For analog signals of high dimension, compression methods relying on the acquisition and quantization of the entire signal may not always be possible. It is hence of interest to develop methods for compressing information directly from the analog to the analog (A2A) domain. Two philosophies have proven successful in addressing the general problem of conveying analog information through an analog medium: the Shannon theoretic approach and the compressed sensing approach. This research investigates a third approach, which blends the generality of the Shannon theoretic approach with the practicality of the compressed sensing approach. The fundamental limits of A2A compression are investigated in single and multi-signal settings, and constructive schemes are proposed to achieve these limits.
The challenge of A2A compression is to achieve the maximal dimensionality reduction, i.e., a bandwidth reduction factor of the signal dimension per measurement, by exploiting prior knowledge about the signal, which may include, but it is not limited to sparsity. Wu-Verdú have shown that the Renyi Information dimension (RID) is the fundamental limit for i.i.d. signals with known distributions. The RID is a very coarse measure of information, as many different signals lead to the same RID. This research investigates other signal features that influence the A2A compression performance beyond RID, particularly in the non-asymptotic regime. It investigates the A2A compression of signals which do not have an i.i.d. known distribution, and of multi-signal settings, where correlations among signals can be exploited. Finally, it proposes constructive schemes to achieve the fundamental limits using polar codes.
