The least mean squares (LMS) algorithm has been widely used for adaptive filtering because of easily implementing at a computational complexity of O(2N) where N is the number of taps. The drawback of the LMS algorithm is that its performance is sensitive to the scaling of the input. The normalized LMS (NLMS) algorithm solves this problem on the LMS algorithm by normalizing with the sliding-window power of the input; however, this normalization increases the computational cost to O(3N) per iteration. In this work, we derive a new formula to strictly perform the NLMS algorithm at a computational complexity of O(2N), that is referred to as the C-NLMS algorithm. The derivation of the C-NLMS algorithm uses the H∞ framework presented previously by one of the authors for creating a unified view of adaptive filtering algorithms. The validity of the C-NLMS algorithm is verified using simulations.

This paper proposes a new adaptive comb filter which automatically designs its characteristics. The comb filter is used to eliminate a periodic noise from an observed signal. To design the comb filter, there exists three important factors which are so-called notch frequency, notch gain, and notch bandwidth. The notch frequency is the null frequency which is aligned at equally spaced frequencies. The notch gain controls an elimination quantity of the observed signal at notch frequencies. The notch bandwidth controls an elimination bandwidth of the observed signal at notch frequencies. We have previously proposed a comb filter which can adjust the notch gain adaptively to eliminate the periodic noise. In this paper, to eliminate the periodic noise when its frequencies fluctuate, we propose the comb filter which achieves the adaptive notch gain and the adaptive notch bandwidth, simultaneously. Simulation results show the effectiveness of the proposed adaptive comb filter.

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.

We have proposed a noise reduction method based on a noise reconstruction system (NRS). The NRS uses a linear prediction error filter (LPEF) and a noise reconstruction filter (NRF) which estimates background noise by system identification. In case a fixed step size for updating tap coefficients of the NRF is used, it is difficult to reduce background noise while maintaining the high quality of enhanced speech. In order to solve the problem, a variable step size is proposed. It makes use of cross-correlation between an input signal and an enhanced speech signal. In a speech section, a variable step size becomes small so as not to estimate speech, on the other hand, large to track the background noise in a non-speech section.

Using a pair of matched square-root-raised-cosine (SRRC) filters in the transmitter and the receiver in a band-limited digital communication system can theoretically achieve zero inter-symbol interference (ISI). In reality, the ISI cannot be zero when both SRRC filters are approximately implemented because of some numerical precision problems in the design phase as well as in the implementation phase. In this paper, the author proposes an iterative method to design the coefficients of SRRC FIR filters. The required ISI of the system can be specified such that both ISI and frequency domain specifications are monitored in the design phase. Since the ISI can be specified beforehand, the tradeoff between performance and the filter length becomes possible in the proposed design algorithm.

In this letter, we present a normalized least-mean-square algorithm of blind estimator for carrier frequency offset estimation of orthogonal frequency division multiplexing systems. In conjunction with the closed-loop estimate structure, the proposed efficient algorithm eliminates the inter-carrier interference for time varying carrier frequency offset. The proposed algorithm offers faster convergence speed and more accuracy to the carrier frequency offset estimate. Several computer simulation examples are presented for illustrating and effectiveness of the proposed algorithm.

The adaptive system design by 16-bit fixed point processing enables to employ an inexpensive digital signal processor (DSP). The narrow dynamic range of such 16 bits, however, does not guarantee the same performance that is confirmed beforehand by computer simulations. A cause of degrading the performance originates in the operation halving the word length doubled by multiplication. This operation rounds off small signals staying in the lower half of the doubled word length to zero. This problem can be solved by limiting the multiplier to only its sign () like the signed regressor algorithm, named 'bi-quantized-x' algorithm in this paper, for the convenience mentioned below. This paper first derives the equation describing the convergence property provided by a type of signed regressor algorithms, the bi-quantized-x normalized least mean square (NLMS) algorithm, and then formulates its convergence condition and the step size maximizing the convergence rate. This paper second presents a technique to improve the convergence property. The bi-qiantized-x NLMS algorithm quantizes the reference signal to 1 according to the sign of the reference signal, whereas the technique moreover assigns zero to the reference signal whose amplitude is less than a predetermined level. This paper explains the principle that the 'tri-qunatized-x' NLMS algorithm employing the technique can improve the convergence property, and confirms the improvement effect by computer simulations.

The signed regressor algorithm, a variation of the least mean square (LMS) algorithm, is characterized by the estimation way of using the clipped reference signals, namely, its sign (). This clipping, equivalent to quantizing the reference signal to 1, only increases the estimation error by about 2 dB. This paper proposes to increase the number of the quantization steps to three, namely, 1 and 0, and shows that the 'tri-quantized-x' normalized least mean square (NLMS) algorithm with three quantization steps improves the convergence property.

The individually normalized least mean square (INLMS) algorithm is proposed as an adaptive algorithm suitable for the fixed point processing. The convergence property of the INLMS algorithm, however, is not yet analyzed enough. This paper first derives an equation describing the convergence property by exploiting the technique of expressing the INLMS algorithm as a first order infinite impulse response (IIR) filter. According to the equation derived thus, the decreasing process of the estimation error is represented as the response of another IIR filter expression. By using the representation, this paper second derives the convergence condition of the INLMS algorithm as the range of the step size making a low path filter of the latter IIR filter. This paper also derives the step size maximizing the convergence speed as the maximum coefficient of the latter IIR filter and finally clarifies the range of the step size recommended in the practical system design.

This paper introduces an acoustic echo canceller system materialized with a 16-bit fixed point processing type DSP (Analog Devices, ADSP-2181). This experimental system uses the tri-quantized-x individually normalized least mean square (INLMS) algorithm little degrading the convergence property under the fixed point processing. The experimental system also applies a small step gain to the algorithm to prevent the double-talk from increasing the estimation error. Such a small step gain naturally reduces the convergence speed. The experimental system compensates the reduction by applying the block length adjustment technique to the algorithm. This technique enables to ceaselessly update the coefficients of the adaptive filter even when the reference signal power is low. The experimental system thus keeps the echo return loss enhancement (ERLE) high against the double-talk.

The filtered-x algorithm, which is widely applied to active noise control system, requires setting a small step gain. Such a small step gain reduces the noise reduction effect when the alogrithm is implemented by fixed point processing. This paper presents an experimental result that the 'polarized-g' individually normalized least mean square (INLMS) algorithm can provide almost the same noise reduction effect even in the fixed point processing of 16 bits as that in floating point processing.