Multirate signal estimation

This article introduces a technique for estimating samples of a random signal based on observations made by several observers and at different sampling rates. We consider a discrete-time mathematical model where an observer sees the original random signal x(n) through a bank of sensors which we model by linear filters and downsamplers. Each sensor, therefore, outputs a measurement signal v_i(n) whose sampling rate is only a fraction of the sampling rate assumed for the original signal under observation. It is straightforward to show that the optimal least-mean-squares estimator for our problem is a linear operator F operating on v_i(n)s. We observe, however, that to find F we need to know the power spectral density P_x(e^jw) of x(n) which is itself not observable. This motivates us to consider the possibility of estimating P_x(e^jw) using the observable low-rate data. We show that the statistical inference problem which addresses estimation of P_x(e^jw) given certain statistics of v_i(n) is mathematically ill-posed. We resolve this ill-posed inference problem using the principle of maximum entropy. We show, moreover, that the proposed maximum entropy inference technique is a continuous mapping. Therefore, one might safely use it to estimate P_x(e^jw) based on approximate statistics of v_i(n) obtained from the samples