Convergence Vectors in System Identification with an NLMS Algorithm for Sinusoidal Inputs

For an adaptive system identification filter with a stochastic input signal, a coefficient vector updated with an NLMS algorithm converges in the sense of ensemble average and the expected convergence vector has been revealed. When the input signal is periodic, the convergence of the adaptive filter coefficients has also been proved. However, its convergence vector has not been revealed. In this paper, we derive the convergence vector of adaptive filter coefficients updated with the NLMS algorithm in system identification for deterministic sinusoidal inputs. Firstly, we derive the convergence vector when a disturbance does not exist. We show that the derived convergence vector depends only on the initial vector and the sinusoidal frequencies, and it is independent of the step-size for adaptation, sinusoidal amplitudes, and phases. Next, we derive the expected convergence vector when the disturbance exists. Simulation results support the validity of the derived convergence vectors.