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The conventional LMS Fourier analyzer has been successfully used to analyze sinusoidal and periodic signals in additive noise. If the user provides the correct signal frequencies, the analyzer produces good estimates for the discrete Fourier coefficients (DFCs) of the signal. However, if the signal frequencies fed to the analyzer vary from the true signal frequencies, i.e., a frequency mismatch (FM) exists, the performance of the conventional LMS algorithm degenerates. We propose a new LMS-based Fourier analyzer that yields superior results to the conventional LMS one. The estimation of DFCs and the reduction of FM in the new algorithm are carried out simultaneously based on the least mean square and the least mean p-power error criteria, respectively. This new algorithm has a simple structure and shows a small increase in computations. Simulations as well as a real-life application to real signals generated by a large-scale factory cutting machine are provided to show the effectiveness of our new algorithm. For the latter, the performance improvement is as high as 8.3 [dB].