Capacity with Outage
Capacity with outage applies to slowly-varying channels, where the instantaneous SNR is constant over
a large number of transmissions (a transmission burst) and then changes to a new value based on the
fading distribution. With this model, if the channel has received SNR during a burst then data can
be sent over the channel at rate B log
2
(1 + ) with negligible probability of error
2
. Since the transmitter
does not know the SNR value , it must fix a transmission rate independent of the instantaneous received
SNR.
Capacity with outage allows bits sent over a given transmission burst to be decoded at the end of the
burst with some probability that these bits will be decoded incorrectly. Specifically, the transmitter fixes
a minimum received SNR
min
and encodes for a data rate C = B log
2
(1 +
min
). The data is correctly
received if the instantaneous received SNR is greater than or equal to
min
[11, 12]. If the received SNR
is below
min
then the bits received over that transmission burst cannot be decoded correctly, and the
receiver declares an outage. The probability of outage is thus P
out
= p( <
min
). The average rate
correctly received over many transmission bursts is C
o
= (1
- P
out
)B log
2
(1 +
min
) since data is only
correctly received on 1
- P
out
transmissions. The value of
min
is typically a design parameter based on
the acceptable outage probability. Capacity with outage is typically characterized by a plot of capacity
versus outage, as shown in Figure 4.2. In this figure we plot the normalized capacity C/B = log
2
(1+
min
)
as a function of outage probability P
out
= p( <
min
) for a Rayleigh fading channel ( exponential) with
= 20 dB. We see that capacity approaches zero for small outage probability, due to the requirement to
correctly decode correctly under severe fading, and increases dramatically as outage probability increases.
Note, however, that these high capacity values for large outage probabilities have higher probability of
incorrect data reception.
10
-4
10
-3
10
-2
10
-1
0
0.5
1
1.5
2
2.5
3
3.5
4
Outage Probability
C/B
Figure 4.2: Normalized Capacity C/B versus outage probability
2
The assumption of constant fading over a large number of transmissions is needed since codes that achieve capacity
require very large blocklengths
89