Communication of Analog Data from a Gaussian Source Over a Noisy Channel

We are interested in the following problem. Suppose we have an analog data source which emits a sequence of statistically independent Gaussian variates at a rate of R per second. We wish to transmit this data through a noisy channel of capacity C nats per second. Our problem is the determination of the minimum possible mean-squared-error. Specifically we shall study the communication system of Figure 1. The output of the analog source is a sequence Xx , X2, X3 · · · of statistically independent Gaussian variates with zero mean and variance a 2 which appear at the coder input at rate of R per second. After N seconds, n = NR source variates have accumulated at the coder input. Let X denote this random n-vector. The channel is a discrete memoryless channel* which we assume accepts one input per second, and the coder contains a mapping of X to an allowable channel input N-vector S. Since it requires N seconds to transmit S, the system can process the data continuously without a "backup" at the coder input. * Actually our results are valid for a broader class of channels. See the remark after Theorem 2 in Section II. 801