TV-NOS. Here we show that cumulant TV-HOS based on. PWVD can preserve important properties of higher-order statistics, for example, elimination of additive ...
TIME-VARYING HIGHER-ORDER CUMULANT SPECTRA: APPLICATION TO THE ANALYSIS OF COMPOSITE FM SIGNALS IN MULTIPLICATIVE AND ADDITIVE NOISE Boualem Boashash and Branko Ristic
Signal Processing Research Centre Queensland University of Technology G. P. 0. Box 2434, Brisbane 4001, AUSTRALIA 2. TV-HOS BASED ON POLYNOMIAL WIGNER-VILLE DISTRIBUTIONS
ABSTRACT Time-varying higher-order spectra (TV-HOS), have recently been proposed for (po1y)spectral analysis of non-stationary signals [l]. Two new results are presented here. First, the cumulant TV-HOS (in particular, cumulant Wigner-Ville trispectrum (WVT)), is shown t o preserve the essential properties of cumulant higher-order spectra (e.g. eliminate Gaussian additive noise) and at the same time characterise time-variations of the signal’s frequency content. Second, when dealing with composite FM signals, a special kind of “non-oscillating cross-terms” is shown to appear in the moment TV-HOS time-frequency subspace. We show that these cross-terms cannot be eliminated by smoothing the WVT, but. rather by an appropriate projection from the full time-multi-frequency space t o a time-frequency subspace. 1. INTRODUCTION
Time-varying higher-order spectra (TV-HOS) have been recently introduced for the analysis and representation of nonstationary signals. They were developed as a hybrid method which is both an extension of the concept of higher order spectra (HOS) for non-stationary signals and an extension of time-frequency distributions (TFDs) from second-order to higher orders. Fundamental concepts and properties of TV-HOS are reviewed in [l], and references given therein.
In this paper, we first briefly define moment and cumulant TV-HOS based on Polynomial Wigner-Vie distributions (PWVDs), so as t o put this work in perspective. Then we perform a comparative analysis of FM signals in multiplicative and additive noise, using moment and cumulant TV-NOS. Here we show that cumulant TV-HOS based on PWVD can preserve important properties of higher-order statistics, for example, elimination of additive Gaussian noise, and a t the same time are able t o characterise timevariatbns of the frequency contents of a signal. Finally, we address the problem of representing multi-component (or composite) signals. For clarity of presentation we focus on a particular member of TV-HOS, the Wigner-Ville trispectrum (WVT).
The Polynomial Wigner-Ville distribution. The question of optimally representing linear FM signals was resolved by defining time-frequency distributions based on the Wigner-Ville distribution (WVD). Then the WVD was extended by defining a Polynomial WVD (PWVD) [2] which can optimally represent non-linear polynomial FM signals, that is signals of the form: P
r ( t ) = n ~ (-t T/2)A exp{j
a,t’)
(1)
$53
where II,(t) is 1 for It1 5 T / 2 and zero elsewhere; A and a, are real constants. The Polynomial WVD, originally defined in [2], and generalised in [3], [l], is given by:
where k is an even integer which indicates the order of nonlinearity of the PWVD, and the coefficients c, and c-, are calculated so as to verify a number of conditions that the PWVD has to meet. For a given class of FM signals with polynomial phase order p, one can design a T F D which behaves optimally by simply choosing the coefficients c,’s and the order k. The choice of coefficients and order is described in [l] and [3].
Time-Varying Higher-Order Moment Spectra The PWVD was defined for finite energy deterministic signal z ( t ) . When dealing with random non-stationary signals, we define the Wigner-Ville polyspectrum (WVP) as the expected value of the PWVD of a random signal z ( t ) as’:
w!”(t,f) = E{W?)(t, f)}
(3)
‘To add clarity to the text, random quantities are shown in boldface letters.
IV-325 0-7803-1775-0/94 $3.00 0 1994 IEEE
~~
~. -
If we interchange expectation operator E with integration, the WVP becomes: WiL)(1, f) =
I {:I: +
E n z ( 1 c , r ) . z’(t
+c-,r)
1
where ~ ( l is) a complex zeremean white Gaussian noise (WGN) and
-
e-’2“’Tdr
(4) Since the RHS of (4) represents the F T of a time-varying k-th order moment function, W!”(t, f ) can be interpreted as a time-varying higher-order moment spectrum. In fact, the WVP are a sub-class of a more general class of moment based TV-HOS defined in [4], since the WVP was defined in the time-frequency subspace of the full timemulti-frequency space. This particular sub-class combines the advantages of classical time-frequmcy analysis and the advantages of higher order spectra [I].
N ( 0 ,u ~ )0 ; U [ - n , T ) . The assumption is with a(t) that w(t), a(t) and 0 are independent. The instantaneous ddeterministic but unknown. frequency (IF) f , ( t ) = f C + is
It was shown [5] that second order time-varying spectral methods, such as the spectrogram, WVD, or any other quadratic time-frequency distribution, are ineffective to r e p resent and analyse such signals. On the contrary, the WVT of this signal produces an impulse (delta function) centred about the FM law of this signal [7], [5]. This is explained below. The M-WVT of z i t ) can be shown to yield:
In this paper, we concentrate on the fourth order member of the WVP, referred to as the moment Wigner-Ville trispectrum (M-WVT) [5]: ~ j 4 = )
J,
my)(t,r ) e - J z n ’ r d r
(5)
where
Time-Varying Higher-Order C u n i u l a n t S p e c t r a A problem with higher order moments, and hence moment based HOS and TV-HOS, is that they never satisfy the superposition principle. On the other hand, the cuniulants verify the superposition principle of independent random variables [6]. We consider the particular case of the time-varying fourthorder cumulant funr:tion of a zero-mean random signal z ( t ) defined as [I]:
W!‘)(t,f) = W p ( t , f) + w&).w,w(t>f) + 4n$fi,,,w(t! f)+
+ w:?.,,,(t,f) + W L q t ,f)
where m!:!1z,.3,z,= E { z l ( t + i ) z z ( t + ~ ) z ; ( t - - ~ ) z ; ( t - f ) ) . If the autocovariance function of a(t) is given by R,(r) = vXe-2Al‘l, then we have [5]:
Wi4)(t,f) = ~ ~ X ~ ~ 6 [ f - f , ( t ) ] + 2 u ’ X
-
is the 4-th order time-varying moment function of a Gaussian signal with the same mean and autocorrelation function as z ( t ) 161. The cumulant WVT (C-WVT) c W ! ” ( t , J ) is hence defined as the F T of c y ) ( t , r )w.r.t. the r variable.
(A
-
-
m) (13)
Since the power of a(t), Pa, is finite and equal to R,(O) = wX, note that if X m, then w 0, and W:”(t,f) = U Z X Z ’ S[f - f , ( t ) ] . For z ( t ) given by (9), it can be shown that the cumulant WVT is given by: cW!’)(t, f ) = cW$’)(t,f)
where
(11)
where W i 4 ’ ( t , f ) and Wk4’(t,f) are the M-WVT of signal and noise respectively. Other terms are the cross M-WVT defined for 4-ordered input signals Z~,ZZ,ZJ,Z~ as:
+ c W c ) ( t , f)
(14)
since y ( t ) and w(t) are both zero-mean and mutually independent. Moreover, since w(t) is Gaussian (of arbitrary ( 1 , f ) = 0. On the other hand, spectral chara~teristic),cWk’~ cW’L4)(t,f ) # 0, since y ( t ) is non-Gaussian with pdf given in [7]. Using definition ( 7 ) , it can be shown that the CWVT of signal z ( t ) yields:
3. COMPARATIVE ANALYSIS OF A LINEAR
FM SIGNAL IN MULTIPLICATIVE AND ADDITIVE WHITE GAUSSIAN NOISE
c W ! ~ )f) ( ~=, [Ra(0)]’6[f - f i ( t ) ]
(15)
Again, for a(t) a real zero-mean Gaussian process with covariance R . ( r ) = vXe-2Alr1, we have [Ra(0)]’ = uZAz = const, and eq.(15) becomes:
cW!‘)(t, f) = W ~ X Z 6 [ /
-
f,(t)]
(16)
Note that if a(t) is Rayleigh distributed, then y ( t ) is a Gaussian process [7] and the C-WVT of z ( t ) is zero (see experimental results).
We consider the following signal model:
1V-326
In theory we have shown that a member of cumulant based TV-HOS, the cumulant WVT, preserves the essential properties of cumulant polyspectra (i.e. eliminating Gaussian additive noise, and the property of superposition for independent processes), and in addition has the property of characterising the time-varying frequency content of nonGaussian signals. Though we have illustrated this statement on the particular example of a linear FM signal in multiplicative and additive WGN, the statement can be shown t o be generally true, as illustrated in Sec5 where we present some experimental results with the C-WVT. 4. COMPOSITE S I G N A L S
constant (non-oscilloting) amplitude in the t-f plane and is expressed by the third term of the RHS of (19). Note that these cross-terms have 4 times greater amplitude than the auto-terms, with frequency contents in the middle of each pair of auto-terms. Since the amplitude of these cross terms is not oscillating, standard methods of smoothing do not affect them. Instead, it can be shown that [3]: If a FED signal has no overlap between its components in the time domain, then the M-WVT is free from constantamplitude (i.e. type 2) cross-terms. If a FED signal has no overlap between its components in the fmquencydomain, then the following form of the MWVT is free from type 2 cross-terms:
Wi')(t, 0)= Consider composite FM signals that can be modelled as follows:
*=1
where y,(t) = a,(t)exp{j[O, + 2 r l f f,(u)du]]
-
(18)
(where
,=I ,='+I
while the cumulant WVT is given by:
,=I
since all components y;(t) are mutually independent. Hence, the C-WVT has no cross-terms, as opposed to the M-WVT, which is affected by the crowterms given by the second and the third summand in (19). F i n i t e e n e r g y d e t e r m i n i s t i c ( F E D ) signals and the M-WVT 'The cross-terms are generally treated as undesirable in timefrequency (t-f) analysis. The cross-terms of the M-WVT for FED signals can be divided into 2 groups. The first group is characterised by the cross-terms with oscillatory amplitude in the t-f plane and is expressed by the second term of the RIIS of (19). These cross-terms can be suppressed by t-f smoothing of the M-WVT using methods equivalent to that of Choi-Williams or Zhao-Atlas-Marks [8],since they correspond to well-studied cross-terms generated by quadratic t-f methods. The second group of the cross-terms have
[Z(O + :)I2
[Z'(0 - ~ ) ] 2 e J 2 n u t d v(21)
Z(0)is the F T of t ( t ) ) .
For general FED composite FM signals with possible time and frequency overlap, one could initially perform an automatic segmentation of data[9] so that the problem is either reduced to the monocomponent case or to one of the two cases presented above.
-
where each y s ( t ) is an FM signal with random amplitude a,(t) N ( p a ua); , 0 , are R.V. 8 , U [ - r , r ) ;and a , and 8 , are all mutually independent. The M-WVT of zr;(t) can be expressed as:
1
5. E X P E R I M E N T A L R E S U L T S
Here we present the results of two experiments with the cumulant WVT (C-WVT), which illustrate and confirm the theoretical derivations presented above. The C-WVT is e s timated using 64 point multiple realizations of random signals in discrete-time form, generated by Monte-Carlo simulations. C h a r a c t e r i s a t i o n of time-varying f r e q u e n c y component: The signal is given by (9) and (10). The estimated M-WVT and C-WVT are shown in Fig.l(a) and (b) respecis equal t o 9 dB. tively. The SNR, defined as lolog &U: We observe that the M-WVT has a "background" above zero, while the C-WVT is zero everywhere except along the instantaneous frequency. The result is in agreement with eq.(ll) and (16). Discrimination b e t w e e n Gaussian and non-Gaussian non-stationary processes: Again the signal is given by (9) and (lo), but this time a(t) is a Rayleigh process, and hence both y(t) and w(t) are Gaussian [7]. The M-WVT and the C-WVT are shown in Fig.2(a) and (b) respectively. As expected, only the M-WVT shows the t-f content, while the C-WVT is zero everywhere. C o m p o s i t e F M signals. In this experiment we consider sum of two independent linear FM signals crossing each uther in the t-f plane. One component is a Gaussian process, and the other is non-Gaussian. The M-WVT and C-WVT of this random signal is shown in Fig.3(a) and (b) respectively. The M-WVT shows both signal components in the t-f plane (creating a shape X) and the type 2 cross-term between them. The C-WVT displays only the non-Gaussian component.
IV-327
6. CONCLUSION
(a) We have defined and applied time-varying higher-order cumulant and moment spectra to the analysis of FM signals affected by multiplicative and additive noise. It was shown that the cumulant Wigner-Ville trispectrum can preserve the essential properties of cumulant higher-order spectra (e.g. eliminates Gaussian additive noise) and at the same time is able to characterise time-variations of the signal frequency content. In dealing with composite independent random FM signals, the cumulant based Wigner-Ville trispectrum, as opposed t o the moment based, has no crossterms. The nature and behaviour of the moment WVT generated cross-terms were discussed.
non-zero amplitude
zero amplitude
Figure 1: T h e non-Gaussian linear FM signal in additive Gaussian noise
7. REFERENCES [I] B. Boashash and B. Ristich. Time-varying polyspectra. In B. Boashash, E. J. Powers, and A. M. Zoubir, editors, Higher Order Statistical Signal Processing. Longman Cheshire, 1994. (In press).
[a]
B. Boashash and P. J. O'Shea. Polynomial WignerVale distributions and their relationship to time-varying higher order spectra. IEEE Trans. on Signal Processing, Jan. 1994.
[3] B. Boashash and B. Ristich. Time-varying higher order spectra and multicomponent signals. In F. Luk, editor, Adu. Signal Proc. Algorithms, Architectures and Implementations, San Diego, July 1993. Proc. of SPIE. [4] J.R. Fonollosa and C.L.Nikias. Wigner-higher-order moment spectra: Definition, properties, computation and application to transient signal analysis. IEEE Trans. Signal Processing. 41(1):245-266, Jan. 1993. [5] B. Boashash and B. Ristich.
Analysis of FM signals affected by gaussian AM using the reduced WignerVille trispectrum. In Proc. of the Int. Conj. Acoustic, Speech and Signal Processing (ICASSP), Mineapolis, Minnesota, April, 1993.
Figure 2: T h e Gaussian linear FM signal in additive Gaussian noise
[6] C. L. Nikias and J. M. Mendel. Signal processing with higher order spectra. IEEE Signal Processing Mag., pages 10-37, July 1993. [7] B. Boashash and B. Ristich. Wigner-Ville trispectrum: Definition and application. In Proc. IEEE Signal Proc. Workshop on Higher- Order Statistics, pages 260-264, South Lake Tahoe, Ca., USA, June, 1993. [8] F. Hlawatsch and G. F. Boudreaux-Bartels. Linear and quadratic time-frequency signal representations. IEEE Signal Processing Mag., 9(2):21-67, April 1991. [9] B. Boashash. Time-frequency signal analysis. In S. Haykin, editor, Advances in Spectral Estimation and A r r a y Processing, vol.], pages 418-517. Prentice Hall, 1991.
,'.""EY
HI
Figure 3: Composite linear FM signal consisting of Gaussian and non-Gaussian component
IV-328
