Fundamental relations between LMS spectrum analyzer and recursive least squares estimation

The least-mean-square (LMS) spectrum analyzer of B. Widrow et al. has been shown to be a means for the calculation of the discrete Fourier transform (DFT) (ibid., vol.CAS-34, no.7, p.814-19, 1987). It uses N periodic complex phasors whose frequencies are equally spaced between DC and the sampling frequency. The phasors are weighted and summed to generate a reconstructed signal, the weights are adapted to provide a least-squares fit between the reconstructed signal and and the input signal whose spectrum is desired. Here, the LMS spectrum analyzer is shown to solve the problem of minimizing the exponentially weighted sum of errors, and results in a recursive estimator. A filter bank model is also deduced