A new LMS-based Fourier analyzer in the presence of frequency mismatch

Adaptive Fourier analyzers estimate the coefficients of the sine and cosine terms of a noisy sinusoidal signal assuming the frequencies are known. In real-life applications though, the frequencies may vary from their supposed values. This is referred to as frequency mismatch (FM). In this paper, we analyze the performance of the conventional LMS Fourier analyzer under existence of the FM. The dynamics and steady-state properties of the LMS algorithm are derived in detail. An optimum step size parameter is also derived, which minimizes the influence of the FM in the mean square error (MSE) sense. Based on the insights provided by the analysis, we then introduce a novel LMS-based Fourier analyzer which simultaneously estimates the discrete Fourier coefficients (DFCs) and accommodates the FM. This new LMS-based algorithm has very simple structure, and hence introduces a small increase in computations compared with the conventional LMS algorithm. However, it can compensate, almost completely, for the performance degeneration due to the FM. Simulations are conducted to show the validity of the analytical results and the excellent performance of the new LMS-based algorithm.