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

We are concerned here with the transmission of a gaussian vector source over an additive gaussian vector channel. The mean square difference between the source and decoder outputs is used to measure the transmission distortion in the system and is, therefore, attempted to be minimized in the design of the encoder and decoder. In this design the encoder is constrained to present only a limited energy to 3075 3088 T H E BELL SYSTEM TECHNICAL J O U R N A L , NOVEMBER 196!) the channel, thus constraining the transmission capacity of the sj^stem.1 It is because the transmission capacity of the system is limited in this way that the given gaussian vector source cannot be transmitted with arbitrarily small error. The distortion which necessarily must exist in the system is prescribed by Shannon's rate-distortion theory. 2 This theory states that when the transmission rate in a system is limited to R, the transmission of the source must include an average distortion of at least clR , which in general is a function of the source statistics and the distortion measure. The theory further states that the distortion level d R is attainable with some modulator-demodulator pair. Unfortunately, the precise form of this modulator and demodulator is not known in general, except that it is nonlinear 3,4 and that it requires the use of arbitrarily long coding block lengths. 2 Since the nonlinearity of the optimum encoder is probably a very complex twisting of the source space locus within the channel input space, the implementation of the optimum encoder, even if it were known, would be extraordinarily complex.