Fundamental relations between the LMS algorithm and the DFT

The digital Fourier transform (DFT) and the adaptive least mean square (LMS) algorithm have existed for some time. This paper establishes a connection between them. The result is the "LMS spectrum analyzer," a new means for the calculation of the DFT. The method uses a set of periodic complex phasors whose frequencies are equally spaced from dc to the sampling frequency. The phasors are weighted and then are summed to generate a reconstructed signal. Weights are adapted to realize a best least squares fit between this reconstructed signal and the input signal whose spectrum is to be estimated. The magnitude squares of the weights correspond to the power spectrum. For a proper choice of adaptation speed, the LMS spectrum analyzer will provide an exact -sample DFT. New DFT outputs will be available in steady flow after the introduction of each new data sample.