourier transformof a convolution of G * ( - w ) and G ( w ) :e'j(W~W').~'jWkTh/$.Wtd W .(9.143)Here the summation is to be performed over the complex exponentials only.Thus, by using(9.144)we achieve(9.145)296Chapter 9. Non-LinearTime-EkequencyDistributionsr= 10‘0Ar = 0.5r= 0.1/l‘T‘T‘0Figure 9.9. Periodic autocorrelation functions andWigner-Ville spectra (raisedcosine filter design with various roll-off factors T ) .Integrating over W yieldsT,,(t + 7,t ) =(9.146)If G ( w ) is bandlimited to x / T , only the term for k = 0 remains, and the2979.4. The Wigner-ViUe SpectrumFigure 9.10. Mean autocorrelation functionsmean power spectral density ( r = 0.5).=so r=,=,(t+T7, t)dtand(9.147)This shows that choosing g ( t ) to be the ideal lowpass with bandwidth 7r/Tyields a Nyquistsystem in which z ( t ) is a wide-sense stationary process.However, if we consider realizable systems we must assume a cyclo-stationaryprocess.Stationarity within a realizable framework can be obtainedby introducinga delay of half a sampling period for the imaginary part of the signal.

Anexample of such a modulation scheme is the well-knownoffset phase shiftkeying. The modified signal readscWd ( t )=I=-R{d(i)} g ( t - iT)+ j9{d(i)} g ( t - iT - T / 2 ) .(9.148)WAssuming that(9.149)298Chapter 9. Non-Linear Time-Ekequency Distributionswe haveTZtZ'c50: c$ c15ui(t + 7, t ) ="g*(t- iT) g(t - iT+7)%=-m+-2loTTg*(t-iT--)g(t-iT+T--)22%=-mloTg*(t- i-) g ( t2%=-moo+ 7- - . T2-)(9.150)for the autocorrelation function. According to (9.146) this can be written asT,','(t + 7,t ) =We see that only the term for k = 0 remains if G(w) is bandlimited to 2lr/T,which is the case for the raised cosine filters.

The autocorrelation functionthen is(9.152)Tzlzr(t7 , t ) = U dZ 1 TEgg(T).THence the autocorrelationfunction T,I,I (t+7-,t ) and themean autocorrelationfunction are identical. Correspondingly, the Wigner-Ville spectrum equals themean power spectral density.+If we regard z'(t) as the complex envelope of a real bandpass processxBP(t),then we cannot conclude from the wide-sense stationarity of z'(t)the stationarity of zBp(t): for this to be true, the autocorrelation functionsT,,,,( t T , t ) and rzrzr( t T , t ) would have to be identical and would haveto be dependent only on 7- (cf.

Section 2.5).++.