Editing Channel capacity (section)

== Channel capacity in wireless communications ==

This section<ref>{{citation | author = David Tse, Pramod Viswanath | title = Fundamentals of Wireless Communication | publisher = Cambridge University Press, UK | year=2005| isbn = 9780521845274 |url=https://books.google.com/books?id=66XBb5tZX6EC&q=%22Channel+capacity%22}}</ref> 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===
{{main|Shannon–Hartley theorem}}
[[File:Channel Capacity with Power- and Bandwidth-Limited Regimes.png|thumb|AWGN channel capacity with the power-limited regime and bandwidth-limited regime indicated. Here, <math>\frac{\bar{P}}{N_0}=1</math>; ''B'' and ''C'' can be scaled proportionally for other values.]]

If the average received power is <math>\bar{P}</math> [W], the total bandwidth is <math>W</math> in Hertz, and the noise [[power spectral density]] is <math>N_0</math> [W/Hz], the AWGN channel capacity is

:<math>C_{\text{AWGN}}=W\log_2\left(1+\frac{\bar{P}}{N_0 W}\right)</math> [bits/s],

where <math>\frac{\bar{P}}{N_0 W}</math> is the received signal-to-noise ratio (SNR). This result is known as the '''Shannon–Hartley theorem'''.<ref>{{cite book|title=The Handbook of Electrical Engineering|year=1996|publisher=Research & Education Association|isbn=9780878919819|page=D-149|url=https://books.google.com/books?id=-WJS3VnvomIC&q=%22Shannon%E2%80%93Hartley+theorem%22&pg=RA1-SL4-PA41}}</ref>

When the SNR is large (SNR ≫ 0&nbsp;dB), the capacity <math>C\approx W\log_2 \frac{\bar{P}}{N_0 W} </math> is logarithmic in power and approximately linear in bandwidth. This is called the ''bandwidth-limited regime''.

When the SNR is small (SNR ≪ 0&nbsp;dB), the capacity <math>C\approx \frac{\bar{P}}{N_0 \ln 2} </math> 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 [[fading|frequency-selective]] channel is given by so-called [[Water filling algorithm|water filling]] power allocation,

:<math>C_{N_c}=\sum_{n=0}^{N_c-1} \log_2 \left(1+\frac{P_n^* |\bar{h}_n|^2}{N_0} \right),</math>

where <math>P_n^*=\max \left\{ \left(\frac{1}{\lambda}-\frac{N_0}{|\bar{h}_n|^2} \right),0 \right\}</math> and <math>|\bar{h}_n|^2</math> is the gain of subchannel <math>n</math>, with <math>\lambda</math> chosen to meet the power constraint.

===Slow-fading channel===

In a [[fading|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, <math>\log_2 (1+|h|^2 SNR)</math>, depends on the random channel gain <math>|h|^2</math>, which is unknown to the transmitter. If the transmitter encodes data at rate <math>R</math> [bits/s/Hz], there is a non-zero probability that the decoding error probability cannot be made arbitrarily small,

:<math>p_{out}=\mathbb{P}(\log(1+|h|^2 SNR)<R)</math>,

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 <math>R</math> such that the outage probability <math>p_{out}</math> is less than <math>\epsilon</math>. This value is known as the <math>\epsilon</math>-outage capacity.

===Fast-fading channel===

In a [[fading|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 <math>\mathbb{E}(\log_2 (1+|h|^2 SNR))</math> [bits/s/Hz] and it is meaningful to speak of this value as the capacity of the fast-fading channel.