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

Signal Analysis: Wavelets, Filter Banks, Time-Frequency Transforms andApplications. Alfred MertinsCopyright 0 1999 John Wiley & Sons LtdPrint ISBN 0-471-98626-7 Electronic ISBN 0-470-84183-4Chapter 9Non-Linear TirneFrequency DistributionsIn Chapters 7 and 8 twotime-frequencydistributions were discussed: thespectrogram and the scalogram. Both distributions are theresult of linearfiltering and subsequent forming of the squared magnitude. In this chaptertime-frequency distributions derived in a different manner will be considered.Contrary to spectrograms and scalograms, their resolution is not restrictedby the uncertainty principle. Although these methods do notyield positivedistributions in all cases, they allow extremelygood insight into signalproperties within certain applications.9.1The AmbiguityFunctionThe goal of the following considerations is to describe the relationship betweensignals and their time aswell as frequency-shifted versions.

We start by lookingat time and frequency shifts separately.Time-Shifted Signals. The distanced ( z , 2,) between an energy signal z ( t )and its time-shifted version z,(t) = z(t T ) is related to the autocorrelationfunction T ~ ~ ( THere) .the following holds (cf. (1.38)):+d(&,Zl.=211412 -2265w%T)},(9-1)266Chapter 9.

Non-Linear Time-Ekequency DistributionswhereLCCTE~,(T)=(zT,z)=z*(t)z(t + .r)dt.(9-2)As explained in Section 1.2, T ; ~ ( T ) can also be understood as the inverseFourier transform of the energy density spectrum S,",(w) = IX(w)I2:In applications in which the signal z(t) is transmitted and the timeshift Tis to be estimated from the received signal z(t T ) , it is important that z ( t )and z(t T) are as dissimilar as possible for T # 0.

That is, the transmittedsignal z(t) should have an autocorrelation functionthat is as Dirac-shaped aspossible. In the frequency domain this meansthat theenergy density spectrumshould be as constant as possible.++Frequency-Shifted Signals. Frequency-shifted versions of a signal z ( t )are often produced due to the Doppler effect. If one wants to estimate suchfrequency shifts in orderto determine the velocity of a moving object, thedistance between a signal z ( t ) and its frequency-shifted version z v ( t )= z ( t ) e j u tis of crucial importance.

The distance is given byd ( z , z v ) = 2 llz112 - 2x{(~,z>}.(9.4)For the inner product (zv,z)in (9.4) we will henceforth use the abbreviation(v). We have,ofzz*(t)z(t) ejutdt=(9-5)J-WWsEz(t) ejVtdt=withsEz(t) = lz(t)I2,J-CCwhere sEz((t) can be viewed as the temporalenergy density.' Comparing (9.5)with (9.3) shows a certain resemblance of the formulae for .Fz(.) and pEz(v),'In (9.5) we have an inverse Fourier transform in which the usual prefactor 1/27r doesnot occur because we integrate over t , not over W .

This peculiarity could be avoided if Ywas replaced by -v and (9.5) was interpreted as a forward Fourier transform.However, thiswould lead to other inconveniences in the remainder of this chapter.9.1. The Ambiguity Function267however, with the time frequency domains being exchanged.

This becomeseven more obvious if pF,(u) is stated in the frequency domain:We see that pF,(u) can be seen as the autocorrelationfunction of X ( w ) .Time and Frequency-Shifted Signals. Let us consider the signalswhich are time andfrequency shiftedversions of one another, centered aroundz ( t ) .With the abbreviationfor the so-called time-frequency autocorrelation function or ambiguity function’ we getThus, the real part of A z z ( v r), is related to thedistance between both signals.In non-abbreviated form (9.8) isWAzz(u,T) = S_,z*(t - 7 ) z(t2+I)Gut dt.2Via Parseval’s relationwe obtain anexpression for computing A,,frequency domainU2X ( w - -) X * ( w2+ -)UejwT dw.(9.10)(U,r) inthe(9.11)‘We find different definitions of this term in the literature.

Some authors also use it forthe term IAZz(u,~)1’[150].268Chapter 9. Non-Linear Time-Ekequency DistributionsExample. We consider the Gaussian signal(9.12)which satisfies 1 1 2 11 = 1. Using the correspondencewe obtainA,, (v,T) = e-- ;?e-&u2(9.14)Thus, the ambiguity function is a two-dimensional Gaussian function whosecenter is located at the origin of the r-v plane.Properties of the Ambiguity Function.1. A time shift of the input signal leads to a modulation of the ambiguityfunction with respect to the frequency shift v:This relation can easily be derived from (9.11) by exploiting the factthat x ( w ) = e-jwtoX(w).2.

A modulation of the input signal leadsto a modulation of the ambiguityfunction with respect to I-:z(t)= eJwotz(t) +AEZ(V,T)= d W o T A,,(~,T).(9.16)This is directly derived from (9.10).3. The ambiguity function has its maximum at the origin,where E, is the signal energy.

A modulation and/or time shift of thesignal z ( t ) leads to a modulation of the ambiguity function, but theprincipal position in the r-v plane is not affected.Radar Uncertainty Principle. The classical problem in radar is to findsignals z(t) that allow estimation of timeandfrequencyshiftswithhighprecision. Therefore, when designing an appropriate signalz(t)the expressiongner9.2. The269is considered, which contains information onthe possible resolution of a givenz ( t ) in the r-v plane.

The ideal of having an impulse located at the origin ofthe r-v plane cannot be realized since we have [l501mIAzz(v,r)I2d r dv = IA,,(O,0)I2 = E:.(9.18)That is, if we achieve that IA,,(v, .)I2 takes on theform of an impulse at theorigin, it necessarily has to grow in other regions of the r-v plane because ofthe limited maximal value IA,,(O, 0)12 = E:. For this reason, (9.18) is alsoreferred to as the radar uncertainty principle.Cross Ambiguity Function. Finally we want to remark that, analogousto the cross correlation, so-called cross ambiguity functions are defined:1,WA?/Z(V, 7)=z(t + f ) y * ( t - f ) ejyt dt(9.19)X ( W - );9.29.2.1+Y * ( w );eJw7 dw.The Wigner DistributionDefinitionand PropertiesThe Wigner distributionis a tool for time-frequency analysis, which has gainedmore and more importance owing to many extraordinary characteristics.

Inorder to highlight the motivation for the definition of the Wigner distribution,we first look at the ambiguity function. From A,, (v,r ) we obtain for v = 0the temporal autocorrelation functionfrom which we derive the energy density spectrum by means of the Fouriertransform:(9.21)W-l m A , , ( O , r ) e-iwT d r .270Chapter 9. Non-Linear Time-Ekequency DistributionsOn the other hand,we get the autocorrelation functionpFz (v) of the spectrumX ( w ) from A z z ( v , 7 )for 7 = 0:The temporal energy density &(-L) is the Fourier transform of &(v):(9.23)These relationships suggest defining a two-dimensional time-frequency distribution W z z ( t ,W ) as the two-dimensional Fourier transform of A,,(v, 7):W22( t ,W ) = 2n-jvt-m-mAzz(v,r)eThe time-frequency distribution W,,(t,ti~n.~W)e-jwr dUdr.(9.24)is known as the Wigner distribu-The two-dimensional Fourier transform in (9.24)can also be viewed asperforming two subsequent one-dimensional Fourier transforms with respectto r and v.

The transformwith respect to v yields the temporal autocorrelationfunction4(9.25)(9.26)=X ( w - g) X * ( w+ 5).3Wigner used W z z ( t , w ) for describing phenomena of quantum mechanics [163], Villeintroduced it for signal analysis later [156], so that one also speaks of the Wigner-Villedistribution.41f z ( t )was assumed to be a randomprocess, E { C J ~ ~ ~ (would~ , T )be} the autocorrelationfunction of the process.9.2. TheWignerDistribution271IWignerdistributionITemporal autocorrelationTemporal autocorrelationFigure 9.1.

Relationship between ambiguity function and Wigner distribution.The function @,,(U, W) is so to say the temporal autocorrelation function ofX(W).Altogether we obtain(9.27)with & , ( t , ~according)to (9.25) and @,,(Y,w) according to (9.26), in full:Figure 9.1 pictures the relationships mentioned above.We speak of W,, (t,W) as a distribution because it is supposed to reflectthe distribution of the signal energy in the time-frequency plane. However,the Wigner distribution cannot be interpreted pointwise as a distribution ofenergy because it can also take on negative values. Apart from this restrictionit has all the properties one would wish of a time-frequency distribution. Themost important of these properties will be briefly listed.

Since the proofs canbe directly inferred from equation (9.28) by exploiting the characteristics ofthe Fourier transform, they are omitted.272Chapter 9. Non-Linear Time-Ekequency DistributionsSome Properties of the Wigner Distribution:1. The Wigner distribution of an arbitrary signal z(t) is always real,(9.29)2. By integrating over W we obtain the temporal energy densitym=L/W,,(t,w)2n -ms,,(t)Edw = lz(t)I2.(9.30)3. By integrating over t we obtain the energy density spectrummW,,(t,w) d t = I X ( W ) ~ ~ .S,”,(w) =(9.31)J -m4. Integrating over time and frequency yields the signal energy:WWW,,(t,w) dw d t =(9.32)5 . If a signal z ( t ) is non-zero in only a certain time interval, then theWigner distribution is also restricted to this time interval:z ( t ) = 0 for t < tl and/or t > t 2UW,,(t,w)(9.33)t < tl and/or t > t z .= 0 forThis property immediately follows from (9.28).6. If X ( w ) is non-zero only in a certain frequency region, then the Wignerdistribution is also restricted to this frequency region:X ( w ) = 0 forW< w1 and/orW> w2UW,,(t,w)=(9.34)0 forW< w1 and/orW> w2.gner9.2.

The2737. A time shift of the signal leads to a time shift of the Wigner distribution(cf. (9.25) and (9.27)):Z ( t ) = z(t - t o )*WEE(t,W ) = W,,(t- to,W ) .(9.35)8. A modulation of the signal leads to a frequency shift of the Wignerdistribution (cf. (9.26) and (9.27)):Z ( t ) = z(t)ejwot+W g g ( t , w ) = W z z ( t , w- W O ) .(9.36)9. A simultaneous time shift and modulation lead to a time and frequencyshift of the Wigner distribution:~ ( t=)z(t - to)ejwOt~ E s ( t , w=) ~ , , ( t- t o , W - W O ) .

(9.37)10. Time scaling leads toSignal Reconstruction. By an inverse Fourier transform of W zz( t,W ) withrespect to W we obtain the function7-+zz(t,7-)=X*(t--127-4 t + 5)'(9.39)cf. (9.27). Along the line t = 7-/2 we get7-2(.) = +zz(5,7-) = X*(O) X(.).(9.40)This means that any z(t) can be perfectly reconstructed from its Wignerdistribution except for the prefactor z*(O).Similarly, we obtain for the spectrumUX * ( u ) = Qzz(-,U) = X ( 0 ) X * @ ) .2(9.41)Moyal's Formula for Auto-Wigner Distributions.

The squared magnitude of the inner product of two signals z ( t ) and y(t) is given by the innerproduct of their Wigner distributions [107], [H]:2749.2.2Chapter 9. Non-Linear Time-Ekequency DistributionsExamplesSignals with Linear Time-Frequency Dependency. The prime examplefor demonstrating the excellent properties of the Wigner distribution in timefrequency analysis is the so-called chirp signal, a frequency modulated (FM)signal whose instantaneous frequency linearlychanges with time:x ( t )= A(9.43),j+Dt2 ejwOt.We obtainW,,(t,w) = 2~ [AI2S(W - WO - pt).(9.44)This means that the Wigner distribution of a linearly modulated FM signalshows the exact instantaneous frequency.Gaussian Signal.