An FIR Notch Filter for Adaptive Filtering of a Sinusoid in Correlated Noise
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An FIR Notch Filter for Adaptive Filtering of a Sinusoid in Correlated Noise Osman Kukrer and Aykut Hocanin Department of Electrical and Electronics Engineering, Eastern Mediterranean University, Gazimagusa, Mersin 10, Turkey Received 26 July 2005; Revised 23 January 2006; Accepted 18 February 2006 Recommended for Publication by Richard Heusdens A novel adaptive FIR filter for the estimation of a single-tone sinusoid corrupted by additive noise is described. The filter is based on an offline optimization procedure which, for a given notch frequency, computes the filter coefficients such that the frequency response is unity at that frequency and a weighted noise gain is minimized. A set of such coefficients is obtained for notch frequencies chosen at regular intervals in a given range. The filter coefficients corresponding to any frequency in the range are computed using an interpolation scheme. An adaptation algorithm is developed so that the filter tracks the sinusoid of unknown frequency. The algorithm first estimates the frequency of the sinusoid and then updates the filter coefficients using this estimate. An application of the algorithm to beamforming is included for angle-of-arrival estimation. Simulation results are presented for a sinusoid in correlated noise, and compared with those for the adaptive IIR notch filter. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
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INTRODUCTION
Estimation of sinusoidal signals and their frequencies from noisy measurements is important in many fields such as angle of arrival estimation, frequency-shift keying (FSK) demodulation, Doppler estimation of radar waveforms, biomedical engineering, sensor array processing, and cancellation of periodic interferences [1]. The observed signal has the following general form: x(k) = a cos(kθ + φ) + q(k).
(1)
The problem in many applications is to recover the signal and/or its frequency θ, from the noisy observations x(k). Various adaptive filtering algorithms have been introduced for solving such problems. The least-mean-square (LMS) algorithm [2] based on the FIR transversal filter has been widely used due to its simplicity and robustness. On the other hand, the performance of this algorithm deteriorates when the input signal is correlated [3]. Transform-domain techniques have been introduced to decorrelate the input signal and achieve faster convergence [3, 4]. Also, in certain applications, the filter length required for a satisfactory performance is large. The adaptive IIR filter, also known as the adaptive notch filter [5–7], has been introduced as an alternative to the LMS FIR filter. The IIR filter has the outstanding advantage of requiring considerably fewer coefficients compared
with its FIR counterpart. However, the performance of the IIR notch filter in correlated noise has not been studied well in the literature. In [8], an adaptive IIR notch filter for suppressing narrow-band interference is described, where the filter’s bandwidth is adaptively controlled to maximize SNR. In this paper, a notch filter based
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