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

Example text

1 in the derivation of complex Newton updates for maximum likelihood independent component analysis. 26) given in Proposition 1 play a key role in the derivation of the complex gradient and Newton update rules. 3 OPTIMIZATION IN THE COMPLEX DOMAIN 39 in the literature to the complex domain such that the limitations of the Newton method can be mitigated. 1), in order to achieve convergence, we require the search direction ¯ R to be a descent direction when minimizing a given cost function. This is the case Dw @2 f is positive deﬁnite.

To completely deﬁne the second-order statistics, as in the case of random variables, we also deﬁne the pseudo-covariance function [81]—also called the complementary covariance [101] and the relation function [91]—as p(n, m) ¼ E{X(n)X(m)} À E{X(n)}E{X(m)}: In the sequel, to simplify the expressions, we assume zero mean random processes, and hence, the covariance and correlation functions coincide. Stationarity and Circularity Properties of Random Processes A random signal X(n) is stationary if all of its statistical properties are invariant to any given time shift (translations by the origin), or alternatively, if the family of 28 COMPLEX-VALUED ADAPTIVE SIGNAL PROCESSING distributions that describe the random process as a collection of random variables are all invariant to any time shift.

2 are related to each other through simple linear transformations, thus making it possible to work in one domain and then transfer the solution ~ zR where to another. Two key transformations are given by z¯ C ¼ U¯zR and z~ C ¼ U~ I U¼ I jI ÀjI ! 1 j ~ and U ¼ diag{U2 , . . , U2 } where U2 ¼ . 3. For trans~ we can use permutation matrices formations between the two mappings, (Á) and (Á), that are orthogonal, thus allowing simple manipulations. 2 PRELIMINARIES 21 region, the Taylor series expression assumes the same form as in the real case given by f (z) ¼ 1 X f (k) (z0 ) (z À z0 )k : k!