Blind Separation of Two Signals by Estimation of Two Fourth-Order Cumulants Ivica Kopriva Laboratory for Electronic Systems and Information Processing Faculty of Electrical Engineering and Computing University of Zagreb, Vukovarska 3 9 10000 Zagreb, CROATIA e-mail: ivica.kopriva@public.srce.hr Abstract - The problem is to recover stochastic signals from an unknown stationary linear mixture. The paper presents analytical solution for blind separation of two statistically independent signals. It requires two fourth-order input sample cumulants to be estimated, contrary to the solution given in references that requires estimation of three fourth- order input sample cross-cumulants. When real-time separation problem is considered this difference can be significant. L INTRODUCTION The so-called blind signal separation problem has been attracted sigdicant attention in last several years. The problem is consisted of separating and estimating generally multiple source signals from an array of sensors. It is assumed that the problem is described with: y = Ax (1) where x,y E R", and AER"", n is number of signals and detA#O. The only assumption made here is that components of x are mutually independent up to at least fourth order. To solve the blind signal separation problem the linear transformation matrix W must be find such that: where s is scaled version of the original source signal vector x. It has been shown that (1)-(2) can be solved by using two approaches: neural network and higher-order (fourth-order) statistics. The neural network based solution was firstly given in [6], while in [7,8] a new algorithms were proposed that allow the extraction of extremely badly scaled signals i.e. the mixing matrix A can be ill-conditioned. The neural network approach requires the source signals x,, j=l ... n, to have even probability density function. The higher-order independence test is introduced indirectly by using nonlinear odd activation functions. The neural network solution enables that the number of signals to be separated can in general case be arbitrarily large, what is not the s = wy (2) case with the cumulant based solution. The potential problems with neural networks arise when input signals are nonstationary. It becomes problematic to determine the values of the convergence control factors. By using cumulants the nonstatioinary case can be easier handled by computing estimates of the input sample cumulants all the time. The cumulant based solution is obtained by equating all three fourth-order o'utput cross-cumulants with zero. Unfortunately, it has been shown in [3,5] that the cumulant based analytical solution is impossible to be found for the number of signals n > 2. II. THE CUMULANT EIASED SOLUTION OF THE N O SIGNALS SEPARATION PROBLEM The k-th order cumulant of the random variable xJ is defined as the k-th coefficient of the Taylor series expansion of the second characteristic function [1,2]: K(o) = In @(w) = In E eJmx I where o=[o 1...02], and @(a) is characteristic function of x. Let the a,, and wg , (i,j=l,2) are elements of the matrices A and W in accordance with (1) and (2). It is usually assumed that the separation signals s1 and s2 represent the source signals XI and x2 up to the scale factors. The sl and s2 are considered to be: separated when mutual higher order statistics is zero. In practice the fourth-order statistics is required to ble zero. The third order statistics is avoiding since most of the real world signals, because of their symmetrical distribution, have the third order statistics nearly zero. The fourth-order cumulants are used instead of moments since their linearity property, [ 1,2], let us work with them easily as operators. According to [l] the zero-lag fourth order cumulant of the random process x, is defined as: and is expressed in terms of moments [1,2]: c'p, = cot,, x, 2 XI, x, 1 (3.1) C,X, = E(x4) - 3E2(x2) (3.2) 0-7803-3694-1 /97/$10.00 @ 1997 IEEE 98 1 Authorized licensed use limited to: Ivica Kopriva. Downloaded on December 21, 2009 at 03:01 from IEEE Xplore. Restrictions apply.