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

We consider the problem of the transmission of analog data from a Gaussian source over a memoryless channel with capacity C nats per second. The source emits R independent zero mean Gaussian variates per second with variance σ2. These digits are block-coded RN at a time into N second channel inputs. The performance criterion is the mean square error. Let ∊2(N) be the smallest attainable mean square error with parameter N(R, C, σ2fixed). Shannon has shown that ∊2(N) ≧ σ2exp (−2C/R) ≜ ∊20and ∊2(N) → ∊20as N → ∞. Hence the ideal error ∊20is attainable in the limit as the coding delay N → ∞. We are concerned with the rate at which ∊2(N) → ∊20, and our principal result is that ∊2(N) — ∊20≦ O[(log N/N)1/2].