In this section we examine the maximum possible transmission rates in a number of various channel settings. First we focus on examples of channels with constant matrix elements. In most examples the channel is known only at the receiver, but not at the transmitter. All other system and channel assumptions are as specified in Section 1.2.
Example 1.1: Single Antenna Channel
Let us consider a channel withnT = nR = 1 andH= h = 1. The Shannon formula gives the capacity of this channel
C =Wlog2
1+ P σ2
(1.37) The same expression can be obtained by applying formula (1.30). Note that for high SNRs, the capacity grows logarithmically with the SNR. Also in this region, a 3 dB increase in SNR gives a normalized capacityC/W increase of 1 bit/sec/Hz. Assuming that the channel coefficient is normalized so that|h|2=1, and for the SNR (P /σ2) of 20 dB, the capacity of a single antenna link is 6.658 bits/s/Hz.
Example 1.2: A MIMO Channel with Unity Channel Matrix Entries For this channel the matrix elementshij are
hij =1, i=1,2, . . . , nR, j =1,2, . . . , nT (1.38)
Coherent Combining
In this channel, with the channel matrix given by (1.38), the same signal is transmitted simultaneously fromnT antennas. The received signal at antennai is given by
ri =nTx (1.39)
and the received signal power at antennai is given by Pri=n2T P
nT =nTP (1.40)
whereP /nT is the power transmitted from one antenna. Note that though the power per transmit antenna isP /nT, the total received power per receive antenna isnTP. The power gain ofnT in the total received power comes due to coherent combining of the transmit- ted signals.
The rank of channel matrixHis 1, so there is only one received signal in the equivalent channel model with the power
Pr =nRnTP (1.41)
Thus applying formula (1.19) we get for the channel capacity C=Wlog2
1+nRnT P σ2
(1.42) In this example, the multiple antenna system reduces to a single effective channel that only benefits from higher power achieved by transmit and receive diversity. This system achieves a diversity gain ofnRnT relative to a single antenna link. The cost of this gain is the system complexity required to implement coordinated transmissions and coherent maximum ratio combining. However, the capacity grows logarithmically with the total number of antennas nTnR. For example, if nT =nR =8 and 10 log10P /σ2=20 dB, the normalized capacity C/W is 12.65 bits/sec/Hz.
Noncoherent Combining
If the signals transmitted from various antennas are different and all channel entries are equal to 1, there is only one received signal in the equivalent channel model with the power ofnRP. Thus the capacity is given by
C=Wlog2
1+nRP σ2
(1.43) For an SNR of 20 dB andnR=nT =8, the capacity is 9.646 bits/sec/Hz.
Example 1.3: A MIMO Channel with Orthogonal Transmissions
In this example we consider a channel with the same number of transmit and receive anten- nas,nT =nR =n, and that they are connected by orthogonal parallel sub-channels, so there is no interference between individual sub-channels. This could be achieved for example,
MIMO Capacity Examples for Channels with Fixed Coefficients 11 by linking each transmitter with the corresponding receiver by a separate waveguide, or by spreading transmitted signals from various antennas by orthogonal spreading sequences.
The channel matrix is given by
H=√ nIn
The scaling by√
nis introduced to satisfy the power constraint in (1.4).
Since
HHH =nIn
by applying formula (1.30) we get for the channel capacity C=Wlog2det
In+ nP nσ2In
=Wlog2det
diag
1+ P σ2
=Wlog2
1+ P σ2
n
=nWlog2
1+ P σ2
For the same numerical valuesnT =nR =n=8 and SNR of 20 dB, as in Example 1.2, the normalized capacityC/W is 53.264 bits/sec/Hz. Clearly, the capacity is much higher than in Example 1.2, as the sub-channels are uncoupled giving a multiplexing gain ofn.
Example 1.4: Receive Diversity
Let us assume that there is only one transmit andnR receive antennas. The channel matrix can be represented by the vector
H=(h1, h2, . . . , hnR)T
where the operator(ã)T denotes the matrix transpose. As nR > nT, formula (1.30) should be written as
C=Wlog2
det
InT+ P nTσ2HHH
(1.44) AsHHH= ni=R1|hi|2, by applying formula (1.30) we get for the capacity
C=Wlog2
1+
nR
i=1
|hi|2P σ2
(1.45) This capacity corresponds to linear maximum combining at the receiver. In the case when the channel matrix elements are equal and normalized as follows
|h1|2= |h2|2= ã ã ã |hnR|2=1
the capacity in (1.45) becomes
C=Wlog2
1+nR
P σ2
(1.46) This system achieves the diversity gain of nR relative to a single antenna channel. For nR=8 and SNR of 20 dB, the receive diversity capacity is 9.646 bits/s/Hz.
Selection diversity is obtained if the best of thenR channels is chosen. The capacity of this system is given by
C=max
i
Wlog2
1+ P
σ2|hi|2
=Wlog2
1+ P σ2max
i {|hi|2}
(1.47)
where the maximization is performed overi,i=1,2, . . . , nR. Example 1.5: Transmit Diversity
In this system there arenT transmit and only one receive antenna. The channel is represented by the vector
H=(h1, h2, . . . , hnT)
AsHHH = njT=1|hj|2, by applying formula (1.30) we get for the capacity
C=Wlog2
1+
nT
j=1
|hj|2 P nTσ2
(1.48)
If the channel coefficients are equal and normalized as in (1.4), the transmit diversity capacity becomes
C =Wlog2
1+ P σ2
(1.49) The capacity does not increase with the number of transmit antennas. This expression applies to the case when the transmitter does not know the channel. For coordinated transmissions, when the transmitter knows the channel, we can apply the capacity formula from (1.35). As the rank of the channel matrix is one, there is only one term in the sum in (1.35) and only one nonzero eigenvalue given by
λ=
nT
j=1
|hj|2
The value foràfrom the normalization condition is given by à=P +σ2
λ
Capacity of MIMO Systems with Random Channel Coefficients 13 So we get for the capacity
C =Wlog2
1+
nT
j=1
|hj|2P σ2
(1.50)
If the channel coefficients are equal and normalized as in (1.4), the capacity becomes C=Wlog2
1+nT
P σ2
(1.51) FornT =8 and SNR of 20 dB, the transmit diversity with the channel knowledge at the transmitter is 9.646 bits/s/Hz.