Channel capacity

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

Proof

We first show that ${\displaystyle C(p_1\times p_2)\ge C(p_1)+C(p_2)}$.

Let ${\displaystyle X_1}$ and ${\displaystyle X_2}$ be two independent random variables. Let ${\displaystyle Y_1}$ be a random variable corresponding to the output of ${\displaystyle X_1}$ through the channel ${\displaystyle p_1}$, and ${\displaystyle Y_2}$ for ${\displaystyle X_2}$ through ${\displaystyle p_2}$.

By definition ${\displaystyle C(p_1\times p_2)=\underset{p_{X_1,X_2}}{sup}(I(X_1,X_2:Y_1,Y_2))}$.

Since ${\displaystyle X_1}$ and ${\displaystyle X_2}$ are independent, as well as ${\displaystyle p_1}$ and ${\displaystyle p_2}$, ${\displaystyle (X_1,Y_1)}$ is independent of ${\displaystyle (X_2,Y_2)}$. We can apply the following property of mutual information: ${\displaystyle I(X_1,X_2:Y_1,Y_2)=I(X_1:Y_1)+I(X_2:Y_2)}$

For now we only need to find a distribution ${\displaystyle p_{X_1,X_2}}$ such that ${\displaystyle I(X_1,X_2:Y_1,Y_2)\ge I(X_1:Y_1)+I(X_2:Y_2)}$. In fact, ${\displaystyle {\pi }_1}$ and ${\displaystyle {\pi }_2}$, two probability distributions for ${\displaystyle X_1}$ and ${\displaystyle X_2}$ achieving ${\displaystyle C(p_1)}$ and ${\displaystyle C(p_2)}$, suffice:

${\displaystyle C(p_1\times p_2)\ge I(X_1,X_2:Y_1,Y_2)=I(X_1:Y_1)+I(X_2:Y_2)=C(p_1)+C(p_2)}$

ie. ${\displaystyle C(p_1\times p_2)\ge C(p_1)+C(p_2)}$

Now let us show that ${\displaystyle C(p_1\times p_2)\le C(p_1)+C(p_2)}$.

Let ${\displaystyle {\pi }_{12}}$ be some distribution for the channel ${\displaystyle p_1\times p_2}$ defining ${\displaystyle (X_1,X_2)}$ and the corresponding output ${\displaystyle (Y_1,Y_2)}$. Let ${\displaystyle {\mathcal{X}}_1}$ be the alphabet of ${\displaystyle X_1}$, ${\displaystyle {\mathcal{Y}}_1}$ for ${\displaystyle Y_1}$, and analogously ${\displaystyle {\mathcal{X}}_2}$ and ${\displaystyle {\mathcal{Y}}_2}$.

By definition of mutual information, we have

${\displaystyle \begin{array}{cc}I(X_1,X_2:Y_1,Y_2)& =H(Y_1,Y_2)-H(Y_1,Y_2|X_1,X_2)\\ & \le H(Y_1)+H(Y_2)-H(Y_1,Y_2|X_1,X_2)\end{array}}$

Let us rewrite the last term of entropy.

${\displaystyle H(Y_1,Y_2|X_1,X_2)=\sum _{(x_1,x_2)\in {\mathcal{X}}_1\times {\mathcal{X}}_2}\mathbb{P}(X_1,X_2=x_1,x_2)H(Y_1,Y_2|X_1,X_2=x_1,x_2)}$

By definition of the product channel, ${\displaystyle \mathbb{P}(Y_1,Y_2=y_1,y_2|X_1,X_2=x_1,x_2)=\mathbb{P}(Y_1=y_1|X_1=x_1)\mathbb{P}(Y_2=y_2|X_2=x_2)}$. For a given pair ${\displaystyle (x_1,x_2)}$, we can rewrite ${\displaystyle H(Y_1,Y_2|X_1,X_2=x_1,x_2)}$ as:

${\displaystyle \begin{array}{cc}H(Y_1,Y_2|X_1,X_2=x_1,x_2)& =\sum _{(y_1,y_2)\in {\mathcal{Y}}_1\times {\mathcal{Y}}_2}\mathbb{P}(Y_1,Y_2=y_1,y_2|X_1,X_2=x_1,x_2)\log (\mathbb{P}(Y_1,Y_2=y_1,y_2|X_1,X_2=x_1,x_2))\\ & =\sum _{(y_1,y_2)\in {\mathcal{Y}}_1\times {\mathcal{Y}}_2}\mathbb{P}(Y_1,Y_2=y_1,y_2|X_1,X_2=x_1,x_2)[\log (\mathbb{P}(Y_1=y_1|X_1=x_1))+\log (\mathbb{P}(Y_2=y_2|X_2=x_2))]\\ & =H(Y_1|X_1=x_1)+H(Y_2|X_2=x_2)\end{array}}$

By summing this equality over all ${\displaystyle (x_1,x_2)}$, we obtain ${\displaystyle H(Y_1,Y_2|X_1,X_2)=H(Y_1|X_1)+H(Y_2|X_2)}$.

We can now give an upper bound over mutual information:

${\displaystyle \begin{array}{cc}I(X_1,X_2:Y_1,Y_2)& \le H(Y_1)+H(Y_2)-H(Y_1|X_1)-H(Y_2|X_2)\\ & =I(X_1:Y_1)+I(X_2:Y_2)\end{array}}$

This relation is preserved at the supremum. Therefore

${\displaystyle C(p_1\times p_2)\le C(p_1)+C(p_2)}$

Combining the two inequalities we proved, we obtain the result of the theorem:

${\displaystyle C(p_1\times p_2)=C(p_1)+C(p_2)}$

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:

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

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 ${\displaystyle {10}^{30/10}={10}^3=1000}$.

Channel capacity estimation

To determine the channel capacity, it is necessary to find the capacity-achieving distribution ${\displaystyle p_X(x)}$ and evaluate the mutual 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, 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 ${\displaystyle I(X;Y)}$.

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 ${\displaystyle \overline{P}}$ [W], the total bandwidth is ${\displaystyle W}$ in Hertz, and the noise power 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 the Shannon–Hartley theorem.

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 the bandwidth-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 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,

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

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.

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, ${\displaystyle {\log }_2(1+|h{|}^{2}SNR)}$, 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,

${\displaystyle p_{out}=\mathbb{P}(\log (1+|h{|}^{2}SNR)<R)}$,

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

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

Feedback Capacity

Feedback capacity is the greatest rate at which information can be reliably transmitted, per unit time, over a point-to-point communication 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 reason C.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 the directed 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. The directed information was coined by James Massey in 1990, who showed that its an upper bound on feedback capacity. For memoryless channels, Shannon showed that feedback does not increase the capacity, and the feedback capacity coincides with the channel capacity characterized by the mutual 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, Ising channel. 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, NOST channel.

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 an alphabet ${\displaystyle \mathcal{W}}$;
• ${\displaystyle X}$ is the channel input symbol (${\displaystyle X^n}$ is a sequence of ${\displaystyle n}$ symbols) taken in an alphabet ${\displaystyle \mathcal{X}}$;
• ${\displaystyle Y}$ 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 a conditional 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 be achievable if there exists a sequence of codes ${\displaystyle (2^{nR},n)}$ such that the average probability of error: ${\displaystyle P_e^{(n)}\triangleq Pr(\hat{W}\ne W)}$ tends to zero as ${\displaystyle n\to \mathrm{\infty }}$.

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

Main results on feedback capacity

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

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

Gaussian feedback capacity

When the Gaussian noise is colored, the channel has memory. Consider for instance the simple case on an autoregressive 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.

Solution techniques

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