Adaptive Frequency-Domain Normalized Implementations of Widely-Linear Complex-Valued Filter

The widely-linear complex-valued least-mean-square (WL-CLMS) algorithm exhibits slow convergence in the presence of non-circular and highly correlated filter input signals. To tackle such an issue with reduced computational complexity, this paper introduces adaptive frequency-domain normalized implementations of widely-linear complex-valued filter. Two normalized algorithms are firstly devised based on the circulant matrices of weight coefficients and the regression vector, respectively. In the design, the normalization matrix using the second-order complementary statistical information of the input signal helps increase the algorithm convergence speed. Then, mean-square and complementary mean-square performance of periodic update frequency-domain widely-linear NLMS (P-FDWL-NLMS) algorithm for non-circular complex signals is analyzed. In addition, by introducing a variable-periodic (VP) mechanism, we propose the VP-FDWL-NLMS method that provides faster convergence than the P-FDWL-NLMS scheme. Computer simulation results show the superiority of the proposed approach over the fullband widely-linear complex-valued least-mean-square, augmented affine projection algorithm and its variable step-size version, in terms of both complexity and convergence rate.