Complex FIR block adaptive digital filtering algorithm with independent adaptation of real and imaginary filter parameters

Complex signal representations are being frequently employed in various adaptive filtering applications such as wireless communications, beamforming, etc. In this paper, a novel complex optimum block adaptive algorithm with individual adaptation of parameters (Complex OBAI-LMS) is presented. The proposed technique effectively utilizes the degrees of freedom of the adaptive filter by individually adapting the real and imaginary components of the complex adaptive finite impulse response (FIR) filter coefficients employing optimally derived convergence factors. In addition, the convergence factors are updated at each block iteration. The formulation of the complex OBAI-LMS shows that the update vectors for the real and imaginary components of the adaptive filter coefficients are estimates of the Wiener solution at each iteration. Furthermore, the matrix inversion operation in the formulation is eliminated by processing the input signal in overlapping blocks and applying a matrix inversion lemma. The convergence properties of the complex OBAI-LMS are compared to the block implementation of the complex LMS algorithm in the estimation of a complex FIR filter. Simulation results show that the complex OBAI-LMS yields a significant improvement in convergence speed over the block complex LMS for different input training signals.