Complex proportionate-type normalized least mean square algorithms

A complex proportionate-type normalized least mean square algorithm is derived 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 the constraint that the adaptive filter a posteriori output is equal to the measured output. The weighting function is assumed positive but otherwise arbitrary and it is directly related to the update gains. No assumptions regarding the input signal are made during the derivation. Different weights (i.e., gains) are used for real and imaginary parts of the estimated impulse response. After additional assumptions special cases of the algorithm are obtained: the algorithm with one gain per impulse response coefficient and the algorithm with lower computational complexity. The learning curves of the algorithms are compared for several standard gain assignment laws for white and colored input. It was demonstrated that, in general, the algorithms with separate gains for real and imaginary parts have faster convergence.