6.4
Fading
In AWGN the probability of bit or symbol error depends on the received SNR or, equivalently, on
s
or
b
. In a fading environment the received signal power varies randomly over distance or time due to
shadowing and/or multipath fading. Thus, in fading
b
and
s
are random variables and therefore so are
P
b
(
b
) and P
s
(
s
). The performance metric for these random variables depends on the rate of change of
the fading. There are three different performance criteria that can be used to characterize the random
variables P
b
or P
s
:
· The outage probability, P
out
, defined as the probability that
b
or
s
falls below a given value
corresponding to the maximum allowable P
b
or P
s
.
· The average error probability, P
b
or P
s
, averaged over the distribution of
b
or
s
.
· Combined average error probability and outage, defined as the average error probability that can
be achieved some percentage of time (or space).
The average probability of symbol error applies when the signal fading is on the order of a symbol
time (T
s
T
c
), so that the signal fade level is constant over roughly one symbol time. Since many error
correction coding techniques can recover from a few bit errors, and end-to-end performance is typically
not seriously degraded by a few simultaneous bit errors, the average error probability is a reasonably
good figure of merit for the channel quality under these conditions.
However, if the signal power is changing slowly (T
s
<< T
c
), then a deep fade will affect many simul-
taneous symbols. Thus, fading may lead to large error bursts, which cannot be corrected for with coding
of reasonable complexity. Therefore, these error bursts can seriously degrade end-to-end performance.
In this case acceptable performance cannot be guaranteed over all time or, equivalently, throughout a
cell, without drastically increasing transmit power. Under these circumstances, an outage probability is
specified so that the channel is deemed unusable for some fraction of time or space. Outage and average
error probability are often combined when the channel is modeled as a combination of fast and slow
fading, e.g. log-normal shadowing with fast Rayleigh fading.
6.4.1
Outage Probability
The outage probability relative to
0
is defined as P
out
= p(
s
<
0
), where
0
typically specifies the
minimum SNR required for acceptable performance (e.g. for a voice signal with binary modulation,
P
b
= 10
-3
is an acceptable error rate since it generally can't be detected by the human ear. Thus, for a
BPSK signal in Rayleigh fading,
b
< 7 dB would be declared an outage). In Rayleigh fading the outage
probability becomes
P
out
=
0
0
1
s
e
-
s
/
s
d
s
= 1
- e
-
0
/
s
.
(6.19)
Inverting this formula shows that for a given outage probability, the required average SNR
s
is
s
=
0
- ln(1 - P
out
)
.
(6.20)
In dB this means that
s
= 10 log
s
must exceed the target
0
= 10 log
0
by F
d
=
-10 log[- ln(1-P
out
)]
to maintain acceptable performance more than 100
(1 - P
out
) percent of the time. The quantity F
d
is
typically called the fade margin.
For example, suppose we want to determine what
b
is needed for BPSK modulation in slow Rayleigh
fading such that 95% of the time (or spatially) we achieve P
b
(
b
) < 10
-4
. We find that for BPSK
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