Feedback coding schemes for an additive noise channel with a noisy feedback link

A coding scheme for an additive Gaussian channel is developed using a noisy feedback link and
D
-dimensional elementary signals with no bandwidth constraint. This allows error-free transmission at a rate
R < R_{c}
where
R_{c}
is slightly less than the channel capacity
C
. When there is no noise in the feedback channel, the coding scheme reduces to a
D
-dimensional generalization of the coding scheme of Schalkwijk and Kailath. In addition, the expression for the probability of error is determined when
T
, the time of transmission rate, is finite. The scheme is also compared with the best codes that use only the forward channel.