13 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Channel capacity | 2/3 | https://en.wikipedia.org/wiki/Channel_capacity | reference | science, encyclopedia | 2026-05-05T14:40:07.588658+00:00 | kb-cron |
C
=
B
log
2
(
1
+
S
N
)
{\displaystyle C=B\log _{2}\left(1+{\frac {S}{N}}\right)\ }
C is measured in bits per second if the logarithm is taken in base 2, or nats per second if the natural logarithm is used, assuming B is in hertz; the signal and noise powers S and N are expressed in a linear power unit (like watts or volts2). Since S/N figures are often cited in dB, a conversion may be needed. For example, a signal-to-noise ratio of 30 dB corresponds to a linear power ratio of
10
30
/
10
=
10
3
=
1000
{\displaystyle 10^{30/10}=10^{3}=1000}
.
== Channel capacity estimation == To determine the channel capacity, it is necessary to find the capacity-achieving distribution
p
X
(
x
)
{\displaystyle p_{X}(x)}
and evaluate the mutual information
I
(
X
;
Y
)
{\displaystyle I(X;Y)}
. Research has mostly focused on studying additive noise channels under certain power constraints and noise distributions, as analytical methods are not feasible in the majority of other scenarios. Hence, alternative approaches such as, investigation on the input support, relaxations and capacity bounds, have been proposed in the literature. The capacity of a discrete memoryless channel can be computed using the Blahut-Arimoto algorithm. Deep learning can be used to estimate the channel capacity. In fact, the channel capacity and the capacity-achieving distribution of any discrete-time continuous memoryless vector channel can be obtained using CORTICAL, a cooperative framework inspired by generative adversarial networks. CORTICAL consists of two cooperative networks: a generator with the objective of learning to sample from the capacity-achieving input distribution, and a discriminator with the objective to learn to distinguish between paired and unpaired channel input-output samples and estimates
I
(
X
;
Y
)
{\displaystyle I(X;Y)}
.
== Channel capacity in wireless communications == This section focuses on the single-antenna, point-to-point scenario. For channel capacity in systems with multiple antennas, see the article on MIMO.
=== Bandlimited AWGN channel ===
If the average received power is
P
¯
{\displaystyle {\bar {P}}}
[W], the total bandwidth is
W
{\displaystyle W}
in Hertz, and the noise power spectral density is
N
0
{\displaystyle N_{0}}
[W/Hz], the AWGN channel capacity is
C
AWGN
=
W
log
2
(
1
+
P
¯
N
0
W
)
{\displaystyle C_{\text{AWGN}}=W\log _{2}\left(1+{\frac {\bar {P}}{N_{0}W}}\right)}
[bits/s], where
P
¯
N
0
W
{\displaystyle {\frac {\bar {P}}{N_{0}W}}}
is the received signal-to-noise ratio (SNR). This result is known as the Shannon–Hartley theorem. When the SNR is large (SNR ≫ 0 dB), the capacity
C
≈
W
log
2
P
¯
N
0
W
{\displaystyle C\approx W\log _{2}{\frac {\bar {P}}{N_{0}W}}}
is logarithmic in power and approximately linear in bandwidth. This is called the bandwidth-limited regime. When the SNR is small (SNR ≪ 0 dB), the capacity
C
≈
P
¯
N
0
ln
2
{\displaystyle C\approx {\frac {\bar {P}}{N_{0}\ln 2}}}
is linear in power but insensitive to bandwidth. This is called the power-limited regime. The bandwidth-limited regime and power-limited regime are illustrated in the figure.
=== Frequency-selective AWGN channel === The capacity of the frequency-selective channel is given by so-called water filling power allocation,
C
N
c
=
∑
n
=
0
N
c
−
1
log
2
(
1
+
P
n
∗
|
h
¯
n
|
2
N
0
)
,
{\displaystyle C_{N_{c}}=\sum _{n=0}^{N_{c}-1}\log _{2}\left(1+{\frac {P_{n}^{*}|{\bar {h}}_{n}|^{2}}{N_{0}}}\right),}
where
P
n
∗
=
max
{
(
1
λ
−
N
0
|
h
¯
n
|
2
)
,
0
}
{\displaystyle P_{n}^{*}=\max \left\{\left({\frac {1}{\lambda }}-{\frac {N_{0}}{|{\bar {h}}_{n}|^{2}}}\right),0\right\}}
and
|
h
¯
n
|
2
{\displaystyle |{\bar {h}}_{n}|^{2}}
is the gain of subchannel
n
{\displaystyle n}
, with
λ
{\displaystyle \lambda }
chosen to meet the power constraint.
=== Slow-fading channel === In a slow-fading channel, where the coherence time is greater than the latency requirement, there is no definite capacity as the maximum rate of reliable communications supported by the channel,
log
2
(
1
+
|
h
|
2
S
N
R
)
{\displaystyle \log _{2}(1+|h|^{2}SNR)}
, depends on the random channel gain
|
h
|
2
{\displaystyle |h|^{2}}
, which is unknown to the transmitter. If the transmitter encodes data at rate
R
{\displaystyle R}
[bits/s/Hz], there is a non-zero probability that the decoding error probability cannot be made arbitrarily small,
p
o
u
t
=
P
(
log
(
1
+
|
h
|
2
S
N
R
)
<
R
)
{\displaystyle p_{out}=\mathbb {P} (\log(1+|h|^{2}SNR)<R)}
, in which case the system is said to be in outage. With a non-zero probability that the channel is in deep fade, the capacity of the slow-fading channel in strict sense is zero. However, it is possible to determine the largest value of
R
{\displaystyle R}
such that the outage probability
p
o
u
t
{\displaystyle p_{out}}
is less than
ϵ
{\displaystyle \epsilon }
. This value is known as the
ϵ
{\displaystyle \epsilon }
-outage capacity.
=== Fast-fading channel === In a fast-fading channel, where the latency requirement is greater than the coherence time and the codeword length spans many coherence periods, one can average over many independent channel fades by coding over a large number of coherence time intervals. Thus, it is possible to achieve a reliable rate of communication of
E
(
log
2
(
1
+
|
h
|
2
S
N
R
)
)
{\displaystyle \mathbb {E} (\log _{2}(1+|h|^{2}SNR))}
[bits/s/Hz] and it is meaningful to speak of this value as the capacity of the fast-fading channel.