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.

The convergence characteristics of the adaptive beamformer with the RLS algorithm are analyzed in this paper. In case of the RLS adaptive beamformer, the convergence characteristics are significantly affected by the spatial characteristics of the signals/noises in the environment. The purpose of this paper is to show how these physical parameters affect the convergence characteristics. In this paper, a typical environment where a few directional noises are accompanied by background noise is assumed, and the influence of each component of the environment is analyzed separately using rank analysis of the correlation matrix. For directional components, the convergence speed is faster for a smaller number of noise sources since the effective rank of the input correlation matrix is reduced. In the presence of background noise, the convergence speed is slowed down due to the increase of the effective rank. However, the convergence speed can be improved by controlling the initial matrix of the RLS algorithm. The latter section of this paper focuses on the physical interpretation of this initial matrix, in an attempt to elucidate the mechanism of the convergence characterisitics.

In this paper the normalized lease mean square (NLMS) algorithm based on clipping input samples with an arbitrary threshold level is studied. The convergence characteristics of these clipping algorithms with correlated data are presented. In the clipping algorithm, the input samples are clipped only when the input samples are greater than or equal to the threshold level and otherwise the input samples are set to zero. The results of the analysis yield that the gain constant to ensure convergence, the speed of the convergence, and the misadjustment are functions of the threshold level. Furthermore an optimum threshold level is derived in terms of the convergence speed under the condition of the constant misadjustment.

Reduction of the complexity of the NLMS algorithm has recceived attention in the area of adaptive filtering. A processing cost reduction method, in which the component of the weight vector is updated when the absolute value of the sample is greater than or equal to an arbitrary threshold level, has been proposed. The convergence analysis of the processing cost reduction method with white Gaussian data has been derived. However, a convergence analysis of this method with correlated Gaussian data, which is important for an actual application, is not studied. In this paper, we derive the convergence cheracteristics of the processing cost reduction method with correlated Gaussian data. From the analytical results, it is shown that the range of the gain constant to insure convergence is independent of the correlation of input samples. Also, it is shown that the misadjustment is independent of the correlation of input samples. Moreover, it is shown that the convergence rate is a function of the threshold level and the eigenvalues of the covariance matrix of input samples as well as the gain constant.

Improvement of the convergence characteristics of the NLMS algorithm has received attention in the area of adaptive filtering. A new variable stepsize NLMS method, in which the stepsize is updated optimally by using variances of the measured error signal and the estimated noise, is proposed. The optimal control equation of the stepsize has been derived from a convergence characteristic approximation. A new condition to judge convergence is introduced in this paper to ensure the fastest initial convergence speed by providing precise timing to start estimating noise level. And further, some adaptive smoothing devices have been added into the ADF to overcome the saturation problem of the identification error caused by some random deviations. By the simulation, The initial convergence speed and the identification error in precise identification mode is improved significantly by more precise adjustment of stepsize without increasing in computational cost. The results are the best ever reported performanced. This variable stepsize NLMS-ADF also shows good effectiveness even in severe conditions, such as noisy or fast changing circumstances.