Complex-valued proportionate affine projection Versoria algorithms and their combined-step-size variants for sparse system identification under impulsive noises

Abstract In real life, many engineering problems are modeled in the complex domain (CD). This paper proposes two complex-valued proportionate adaptive filtering (CPAF) algorithms in the CD for identifying the complex sparse systems in impulsive noises. The proposed CPAF algorithms are derived by maximizing the reuse of Versoria cost subjected to the weighted squared Euclidean norm of the filter tap-weight vector difference with the proportionate matrices as the weight of the weighted Euclidean norm. To address the tradeoff problem between the fast convergence rate and low steady-state misadjustment of the CPAF algorithms, two combined-step-size (CSS) CPAF algorithms are also proposed in the CD, which are obtained by introducing a time-varying step-size bound (TVSSB) into the constraint condition of the cost function of the CPAF algorithms. The TVSSB is achieved adaptively by using a modified sigmoidal function (MSF) to combine a large step-size and a small one. In order to be robust against impulsive noises, the MSF is adapted with the aid of a stochastic gradient ascent algorithm which is obtained by maximizing the Versoria cost function with respect to the real part of the a priori prediction error. Simulations on complex sparse system identification under impulsive noises in the CD demonstrate that the performance of the proposed CPAF algorithms is superior to that of the complex-valued variants of the affine projection Versoria, affine projection sign (APS), proportionate APS (PAPS), and improved PAPS (IPAPS) algorithms, also validate the effectiveness of the proposed CSS-CPAF algorithms.