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 σ
²
. 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, σ
²
fixed). Shannon has shown that ∊
²
(N) ≧ σ
²
exp (−2C/R) ≜ ∊
²
0
and ∊
²
(N) → ∊
²
0
as N → ∞. Hence the ideal error ∊
²
0
is attainable in the limit as the coding delay N → ∞. We are concerned with the rate at which ∊
²
(N) → ∊
²
0
, and our principal result is that ∊
²
(N) — ∊
²
0
≦ O[(log N/N)
^1/2
].