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

Example text

Also, a quasi-Newton update is given in [117] by setting the matrix H1 to a zero matrix, which might not deﬁne a descent direction for every case, as also noted in [64]. 3 Matrix Optimization in CN Complex Matrix Gradient Gradient of a matrix-valued variable can also be written similarly using Wirtinger calculus. For a real-differentiable f (W, WÃ ): CNÂN Â CNÂN 7 ! 20) ) ( ) ( @f Ã @f þ DW Df % DW, , @WÃ @W &( )' @f ¼ 2Re DW, @WÃ (1:33) @f is an N Â N matrix whose (m, n)th entry is the partial derivative of f with @W respect to wmn.

When deriving complex-valued signal processing algorithms, however, the functions of interest are real valued and have complex arguments z, hence are not analytic 22 COMPLEX-VALUED ADAPTIVE SIGNAL PROCESSING on the complex plane. In this case, we can use Wirtinger calculus and write the expansions by treating the function f(z) as a function of two arguments, z and zÃ . In this approach, the main idea is treating the two arguments as independent from each other, when they are obviously dependent on each other as we discussed.

We can calculate the derivatives of g with respect to Z and ZÃ by simply treating one variable as a constant and directly using the results from real-valued matrix differentiation as @g @Trace[Z(ZÃ )T ] ¼ ¼ ZÃ @Z @Z and @g ¼Z @ZÃ A good reference for real-valued matrix derivatives is [88] and a number of complex-valued matrix derivatives are discussed in detail in [46]. For computing matrix derivatives, a convenient tool is the use of differentials. In this procedure, ﬁrst the matrix differential is computed and then it is written in the canonical form by identifying the term of interest.