Variable step-size μ-law memorised improved proportionate affine projection algorithm for sparse system identification

System identification has wide applications in adaptive echo/feedback cancellation, active noise control, adaptive channel equalization, etc. When the system is sparse, recently, a μ-law memorised improved proportionate affine projection algorithm (MMIPAPA) has been proposed to improve the misalignment performance remarkably. However, the MMIPAPA with constant step-size has the conflicting requirement of fast convergence rate and low steady-state error. To solve this problem, a variable step-size version of the MMIPAPA (VSS-MMIPAPA) has been extended by setting each component of the a posterior error energy vector equal the system noise energy. Furthermore, through an alternative method to compute the variable step-size, when the estimated component of the a prior error energy vector is smaller than the system noise energy, further lower steady-state misalignment is achieved. This method leads to an improved VSS-MMIPAPA (IVSS-MMIPAPA). The computational complexity of the IVSS-MMIPAPA is O(P2L), which may be too high for many applications. Therefore, an approximate algorithm with little performance degradation has been implemented using recursive filtering and dichotomous coordinate descent iteration techniques, by which the complexity has been reduced to O(PL). Finally, the efficiency of both the exact and approximate algorithms developed here has been verified by simulation results.