Complex-Valued Adaptive Filtering Based on the Random Fourier Features Method

Abstract

The complex kernel adaptive filters (CKAFs) developed in complex reproducing kernel Hilbert space (RKHS) improve the performance of complex linear adaptive filters, but result in large burdens of computation and memory. To address these issues, a novel complex random Fourier features mapping (CRFFM) is proposed to approximate the kernel-induced mapping by applying the complexification of real RKHSs, and thus projects the complex-valued data from the original data space into the fixed-dimensional complex random Fourier features space (RFFS). To combat complex-valued non-Gaussian noises, a complex Cauchy loss function is presented, and the complex random Fourier features recursive complex Cauchy (CRFFRCC) algorithm is therefore proposed by the combination of CRFFM and the stochastic recursive method. The proposed CRFFRCC with a linear filter structure can reduce the computational and space complexities of CKAFs, significantly. Monte Carlo simulations conducted in the complex-valued nonlinear channel equalization validate the superiorities of CRFFRCC.