Statistical properties of the LMS fourier analyzer in the presence of frequency mismatch

TitleStatistical properties of the LMS fourier analyzer in the presence of frequency mismatch
Publication TypeJournal Article
Year of Publication2004
AuthorsXiao, Y., A. Ikuta, L. Ma, L. Xu, and R. K. Ward
JournalCircuits and Systems I: Regular Paper s, IEEE Transactions on
Pagination2504 - 2515
Date Publisheddec.
Keywordsadaptive Fourier analyzers, adaptive signal processing, closed-form steady-state expression, difference equations, discrete Fourier coefficient, discrete Fourier transforms, Fourier analysis, frequency mismatch, least mean square algorithms, least mean squares methods, linear combiner, LMS Fourier analyzer, mean square error, optimum step size parameters, recursive least square algorithms, signal frequencies, sinusoidal signal, statistical properties

The statistical performances of the conventional adaptive Fourier analyzers, such as the least mean square (LMS), the recursive least square (RLS) algorithms, and so forth, may degenerate significantly, if the signal frequencies given to the analyzers are different from the true signal frequencies. This difference is referred to as frequency mismatch (FM). We analyze extensively the performance of the conventional LMS Fourier analyzer in the presence of FM. Difference equations governing the dynamics and closed-form steady-state expression for the estimation mean square error (MSE) of the algorithm are derived in detail. It is revealed that the discrete Fourier coefficient (DFC) estimation problem in the LMS eventually reduces to a DFC tracking one due to the FM, and an additional term derived from DFC tracking appears in the closed-form MSE expression, which essentially deteriorates the performance of the algorithm. How to derive the optimum step size parameters that minimize or mitigate the influence of the FM is also presented, which can be used to perform robust design of step size parameters for the LMS algorithm in the presence of FM. Extensive simulations are conducted to reveal the validity of the analytical results.


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