Unbiased normalized least-mean-square algorithm with input and output noise

Bias compensation algorithms have been extensively studied to improve the performance of adaptive filters in error-in-variable models. However, the performance of these algorithms degrades when the input and output noise are correlated. To address this limitation, we propose a new unbiased normalized least-mean-square algorithm that considers the correlation between input and output noise, which is not addressed by conventional bias-compensated algorithms. Our approach is based on a mean performance analysis framework of the weight error vector. The algorithm was derived by eliminating the bias caused by noisy input and accounting for the correlation between input and output noise. As a result, the proposed algorithm achieves unbiased estimation under these conditions. Additionally, we propose an estimation method to handle correlation between input and output noise when it is unknown. Simulations in system identification demonstrate that the proposed algorithm achieves improved steady-state performance and faster convergence in tracking scenarios compared to existing conventional algorithms, particularly with smaller step sizes.