A Widely Linear Complex-Valued Affine Projection Sign Algorithm with Its Steady-State Mean-Square Analysis

Abstract

In this article, a widely linear complex-valued affine projection sign algorithm is proposed for handling nonstationary noise in the complex domain. Apart from using the second-order statistical information in the complex domain adequately, the proposed algorithm can succeed in obtaining the robustness against the impulsive noise. With the help of the Price’s theorem and some rational assumptions, the steady-state mean-square analysis of the proposed algorithm has been shown in this paper. In addition, we achieve a low computational complexity combining the l₁-norm operation of the error signal. Moreover, we still analyze the stability of our finding, providing the range of the step size of its widely linear model. In the end, the results of experiments with the impulsive noise make clear that the proposed algorithm obtains the robustness for both complex circular and noncircular signals. However, simulation results without the impulsive noise indicate that the proposed algorithm gets a worse behavior in terms of normalized mean-square deviation compared with various existing algorithms. In the stereophonic acoustic echo cancellation, our finding still achieves a good performance.

Appendix

Consider the Price’s theorem and Assumptions 1–2, $E[{\mathbf{e}}_{a}^{H} (i){\rm sign}({\mathbf{e}}(i))]$ can be calculated as

$$ \begin{aligned} E\left[ {{\mathbf{e}}_{a}^{H} (i){\text{sign}}({\mathbf{e}}(i))} \right] & = PE\left[ {e_{a} (i){\text{sign}}(e(i))} \right] \\ & = P\sqrt {\frac{2}{\pi }} E\left[ {\left| {e_{a} (i)} \right|^{2} } \right]\left( {\frac{\Pr }{{\sqrt {E\left[ {\left| {e_{a} (i)} \right|^{2} } \right] + (\lambda + 1)\sigma_{v}^{2} } }} + \frac{1 - \Pr }{{\sqrt {E\left[ {\left| {e_{a} (i)} \right|^{2} } \right] + \sigma_{v}^{2} } }}} \right) \\ & \approx P\sqrt {\frac{2}{\pi }} E\left[ {\left| {e_{a} (i)} \right|^{2} } \right]\frac{1 - \Pr }{{\sqrt {E\left[ {\left| {e_{a} (i)} \right|^{2} } \right] + \sigma_{v}^{2} } }}. \\ \end{aligned} $$

(40)

In the same way, $E[{\text{sign}}({\mathbf{e}}^{H} (i)){\mathbf{e}}_{a} (i)]$ also can be got

$$ E\left[ {{\text{sign}}({\mathbf{e}}^{H} (i)){\mathbf{e}}_{a} (i)} \right] \approx P\sqrt {\frac{2}{\pi }} E\left[ {\left| {e_{a} (i)} \right|^{2} } \right]\frac{1 - \Pr }{{\sqrt {E\left[ {\left| {e_{a} (i)} \right|^{2} } \right] + \sigma_{v}^{2} } }}. $$

(41)

Moreover, this subsection still is employed in (36).