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Chapter 7. Second-order methods based on color > 7.2. WSS processes - Pg. 253

228 CHAPTER 7 Second-order methods frequency domains, which can be conveniently exploited for separation of their mix- tures. Thus, the use of SOS for separation of WSS processes (often also called "colored" sources) is the main theme of this chapter. SOS-based separation approaches for WSS sources can roughly be divided into two cat- egories: approaches exploiting the special structure of the correlation matrices through (approximate) joint diagonalization; and approaches based on the principle of Maximum Likelihood (ML). While in general ML estimation is based on more than SOS (and is rather involved), under the assumption of Gaussian sources the ML estimate takes a rel- atively simple form (asymptotically) and is indeed based on SOS alone. Using the Gaus- sianity assumption, it is also possible to apply (asymptotically) optimal weighting to the joint-diagonalization based approach, thus obtaining estimates which are asymptotically optimal, and are thus asymptotically equivalent to the ML estimate. The chapter is structured as follows. In the next section we provide some prelimi- nary definitions and concepts associated with discrete-time WSS processes in general, and with parametric (Autoregressive (AR), Moving-Average (MA) and Autoregressive Moving-Average (ARMA)) processes in particular. In section 7.3 we outline the prob- lem formulation and discuss issues of identifiability and of performance measures and bounds. In section 7.4 we present joint-diagonalization based methods, whereas in sec- tion 7.5 we present ML-based methods. In section 7.6 we discuss some supplementary issues, such as the effect of additive noise; some particular cases of nonstationary sources; and the framework of complex-valued signals. 7.2 WSS PROCESSES Let s(t ), t denote a real-valued discrete-time random process, assumed for simplicity (and without loss of generality, for our purposes) to have zero mean. With vanishing first- order statistics, its second-order statistics are fully described by its correlation function R s (t , ) {s (t + )s (t )} t , . (7.1) When R s (t , ) depends only on , the process is called WSS, and we may simply denote R s () R s (t , ). The Fourier transform of the correlation R s () is the spectrum (or spectral power density), S s () =- R s ()e - j 2 (7.2) An appealing property of WSS processes is that since the covariance matrix of any set of T consecutive samples is a T × T Toeplitz matrix, it is asymptotically diagonalized by the Fourier matrix. Indeed, define the (zero-mean) normalized Fourier coefficients of k the series s (1), . . . , s(T ) at Fourier frequencies k T , k T 1 T T t =1 s(t )e - j 2 T , kt k = 0, 1, . . . , T - 1. (7.3)