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.
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Notes
This result was proved in [4] with the assumption that the starting frame is unit norm.
By condition number of a frame is meant the ratio of the maximum and the minimum eigenvalue of the frame operator, i.e., the ratio of the optimal upper and lower frame bounds.
As a one dimensional sequence, if \(\{ Y_{k}\} _{k \in \mathbb{Z}}\) are i.i.d. random variables following a Gaussian distribution with mean zero and variance \(\sigma ^{2}\), then \(X: \mathbb{Z} \to \mathbb{C}\) with \(X(k) = e^{\frac{2\pi }{\epsilon } i Y_{k}}\) is a sequence whose autocorrelation can be made arbitrarily small, depending on \(\epsilon \), everywhere except at the origin [2].
Even though the calculations here are with the Gaussian distribution, some other distribution may be assumed.
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Acknowledgements
The author would like to thank Doug Cochran for many useful discussions on the topic of this work. The author was supported by the National Science Foundation under Award No. CCF-1422252.
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Datta, S. Frames, Erasures, and Signal Estimation with Stochastic Models. Acta Appl Math 169, 411–431 (2020). https://doi.org/10.1007/s10440-019-00304-x
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DOI: https://doi.org/10.1007/s10440-019-00304-x