Complex proportionate-type affine projection algorithms

An extension of complex proportionate-type normalized least mean square algorithms is proposed and derived. This new algorithm called the complex proportionate-type affine projection algorithm helps the estimation of unknown impulse responses when the input signal is colored. The derivation of the complex proportionate-type affine projection algorithm is performed by minimizing the second norm of the weighted difference between the current estimate of the impulse response and the estimate at the next time step under constraints that multiple a posteriori output errors are zero. Several variants of the algorithm are obtained as in the case of the complex proportionate-type normalized least mean square algorithm. It is shown how to use the algorithm in the case of widely linear systems. The learning curves of the algorithms are compared for several standard gain assignment laws for colored and speech input. Through simulation it was demonstrated that the complex proportionate-type affine projection algorithm offers superior convergence performance for colored input signals relative to complex proportionate-type normalized least mean square algorithms and that using separate gains to update the real and imaginary parts of the estimated impulse response, as opposed to the same gain, improves the convergence performance.