1
3
2
M
1
j
1
r e s(t)
j
2
r e s(t)
2
3
r e s(t)
j
3
j
M
M
r e s(t)
i
2
ith Branch SNR: r /N
i
Combiner
Output
SNR:
Figure 7.1: Linear Combiner.
random complex amplitude term
=
i
i
r
i
e
j
i
that results from the path combining. This complex
amplitude term results in a random SNR
at the combiner output, where the distribution of
is
a function of the number of diversity paths, the fading distribution on each path, and the combining
technique, as shown in more detail below. Since the combiner output is fed into a standard demodulator
for the transmitted signal s(t), the performance of the diversity system in terms of P
b
and P
out
is as
defined in Section 6.4.1, i.e.
P
b
=
0
P
b
(
)p(
)d
,
(7.1)
where P
b
(
) is the probability of bit error for demodulation of s(t) in AWGN with SNR
, and
P
out
= p(
0
),
(7.2)
for some target SNR value
0
.
In the following subsections we will describe the different combining techniques in more detail. These
techniques entail various tradeoffs between performance and complexity.
7.3
Selection Combining
In selection combining, the strategy is to choose the branch with the highest SNR=r
2
i
/N
i
or, more
practically, the highest S+N, since the noise N
i
= N is assumed to be the same on all branches. This
method requires each branch to have its own dedicated receiver, since S +N must be computed separately
for each branch. With this selection technique, the path output from the combiner has an SNR equal to
the maximum SNR of all the branches.
Assume that we have M branches with uncorrelated Rayleigh fading amplitudes r
i
. The instanta-
neous SNR on the ith branch is therefore given by
i
= r
2
i
/N . Defining the average SNR on the ith
branch as
i
= E[
i
], the SNR distribution will be exponential:
p(
i
) =
1
i
e
-
i
/
i
.
(7.3)
147