Variable Step-Size Widely Linear Complex-Valued Affine Projection Algorithm and Performance Analysis

In this paper, a variable step-size widely linear complex-valued affine projection algorithm (VSS-WLCAPA) is proposed for processing noncircular signals. The variable step-size (VSS) is derived by minimizing the power of the augmented noise-free a posteriori error vector, which speeds up the convergence and reduces the steady-state misalignment. By exploiting the evolution of the covariance matrix of the weight error vector, we provide insight into the theoretical behavior of the VSS-WLCAPA algorithm. In the analysis, we take into account the dependency between the weight error vector and the noise vector, which is useful for accuracy of the theoretical prediction. To evaluate the mean step-size, the probability density function of the magnitude of the error is derived by employing polar coordinate transformation. Moreover, a special case when the projection order reduces to one is analysed in detail. The presented theoretical analysis is different from existing methodologies for analyzing affine projection algorithms due to the use of the Kronecker product. Simulation results for system identification scenarios demonstrate the merits of the proposed algorithm and verify the accuracy of the theoretical analysis. Wind prediction experiments support the superiority of the proposed VSS-WLCAPA as well.