Complex-Valued Random Fourier Geometric Algebra Adaptive Filtering

Abstract

The complex-valued fixed dimensional adaptive filters (CFDAFs) can improve the performance and computational efficiency of complex-valued kernel adaptive filters (CKAFs), but have the challenge of over-coupling for the real and imaginary parts of complex-valued signals. To this end, this brief proposes a novel fixed dimensional adaptive filter for complex-valued signals named complex-valued random Fourier geometric algebra least mean square (CRFGALMS). On the basis of the framework of geometric algebraic adaptive filtering, the real and imaginary parts of complex-valued signals are mapped into random Fourier features space (RFFS) to improve the efficiency of the nonlinear mapping for complex-valued signals, respectively. The proposed random Fourier feature mapping based on the geometric algebra is endowed with superior presentation abilities in complex-valued domain as it can decouple the nonlinear mapping of the real and imaginary parts of complex-valued signals, efficiently. Simulations on nonlinear channel equalization are used to illustrate the superiority of the proposed CRFGALMS algorithm in terms of dimensionality and filtering performance.