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

784 INTRODUCTION TO INFORMATION THEORY
Consider the eigendecomposition of
where U is a N x N square unitary matrix such that U • U H = IN and D is a diagonal matrix
with nonnegative diagonal elements in descending order:
D = Diag (d1 , d2, ... , dr, 0, · 0)
Notice that d, > 0 is the smallest nonzero eigenvalue of H T c; 1 H whose rank is bounded by
r =S min(N, M). Because det (/ +AB) = det (/ +BA) and u H u = I , we have
C = Blog det (1 + a; · UDU H )
(13.103a)
= Blog det (1 +a;· DU H U)
=Blog det (I +a;n)
=Blog n(l +a;di)
i=l
(13.103b)
= B L log(l + a;di)
i=l
(13. 103c)
In the special case of channel noise that is additive, white, and Gaussian, then Cw = a;l
and
YI
Y2
Yr (13.104)
0
where Yi is the ith largest eigenvalue of H T H, which is assumed to have rank r. Consequently,
the channel capacity for this MIMO system is
0
t C = B log ( 1 + : Yi)
i=I
w
(13.105)
In short, this channel capacity is the sum of the capacity of r parallel AWGN channels. Each
subchannel SNR is a; • Yi/a;,. Figure 13.14 demonstrates the equivalent system that consists
of r parallel AWGN channels with r active input signals x1 , ... , x,.
In the special case when the MIMO channel is so well conditioned that all its nonzero
eigenvalues are identical Yi = y, the channel capacity is
C MIMO = r · Blog ( 1 + :i y) (13.106)