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Statistics for complex variables and signals — Part II: signals
SIGNAL
PROCESSING ELSEVIER
Signal Processing
53
( 1996) 15-25
Statistics for complex variables and signals Part II: signals P.O. Amblard*, M. Gaeta, J.L. Lacoume CEPHAG-ENSIEG
URA CNRS Received
346, BP 46, 38402 S&t-Martin
27 January
d’Ht?res CCdex, France
1995; revised 28 March 1996
Abstract
In this paper, we study the higher-order statistics of complex stationary signals. In a first part, we define precisely the multicorrelations and multispectra of complex signals. Different properties of these tools are put in evidence for certain classes of signals, such as analytic, band limited or circular signals. We show for example that band limited complex signals are circular up to a certain order. We then give the definition for the practical case of discrete time and discrete frequency processes. Finally, some extension of linear filtering relations are provided. Zusammenfassung
In diesem Beitrag studieren wir Statistiken h6herer Ordnung von komplexen stationkren Signalen. In einem ersten Teil definieren wir pr;izise die Multikorrelationen und Multispektren von komplexen Signalen. Verschiedene Eigenschaften dieser Werkzeuge werden fir gewisse Klassen von Signalen in Augenschein genommen, wie z.B. analytische, bandbegrenzte oder zirkulare Signale. Wir zeigen z.B. daR bandbegrenzte komplexe Signale bis zu einer gewissen Ordnung zirkular sind. Wir geben sodann die Definition ftir den praktischen Fall von diskreten Zeit- und diskreten Frequenz-Prozessen. SchlieRlich werden einige Erweiterungen linearer Filterbeziehungen vorgestellt. R&urn&
L’objet de cet article est 1’Ctude des statistiques d’ordre supkrieur pour les signaux B valeurs complexes, non gaussiens, stationnaires. Dans une premikre partie, nous donnons des dkfinitions prkcises pour les multicorr&lations et les multispectres de signaux B valeurs complexes. Diffkrentes propri&&s de ces outils sont mises en kvidence pour certaines classes de signaux, comme par exemple les signaux analytiques, bande ktroite ou circulaires. Nous montrons par exemple que les signaux complexes g bande Ctroite sont circulaires jusqu’8 un certain ordre. Les definitions sont alors &endues aux cas pratiques des signaux g temps et frtquence discrets. Quelques gCnCralisations simples des relations de filtrage 1inCaire sont enfin propos6es. Keywords:
Complex random signals; Multicorrelations;
* Corresponding
author. Tel: (33)-76-82-71-06;
Multispectra;
fax: (33).76-82-63-84;
0165-I 684/96/$15.00 @ 1996 Elsevier Science B.V. All rights reserved PII SO1 65- 1684(96)00072-2
Circular signals; Band limited signals
e-mail: [email protected].
16
P. 0. Amblurd
rf al. / Signul Processing
1, Introduction In [l], higher-order statistics for complex valued random variables have been presented. The logical extension to this work concerns the development of higher-order tools for complex valued signals. The most important features concerning higher-order statistics for complex quantities have been shown in [l]: a variable and its complex conjugate may give different statistical information. Therefore, a theory of higher-order statistics for complex variables must include higher-order cross-statistics between the variable and its complex conjugate. For signals, in the real case, the theory of higherorder statistics is now well-known. The first work goes back to 1953 with the book by Blanc-Lapierre and Fortet [4]. In this work, the tools were presented in terms of moments. In the beginning of the 1960s the cumulants were introduced in the theory by Shirayev [26]. Later, Brillinger and Rosenblatt and others made considerable efforts in the theory of higherorder ‘spectra’, especially in the domain of estimation. Thus, for real valued signals, the theory was almost complete at the end of the 1960s. We had to wait until the beginning of the 1980s for the Signal Processing community to discover (or rediscover) higher-order statistics and their interesting features. In the past ten years, many applications have grown around higher-order statistics. They are as different as blind deconvolution or equalization, separation of sources, detection of quadratic phase coupling, Volterra filtering, etc. In several fields of Signal Processing, from communication to array processing, many problems use complex signals in modeling and their solution requires higher-order statistics. However, no general theory concerning higher-order statistics for complex signals is available. Some work has been done [ 19,231 but to our knowledge, a unifiying view of higher-order statistics for complex signals does not exist. The aim of this paper is to present a theory of higherorder tools for complex signals. These tools will be multicorrelations and multispectra for stationary signals and may be higher-order time-frequency distributions for nonstationary signals. We however restrict the scope of the paper to the case of stationary signals and we try to highlight some features specific to complex signals.
53 (1996)
15-25
The organization of the paper is as follows. After the definition of some general objects, we present the definitions of the multicorrelations and the multispectra of a signal, insisting on the notion of stationary manifold. Section 4 presents the traditional debate ‘moments or cumulants’. We use an example to present the notion of Gaussian manifold and to plead in favor of the cumulants. We then examine particular signals (Section 5) such as analytic, circular or band limited signals for which higher-order tools have an interesting form. Then, Section 6 introduces higher-order cross statistics between two complex valued signals. These tools find an application in the derivation of input-output relations in linear filtering.
2. General definitions This first section gives precise and general definitions for the higher-order tools for complex valued random signals. We begin in the time domain and then examine equivalent definitions in the frequency domain. But before, let us introduce classes f@) and CD(‘) of Blanc-Lapierre and Fortet [4]. A signal X(v) is in @ck) ifV1 < k, ~‘E,,...,EI
J J .‘.
IE[dX,,(Vl)“~dX,,(Vl)ll
<
c < +m
(1)
where the &I’S correspond to a complex conjugated term if their value is - 1 and to a nonconjugated term if their value is + 1. Now, a signal x(t) is in fCk) if its Fourier transform belongs to @ck). The principal result we use here is then ([4, p. 4271) the following. If x(t) E ftk) and if k = l(Z - 1), then
= S-_
Jexp[2in(C
Eiti)Vi)]dX,,(vI).
” dx,,Cvj)
(2) exists in the quadratic mean sense. As a consequence, E[x,, (tl ) . .x,! (tj)] exists and satisfies
X
-f3d&f(Vl>. . d&,(Vj)l.
(3)
17
P. 0. Amblurd et al. / Signal Processing 53 (1994) 15-25
This result will be useful since the existence of higherorder tools for stationary and non-stationary signals relies on it.
Remark 1. The class f@) is the class of signals which belong to fck) for all k.
2.2. Symmetric multispectra Since the signal belongs to the class f’(‘“) of BlancLapierre and Fortet, it admits the Cramer representation x(t) =
exp(2irrvt)dX(
results were historically given in terms of moments. However, since the cumulants are nonlinear combinations of moments of lower or equal order, these results are also true when given in terms of cumulants. 2.1. hfulticorrelutions
(5)
Inserting this representation in (4), and making use of the multilinearity of the cumulant operator leads to Cx,p+y.p(t) =
Let x(t) be a complex valued random signal which of order p + q is belongs to fcoo). Its multicorrelation defined by’
v).
J’
Remark 2. The previous
J’
Cum[dX( vo), . . . , dx(v,-I
>M”(Q,.
(6)
Cx,p+‘,.p(r) =
Cumb(f0>,
“,
. . ,x(t,-~
),x*(f,),
,x*(t,+,-I
>I, (4)
where Cum[.] is the cumulant operator and where t = (to,. , t,+,_l ). Note that in this definition, q conjugated components and p nonconjugated components appear. The notation is then clear: p + q is the order of the multicorrelation whereas p represents the number of nonconjugated components. Note that as discussed in the companion paper [2], we have to include in the definitions all possible forms of (p + q)th order cumulants with distinct selections and conjugations in order to alofl= (to,...,t,+,_,) low the complete extraction of all the statistical information contained in the signal. Of course, this implies that, at a fixed order p + q, some of the multicorrelations may be redundant (relations of conjugation). The properties of the multicorrelation are direct consequences of the properties of the cumulants. It is multilinear, identically zero for p + q > 2 if the signal is Gaussian. Furthermore, if the signal is white in the strict sense (the random variables X(Q), . . , ~(t~+~_ I ) are statistically independent for all p + q), the multicorrelation vanishes except when the instants tj are all equal. We now turn to the definitions in the frequency domain.
Furthermore, if we assume that C.X.p+q.p(.t)possesses the following Fourier transform:
CX,p+q,p(f) =
.i’
&,+,,(v)
(7) where v = (~0,. . . , vp+_ i ), we obtain the definition of the symmetric multispectrum of order p + q: Cum[dX(vo),
,dX(~,_~),dX*(v,),. ,
M*(Q+~-I >I= G,p+,,,(vW.
(8)
These definitions are general and valid for all signals in f(“). However, their use is difficult, and we now assume that the signals under study are stationary.
3. Multicorrelations and multispectra for stationary
signals A signal is stationary if the statistics of all the random vectors it induces are invariant under a time shift.
P. 0. Amblard et al. /Signal
18
More precisely, let x(t) be a complex valued random signal. Let xn = (x(tt ), . . . ,x(t,)) be an n-dimensional induced vector. Then the signal is said to be stationary of order k if the probability density function of X, is invariant under a time shift for all n
of order p + q, the station-
Cx,p+q,p(t + rl) = Cx,p+q,p(t)
(9)
for all (p, q) such that p fq ,< PIand all lags Z. This last equation shows that the multicorrelation is no longer p + q-dimensional but indeed ( p+q - 1 )-dimensional. This is true since this last equation shows that the differential of Cx,p+4,p(t) is zero along a certain direction. Consider the following change of variables: Vi = 1,...,p1 Vi = p, . . , p + q - 1 I
Zi = tj - to, Zi = -tj + to.
x*(t - 2,),
+ r~ ), . . At
(11)
Remark. The particular role given to t shows that the term x(t) cannot be conjugated in our definition. This may be taken into account by defining
x(t
+ Zp-_I),X*(t
-
Tp) ,..‘)
+
71),
x*(t
=
Cx,p+q,q+1(-71,
which shows the redundancy
x exp (-2irt
-
vjtj ‘g\jtj))
fg
dt.
(14)
Making the change of variables ( 10) under the integral shows that
c&P+qsP
o=JJ Cx,p+q,p(r) exp(-2inrTv’) v
Pfq- 1
P-1 ~0 +
C
vj
/=I
-
C
vj
j=P
(15)
sx,p+q,p(vv =J
7p+q-l)J.
(7) exp( -2ircrTv’)dr cr>,+,,
(16)
cx,P+q,P(v)
= (
P-1 Sx,p+q,p(v’)6 VO+CVj-
(12)
(1 1 ), since (13)
of the second definition.
j=l
p+4-'
C i=P
Vj
1 .
(17)
This relation means that if the analyzed signal is stationary, the symmetric multispectrum takes nonzero values only on the (p + q - 1)-dimensional manifold defined by P--l
Ph-
V()+CVj-'2 j=l
But this form may be related to definition C:;+&)
s
leads to
+ zp-l),
. . ,x*(t - zp+q_l )].
= Cum[x*(t),x(t
We have seen before that the multicorrelation and the symmetric multispectrum may be related via a particular multidimensional Fourier transform. More precisely
where v’ = (VI,. . . , vpfq_l ). Letting
Let us point out that for a nonconjugated term, the lag is added, whereas it is substracted for a conjugated term. This is done to extend the well-known WienerKhintchine theorem to higher orders. This will appear later.
C:,,+,,(r)
3.2. Symmetric multispectra and stationarity: multispectrum
(10)
In this change, to plays a particular role, and we will denote it now by t. The multicorrelation is then a function of only t = (71,. . , z~+~__I) and reads c X,P+q,P(r) = Cum[x(t),x(t
15-25
cx,p+q,p(v) = C*,p+q,p(Q
3.1. Multicorrelations and stationarity For the multicorrelation arity property leads to
Processing 53 (1996)
1
Vj=O
(18)
i=p
and called the stationary manifold [4]. The value of the symmetric multispectrum on this manifold is then s X,p+q,p(v’). We will call this function multispectrum of order p + q of the signal x(t). Note that, like the multicorrelation, the multispectrum also depends on p + q - I frequency variables.
P. 0. Amblurd et al. / Siynal Processing
We now make use of (8) to obtain an explicit expression of the multispectrum. Let us recall that
. ,dX(v,_~),dX*(v,),. . ,
Cum[dX(vo),.
dX*@,+,-I )I = &,+,,p(v)dv.
(19)
Then, using (17) leads to Cum[dX(vo) ,..., dX(v,_,),dX*(v,)
,....
cM*( I’p+y- I )I
.I- I
/
I’+Y-l \
53 11996) 15-25
19
The physical interpretation of the multipectrum is then clear. It examines the statistical interactions between the p + 4 - 1 frequencies (VI,. . . , v.+,._ 1) and the v,. Thus, a frequency vp+q = l Cy?,’ vJ + Cpki’ stochastic process whose frequency ;d+, comes from the interaction between the p + q - I others will be perfectly analyzed by the multispectrum 2 S,,+,,,(v). Let us conclude this section by noting that the theory developed above allows the conservation (or the extension) of the Wiener-Khintchine theorem. At the second order, it is well-known that the power spectrum is the Fourier transform of the correlation function. We have constructed the tools here to respect this one-to-one correspondence again.
Fundamental remark. We use here the Cramer representation since a stochastic process cannot be represented as a standard Fourier integral (even in the mean square sense). But for physical situations, it is hard to work on the spectral process U(v). Furthermore, in practical applications, we observe the signals of interest on a short duration. But, the realizations of a time limited stochastic process have a Fourier transform. Hence, to understand clearly what is the physical meaning of multispectra, we now replace the harmonic increments U(v) by the Fourier transform of x(t) that we also write X(v) ’ . The reader must be aware that all the following is subject to this remark. Then, if X(v) is the Fourier transform of interest, the last equation reduces to Cum[X(va) . . . . . X(v,_r),X*(v,) / 7
+
VO = ~L.p+q.p(v’)~
,..., X*(vpfq_r)]dv P+4-I
P-1
C
Vj -
J=I
C
\ Vj
dv,
(21)
i=p
and therefore Sx,p+q,p(v’)
r
/
p-1
p+4-I
\
’ We symbolically write here U(v) = X(v)dv to mean that the dX(v) are the harmonic increments of X(\s).
4. Cumulants or moments We have defined the multicorrelations and multispectra in terms of cumulants. But the preceding definitions may have been given in terms of moments. The question is then why do we use cumulants’? What are the advantages of the cumulants over the moments? A rapid answer is provided by the arguments given in the companion paper [2]: 1. Cumulants provide a measure of independence. 2. Cumulants of order greater than two are zero for Gaussian signals. 3. Cumulants are cumulant: the cumulant of the sum of two independent signals equals the sum of their respective cumulants. These properties are the standard arguments which make the community prefer the cumulants to the moments. 3 We now present another advantage which is specific to signals. This advantage relies on the socalled Gaussian manifolds. For the sake of clarity, we describe these manifolds on the fourth order with an equal number of conjugated terms and nonconjugated terms.
2 In the sequel, the vector V’ is simply denoted by 11. ’ Note however that moments are used for deterministic signals. In particular, the ouptput multicorrelation, in term of cumulant, of a finite impulse response filter driven by a pure white noise is directly proportional to the multicorrelation, in term of moment, of the impulse response of the filter.
P. 0. Amblard et al. / Signal Processing 53 (1996)
20
We recall here the cumulant-based definitions, and give the equivalent definitions for the multicorrelations and spectra based on moments. They read cx,4,2(z) =
Cum[x(t),x(t
+ r]),.~*(t - rz),~*(t
-
z3)],
cA4~,4,2(2) =
E[x(t)x(t
+
q)x*(t
-
T2)x*(t
-
z3)],
(23)
15-25
are called Gaussian manifolds [25]. They belong to the stationary manifold and represent the domains of existence of SM,,x,4,2(v)in the zero mean Gaussian case. But they also appear in non-Gaussian cases. To explain this, let us consider the example of a zero mean non-Gaussian white stochastic process, of symmetric probability density function. Furthermore, suppose that the second and fourth-order statistics are given by
&4,2(v)
=
Cum[X(-h
+v2
+
v3),x(vl
),~*(v2),~*(v3)1,
h’,x,4,2(v) = ‘%f(-v]
+
“2 +
v3)x(vl)x*(v2)x*(v3)1,
=
=
a2w,
G,2,2(7)
=
Y2W,
&2,1(v)
=
02,
&,2,2(v)
=
Y2,
cx,4,2(z)
=
K4&7l,
&,4,2(v)
=
K4.
(27)
where we have added the subscript M to mean moment-based. Suppose that x(t) is zero mean stationary signal. We can write the moment-based multispectrum as a function of the cumulant-based multispectrum via &,x,4,2(v)
G,Z,l(~)
T2,73 ),
Using the explicit values of the statistics involved, we obtain for the moment-based multispectrum
&4,2(v) &4,x,4,2(v)
+E[X(-vl
+
v2 +
v3)x(vI
+E[X(-vl
+
v2 +
v3)X*(v2)I@X(vl
+EK(-VI
+ ‘9 +
=
K4 + a4@(v3 +02y*26(v2
)x*(v3)1 whereas
v3)X*(v3)IE[~(vl)X*(v2)1.
The second-order moments that appear in the last equation are of the type &Q(V) and &,2,](v); they are respectively equal to E[X(-v)X(v)] and E[X(v)X*(v)]. Thus, the terms of the first line of the last expression are identically zero if v] = -(-v] + v2 + ~2) and v3 = -vz. These two conditions are identical and equivalent to v?; + v3 = 0. This is a direct application of the concept of stationary manifold. Reasoning in the same way for the remaining terms leads to = &,4,2(v)
+ &2,2(vl
)$2,2(v2
)6(v2
-t&,2,1
(v2 )&,2,1 (v3 )&v3
-
VI )
+&,2,t
(v3 )&,2,1 (v1 P(v2
-
VI).
+ v3)
(25)
The manifolds Q-v] V3_V] v2 +
=o, =o, v3 =
(26) 0
vl)
+ &v2
-
vl))
the
+
v3),
(28)
cumulant-based multispectrum reads This shows that the moment-based multispectrum, SM,~,~,~(V) equals ~4 on all the stationary manifold, except on the Gaussian manifolds where pikes of magnitude cr4 or o2y*2 appear. On the opposite side, the cumulant-based multispectrum is completly flat, and reflects the whiteness of the analyzed signal. This result comes from the fact that the fourth-order cumulants measure the fourth-order statistical interaction when the second-order cumulants have been cancelled. In a general manner, an nth-order cumulant measures the nth-order interaction when the interactions of lower order have been eliminated. This principle is equivalent to the methods of orthogonalization of polynoms. At the second order, this is well-known: a pike in the spectrum appears at the zero frequency if the spectrum is moment based and if the signal is not zero mean. If the spectrum is constructed on the covariance (second-order cumulant), this pike does not exist. Therefore, these results provide in our opinion another great advantage of the cumulants over the moments, since they cannot present pikes on the $4,2(v)
(24)
&,x,4,2(v)
-
)]-‘%f*(v2)X*(v3)1
=
7~4.
P. 0. Amblard
et al. / Signal Processiny
Gaussian manifolds. This claim may take all its importance in estimation contexts.
5. Particular signals We examine in this section the multicorrelations and/or multispectra for some special signals of interest in different fields of Signal Processing, from communication to array processing.
A signal x(t) is said to be analytic4 negative frequencies, i.e.
if it has no
if v < 0.
(29)
If we examine (22), the multispectrum non zero if the intersection Vi =
1,...,p+q-
1
It was shown in the first paper that if a random vector is circular of order n, then the statistics of lower or equal order, containing a number of conjugated terms different from that of nonconjugated terms are zero. Therefore, this characterization of circularity suffices to proove that, for circular signals of order n,
Cx,p+q.p(r)= 0 ‘Jq,P
S,,,+,,,(v)
is
Vi > 0, (30)
v, > 0
is not empty. But, the case q = 0, which corresponds to a definition with no complex conjugate, leads the preceding system to
,I-
_
1
>
(32) by a pure jkquency
Let x(t) be a signal satisfying x(l) = z(t) x exp(2irv$), z(t) being stationary. Note that x(t) is not stationary. To see this, suppose that E[z(t)] = m # 0. Then E[x(t)] = mexp(2i7tvot) which does depend on time. Therefore, x(t) is not stationary, but here at least cyclostationary of order 1 (its mean is periodic). Thus, we a priori cannot define multispectra of x(t). However, let us evaluate &p+q,p(v), even if it has no sense. 5 In the frequency domain, the definition of x( t ) is X(v) = Z(v - VO),and then a multispectrum of x(t) is written as a function of a multispectrum of z(t) via
sx,p+q,p(v) ~-~ViJffj’vi)
-hl)
\‘j > 0,
3
i=p
I
L7 v,
that
ptqdnandqfp.
= Cum [ Z ( Vi = 1,...,p+q-
such
.3. Signals modulated
5. I. Anulytic siynuls
X(v) = 0
21
53 11996) IS-25
Z(Vl
(31)
-
vo),...,Z(vp-I
-
vo),
0,
i= I
zyv,
which obviously has no solution. Therefore, in the case of an analytic signal, multispectra of the type S,,,,(v) are identically zero.
-
VO),...,Z*(~~p+q-l
-
.
vo)
(33)
I But, &,+,,, (v) is zero if v escapes from the stationary manifold. Thus, this multispectrum of x(t) is not zero if
5.2. Circular random signals In the companion paper, the notion of circular random vector was introduced. This notion may be extended to random signals in the following way [24]. We say that a signal is circular if the induced random vectors of all orders are circular. This makes it possible to define circularity of order n: a signal is circular of order n if the induced vectors of order lower or equal to IZare circular. 4 For a discussion
on analytic
signals,
see [27]
(-~v~+pq
p+a-
P-l
=-
-v.
i=p
!=I
C(Vi i=l
1’0)+
.C
I
(vi - V(j)
(34)
i=p
‘This is of no sense in the context of this paper. However, evaluating this multispectrum is full of sense when working, for example, with cyclostationary signals.
P. 0. Amblard et al. /Signal
22
(35)
4 = P.
Therefore, S,,,,,, (v) is not zero if it has an equal number of conjugated terms and nonconjugated terms. Furthermore = &zp,p(v
-
vol).
(36)
Hence, for stationary signals modulated by pure frequencies, the notion of nonstationarity has to be moderated, since statistics of the type SX,~P,P(v) are stationary. 5.4. Band limited signals Let x(t) be a band limited signal, i.e. a signal obeying X(v) = 0
15-2.5
6. Crossmulticorrelations and spectra
or
Sx,2p,p(v)
Processing 53 (1996)
if Jv i VO]> Av.
5.5. Some remarks Gaussian analytic signal is stictly circular (i.e. circular at all orders). The only nonzero multispectra of a Gaussian signal are &,2,0(v), &,2,1(v) and ,&,2(v). But, if the signal is also analytic, only &,JJ (v) is nonzero, which proves the assertion. A (2) band limited signal is circular of order N = [vo/Av]. This immediately comes from the preceding sections. Up to this point, we have examined the higherorder ‘auto’-statistics for random signals. We now give some ideas to extend the theory to the case of higher-order ‘cross’-statistics. 6 [,I stands for integer part of.
cx,y,p+q,e(r) = CumMtk, (t + EI~I>,. . . , XE,,_,(t+Ep--1+-l),
(37)
These signals are important in array processing since the quasi-monochromatic assumption is often made. We show in Appendix A the following result: For all orders lower than6 N = [vo/Av], the only nonzero multispectra are of the type &,J~,~(v). This result is the extension to the complex case of Theorem 1 of [26, p. 2741 proved by Shirayev. An analog of this result is proposed in [12] where the authors show that the moment-based multispectra with one or no complex conjugate equal zero.
(1) A
In this section, we enter the study of the statistics of interaction between random signals. It is possible to provide a general theory, i.e. giving tools that examine the interaction between 12different signals. However, this complicates the formalism and is straightforward when the definitions for two signals have been given. Therefore, we provide here the study of the interaction between two random signals x(t) and y(t). Moreover, we only give definitions; the derivations are made as before. Let x(t) and y(t) be two signals off(“). The crossmulticorrelation between x(t) and y(t) of order p + q is
Y&
+ EpZp),.. . >
YE,,+y_, (t + Ep+q- I zp+q-
I119
(38)
where the &i’s equal +l for a nonconjugated term and - 1 for a conjugated one, and where x+1(t) = x(t) and x-l(t) =x*(t). In the notations concerning the ‘auto’-statistics, the information relative to the position of the conjugated terms appeared in p whereas here this information is contained in vector E. Furthermore, note that x(t) is again taken as a reference and then the order x, y in the notation is of importance. The crossmultispectrum of order p + q is defined as the Fourier transform of the crossmulticorrelation or explicitly (z) sx,Y,P+q,E(v) = j Gv,,+%~
exp(-2ixzTv))dz.
(39)
It is again possible to show that
sx,y,p+q,E(v)
= Cum
[x
&, (VI ),
(-
9..
p$‘Ejvj)2
.
,x,,>_, (VP-1
h
y&$J>, 7.. .) YE,+,_,(vp+q-l>
1 .
(40)
This provides a clear physical interpretation of the ‘cross’-statistics. They examine the links between q
P. 0. Amblurd et al. / Signal Processing 53 (1996) IS--25
frequencies of y(t), p - 1 of x(t) and the particular frequency of x(t) which defines the stationary manifold. Remark 1. In the real case, the definitions Cx,y.p+y(z)= Cum[x(l),x(t Y(t+
become
+ TI 1,. . . ,x(t + T~~-I1,
Tph.. y(t + ?
23
I
~,,(~~p),...,~E,,+,~,(vp+q-l) The cumulant operator being multilinear, ing equation turns into
the preced-
Tp+q-I 113
(41) ffc,,_,(Vp-I
The same reasoning
Kr
. , p+q,Av).
(45)
leads to
X(v,-l),~~~~,~,...,~~~~,+,-l~ I
Remark 2. Letting
H(v~-I )H*(l’p)...H*(~~+q--)S,,,+q,p(v>. (46)
& = (l,..., 1, -l,..., -1) -p-l 4
Remark. are
we then have Gr,p+y,t(r)
=
relations in the the real case
/P+4--l
= Cr.p+q,p(r), (42)
SI.l.p~-qJv)
The equivalent
\
Sw,p+q(v)= H*
s,.,+q.p(v>.
The interest of ‘cross’-statistics especially appears when dealing with the transformation of a signal into another. Of particular importance are linear filters. We thus provide input-output relations for higher-order statistics. Consider a linear, time invariant system, of impulse response h(t) and of complex gain H(v), driven by a signal x(t) off(=). We denote the output by y(t). Therefore, we have
ff(~y-
I )&,p+q(v),
(47) S v,p+q(~)
=
H’ \ H(~p+q-I
I==’
/ )&p+q(v>.
7. Summary and discussion y(t)
=
I
h(t - r)x(T)dz, (43)
Y(v) = H(v)X(v). Let us then examine7 (43) leads to &.x,p
=
Sy,X,p+q,E(v). Using (40) and
+y,s(V>
ChtIl!,,i
‘Note
-
‘$‘EjVj)X(
-
‘E’EjVj),
that y(t) is taken here as the reference.
The aim of this part was the introduction of precise definitions of higher-order tools for complex valued signals. Based on the comments of [2], we had to include in the definitions a certain number of conjugated and nonconjugated terms, in order to extract from the signal all the statistical information. Moreover, we have performed the construction of multicorrelations and multispectra in order to extend the Wiener-Khintchine theorem to higher orders. This implies a particular choice of the sign of the lags in the definition of the multicorrelation. But let us point out that this is our choice, and as such this is arbitrary.
P. 0. Amblard et al. /Signal
24
Two other important points have been presented: the known stationary and Gaussian manifolds. We insist on the fact that Gaussian manifolds are in our opinion the basis of arguments to prefer cumulants to moments. In this paper, we did not present any application. However, we showed several results concerning the higher-order statistics of useful types of signals. For example, we have shown that band limited signals which are often used in array processing are circular up to an order which depends on the characteristics of the band. Finally, the problem of input-output relations in linear filtering was rapidly discussed, but we want to insist on the fact that in this problem, more general theories may be developed (for example multi-input, multi-output relations). However, we claim that such theories are straightforward extensions of ours, and essentially rely on a complication of notations. The next extension of this theoretical work concerns the definition of tools for complex valued nonstationary signals. This work has been done for the third-order case in [3], and for the general case in [ 11. The results concerning nonstationary signals will be considered in another paper.
Processing 53 (1996)
-vo-Av
vo -
AV < I- CztA1 vi + Cp_fp4-’ vi) < vo + Av. Let 11,
12,kl and k2 be four integers such that 11+ 12 = p - 1 and kl + kz = q, Consider then the following p +q - 1 frequencies: Vi=
~o-Av
-VO+AV
p+kl
+ I,...,p+q-
D-1
Vi = II + l,...,p-
i=l <(q
~o-Av
+ p -
1 - 2(11 + k2))vo + (p + q - l)Av,
(-4.2) (1)
Suppose that the order is odd, or q + p = 2n + 1. Then (A.2) becomes p-1
2(n - (I1 + k2))vo - 2nAv<
-C
p+4-
1
vi + C i=l
vi
i=p
<2(n - (I, + k2))vo + 2nAv.
(A.3)
Moreover, suppose that n - (11 + k2) > 1. Therefore 2(n - (11 + k2))vo 22~0. But, the order is chosen such that 2n + 1~ vo/Av, and thus ve 2 (2n + 1)Av. Then 2(n - (II + k2))vo + vo >2vo + (2n + 1)Av (-4.4)
(2)
or 2(n - (I, + k2))vo - 2nAvavo + Av. This expression is, accordin to (A.3), in contradiction f with VO-AV< -CltY vi+Cp_f,4-’ vi
1 -2(Z, p--l
+kz))vo-(2p+n-
+(2p+n-
l)Av
Pf9-1 C
<(2p+nVi=p,...,p+kl,
y, i=p
i=l
1,
the first
D+a- 1
-&+‘E
6
1,...,11,
-VO+AV
(A.1)
(q + p - 1 - 2(11 + k2))vo - (p + q - l)Av
<-Cvi+
-vo-Av
1.
Summing the last two lines and sustracting two leads to
Appendix A. Band-limited signals are circular In this first section, we prove the following result: Let x(t) be a band limited signal, i.e. X(v) = 0 if Iv i v. 1 > Av; then for all orders lower than N = [vo/Av], the only nonnull multispectra are of the type Sx,2p,p(v). Consider multispectra of order p + q d vo/Av. Such a multispectrum is nonzero if vo - Av< [vi]
15-25
Vi
i=p
1 -2(Z1 +kz))vo 1)Av.
(A.5)
P. 0. Amblard
et al. / Signal Processing
But the order is such that vg >,(2p + n)Av, and supposing again that p - (11 + kz) 3 1, it follows 2( p - (2, + k2))vo - (2~ + n - l>Av 3vo + Av which immediately
(A.6) implies, since n > 1,
(2~ -t n - 1 - 2(1, + kz))vo - (2~ + y1- 1)A 3 vo + Av.
(A.7)
This is again in contradiction with v. - Av< - CL<’ vi + C!‘+J-’ vi < vo + Av, and thus p = (Remark. If n’< 1, the roles of p and q are exchanged). (3) The proof is made in the same way to examine the contradictions between the hvDothesis and -vn q.
References [I] P.O. Amblard,
[2]
[3]
[4] [S] [6]
[7] [8] [9]
Statistiques d’Ordre Superieur pour les signaux nongaussiens, nonlintaires, nontationnaires, These de Doctorat, INP Grenoble, January 1994 (in French). P.O. Amblard, M. Gaeta and J.L. Lacoume, “Statistics for complex random variables and signals ~ Part I: Variables”, Signul Processing, Vol. 53, No. 1, August 1996, pp. l-13. P.O. Amblard and J.L. Lacoume, “A deductive construction of third-order time-frequency distributions”, Signal Processing, Vol. 36, No. 3, April 1994, pp. 277-386. A. Blanc-Lapierre, and R. Fortet, Theorie des Fonctions Aleatoires, Masson, 1953. P. Bondon, Statistiques d’ordre superieur et modelisation en traitement du signal, These de Doctorat, Paris XI Orsay, 199 I. P. Bondon and B. Picinbono, “De la blancheur et de ses transformations”, Truitement du Signal, Special NonLineaire, Non-Gaussien, Vol. 7, 5, 1990. D. Brillinger, “An introduction to polyspectra”. Ann. Math. Statist., Vol. 36 , 1965, pp. 1351-1374. D. Brillinger, Time Series Analysis : Data Analysis und Theory, Ma&raw-Hill, New York, 1981. D Brillinger and M. Rosenblatt, “Asymtotic theory of kth order spectra”, in: Spectral Analysis of Time Series, Wiley. New York, 1967, pp. 153-188.
53 (1996)
15-25
25
[IO] D. Brillinger and M. Rosenblatt, “Computation and interpretation of kth order spectra”, in: Spectral Analysis of Time Series, Wiley, New York, 1967, pp. 189-232. [l I] J.F. Cardoso, “Higher-order narrow-band array processing”, in: J.L. Lacoume. ed., Higher-Order Statistics, Elsevier, Amsterdam, 1992. [12] P. Chevalier, B. Picinbono and P. Duvaut, “Le filtrage de Volterra transverse reel et complexe”, Traitement du Siynul, Special Non-Lineaire, Non-Gaussien, Vol. 7, 5, 1990. en composantes indtpendantes et [I31 P. Comon, “Analyse identification aveugle”, Truitement du Sigtnai, Special NonLineaire, Non-Gaussien, Vol. 7, 5, 1990. [I41 P. Comon, “Separation de melanges de signaux”, Proc. GRETSI 1989, Juan-les-Pins. component analysis”, in: J.L. [15] P. Comon, “Independent Lacoume, ed., Higher-Order Statistics, Elsevier, Amsterdam, 1992. [ 161 P. Comon and J.L. Lacoume, Statistiques d’ordre superieur, Cours de I’Ecole d’ett de physique des Houches, September 1993. [17] P. Durand, Theorie et estimation conventionnelle du bispectre. Application aux processus lintaire-quadratiques a spectres de raies, These de Doctorat, INP Grenoble, I99 I. [IX] P. Duvaut, Truitement du Sitpntl: Concepts et Applications, Hermes, Paris, 199 I, [19] 1.1. Jouny and R.L. Moses, “The bispectrum of complex signals: definitions and properties”, IEEE Trans. Signal Proc,ess.. Vol. 40, November 1992, pp. 283332836. [20] J.L. Lacoume and M. Gaeta, ‘Complex random variables: A tensorial approach”, in: J.L. Lacoume. ed.. Hiyher-Order Statistics, Elsevier, Amsterdam, 1992. [21] V. Leonov and A. Shirayev, “On a method of calculation of semi-invariants”, Th. Prob. Appl., Vol. 4, No. 3, 1959. [22] P. MacCullagh, Tensor Methocls in Statistics, Chapman and Hall, London, 1987. [23] F.D. Neseer and J.L. Massey, “Proper complex random processes with applications to information theory”, IEEE Truns. on Injorm. Theory. Vol. 39. July 1993, pp. I2933 1302. [24] B. Picinbono, Variables et signaux aleatoires circulaires, Communication au GDR TSI, groupe 9, reunion du ler octobre 1990. [25] B. Picinbono, “Geometrical concepts in higher-order statistics”. in: J.L. Lacoume, ed., Higher-Order Statistics, Elsevier, Amsterdam, 1992. [26] A. Shirayev, “Some problems in the spectral theory of higherorder moments”, Th. Prob. and Appl., Vol. 5, No. 3, 1960. [27] J. Ville, Theorie et applications de la notion de signal analytique. Cable et Transmission, 2eme annee, 1, 1948.
PROCESSING ELSEVIER
Signal Processing
53
( 1996) 15-25
Statistics for complex variables and signals Part II: signals P.O. Amblard*, M. Gaeta, J.L. Lacoume CEPHAG-ENSIEG
URA CNRS Received
346, BP 46, 38402 S&t-Martin
27 January
d’Ht?res CCdex, France
1995; revised 28 March 1996
Abstract
In this paper, we study the higher-order statistics of complex stationary signals. In a first part, we define precisely the multicorrelations and multispectra of complex signals. Different properties of these tools are put in evidence for certain classes of signals, such as analytic, band limited or circular signals. We show for example that band limited complex signals are circular up to a certain order. We then give the definition for the practical case of discrete time and discrete frequency processes. Finally, some extension of linear filtering relations are provided. Zusammenfassung
In diesem Beitrag studieren wir Statistiken h6herer Ordnung von komplexen stationkren Signalen. In einem ersten Teil definieren wir pr;izise die Multikorrelationen und Multispektren von komplexen Signalen. Verschiedene Eigenschaften dieser Werkzeuge werden fir gewisse Klassen von Signalen in Augenschein genommen, wie z.B. analytische, bandbegrenzte oder zirkulare Signale. Wir zeigen z.B. daR bandbegrenzte komplexe Signale bis zu einer gewissen Ordnung zirkular sind. Wir geben sodann die Definition ftir den praktischen Fall von diskreten Zeit- und diskreten Frequenz-Prozessen. SchlieRlich werden einige Erweiterungen linearer Filterbeziehungen vorgestellt. R&urn&
L’objet de cet article est 1’Ctude des statistiques d’ordre supkrieur pour les signaux B valeurs complexes, non gaussiens, stationnaires. Dans une premikre partie, nous donnons des dkfinitions prkcises pour les multicorr&lations et les multispectres de signaux B valeurs complexes. Diffkrentes propri&&s de ces outils sont mises en kvidence pour certaines classes de signaux, comme par exemple les signaux analytiques, bande ktroite ou circulaires. Nous montrons par exemple que les signaux complexes g bande Ctroite sont circulaires jusqu’8 un certain ordre. Les definitions sont alors &endues aux cas pratiques des signaux g temps et frtquence discrets. Quelques gCnCralisations simples des relations de filtrage 1inCaire sont enfin propos6es. Keywords:
Complex random signals; Multicorrelations;
* Corresponding
author. Tel: (33)-76-82-71-06;
Multispectra;
fax: (33).76-82-63-84;
0165-I 684/96/$15.00 @ 1996 Elsevier Science B.V. All rights reserved PII SO1 65- 1684(96)00072-2
Circular signals; Band limited signals
e-mail: [email protected].
16
P. 0. Amblurd
rf al. / Signul Processing
1, Introduction In [l], higher-order statistics for complex valued random variables have been presented. The logical extension to this work concerns the development of higher-order tools for complex valued signals. The most important features concerning higher-order statistics for complex quantities have been shown in [l]: a variable and its complex conjugate may give different statistical information. Therefore, a theory of higher-order statistics for complex variables must include higher-order cross-statistics between the variable and its complex conjugate. For signals, in the real case, the theory of higherorder statistics is now well-known. The first work goes back to 1953 with the book by Blanc-Lapierre and Fortet [4]. In this work, the tools were presented in terms of moments. In the beginning of the 1960s the cumulants were introduced in the theory by Shirayev [26]. Later, Brillinger and Rosenblatt and others made considerable efforts in the theory of higherorder ‘spectra’, especially in the domain of estimation. Thus, for real valued signals, the theory was almost complete at the end of the 1960s. We had to wait until the beginning of the 1980s for the Signal Processing community to discover (or rediscover) higher-order statistics and their interesting features. In the past ten years, many applications have grown around higher-order statistics. They are as different as blind deconvolution or equalization, separation of sources, detection of quadratic phase coupling, Volterra filtering, etc. In several fields of Signal Processing, from communication to array processing, many problems use complex signals in modeling and their solution requires higher-order statistics. However, no general theory concerning higher-order statistics for complex signals is available. Some work has been done [ 19,231 but to our knowledge, a unifiying view of higher-order statistics for complex signals does not exist. The aim of this paper is to present a theory of higherorder tools for complex signals. These tools will be multicorrelations and multispectra for stationary signals and may be higher-order time-frequency distributions for nonstationary signals. We however restrict the scope of the paper to the case of stationary signals and we try to highlight some features specific to complex signals.
53 (1996)
15-25
The organization of the paper is as follows. After the definition of some general objects, we present the definitions of the multicorrelations and the multispectra of a signal, insisting on the notion of stationary manifold. Section 4 presents the traditional debate ‘moments or cumulants’. We use an example to present the notion of Gaussian manifold and to plead in favor of the cumulants. We then examine particular signals (Section 5) such as analytic, circular or band limited signals for which higher-order tools have an interesting form. Then, Section 6 introduces higher-order cross statistics between two complex valued signals. These tools find an application in the derivation of input-output relations in linear filtering.
2. General definitions This first section gives precise and general definitions for the higher-order tools for complex valued random signals. We begin in the time domain and then examine equivalent definitions in the frequency domain. But before, let us introduce classes f@) and CD(‘) of Blanc-Lapierre and Fortet [4]. A signal X(v) is in @ck) ifV1 < k, ~‘E,,...,EI
J J .‘.
IE[dX,,(Vl)“~dX,,(Vl)ll
<
c < +m
(1)
where the &I’S correspond to a complex conjugated term if their value is - 1 and to a nonconjugated term if their value is + 1. Now, a signal x(t) is in fCk) if its Fourier transform belongs to @ck). The principal result we use here is then ([4, p. 4271) the following. If x(t) E ftk) and if k = l(Z - 1), then
= S-_
Jexp[2in(C
Eiti)Vi)]dX,,(vI).
” dx,,Cvj)
(2) exists in the quadratic mean sense. As a consequence, E[x,, (tl ) . .x,! (tj)] exists and satisfies
X
-f3d&f(Vl>. . d&,(Vj)l.
(3)
17
P. 0. Amblurd et al. / Signal Processing 53 (1994) 15-25
This result will be useful since the existence of higherorder tools for stationary and non-stationary signals relies on it.
Remark 1. The class f@) is the class of signals which belong to fck) for all k.
2.2. Symmetric multispectra Since the signal belongs to the class f’(‘“) of BlancLapierre and Fortet, it admits the Cramer representation x(t) =
exp(2irrvt)dX(
results were historically given in terms of moments. However, since the cumulants are nonlinear combinations of moments of lower or equal order, these results are also true when given in terms of cumulants. 2.1. hfulticorrelutions
(5)
Inserting this representation in (4), and making use of the multilinearity of the cumulant operator leads to Cx,p+y.p(t) =
Let x(t) be a complex valued random signal which of order p + q is belongs to fcoo). Its multicorrelation defined by’
v).
J’
Remark 2. The previous
J’
Cum[dX( vo), . . . , dx(v,-I
>M”(Q,.
(6)
Cx,p+‘,.p(r) =
Cumb(f0>,
“,
. . ,x(t,-~
),x*(f,),
,x*(t,+,-I
>I, (4)
where Cum[.] is the cumulant operator and where t = (to,. , t,+,_l ). Note that in this definition, q conjugated components and p nonconjugated components appear. The notation is then clear: p + q is the order of the multicorrelation whereas p represents the number of nonconjugated components. Note that as discussed in the companion paper [2], we have to include in the definitions all possible forms of (p + q)th order cumulants with distinct selections and conjugations in order to alofl= (to,...,t,+,_,) low the complete extraction of all the statistical information contained in the signal. Of course, this implies that, at a fixed order p + q, some of the multicorrelations may be redundant (relations of conjugation). The properties of the multicorrelation are direct consequences of the properties of the cumulants. It is multilinear, identically zero for p + q > 2 if the signal is Gaussian. Furthermore, if the signal is white in the strict sense (the random variables X(Q), . . , ~(t~+~_ I ) are statistically independent for all p + q), the multicorrelation vanishes except when the instants tj are all equal. We now turn to the definitions in the frequency domain.
Furthermore, if we assume that C.X.p+q.p(.t)possesses the following Fourier transform:
CX,p+q,p(f) =
.i’
&,+,,(v)
(7) where v = (~0,. . . , vp+_ i ), we obtain the definition of the symmetric multispectrum of order p + q: Cum[dX(vo),
,dX(~,_~),dX*(v,),. ,
M*(Q+~-I >I= G,p+,,,(vW.
(8)
These definitions are general and valid for all signals in f(“). However, their use is difficult, and we now assume that the signals under study are stationary.
3. Multicorrelations and multispectra for stationary
signals A signal is stationary if the statistics of all the random vectors it induces are invariant under a time shift.
P. 0. Amblard et al. /Signal
18
More precisely, let x(t) be a complex valued random signal. Let xn = (x(tt ), . . . ,x(t,)) be an n-dimensional induced vector. Then the signal is said to be stationary of order k if the probability density function of X, is invariant under a time shift for all n
of order p + q, the station-
Cx,p+q,p(t + rl) = Cx,p+q,p(t)
(9)
for all (p, q) such that p fq ,< PIand all lags Z. This last equation shows that the multicorrelation is no longer p + q-dimensional but indeed ( p+q - 1 )-dimensional. This is true since this last equation shows that the differential of Cx,p+4,p(t) is zero along a certain direction. Consider the following change of variables: Vi = 1,...,p1 Vi = p, . . , p + q - 1 I
Zi = tj - to, Zi = -tj + to.
x*(t - 2,),
+ r~ ), . . At
(11)
Remark. The particular role given to t shows that the term x(t) cannot be conjugated in our definition. This may be taken into account by defining
x(t
+ Zp-_I),X*(t
-
Tp) ,..‘)
+
71),
x*(t
=
Cx,p+q,q+1(-71,
which shows the redundancy
x exp (-2irt
-
vjtj ‘g\jtj))
fg
dt.
(14)
Making the change of variables ( 10) under the integral shows that
c&P+qsP
o=JJ Cx,p+q,p(r) exp(-2inrTv’) v
Pfq- 1
P-1 ~0 +
C
vj
/=I
-
C
vj
j=P
(15)
sx,p+q,p(vv =J
7p+q-l)J.
(7) exp( -2ircrTv’)dr cr>,+,,
(16)
cx,P+q,P(v)
= (
P-1 Sx,p+q,p(v’)6 VO+CVj-
(12)
(1 1 ), since (13)
of the second definition.
j=l
p+4-'
C i=P
Vj
1 .
(17)
This relation means that if the analyzed signal is stationary, the symmetric multispectrum takes nonzero values only on the (p + q - 1)-dimensional manifold defined by P--l
Ph-
V()+CVj-'2 j=l
But this form may be related to definition C:;+&)
s
leads to
+ zp-l),
. . ,x*(t - zp+q_l )].
= Cum[x*(t),x(t
We have seen before that the multicorrelation and the symmetric multispectrum may be related via a particular multidimensional Fourier transform. More precisely
where v’ = (VI,. . . , vpfq_l ). Letting
Let us point out that for a nonconjugated term, the lag is added, whereas it is substracted for a conjugated term. This is done to extend the well-known WienerKhintchine theorem to higher orders. This will appear later.
C:,,+,,(r)
3.2. Symmetric multispectra and stationarity: multispectrum
(10)
In this change, to plays a particular role, and we will denote it now by t. The multicorrelation is then a function of only t = (71,. . , z~+~__I) and reads c X,P+q,P(r) = Cum[x(t),x(t
15-25
cx,p+q,p(v) = C*,p+q,p(Q
3.1. Multicorrelations and stationarity For the multicorrelation arity property leads to
Processing 53 (1996)
1
Vj=O
(18)
i=p
and called the stationary manifold [4]. The value of the symmetric multispectrum on this manifold is then s X,p+q,p(v’). We will call this function multispectrum of order p + q of the signal x(t). Note that, like the multicorrelation, the multispectrum also depends on p + q - I frequency variables.
P. 0. Amblurd et al. / Siynal Processing
We now make use of (8) to obtain an explicit expression of the multispectrum. Let us recall that
. ,dX(v,_~),dX*(v,),. . ,
Cum[dX(vo),.
dX*@,+,-I )I = &,+,,p(v)dv.
(19)
Then, using (17) leads to Cum[dX(vo) ,..., dX(v,_,),dX*(v,)
,....
cM*( I’p+y- I )I
.I- I
/
I’+Y-l \
53 11996) 15-25
19
The physical interpretation of the multipectrum is then clear. It examines the statistical interactions between the p + 4 - 1 frequencies (VI,. . . , v.+,._ 1) and the v,. Thus, a frequency vp+q = l Cy?,’ vJ + Cpki’ stochastic process whose frequency ;d+, comes from the interaction between the p + q - I others will be perfectly analyzed by the multispectrum 2 S,,+,,,(v). Let us conclude this section by noting that the theory developed above allows the conservation (or the extension) of the Wiener-Khintchine theorem. At the second order, it is well-known that the power spectrum is the Fourier transform of the correlation function. We have constructed the tools here to respect this one-to-one correspondence again.
Fundamental remark. We use here the Cramer representation since a stochastic process cannot be represented as a standard Fourier integral (even in the mean square sense). But for physical situations, it is hard to work on the spectral process U(v). Furthermore, in practical applications, we observe the signals of interest on a short duration. But, the realizations of a time limited stochastic process have a Fourier transform. Hence, to understand clearly what is the physical meaning of multispectra, we now replace the harmonic increments U(v) by the Fourier transform of x(t) that we also write X(v) ’ . The reader must be aware that all the following is subject to this remark. Then, if X(v) is the Fourier transform of interest, the last equation reduces to Cum[X(va) . . . . . X(v,_r),X*(v,) / 7
+
VO = ~L.p+q.p(v’)~
,..., X*(vpfq_r)]dv P+4-I
P-1
C
Vj -
J=I
C
\ Vj
dv,
(21)
i=p
and therefore Sx,p+q,p(v’)
r
/
p-1
p+4-I
\
’ We symbolically write here U(v) = X(v)dv to mean that the dX(v) are the harmonic increments of X(\s).
4. Cumulants or moments We have defined the multicorrelations and multispectra in terms of cumulants. But the preceding definitions may have been given in terms of moments. The question is then why do we use cumulants’? What are the advantages of the cumulants over the moments? A rapid answer is provided by the arguments given in the companion paper [2]: 1. Cumulants provide a measure of independence. 2. Cumulants of order greater than two are zero for Gaussian signals. 3. Cumulants are cumulant: the cumulant of the sum of two independent signals equals the sum of their respective cumulants. These properties are the standard arguments which make the community prefer the cumulants to the moments. 3 We now present another advantage which is specific to signals. This advantage relies on the socalled Gaussian manifolds. For the sake of clarity, we describe these manifolds on the fourth order with an equal number of conjugated terms and nonconjugated terms.
2 In the sequel, the vector V’ is simply denoted by 11. ’ Note however that moments are used for deterministic signals. In particular, the ouptput multicorrelation, in term of cumulant, of a finite impulse response filter driven by a pure white noise is directly proportional to the multicorrelation, in term of moment, of the impulse response of the filter.
P. 0. Amblard et al. / Signal Processing 53 (1996)
20
We recall here the cumulant-based definitions, and give the equivalent definitions for the multicorrelations and spectra based on moments. They read cx,4,2(z) =
Cum[x(t),x(t
+ r]),.~*(t - rz),~*(t
-
z3)],
cA4~,4,2(2) =
E[x(t)x(t
+
q)x*(t
-
T2)x*(t
-
z3)],
(23)
15-25
are called Gaussian manifolds [25]. They belong to the stationary manifold and represent the domains of existence of SM,,x,4,2(v)in the zero mean Gaussian case. But they also appear in non-Gaussian cases. To explain this, let us consider the example of a zero mean non-Gaussian white stochastic process, of symmetric probability density function. Furthermore, suppose that the second and fourth-order statistics are given by
&4,2(v)
=
Cum[X(-h
+v2
+
v3),x(vl
),~*(v2),~*(v3)1,
h’,x,4,2(v) = ‘%f(-v]
+
“2 +
v3)x(vl)x*(v2)x*(v3)1,
=
=
a2w,
G,2,2(7)
=
Y2W,
&2,1(v)
=
02,
&,2,2(v)
=
Y2,
cx,4,2(z)
=
K4&7l,
&,4,2(v)
=
K4.
(27)
where we have added the subscript M to mean moment-based. Suppose that x(t) is zero mean stationary signal. We can write the moment-based multispectrum as a function of the cumulant-based multispectrum via &,x,4,2(v)
G,Z,l(~)
T2,73 ),
Using the explicit values of the statistics involved, we obtain for the moment-based multispectrum
&4,2(v) &4,x,4,2(v)
+E[X(-vl
+
v2 +
v3)x(vI
+E[X(-vl
+
v2 +
v3)X*(v2)I@X(vl
+EK(-VI
+ ‘9 +
=
K4 + a4@(v3 +02y*26(v2
)x*(v3)1 whereas
v3)X*(v3)IE[~(vl)X*(v2)1.
The second-order moments that appear in the last equation are of the type &Q(V) and &,2,](v); they are respectively equal to E[X(-v)X(v)] and E[X(v)X*(v)]. Thus, the terms of the first line of the last expression are identically zero if v] = -(-v] + v2 + ~2) and v3 = -vz. These two conditions are identical and equivalent to v?; + v3 = 0. This is a direct application of the concept of stationary manifold. Reasoning in the same way for the remaining terms leads to = &,4,2(v)
+ &2,2(vl
)$2,2(v2
)6(v2
-t&,2,1
(v2 )&,2,1 (v3 )&v3
-
VI )
+&,2,t
(v3 )&,2,1 (v1 P(v2
-
VI).
+ v3)
(25)
The manifolds Q-v] V3_V] v2 +
=o, =o, v3 =
(26) 0
vl)
+ &v2
-
vl))
the
+
v3),
(28)
cumulant-based multispectrum reads This shows that the moment-based multispectrum, SM,~,~,~(V) equals ~4 on all the stationary manifold, except on the Gaussian manifolds where pikes of magnitude cr4 or o2y*2 appear. On the opposite side, the cumulant-based multispectrum is completly flat, and reflects the whiteness of the analyzed signal. This result comes from the fact that the fourth-order cumulants measure the fourth-order statistical interaction when the second-order cumulants have been cancelled. In a general manner, an nth-order cumulant measures the nth-order interaction when the interactions of lower order have been eliminated. This principle is equivalent to the methods of orthogonalization of polynoms. At the second order, this is well-known: a pike in the spectrum appears at the zero frequency if the spectrum is moment based and if the signal is not zero mean. If the spectrum is constructed on the covariance (second-order cumulant), this pike does not exist. Therefore, these results provide in our opinion another great advantage of the cumulants over the moments, since they cannot present pikes on the $4,2(v)
(24)
&,x,4,2(v)
-
)]-‘%f*(v2)X*(v3)1
=
7~4.
P. 0. Amblard
et al. / Signal Processiny
Gaussian manifolds. This claim may take all its importance in estimation contexts.
5. Particular signals We examine in this section the multicorrelations and/or multispectra for some special signals of interest in different fields of Signal Processing, from communication to array processing.
A signal x(t) is said to be analytic4 negative frequencies, i.e.
if it has no
if v < 0.
(29)
If we examine (22), the multispectrum non zero if the intersection Vi =
1,...,p+q-
1
It was shown in the first paper that if a random vector is circular of order n, then the statistics of lower or equal order, containing a number of conjugated terms different from that of nonconjugated terms are zero. Therefore, this characterization of circularity suffices to proove that, for circular signals of order n,
Cx,p+q.p(r)= 0 ‘Jq,P
S,,,+,,,(v)
is
Vi > 0, (30)
v, > 0
is not empty. But, the case q = 0, which corresponds to a definition with no complex conjugate, leads the preceding system to
,I-
_
1
>
(32) by a pure jkquency
Let x(t) be a signal satisfying x(l) = z(t) x exp(2irv$), z(t) being stationary. Note that x(t) is not stationary. To see this, suppose that E[z(t)] = m # 0. Then E[x(t)] = mexp(2i7tvot) which does depend on time. Therefore, x(t) is not stationary, but here at least cyclostationary of order 1 (its mean is periodic). Thus, we a priori cannot define multispectra of x(t). However, let us evaluate &p+q,p(v), even if it has no sense. 5 In the frequency domain, the definition of x( t ) is X(v) = Z(v - VO),and then a multispectrum of x(t) is written as a function of a multispectrum of z(t) via
sx,p+q,p(v) ~-~ViJffj’vi)
-hl)
\‘j > 0,
3
i=p
I
L7 v,
that
ptqdnandqfp.
= Cum [ Z ( Vi = 1,...,p+q-
such
.3. Signals modulated
5. I. Anulytic siynuls
X(v) = 0
21
53 11996) IS-25
Z(Vl
(31)
-
vo),...,Z(vp-I
-
vo),
0,
i= I
zyv,
which obviously has no solution. Therefore, in the case of an analytic signal, multispectra of the type S,,,,(v) are identically zero.
-
VO),...,Z*(~~p+q-l
-
.
vo)
(33)
I But, &,+,,, (v) is zero if v escapes from the stationary manifold. Thus, this multispectrum of x(t) is not zero if
5.2. Circular random signals In the companion paper, the notion of circular random vector was introduced. This notion may be extended to random signals in the following way [24]. We say that a signal is circular if the induced random vectors of all orders are circular. This makes it possible to define circularity of order n: a signal is circular of order n if the induced vectors of order lower or equal to IZare circular. 4 For a discussion
on analytic
signals,
see [27]
(-~v~+pq
p+a-
P-l
=-
-v.
i=p
!=I
C(Vi i=l
1’0)+
.C
I
(vi - V(j)
(34)
i=p
‘This is of no sense in the context of this paper. However, evaluating this multispectrum is full of sense when working, for example, with cyclostationary signals.
P. 0. Amblard et al. /Signal
22
(35)
4 = P.
Therefore, S,,,,,, (v) is not zero if it has an equal number of conjugated terms and nonconjugated terms. Furthermore = &zp,p(v
-
vol).
(36)
Hence, for stationary signals modulated by pure frequencies, the notion of nonstationarity has to be moderated, since statistics of the type SX,~P,P(v) are stationary. 5.4. Band limited signals Let x(t) be a band limited signal, i.e. a signal obeying X(v) = 0
15-2.5
6. Crossmulticorrelations and spectra
or
Sx,2p,p(v)
Processing 53 (1996)
if Jv i VO]> Av.
5.5. Some remarks Gaussian analytic signal is stictly circular (i.e. circular at all orders). The only nonzero multispectra of a Gaussian signal are &,2,0(v), &,2,1(v) and ,&,2(v). But, if the signal is also analytic, only &,JJ (v) is nonzero, which proves the assertion. A (2) band limited signal is circular of order N = [vo/Av]. This immediately comes from the preceding sections. Up to this point, we have examined the higherorder ‘auto’-statistics for random signals. We now give some ideas to extend the theory to the case of higher-order ‘cross’-statistics. 6 [,I stands for integer part of.
cx,y,p+q,e(r) = CumMtk, (t + EI~I>,. . . , XE,,_,(t+Ep--1+-l),
(37)
These signals are important in array processing since the quasi-monochromatic assumption is often made. We show in Appendix A the following result: For all orders lower than6 N = [vo/Av], the only nonzero multispectra are of the type &,J~,~(v). This result is the extension to the complex case of Theorem 1 of [26, p. 2741 proved by Shirayev. An analog of this result is proposed in [12] where the authors show that the moment-based multispectra with one or no complex conjugate equal zero.
(1) A
In this section, we enter the study of the statistics of interaction between random signals. It is possible to provide a general theory, i.e. giving tools that examine the interaction between 12different signals. However, this complicates the formalism and is straightforward when the definitions for two signals have been given. Therefore, we provide here the study of the interaction between two random signals x(t) and y(t). Moreover, we only give definitions; the derivations are made as before. Let x(t) and y(t) be two signals off(“). The crossmulticorrelation between x(t) and y(t) of order p + q is
Y&
+ EpZp),.. . >
YE,,+y_, (t + Ep+q- I zp+q-
I119
(38)
where the &i’s equal +l for a nonconjugated term and - 1 for a conjugated one, and where x+1(t) = x(t) and x-l(t) =x*(t). In the notations concerning the ‘auto’-statistics, the information relative to the position of the conjugated terms appeared in p whereas here this information is contained in vector E. Furthermore, note that x(t) is again taken as a reference and then the order x, y in the notation is of importance. The crossmultispectrum of order p + q is defined as the Fourier transform of the crossmulticorrelation or explicitly (z) sx,Y,P+q,E(v) = j Gv,,+%~
exp(-2ixzTv))dz.
(39)
It is again possible to show that
sx,y,p+q,E(v)
= Cum
[x
&, (VI ),
(-
9..
p$‘Ejvj)2
.
,x,,>_, (VP-1
h
y&$J>, 7.. .) YE,+,_,(vp+q-l>
1 .
(40)
This provides a clear physical interpretation of the ‘cross’-statistics. They examine the links between q
P. 0. Amblurd et al. / Signal Processing 53 (1996) IS--25
frequencies of y(t), p - 1 of x(t) and the particular frequency of x(t) which defines the stationary manifold. Remark 1. In the real case, the definitions Cx,y.p+y(z)= Cum[x(l),x(t Y(t+
become
+ TI 1,. . . ,x(t + T~~-I1,
Tph.. y(t + ?
23
I
~,,(~~p),...,~E,,+,~,(vp+q-l) The cumulant operator being multilinear, ing equation turns into
the preced-
Tp+q-I 113
(41) ffc,,_,(Vp-I
The same reasoning
Kr
. , p+q,Av).
(45)
leads to
X(v,-l),~~~~,~,...,~~~~,+,-l~ I
Remark 2. Letting
H(v~-I )H*(l’p)...H*(~~+q--)S,,,+q,p(v>. (46)
& = (l,..., 1, -l,..., -1) -p-l 4
Remark. are
we then have Gr,p+y,t(r)
=
relations in the the real case
/P+4--l
= Cr.p+q,p(r), (42)
SI.l.p~-qJv)
The equivalent
\
Sw,p+q(v)= H*
s,.,+q.p(v>.
The interest of ‘cross’-statistics especially appears when dealing with the transformation of a signal into another. Of particular importance are linear filters. We thus provide input-output relations for higher-order statistics. Consider a linear, time invariant system, of impulse response h(t) and of complex gain H(v), driven by a signal x(t) off(=). We denote the output by y(t). Therefore, we have
ff(~y-
I )&,p+q(v),
(47) S v,p+q(~)
=
H’ \ H(~p+q-I
I==’
/ )&p+q(v>.
7. Summary and discussion y(t)
=
I
h(t - r)x(T)dz, (43)
Y(v) = H(v)X(v). Let us then examine7 (43) leads to &.x,p
=
Sy,X,p+q,E(v). Using (40) and
+y,s(V>
ChtIl!,,i
‘Note
-
‘$‘EjVj)X(
-
‘E’EjVj),
that y(t) is taken here as the reference.
The aim of this part was the introduction of precise definitions of higher-order tools for complex valued signals. Based on the comments of [2], we had to include in the definitions a certain number of conjugated and nonconjugated terms, in order to extract from the signal all the statistical information. Moreover, we have performed the construction of multicorrelations and multispectra in order to extend the Wiener-Khintchine theorem to higher orders. This implies a particular choice of the sign of the lags in the definition of the multicorrelation. But let us point out that this is our choice, and as such this is arbitrary.
P. 0. Amblard et al. /Signal
24
Two other important points have been presented: the known stationary and Gaussian manifolds. We insist on the fact that Gaussian manifolds are in our opinion the basis of arguments to prefer cumulants to moments. In this paper, we did not present any application. However, we showed several results concerning the higher-order statistics of useful types of signals. For example, we have shown that band limited signals which are often used in array processing are circular up to an order which depends on the characteristics of the band. Finally, the problem of input-output relations in linear filtering was rapidly discussed, but we want to insist on the fact that in this problem, more general theories may be developed (for example multi-input, multi-output relations). However, we claim that such theories are straightforward extensions of ours, and essentially rely on a complication of notations. The next extension of this theoretical work concerns the definition of tools for complex valued nonstationary signals. This work has been done for the third-order case in [3], and for the general case in [ 11. The results concerning nonstationary signals will be considered in another paper.
Processing 53 (1996)
-vo-Av
vo -
AV < I- CztA1 vi + Cp_fp4-’ vi) < vo + Av. Let 11,
12,kl and k2 be four integers such that 11+ 12 = p - 1 and kl + kz = q, Consider then the following p +q - 1 frequencies: Vi=
~o-Av
-VO+AV
p+kl
+ I,...,p+q-
D-1
Vi = II + l,...,p-
i=l <(q
~o-Av
+ p -
1 - 2(11 + k2))vo + (p + q - l)Av,
(-4.2) (1)
Suppose that the order is odd, or q + p = 2n + 1. Then (A.2) becomes p-1
2(n - (I1 + k2))vo - 2nAv<
-C
p+4-
1
vi + C i=l
vi
i=p
<2(n - (I, + k2))vo + 2nAv.
(A.3)
Moreover, suppose that n - (11 + k2) > 1. Therefore 2(n - (11 + k2))vo 22~0. But, the order is chosen such that 2n + 1~ vo/Av, and thus ve 2 (2n + 1)Av. Then 2(n - (II + k2))vo + vo >2vo + (2n + 1)Av (-4.4)
(2)
or 2(n - (I, + k2))vo - 2nAvavo + Av. This expression is, accordin to (A.3), in contradiction f with VO-AV< -CltY vi+Cp_f,4-’ vi
1 -2(Z, p--l
+kz))vo-(2p+n-
+(2p+n-
l)Av
Pf9-1 C
<(2p+nVi=p,...,p+kl,
y, i=p
i=l
1,
the first
D+a- 1
-&+‘E
6
1,...,11,
-VO+AV
(A.1)
(q + p - 1 - 2(11 + k2))vo - (p + q - l)Av
<-Cvi+
-vo-Av
1.
Summing the last two lines and sustracting two leads to
Appendix A. Band-limited signals are circular In this first section, we prove the following result: Let x(t) be a band limited signal, i.e. X(v) = 0 if Iv i v. 1 > Av; then for all orders lower than N = [vo/Av], the only nonnull multispectra are of the type Sx,2p,p(v). Consider multispectra of order p + q d vo/Av. Such a multispectrum is nonzero if vo - Av< [vi]
15-25
Vi
i=p
1 -2(Z1 +kz))vo 1)Av.
(A.5)
P. 0. Amblard
et al. / Signal Processing
But the order is such that vg >,(2p + n)Av, and supposing again that p - (11 + kz) 3 1, it follows 2( p - (2, + k2))vo - (2~ + n - l>Av 3vo + Av which immediately
(A.6) implies, since n > 1,
(2~ -t n - 1 - 2(1, + kz))vo - (2~ + y1- 1)A 3 vo + Av.
(A.7)
This is again in contradiction with v. - Av< - CL<’ vi + C!‘+J-’ vi < vo + Av, and thus p = (Remark. If n’< 1, the roles of p and q are exchanged). (3) The proof is made in the same way to examine the contradictions between the hvDothesis and -vn q.
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[2]
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[4] [S] [6]
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[IO] D. Brillinger and M. Rosenblatt, “Computation and interpretation of kth order spectra”, in: Spectral Analysis of Time Series, Wiley, New York, 1967, pp. 189-232. [l I] J.F. Cardoso, “Higher-order narrow-band array processing”, in: J.L. Lacoume. ed., Higher-Order Statistics, Elsevier, Amsterdam, 1992. [12] P. Chevalier, B. Picinbono and P. Duvaut, “Le filtrage de Volterra transverse reel et complexe”, Traitement du Siynul, Special Non-Lineaire, Non-Gaussien, Vol. 7, 5, 1990. en composantes indtpendantes et [I31 P. Comon, “Analyse identification aveugle”, Truitement du Sigtnai, Special NonLineaire, Non-Gaussien, Vol. 7, 5, 1990. [I41 P. Comon, “Separation de melanges de signaux”, Proc. GRETSI 1989, Juan-les-Pins. component analysis”, in: J.L. [15] P. Comon, “Independent Lacoume, ed., Higher-Order Statistics, Elsevier, Amsterdam, 1992. [ 161 P. Comon and J.L. Lacoume, Statistiques d’ordre superieur, Cours de I’Ecole d’ett de physique des Houches, September 1993. [17] P. Durand, Theorie et estimation conventionnelle du bispectre. Application aux processus lintaire-quadratiques a spectres de raies, These de Doctorat, INP Grenoble, I99 I. [IX] P. Duvaut, Truitement du Sitpntl: Concepts et Applications, Hermes, Paris, 199 I, [19] 1.1. Jouny and R.L. Moses, “The bispectrum of complex signals: definitions and properties”, IEEE Trans. Signal Proc,ess.. Vol. 40, November 1992, pp. 283332836. [20] J.L. Lacoume and M. Gaeta, ‘Complex random variables: A tensorial approach”, in: J.L. Lacoume. ed.. Hiyher-Order Statistics, Elsevier, Amsterdam, 1992. [21] V. Leonov and A. Shirayev, “On a method of calculation of semi-invariants”, Th. Prob. Appl., Vol. 4, No. 3, 1959. [22] P. MacCullagh, Tensor Methocls in Statistics, Chapman and Hall, London, 1987. [23] F.D. Neseer and J.L. Massey, “Proper complex random processes with applications to information theory”, IEEE Truns. on Injorm. Theory. Vol. 39. July 1993, pp. I2933 1302. [24] B. Picinbono, Variables et signaux aleatoires circulaires, Communication au GDR TSI, groupe 9, reunion du ler octobre 1990. [25] B. Picinbono, “Geometrical concepts in higher-order statistics”. in: J.L. Lacoume, ed., Higher-Order Statistics, Elsevier, Amsterdam, 1992. [26] A. Shirayev, “Some problems in the spectral theory of higherorder moments”, Th. Prob. and Appl., Vol. 5, No. 3, 1960. [27] J. Ville, Theorie et applications de la notion de signal analytique. Cable et Transmission, 2eme annee, 1, 1948.