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

13.8 Multiple-Input-Multiple-Output Communication Systems 783
As a result, we can use the result of Eq. (13.94) to obtain
1
max l(x, y) = max H (y) - 2 [N • log 2 (2;re) + log det (C w ))
(1 3.99a)
1 1
= 2
[N - log 2 (2ne) + log det (C y
)) - 2 [N • log 2 (2;re) + log det (C w ))
1
= 2
[log det (Cy) - log det (Cw)]
(13.99b)
= [1og det (Cy · c; 1 )]
( 1 3.99c)
Since the channel input x is independent of the noise vector w, we have
C y
= Cov(y, y) = H · C x H T + C w
Thus, the capacity of the channel per vector transmission is
C s = max l(x, y)
p(X)
= log det (lN +HC x H T C; 1 )
(13.100)
Given a symmetric low-pass channel with B Hz bandwidth, 2B samples of x can be
transmitted to yield provide channel capacity of
C(H) = Blog det (r +HC x H T C; 1 )
= Blog det (r + C x H T C; 1 H) (13.101)
where we have invoked the equality that for matrices A and B of appropriate dimensions,
det (/ + A • B) = det (/ + B • A). We clearly can see from Eq. (13.101 ) that the channel
capacity depends on the covariance matrix Cx of the Gaussian input signal vector. This result
shows that, given the knowledge of the MIMO channel (H T c;, 1 H) at the transmitter, an
optimum input signal can be determined by designing Cx to maximize the overall channel
capacity C(H).
We now are left with two scenarios to consider: (1) MIMO transmitters without the
MIMO channel knowledge and (2) MIMO transmitters with channel knowledge that allows
C x to be optimized. We shall discuss the MIMO channel capacity in these two separate
cases.
13. 8.2 Transmitter without Channel Knowledge
For transmitters without channel knowledge, the input covariance matrix C x should be chosen
without showing any preference. As a result, the default C x = a}I should be selected. In this
case, the MIMO system capacity is simply
(1 3.102)