Frames, Erasures, and Signal Estimation with Stochastic Models

Abstract

Frame properties and conditions are determined that would minimize the error in signal reconstruction or estimation in the presence of noise and erasures. The special focus here is on stochastic models. These include estimating a random signal with zero mean and a general covariance matrix, minimizing the mean-squared error (MSE) when the frame coefficients are erased according to some a priori probability distribution in the presence of random noise, and also studying the use of stochastic frames in estimating a random signal. In estimating a random signal from noisy coefficients, when a frame coefficient is lost or erased, it is established that the MSE is minimized under certain geometric relationships between the frame vectors and the signal. When the coefficients are erased according to some a priori distribution, conditions are found for the norms of the frame vectors in terms of the probability distribution of the erasure so that the MSE is minimized. Results obtained here also show how using stochastic frames can lead to more flexibility in design and greater control on the MSE.