Non-Linear Time-Frequency Distributions (Mertins - Signal Analysis (Revised Edition)), страница 2

We consider the signal(9.45)with(9.46)The Wigner distribution W,,(t,W)isW,, (t,u)= 21e-at'e-nW',(9.47)and for WZZ(t,W ) we getwZE((t,W ) = 2 e-"(t1-to)' e-a[W- WO]'.(9.48)Hence the Wignerdistribution of a modulatedGaussiansignal is a twodimensional Gaussian whose center is located at [to,W O ] whereas the ambiguityfunction is a modulated two-dimensional Gaussiansignal whose center islocated at the origin of the 7-v plane (cf.

(9.14), (9.15) and (9.16)).Signals with Positive Wigner Distribution. Only signals of the form(9.49)have a positive Wigner distribution [30]. The Gaussian signal and the chirpare to be regarded as special cases.For the Wigner distribution of z ( t ) according to (9.49) we get(9.50)with W,,(t,W)2 0 V t ,W .276Chapter 9. Non-Linear Time-Ekequency DistributionsIt canberegardedas a two-dimensionalFouriertransformof the crossambiguity function AYz(v,7). As can easily be verified, for arbitrary signalsz ( t ) and y ( t )we haveW,, ( 4 W ) = W:, (t,W ) .(9.54)We now consider a signaland the corresponding Wigner distribution=WZZ(t,W)+ 2 WW,z(t,41 +W&,W).(9.56)We see that the Wigner distribution of the sum of two signals does notequal the sum of their respective Wigner distributions.The occurrence ofcross-terms WVZ(t,u)complicates the interpretation of the Wigner distribution of real-world signals. Size and location of the interference terms arediscussed in the following examples.Moyal'sFormulafor Cross Wigner Distributions.

For the innerproduct of two cross Wigner distributions we have [l81with ( X ,y) = J z ( t )y * ( t ) dt.Example. We consider the sum of two complex exponentials(9.58)For W,, (t,W ) we getW z z ( t ,W )=A: S(W+2A1Az-wI)+ A$ S(W-wZ)C O S ( ( W ~- W 1 ) t )S(W-+(9.59)~ 2 ) )Figure 9.3 shows W,,(t,w) andillustrates the influence of the cross-term2 A 1 A 2 C O S ( ( W ~- W 1 ) t ) S(W - ~ ( w I ~ 2 ) ) .+2779.2. TheWignerDistributionFigure 9.3. Wigner distribution of the sum of two sine waves.Example. In this example the sumconsidered:of two modulated Gaussian signals4 t ) = 4 t ) + Y(t>is(9.60)withz ( t ) = ,jw1(t- tl),-fa(t-(9.61)and(9.62)Figures 9.4 and 9.5 show examples of the Wigner distribution.

We see that theinterference term lies between the two signal terms, and themodulation of theinterference term takes place orthogonal to theline connecting the two signalterms. This is different for the ambiguity function, also shown in Figure 9.5.The center of the signal term is located at the origin, which results from thefact that the ambiguity function is a time-frequency autocorrelation function.The interference terms concentrate around278Chapter 9. Non-LinearTime-EkequencyDistributionstt(b)(a)Figure 9.4. Wigner distribution of the sum of two modulated and time-shiftedGaussians; (a) tl = t z , w1 # wz; (b) tl # t z , w1 = w2.tSignal (real part):ldistributionm Wignerwz----j-*Ambiguity function(c)Figure 9.5.

Wigner distribution and ambiguity function of the sum of two modulated and time-shifted Gaussians (tl # t z , w1 # w z ) .9.2. TheWignerDistribution9.2.4279Linear OperationsMultiplication in the Time Domain. We consider the signalZ ( t ) = z(t) h(t).(9.63)For the Wigner distribution we getLW-The multiplication of $==(t,T) and $ h h ( t , r ) with respect to r can be replacedby a convolution in the frequency domain:WE ( t ,W )=lW,, ( t ,W )271-:W h h ( t ,W)That is, a multiplication in the time domain is equivalent to a convolution ofthe Wigner distributions W,,(t,w) and W h h ( t ,W ) with respect to W .Convolution in the Time Domain. Convolving z ( t ) and h ( t ) , or equivalently, multiplying X ( W )and H ( w ) , leads to a convolution of the Wignerdistributions W,,(t,w) and W h h ( t ,W ) with respect to t .

ForZ(t) = x(t) * h(t)=I(9.66)(9.67)Wzz(t’,~W’h(t)-t’ , ~dt’.)Pseudo-Wigner Distribution. A practical problem one encounters whencalculating the Wigner distribution of an arbitrary signal x ( t ) is that (9.28)canonlybeevaluatedfor a time-limited z(t). Therefore, the concept ofwindowing is introduced. For this, one usually does not apply a single window280Chapter 9. Non-Linear Time-Ekequency Distributionsh(t) to z ( t ) ,as in (9.65), but one centers h ( t ) around the respective time ofanalysis:Mz*(t- 7) x ( t + -) h ( ~e-jwT)dT.722(9.68)Of course, the time-frequency distribution according to (9.68) correspondsonly approximately to theWigner distributionof the original signal.

Thereforeone speaks of a pseudo- Wigner distribution [26].Using the notationMh ( r ) &(t, r) e-jwT d rit is obvious that the pseudo-WignerdistributioncanbeW,, ( t ,W) as(9.69)calculated from1WArW)(t,W)= 2?rWzz(t,W ) * H(w)(9.70)with H ( w ) t)h@).This means that the pseudo-Wigner distribution is asmoothed version of the Wigner distribution.9.3GeneralTime-Frequency DistributionsThe previous section showed that the Wigner distribution is a perfect timefrequency analysis instrument as long as thereis a linear relationship betweeninstantaneous frequency and time.For general signals, the Wigner distributiontakes on negative values as well and cannot be interpreted asa “true” densityfunction. A remedy is the introductionof additional two-dimensional smoothing kernels, which guarantee for instance that the time-frequency distributionis positive for all signals.

Unfortunately, depending on the smoothing kernel,other desired properties mayget lost. To illustrate this,we will consider severalshift-invariant and affine-invariant time-frequency distributions.2819.3. General Time-frequency Distributions9.3.1Shift-InvariantTime-Frequency DistributionsCohen introduced a general class of time-frequency distributions of the form~ 9 1///rr.ej'(u - t , g(v,r ) X* (U - -) ~ ( u-) e-JWTdv du dr.22(9.71)This class of distributions is also known as Cohen's class. Since the kernelg(v,r ) in (9.71) is independent of t and W , all time-frequency distributions ofCohen's class are shift-invariant. That is,TZZ(t,W)=2T+By choosing g ( v , T) all possible shift-invariant time-frequency distributionscan be generated. Depending on the application, one canchoose a kernel thatyields the required properties.If we carry out the integrationover u in (9.71), we getT,,(t,w) = 271ssg ( u , r ) AZZ(v,r)eCjyt eCjWT du d r .(9.73)This means that the time-frequency distributions of Cohen's class are computed as two-dimensional Fourier transforms of two-dimensionally windowedambiguity functions.

From (9.73) we derive the Wignerdistribution forg ( v , r ) = I. For g ( v , r ) = h ( r ) we obtain the pseudo-Wignerdistribution.The productM ( v ,). = g(v,). A&, ).(9.74)is known as the generalized ambiguity function.Multiplying A z z ( v , r )with g(v,r)in (9.73) can also be expressed as theconvolution of W,, (t,W ) with the Fourier transform of the kernel:with1G(t,W ) = 2T//g(v,r ) ,-jute-jWT dv dr.(9.76)282Chapter 9. Non-Linear Time-Ekequency DistributionsThat is, all time-frequency distributions of Cohen’s class can be computedby means of a convolution of the Wigner distribution witha two-dimensionalimpulse response G ( t ,W ) .In general the purpose of the kernel g(v,T ) is to suppress the interferenceterms of the ambiguity function which are located far from the origin of theT-Y plane (see Figure 9.5); this again leads to reduced interference termsin the time-frequency distribution T,,(t,w).

Equation (9.75) shows that thereduction of the interference terms involves “smoothing” and thus results ina reduction of time-frequency resolution.Depending on the type of kernel, some of the desired properties of thetime-frequency distribution are preserved while others get lost. For example,if one wants to preserve the characteristic(9.77)the kernel must satisfy the conditiong(u,O) = 1.(9.78)We realize this by substituting (9.73) into (9.77) and integrating over dw, dr,du. Correspondingly, the kernel must satisfy the conditionin order to preserve the property(9.80)A real distribution, that isis obtained if the kernel satisfies the conditiong(Y,T) = g*(-V, - T ) .(9.82)Finally it shall be noted that although(9.73) gives a straightforward interpretation of Cohen’s class, the implementationof (9.71) is more advantageous.For this, we first integrate over Y in (9.71).

With(9.83)9.3. General Time-frequency Distributions283Convolutionwith r(4,r)40Fouricr transformT,(44Figure 9.6. Generation of a general time-frequency distribution of Cohen’s class.we obtainT,,(t,W)=//7-T(U- t ,r ) z*(u - -) z(u2+ -)72’,-JUTdu dr.(9.84)Figure 9.6 shows the corresponding implementation.9.3.2Examples ofShift-InvariantTime-FrequencyDistributionsSpectrogram. The best known example of a shift-invariant time-frequencydistribution is the spectrogram, described in detail in Chapter7.

An interesting relationship between the spectrogram and the Wigner distribution can beestablished [26].In order to explain this, the short-time Fourier transform isexpressed in the formCCz(t‘) h*(t- t’) ,-jut‘Fz(t,w) =dt’.(9.85)J-CCThen the spectrogram isAlternatively, with the abbreviationX&’)= X@’) h*(t- t’),(9.87)= l X t ( 4 I 2.(9.88)(9.85)can be written asS,,(W)Furthermore, the energy density lXt(w)12 can be computed from the Wignerdistribution W,,,, (t’,W ) according to (9.31):(9.89)284Chapter 9.