A proof of the separability of prefiltering and coding (Corresp.)

A random signal observed in additive noise is to be linearly preprocessed for transmission via a channel of dimensionality less than that of either the observations or the signal. At the receiver, the signal is to be linearly reconstructed under a weighted-mean-square-error criterion. It is shown that there is no loss of overall optimality if the operation on the raw observations is partitioned into an optimal (least mean-square-error) prefilter followed by a linear coder.