2.Continuous Wavelet Techniques

8/13/2019 2.Continuous Wavelet Techniques

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Tapan K. Sarkar and C. SuDepartment o f Electrical and Computer EngineeringSyracuse University121 Link HallSyracuse, New York 13244-1240 USATel: +1 (315) 443-3775Fax: +1 (315) 443-2583E-mail: [email protected]

Keywords: Wavelets, wavelet transforms, Fourier transform,Gabor transform

1. AbstractThe wavelet transform is described from the perspective of aFourier transform. The relationships among the Fourier transform,the Gab or transform (windowed Fourier transform), and the wave-let transform are described. The differences are also outlined, tobring out the characteristics of the wavelet transform. The limita-tions of the wavelets in localizing responses in various dom ains arealso delineated . Finally, an adaptive window is presented that maybe optimally tailored to suit ones needs, and hence, possibly, thescaling functions and the wavelets.

2. Introductionn nature, one does not encounter pure single-tone signals, butsignals the frequency content of which varies with time. G en-erally, frequency is defined as a phenomenon where the period ofzero crossings of a signal has a fixed duration for all times, andfrequency is related to the period. However, one may define theterm instantaneous frequency by defining signals the frequencycontent of which changes as a function of the time or duration ofthe signal. A good exa mple of this is speech, where the instantane-ous frequency may change from 20 Hz to 20 kHz, depending onthe system. Hence, it is interesting to develop methodologies thatcan be introduced to analyze such signals.

In modern times, such concepts have proven to be useful inmodem radar-system analysis. Conventionally, radar has dealt withpure CW signals, which are either frequency modulated or tumedoff and on to generate pulses. However, there are other radar sys-tems that deal with wideband pulses. Hence, in understanding howsuch radar systems work, it is necessary to understand the time-frequency representation of waveforms, and how they are charac-terized and analyzed.

In this stud y, we look at the classical Fourier transforms, theGabor transforms, and the wavelet transforms, which have been

This s part 2 of two-part article. Part 1 , which treated d iscretewavelet tech niques , appeared in the October issue.

utilized for characterizing waveforms the frequency content ofwhich changes with time. One of the main features that distin-guishes the three transforms is the choice of the window function.The purpose of the window function is to localize the informationabout the signal, in both space and time. To this end, the T-pulsetechnique provides a very flexible window function, which can begenerated a priori, utilizing computerized optimization techniques.Once such methodologies are known, one can then utilize specialpulse shapes to enhance detection performance. Hence, waveformshaping and analysis of wave shapes is the main theme of thispaper.3. Mathematical background

This se ction summarizes the various analytical tools that per-form time-frequency analysis. The three techniques that are mostpopular are the Fourier transform, the Gabor transform, and thewavelet transform. These techniques are presented both from acontinuous and a discrete-time signal point of view.The basic difference that distinguishes the three transfonntechniques as time-frequency localization tools is the choice OP awindow function. The window function is essentially an arti-fact through wh ich we o bserve the data. For each of the three trans-forms, the window function has certain fixed properties. As aresult, each of these analysis techniques has preferred domainsover which their application provides good results. One of theobjectives of this paper is to generalize these analysis techniques,and to describe a window function that is very flexible and can dealwith a very broad class of practical target-identification problems.

4. Continuous transforms4.1 Fourier transform

The classical Fourier-transform technique is utilized to findthe frequency content of a particular wave shape, p ( t ) ,which hasoccurred just once, has existed for a time interval [O , T] ,and isnonexistent fo r any other times. It is well known that the frequencycontent of such a signal p t ) is given by the Fourier transformP ( w )

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Equation (1) can be interpreted as modulating the function p ( t ) bye-wf and then integra ting it. The Fourier transform is define d forall values of the variable t [Le., from --co < t < a ] , w h ereas th eFourier series of the function p ( t ) is defined to be periodic. Thatis, p ( t ) repeats itself after every period (e.g., 2 7 ~ ) . ence, p F ( t ) ,the Fourier series of p t ) s defined as

mp F ( )= c e+jnfn=-m

where

c , = L T p ( t ) e - j d t .27T0 (3)

So, for a Fourier series, the function p t ) is decom posed into asum of orthogonal functions c , eln f Observe that the orthogonalfunctions into which p ( t ) is decomposed in Equation 2) is gener-ated by integer dilations of a single function e Jt . Also, note thatthe function p F( t ) s periodic with period 2n . In contrast, in theFourier Transform, the spectrum is decomposed into non-integerdilations of the function e Jf , s we know:

. m(4)

The problem in ana lyzing signals that are limited in time-Le., theyexist over a finite time window and are zero elsewhere-by Fouriertransform techniques is that such functions can not simultaneouslybe bandlimited. Hence, to represent signals p ( t ) , which exist for0 5 t 2 T , he Fourier transform is generally not the best w ay tocharacterize such signals. However, a short-time Fourier transformhas been utilized to analyze such signals in the time-frequencyplane.

By its very definition, the Fourier transformation uses theentire signal, and permits analysis of only the frequency distribu-tion of energy of the signal as a whole [ l]. To solve this problem,many who do spectral analysis have taken a piecewise approach.The signal is broken up into contiguous pieces, and each piece isseparately Fourier transformed. The resulting family of Fouriertransformations is then treated as if it were the basis for a jointtime-frequency energy distribution [1-41.In the short-time Fourier transform, the function p ( t ) is mul-

tiplied by a window function w ( t ) , and the Fourier transform iscalculated. The window function is then shifted in time, and theFourier transform of the product is computed again. So, for a fixedshift P of the window w(t ) , he w indow captures the features ofthe signal p ( t ) around P The window helps to localize the time-domain data, before obtaining the frequency-domain information.Hence, the short-time Fourier transform is given by

mP S T F T ( W P ) = j p ( t ) w ( t -P)e- dt > (5)

-m

and when w ( t )= 1 for all t , one obtains the classic Fourier Trans-form.

By observing Equation (5), it is clear that the short-time Fou-rier transform is the convolution of the signal p ( t ) with a filterhaving an impulse response of the form h ( t )= w(-t)e@, so that

The problem here is that as the window function gets nar-rower, the localization information in the frequency domain iscompromised. In other words, as the frequency-domain localiza-tion gets narrower, the window gets wider, so that localizationinformation in the time domain gets compromised due to theuncertainty principle. [The uncertainty in time, A t , and the uncer-tainty in angular frequency, A w , re related by AwAt 2 0.5 1.

Since the requirements in the time localization and frequencyresolution are conflicting, one has to make some judicious choices.The best window w(t) depends on what is the meaning of the termbest.

The above problem has solutions only under certain condi-tions. For example, if we require the function w(t) to be symmet-ric, and we require that its energy be confined to a certain bandIwI 5 B , hen we know the optimum function w ( t ) is given by theprolate-spheroidal functions. Other criteria, such as the signal-to-noise ratio, can also be utilized in designing the function w( t ) orutilizing other criteria for best [4-71.

To obtain the original signal back from the short-time Fouriertransform, we observe that [4]

If we set p = t , hen

So, we can recover the original signal p ( t ) for all t as long asw 0) 0 . If w 0) 0 , then choose some other value of p that willguarantee w 0)+ 0. Observe that it is not necessa ry to know w(t)for all t in order to recover p ( t ) from its shor t-time Fourier trans-form.

Altemately, one can also recover p ( t ) from Equation (7) bymultiplying both sides by G(t-p) , nd integrating with respect top . The over-bar denotes the complex conjugate. Then,

m 2This, of course, assumes that J\w t p)I d p is finite. However , ifthe window f kc ti on is infinite, then one can multiply both sides ofEquation 7) by another sequence ~ ( t )uch that

-m

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-m

An attempt was made by Gabor [8] in 1946, to develop amethodology where a function can be simultaneously localized intime and frequency. If that is possible, then the frequency con-tent of any signal can easily be obtained, by observing theresponse in ce rtain narrow frequenc y bands. Hence, it is possible totrack the instantan eous frequency of a signal.In order to sim plify the presentation, we introduce the gener-

alized form of Parsevals theorem for two functions, p ( t ) andq ( t ) nd their Fourier transforms, P ( w ) and e(@).hus,

l mP; ) p ( t ) g ( t ) d t= P ( w ) e ( o ) d o - Pi Q) (11)2 n-m 2n mwhere (*;a) denotes the inner product between two functions, andthe over-bar denotes the complex conjugate. Note that if [l]

then Equation (11) along with Equation (12) defines the inversetransform. Ife(@)2 n 6 ( w -wo) and q t)= e lm o t , 13)

then Equation (13), when substituted into Equation (1 I), definesthe forward Fourier transform.In the next se ction, we see how Ga bor, through the choice ofcertain window functions, made it possible not only to localize asignal in the time domain, but was also able to localize its fre-quency content in a narrow heq uency band.

4.2 Gabor transform [l 8 9]The objective of the G abor transform is to expand p ( t ) into aset of functions that are simultaneously limited in both time andfrequency. This is in contrast to the Fourier transform, where the

expansion is done by functions e-Jw t that are not time limited, buthighly localized in frequency.Even though, ho m a strictly mathematical point of view, it isnot possible to localize a function simultaneou sly both in the timeand frequency domain, this can, however, be achieved from apractical standpoint. Let us illustrate this by considering a family

of functions q( t ) such that a member q a ( t ) is defined ast 21 --q , ( t ) = - e 4a2&

Thus, ,B is a shift parameter, and now the functions ~ , , ~ ( t )anspan any function p ( t ) for all possible choices of the param eter aand ,B. Now, if we look at the Fourier transform of the product ofthe two functions p ( t ) ~ ~ , ~ ( t ) ,hich is called the Gabor transform

G,,p(w) 1 3 then

Note that (here, ~ , , ~ ( t )s assum ed to be a real function)

Hence, integrating Equation (16) with respect to ,O and substitut-ing Equation (17), one obtains

So, the Fourier transform of the function p ( t ) given by P w),results from integrating the Gabor transform with respect to allpossible delay parameters ,B.Since (p;q)=-(P;Q), a nd if q ( t ) = ~ , , ~ ( t ) e @ ,hen2 n

o mJ p ( t ) q ( t ) d t jp(t)w,,p(t)e-jox dt

-m -m

- 1= G,,p w) = -P( r r l )Q(Q)dQ.2 z-m

And since ~ , , ~ ( t )s real, we have

-( wn CrejP(0-n)= e

Then the product p( t)q , ( t ) can be localized in time from a practi-cal standpoint if a>O. This is because, beyond a certain valuet = q , he function qa(t) decays down to practically zero, and sowill the product p( t )q , ( t ) . Next, we introduce another parameter,,L? so that the function ~ ~ , ~ ( t )which is real) is defined by

Also, since

we have

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4.3 Wavelet transform

It is clear from Equation 22) that the Gabor transform also local-izes the Fourier transform, P ( w ) o f p( t ) (it in fact, the windowfunction wa,B(t) is highly localized), to give its local spectralinformation. Hence, not only is the function p( t) localized in timeby the function ~ ~ , ~ ( t ) ,ut also its transform P ( w ) is localized infrequency by the function W ~ / ( ~ ~ ) , ~ ( W ) .he width of the windowfunction in th e time domain, A t , is then obtained as

We have

and

Hence,

and the width of the spectrum of the window function in the fre-quency domain is given by A. :

So the product of the functions p(t)wa,s(l) is localized in time atp 9 and their spectrum [it is the spectrum of the windowed2

1version o f p ( t ) ] s localized in frequency at R * In addition,please note that 4

A property not possessed by the window functions of theGabor transform is the additional condition that the windows haveno "dc" component,

namely, the average value of the window is zero. This propertygives an extra degree of freedom for introducing a dilation (orscale) parameter in the window, in order to make the time-fre-quency window flexible. With this dilation parameter, the integralwavelet transform provides a flexible time-frequency window,which automatically narrows when observing high-frequency com-ponents, and widens when studying low-frequency components.Hence, it is in tune with our auditory and visual sense perceptions,which, at least in the first step, process signa ls in this fashion. Thisexplains the evolution of the musical scale in the West. For exam-ple, Figure 1shows the location of the notes C and G in the m ajordiatonic scale, for several octaves [4]. On a logarithmic scale, theywould appear to be nearly equally spaced. Thus, the notes C and Gbecome sparser and sparser as the frequency increases.The window function in the wavelet transform takes the formof [for both positive an d negative scale values of a ]

wa,$(t) = -w __ .The above function is admissible as a window function in thewavelet transform (or succinctly, a wavelet) provided Equation(30) is satisfied. The integral wavelet transform of p( t ) is definedby W T p :

with a 8 0 .The shortcom ing of the short-time Fourier techniques is taken

care of by utilizing the scaled window ~ , , ~ ( t ) .or a fixed timewidth of the window ~ ( t )n Equation ( 7 ) ,when p t ) is a high-frequency signal, many cycles are captured by the window,whereas if p ( t ) is a low-frequency signal, only very few cycles arewithin the same window. Thus, the accuracy of the estimate of theFourier transform is poor at low frequencies, and improves as thefrequency increases. One way to correct this problem is to replacethe window w t) with a function of both frequency and time, sothat the time-domain plot of the window gets wider as frequencies

1 1 1 1 1 1p C Q C 0 c 0 c28)12AtAW= -. ile wz :,.cu.nn - 1 2 2 1 HI 116 2So, the width of the time-frequency window is unchanged for aw H I

628 H tobserving the spectrum of p(t) at all frequencies. In fact, it is seenthat the Gabor transform is essentially a short-time Fourier trans-form with the smallest time-frequency window. It is the smallesttime-frequency window because the principle of uncertainty isexactly equal to 1/2, as given by Equation (28).

Figure 1. The pitch frequencies corresponding to the keys of Cand G on a piano. These correspond to the major diatonic scaleof Western music. The spacing is very nonuniform, and willappear to be almost uniform on a logarithmic scale.

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decrease: so that the window adjusts its width according to fre-quency. This is accomplished by the wavelet transform, as outlinedabove. For example, if

then in the frequency domain,

(33)Thus, all the possible states of the window are obtained by fre-quency scaling the response of the window function. This is incontrast to the STFT (short-time Fourier transform), where thevarious window functions were obtained as W(w)eJpW , Le., bymodulating the spectrum of the window function, which is fixed.

Hence, if the center of the window function w ( t ) is given byt * , and the width of the window in the time domain is given byA then the function ~ , , ~ ( t )s a window function with its centerat p + at*, nd its width is aA 7 . In addition, its spectrum is givenby

where W ( o ) s the Fourier transform of w ( t )From Parseval's relation in Equation (19), and utilizing

Equation (34), the wavelet transform WTp ofp is given by

(35)

The window function in the frequency domain is centered at w *and has a width -, with the exception of the multiplicativefactor ~ , and the phase factor eJpW The wavelet transform

2 Aa

a274&/

provides local information in the frequency window

From the wavelet transform given by Equations (31) or (35),the original function can be recovered, utilizing

where a > 0 . The constant C, is given by

Thus, Equation (39) implies that only certain classes of windowfunctions can be utilized in the wavelet transform, namely, those1responses that decay at least as fast as

The convergence of the integral in Equation (38) is defined ina weak sense [2], Le., taking the inner product of both sides ofEquation (38) with any function g ( x ) L 2 and commuting theinner product with the integral over a,P on the right-hand sideleads to the true formula. Since for any absolutely integrable func-tion W ( o ) the Fourier transform of w(t)-is continuous, Equation(39) can only b e evaluated provided

w(0)= 0 , 40)m

or, equivalently, j w ( t ) d t = 0 , i.e., the wavelet w ( t ) has no dcvalue, as mentioned in Equation (29).-m

Implementation of the wavelet transform is equivalent to fil-tering the signal p ( t ) by a bank of bandpass filters that are of con-stant Q . The center hequencies of the filters are offset from oneanother by an octave and, in the limit of Equation (39), this guar-antees that the center frequencies of the bandpass filters will neverbe zero.

In the continuous domain, all three techniques (Fourier,Gabor, and wavelet) have good theoretical properties. The questionis, what happens in the discrete domain? Do all these propertiescany over to the discrete domain, or do certain additional con-straints need to be imposed?

Note that in the wavelet type of analysis, if w * of W ( a w ) isassumed to be positive, then the ratio

(37)center frequency -w./a = w ]bandwidth 2A:ia 2Az .is independent of the scaling factor, a . The class of bandpass fil-ters represented by Equation (36) as a function of a has the prop-erty given in Equation (37), and these are called constant-Q filters.This type of processing is done by the human ear, at least in thefirst stage of signal detection [4].

(39)

5. Discrete transformsThe problem of dealing with the discrete transforms has someadvantages and disadvantages. For example, the Fourier transformfor the discrete case changes over to the Fourier-series representa-

tion. The Gabor transform, on the other hand, provides a non-physical representation of the spectrum for a certain class of win-dow functions. The discrete-wavelet transform provides a fastalgorithm for computation of the wavelet coefficients from a multi-rate digital-filtering point of view, without even introducing theconcept of wavelets.

5.1 A discrete short-time Fourier transform DSTFT )The discrete representation of Equation 5 ) is given by [4]

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where

and the p ( n ) are the sampled versions of p ( t ) ,with p as an inte-ger. So, the window function is represented by values that are sa m-pled at w n- p) The inverse DSTFT of PDL r ~FTs given by

If we set f l = n , hen(43)

(44)

Hence, one can recover p ( n ) as long as w(0) 0 . If w 0) 0 ,then one chooses a value of /3 for which w ) w(n -p)z 0 , ndthe procedure continues.

An altema tive representation can be made, provided we have

For Equation (44), p ( n ) is recovered by

It is interesting to note that the invers ion formula is not unique. Forexample, if zo is a zero of the z transform of the conjugate of thewindow function, i.e., zo is a zero of the polynomial

m& ( z ) = C ( k ) z - k ,k=-m

(47)

then if we replace P ' ~ T F T ( z , ~ ) in Equation (46) byP'sTFT(z,m) + z: , then Equation (46) is still satisfied. This is incontrast to the conventional, discrete Fourier transform (or,equivalen tly, the Fourier series ), which provides a unique inver se.

5.2 Discrete Gabor transformIn the discrete Gabor transform, the objective is to representthe signal, p ( t ) y a series,

where

(49)

Gabor presented a h euristic argument for itera tively estimating thecoefficients ake so as to obtain an approximation j ( t ) . However,no justification of this process was given [8,9 ].

Martin Bastiaans [lo] used a different set of expansions forp ( t ) , given by

He demonstrated that when gke t) replaces Y k t ( t ) in Equa-tion (48), then, indeed, th e repre sentation of Equations (48) and(SO) form a complete set, and the representation error, p ( t )- ( t )goes to zero for a sufficiently large number of basis functionsg k e ( t ) .The important difference between the representations ofEquations (49) and (SO) are the choices

and

where Tis he sampling interval.One of the objectives of this representation is to look at thecell centered at the point ( k ,f ) n the time-frequency plane, and

how its su rrounding regions are distributed. The main objectives ofthe representation of Equation (48) is that if the signal within theT Twindow NT -- t 5 NT + contains a pure sinusoid of fie-2 2

quency fo,hen we would hope that the expression for the time-limited spectrum would be

and that IP (N T, f , T ) I2 would have a significant magnitude, pri-marily near f0 .Howe ver, if one looks at the representation of p ( t )given by (48) and (49), then it is not at all clear that there wouldnot be any leakage from nearby cells, where k = N +1 N + 2 , orN -1, N - , and so on, to IP(NT,f, )I . Indeed, the leakagefrom neighboring cells can produce erroneous interpretations, asdescribed in [9]. This is because neither Equation (49) nor (SO) hasa finite support in the time domain. They extend for all times. Onewould expect the signals centered around the cell NT to contributeto P ( N T , ,f) f and only if the time-domain support in the repre-sentation in Equation (48) is finite. Otherwise, the results are notrelevant. This is why Lemer [111 extended the representation inEquation (SO) to have the basis signals of the general form

2

v ( t ) is considere d to be a "convenient" finite-ene rgy function, theenergy of which is concentrated near t = 0 , and the energy spec-trum of which, IV(w)l , is concentrated near w = 0 . This was later

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modified by Roach [SI, who demonstrated that unless the functionv ( t ) is of finite support in the time domain, representations of theform in Equation (48) produce an energy spectrum IP NT,f,T)Ithat has no clear relationship to the energy of the signal concen-trated in the corresponding interval in time. Note that this pre-cludes the Gaussian fimction or the prolate-spheroidal functions aspossible window functions, since they are of infinite duration

82

Equation (62) guara ntees that the coefficien t ckf for a givensegment centered at t = tk is com pletely independent of the signaloutside the Mh segment. To confirm this, observe that

Now note that, always, k = m : otherwise, the window functionsw(t- k) and w(t- m) are disjoint. We havet has been shown that for a proper time-hequency represen-tation, the expansion must be of the form [9]

-m tk -T i 2where

and finallyThe window function ~ ( t )n Equation (56) is chosen in such away that

Ti2W T ( W ) = ~ ( t ) dt ,

-T i 257) Utilizing the property of Equations (58) and (59) of the windowfunction, it is seen that Equation (65) becomes unity for = n andzero otherwise, satisfying Equation (62), and hence Equation (61)holds. The interesting point in Equation (61) is that the coefficientfor a given segment in time is completely independent of the signaloutside the kth cell, since the coefficients in different time seg-ments a re orthonorm al, Le., the total energy in the signal is then the

sum of all the individual coefficients squared.

and

withIt is interesting to observe that Equations (56) and (61) haveclose resemblance to the w indowed Fourier transform, presented inthe earlier sections, when the window function becomes a rectan-gular window. However, any window function that results from theconvolution of any rectangular pulse with a symmetric window

function will provide the m athematical requirements for ~ ( t )nEquations (56) to (59).

(59)The class of window functions that possesses this property isformed by convolving a rectangular pulse of duration ( T - p) witha symmetric positive pulse a ( t ) having duration p and area

1__T - p ' 5.3 Discrete wavelet transformIf, in the continuous wavelet transform, one uses integer val-ues for some integers k,n in Equation (32), and usually assumes

p = 2k nT (one can assume T = 1 witho ut loss of generality for thediscrete case) and a = 2 k , then the discrete wavelet transform ofp ( t ) s given by [4]

Here,

where Q is the integer frequency-spacing factor. The coefficientseke in Equation (55) can be given by

The inverse transform is given byc n mprovided p ( t )= 2 WT,(k ,n )2 -k 2w(2 -k t -n~) , (67)

under the condition that the window functions, given by

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are orth onormal, i.e.,

The shift integers n are chosen in such a way that ~ 2 - ~ tT )covers the whole line for all values of t . The wavelet transformthus separates the object into different components in its trans-form domain, and studies each component with a resolutionmatche d to its scale . Another interesting property to observe is thatthe original signal in Equation (67) is recovered from the discrete-wavelet-transform coefficients by filtering with the window~ , , ~ ( t )f Equation (68). Equivalently, the discrete wavelet transform is obtained by filtering the signal p ( t ) by ~ ~ , ~ ( - t ) ,r2-kZw(-2-kt).

The wavelet series of Equation (67) amounts to expandingthe function p ( t ) in terms of wavelets wk,n(t) or, equiva lently, bythe shifted -dilated versions of the sam e window function, so that

If w e further assume that the wavelets w k , n ( X ) are ortho gonal (i.e.,that Equation (69 ) holds), then

By comparing Equations (67) and (71), it is apparent that the(k , n)th wavelet coefficient of p is given by the integral wavelettransform ofp , if the sam e orthogonal wavelets are used in both theintegral wavelet transform and in the wavelet series. So the waveletseries provides an approximation for p that is not necessarily theleast-squares l?) orthogonal projection that a Fourier Series pro-vides.

Also the wavelets provide an unconditional basis for f for1< < o .Since 2 as no unc onditional bases, wav elets cannot dothe impossible. However, they can still do a better job than theFourier expansion, by displaying no Gibbs phenomenon forapproximating functions that are discontinuous, by utilizing theHaar wavelets, for example. However, if the wavelets used in theapproximation are not discontinuous, like the Haar wavelets (orWalsh functions), but utilize continuous functions, instead, thenthe Gibbs phenomenon is visible in the wavelet approximation.The problem now at hand is to determine if are there any numeri-cally stable algorithms to compute the wavelet coefficients Ck,* inEquation (71). Specifically, in real life, p is not a given function,but is a sampled function. Computing the integrals ofthen requires a quadrature formula. For the smallest value of k-often referred to by the scale parameter, Le., most negative k-Equation (71) will not involve many samples ofp , and one can dothe computation quickly. For arge scales, however, one faces largeintegrals, which might considerably slow down the com putation ofthe wavelet transform o f any given func tion. Especially for on-lineimplementations, one should avoid having to compute these longintegrals. One way out is the technique used in multirate/multi-resolution analysis, by introducing an auxiliary function, @ x), sothat [ 2 ]

mW ( X )= dm 4(2x-m) .m=-m

The + ( x ) are called the sc aling functions, and are gener ated by thedilation equation

(73)

where, in each case, only a finite number of coefficients cm anddm are different from zero.

Here, 4 does not have integral zero, but w does, and q isnormalized such that

mJ #(x )dx = 1 .-m

We define (bk,n even though 4 is not a w avelet, i.e.,(74)

(75)5 k n = 2-k24(2-% - n ) .The inner products of Equation (71) can be evaluated using

So, the problem of finding the wavelet coefficients is reduced tothat of com puting ( p ; j , k ) . Note that

(77)

so that ( P ; $ ~ , ~ )an be computed recursively, starting from thesmallest scale (most-negative k ) to the largest. The advantage ofthis procedure is that it is numerically robust, namely, even thoughthe wavelet coefficients ck,nn Equation (71) are computed withlow precision-say, with a couple of bits-one can still reproducep ( t ) with comparatively much higher precision [ 2 ] .

In summary, what we have done is as follows. Consider p ( t )to be a function of time. We have taken the spectrum of p ( t ) , andhave separated the spectrum into octaves of widths A u k , uch thatits frequency band w has been divided into [ 2 k ~o 2 k + z ] , orall values of k . We now define wavelets in each frequency binAmk , and approximate p ( t ) by them. If we choose

sinntnt4 t)= 3

then

and the w avelet expansion of p t ) is given by~ ~ , ~ ( t )2k2y1(2kt- ) (with T = 1),

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The functions ~ ~ , ~ ( t )re orthonormal because their bandwidthsare non-overlapping, namely, for a fixed k, P k ( 0 ) has the band-width A m k , w hich i s [ 2 k n , 2 k +n ] .So, the wavelet expansion ofa function is complete in the sense that it makes an approximationby orthogonal functions that have non-overlapping bandwidth.

As concluded by Vaidyanathan [4] ven though the continu-ous wavelet transform has a wider scope with dee per mathematicalissues, the discrete wavelet transform is quite simple, and can beexplained in terms of basic filter theory. Even before the develop-ment of wavelets, nonuniform filter banks had been used in speechprocessing by [12, 131. The motivation was that the nonuniformbandwidths could be used to exploit the nonuniform frequencyresolution of the human ear [14]. So if the wavelet application isalready in the digital domain, it is really not necessary to under-stand the deeper results of the scaling function 4( t ) and wavelets~ ( t )In this case, all we need to focus on is the dilation equationand the shift principle to generate a com plete basis.

5.3.1 First word of caution. Often, over-zealous researchers,in order to push the advantages of this newly developed tool(which, of course, is quite useful for selective problems), overlooktwo important factors.The first one is termed a wavelet crime by Strang and

Nguyen. The problem has to do with the selection of sample valuesof the function as the wavelet coefficients. As Strang and Nguyenpoint out, ...start with a function ~ ( t ) .ts samples x ( n ) are oftenthe input to the filter bank (to generate the wavelet decom position).Is this legal? NO. IT IS A WAVELET CRIM E. Some cant imag-ine doing it, others cant image not doing it. Is this crime conven-ient? YES. We may not know the whole function n(t) t may notbe a combination of 4(t - k ) , and computing the true coefficientsin x a k ) 4 t - k ) may take too long. But the crime cannot gounnoticed-we have to discuss it [20 ,p. 2321.

The problem is in Equations (71) and (76) ,where the samplesof x ( n ) are used at the zeroth scale to initiate the pyramid algo-rithm for doing a discrete wavelet decomposition. As Strang andNguyen point out, [20,p. 2321 ...the pyramid algorithm acts onthe numbers x ( n ) as if they were expansion coefficients of itsunderlying function x s ( t )= C x ( n ) @ ( t n ) , .e.,

Does xs t) have the correct sample values x ( n ) ? This seems aminimum requirement. They further point out the correct way tosolve the problem. There is a perennial problem of relating thesampled values of the function to the continuous function, as themapping from the sampled z domain to the continuous s domain isnot unique [21].

5.3.2 Second word of caution. It is possible to generate alocal orthogonal basis, by using the Malvar wavelets, for example[22]. The Malvar wavelets are basically windowed cosine and,

possibly, sine functions. Even though they generate orthogonalbases with som e localization, there is a problem, as with the Gaborbasis. Unless the windows have the specific properties as outlinedby Equations (56) to (59) the windowed localized bases of Equa-tion (61), which are similar to the Malvar wavelets, do not alwaysprovide the correct localization picture in the transformed domain

6. A discussion of possible choices of the window functionThe claim that a particular transform provides simultaneouslocalization, strictly in time and frequency, is preposterous. Whatwe have seen is that if a function is limited in time, it cannotsimultaneously be limited in frequency, Le., bandlimited. Hence,resolution in time is accom plished at the expense of the frequency,such that for the best window, we have the situation that theHeisenberg principle of uncertainty, A t A o 5 112 is satisfied with

the equality. The equality is achieved only for a Gaussian windowfunction. However, as we have seen for the Gabor transform, theGaussian window provides nonphysical power spectral density fora region of the function localized in time. The wavelet transformcannot do any better For all other choices of the window function,A t A , is > 112.

Other possible choices for the window may be the prolate-spheroidal functions. Unfortunately the prolate-spheroidal func-tions are not limited in time, but they are strictly bandlimited. Analtemate choice is to deal with the truncated prolate-spheroidalfunctions, which are strictly time limited, but one does not haveany control over the spectrum of such functions [15-171. That iswhy a flexible methodology has been developed, where one candesign a window shape that it is finite in time and, in addition, canfocus the energy into a pre-specified band [18, 191.However, what one may try to do is to have a finite timewindow for localizing a process in time. However, its shape can bemanipulated in such a way that it is approximately bandlimited formost practical purposes, Le., 99.9% of its energy is within a verynarrow prescribed band. Such a construct is called a T pulse. A Tpulse is a pulse limited in time T. Moreover, its spectral content,Le., 99.9% of its energy, is focussed within a few band times (i.e.,z ,here n is about 2 or 3.TThe design of the T pulse is carried out in the discrete

domain. Let us assume a discrete signal sequence f m) which isdefined for m = 0, 1,2, ... N , - 1, and is identically zero outsidethese N m values. Let us assume there are N amples in one baudtime (in an approximate way, the baud time is the time durationbetween z ero crossings of a signal). Then, the total number of baudtimes N s

The discrete Fourier transform (DFT) of the signal f m) s givenby

for k = 0, 1,... Nk -1So, in the frequency domain, N k is the total number of sam-

ples of the DFT sequence F ( k ) In the frequency domain, if we44 IEEE Antennas and Propagation Magazine, Vol. 40, No.6 ecember 1998


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assume there are N,. samples per baud rate (the inverse of baudtime), then

and increasing N,. increases the resolution in the frequencydomain.

In the T pulse construction, the objective is to maximize thein-band energy within the set 4 = {-Nb I k I Nb} Or, equiva-lently, it minimizes the energy outside the N b samples. In addi-tion, the wave shape has to be orthogonal to its shifted version, andthis will minimize the inter-symbol interference. This implies that

In summary, the following observations are of importance.I. Note that the objective is to m inimize the cost function, J, suchthat E,,, is minimum with em = 0 , and ep = 0 forp = O , l , ..., N , - l .2. The weights w e , w m , and w p p = 0, 1,...,N , - 1 should beadjusted in a search procedure to ac hieve the above goal.=O

for k = 0 , 1 , .., N c - I , The minimization process can be outlined as follows:Step 1. Choose an initial guess for f m) nd an initial guess

for the weights w e , w m, nd w p , p = 0 , 1 ..., N e - .where 6( k ) is the impulse function. This guarantees that if thewaveform is shifted by a baud time or its multiples, then the waveshape is orthogona l to itself. This is termed zero inter-sym bol inter-ference. Note that when k = 0, it is the square of the functionitself, and no constraint needs to be put on that. In addition, w eneed to put in a dc constraint, i.e., the wave shape should have nodc com ponent. Hence,

Step 2. Compute the gradient of the functional Jwi th respectto f ) . This is give n by

So, the cost function, J , hat will be minimized isN 1

p = o

where w e , w m, nd w p are various weights to the errors E,,, , e m ,e p The weights should be adjusted in a search procedure that hasbeen designed to minimize J .

assuming f ( m ) s real.Next, an optimum step length to update the signal sequence

f ( ) s chosen, through one-dimensional searches.In addition,E,,, = out-of-band energy

and

m=O-0 .4

-0.61

-0.8

N,-1m=O

e, = C f ( m ) f ( m + p N s ) - 6 ( p ) or p = l , 2,..., N c - l . 90)

Equivalently, 10 10 20 30 40 50 60 70 BO 90 100Time Index

Figure 2. A baseband T pulse with zero mean and uon inter-symbol interference.=OIEEE Antennas and Propaga tion Magazine,Vol. 40 o. 6 December 1998 45


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20

Frequency IndexFigure 3. The spectrum of the zero-mean baseband Tpulse.

0 10 20 30 0 50- 1 .5

Time light-meters)Figure 4. The curren t at the center point of a thin-wire antenna2.0 m in length and with a radius of 0.01 m excited by a G aus-s ian-s~ aped u lse.

Frequency IndexFigure 5. The Fo urier transform of the current in Figure 4.

Step 3. If the norm of the previous gradient vector is notsmall enough, go to Step 2. Otherwise, see whether theorthogonality errors lepl, for p = 1,2,... N, , are small enough.If the errors are small enough, stop the process. If the errors arestill considered to be large, increase each w p p = 1, 2, ..., N 1)by a fac tor, and then go to Step 2.

Note that throughout the process, we is a fixed nonzerovalue, since the abso lute value of the total energy is not important.However, we an be increased during the process, if the in-bandenergy to generate the T pulse is unexpe ctedly low. This may hap-pen since the cost function may have more than one local mini-mum, and increasing we an make one jump out of an undesiredlocal minimum.

An example is presented, showing the synthesized T pulse inthe time domain, in Figure 2. Its spectrum is in Figure 3. The Tpulse has been designed for Nbaud= 4 , Nband= 130 , and for 100samples. Note that the T pulse has zero mean, and is orthogonal toits shifted versions. The second figure shows the spectrum of the Tpulse. It is seen that the bandwidth of the Tp ulse is 13 units, asseen from the FFT. As seen, one can design a window function asshown in Figure 2 that has zero me an and, in addition , has no inte r-symbol interference, i.e., the waveform is orthogonal to its shiftedversion. This window function is called the T pulse or the time-limited pulse. The spectrum of the T pulse is illustrated in Figure 3,and is shown to be very highly localized.To further illustrate the application of time limiting and, practi-cally, band-limiting a signal, we consider the electromagneticscattering from a thin wire. If a Gaussian-shaped pulse excites athin-wire antenna 2.0 m in length and with a radius of 0.01m, thenthe curre nt at the cen ter point of the wire is given in Figure 4. If we

0 10 20 30 40 50 60 ,o 80 90 100 ,io 120 130

Time IndexFigure 6. A T pulse with a dc value that is not zero.

-30-40

\ , , , , -

- ' t o

-120 - . . . . . - , . . . . . . . . . - - . . . . I . . . . .0 IO 20 30 0 50 SO 0

Frequency IndexFigure 7. The spectral content of the Tpulse shown in Figure 6.

46 /A nten nas and Propagation Magazine,Vol. 40 o. 6, ecember 1998


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1.00 . 90 . 8 .0 . 70.60 . 10 40 . 3 .0.20 .

-0.0-0 .1 .w8 0.1.6 0 . a

VI - 0 , s ..CI - 0 . 4-0 . 6 .- 0 . 7- 0 . 0

-1.00 10 20 30 40 110 70 8 90 100 110 120 130

0.0170 0150 . 0 1 3 .O . O I ,0. 00s.0 . 0 0 7 .0.00S.0 . 0 0 3 .0.001.-0.001.-0.003.- 0 . 005-0.007.0.009.

-0.0,1.-0.013'- 0 .O ,S,

-O ----- , . . ~~. . . .0 IO zo 30 0 70

Time Index Frequency IndexFigure 8. The amplitude-modulated carrier with the T pulse. Figure 11. The spectral content of the signal in Figure 11.

EeYQ)arA

YB

observe the Fourier transform of the current, it is given in Figure 5 .Now, th e question is, it is possible to excite the second resonancein the wire structure, which is 20 dB down, without exciting thefirst resonance? To this effect, we amplitude modulate a carrierfrequency that is at the second resonance by the T pulse. In thisexample, we consider a T pulse with a dc value that is not zero, asis shown in Figure 6. The spectral content of the T pulse is givenby Figure 7. The am plitude-modulated carrier with the T pulse isshown in Figure 8.Its spectral content is given in Figure 9.Observe that the spectral energy is dominant around the secondresonance. Now , if we excite the dipole antenna with this incidentpulse of Figure 8, the current at the center is given in Figure 10.The spectral content is given in Figure 11. From Figure 11, it isquite clear that it is possible to generate the second resonancewithout exciting the dominant first resonance. Hence, this multi-resolution analysis can have significant potential application in tar-get identification.

0 10 20 30 0 50 60

Frequency Index Finally, we conclude this section by pointing out the differ-ences between a T pulse and a wavelet. In the T pulse, a windowhas been created that is not only limited in time, but 99% of theenergy is focused in a narrow band and, in this way, an attempt ismade to satisfy the equality in the Heisenberg principle of uncer-tainty, i.e.,

.IO0

Figure 9. The spectral content of the signal in Figure 8.

-0.017 \-.-.-..-~,-~.-- . , . . . . . . . . , . . . . . . . .0 IO 20 30 0 HITime light-meters)

A,A, 4 0.5.The expansion in Equation 70) can be done in terms of a dilatedand shifted version of the T pulse in an approximate fashion. Forthe wavelets. the decomposition by Equation 70) is essentiallyexact, and perfect reconstruction can be done, even after the func-tions have been down-sampled [4]. Also, in Equation (70), there isa desire to limit the number of nonzero coefficients in Equation(70), to improve the efficiency of the decomposition. This isachieved by enforcing that the pth derivative and all its lowerderivatives are zero at co = 0 , Le.,

(93)

Depending on the nature of the requirements for the problem ofFigure 10. The current at the center of the dipole antenna,when excited by the incident pulse of Figure 8. interest, one can use either of the constraints on the window fiinc-tions.

IEEE Antennas and Propagation Magazine, Vol. 40, No.6, December 1998 47


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7. ConclusionThis paper provided a survey of the short-time Fourier tech-niques, the Gabor transform, and the wavelet transform. Thestrengths and the weaknesses of each method have been identified.The unity in diversity between the three transforms is the choice ofthe appropriate window function. In addition, it has been shownhow to apply num erical-optimization techniques to design a flexi-ble window function, such that it is strictly time limited, yet 99.9%

of its energy can be focused in a narrow band.

8. AcknowledgmentThe authors would like to acknowledge the suggestions of theEditor-in-Chief regarding ways to improve the readability of thearticle.

9. References1. C. K. Chui, An Introduction to Wavelets, New York, AcademicPress, 1992.2. I. Daubechies, Ten Lectures on Wavelets, Philadelphia, Societyof Industrial and Applied Mathematics (CBMS , 61), 1992.3. R. Gopinath and C. S. B um s, Wavelet Transforms and FilterBanks, in C. K. Chui (ed.), Wavelets-A Tutorial in Theoly andApplications, New No rk, Academic Press, 1992, pp. 603-654.4. P. P. V aidyanathan, Multirate Systems and Filter Banks, NewJersey, Prentice Hall, 1993.5 . F. I. Tseng, T. K. Sarkar, and D. D. Weiner, A Novel Windowfor Harmonic Analysis, IEEE Transactions on Acoustics, Speechandsign al Processing, April 1981, pp. 177-188.6. W. F. Walker, T. K. Sarkar, F. I. Tseng, and D . D. Weiner, Car-rier Frequency Estimation Based on the Location of the SpectralPeak of a Windowed Sample of Carrier Plus Noise, IEEE Trans-actions on Instrumentation and Measurements, IM-31, December1982, pp. 239-249.7. W. F. Walker, T. K. Sarkar, F. I. Tseng, and J. Cross, OptimumWindows for Carrier Frequency Estimation, IEEE Transactionson Geoscience Electronics, GE-21, October 198 3, pp. 446-454.8. D. Gabor, Theory of Communications, Journal of the Institutefo r Electrical Engineers, Novemb er 194 6, pp. 429-457.9. J . E . Roach, A Vector Space Approach to Time-Variant EnergySpectral Analysis, PhD thesis, Syracuse University, Syracuse, NewYork, 1982.10. M. Bastiaans, Gabors Expansion of a Signal into GaussianElementary Signals, Proceedings oft he IEEE, 68, April 1980, pp.538-539.11. R. Lemer, Representation of Signals, in E . Baghdady (ed.),Lectures on Communrcation Svstem The ow, New York, McGraw-Hill, 1961, Chapter 10.12. G. A. Nelson, L. L. Pfeiffer, and R. C. Wood, High SpeedOctave Band Digital Filtering, IEEE Transactions on Audio andElectroacoustics, AU-20, March 1972, pp. 8-65.

13. R. W . Schafer, L. R. Rabiner, and 0 Herrmann, FIR D igitalFilter Banks for Speech An alysis, Bell System Technical Journal,54, March 1975, pp. 531-544.14. J. L. Flanagan, Speech Analysis, Synthesis and Perception,New Y ork, Springer-Verlag, 1972.15. I. Gerst and J. Diamond, The Elimination of IntersymbolInterference by Input S ignal Shaping, Proceedings of the IRE, 9,1963, pp. 1195-1203.16. P. H. Halpern, Optimum Finite Duration Nyquist Signals,IEEE Transactions on Communications, COM-27, June 1979, pp.886-888.17. E. Panayirci and N. Tugbay, O ptimum Design of Finite Dura-tion Nyquist Signals, Signal Processing, 7 984, pp. 57-64.18. Y. Hua and T. K. Sarkar, Design of Optimum D iscrete FiniteDuration Orthogonal Nyquist Signals, IEEE Transactions onAcoustics, Speech and Signal Processing, ASSP-36, 4, April 1988,pp. 606-608.19. T. K. Sarkar, S. M. Naryana, H. Wang, M. Wicks, and M.Salazar-Palma, Wavelets and T-pulses, in L. Carin and L. B .Felsen (eds.), Ultra- Wideband Short-Pulse Electromagnetics 2,New Y ork, Plenum Press, 1995, pp. 475-485.20. G. Strang and T. Nguyen, Wavelets and Filter Banks,Wellesley Cambridge Press, 1996.21. T. K. Sarkar, N. Rad haknshna, and H. Chen, Survey of Vari-ous z-Domain to s-Domain Transformations, IEEE Transactionson Instrumentation and Measurements, IM-35, December 1986,pp. 508-520.22. M . Vettereli and J. Kovacevic, Wavelets and Subband Coding,New Y ork, Prentice Hall, 1995.

Introducing Feature Article AuthorsTapan Kum ar Sarkar received the B. Tech. degree from theIndian Institute of Technology, Kharagpur, India, the MScE degreefrom the U niversity of New Brunswick, Fredericton, Canada, andthe MS and PhD degrees from Syracuse University, Syracuse, NewYork in 1969, 1971, and 1975 , respectively.From 1975 to 1976, he was with the TA CO Division of theGeneral Instruments Corporation. From 1976 to 1985, he was w iththe R ochester Institute of Technology, Rochester, N Y . From 1977to 1978, he wa s a Research Fellow at the Gordon McKay Labo ra-tory, Harvard University, Cambridge, MA. He is now a Professor

in the Department of Electrical and Computer Engineering, Syra-cuse University; Syracuse, N Y . He has authored or co-authoredmore than 170 joum al articles, and has written chapters in tenbooks. H is current research interests deal with numerical solutionsof operator equations arising in electromagnetics and signal proc-essing, with application to system d esign.Dr. Sarkar is a registered Professional Engineer in the Stateof New York. He w as an A ssociate Editor for Feature Articles ofthe IEEE Antennas and Propagation Society Newsletter, and he

48 IEEE Antennas and Propagation Magazine Vol. 40,No. 6 ,December 1998


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was the Technical Program Chairman for the 1988 IEEE Antennasand Propagation Society International Symposium and URSIRadio Science Meeting. He was the Chairman of the Inter-Com-mission Working Group of URSI on Time-Domain Metrology. Heis a member of Sigma Xi and USNC/URSI Commissions A and B.He received on e of the best solution awards in M ay, 1977, at theRome Air Development Center (RADC) Spectral EstimationWorkshop. He received the Best Paper Award of the IEEE Truns-actions on Electromagnetic Compatibility in 19 79, and at the 1997National Radar Conference. He is a Fellow of the IEEE. Hereceived the degree of Docteur Honoris Causa from the UniversitCBlaise Pascal, Clermont-Ferrand, France, in 1998.

Chaowei Su was bom in Jiangsu, Peoples Republic ofChina, on April 23, 1961. He received the BS and MSc degreesfrom Northwestem Polytechnical University (NPU), both in theDepartment of Applied Mathematics, in 1981 and 1986, respec-tively. H e joined the D epartment of Applied Mathematics at NPUin 1982. Mr. Su was appointed as an Associate Professor and a fullProfessor at NPU in 1993 and 1996, respectively. He was a Visit-ing Scholar at S U N Y , Stony Brook, New York, USA, betweenDecember, 1990, and June, 1992, supported by a Grumman Fel-lowship. Since August, 1996, he has been a Visiting Professor atSyracuse University, U SA. He is an author of Numerical Methodsin Inverse Problems of Partial Differential Equations and TheirApplications (published by NPU Press). His ma in research interestsare in the area of numerical methods of electromagnetic scatteringand inverse scattering, and inverse problems of partial differentialequations.

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