Page 279 - Antennas for Base Stations in Wireless Communications

252 Chapter Seven

2
random variables with equal variance s . Therefore, the channel capacity
can be expressed as
HR H H
C = max log I + x (7.9)
)
x {R : Tr( R ≤ P T} 2 σ 2
x
The diagonal elements of the transmit covariance matrix represent
the transmit power from each antenna. The off-diagonal elements of R
x
represent the correlation between the transmitted signal streams. An
increased correlation will result in a decreased capacity. Also, the term
H
HR H represents the covariance of the received signal in the absence
x
of noise, such that the l eigenvalue represents the received signal level
ii
th
in the i eigenchannel.
When the transmitter has no CSI, it divides its power equally among
the transmit antennas to form N independent streams, or R = (P /N )I.
T
T
x
T
The capacity can be given by
P
T
C = log I + N σ 2 HH H (7.10)
2
T
When the transmitter has CSI, the use of equal power allocation is
suboptimal. The optimal solution can be obtained by applying the water-
10
filling principle. In order to maximize Eq. 7.11, R′ x,ii must be diagonal.
Capacity is given by
2
N R  S R′ 
,
C = max ∑ log 2  1 + ii x ii  (7.11)
   σ 2 2 
x ∑
′
 R : R′ x ii ≤ P  =i 1
T
,
 i 
where R′ x,ii represents the optimal transmit power on the i th unencoded
stream and S is the power gain of the i th eigenchannel. The values of R′ x,ii
2
ii
that maximize Eq. 7.11 can be determined using Lagrange multipliers
to obtain the water-filling solution. 9−12 This method allocates power to
the high-gain channels and generally does not use weaker channels.
7.2.2.3 Antenna Effects on MIMO Capacity Antenna properties, such as
impedance matching, pattern, polarization, array configuration, and
mutual coupling affect the correlation. Angle (pattern) diversity occurs
when the antennas have distinct radiation patterns. Large capac-
ity gains are possible when the element patterns are appropriately
designed to minimize the correlation. Also, by directing the majority
of the radiation in the direction where most of the multipath compo-
13
nents are concentrated, higher capacity can be achieved. The correla-
14
tion can be calculated from the S-parameters according to Eq. 7.12 or