Let and be modeled as random variables. Furthermore, let be the conditional probability distribution function of given, which is an inherent fixed property of the communication channel. Then the choice of the marginal distribution completely determines the joint distribution due to the identity which, in turn, induces a mutual information. The channel capacity is defined as where the supremum is taken over all possible choices of.
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 and be two independent channels modelled as above; having an input alphabet and an output alphabet. Idem for. We define the product channel as This theorem states:
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 channel with B Hz bandwidth and signal-to-noise ratioS/N is the Shannon–Hartley theorem: 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. 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.
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 and the noise power spectral density is , the AWGN channel capacity is where is the received signal-to-noise ratio. This result is known as the Shannon–Hartley theorem. When the SNR is large, the capacity is logarithmic in power and approximately linear in bandwidth. This is called the bandwidth-limited regime. When the SNR is small, the capacity 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, where and is the gain of subchannel, with 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,, depends on the random channel gain, which is unknown to the transmitter. If the transmitter encodes data at rate , 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 such that the outage probability is less than. This value is known as the -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 and it is meaningful to speak of this value as the capacity of the fast-fading channel.
