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Information theory
Entropy Differential entropy Conditional entropy Joint entropy Mutual information Directed information Conditional mutual information Relative entropy Entropy rate Limiting density of discrete points
Asymptotic equipartition property Rate–distortion theory
Shannon's source coding theorem Channel capacity Noisy-channel coding theorem Shannon–Hartley theorem
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Channel capacity, inelectrical engineering,computer science, andinformation theory, is the theoretical maximum rate at whichinformation can be reliably transmitted over acommunication channel.

Following the terms of thenoisy-channel coding theorem, the channel capacity of a givenchannel is the highest information rate (in units ofinformation per unit time) that can be achieved with arbitrarily small error probability.^[1][2]

Information theory, developed byClaude 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 themutual information between the input and output of the channel, where the maximization is with respect to the input distribution.^[3]

The notion of channel capacity has been central to the development of modern wireline and wireless communication systems, with the advent of novelerror correction coding mechanisms that have resulted in achieving performance very close to the limits promised by channel capacity.

The basic mathematical model for a communication system is the following:

${\displaystyle \underset{\text{Message}}{\overset{W}{\to }}\begin{array}{|c|}\hline \text{Encoder}\\ f_n\\ \hline\end{array}\underset{\genfrac{}{}{0ex}{}{\mathrm{E}\mathrm{n}\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{d}}{\mathrm{s}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}}}{\overset{X^n}{\to }}\begin{array}{|c|}\hline \text{Channel}\\ p(y|x)\\ \hline\end{array}\underset{\genfrac{}{}{0ex}{}{\mathrm{R}\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{d}}{\mathrm{s}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}}}{\overset{Y^n}{\to }}\begin{array}{|c|}\hline \text{Decoder}\\ g_n\\ \hline\end{array}\underset{\genfrac{}{}{0ex}{}{\mathrm{E}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}}{\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{a}\mathrm{g}\mathrm{e}}}{\overset{\hat{W}}{\to }}}$

where:

• ${\displaystyle W}$ is the message to be transmitted;
• ${\displaystyle X}$ is the channel input symbol (${\displaystyle X^n}$ is a sequence of${\displaystyle n}$ symbols) taken in analphabet${\displaystyle \mathcal{X}}$;
• ${\displaystyle Y}$ is the channel output symbol (${\displaystyle Y^n}$ is a sequence of${\displaystyle n}$ symbols) taken in an alphabet${\displaystyle \mathcal{Y}}$;
• ${\displaystyle \hat{W}}$ is the estimate of the transmitted message;
• ${\displaystyle f_n}$ is the encoding function for a block of length${\displaystyle n}$;
• ${\displaystyle p(y|x)=p_{Y|X}(y|x)}$ is the noisy channel, which is modeled by aconditional probability distribution; and,
• ${\displaystyle g_n}$ is the decoding function for a block of length${\displaystyle n}$.

Let${\displaystyle X}$ and${\displaystyle Y}$ be modeled as random variables. Furthermore, let${\displaystyle p_{Y|X}(y|x)}$ be theconditional probability distribution function of${\displaystyle Y}$ given${\displaystyle X}$, which is an inherent fixed property of the communication channel. Then the choice of themarginal distribution${\displaystyle p_X(x)}$ completely determines thejoint distribution${\displaystyle p_{X,Y}(x,y)}$ due to the identity

${\displaystyle p_{X,Y}(x,y)=p_{Y|X}(y|x)p_X(x)}$

which, in turn, induces amutual information${\displaystyle I(X;Y)}$. Thechannel capacity is defined as

${\displaystyle C=\underset{p_X(x)}{sup}I(X;Y)}$

where thesupremum is taken over all possible choices of${\displaystyle p_X(x)}$.

Channel capacity is additive over independent channels.^[4] It means that using two independent channels in a combined manner provides the same theoretical capacity as using them independently. More formally, let${\displaystyle p_1}$ and${\displaystyle p_2}$ be two independent channels modelled as above;${\displaystyle p_1}$ having an input alphabet${\displaystyle {\mathcal{X}}_1}$ and an output alphabet${\displaystyle {\mathcal{Y}}_1}$. Idem for${\displaystyle p_2}$. We define the product channel${\displaystyle p_1\times p_2}$ as${\displaystyle \mathrm{\forall }(x_1,x_2)\in ({\mathcal{X}}_1,{\mathcal{X}}_2),(y_1,y_2)\in ({\mathcal{Y}}_1,{\mathcal{Y}}_2),(p_1\times p_2)((y_1,y_2)|(x_1,x_2))=p_1(y_1|x_1)p_2(y_2|x_2)}$

This theorem states:$${\displaystyle C(p_1\times p_2)=C(p_1)+C(p_2)}$$

IfG is anundirected 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, theLovász number.^[5]

Thenoisy-channel coding theorem states that for any error probability ε > 0 and for any transmissionrateR less than the channel capacityC, there is an encoding and decoding scheme transmitting data at rateR 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.

An application of the channel capacity concept to anadditive white Gaussian noise (AWGN) channel withB Hzbandwidth andsignal-to-noise ratioS/N is theShannon–Hartley theorem:

${\displaystyle C=B{\log }_2\left(1+\frac{S}{N}\right)}$

C is measured inbits per second if thelogarithm is taken in base 2, ornats per second if thenatural logarithm is used, assumingB is inhertz; the signal and noise powersS andN are expressed in a linearpower unit (like watts or volts²). SinceS/N figures are often cited indB, a conversion may be needed. For example, a signal-to-noise ratio of 30 dB corresponds to a linear power ratio of${\displaystyle {10}^{30/10}={10}^3=1000}$.

To determine the channel capacity, it is necessary to find the capacity-achieving distribution${\displaystyle p_X(x)}$ and evaluate themutual information${\displaystyle I(X;Y)}$. 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,^[6] relaxations^[7] and capacity bounds,^[8] have been proposed in the literature.

The capacity of a discrete memoryless channel can be computed using theBlahut-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,^[9] a cooperative framework inspired bygenerative 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${\displaystyle I(X;Y)}$.

This section^[10] focuses on the single-antenna, point-to-point scenario. For channel capacity in systems with multiple antennas, see the article onMIMO.

If the average received power is${\displaystyle \overline{P}}$ [W], the total bandwidth is${\displaystyle W}$ in Hertz, and the noisepower spectral density is${\displaystyle N_0}$ [W/Hz], the AWGN channel capacity is

${\displaystyle C_{\text{AWGN}}=W{\log }_2\left(1+\frac{\overline{P}}{N_{0}W}\right)}$ [bits/s],

where${\displaystyle \frac{\overline{P}}{N_{0}W}}$ is the received signal-to-noise ratio (SNR). This result is known as theShannon–Hartley theorem.^[11]

When the SNR is large (SNR ≫ 0 dB), the capacity${\displaystyle C\approx W{\log }_2\frac{\overline{P}}{N_{0}W}}$ is logarithmic in power and approximately linear in bandwidth. This is called thebandwidth-limited regime.

When the SNR is small (SNR ≪ 0 dB), the capacity${\displaystyle C\approx \frac{\overline{P}}{N_0\ln 2}}$ is linear in power but insensitive to bandwidth. This is called thepower-limited regime.

The bandwidth-limited regime and power-limited regime are illustrated in the figure.

The capacity of thefrequency-selective channel is given by so-calledwater filling power allocation,

${\displaystyle C_{N_c}=\sum _{n=0}^{N_c-1}{\log }_2\left(1+\frac{P_n^{\ast }|{\overline{h}}_n{|}^2}{N_0}\right),}$

where${\displaystyle P_n^{\ast }=max\left\{\left(\frac{1}{\lambda }-\frac{N_0}{|{\overline{h}}_n{|}^2}\right),0\right\}}$ and${\displaystyle |{\overline{h}}_n{|}^2}$ is the gain of subchannel${\displaystyle n}$, with${\displaystyle \lambda }$ chosen to meet the power constraint.

In aslow-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,${\displaystyle {\log }_2(1+|h{|}^{2}SNR)}$, depends on the random channel gain${\displaystyle |h{|}^2}$, which is unknown to the transmitter. If the transmitter encodes data at rate${\displaystyle R}$ [bits/s/Hz], there is a non-zero probability that the decoding error probability cannot be made arbitrarily small,

,

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${\displaystyle R}$ such that the outage probability${\displaystyle p_{out}}$ is less than${\displaystyle ϵ}$. This value is known as the${\displaystyle ϵ}$-outage capacity.

In afast-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${\displaystyle \mathbb{E}({\log }_2(1+|h{|}^{2}SNR))}$ [bits/s/Hz] and it is meaningful to speak of this value as the capacity of the fast-fading channel.

Feedback capacity is the greatest rate at whichinformation can be reliably transmitted, per unit time, over a point-to-pointcommunication channel in which the receiver feeds back the channel outputs to the transmitter. Information-theoretic analysis of communication systems that incorporate feedback is more complicated and challenging than without feedback. Possibly, this was the reasonC.E. Shannon chose feedback as the subject of the first Shannon Lecture, delivered at the 1973 IEEE International Symposium on Information Theory in Ashkelon, Israel.

The feedback capacity is characterized by the maximum of thedirected information between the channel inputs and the channel outputs, where the maximization is with respect to the causal conditioning of the input given the output. Thedirected information was coined byJames Massey^[12] in 1990, who showed that its an upper bound on feedback capacity. Formemoryless channels, Shannon showed^[13] that feedback does not increase the capacity, and the feedback capacity coincides with the channel capacity characterized by themutual information between the input and the output. The feedback capacity is known as a closed-form expression only for several examples such as the trapdoor channel,^[14] Ising channel,.^[15][16] For some other channels, it is characterized through constant-size optimization problems such as the binary erasure channel with a no-consecutive-ones input constraint,^[17] NOST channel.^[18]

The basic mathematical model for a communication system is the following:

Here is the formal definition of each element (where the only difference with respect to the nonfeedback capacity is the encoder definition):

• ${\displaystyle W}$ is the message to be transmitted, taken in analphabet${\displaystyle \mathcal{W}}$;
• ${\displaystyle X}$ is the channel input symbol (${\displaystyle X^n}$ is a sequence of${\displaystyle n}$ symbols) taken in analphabet${\displaystyle \mathcal{X}}$;
• ${\displaystyle Y}$ is the channel output symbol (${\displaystyle Y^n}$ is a sequence of${\displaystyle n}$ symbols) taken in an alphabet${\displaystyle \mathcal{Y}}$;
• ${\displaystyle \hat{W}}$ is the estimate of the transmitted message;
• ${\displaystyle f_i:\mathcal{W}\times {\mathcal{Y}}^{i-1}\to \mathcal{X}}$ is the encoding function at time${\displaystyle i}$, for a block of length${\displaystyle n}$;
• ${\displaystyle p(y_i|x^i,y^{i-1})=p_{Y_i|X^i,Y^{i-1}}(y_i|x^i,y^{i-1})}$ is the noisy channel at time${\displaystyle i}$, which is modeled by aconditional probability distribution; and,
• ${\displaystyle \hat{w}:{\mathcal{Y}}^n\to \mathcal{W}}$ is the decoding function for a block of length${\displaystyle n}$.

That is, for each time${\displaystyle i}$ there exists a feedback of the previous output${\displaystyle Y_{i-1}}$ such that the encoder has access to all previous outputs${\displaystyle Y^{i-1}}$. An${\displaystyle (2^{nR},n)}$ code is a pair of encoding and decoding mappings with${\displaystyle \mathcal{W}=[1,2,\dots ,2^{nR}]}$, and${\displaystyle W}$ is uniformly distributed. A rate${\displaystyle R}$ is said to beachievable if there exists a sequence of codes${\displaystyle (2^{nR},n)}$ such that theaverage probability of error:${\displaystyle P_e^{(n)}\triangleq Pr(\hat{W}\ne W)}$ tends to zero as${\displaystyle n\to \mathrm{\infty }}$.

Thefeedback capacity is denoted by${\displaystyle C_{\text{feedback}}}$, and is defined as the supremum over all achievable rates.

Let${\displaystyle X}$ and${\displaystyle Y}$ be modeled as random variables. Thecausal conditioning${\displaystyle P(y^n||x^n)\triangleq \prod _{i=1}^{n}P(y_i|y^{i-1},x^i)}$ describes the given channel. The choice of thecausally conditional distribution${\displaystyle P(x^n||y^{n-1})\triangleq \prod _{i=1}^{n}P(x_i|x^{i-1},y^{i-1})}$ determines thejoint distribution${\displaystyle p_{X^n,Y^n}(x^n,y^n)}$ due to the chain rule for causal conditioning^[19]${\displaystyle P(y^n,x^n)=P(y^n||x^n)P(x^n||y^{n-1})}$ which, in turn, induces adirected information${\displaystyle I(X^N\to Y^N)=\mathbf{E}\left[\log \frac{P(Y^N||X^N)}{P(Y^N)}\right]}$.

Thefeedback capacity is given by

${\displaystyle C_{\text{feedback}}=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}\underset{P_{X^n||Y^{n-1}}}{sup}I(X^n\to Y^n)}$,

where thesupremum is taken over all possible choices of${\displaystyle P_{X^n||Y^{n-1}}(x^n||y^{n-1})}$.

When the Gaussian noise is colored, the channel has memory. Consider for instance the simple case on anautoregressive model noise process${\displaystyle z_i=z_{i-1}+w_i}$ where${\displaystyle w_i\sim N(0,1)}$ is an i.i.d. process.

The feedback capacity is difficult to solve in the general case. There are some techniques that are related to control theory andMarkov decision processes if the channel is discrete.

• Bandwidth (computing)
• Bandwidth (signal processing)
• Bit rate
• Code rate
• Error exponent
• Nyquist rate
• Negentropy
• Redundancy
• Sender,Data compression,Receiver
• Shannon–Hartley theorem
• Spectral efficiency
• Throughput
• Shannon capacity of a graph

• MIMO
• Cooperative diversity

• "Transmission rate of a channel",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• AWGN Channel Capacity with various constraints on the channel input (interactive demonstration)

This articleneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Channel capacity" – news ·newspapers ·books ·scholar ·JSTOR(January 2008) (Learn how and when to remove this message)

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