Canonical coordinate geometry of precoder and equalizer designs for multichannel communication

For vector communication over a matrix channel, precoders and equalizers are now known for designs that minimize MSE, maximize SNR, minimize BER, or maximize information rate, under a power constraint. Some of these designs may be made to be zero-ISI. For reduced-rank signal processing, designs are known for estimation at minimum MSE and maximum information rate. These signal processing designs show that reduced-rank estimation must be done in a system of canonical coordinates. What connection is there, then, between reduced-rank signal processing and precoder/equalizer design under a power constraint? In this paper we show that all known designs for precoder/equalizer design are, in fact, decompositions of a virtual two-channel problem into a system of canonical coordinates, wherein whitened variables in the canonical message channel are correlated only pairwise with whitened variables in the canonical measurement channel. This finding clarifies the geometry of known precoder/equalizer designs and illustrates that these designs decompose the two-channel communication problem into the Shannon channel [L. L. Scharf et al. (2000)], where its geometry is revealed.