Non-Linear Time-Frequency Distributions (Mertins - Signal Analysis (Revised Edition)), страница 3

Non-Linear Time-Ekequency DistributionsObserving (9.35) and (9.65), we finally obtain from (9.89):1Sx(t,w) = 27F //Wxx(t',w') W h h ( t - t',w-W')dt' dw'(9.90)1=2 Wzz(t,w)* *27FWhh(t,W).Thus the spectrogram results from the convolutionof Wzz(t,W ) with theWigner distribution of the impulse response h@).Therefore, the spectrogrambelongs to Cohen's class. The kernel g(v, r ) in (9.73) is the ambiguity functionof the impulse response h(t) (cf.

(9.75)):Although the spectrogram has the properties(9.81) and (9.72), the resolutionin the time-frequency plane is restricted in such a way (uncertainty principle)that (9.77) and (9.80) cannot be satisfied. This becomes immediately obviouswhen we think of the spectrogram of a time-limited signal (see also Figure9.2).Separable Smoothing Kernels. Using separable smoothing kernelsdv,4 Q2 (.l,=G 1((9.92)means that smoothingalongthetimeandfrequencyaxis is carried outseparately.

This becomes obvious in (9.75), which becomes1Tzz(t,W ) = - G(t,W )21r* * wzz(t,W )(9.93)1= - g1(t)21r*[ G z ( w )*wzz(t,W )]whereG ( ~ , w=)g 1 ( t ) G Z ( W ) ,g1(t) WGI(w), G ( w )gz(t).(9.94)From (9.83) and (9.84) we derive the following formula for thetime-frequencydistribution, which can be implemented efficiently:Txx(t1w)=/ [/Z*(U -775)Z(U + 5)g l ( u-1t ) dug2(T),-jwTdT.(9.95)Time-frequency distributions which are generated by means of a convolutionof a Wigner distribution with separable impulse responses canalso be understood as temporally smoothed pseudo-Wigner distributions.

The window9.3. General Time-frequency285DistributionsgZ(T) in (9.95) plays the role ofh ( ~in) (9.68). Temporal smoothingis achievedby filtering with g1 ( t ) .An often usedsmoothing kernel (especially in speech analysis) is theGaussianff,p E R,ff,p > 0.g ( v , T ) = - e-a2u2/4 e - p 2 ~ 2 / 4 ,2(9.96)Thus we derive the distributionr2z*(u - -) z(u+ -)r2du dT.(9.97)For the two-dimensional impulse responseG ( t ,U ) we havewith1g1(t) = ;e$/a2(9.99)and1 e- w 2 / pGz(w)= -P(9.100)It can be shownthat for arbitrary signals a positive distribution is obtainedif [75](9.101) ffp 2 1.For aP = 1, T,,( t , ~is) equivalent to a spectrogramwithGaussian, ~ )more smoothed than a spectrogram.window.

For crp > 1, T i ~ a u s B )is( teven(Gauss)Since T,,( t ,W ) for aP 2 1 can be computedmuch more easily and moreefficiently via a spectrogram, computing with the smoothed pseudo-Wignerdistribution is interesting only for the case(Gauss)(9.102)ffp < 1.The choice of (Y and p is dependent on the signal in question. In order togive a hint, consider a signal z ( t ) consisting of the sum of two modulatedtime-shifted Gaussians.

It is obvious that smoothing should be carried outtowards the direction of the modulationof the cross term (compare Figures9.4and 9.5). Although the modulation may occur in any direction, we look at288Chapter 9. Non-Linear Time-Ekequency DistributionsExamples of Time-Frequency Distributions of Cohen's Class. Inthe literature we find many proposals of shift-invariant time-frequency distributions. A survey is presented in [72] for instance. In the following, threeexamples will briefly be mentioned.Rihaczek distribution.

The Rihaczek distribution is defined as [l241T,(f)(t,w) =Sx * ( t )x ( t+ r ) e-jwTdr(9.103)= x * ( t )X ( W )ejwt.This type of distribution is of enticing simplicity, but it is not real-valued ingeneral.Choi- Williarns Distribution. For the Choi-Williams distribution the following product kernel is used [24]:g(v,= -W(47W,g> 0.(9.104)We see that g(v,0) = 1 and g(0,r ) = 1 are satisfied so that theChoi-Williamsdistribution has the properties (9.77) and (9.80).The quantity g in (9.104) maybeunderstoodasa free parameter.

Ifa small g is chosen, the kernel concentrates around the origin of the r-vplane, except for the r and the v axis. Thus we get a generalized ambiguityfunction M ( v ,r ) = g(v,r ) A,,(v, r ) with reduced interference terms, and thecorresponding time-frequency distribution has reduced interference terms aswell. From (9.71), (9.83), and (9.84) we getT g W )( t ,W ) =7-re-r2a(U - t12/T2~*(u- -) ~ ( u -)+22'e-J'JT du dr.(9.105)Zhao-Atlas-Marks Distribution. Zhao, Atlas and Marks [l681 suggestedthe kernel(9.106)This yields the distribution9.3. General Time-frequency Distributions9.3.3289Affine-InvariantTime-Frequency DistributionsAffine smoothing is an alternative to regular smoothing of the Wigner distribution (Cohen’s class). A time-frequency distribution that belongs to theaffine class is invariant with respect to time shift and scaling:Any time-frequency distribution that satisfies (9.108) can be computed fromthe Wigner distribution by means of an affine transform [54], [126]:‘ s sK(W(t‘T,,(t,W) = 2n- ~ ) , w ’ / w )Wzz(t‘,W’)dt’ h‘.(9.109)This can be understood as correlating the Wigner distribution withkernel Kalong the time axis.

By varying W the kernel is scaled.Since (9.108) and (9.72) do not exclude eachother, there exist other timefrequency distributions besides the Wigner distribution which belong to theshift-invariant Cohen class as well as to theaffine class. These are, for instance,all time-frequency distributions that originate from a product kernel, such asthe Choi-Williams distribution.Scalogram. An example of the affine class is the scalogram, that is, thesquared magnitude of the wavelet transform of a signal:2(9.110)with(9.111)(9.112)where(9.113)Thus, from (9.112) we deriveI W , ( ~ , U=) L2n~ /~/ W $ , $(-,a.‘)t’ - bWzz(t‘,w‘) dt’ h’.(9.114)290Chapter 9. Non-Linear Time-Ekequency DistributionsThe substitutions b = t and a = wo/w finally yield(9.115)=(%(P-t),-1I W $ , $:U’)2nW , , ( ~ ’ , W dt’’ ) h’.The resolution of the scalogram is,just like that of the spectrogram,limitedby the uncertainty principle.9.3.4Discrete-Time Calculation of Time-FrequencyDistributionsIf we wish to calculate the Wigner distribution or some other time-frequencydistribution on a computer we are forced to sample our signal and the transform kernel and to replace all integrals by sums.

If the signal and thekernel arebandlimited and if the sampling rate is far above the Nyquist rate for bothsignal and kernel, we do not face a substantial problem. However, in somecases, such as the Choi-Williams distribution, sampling the kernel alreadyposes a problem. On the other hand, the testsignal may be discrete-timeright away, so that discrete-time definitions of time-frequency distributionsare required in any case.Discrete-Time Wigner Distribution [26].

The discrete-timeWignerdistribution is defined as(9.116)mHere, equation (9.116) is the discrete version of equation (9.28), which, usingthe substitution r’ = 3-12, can be written asWX* (t - r‘) z ( t+ r’) e-i2wr‘dr’.(9.117)As we know, discrete-time signals have a periodic spectrum, so that onecould expect the Wignerdistribution of a discrete-time signal to have aperiodic spectrum also. We have the following property: while the signal z(n)has a spectrum X ( e j w ) t)z(n) with period 2n, the period of the discretetime Wigner distribution is only n.

Thus,W,, (n,e j w > = W,, (n,ejw +),kEZ.(9.118)9.3. General Time-frequency Distributions291The reason for this is subsampling by the factor two with respect to T . Inorder to avoid aliasing effects in the Wigner distribution, one hasto take carethat the bandlimited signal z ( t ) is sampled with the ratefa24(9.119)fmazand not with f a 2 2 f m a z , whereX ( w ) = 0 for IwI > 27~f m a z .(9.120)Because of the different periodicity of X ( e j w ) and W z z ( n , e j w )it is notpossible to transfer all properties of the continuous-time Wigner distributionto the discrete-time Wigner distribution. A detailed discussion of the topiccan be found in [26], Part 11.General Discrete-Time Time-Frequency Distributions.

Analogousto (9.84) and (9.116), a general discrete-time time-frequency distribution ofCohen’s class is defined asccMTzz(n,k) = 2Np ( [ , m) 2 * ( C+n-m) 2 ( C+ n + m) e-j47rkmlL.m=-Me=-N(9.121)Here we have already taken into account that in practical applications onewould only consider discrete frequencies 2 ~ k / L where,L is the DFT length.Basically we could imagine the term p(C,m) in (9.121) to be a 2M + 1 X2 N + 1 matrix which contains sample values of the function ~ ( u7),in (9.84).However, for kernels that are not bandlimited, sampling causes a problem.For example, for the discrete-time Choi-Williams distribution we thereforeuse the matrixwithNe-uk2/4m2, n = -N,.

. . ,N ,m = -M,... , M .(9.123)h=-NThe normalization C , p(n, m ) = 1 in (9.122) is necessary in order to preservethe properties [l11C T , ( F W ) ( n , k=) IX(k)12 = IX($W”)12n(9.124)292Chapter 9. Non-Linear Time-Ekequency Distributionsand9.4The Wigner-Ville SpectrumSo far the signals analyzed have been regarded as deterministic. Contrary tothe previous considerations, z ( t )is henceforth defined as a stochastic process.We may view the deterministicanalyses considered so far as referring to singlesample functions of a stochastic process. In order to gain information onthe stochastic process we define the so-called Wigner-Vdle spectrum as theexpected value of the Wigner distribution:with~ z z ( t r+ - , t r- - ) = E { 4 z z ( t , r ) } = E { ~r* ( t - ~ )r~ ( t + - (9.127))}22This means that the temporal correlation function(t,r ) is replaced by itsexpected value, which is the autocorrelation function r z z(t 5, t - 5) of theprocess z ( t ) .+The properties of the Wigner-Ville spectrum are basically the same asthose of the Wignerdistribution.Butby forming the expected value itgenerally contains fewer negative values than the Wigner distribution of asingle sample function.The Wigner-Ville spectrum is of special interest when analyzing nonstationary or cyclo-stationary processes because here the usual terms, suchas power spectral density, donot give anyinformationonthetemporaldistribution of power or energy.