Maximum entropy spectral analysis of multiple sinusoids in noise

Abstract

An analytical technique based on the method of undetermined coefficients is applied to the problem of computing the theoretical spectral estimate by the maximum entropy method (MEM) when the autocorrelation function of the data is known exactly and corresponds to N sinusoids in additive white noise and to N sinusoids in additive 1-pole, low‐pass noise. For the white noise case, the L prediction filter coefficients are expanded directly in terms of the input sinusoids. This expansion leads to a transformation of the L × L normal equations for the prediction filter coefficients to a set of 2N × 2N equations. The transformed equations are a smaller set of equations to be solved whenever L > 2N and provide a convenient description of the interaction between the various frequency components of the sinusoids which occurs in the MEM estimate. Further, for certain cases where there is little interaction between some of the frequency components of the sinusoids, the solution of the 2N × 2N equations may be approximated (to zeroth order) by the solution of a smaller set of coupled equations. A better approximation to the exact solution of the 2N × 2N equations can then be obtained from a perturbation expansion of the exact solution about the zeroth order approximation. For the case of N sinusoids in l‐pole, low‐pass noise, the L prediction filter coefficients are expanded in terms of the input sinusoids as well as two delta functions which occur at the beginning and end of the filter. This expansion also leads to a set of 2N × 2N equations. For this case the values of the MEM estimate evaluated at the frequencies of the sinusoids are shown to be a function of the frequencies of the sinusoids. This result is reasonable since the signal‐to‐noise ratio per unit bandwidth is also a function of frequency.