IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 39, NO. zyxwvutsrqp 4, APRIL 1991 1001 [7] A, Dembo, “Bounds on the extreme eigenvalues of positive-definite Toeplitz matrices,” IEEE Trans. zyxwvutsrqpon Inf, Theory, vol. 34, no. 2, pp. zyxwvutsr 352- 355, Mar. 1988. zyxwvutsrqp [8] D. Slepian and H. J. Landau, “A note on the eigenvalues of Hermitian matrices,” SIAM J. Math. Anal., vol. 9, no. 2, pp. 291-297, Apr. 1978. Cumulant Series Expansion of Hybrid Nonlinear Moments of Complex Random Variables Gaetano Scarano Abstract-In this correspondence a general theorem for zero-mem- ory nonlinear (ZNL) transformations of complex stochastic processes is presented. It will be shown that, under general conditions, the cross covariance between a stochastic process and a distorted version of an- other process can be represented by a series of cumulants. The coeffi- cients of this cumulant expansion are expressed by the expected values of the partial derivatives, appropriately defined, of the function de- scribing the nonlinearity. The theorem includes as a particular case the well-known invariance property (Bussgang’s theorem) of Gaussian processes, while holding for any joint distribution of the processes. The expansion in cumulants constitutes an effective means of analysis for higher order moment based estimation procedures involving non-Gaussian complex pro- cesses. I. INTRODUCTION Bussgang’s theorem zyxwvutsrqpo [ 11 (as extended to the complex case in [2]) states that the cross correlation between two jointly normal, zero-mean, stationary complex stochastic processes x ( t) and y (t) is proportional to the cross correlation between zyxwvutsrq x(t) and zyxwvutsrq z(t), a (complex) zero-memory nonlinear (ZNL) transformation of y (t) Rz(7) = E{x(t) . 2(t - zyxw T)} = K . Rxy(~) = K. E{x(t) . Y(t - 7)) in which z(t) = g[y(t)]. An overbar denotes complex conjuga- tion. The proportionality factor was derived in [2], in which U: = E { I y 1’ } is the variance of y ( t). This is also referred to as the invariance property of complex Gaussian processes. In this correspondence, it is shown that the invariance property is a special case for Gaussian processes of a more general theorem that holds for any distribution of the processes. In fact, it will be demonstrated that, when the function g ( . ) that can be expressed in the formg(x) =f(u, w)\~=~ wheref(u, w) is analytic both in U and w, the cross covariance E { x( t) . Z( t - T) } can be expanded in a series of cumulants weighted by the expected values of the partial derivatives of the function f (U, w). This expansion in cu- mulants constitutes a useful means of analysis for higher order mo- ment based estimation procedures involving complex, non-Gaussian processes. w=E Manuscript received September 10, 1990. The author is with C.N.R., Istituto di Acustica “0. M. Corbino,” IEEE Log Number 9042268. 1-00189 Rome, Italy. 11. THE CUMULANT EXPANSION The class of analytic conjugate functions in a field A is intro- duced and defined as follows: xx = {g(x): g(x) =f(u, w)Ju=._, W=X f(u, w) analytic in (A, A)}. For analytic conjugate functions, the differentiation is defined in terms of differentiation of the analytic function f (U, w), Le., aP+q axpaxq g(x) = g(p3q)(x) The expected value is denoted by E { . } , complex conjugation by the overbar, and the random variables (RV’s) extracted at the in- stants t and t - 7 from processes x(t) and y(t) by X and Y, re- spectively. In order to simplify the notations in the following, the case X = Y is considered first. The extension to the bivariate case is straight- forward and does not affect the essence of the development that follows. Theorem 1: Let g ( . ) be a conjugate analytic ZNL transforma- tion in a field A, on which is defined a circularly symmetric com- plex random variable ( CRV )XI. Then where kiq+13q+l) is the comples cumulant of order zyxw (q + 1, q + 1 ) of the bidimensional CRV(X, X). be the joint probability density function (p.d.f.) of the real and the imaginary part of X = X, + jx; ; the (complex) moment generat- ing function (rn.g.f.) of (X, X); the cumulant generating func- tion (c.g.f.) of (X, X). Differentiating with respect to s both sides of Cx(s, U) = log Px(s, U) Px(s, U) . Cp’(s, U) = Pjll-O’(s, U) yields where, generically, the subscript (P.4) denotes partial differentiation p times with respect to s and q times with respect to U. Again, differentiating r times with respect to s and t times with respect to U, the Leibnitz theorem for functions of two variables is obtained : ’For simplicity, moments and cumulants are taken around the origin, which is supposed to be enclosed in the field A. More generally, the theo- rem holds replacing x by x - xo (for xo E A), and considering moments and cumulants around xo. 1053-587X/91/0400-1001$01.00 zyxwvu 0’ 1991 IEEE