It has been shown in Chapter 12 that the full-second order statistical description of a general complex valued process can be obtained only by using the augmented complex statistics, that is, by considering both the covariance and pseudocovariance functions. It is therefore natural to ask how much we can gain in terms of the performance of statistical signal processing algorithms by doing so. To that end, this chapter addresses linear estimation for both circular and noncircular (proper and

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... oper) complex signals; this is achieved based on a finite impulse response (FIR) system model and for both the second-order regression modelling with fixed coefficients (autoregressive modelling) and for linear adaptive filters for which the filter coefficients are adaptive. Based mainly on the work by Picinbono [239, 240] and Schreier and Scharf [268], Sections 13.1 -13.3 show that for general complex signals (noncircular), the optimal linear model is the 'widely linear' (WL) model, which is linear both in z and z * . Next, based on the widely linear model, for adaptive filtering of general complex signals, the augmented complex least mean square (ACLMS) algorithm is derived, and by comparing the performances of ACLMS and CLMS, we highlight how much is lost by treating improper signals in the conventional way.