Analysis of the Unconstrained Frequency-Domain Block LMS for Second-Order Noncircular Inputs

Abstract

The unconstrained frequency-domain block least mean square (UFBLMS) algorithm is a de facto standard in frequency-domain adaptive filtering, owing to its fast convergence for correlated input signals. However, although many complex-valued signals in real-world applications are second-order noncircular (improper), for computational convenience, existing mean square analyses of UFBLMS assume second-order circular (proper) input data. This, in turn, makes conventional mean square evaluations of UFBLMS inadequate. To address this issue, we propose a novel performance evaluation framework of UFBLMS, which makes it possible to investigate how the full second-order statistics of second-order noncircular correlated input signals, i.e., both their covariance and complementary covariance, influence the output error and the weight error of UFBLMS. A unified stability bound on the step-size of UFBLMS is next provided, enabling intuitive closed-form relations, in the steady-state stage, between the terms within full second-order error statistics, namely, the mean square error (MSE) and complementary MSE, and the input noncircularity. Furthermore, this proposed full second-order statistical framework is shown to enable more physical insight into the underlying physics of the contribution of the real and imaginary channels of UFBLMS, which cannot be achieved using the standard MSE analysis due to its insufficient degrees of freedom. Simulations and real-world experiments in the system identification setting support the analysis.