On the Sliding-Window Representation in Digital Signal Processing

Transcription

1 868 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, SIGNAL AND PROCESSING, VOL. ASSP-33, Nb. 4, AUGUST 1985 On the Sliding-Window Representation in Digital Signal Processing MARTIN J. BASTIAANS Abstract-The short-tie Fourier transfor of a discrete-tie signal, which is the Fourier transfor of a windowed version of the signal, is interpreted as a sliding-window spectru. This sliding-window spectru is a function of two variables: a discrete tie index, which represents the position of the window, and a continuous frequency variable. It is shown that the signal can be reconstructed fro the sapled sliding-window spectru, i.e., fro the values at the points of a certain tie-frequency lattice. This sapling lattice is rectangular, and the rectangular cells occupy an area of 27r in the tie-frequency doain. It is shown that an elegant way to represent the signal directly in ters of the saple values of the sliding-window spectru, is in the for of Gabor s signal representation. Therefore, a reciprocal window is introduced, and it is shown how the window and the reciprocal window are related. Gabor s signal representation then expands the signal in ters of properly shifted and odulated versions of the reciprocal window, and the expansion coefficients are just the values of the sapled sliding-window spectru. S INTRODUCTION HORT-TIME Fourier analysis [l] of discrete-tie signals is of considerable interest in a nuber of signalprocessing applications. In order to study spectral properties of speech signals, for instance, the concept of a short-tie Fourier transfor of the signal is very convenient [ 11, [2]. Such a short-tie Fourier transfor is usually constructed by first ultiplying the signal by a window function that is slided to a certain position, and then Fourier transforing the windowed signal. Therefore, we like to consider the short-tie Fourier transfor as a sliding-window representation of the signal. There are other interpretations, of the short-tie Fourier transfor, including a well-knowpfilter bank interpretation [ 11. However, for the purpose of this paper, we find the slidingwindow interpretation to be the ost appropriate, and, to ephasize this, we shall call the short-tie Fourier transfor the sliding-window spectru of the signal. The sliding-window representation of a signal, which is a signal description in tie and frequency siultaneously, is coplete in the sense that the signal can be reconstructed fro its sliding-window spectru [ 11. However, to reconstruct the signal, we need not know the entire slidingwindow spectru. In this paper we show that it suffices to know the values of the sliding-window spectru only at the points of a certain rectangdar lattice in the tie-frequency doain, and we describe how the signal Manuscript received January 17, 1984; revised January 2, The author is with the Technische Hogeschool Eindhoven, Afdeling der Elektrotechniek, Postbus 513, 5600 MB Eindhoven, The Netherlands. can be expressed directly in ters of the values of this sapled sliding-window spectru. This will lead us in a natural way to Gabor s representation [3] of a signal as a superposition of properly shifted and odulated versions of a function that is related to the window. We show a way to deterine this function fro the knowledge of the window, and we elucidate this with soe siple exaples of window functions. SLIDING-WINDOW REPRESENTATION OF DISCRETE-TIME SIGNALS -, - 1, 0, 1, - * a) denote a one-di- Let x(n) (n + =. ensional discrete-tie signal and let w(n) represent a window sequence; the signal and the window ay take coplex values and they need not have a finite extent. We ultiply the signal by a shifted and coplex conjugated version of the window and take the Fourier transfor of the product, thus constructing the function [cf. [l],(6.1)] 03 f(~!, n) = C i(> w*( - n) exp [-j~!]. (1) /85/ $ IEEE Unlike (6.1) in [l], (1) uses a coplex conjugated version of the window; oreover, the window has not been tiereversed. The only reason for doing this is to get ore elegant forulas in the reainder of the paper. We shall callf(q, n) the sliding-window spectru [cf. [4], Section 4.11 of the discrete-tie signal; it is clearly a function of two variables: the tie index n, which is discrete and represents the position of the window, and the frequency variable L!, which is continuous. Of course, as in the case of noral Fourier transfors of discrete-tie signals, the sliding-window spectru f(q, n) is periodic in L! with period 2n. Two choices of window sequences are of special interest. If w(n) vanishes for n # 0, then f(0, n) is proportional to the signal x(n); the sliding-window spectru thus reduces to a pure tie representation of the signal. If, on the other hand, w(n) does not depend on n, thenf(q, 0) is proportional to the Fourier transfor of x(n); the sliding-window spectru then reduces to a pure frequency representation of the signal. In general, however, the sliding-window spectru is an interediate signal description between the pure tie and the pure frequency representation. We can reconstruct the signal x(n) fro its sliding-win- dow spectru f(q, n) in the usual way [cf. [l], (6.6)] by inverse Fourier transforing and taking = n, which

2 BASTIAANS: SLIDING-WINDOW REPRESENTATION 869 yields the inversion forula in which j2= dq * represents integration over one period 2s; of course, the rather ild requireent that w(0) be nonzero should be satisfied. There exists another way of reconstructing the signal fro its sliding-window spectru, viz. by eans of the,inversion forula [cf. [5],(2) and [6],( )l x() = n= -- Iw(n>I 03 cls dq 2n=- 2~ 2n f(q, n) w( - n) exp UQ], (3) which represents the signal as a linear cobination of shifted and odulated versions of the window. However, this linear cobination is not' unique [cf. [6], Section ; indeed, there are any kernels@, n), periodic in Q with period 27r, that satisfy the relationship 1 x() = 'i dq 2n=-co 2~ 2a Iw(n)l n=- (4) - p(fl, n) w( - n) exp ufl]. The representation (3), i.e., choosing the kernel p(q, n) in (4) equal to the sliding-window spectruf(q, n), is the best possible one in the sense that for this choice the L2- nor of p(q, n) takes its iniu value. To see this, we ultiply both sides of (3) and (4) by x*(), su up over all, and conclude fro the equivalence of the right-hand sides of the resulting equations that f (Q, n) and p(q, n) - f(q, n) are orthogonal in the sense 5 -!- 1 dfl {p(q, n) - f(q, n)) f*(q, n) = 0; n=-a 2~ 2a hence, the relationship = -!-. dq - If(fl, n)i2 n=- 21r 2* holds. It will be clear that the L2-nor of p(q, n), i.e., the left-hand side of (6), takes its iniu value if we choose the kernel p(q, n) equal to the sliding-window spectru f(q, n). We can reconstruct the signal fro its sliding-window spectru via the inversion forulas (2) or (3). However, in order to reconstruct the signal, we need not know the entire sliding-window spectru; it suffices to know its values at the points of a certain lattice in the fl-n doain. This will be shown in the next section. SIGNAL RECONSTRUCTION FROM ITS SAMPLED SLLDING-WINDOW SPECTRUM Let N be a positive integer, let the sliding-window spectruf(q, n) be known at the points {Q = k(27r/n), n = N) (k, = - *, -1, 0, 1, * * e), and let the values at these points be denoted by&,; hence, n=- Of course, the array of coefficientsh is periodic in'k with period N. Note that the sapling lattice ($2 = k(27r/n), n = N) is rectangular, and that the rectangular cells occupy an area of 27r in the tie-frequency doain. We shall now deonstrate how the signal can be found when weknow the values fk of the sapled sliding-window spectru (cf. [4, Section 4.21). We first define the functionf(n, w) by a Fourier series with coefficients fk, f(n, w) = =- k=(n) fk exp [-j(wn - k- N n where represents suation over one period N. Note that the functionf(n, w) is periodic in n and w, with periods Nand 27rlN, respectively. The'inverse relationship has the for wn - k N Furtherore, we define the function Z(n, w) by 2(n, w) = 00 = -w x(n f N) exp [-jwn]. (10) Note that the function X(n, w) is periodic in w, with period 2alN, and quasi-periodic in n, with quasi-period N: (8) X(n + N, w) = Z(n, w) exp IjwN]. (1 1 ) Equation (10) provides a eans of representing a one-diensional discrete-tie signal x(n) by a two-diensional tie-frequency function a(n, w) on a rectangle withjinite area 27r. The inverse relationship has the for n(n + N) = - do * Z(n, w) exp UwN]. (12) It will be clear that the variable n in (12) can be restricted

3 870 TRANSACTIONS IEEE ON ACOUSTICS. SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-33, NO. 4, AUGUST 1985 to an interval of length N, with taking on all integer values. With the help of the functions f(n, w), X(n, w) and a siilar w) associated with the window w(n), (7) can be rewritten as f(n, w) = NX(n, w). (13) In fact, we have now solved the proble of reconstructing the signal fro its sapled sliding-window spectru: 1) fro the saple values fk we deterine the function f(n, w) via (8); 2) fro the window w (n) we derive the associated w) by (10); 3) under the assuption that division w) is allowed, the function X(n, w) can be found with the help of (13); and 4) finally, the signal follows fro X(n, w) by eans of the inversion forula (12). A sipler reconstruction ethod will be derived in the next section. Probles ay arise in the case w) has zeros. In that case hoogeneous solutions h"(n, w) ay occur, for which the relation Nh"(n, w) = 0 (14) holds. Equation (14),whichis siilar to (13) with f(n, w) = 0, can be transfored into the relation which is siilar to (7) with fk = 0. Equation (15) shows that the sliding-window spectru of a hoogeneous solution h(n) vanishes at the sapling points (Q = k(2~/n), n = N f. We conclude that the existence of hoogeneous solutions akes the reconstruction of the signal fro its sapled sliding-window spectru nonunique: if x(n) is a possible reconstruction, then x(n) + h(n) is a possible reconstruction, too. RECONSTRUCTION VIA GABOR'S SIGNAL REPRESENTATION In the previous section we showed a way to reconstruct the signal fro its sapled sliding-window spectru; in this section we shall elaborate this a little further, and show a different way of signal reconstruction [cf. [4], Section We therefore introduce a reciprocal window sequence g(n), say, which is defined via its associated function g(n, w) by Ng(n, w) = 1. (16) On substituting fro (16) into (13) we get w, w) = f@, w)g(n, w), (17) which relationship can be transfored into =--Qi k=(n) by expressing f(n, w) in ters of fk via (8), expressing g(n, w) in ters of g(n) via (lo), and using the inversion relationship (12). Note the strong reseblance between (18) and the inversion forula (3). Equation (18), which expresses the signal as a cobination of properly shifted and odulated versions of the reciprocal window, is in the for of Gabor's signal representation [cf. [3], (1.29)]. Gabor's signal representation thus provides a way to express the signal directly in ters of the saple values of the sliding-window spectru. Equation (16) can be transfored into n= - g(n) w*(n - N) exp [-jkzn] 2T = 1 for k==o 0 elsewhere. (19) Fro (19) we conclude that the discrete set of shifted and odulated versions of the window, w(n - N) exp [jk(2n/n) n], and the corresponding set of versions of the reciprocal window, g(n - N) exp [jk(2a/n) n], are, in a certain sense, biorthonoral. Gabor's signal representation ay be nonunique in the case that g(n, w) has zeros. In that case functions Z(n, w) ay occur, for which the relation 2(n, w) g(n, w) = 0 (20) holds. Equation (20), which is siilar to (17) with T(n, w) = 0, can be transfored into the relation which is siilar to (18) with x(n) = 0. Equation (21) shows that certain arrays of nonzero coefficients in Gabor's signal representation ay yield a zero result. We conclude that Gabor's signal representation ay be nonunique: if the array of coefficients fk yields the signal x(n), then fk + zk yields the sae signal. It is easy to forulate Parseval's energy theore which follows directly fro (10) or (12). When we apply Parseval's energy theore to the reciprocal window g(n) and substitute fro (16), we get the relationship N c lg()i2 = c - = - n=<n) 2T Fro (23) we conclude that in the case w) has zeros, the reciprocal window ay not be quadratically suable. This consequence of the occurrence of zeros

4 BASTIAANS: SLIDING-WINDOW REPRESENTATION 871 w) is even worse than the fact that hoogeneous solutions ay be present; it ay cause a very bad convergence of Gabor s signal representation. We conclude this section with an. interpolation forula that enables us to express the sliding-window spectru f(q, n) directly in ters of its saple valuesfk. On substituting fro (18) into (l), we get indeed the relationship T - exp [ -jqn], (24) where we have introduced the interpolation jknction rn -. q(q, n) = g() w*( - n) exp [-jq], (25) =-a which is, in fact, the sliding-window spectru of the reciprocal window. SPECIAL CASES: MAXIMUM AND MINIMUM OVERLAP The cases of axiu and iniu overlap deserve special attention. Maxiu overlap occurs for N = 1: in that case there is axiu overlap between the window w (n) and its direct neighbors w (n N). In the axiuoverlap case, the forulas of the previous two sections can be siplified. Without loosing any inforation, we can take k = 0 in (7), (15), (18), (19), and (21), and take = 0 in (81,(91, (lo), (111,(121,(131, (W, (W, (1% (20), (22), and (23). Equation (7), for instance, then reduces to a siple correlation, h = x(n) w*(n - >, (26) n= -a and so do (15) and (19); note that, oreover, the coefficients fo becoe real when the signal x(n) and the window w (n) are real. Equation (18), on the other hand, reduces to a siple convolution, x(n> = f~g(n - ), (27) =- and so does (21). Furtherore, (8) and (9) then constitute EXAMPLES a noral Fourier transfor pair, and so do (10) and (12). Note that if in the axiu-overlap case the window w (n) To elucidate the concepts of this paper, we consider two vanishes for n # 0, then (7) or (26) tell us that the array siple exaples of window sequences, and deterine the of coefficientsfo is proportional to the signal x@), as can corresponding reciprocal window sequences for different be expected. values of the shifting distance N. Our first exaple is the Miniu overlap occurs when the window has afinite three-point, syetrical window [see Fig. ](a)] extent and the shifting distance N is chosen equal to this finite extent. In that case the forulas of the previous two sections again siplify drastically, and the relationship between the window w (n) and the reciprocal window g(n) takes the siple for If elsewhere; inside the extent of w(n) (29) lo outside the extent of w(n). I (b) 2 see Eq. (32) (dl Fig. 1. Sketches of (a) a three-point, syetrical window w(n), and its corresponding reciprocal window g(n) in the case of (b) axiu overlap, (c) iniu overlap, and (d) partial overlap. It will be clear that inside its finite extent,.the window w(n) should take nonzero values. Note that if in the iniu-overlap case the window is unifor inside its finite extent, and if the signal x(n) vanishes outside the extent of the window, then (7) tells us that the array of coefficients fko is proportional to the discrete Fourier transfor of the signal, as can be expected. note that for a = 0.16, we are dealing with a three-point Haing window. For the axiu-civerlap case (N = l), we find..

5 872 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-33, NO. 4, AUGUST 1985 *(O, 0) = 1 + a cos w, (30) and, hence, the reciprocal window [see Fig. I(b)] g(n) takes the for For the iniu-overlap case (N = 3) the reciprocal window becoes [see Fig. l(c)] - for n = 0 (b) Fig. 2. Sketches of (a) a one-sided, exponential window w(n), and (b) its corresponding reciprocal window g(n). for n = +1 0 elsewhere. For the case of partial overlap (N = 2) we find a \ *(I, w> = - (1 + exp ~2wl), (33) Our second exaple is the one-sided, exponential window [see Fig. 2(a)] exp [an] for n I 0 (a > 0) w(n) = (39) (0 for n > 0. In the interval -(N - 1) 5 n 5 0, the associated function N(n, w) takes the for 2 (34) 1 N(n, w) = exp [an] 1 - exp [-(a - jw)n] (40) (2g(O, w) = 1 and the function Ng(n, w) in this interval thus reads 2 1 2f(l, w) = - a 1 + exp [-j2w] (35) Ng(n, w) = exp [-an] (1 - exp [-(a + jw)n]). (41) and the reciprocal window now takes the for [see Fig. 1(d)l for = 0 for # 0 The reciprocal window g(n) now takes the for [see Fig. 2(b) 1;- exp [-an] for -(N - 1) I n 5 0 (36) Note that in the case of partial overlap, the function N(n, w)haszerosforw = a/2 + (r=, -1,0, 1, * e), and hence a hoogeneous solution h(n) arises. Its associated function &, w) is given by 240, w) = 0 QD 2q1, 0) = Rh c 6 w PR, (37) r= -CQ ( ; ) where 6( -) represents the Dirac delta function. The hoogeneous solution h(n) thus takes the for h(2) = 0 h(2 + 1) = (-1) h. (38) \O elsewhere. (42) We use this exaple to show the possible nonuniqueness of Gabor s signal representation. In the liiting case CY = 0, the function g(n, w) has zeros for w = r(2a/n) (r =..., - 1, 0, 1, e), and an array of coefficients Zk arises whose associated function Z(n, w) in the interval 0 I n 5 N - 1, say, is given by The array Zk,,, thus takes the for and yields a zero result when substituted in Gabor s signal representation. CONCLUSION In this paper we have studied the short-tie Fourier transfor of a discrete-tie signal, or, as we prefer to call

6 BASTIAANS: SLIDING-WINDOW REPRESENTATION 873 it, the sliding-window spectru. Thisliding-window spectru is a function of two variables: a discrete tie index, which represents the position of the window, and a continuous frequency variable. We have shown that the signal can be reconstructed fro the sliding-window spectru, when we know its values at the points of a certain tie-frequency lattice. This lattice is rectangular, and the rectangular cells occopy an area of 27r; hence, the coarser the sapling in tie, the finer the sapling in frequency, and vice versa. The ost elegant for to represent the signal in ters of the saple values of the sliding-window spectru, is by eans of Gabor s signal representation. We therefore had to introduce a reciprocal window, and we have shown how the window and the reciprocal window are related. Gabor s signal representation then expresses the signal as a superposition of properly shifted and odulated versions of the reciprocal window. REFERENCES [3] D. Gabor, Theory of counication, J. insf. Elec. Eng., vol. 93, part 111, pp , [4] M. J. Bastiaans, Signal description by eans of a local frequency spectru, in Transforations in Optical Signal Processing, W. T. Rhodes et al. Eds. Bellingha: Society of Photo-Optical Instruentation Engineers, Proc. SPIE, vol. 373, pp , [SI C. W. Helstro, An expansion of a signal in Gaussian eleentary signals, ieee Trans. Infor. Theory, vol. IT-12, pp , [6] N. G. de Brujin, A theory of generalized functions, with applications to Wigner distribution and Weyl correspondence, Nieuw Archiefvoor Wiskunde vol. 21, no. 3, pp , Martin J. Bastiaans was born in Helond, The Netherlands, in He received the M.Sc. degree in electrical engineering and the Ph.D. degree in technical sciences fro Eindhoven University of Technology, Eindhoven, The Netherlands, in 1969 and 1983, respectively. Since 1969 he has been an Assistant/Associate Professor with the Departent of Electrical Engineering, Eindhoven University of Technology, where he teaches electrical network theory. His research includes a syste-theoretical approach of all kinds of probles that arise in Fourier optics, such as partial coherence, coputer holography, iage processing and optical coputing; his current interest is in describing signals by eans of a local frequency spectru. Dr. Bastiaans is a eber of the Institute of Electrical and Electronics [l] L. R. Rabiner and R. W. Schafer, Digital Processing of Speech Signals. Englewood Cliffs, NJ: Prentice-Hall, 1978, chap. 6. [2] A. V. Oppenhei, Digital processing of speech, in Applications of Digital Signal Processing, A. V. Oppenhei,.. Ed. Englewood Cliffs, NJ: Prentke-Hall, 1978,-chap. 3. Engineers and of the Optical Society of Aerica.