The Optimum Linear Modulator for a Gaussian Source Used with a Gaussian Channel

The optimum linear modulator and demodulator which provide transmission of a gaussian vector source through an additive gaussian vector channel are derived in this paper. The measure of performance that is used is the transmission distortion, which is defined here as the mean square error between the source output and the decoder output. It is assumed that the source and channel are mutually independent but that correlations can exist among the components of each. The performance of the best linear system is then compared with the distortion shown by Shannon to be theoretically obtainable when no functional constraint is imposed at the modulator other than an energy constraint. Although the precise form of this optimum modulator is not known for general gaussian vector sources and channels, it is known to be nonlinear and to require arbitrarily long coding block lengths. However, it is a commonly held notion that when the source and channel dimensionalities are equal the optimum modulator is linear and requires a block length of only one. It is shown here that this belief is incorrect except in very particular situations which are described. Some relations between the optimum linear modulator-demodulator pair and Shannon's test channel are discussed, and an example is included which shows that the nonoptimality of linear devices can be quite small.