Variable step-size widely linear complex-valued NLMS algorithm and its performance analysis

The shrinkage widely linear complex-valued least mean square (SWL-CLMS) algorithm with a variable step-size (VSS) overcomes the tradeoff between fast convergence and low steady-state misalignment, but meanwhile suffers from instability for highly correlated input signals because of the gradient noise amplification problem. To obtain a VSS that is also applicable to the case of highly correlated input signals, in this paper, we propose the VSS widely linear complex-valued normalized least mean square (VSS-WL-CNLMS) algorithm, where the VSS is derived by minimizing the mean-square deviation (MSD). Owing to the normalization, the VSS-WL-CNLMS algorithm is convergent in the mean square sense. By using the Rayleigh distribution, we calculate the mean step-size, which is then combined with the approximate uncorrelating transform to analyze the transient and steady-state mean square error (MSE) behaviors. Simulations for system identification scenario show that the proposed VSS-WL-CNLMS algorithm outperforms some well-known techniques and verify the accuracy of the theoretical analysis.