This section shows the error-performance improvements that can be obtained with the use of diversity techniques.
The bit-error-probability,
, averaged through all the “ups and downs” of the fading experience in a slow-fading channel is as follows:
where
is the bit-error probability for a given modulation scheme at a specific value of
, where
, and
is the
of
due to the fading conditions. With
and
constant,
is used to represent the amplitude variations due to fading.
For
Rayleigh fading ,
has a
Rayleigh distribution so that
, and consequently
, have a
chi-squared distribution :
where
is the
averaged through the “ups and downs” of fading. If each diversity (signal) branch,
, has an instantaneous
, and we assume that each branch has the same average
given by
, then
The probability that a single branch has
less than some threshold
is:
The probability that all
independent signal diversity branches are received simultaneously with an
less than some threshold value
is:
The probability that any single branch achieves
is:
This is the probability of exceeding a threshold when selection diversity is used.
Example: Benefits of Diversity
Assume that four-branch diversity is used, and that each branch receives an independently Rayleigh-fading signal. If the average
is
, determine the probability that all four branches are received simultaneously with an
less than
(and also, the probability that this threshold will be exceeded).
Compare the results to the case when no diversity is used.
Solution
With
, and
, we solve for the probability that the
will drop below
, as follows:
or, using selection diversity, we can say that
Without diversity,