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.

A parallel notch filter (PNF) for eliminating a sinusoidal signal whose frequency and phase are unknown, has been proposed previously. The PNF achieves both fast convergence and high estimation accuracy when the step-size for adaptation is appropriately determined. However, there has been no discussion of how to determine the appropriate step-size. In this paper, we derive the convergence condition on the step-size, and propose an adaptive algorithm with variable step-size so that convergence of the PNF is automatically satisfied. Moreover, we present a new filtering structure of the PNF that increases the convergence speed while keeping the estimation accuracy. We also derive a variable step-size scheme for the new PNF to guarantee the convergence. Simulation results show the effectiveness of the proposed method.

In this paper, we propose a real-time BSS (Blind Source Separation) system with two microphones that extracts only desired sound sources. Under the assumption that the desired sound sources are close to the microphones, the proposed BSS system suppresses distant sound sources as undesired sound sources. We previously developed a BSS system that can estimate the distance from a microphone to a sound source and suppress distant sound sources, but it was not a real-time processing system. The proposed BSS system is a real-time version of our previous BSS system. To develop the proposed BSS system, we simplify some BSS procedures of the previous system. Simulation results showed that the proposed system can effectively suppress the distant source signals in real-time and has almost the same capability as the previous system.