# Compressive Estimation of a Stochastic Process with Unknown Autocorrelation Function

## TL;DR

This paper introduces a blind spectral estimation method for predicting stationary Gaussian processes without prior knowledge of the autocorrelation function, achieving performance comparable to traditional MMSE predictors.

## Contribution

It proposes a novel spectral estimation approach using atomic-norm minimization for blind prediction of Gaussian processes with unknown autocorrelation.

## Key findings

- The blind predictor performs comparably to the MMSE predictor with known PSD.
- Atomic-norm minimization effectively quantizes the spectral content from observed samples.
- Simulation results validate the proposed method's effectiveness.

## Abstract

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