1、106 IEEE SIGNAL PROCESSING LEITERS, VOL. 1, NO. 7, JULY 1994 A Multitime Definition of the Wigner Higher Order Distribution: L-Wi ner Distribution Abstruct-A dual form of the Wigner higher order spectra is introduced. Its analysis in the case of multicomponent signals is performed. An efficient dist
2、ribution for time-frequency signal analysis (L-Wigner distribution) is derived from that analysis. The theory is illustrated on a numerical example. I. INTRODUCTION IGHER order spectral analysis has been intensively H studied during last few years. Higher order statistics known as cumulants and thei
3、r Fourier transforms (FT), which are known as higher order spectra, are often considered, but we refer here only to the review paper l and the references therein. Recently, higher order time-varying spectra have been defined and analyzed 2. The basic representation in the time-varying higher order s
4、pectral analysis is the Wigner higher order spectra. Its definition, of order IC, of a complex deterministic signal x(t) is 2 s, J, . . s, x*(t - Wk(t,WI,&?,.,Uk) = E x nx(t - cy + ;)e-jdri i=l lk IC + 1 i=l with Q = - Ti. The Wigner higher order spectra, expressed in terms of the FT X(w) of signal
5、x(t), is 2 (2) We will introduce and analyze a distribution dual to (1) and (2). It will be referred to as the multitime Wigner higher order distribution (MTWHOD); it is defined by L-1 x n X*(w - A + 6i) i=l Manuscript received March 18, 1994; approved May 2, 1994. The associate editor coordinating
6、the review of this letter and approving it for publication was Prof. G. F. Boudreau-Bartels. The author is with Elektrotehnicki Fakultet, Podgorica, Montenegro, Yu- goslavia. IEEE Log Number 9403532. k k i=L i=l .k 1 with A = - 1 Oi + 1 (3) and 1 5 L 5 IC. Equation (3) is dual to (1) when L = 1. The
7、 MTWHOD in terms of x(t) dual to (2) is k All properties of the MTWHOD are just dual to the ones described in 2 and 3 for the Wigner higher order spectra. Our intention is to analyze the MTWHOD in the case of multicomponent signals. This analysis will serve as a basis for the derivation of a distrib
8、ution with interesting properties. 11. AUTOTERMS AND CROSSTERMS IN THE MTWHOD Consider a multicomponent signal formed as a sum of short duration (pulse) signals M m=l where xm(t) (m = 1,2, . . . , M) are such that xm(t) = 0 for It) E, with t being small as compared with the considered time interval.
9、 The integrand in (4) is different from zero only if the following inequalities are satisfied: k I - I where i,j,Z, = 1,2 ,., M; m = 1,2 ,., L - 1; n = In order to analyze the autoterms, consider the case M = 1 with dl = d, when the crossterms do not exist. Eliminating t, (q = 1, . . . , IC) from th
10、e first inequality in (6) and T from L,L+l, ., IC. 1070-9908/94$04.00 0 1994 IEEE Authorized licensed use limited to: IEEE Customer. Downloaded on April 17,2021 at 09:41:11 UTC from IEEE Xplore. Restrictions apply. STANKOVIC: MULTITIME DEFINITION OF THE WGNER DISTRIBUTION 107 A the remaining ones, w
11、e get (2L - k - l)d k+l - d 2T -tm+ T (2L- k- l)d k+l - d (2L - k - 1)d k+l 1, then for L = (k + 1)/2, considering only line s, the regions where the integrand in (4) is different from zero may be obtained from (6) with j, = 1, = j: (9) It follows from (9) that the components of the integrand in (4)
12、, corresponding to the crossterms, are dislocated from the r axis origin. They lie around r = di - dj, i, j = 1,2,. . . , M and i # j. From (9), one may conclude that the integration over autoterms (i = j) is completely performed, and at the same time, the crossterms on s are removed if we use the w
13、indow wL(t) in (4) of width T,(wL(T) = 0 for 171 2): (k + 1) 5 T, 1 and N = M(M + 1)/2 for IF = 1. The maximal number of terms in the L-Wigner distribution may be obtained from the recursive formula Nb = Nb-I(Nb-1 + 1)/2. The starting value is No = M, and the final one is N = N, for L = 2-. This for
14、m of L is used in the realization; see Section IV. For example, taking A4 = 4 and IC = 3 (L = 2), we get, for MTWHOD, N = 256, whereas for the L-Wigner distribution, N = 55. Authorized licensed use limited to: IEEE Customer. Downloaded on April 17,2021 at 09:41:11 UTC from IEEE Xplore. Restrictions
15、apply. IEEE SIGNAL PROCESSING LETERS, VOL. 1, NO. 7, JULY 1994 108 locations, we have shown in Section I1 that these may be eliminated in the case of nontime-overlapping pulses. In the section that follows, we will show that crossterm elimination is possible in the case of nonfrequency-overlapping l
16、ong signals (tones) as well. Expanding q5m(t f 7/2L) into a Taylor series around t up to the third order term, we get .M 1 LWDL(, t) = - 2.rr rz(t)6( - $h(t) m=l 8 WL(W) + CT(w, t) (14) where 8 denotes the frequency domain convolution 0 5 111,1721 5 I&/, and FTO is the lT operator. From (14), one ma
17、y conclude that the generalized power r$L (t ) is concentrated at the instantaneous frequencies & (t) . The distortions caused by the shape of the phase function are due to the existence of its third and higher order deriva- tives. If the instantaneous frequency is a linear function of time, then th
18、e Wigner distribution (L = 1) produces the ideal concentration. However, if that is not the case, then L 1 dramatically reduces the distortion. In other words, the L-Wigner distribution locally linearizes the instantaneous frequency function (some interesting results dealing with the polynomial phas
19、e function are reported in 8 and 9). Iv. A METHOD FOR THE L-WIGNER DISTRIBUTION REALIZATION The realization of L-Wigner distribution may be efficiently done using the recursive formula (15) Note that LWDPL(W, t) will be crossterm free provided that 1) the starting transform is crossterm free, and 2)
20、 the recursions do not introduce crossterms. We will show that these require- ments may be met under certain conditions. Let us prove that assumption 2) may hold. Suppose that LWDL(W, t) in (15) is crossterm free. Its autoterms are located around &(t), m = 1,2,. . . , M; see (14). The terms in LWDL(
21、W, t) are located along the B axis at 10 - &(t) - 4i(t)/21 W,) satisfies The modified L-Wigner distribution is The distortion due to the higher order-phase derivatives in (14) is neglected in this analysis. The starting iteration (2L = 1) is MWD(w, t) = - P(B)STFT(w + 0, t)STFT*(w - 0, t)dB (18) whe
22、re STFT(w, t) is the short-time Fourier transform defined as STFT(w, t) = FT,z(t + T)w(T). This way, the resulting modified L-Wigner distribution is crossterms free (terms denoted by CT(w,t) in (13) do not exist) if the starting transform STFT(w,t) is crossterm free (which is the case if the signal
23、components do not overlap in the time-frequency plane) and if, at the same time, (16) is satisfied in each iteration. Note that if (16) cannot be satisfied for some i,j, and t (i.e., I$;(t) - 4(t)(/2 - Ww Ww), then the crossterms will appear at that instant t between the ith and jth signal component
24、s. The discrete forms of (17) and (18), with a rectangular window P(B), are re J MWD(n,lc) = ISTFT(n,lc)I2 NP +2 ReSTFT(n,k+i)STFT*(n,k-i) i=l MLWDL(,)=MLWD(,) NP i=l +CMLWD(,+)MLWD(,-) (19) where 2Np + 1 is the width of the discrete form of P(0). In 4, it is shown that the realization of MWD(n, IC)
25、 may be computationally very efficient. Here, we will only indicate that the oversampling in the modified Winger distribution (18) is not necessary because the aliasing components are removed in the same way as the crossterms are3 4, lo. The same conclusions are valid for the L-Wigner distribution.
26、V. NUMERICAL EXAMPLE As a numerical example, consider the multicomponent sig- nal .(t) = ej8x(2-t)2 + ej12 sin3/2(t+l)-j20t + n(t). + 2e-25(t-0.3)2-j407rt The first two components are of form (12), whereas the second two are of form (5). The last component n(t) is a Gaussian white noise with the var
27、iance on = 0.3. The spectrogram of z(t) is given in Fig. 2(a). In the STFT calculation, a window W(T) = h2(.) (where h(7) is a Harming window) of the width 2Tm = 1 is used (the window selection in the STFT is very widely studied). The number of samples A sampled signal has a periodic spectrum. It ma
28、y be formally treated as a multicomponent signal with an infinite number of components equally spaced with the distance up = 2n/At, where At is the sampling interval. Note that the distance up. which is considered from the point of view of (16), is usually less demanding than the minimal distance be
29、tween the autocomponents in the original multicomponent signal. 1 - Authorized licensed use limited to: IEEE Customer. Downloaded on April 17,2021 at 09:41:11 UTC from IEEE Xplore. Restrictions apply. STANKOVI: MULTITIME DEFINITION OF THE WIGNER DISTRIBUTION 1 0 -1 n 4on Fig. 2. Time-frequency repre
30、sentation of a multicomponent signal: (a) Spectrogram, (b) Wigner distribution; (c) modified Wigner distribution, (d) modified L-Wigner distribution with L = 4. Sampling interval At = 1/64, rectangular window P(), and Hanning squared window W(T) are used. is N = 64. Note that the number of nonzero s
31、amples in the Fourier transform of W(T) is N = 2Nw + 1 = 5, with N, = 2 and W, = 47r. The Wigner distribution, which is calculated by the standard routines 4, is shown in Fig. 2(b). The modified Wigner distribution, which was obtained from the STFT and (19), is presented in Fig. 2(c). A rectangular
32、window P(B), whose width in the discrete domain is defined by Np = 3, is used in (19). In order to ensure the integration over autocomponents (W,L 5 W, in (16), a very narrow window P(8) is usually sufficient. In our example, with the described window w(T), Np = N, = 2 is sufficient, but we will tak
33、e a margin with Np = 3 because the distortion due to the higher order phase derivatives is not included in (16). The value Np = 3 i.e., W, = 67r in the analog domain, provides the complete elimination of crossterms (as well as the aliasing effects) between the components whose instantaneous frequenc
34、ies are more than 207r apart along the frequency axis. This holds for all considered components in the example (otherwise, the crossterms would appear but still in a very reduced form). The MTWHOD along line s, i.e., the modified L-Wigner distribution with L = 4, calculated with the same window P(0)
35、 and using (19) in two iterations (L = 2 and L = 4) is given in Fig. 2(d). The analysis of the crossterm elimination is the same as in case of the modified Wigner distribution. VI. CONCLUSION An efficient distribution (L-Wigner distribution) for time- frequency analysis is derived from the dual defi
36、nition of the Wigner higher order spectra. This distribution has the follow- ing advantages over the standard Wigner distribution: There is a very high distribution concentration at the instantaneous frequency, crossterms are removed (or reduced), and signal oversampling is not necessary REFERENCES
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