Complex-Valued Adaptive Signal Processing Using Nonlinear Functions

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.