Unitary Precoding and Basis Dependency of MMSE Performance for Gaussian Erasure Channels

Details Unitary Precoding and Basis Dependency of MMSE Performance for Gaussian Erasure Channels Unitary Precoding and Basis Dependency of MMSE Performance for Gaussian Erasure Channels

2011-11-10

arxiv.org/abs/1111.2451v1

Computer Science

Information Theory

Submitted to IEEE

Scientific paper

We consider the transmission of a Gaussian vector source over a multi-dimensional Gaussian channel where a random or a fixed subset of the channel outputs are erased. We consider the setup where the only encoding operation allowed is a linear unitary transformation on the source. For such a setup, we consider the minimum mean-square error (MMSE) as the performance criterion and investigate the MMSE performance both in average and in terms of guarantees that hold with high probability as a function of system parameters. Necessary conditions for optimal unitary encoders are established, and explicit solutions for a class of settings are presented. Although there are observations (including evidence provided by the compressed sensing community) that may suggest the result that the discrete Fourier transform (DFT) matrix may be indeed an optimum unitary matrix for any eigenvalue distribution, we provide a counterexample. Finally, we consider equidistant sampling of circularly wide sense stationary (c.w.s.s.) signals, and present an upper bound that summarizes the effect of the sampling rate and the eigenvalue distribution. These findings may be useful in understanding the geometric dependence of signal uncertainty in a stochastic process. In particular, unlike information theoretic measures such as entropy, we wish to highlight the basis dependence of uncertainty in a signal with another perspective. The unitary encoding space restriction allows us to extract the most and least favorable signal bases for estimation.

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