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We describe a framework based on Wirtinger calculus for adaptive signal processing
that enables efficient derivation of algorithms by directly working in the complex
domain and taking full advantage of the power of complex-domain nonlinear processing.
We establish the basic relationships for optimization in the complex domain and the
real-domain equivalences for first- and second-order derivatives by extending the
work of Brandwood and van den Bos. Examples in the derivation of first- and second-order
update rules are given to demonstrate the versatility of the approach.