Tracking the Frequency Translation of a Sum of Orthogonal Sinusoids

The maximum likelihood approach is used to derive a method for estimating and tracking the frequency translation of a signal consisting of a sum of orthogonal sinusoids corrupted by additive white noise. The likelihood function is reduced to an equivalent statistic expressed in terms of the squared magnitude of the finite Fourier transform of the received signal. A function that generates an error signal for a frequency translation tracking loop is derived, and a method of generating the error signal using the discrete Fourier transform (DFT) of the received signal weighted by a linear ramp is suggested. Two noise-free examples are presented.