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Adaptive frequency estimation and new convergence properties for the least mean square algorithm.
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1980
[PDF-309.86 MB]
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Description:The convergence properties of the least mean square (LMS) algorithm are interpreted in terms of a vector space associated with the coefficients of the adaptive linear prediction filter (ALPF). Signal planes defined in this weight vector space are used to describe the frequency tracking by characteristics of spectral estimators based on the ALPF and are used to explain the effects of both the filter parameters and the algorithm on tracking speed. The performances of three different adaptive frequency estimators derived from the ALPF are compared. Two of these employ Fourier transforms of the coefficients and the third is based on a transform of the ALPF output. Comparisons with the conventional periodogram spectrum estimator are presented in terms of a signal-to-noise
ratio (SNR) defined in frequency domain parameters. Specific calculations for one ALPF frequency estimator (the maximum entropy estimator) are used to demonstrate a bias in this estimator. It is known that continuously adapting filter coefficients fluctuate about their mean steady-state values and thereby generate "misadjustment noise" at the filter output. Freezing the converged weight vector avoids these fluctuations and is used in some adaptive applications. It is shown that the expected value of the excess mean square error (mse) for the frozen weight case is identical to the excess mse for the continuously adapting weight case. Closed form expressions for the variance of the frozen weight excess mse are also given. For an ALPF having a noisy sinusoidal input at frequency f, a periodic oscillation at frequency 2f on the adapting weight vector is predicted and verified experimentally. It is also shown that at convergence the random weight vector fluctuations mask these cyclic perturbations. For an ALPF having an input of a deterministic signal in white noise, it is shown that the steady state weight vector is nonoptimal when the filter coefficients are adapted at intervals less than or equal to the filter length. That is, adaptive filtering does not yield minimum mse at the filter output. Non-optimal convergence is shown to be caused by the correlation between the data vector and the weight vector that exists under these conditions. (This NOAA Technical Memorandum is essentially a copy of the Ph.D. Dissertation by R. Jeffrey Keeler, approved at the Department of Electrical Engineering, University of Colorado, December, 1979.)
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Rights Information:CC0 Public Domain
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