B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

782 INTRODUCTION TO INFORMATION THEORY
Among all the real-valued random variable vectors that have zero mean and satisfy the condition
we have 11 C x = Cov(x, x) = E{xx T }
max H(x) = - [N - log(2ne) + log det (C x )] .
Px(X): Cov(x, x T )=C x 2
(13.94)
This means that Gaussian vector distribution has maximum entropy among all real random
vectors of the same covariance matrix.
Now consider a flat fading MIMO channel with matrix gain H. The N x M channel matrix
H connects the M x 1 input vector x and N x 1 output vector y such that
y=H•x + w (13.95)
where w is the N x 1 additive white Gaussian noise vector with zero mean and covariance
matrix C w , As shown in Fig. 13.13, a MIMO system consists of M transmit antennas at the
transmitter end and N receive antennas at the receiver end. Each transmit antenna can transmit
to all N receive antennas. Given a fixed channel H of dimensions N x M (i.e., M transmit
antennas and N receive antennas), the mutual information between the channel input and output
vectors is
/(x, y) = H(y) - H(ylx)
= H(y) - H(H · x + wlx)
(13.96a)
(13.96b)
Recall that under the condition that x is known, H • x is a constant mean. Hence, the conditional
entropy of y given x is
H(ylx) = H(H · x + wlx) = H(w)
(13.97)
and
/(x, y) = H(y) - H(w)
1
= H(y) - 2 [N - log2 (2ne) + log det (Cw)]
(13.98a)
(13.98b)
Figure 13. 13
MIMO system
with M transmit
antennas and N
receive
antennas.
Transmitter