Joint Time-frequency Analysis

8/3/2019 Joint Time-frequency Analysis

1/16


2/16

t has been well understood that a given signal can berepresentedin an infinite number of different ways.Different signal representations can be used for dif-ferent applications. For example, signals obtainedfrom most engineering applications are usuallyfimctions of time. But when studying or designing the sys--

tem, we often llke to study signals and systems in the fre-quency domain. Ths is because many important features ofthe signal or system are more easily characterized in the fre-quency domain than in the time domain.

Although the number ofways of describing a given sig-nal are countless, the most important and fundamentalvariables in nature are time andpeguency.While the timedomain function indicates how a signals amplitudechanges over time, the frequency domain function tellshow often such changes take place. The bridge betweentime and frequency is the Fourier wansfom.

The fundamental idea behind Fouriers original workwas to decompose a signal as the sum ofweighted sinusoi-dal functions. Despite their simple interpretation of purefrequencies, the Fourier transform is not always the besttool to analyze real life signals, which are usually of fi-nite, perhaps even relatively short duration. Common ex-amples include biomedical signals [381, [501, [511, [531,[62] and seismic signals (Fig. 1).The seismic signals arenot like the sinusoidal functions, extending from negativeinfinity to positive infinity in time. For such applications,the sinusoidal functions are not good models.

In addition to linear transformations, the square of theFourier transform (power spectrum) is also widely used inmany applications. However, the power spectrum in gen-eral is only suitable for those signals whose spectra do notchange with time.

One common example is the speech signal. The bottomplot of Fig. 2 is the time waveform of the word hoodspoken by a five-year-oldboy. The plot on the right is thestandard power spectrum, which reveals three frequencytones. From the spectrum alone, however, we cannot tellhow those frequencies evolve over time. The larger plot inthe upper left is the time-dependent spectrum, as a func-tion of both time and frequency, which clearlyindicates he pattern of the formants. From thetime-dependent spectrum, not only can we seehow the frequencies change, but we also cansee the intensity of the frequencies as shown bythe relative brightness levels of the plot. Inother words, by analyzing the signal in timeand frequency ointly, we are able to better un-derstand the mechanism of human speech.Although the frequency content of the ma-jority of signals in the real world evolves overtime, the classical power spectrum does not re-veal such important information. In order toovercome this problem, many alternatives,such as the Gabor expansion, wavelets 1211,

classical time and frequency analysis, we name these newtechniques as joint time-peguency anabsis.

In this article, we introduce the basic concepts andwell-tested algorithms for joint time-frequency analysis.Analogous to the classical Fourier analysis, we roughlypartition this article into two parts: the linear (e.g.,short-time Fourier transform, Gabor expansion) and thequadratic transforms (e.g., Wigner-Ville distribution).Finally, we briefly introduce the so-called model-based (orparametric) time-frequency analysis method.

Expansion and Inner ProductIn what follows, we will first review the fundamentalmathematical tools to perform the joint time-frequencyanalysis, that is, signal expansion and inner product. As weall understand, for a given signal s from the domain Y,where Y can be of finite dimension or infinite dimension,we may write a signals in terms of a linear combination ofthe set of elementary functions {tp .>E Z ,where2 enotesthe set of integers for the Y domain, i.e.,

ns Z

1. Seismic signal.

r 1 ~ n n - .

0.000 0.100 0.200 0.3000.400 0.500 0.600 0.700 0.819Seconds2. Hood (data courtesy of Y. Zhao, the Beckman institute at the University of

1 1 . 1 , 1 . 1 T .... , .opea ana wiaeiy sruaiea. i n contrast to me iiiinois).MARCH 1999 IEEE SIGNAL PROCESSING M AGAZIN E 53


3/16

If a physician uses a ruler whsmallest scale is the decimeterto measure a patients height,then there is no way to tellthe patients height in termsof centimeters.which is known as an expansion or series.

When the set of {U/,} n s Z forms a frame [21] , a familyof vectors that characterizes any signal from its innerproducts (or scalar product) as in (2) and (3) , there willbe a dual set {$.}sZ such that the expansion coefficientscan be computed by

for continuous-time signals, or

(3)for dscrete-time signals. * in (2) and (3) denote thecomplex conjugate operation. In electrical engineering,formulae (2) and (3) are known as transformations and$,,( t )s called the analysis function. Accordingly, (1) sknown as an inverse transform and ~ / , ( t )s called the syn-thesis function.

The inner product has an explicit physical interpreta-tion, which reflects the similarity between the signals(t)and the dual function$ (t). n other words, the larger theinner product a,, th e closer the signal s(t) is to the dualfunction$ , t ) .a.>epicts the signals behavior in the Ydomain.The operation of the inner product in (2)and (3 ) maybe thought of as using a ruler, constituted by a set offunctions {$.}, o measure the signal under investiga-tion. Each individual function can be considered as thetick mark of the ruler. The expansion coefficient {a,,} n-dicates the weight of the signals projection on the tickmark determined by @,We all know tha t the precisionof any measurement largely depends on the smallest unitused. If a physician uses a ruler whose smallest scale i:ithe decimeter to measure a patients height, then there i!ino way to tell the patients height in terms ofcentimeters.The goodness of the ruler is measured by the fineness ofthe unit. So, elementary functions for the signal expan-sion should be chosen such that the resulting tick mark isthe finest.

For a given frame { u / , ~ } ~ ~ ~ ,ts corresponding dualframe is generally not unique. One particularly interest-ing scenario is$ ( t ) = c ~ /. t ) , here c denotes a constantnumber. In this case, we name {U/ ns Z as a tight frame.For the tight frame, the expansion coefficients c{a,} areexactly the signals projection of the synthesis functions,

up to a constant multiplier c . When the dual frame is notunique, the resulting expansion is redundant.

When the dual frame {$ } nt z is unique, then the setof {U/,,} tZ forms a basis and there is no redundancy. Inthis case,

where1 n=O0 otherwise6[n]= ( 5 )

I f $ , = ~ ~ / , , t h ~ n ( ~ , , $ , ) = c 8 [ n - - n ]nd{v.},,,z areorthogonal. Otherwise, {U/ } ns % and {$ } nt Z arebiorthogonal, when { u / ~ } ~ ~ ~orms a basis. For thebiorthogonal expansion, the signals feature described bythe coefficients {a,} may not have obvious relations withthe set of the basis function {U/ n } ns Z [Fig. 3(a)] . For theorthogonal expansion, however, the expansion coeffi-cients {a,} must be the signals projection on the basisfunction {U/ nt Z [Fig. 3(b)]. Moreover, in all cases, thepositions of U/ and u rn are interchangeable. That is, ei-ther of them could be used for analysis or synthesis.

The concepts introduced above can also be understoodfrom the matrix operation point ofview. Suppose that thelength of the signal 5 isL. We have the matrix G with di-mensionM x L and the matrix H with dimensionL xM ,where M 2 L. IfH,G,S=S (6)for all vectors S with L elements, then we say that thespace constituted by the matrix H i s complete. Note, thatin general, the matrix G could not be unique. If M = L (Hand G are square matrices), then G must be equal toIT.In this case, H and G are said to be biorthogonal to eachother. If G =H-= HH,Hand G are orthogonal. The su-perscript H enotes the Hermitian conjugate. In bothbiorthogonal and orthogonal cases, the matrix G isunique.

BecauseHG = I (6), aking the transpose, with respectto HG, yields,

/ y3ey2(a) Biorthogonal Expansion (b) Orthogonal Expansion3. In both cases,{w,),,, is complete for three-dimensionalspace, but in orthogonal expansion, expansion coefficients a,are exactly the signals projection on th e ele men tary functions I+

54 IEEE SIGNAL PROCESSING MAG AZIN E MARCH 1999


4/16

The elementary functionsemployed in the Fourier seriesextend from negative infinitetime to positive infinite time,and are not associated with aparticular instant in time.H,G,~=(G,)~ ( H , ) ~ s=s (7)which indicates that eitherH or G can be used as analysis(or synthesis). The twomost fundamental questions forthe expansion areA Ho w to build a meaningful set of functions {U/ - } sZ

Ho w to compute the corresponding set of dual func-One of the most well-known expansions is the Fourier se-ries. For a periodic signalT ( t ) ith a period T, he Fou-rier series is defined as

tions { C x } s e z .

Because the elementary functions e j2m r / T in (S), har-monically related complex sinusoidal functions, areorthonormal with respect to the scalar product, the dualfunction andthe elementary function have the same form.Hence, we can readily obtain the expansion coefficients ofthe Fourier series in (8 ) by the regular inner product op-eration ( 2 ) ,e.g.,

( 9 )which indicates the similarity between the signal and a setof harmonically related complex sinusoidal functions{eiznnr'*}. Because eizNlt'Tcorrespon&o impulses in thefrequency domain, the ruler used for the Fourier trans-form possesses the finest frequency ick marks (that s, thefinest frequency resolution). Consequently, the measure-ments, the Fourier coefficients { f i n } , precisely describethe signal's behavior at the frequency 2nn/T.

For a non-periodic signals ( t ) in L2,we have1s( t =-I- S(o)eitod o27c -- (10)

and the Fourier transform

Conventionally,the square of Fourier transform is namedthe power spectrum. According to the Wiener-Khinchintheorem, the power spectrum can also be considered asthe Fourier transform of the correlation function, i.e.,

where the correlation function R ( T )s the average of theinstantaneous correlation s ( t ) s ( t -T), i.e.,R(T)=jl ( t ) s * t T)dt =J- s ( t +2/2)s* ( t 2 l 2 ) d t . ( 1 3 )Note that the averaging process removes all time infor-mation from R(2) .

Finally, the reader should bear in mind that a func-tion's time and frequency properties are not independent.They are linked by the Fourier transform. For example,we can't find a function that has an arbitrarily short timeduration and narrow frequency bandwidth at the sametime. If a function has a short time duration, its frequencybandwidth must be wide, or vice versa. From the uncer-tainty principle point of view, it is the Gaussian-type sig-nal, such as

that achieves the optimal joint time-frequency concentra-tion. The balance of the time and the frequency concen-tration is controlled by the parameter a. The smaller thevalue of a, he narrower the frequency bandwidth (longertime duration), or vice versa.

Cabor ExpansionAs mentioned in the preceding section, the classical Fou-rier series is unsuitable for characterizing signals that onlylast for a short time or whose frequency contents changeover time. The reason for this is that the elementary fimc-tions employed in the Fourier series extend from negativeinfinite time to positive infinite time, and are not associ-ated with a particular instant in time. I n other words, thetick mark adopted by the classical Fourier transform doesnot contain time information.

T4. Cabor ex pansion sampling grid.

MARCH 1999 IEEE SIGNAL PROCESSING MAGA ZINE 55


5/16

Gabor Expansion and Short-TimeFourier TransformIn 1946, Dennis Gabor, a Hungarian-born British physi.-cist, suggested expanding a signal into a set of function:;that are concentrated in both the time and frequency do-.mains [26J. (In 1970, Gabor was awarded the NobelPhysics Prize for his dscovery of the principles underlyingthe science of holography.) Then, use the coefficients asthe description of the signals local property. The resultingrepresentation is now known as the Gabor expansion,

where Cm,nre called the Gabor coefficients. The set of el-ementary functions { L I % , ~ ( ~ ) }onsists of a time- and fre-quency-shifted function h( t ) , .e.,b,.n (t)=h(t-mT)eQm. (l6;1In Gabors original paper, h ( t ) s the normahzed Gaussianfunctiong(t), defined in (14),because it is optimally c o ncentrated in the joint time-frequency domain, in terms ofthe uncertainty principle. The parameters T and Q aretime and frequency sampling steps, respectively. Theproduct of TQ determines the density of the samphggrid. The smaller the product, the denser the sampling.

Fig. 4 llustrates the elementary functions used in theGabor expansion and the Gabor expansion-samplinggrid. Because the frequency-shifted Gaussian functionshave a short time duration, the Gabor expansion is moresuitable for characterizing transients and signals withsharp changes (Fig. 1).The time and the frequency reso-lutions of bm,n(~)an be adjusted by the parameter a in(14).The smaller the value of a, he better the frequencyresolution (poorer time resolution), or vice versa. Be-cause h,,,(t) are concentrated in [mT,nQ],Gabor sug-gested that the coefficient Cm,neflected a signalsbehavior in the vicinity of [mT,nQ] As a matter of fact,that is not really true, because {~? , ,~ ( t ) }n general doesnot form a tight frame. Consequently, the Gabor coeffi-cients C,,%are not a signals projection onh,,,(t). Weshallelaborate on this issue later.

Gabor had the clear insight to see the potential of thisexpansion, and it was to a great extent, his work and hi:;claims that spurred much of the subsequent work onjoini:time-frequency analysis. However, he did not actuallycontribute to the mathematical theory of the Gabor e xpansion. Many important aspects of the Gabor expansionwere not known during his lifetime [24], [29]-[31].

Moreover, Gabor limi tedh( t) o be the Gaussianf u n ction and TQ = 2n. In fact, as long as the sampling grid i:;dense enough, for instance,TQ 227~ (17)all those commonly used analysis fbnctions, such as theexponential, rectangular, and Gaussian-type functions,

great extent, his work and hisclaims that spurred much ofthe subsequent work on jointtime-frequency analysis.can be used in the Gabor expansion. When the productTQ= 2n , it is considered critical sampling. When TQ


6/16

is determined by the time sampling step T and the lengthof the analysis window function y(t )Let us rewrite (18) n the regular inner product form,i.e.,STFT[mT nQ]=jw(t ) y* , , t ) d t

-m

wherey,,, (t)=y(tmT)einRt. (20)Note that (20) and (16)have exactly the same form; bothare time- and frequency- shifted versions of a single pro-totype function. Let C,,+ = STFT[mT, nQ], hen (15)and (19) form a pair of Gabor expansions. The relation-ship between y ( t ) and h( t ) s

which is known as a formal identity. Equations (19) and(15) indicate that the STFT is, in fact, the Gabor coeffi-cient. Conversely, the Gabor expansion can be thought ofas the inverse of the STFT. However, these relationshipswere not widely understood until the early 80s [3]-[SI.

Traditionally, we investigate signals and systems in thetime or the frequency domain separately. The Gabor ex-pansion pair, (19) and ( lS ), now allow us to maptime-domain functions into the joint time-frequency do-main. This has been found extremely helpful when thesystem to be analyzed is time variant.

Importantly, for a given time functions(t) and analysiswindow function y( t ) , we are always able to find the jointtime-frequency function C,,,%.However, for an arbitrary2D function, there may be no corresponding signals ( t )and window function y ( t ) . The set of {C,n,,>s a subspaceof the entire set of 2D functions (Fig. 6) .

The key issue of the Gabor expansion is how to com-pute the dual function y ( t ) for a given h(t) .Except for afew functions at critical sampling [4] , 22], [2S], the ana-

6. The Cabor coefficients are a subset of the set of 2 0 func-tions. An arbitrary 20 time-frequency function may not be avalid C , ,

lytical solutions of (21), n general, do not exist. This factinspired many researchers to seek the discrete version ofthe Gabor expansion. Recently, one of the major ad-vances in the study of the Gabor expansion is the discov-ery of the discrete version [33], 34], 37], 44], [63],[721. Although there were some discrete versions of theinverse STFT [40] suggested before, they were no t moti-vated by expansion and inner product operations. Someimportant aspects, such as the relationship betweenoversampling and perfect reconstruction were not clear.In particular, the summation of computing the dual func-tion derived previously, was generally unbounded, whichis not directly suitable for numerical implementation. Byusing digital computers, we can now readily compute theGabor expansion and apply it to real-world problems.

Discrete Cu6or ExpansionApplying the sampling theorem and the Poisson-sum for-mula [44], 63],we derive the discrete Gabor expansionas

N-1s [i]=c -E ,, 1h[i-mAM]W,

nz n-I)

where

andc,, =c [iB* i- ]W; nr (23)

c

where AM denotes the discrete time sampling interval.Nindicates the number of frequency channels. The ratioN/M4 s considered as the oversampling rate. For a stablereconstruction, the oversampling rate must be more thanor equal to one. Usually, we require that L is evenly divisi-ble by N and AM. The number of frequency bins N is apower of two.

The dual function can be solved by AM independentlinear systems [47], i.e.,A - _ -where the elements of the matrixA, and vectors7, andp,are defined as

k = 0,1,2,...A A4-1 (24)y * b - p k

where O


7/16

When the signal length is equal to th e length of h [ i ]ancly[i] [63], we can also simply let

LNZ[i+nL]=h[i] o


8/16

can be substantially mproved in the joint time-frequencydomain [71].

Fig. 9 depicts the impulse signal received by the U.S.Department of Energy ALEXIS/BLACKBREAD satel-lite. After passing through dispersive media, such as theionosphere, the impulse signal becomes a nonlinear chirpsignal. While the time waveform is severely corrupted byrandom noise, the power spectrum is dominated mainlyby radio carrier signals that are basically unchanged overtime (Fig. 9, again). In this case, neither the time wave-form nor the power spectrum indicate he existence of theimpulse signal. However, when looking at the plot ofIC,,,,,, we can immedately identify the presence of thechirp-type signal arching across the joint time-frequencydomain. This observation suggests that we canuse a maskto filter out the desired Gabor coefficients {C,,,,} fromthe noise background and then reconstruct the timewaveform via the Gabor expansion.The problem here is that the Gabor coefficients are asubset of the set of 2D functions. Usually, the modified{C,,fl} are no longer in the set of valid Gabor coeffi-cients, In this case, for a given modified {C,,,}, there

may not be an existing signal that corresponds to it. Forthe sake of presentation simplicity, let us write (22) inmatrix form, i.e.,F=HS (28)where T, H , and E denote the signal vector, synthesis ma-trix, and Gabor coeacients vector. The matrix form of(23) iss = G 5 (29)where G denotesthe analysis matrix. If Hand G satisfy thedual relation determined by (24), hen HG = I andS=H 5= HG S.Note that H and G, in general, are not square matrices ex-cept for critical sampling. Let 5, denote the modified anddesired Gabor coefficients, that is,

(30)

Z = E = @ G F (31)

0.60.50.40.30.20.1

0-0.1

(a) L = 56, N = 8, AM= 7,Oversampling= 8/7,err = 0.4018 (b)L=48, N = 8 , AM = 6,Oversampling= 4/3, err = 0.2598

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1(C) L = 40, N = 8, AM = 5, (d) L = 32, N = 8, AM = 2,Oversampling= 8/5, err = 0.1628 Oversampling= 2, err = 0.0865

8. h[i] (dotted line) and yJi] (solid line) (as the oversampling rate increases, y,,,[i] becomes more and m ore close to h[i]).MARCH 1999 IEEE SIGNAL PROCESSING MAGAZINE 59


9/16

where @ is a diagonal matrix whose diagonal elements areeither one or zero. Based on Ed we compute a time wave-form s,,SI=WEd.Because the modified Gabor coefficientsare generally notvalid Gabor coefficients,GF, #Ed (31)which says that the joint time-frequency properties of T,are not the same as the desired Gabor coefficients Ed .

such that the dis-tance between the Gabor coefficients of Soprand the de-sired Gabor coefficients zd is minimum, that is,

A common approach is to seeka S opt

(3;!)minj/zd G S ~ ~ ~1which is known as the least square error (LSE)method. ttis well known that the solution of (32) is thepseudoinverse of G, that is,S ~ ~ ~ = ( G ~ G ) - I G ~ E ~ . ( 3 3 )Fig. 1 0 illustrates the LSE algorithm.

The LSE has been widely used and extensively stud -ied for many years [7], [23], [27], [65]. But it may notbe the best solution for many applications. In particular,to solve the LSE, we need to compute thepseudoinverse, which is computationally intensive andmemory demanding. When the data includes over10,000 samples, it would be difficult to apply an LS E al-gorithm with conventional PCs. A more effcienmethod developed recently is an iterative algorithm inwhich we continuously apply the same mask and recon-struct the time waveform. The procedure can be summa-rized as follows:S, =HEdEl =@GF, =@GHEds, =HE,E, =@GF, =@GHEl=(@GH) dC;, =@G?, =(@GH)hd .The sufficient condition for the iteration above to con-verge can be found in [69]. Under that condition,z, onverges to inside of @;The result of the first iteration S, is equivalent to rapt

computed by the LSE method.Two trivial cases, h[i]= y[ i]or critical sampling ({hPn+}forms a basis), satisfy the sufficient condition [69]. Forcritical sampling, the modified Gabor coefficients are al-ways valid Gabor coefficients, that is,GF , = E d . (34)

In this case, the iteration convergesa t th e first step. HOW-ever, critical sampling often leads to bad localization ofthe dual function y[i], and, as pointed o ut earlier, it is lessinteresting in applications.On the other hand, h[i] = ~ [ z ]usually implies high oversampling and thereby heaviercomputation. To reduce the computational burden inreal applications, we employ the orthogonal-like Gaborexpansion introduced in the preceding section. Althoughy[i] is hardly exactly equal to h [i], he performance of theorthogonal-like, Gabor-expansion-based algorithm hasbeen found to be good enough for most applications.

Fig. 1 1 plots the simulation results for the example il-lustrated in Fig. 9. Fig. l l ( a ) depicts the error vs. thenumber of iterations, which shows the iteration exponen-tially converging. Note that there is a substantial im-provement between the first and second iterations. Theresult obtained by the first iteration is closer to that com-puted by the LSE method. But, they are generally notequal unless the convergence condition [69] is truly met.In many applications, the LSE solution often is not thebest one. Fig. 11( ) depicts the reconstructed signal after3 0 iterations. In this example, the number of samples is9000, which is impossible to be solved by the LSE de-

SpectrumGabor Coefficients

_It i t I I ,0 0.013 0.027 0.040 0.0530.060 (ms)9. Due to the low SNR, the ionized impulse signal cannot berecognized in either time or frequency domains. However, fromlC,J we can readily distinguish t. (Data courtesy of theNonproliferation and international security division, LosAlamos National Laboratory.)

II - 2-D Functions

IIO.Map of LSE filtering.

60 IEEE SIGNAL PROCESSING MAGAZINE MARCH 1999


10/16

scribed by ( 3 3 )due to the computational complexity andmemory limitations. The iterative algorithm not onlyyields a better SNR, but also is practical for online imple-mentation.

Time-Dependent SpectrumBesides the linear transforms, such as the Gabor expan-sion and wavelets, another important method for thejoint time-frequency analysis is the time-dependent spec-trum. The goal here is to seek a representation that de-scribes a signal's power spectrum changing over time.Analogous to the conventional power spectrum, astraightforward approach of computing thetime-dependent spectrum is to take the square of theSTFT, the counterpart of the classical Fourier transform,i.e.,

We call it the STFT-based spectrogram to distinguish itfrom the Gabor-expansion-based spectrogram intro-duced later.

Fig. 12 illustrates the STFT-based spectrogram for alinear chirp signal. While the time waveform (bottomplot) and classical power spectrum only tell a part of thesignal's behavior, the STFT-based spectrogram indi-cates how the linear chirp signal's frequencies changeover time. The main problem of the STFT-based spec-trogram is that it suffers from the so-called window ef-fect. As shown in Fig. 12, the width of y ( t ) governs theresulting time and frequency resolutions. I n the left plot,a short time duration window function is used, whereasthe right plot uses a long time duration window. Obvi-ously, the left plot has better time resolution (poor fre-quency resolution). On the other hand, the right one haspoor time resolution (better frequency resolution). Al-though the STFT-based spectrogram is simple and eas-ily implemented, it has been found inadequate for theapplications where both high time and frequency resolu-tions are required.

Obviously, there is no window effect. Compared to theSTFT-based spectrogram, as we shall see later, ( 3 7 ) canbetter characterize a signal's properties in the jointtime-frequency domain. Formula (37) is known as theWigner-Ville distribution (WVD),which was originallydeveloped in the area of quantum mechanics by a Hun-

1 5 10 15 20 25 30(a) Numberof Iterationsvs. ICk-Ck-,I2

Noise TF

I I I 1 10 0.013 0.027 0.040 0.053 0.060 (ms)(b) Reconstruction After 30 Iterations)

1 1. Error (a) and reconstruction (b).

Wigner-Ville DistributionAs mentioned in the preceding section (12),the conventional correlation, in fact, is the av-erage of instantaneous correlat ionss ( t + z / 2 ) s * ( t - ~ / 2 ) ,.e.,

By averaging the instantaneous correlations,we lose the time information. How about us-ing R(t,z) o replaceR(T)?or instance,

Seconds

12. The STFT-based spectrogram for a linear chirp signal (The results aresubject to the selection of the analysis window function. The short time dura-tian window (left) leads to good time resolution and poor frequency resolu-tion, or vice versa).

MARCH 1999 IEEE SIGNAL PROCESSING MAGAZIN E 61


11/16

It is the crossterm interferencethat prevents the Wdistribution from breal applications, though itpossesses many desirableproperties for signal analysis.garian-born American physicist, EugeneP.Wigner [64],in 1932.Fifteen years later, it was introduced into the sig-nal processing area by a French scientist J. Ville 11591.(Wigners pioneering application of group theory to ;inatomic nucleus established a method for discoveringaridapplying the principles of symmetry to the behavior ofphysical phenomena. In 1963,he was awarded the NobelPhysics Prize.)

Let S( t )=A( t )e ) , where both A( t ) and $ ( t )arereal-valued time functions. Then, the first derivative ofthe phase@( t )epresents the weighted average of the in-stantaneous frequency. Traditionally, the first derivativeof the phase is called the instantaneous frequency, whichis actually incorrect for several reasons. For example, $(t)is a single-valued fimction, whereas at each time instant:,

time waveform and the power spectrum, we can easilydistinguish those two components. Essentially, there isnothing in the vicinity of 2 kHz during the time intervalfrom 0.01 - 0.015 s. However, the WVD plot shows astrongly oscillated term in that area. Because this term re-flects the correlation of two signal components [47], it isnamed the crossterm. Although the average of thecrossterm in this example is near to zero (in other words,it does not contribute energy to the signal), the magni-tude of the crossterm can be proven to be twice as large asthat of the signal terms. It is the crossterm interferencethat prevents the Wigner-Ville distribution from beingused for real applications, hough it possesses many desir-able properties for signal analysis.

In what follows, we shall introduce two alternativemethods, the Cohens class and the Gabor spectrogram,to reduce the crossterm interference with limited affectson the useful properties. While the Cohens class can bethought of as 2D linear filtering of the Wigner-Ville dis-tribution, the Gabor spectrogram is a truncatedWigner-Ville Ist ribu tion .

Cohens CIassAs mentioned previously, the crossterm is almost alwaysstrongly oscillated.So, an intuitive approach of removing

gene ray there are multiple frequencies. 7We can prove that the mean frequency ofthe WVD at time t is truly equal to the sig-nals weighted average instantaneous fre-quency, i.e.,

Therefore, the WVD indeed describes the sig- 1 0.000 0.005 0.010 0.015 0.020 0.025Secondsnays joint time-frequency behavior. More-energy content in the signal s ( t ) , i.e.,- WVD(to)dtdo=J - (s ( t ) l d t tion, or vice versa).

Iover, the energy ofthe WVD is the Same as the 12. The STFT-based spectrogram for a linear chirp signal (The results oresubject to the selection of the analysis wind ow function. The short time dur a-tion window (left) leads to good time resolution and poor frequency resolu--=-j- p(o) / dw

I 5 n% -- (39)As a result of this property, the WVD is often thought ofas a signals energy distr ibut ion in the joi nttime-frequency domain. It is interesting to note that alluseful properties of the WVD [i.e., ( 3 8 ) and (39)]are as-sociated with certain types of averages. The STFT-basedspectrogram possesses neither (38) nor (39).Fig. 13 il-lustrates the WVD of the linear chirp signal. Comparedto the STFT-based spectrogram in Fig. 12, he WVDhasmuch better time and frequency resolutions.

The problem of the WVD is the so-calledcrossterm in-terference. As an example, lets look at two time- and fre-quency-shifted Gaussian functions in Fig. 14. From the

0.000 0.005 0.010 0.015 0.020 0.025Seconds

62 IEEE SIGNAL PROCESSING MAGAZIN E MARCH 1999


12/16

the crossterm is to apply a 2D low-pass filter@( ,t ) o theWigner-Ville distribution, i.e.,

Expanding the term of W D n the above equation yields

where the function Q(t, T) denotes the Fourier transformof@(p, T) .Formula (41) was first developed in the fieldofthe quantum mechanics by Leon Cohen [17] and, it isnamed as Cohens class. Because the kernel hnc tion@( t ,T) [or +( t, o) ] determines the property of (41), Cohensclass greatly facilitates the selectionof our desired trans-formation. It is interesting to note that almost all previ-ously known time-dependent spectra [38], [49] can bewritten in the form (41).The prominent members of Co-hens class include the STFT-based spectrogram, theChoi-Williams distribution [141, the cone-shaped distri-bution [741, the adaptive kernel representation [61, andthe radial Butterworth kernel representation [67] and[68]. Comprehensive discussions of Cohens class can befound in [16], [18], [i9 ], [28], [47], and [66].

Crossterm

1OE+2OOE+O1

i

8 IO.OE+O 5.OE-2 1 OE-1 1.5E-1 2.OE-1 2.6E-1

14. W VD of the sum of two Gaussian functions.

15. The signal energy in the joint time-frequency domain canbe represented in terms of an infinite number of elementaryenergy atoms. All those atoms are concentrated, symmetrical,and oscillated. The amou nt of energy contained in each indi-vidual atom is inversely proportional to the rate of oscillation.

Cabor SpectrogramAnother way of separating a signals components in timeand frequency is to apply the Gabor expansion [42] , [46]as introduced in the preceding section. For example, wefirst expand the signal into,

Then take the Wigner-Ville distribution on both sides toobtainm ( t , o )

where WVD, ( t , ~ )enotes the WVD of two time- andfrequency-shifted Gaussian functions defined in (16).Asillustrated in Fig. 15,WVD,,,, (t ,o)s concentrated, sym-metrical, and oscillated.It can be shown that the energyof WTD,, , , (t ,o) s inversely proportional to the rate ofoscillation. If we consider WVD,,,,(t,o) s an energyatom, then (42) indicates that a signals energy can bethought of as the sum of an infinite number of energy at-oms. The contribution of each individual energy atom tothe signals energy not only depends on its weight,Cn2,nC*wt,,n,,ut also on the oscillation rate. The highlyoscdlated atoms are directly associated with crossterm in-terference, but have negligible influence to the signal en-

Based on the closeness of hIn,=(t)nd hllc, , , , ,t) n theergy [46l, [47].time and frequency domains, we rewrite (42) asGS, @,U)= Cm,nC*m,n, , l j , ( t ,o)

(43)m-m l+(n-n l i 11which is known as the Gabor spectrogram (GS), becauseit is a Gabor expansion-based-spectrogram. The parame-ter U in (43) denotes the order of the GS. When D = 0,the GS only contains those terms in which m = m and n= n. In other words, in this case, we only consider thecorrelation of two identical components. As the order Dincreases, increasingly more less-correlated componentsare included. When D goes to infinity, the GS convergesto the WVD.Fig. 16 depicts the WVD of an echo-location pulseemitted by the large brown bat, Eptesicusfiscus. In addi-tion to the true signals, the WVD also exhibits strongcrossterm interference.Fig. 17 llustrates he STFT-basedspectrogram and Gabor spectrogram for the same batsound. Obviously, as the order increases, the resolutionimproves. The best selection usually is D = 3 - 4. WhenD gets larger, the GS converges to the WVD. While theinstantaneous frequencies computed by the STFT-basedspectrogram are subject to the selection of the analysiswindow function, they are much closer to th e real value inthe Gabor spectrogram cases.It is worth noting that the energy atom,WTD,,,(t,o)in (43), has a closed form that can be saved as a table.

MARCH 1999 IEEE SIGNAL PROCESSING MAGAZINE 63


13/16

Consequently, once we o b -tain the Gabor coefficientsC*t,,zwhich can be effi-ciently computed by thewindowed FFT), the rest ofthe process is nothing morethan repeatedly using thelook-up table.In principle, we can alsodecompose other trans-forms, such as theChoi-Williams distribution,via a signal's Gabor expan-sion. It has been found, how-ever, that the Wigner-Ville-distribution-based de-composition (42) has thebest performance-a result

3%--


14/16

It is easy to see that adding (47), (48), and (49) yields(44). Based on (46) and (48), we have

... ...and finally,

(53)It can be shown that the residue (t)lJ2onverges asthe number of terms Nincreases. In real applications, weusually neglect the residue when i t is small enough, to re-write (44) as

and (53) as

Taking the WVD on the both sides of (54) yieldsm, v4=C14J2m,,C P )

(54)

(55)

The first summation represents the set of autoterms,whereas the second indicates the set of crossterms. Be-cause the Wigner-Ville distribution conserves energy(39) and is a using relationship (55),we have the adaptivespectrogram as

(57)Formula (54) is a signal adaptive approximation, whichwas independently developed by the authors of this articleand Mallat et a1 [36],[41], 1431, [45]. When the elemen-tary function h,(t) has a form (45), we name (57) as achirplet-based adaptive spectrogram.

The chirplet-based adaptive spectrogram has very highresolution without crossterm interference. Compared tothe nonparametric methods, the parametric algorithmsare computationally intensive. The key issue of signaladaptive approximation and chirplet-based adaptivespectrogram [73] is the estimation of the optimal elemen-tary function h b ( t ) n (50).Usually, they are only suitablefor certain types of signals. However, they could performextremely well if a parametric model indeed fits the ana-lyzed signal [57], [60].

While the STFT and Gaborexpansion provide vehicles tomap a signal between the timedomain and time-frequencydomain, the time-dependentspectrum is a powerful tool forunderstanding the nature ofthose signals whose powerspectra change with time.SummaryIn this article, we briefly introduced the concepts andmethods of joint time-frequencyanalysis. Like the classicalFourier analysis, the mathematical tools of jointtime-frequency analysis are also inner product and expan-sion. Moreover, the joint time-frequency algorithms alsofall into two categories: h ea r and quadratic. While theSTFT and Gabor expansion provide vehicles to map a sig-nal between the time domain and time-frequency domain,the time-dependent spectrum is a powerful tool for under-standmg the nature of those signals whose power spectrachange with time. The material presented in this article ap-pears to have reached a level of maturity for real applica-

Note that all methods introduced in this section aremainly designed for signals whose frequency contentschange rapidly over time. But their high performance is atthe cost of computational complexity. Many applicationsmay start with the STFT-based spectrogram when a sig-nals frequencies do not change dramatically because theSTFT-based spectrogram is simple and easily imple-mented. When high resolution is a must, the Gabor spec-trogram is usually a favorite because it is relatively robustand computationally efficient.

tions [91, [111-[131,~ 5 1 ,521,551, [561,[601, ~ 1 .

AcknowledgmentsThe authors wish to express their deep appreciation toProfessors Piotr J. Durka, Hans G. Feichtinger, ShidongLi, Flemming Pedersen, and XangGenXia for their manyconstructive comments as well as advice. The authorswould also like to thank their colleague, Dr. MaheshChugani, for his careful reading of manuscripts and valu-able suggestions.

Shie Qiun is a Senior Research Scientist in the AdvancedSignal Processing Department at National InstrumentsCo. ([email protected]). Dr. Dupang Chen isManager of the Advanced Signal Processing Departmentat National InstrumentsC O . [email protected]).

MARCH 1999 IEEE SIGNAL PROCESSING MA GAZ INE 65


15/16

References[ I ] .B. Allen and L .R. Rabiner, A unified approach to short-time Fourier

transform analysis and synthesis, Pruc. IEEE , vol. 65, no, 1 1, pp.1558-1564, 1977.

[2 M. Amin, Time-frequency spectrum analysis and estimation fortionstationary random processes, in Time-FrequencySignal Analysis:Methods and Applications,B. Boashash, Ed., pp. 208-232 , Australia: WileyHalstcd Press, 1992 .

[3] M.J . Bastiaans, Gabors expansion of a signal into Gabor clcmentary sig-nals, Pruc. IEEE, vol. 68, pp. 538-539, 19 80.

[4] M .J. Bastiaans, A sampling theorem for the complex spectrogram, andGabors expansion of a signal in Gaus sian elementary signals, Optical Erg i-neeriizg, vol. 20, no. 4, pp . 594-598, July/August 1981.

[S I 1M.J.Bastiaans, On thc sliding-window reprcscntation in digital signal pr ocessing, IEEE Trans.Acoust., Speech, Signal Processing, vol. ASSP-33, no. 4,pp . 868-873 , August 1985.

[6 ] R.G. Baraniuk and D.L. Jones, A signal-dependent time-frequency repre-sentation, lEEE Tr am Signal Processing,vol. 41, no . 1, pp. 1589-1602,January 1994.

[71 G.F. RoudreaiwB artcls and T.W. Parks, Time-varying filtering and signalestimation using Wigiier distribution synthesis techniques, IEEE Tram.Aco ust., Speech, Signal Processing, vol. 34, no. 6, pp. 442- 451, June 1 986.[SI V. C hcn, Radar ambiguity function, time-varying matched filter, and o pti-mal wavelet correlator, OpticalEngineering, ol. 33, no. 7, pp. 2212-2217,1994.

[9] V. Chen and S.Qian, Time-frequency transform vs. Fourier transform forradar imaging, lroc. IEEE-SP Intemation al Symposium on Time-Frequencyand Time-ScaleAnalysis, Paris, France, pp. 389 -392, June 19 96.

[ 101 V. Chen, CFAR detection an d extraction of unknown signal in noisewith time-frequency Gabor transform, lruc. S H E un WaveletApplications,vol. 2762, pp. 285-294, April 1996.

[111 V. Chcn, Applications of time-frequency processing to radar imaging,OpticalErgineeiing, vol. 36, no. 4, pp. 1152-1161, Apr. 1997.

[121 V. Chen, Time-frequency-based ISAR image formation technique, Proc.SPIE on AboYithmJ fov Synthetic Aperture k d a r Imapvy, vol. 3070, p p .43-54, April 199 7.

[131 V. Chen and S. Qian, Jo int time-frequency transform for radarrange-doppler imaging, IEEE Trans.Aerosp. Elecwun. Syst., vol. 34, no. 2,pp. 486-499, April 1998.

[ 14 ) H. Cho i and W .J. Williams, Improved time-frequency representation ofmulticom ponent signals using exponential kernels, IEEE Trans.Acoust,Speecb, Signallrocessifig, vol. 37, no. 6, pp. 862-871, June 1989.

[151 M . Chugan i, Feature analysis of Dopp ler ultrasound signals obtainedfrom mammalian arteries, 1h.D. Thesis, Rensselaer Polytechnic Institute,August 1996.

[161 T. Claasen and W . M ecklenbrauker, The W igner-distribution - a tool fortime-frequency signal analysis - Part I: Continu ous-time signals, Part 11:Discrete - ime signals, Part 111: Relations with oth er time-frequency signaltransformations, Philips/. Res., vol. 35, no. 3, pp. 217-250, 1980.

[171 L. Cohen, Generalized phase-spacc distribution functions,/. Math.P ~ J ~ s . ,ol. 7, pp. 781 -806, 1966.

[ 181 L. Cohen, Time-frequency distribution-a review, Proc. IEEE, vol. 77,no . 7, pp. 941-981, July 1989 .

[ 191 L. Cohen, Time-FrequencyAnalysis,Englewood Cliffs,NJ : Prentice Hall,1995.

[ 2 0 ] R.E. Crochiere aiid L. R. Rabiner, Mulctrate Digital Signal Prucessing,Englcwood Cliffs, NJ: Prentice-Hall, 19 83.

[21] I. Daubechies, The wavelet transform, time-frequency localization, andsignal analysis, IEEE Trans. lnfinn. Theoiy,pp. 9 61-1005 , September1990.

[22] P.D. Einzigcr, Gabor expansion of an aperture field in exponential ele-mentary beams, IEE Electrun. Lett., vol. 24, pp. 665-666, 1988.

[23] S.Farkash and S.Raz, Time-variant filtering via the Gabor expansion,Signal Prucessing I/: Tbeories and Applications,pp. 509-5 12, New York:Elsevier, 1990.

[24] H. Feichtiiiger and T . Stromas, GaburAnalysysb and A lgor ithm s: Theoiy andApplications,Birkhauser, 1998.

[25] B. Friedlander and B. Porat, Detection of transient signal by thc Gaborrepresentation, IEEE Trans.Acoust., Speech, Sign al Processirg, vol. 37, no.2, pp, 169-180, February 198 9.

1261 D. Gabor, The ory of commu nication,/. IEE, vol. 93, no. 111, pp .429-457 , November 1946.

[27] F. Hlawatsch and W. Krattcnthaler, Bilinear signal synthesis, IEEETran s. Signallrocessing, vol. 40, no. 2, pp. 352-363, February 199 2.

[28] F. Hlawatsch and G.F. Boudreaux-Bartels, Linear and quadratictimc-frequency signal representations, IEEE Signal Processing Magazine,vol. 9, no. 2 , pp. 21-67, March 19 92.

[29] A. J.E.M . Janssen, Gabor representation o f generalized functions,J.Math. Anal. Appl.,vol. 38, pp. 377-394, 1981.

[30] A.J.E.M. Janssen, Optimality property of the G aussian window spectro-gram,IEEE Trans. Signal Processing,vol. 39, no. 1, pp. 202-204 , January1991.

[3 11 A.J.E.M. Janss en, Duality and biorthog onality for Weyl-Heisen bcrgframe, to appear inJ. FourierAnal.Appl.

[32] R. Koenig, H.K . Dun n, and L.Y. Lacy, The soun d spectrograph,].Acoust. Soc. Amer., vol. 18, pp. 19-49, 1946.

[33] S. i, A general theory of discrete Gabor expansion, Proc. SPIE94Mathematical Imaging: WaveletApplications,San Diego, CA, July 1 994.

[34] S. Li and D .M. Healy Jr. A parametric class of discrcte Gabor expan-sions,IEEE Trans. Signal Processing,vol. 44, no. 2, pp. 201-211, February1996.

[35] Y . Lu and J.M. Morris, Gabor Expansion for Echo Cancellation, SignalProcessing Ma ga zim , March 1999.

[36] S . Mallat and 2. Zhang, M atching pursuit with time-frequency dictionar-i e s , I E E E Trans. Signal Processing,vol. 41, no . 12, pp. 3397-3415, Deccm-her 1993.

[37] J.M. Morris and Y. Lu, Discrete Gab or expansion of discrete-time signalsin 12 (Z) via frame theory, IEEE S&al Processing M aga zine , vol. 40, no . 2pp.155-181, March. 1994 .

[38] C .H. Page, Instantaneous power spectrum,/. Appl. Phys., vol. 23, pp.103-106, 1952.

[39] F. Iedersen, The Gabor-expansion-based time-frequency spectra, Pruc.ICASSP98, Seattle, WA, May 199 8.

[40] M .R. Protnoff, Time-frequency representation of digital signal and sys-tems based on short-time Fourier analysis, EEE Trans.Acoust., Speech, S&-nul Pmcessing, vol. 28 , no . 1, pp. 55-69,February 1980.

[41] S.Qian, D. Chen, an d K. C hen, Signal approximation via data-adaptivenormalized Gaussian functions and its applications for speech processing,Pruc. ICASSP92, San Francisco, CA, pp. 1 41-1 44, March 1 992.

[42] S. Qian aiid J.M. Morris, W ip er distr ibution decomposition aiidcross-term deleted representation, Signal Processing,vol. 25, no. 2, pp.125-144, May 19 92.

[43] S. Qian and 1). Chen, Signal representation in adaptive Gaussian func-tions an d adaptive spectrogram, Pruc. The 27th Annual Conference onInfor-mation Sciences and Systems, pp. 59-65, Baltimore, MD, March 1993.

[44] S. ian and D. Chen, Discrete Gab or transform, IEEE Trans. SignalProcessing,vol. 41, no. 7, pp. 2429-2439, July 1993.

[45] S.Qian and D. Chcn, Signal representation using adaptive normalizedGaussian functions, Shnal Processing,vol. 36, no. 1,pp. 1-11, March1994.

66 IEEE SIGNAL PROCESSING MAGAZINE MARCH 1999


16/16

[46].Qian and D. Chen, Decomposition of the Wigner-Ville distributionand time-frequencydistribu tion series IEEE Trans. Sgn al lmessing, vol.42, no. 10, pp. 2836-2841, October 1994.

[47] S.Qian and D. ChenJoint Time-FreyuencyAna&, Englewood Cliffs,NJ: Prentice-Hall, 1996 .

[48] S.Qiu and H.G . Feichtinger, Discrete Gabor structure and optimal rep-resentation, EEE Trans. Signal Processing, vol. 43, no. 10, pp. 2258-2268,October 1995.[49] W . Rihaczek, Signal energy distribution in time and frequency, IEEETrans.Inst.RadioEngineers, vol. rr-14 , pp. 369-374, 1968.

[50] M. Savic, M. Ch ugani, Z. Macek, M. Tang , and A. Hu sain, Detection ofstenotic plaque in arteries,Proc. 15th Annual InternationalConference ofrbelEEE Engineering inMedicine and Siolay Soc ie ty , San Diego, CA, pp.317-318, October 1993.

I511 M. Savic, M. Chugani, K. Peabody, A. Husain , M. Tang, and Z. Macek,Digital signal processing aids cholesterolplaque detection,Proc.ICASSP95, Detroit, M I, pp. 1173.1176, May 1995.

[52] R .J. Sclabassi, M. Sun, D.N . Krieger, P. Jasiukaitis, and M .S. Scher,Time-frequencyanalysis of the EEG signal, in Proc. ISSP90, Sknal Pro-cessing, Theories, Implementa tions and Applications, Gold Coast, Australia, pp.935-942, 1990.

[53 ] R.J. Sclabassi, M. S un, D.N . Krieger, P. Jasiukaitis, an dM S. Scher,Time-frequency omain problems in the neurosciences, in Time-FrequencySignal Analysis:Methods and Applications, B. Boashash, Ed.,Longman-Cheshire,England: Wiley Halsted Press, pp. 498-5 19, 199 2.

[54] G. Strang and T.Q. Nguyen, Waveletand Filter Banks. Wellesley, MA :Wellesley-Cambridge Press, 1995.

[55] M. Sun, S.Qian, X. Yan, S.Baumann, X:G Xia, R. Dehl, N. Ryan, andR. Sclabassi, Localiz ing u nctiona l activity in the b rain throughtime-frequencyanalysis and synthesis of the EEG , Proc. IEEE, vol. 85, no.9, pp . 1302-1311,November 1996.

[56] M. Sun, M.L. Scheuer,S.Qian, S.B. Baumann, P.D. Adelson, and R.J.Sclabassi, Time- frequenc y nalysis of high-freq uency activity at the start ofepileptic seizures,Proc. IEEE EMBC97, Chicago, IL, October 1997.

[57] L. Trintinalia and H. Ling, Joint time-frcquency SA R using adaptiveprocessing, IEEE Trans.AntennasPropajat., vol. 45 , no. 2, pp. 221-227,February 1997.[58] P.P. Vaidyanathan,Mu ltirat e Systems and fiilter Banks, Englewood Cliffs,NJ: Prentice Hall, 1993.

1591 J. Ville, Thovrie et applicationsde la notio n de signal analylique, (inFrench) cables et Transmission,vol. 2, pp. 61-74, 1948.

[60] J. Wang and J. Zhou, Chirplet-based ignal approximation for strongearthquake ground m otion model, this issue, pp. 9 4.

[61] Y.Wang, H.Ling, and V.C.Chen, ISAR motion compensation via adap-tive joint time-frequency echnique, IEEE Trans.AES -34, o appear.

[62] T.P. Wang, M . Sun, C .C. Li and A.H . Vagnucci, Classification of abnor-mal cortisol patterns by feat ures from Wign er spectra, in Proc. 10th Inter-national ConferenceonPattem Recognition,Atlantic City, NJ, pp. 228 -230,June 1990.

[63] J. Wexler and S. Raz, Discrete Gabor expansions,Signal Irocessing, vol.21, no. 3, pp. 207-221, November 1990.

1641 E.P. Wigner, On the quantum correction for thermodynamic equilib.rium, Phys. Rev., vol. 40, pp. 749, 19 32.

[65] C . Wilcox, The synthesis problem for radar amb iguity functions,Ttch.Summay Rep. 157, Math. Res. Center, University of Wisconsin at M adison,April 1960.

[66] W.J. Williams, Reduced interferencedistributions: biological applicationsand interpretations,Proc. IEEE, vol. 84, no. 9, pp. 126 4-1280 , September1996.

[67] D. Wu and J.M. M orris, Time-frequency epresentations using a radialButtenvorth kernel, lroc. IE EE -SI] Zntem ation al SymponnmonTime-Freyuencyand Time-ScaleAnalysis,pp. 6 0-63, Philadelphia, PA, Octo-ber 1994.

[ 6 8 ]D. Wu, T ime-freq uency ignal analysis via Coheils class of distribut ionsand implementation algorithms, Ph.D. T hesis, Universityof Maryland atBaltimore, May 19 98.

[69] X.-G. Xia and S . Qian, A n iterative algorithm for time-varying iltering inthe discrete Gabor transform domain, IEEE Trans.on Sgnal Processing, tobe published.

[70] X.-G. Sa , System identificationusing chirp signals and time-varying il-ters in the joint time-frequencydomain, IEEE Tran s. Signal Processing,vol.45, no. 8, pp. 2072-2084, August 1997.

[71] X.-G. %a , A quantitative analysis of SNR in the short-time Fourier trans-form domain for multicomponent signals, IEEE Trans.S&al P m ~ ~ e ~ ~ i n g ,vol. 46, pp. 200-203, January 1998.

[72] J . Yao, Complete Gabor transformation for signal representation, EEETrans.Image ProcessinJ, vol. 2, pp. 152-159, April 1993.

[73] Q. Yin, 2. Ni , S. Qian, and D. Chen, Adaptive oriented orthogonal pro-jective decomposition,]. Electron. (Chinese), vol. 25,110.4, pp. 52-58,April 1997.

[74] Y. Zhao, L.E. Atlas, and R.J.Marks, The use of cone-shaped kernels forgeneralized time-freq uency epresent ations of nonstati onary signals, IEEETrans.Acoust.,Speech, SignalProcessinj,vol. 38 , no. 7, pp. 108 4-1091 , July1990.

MARCH 1999 IEEE SIGNAL PROCESSING MAGAZINE 67

Documents

Joint Time-frequency Analysis