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The performance of the conventional least mean square (LMS) Fourier analyzer may degenerate significantly, if the signal frequencies given to the analyzer are different from the true signal frequencies. This difference is referred to as frequency mismatch (FM). We first analyze the performance of the conventional LMS Fourier analyzer for a single sinusoid in the presence of FM. We derive the dynamics and steady-state properties of this analyzer as well as the optimum step size parameter which minimizes the influence of the FM. Extensive simulations reveal the validity of the analytical results. Next, a new LMS-based Fourier analyzer is proposed which simultaneously estimates the discrete Fourier coefficients (DFCs) and accommodates the FM. This new analyzer can very well compensate for the performance degeneration due to the FM. Applications to estimation/detection of dual-tone multiple frequencies (DTMF) signals and analysis of real-life noise signals generated by a large-scale factory cutting machine are provided to demonstrate the excellent performance of our new Fourier analyzer.