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Additional info for Adaptive Signal Processing: Next Generation Solutions (Adaptive and Learning Systems for Signal Processing, Communications and Control Series)

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Vector-Concatenation Type Mappings The two mappings in this class, the (Á)R and (Á)C mappings have very different uses. The most commonly used mapping CN 7 ! R2N takes a very simple form and is written such that z [ CN 7 ! z¯ R ¼ zr zi ! [ R2N (1:10) 18 COMPLEX-VALUED ADAPTIVE SIGNAL PROCESSING and for a matrix A as A[C MÂN ¯ R ¼ Ar 7! A Ai ÀAi Ar ! [ R2MÂ2N : (1:11) ¯ R z¯ R : It can be easily shown that (Az)R ¼ A The mapping provides a natural isomorphism between CN and R2N, and thus is a practical approach for derivations in the complex domain.

5 Statistics of Complex-Valued Random Variables and Random Processes Statistical Description of Complex Random Variables and Vectors A complex-valued random variable X ¼ Xr þ jXi is deﬁned through the joint probability density function (pdf) fX (x) W fXr Xi (xr , xi ) provided that it exists. For a pdf fXr Xi (xr , xi ) that is differentiable with respect to xr and xi individually, we can write fXr Xi (xr , xi ) ¼ f (x, xÃ ) where x ¼ xr þ jxi, and use the expression written in terms of x and xÃ in the evaluations to take advantage of Wirtinger calculus.

A random process is called second-order circular if its pseudo-covariance function p(k) ¼ 0, a condition that requires the process to be SOS. 2 PRELIMINARIES 29 Also, it is easy to observe that an analytic signal constructed from a WSS real signal is always second-order circular, since for an analytic signal we have C( f ) ¼ 0 for f , 0, which implies that P( f ) ¼ 0. An analytic signal corresponding to a nonstationary real signal is, on the other hand, in general noncircular [93]. 3, we show scatter plots of three random processes: (1) a circular complex autoregressive (AR) process driven by a circular Gaussian signal; (2) a 16 quadrature amplitude modulated (QAM) signal; and (3) a noncircular complex AR process driven by a circular Gaussian signal.