10.1.4
MIMO Channel Capacity
The MIMO decomposition described above allows a simple characterization of the MIMO channel capacity
when both transmitter and receiver have perfect knowledge of the channel matrix H. The capacity formula
is [45]:
C =
max
Q:T r(Q)P
log
|I + HQH
|,
(10.2)
where the maximum is taken over all matrices Q that satisfy the average power constraint.
By substituting the matrix SVD (10.1) into (10.2) and using properties of unitary matrices yields
C =
max
{P
i
}:
i
P
i
P
i
B log 1 +
2
i
P
i
N
0
B
,
(10.3)
which is similar to the capacity formula in flat fading (4.10) or in frequency-selective fading with constant
channel gains (??). We therefore get a similar water-filling power allocation for the MIMO channel with
the channel gain given by the eigenvalues:
P
i
P
=
1
0
-
1
i
i
0
0
i
<
0
(10.4)
for some cutoff value
0
, where
i
=
2
i
P/(N
0
B). The resulting capacity is then
C =
i=1(
i
0
)
B log(
i
/
0
).
(10.5)
10.1.5
Beamforming
In this section we consider the case when the transmitter does not know the instantaneous channel.
It is no longer possible to transform the MIMO channel into non-interfering SISO channels. Since the
decoding complexity is exponential in r, we can keep the complexity low by keeping r small. Of particular
interest is the case where r = 1. A transmit strategy where the input covariance matrix has unit rank
is called beamforming. This corresponds to the precoding matrix being just a column vector M = c, the
beamforming vector, as shown in Figure 10.4
Spatial matched filtering yields a single SISO AWGN channel as follows.
~
y
=
c
H
||c
H
||
y
=
c
H
||c
H
||
Hcx +
c
H
||c
H
||
N
=
||Hc||x + ~
N
where ~
N is zero-mean, unit-variance AWGN.
The optimal demodulation complexity with beamforming is of the order of
|X |, the size of the
modulation symbol alphabet. Recall that c does not change with time. For a given choice of c and a
given channel matrix H the SNR becomes
SNR
=
c
H
HcE[xx
]
Define the optimal choice of c as one that maximizes the average SNR (averaged over the distribution
of H). Note that optimality can also be defined so that the information theoretic capacity of this fading
channel is maximized. However, for now, we are interested in uncoded systems and therefore we choose
the average SNR as the optimality criterion.
240
Summary :
10.1.4 MIMO Channel Capacity The MIMO decomposition described above allows a simple characterization of the MIMO channel capacity when both transmitter and receiver have perfect knowledge of the channel matrix H. For a given choice of c and a given channel matrix H the SNR becomes SNR = c H HcE[xx ] Define the optimal choice of c as one that maximizes the average SNR (averaged over the distribution of H).
Tags :
capacity,matrix,mimo,beamforming,snr,fading,gien,aerage,complexity,log,does,choice,similar