15 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Channel capacity | 1/3 | https://en.wikipedia.org/wiki/Channel_capacity | reference | science, encyclopedia | 2026-05-05T14:40:07.588658+00:00 | kb-cron |
Channel capacity, in electrical engineering, computer science, and information theory, is the theoretical maximum rate at which information can be reliably transmitted over a communication channel. Following the terms of the noisy-channel coding theorem, the channel capacity of a given channel is the highest information rate (in units of information per unit time) that can be achieved with arbitrarily small error probability. Information theory, developed by Claude E. Shannon in 1948, defines the notion of channel capacity and provides a mathematical model by which it may be computed. The key result states that the capacity of the channel, as defined above, is given by the maximum of the mutual information between the input and output of the channel, where the maximization is with respect to the input distribution. The notion of channel capacity has been central to the development of modern wireline and wireless communication systems, with the advent of novel error correction coding mechanisms that have resulted in achieving performance very close to the limits promised by channel capacity.
== Formal definition == The basic mathematical model for a communication system is the following:
→
Message
W
Encoder
f
n
→
E
n
c
o
d
e
d
s
e
q
u
e
n
c
e
X
n
Channel
p
(
y
|
x
)
→
R
e
c
e
i
v
e
d
s
e
q
u
e
n
c
e
Y
n
Decoder
g
n
→
E
s
t
i
m
a
t
e
d
m
e
s
s
a
g
e
W
^
{\displaystyle {\xrightarrow[{\text{Message}}]{W}}{\begin{array}{|c|}\hline {\text{Encoder}}\\f_{n}\\\hline \end{array}}{\xrightarrow[{\mathrm {Encoded \atop sequence} }]{X^{n}}}{\begin{array}{|c|}\hline {\text{Channel}}\\p(y|x)\\\hline \end{array}}{\xrightarrow[{\mathrm {Received \atop sequence} }]{Y^{n}}}{\begin{array}{|c|}\hline {\text{Decoder}}\\g_{n}\\\hline \end{array}}{\xrightarrow[{\mathrm {Estimated \atop message} }]{\hat {W}}}}
where:
W
{\displaystyle W}
is the message to be transmitted;
X
{\displaystyle X}
is the channel input symbol (
X
n
{\displaystyle X^{n}}
is a sequence of
n
{\displaystyle n}
symbols) taken in an alphabet
X
{\displaystyle {\mathcal {X}}}
;
Y
{\displaystyle Y}
is the channel output symbol (
Y
n
{\displaystyle Y^{n}}
is a sequence of
n
{\displaystyle n}
symbols) taken in an alphabet
Y
{\displaystyle {\mathcal {Y}}}
;
W
^
{\displaystyle {\hat {W}}}
is the estimate of the transmitted message;
f
n
{\displaystyle f_{n}}
is the encoding function for a block of length
n
{\displaystyle n}
;
p
(
y
|
x
)
=
p
Y
|
X
(
y
|
x
)
{\displaystyle p(y|x)=p_{Y|X}(y|x)}
is the noisy channel, which is modeled by a conditional probability distribution; and,
g
n
{\displaystyle g_{n}}
is the decoding function for a block of length
n
{\displaystyle n}
. Let
X
{\displaystyle X}
and
Y
{\displaystyle Y}
be modeled as random variables. Furthermore, let
p
Y
|
X
(
y
|
x
)
{\displaystyle p_{Y|X}(y|x)}
be the conditional probability distribution function of
Y
{\displaystyle Y}
given
X
{\displaystyle X}
, which is an inherent fixed property of the communication channel. Then the choice of the marginal distribution
p
X
(
x
)
{\displaystyle p_{X}(x)}
completely determines the joint distribution
p
X
,
Y
(
x
,
y
)
{\displaystyle p_{X,Y}(x,y)}
due to the identity
p
X
,
Y
(
x
,
y
)
=
p
Y
|
X
(
y
|
x
)
p
X
(
x
)
{\displaystyle \ p_{X,Y}(x,y)=p_{Y|X}(y|x)\,p_{X}(x)}
which, in turn, induces a mutual information
I
(
X
;
Y
)
{\displaystyle I(X;Y)}
. The channel capacity is defined as
C
=
sup
p
X
(
x
)
I
(
X
;
Y
)
{\displaystyle \ C=\sup _{p_{X}(x)}I(X;Y)\,}
where the supremum is taken over all possible choices of
p
X
(
x
)
{\displaystyle p_{X}(x)}
.
== Additivity of channel capacity == Channel capacity is additive over independent channels. It means that using two independent channels in a combined manner provides the same theoretical capacity as using them independently. More formally, let
p
1
{\displaystyle p_{1}}
and
p
2
{\displaystyle p_{2}}
be two independent channels modelled as above;
p
1
{\displaystyle p_{1}}
having an input alphabet
X
1
{\displaystyle {\mathcal {X}}_{1}}
and an output alphabet
Y
1
{\displaystyle {\mathcal {Y}}_{1}}
. Idem for
p
2
{\displaystyle p_{2}}
. We define the product channel
p
1
×
p
2
{\displaystyle p_{1}\times p_{2}}
as
∀
(
x
1
,
x
2
)
∈
(
X
1
,
X
2
)
,
(
y
1
,
y
2
)
∈
(
Y
1
,
Y
2
)
,
(
p
1
×
p
2
)
(
(
y
1
,
y
2
)
|
(
x
1
,
x
2
)
)
=
p
1
(
y
1
|
x
1
)
p
2
(
y
2
|
x
2
)
{\displaystyle \forall (x_{1},x_{2})\in ({\mathcal {X}}_{1},{\mathcal {X}}_{2}),\;(y_{1},y_{2})\in ({\mathcal {Y}}_{1},{\mathcal {Y}}_{2}),\;(p_{1}\times p_{2})((y_{1},y_{2})|(x_{1},x_{2}))=p_{1}(y_{1}|x_{1})p_{2}(y_{2}|x_{2})}
This theorem states:
C
(
p
1
×
p
2
)
=
C
(
p
1
)
+
C
(
p
2
)
{\displaystyle C(p_{1}\times p_{2})=C(p_{1})+C(p_{2})}
== Shannon capacity of a graph ==
If G is an undirected graph, it can be used to define a communications channel in which the symbols are the graph vertices, and two codewords may be confused with each other if their symbols in each position are equal or adjacent. The computational complexity of finding the Shannon capacity of such a channel remains open, but it can be upper bounded by another important graph invariant, the Lovász number.
== Noisy-channel coding theorem == The noisy-channel coding theorem states that for any error probability ε > 0 and for any transmission rate R less than the channel capacity C, there is an encoding and decoding scheme transmitting data at rate R whose error probability is less than ε, for a sufficiently large block length. Also, for any rate greater than the channel capacity, the probability of error at the receiver goes to 0.5 as the block length goes to infinity.
== Example application == An application of the channel capacity concept to an additive white Gaussian noise (AWGN) channel with B Hz bandwidth and signal-to-noise ratio S/N is the Shannon–Hartley theorem: