# Shannon–Hartley theorem

## Description

In information theory, the Shannon–Hartley theorem tells the maximum rate at which information can be transmitted over a communications channel of a specified bandwidth in the presence of noise. It is an application of the noisy-channel coding theorem to the archetypal case of a continuous-time analog communications channel subject to Gaussian noise. The theorem establishes Shannon’s channel capacity for such a communication link, a bound on the maximum amount of error-free digital data (that is, information) that can be transmitted with a specified bandwidth in the presence of the noise interference, assuming that the signal power is bounded, and that the Gaussian noise process is characterized by a known power or power spectral density. The law is named after Claude Shannon and Ralph Hartley.

Considering all possible multi-level and multi-phase encoding techniques, the Shannon–Hartley theorem states the channel capacity C, meaning the theoretical tightest upper bound on the information rate (excluding error correcting codes) of clean (or arbitrarily low bit error rate) data that can be sent with a given average signal power S through an analog communication channel subject to additive white Gaussian noise of power N, is as shown here.

Related formulas## Variables

C | channel capacity in bits per second (1/sec) |

B | bandwidth of the channel (passband bandwidth in case of a modulated signal); (hz) |

S | average received signal power over the bandwidth (in case of a modulated signal, often denoted C, i.e. modulated carrier) (watt) |

N | average noise or interference power over the bandwidth (watt) |