Compressive Estimation of a Stochastic Process with Unknown\n Autocorrelation Function

Abstract

In this paper, we study the prediction of a circularly symmetric zero-mean\nstationary Gaussian process from a window of observations consisting of\nfinitely many samples. This is a prevalent problem in a wide range of\napplications in communication theory and signal processing. Due to\nstationarity, when the autocorrelation function or equivalently the power\nspectral density (PSD) of the process is available, the Minimum Mean Squared\nError (MMSE) predictor is readily obtained. In particular, it is given by a\nlinear operator that depends on autocorrelation of the process as well as the\nnoise power in the observed samples. The prediction becomes, however, quite\nchallenging when the PSD of the process is unknown. In this paper, we propose a\nblind predictor that does not require the a priori knowledge of the PSD of the\nprocess and compare its performance with that of an MMSE predictor that has a\nfull knowledge of the PSD. To design such a blind predictor, we use the random\nspectral representation of a stationary Gaussian process. We apply the\nwell-known atomic-norm minimization technique to the observed samples to obtain\na discrete quantization of the underlying random spectrum, which we use to\npredict the process. Our simulation results show that this estimator has a good\nperformance comparable with that of the MMSE estimator.\n