Optimal Linear Coding for Vector Channels

This paper is concerned with the problem of obtaining the optimal linear vector coding (transformation) method that matches an
r
-dimensional vector signal and a
k
-dimensional channel under a given channel power constraint and mean-squared-error criterion. The encoder converts the
r
correlated random variables into
r
independent random variables and selects at most
k
independent random variables which correspond to the
k
largest eigenvaiues of the signal covariance matrix
Q
. The encoder reinserts cross correlation into the
k
random variables in such a way that the largest eigenvalue of
Q
is assigned to the smallest eigenvalue of the channel noise covariance matrix
R
and the second largest eigenvalue of
Q
to the second smallest eigenvalue of
R
, etc. When only the total power for all
k
channels is prescribed, the optimal individual channel power assignments are obtained in terms of the total power, the eigenvalues of
Q
, and the eigenvalues of
R
. When the individual channel power limits are constrained by
P_{1}, ..., P_{k}
and
R
is a diagonal matrix, the necessary conditions of an inverse eigenvalue problem must be satisfied to optimize the vector signal transmission system. An iterative numerical method has been developed for the case of correlated channel noise.