Multirate spectral estimation

This article introduces a mathematical theory for estimating the power spectral density (PSD) of a random signal based on low-sampling-rate measurements. We formulate the problem using a mathematical model where an observer sees a discrete-time WSS (wide-sense stationary) random signal x(n) through a bank of measurement devices or sensors. Each sensor outputs a measurement signal v_i (n) whose sampling rate is only a fraction of the sampling rate assumed for the original non-observable signal. Knowing statistics of v _i(n) is not, in general, sufficient to specify the PSD of x(n) uniquely. Therefore, the problem of multirate spectral estimation is mathematically ill-posed. We show that it is possible to convert the multirate spectral estimation problem into a mathematically well-posed one using the maximum entropy principle. Moreover, we obtain a closed-form expression for the PSD estimate that results from applying this principle and show that it is unique