Page 271 - Antennas for Base Stations in Wireless Communications

244 Chapter Seven

reliability (or decrease the frame-error rate, or FER). A MIMO channel
with N T transmitting antennas and N R receiving antennas potentially
offers N T N R independent fading links.
On the other hand, the data rate can be maximized through spatial
multiplexing where multiple independent signals are transmitted but
at the expense of an increase in the FER. In general, the number of data
streams that can be reliably supported by a MIMO channel is equal
to the minimum number of transmitting and receiving antennas, i.e.,
min{N T , N R }. Capacity is enhanced by a multiplicative factor equal to the
number of data streams. Therefore, a tradeoff has to be made between the
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data rate and reliability. In a MIMO context, diversity gain (d) is often
associated with reducing the FER while multiplexing gain (r) is associ-
ated with increasing the data rate. The maximum multiplexing gain r max,
as shown in Eq. 7.1, is given by the slope of the outage capacity (for a fixed
FER) plotted as a function of the SNR (g ) on a log-linear scale
C ) γ (
out,
p
r max = lim log γ (7.1)
γ →∞
2
where C out,p (g) is the outage capacity defined as the data rate that can be
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supported by (100 – p)% of the fading realizations of the channel. For a
fixed FER, the transmission rate can be increased by min{N T , N R } bps/Hz
for every 3-dB increase in the SNR.
The maximum diversity gain d max , shown in Eq. 7.2, that can be
achieved is given by the negative of the asymptotic slope of FER for
a fixed data rate, plotted as a function of SNR on a log-linear scale.
P e denotes the probability that the frame will be decoded incorrectly.
With every 3-dB increase in the SNR for a fixed transmission rate, the
FER decreases by a factor of 2 −N N R .
T
log P ( , R)
γ
d max = − lim 2 log γ (7.2)
e
γ
→∞
2
It may be desirable that the increase in the SNR be a combination of
an increase in transmission rate and decrease in FER. The optimal trade-
off curve for the H MIMO channel, d(r), is piecewise linear such that
w
(
(
)
d r) = ( N − r N − r) (7.3)
R
T
This equation implies that if the data rate is increased by r bps/Hz
over a 3-dB increase in SNR, the corresponding reduction in the FER
will be 2 –d(r) .
Coherent combining of the wireless signals at the receiver through
spatial processing at the receiving antenna array and/or preprocessing
at the transmitting antenna array is able to achieve an increase in the