Optimum Compression of a Noisy Measurement for Transmission Over a Noisy Channel

This paper is motivated by sensing and wireless communication, where data compression or dimension reduction may be used to reduce the required communication bandwidth. High-dimensional measurements are converted into low-dimensional representations through linear compression. Our aim is to compress a noisy sensor measurement, allowing for the fact that the compressed measurement will then be transmitted over a noisy channel. We give the closed-form expression for the optimal compression matrix that minimizes the trace or determinant of the error covariance matrix. We show that the solutions share a common architecture consisting of a canonical coordinate transformation, scaling by coefficients which account for canonical correlations and channel noise variance, followed by a coordinate transformation into the sub-dominant invariant subspace of the channel noise. Furthermore, we explore the design problem with respect to more general criteria and provide a unified factorization for the corresponding optimal compression matrix. A necessary condition is obtained for the optimal compression.