Widely Linear Momentum LMS Algorithm for Second Order Noncircular Signals and Performance Analysis

In this letter, we propose a widely linear momentum least mean squares (WLMLMS) algorithm by incorporating the momentum term into the widely linear framework. This approach effectively handles noncircular signals and achieves faster convergence compared to the conventional augmented complex LMS, with minimal additional computational cost. Since the lag term of momentum complicates the analysis, we perform the Schur-Cohn test in z domain, and the direct bound without mixing the momentum factor is derived, for the first time, which further simplifies the parameter selection. Our derived expressions elucidate that, although augmented statistics integrate into mean-square error (MSE) iterations, the stability threshold is predominantly influenced by the covariance of the input signal and the momentum factor. Additionally, the upper bound does not exhibit a monotonic decrease with the momentum factor. Finally, the theoretical findings of this study are corroborated through numerical simulations.