A variable step size total least squares affine-projection-like algorithm: Formula derivation and performance analysis.

• The TLS-APL algorithm is proposed by using the gradient descent method and the unconstrained optimization method, which solves the problem of overall performance degradation of the APL algorithm in the EIV model. • By using some common assumptions, the local extremum of the TLS-APL algorithm is analyzed. Therefore, the expected parameter is the optimal value of the iterative estimation and fully demonstrates the correctness of the iterative update. • The mean, mean square, and steady-state MSD performance of the TLS-APL algorithm is derived by using the Kronecker product and vec operator respectively. In addition, the fitting effects of theoretical steady-state MSD curves and actual steady-state MSD curves under different parameters are analyzed, which proves the rationality of theoretical derivation. • The VSS method is presented from the maximum decrease of MSD from the current iteration to the next iteration. Compared to the TLS-APL algorithm with a fixed step size, the VSS-TLS-APL algorithm with the VSS method has both fast convergence speed and low steady-state error. In the errors-in-variables (EIV) model, the steady state characteristics of the affine-projection-like (APL) algorithm are poor. Hence, the total least squares APL (TLS-APL) algorithm different from the bias-compensated is proposed, which can reduce the adverse influence of input and output noise on weight update. Then, the local extremum of the TLS-APL algorithm is verified, that is, the expected parameter is the optimal value of the iterative estimation. Moreover, the mean, mean square, and steady-state mean square deviation (MSD) performance of the proposed TLS-APL algorithm are deduced by using the Kronecker product and vec operator, respectively. To balance the convergence characteristics and steady-state characteristics, the step size optimization of the TLS-APL algorithm, namely the variable step size TLS-APL (VSS-TLS-APL) algorithm, is derived from the analysis of MSD. Considering the existence of gradient errors, the fitting effects of theoretical steady-state MSD curves and actual steady-state MSD curves under different parameters are analyzed separately, which proves the rationality of theoretical derivation. In addition, the VSS-TLS-APL algorithm with optimized step size is compared with the TLS-APL algorithm with fixed step size in the application of system identification, and the effect of the variable step size (VSS) is verified. The experiments of system identification and acoustic echo cancelation prove that the VSS-TLS-APL algorithm is superior to the existing known algorithms. [ABSTRACT FROM AUTHOR]