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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 information per time unit 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.

Statement of the theorem

The Shannon–Hartley theorem states the channel capacity C, meaning the theoretical tightest upper bound on the information rate of data that can be communicated at an arbitrarily low error rate using an average received signal power S through an analog communication channel subject to additive white Gaussian noise (AWGN) of power N: C=Blog ₂(1+S/N)

Historical development

During the late 1920s, Harry Nyquist and Ralph Hartley developed a handful of fundamental ideas related to the transmission of information, particularly in the context of the telegraph as a communications system. At the time, these concepts were powerful breakthroughs individually, but they were not part of a comprehensive theory. In the 1940s, Claude Shannon developed the concept of channel capacity, based in part on the ideas of Nyquist and Hartley, and then formulated a complete theory of information and its transmission.

Implications of the theorem

Comparison of Shannon's capacity to Hartley's law

Comparing the channel capacity to the information rate from Hartley's law, we can find the effective number of distinguishable levels M: 2Blog ₂(M)=Blog ₂(1+S/N)

M=√(1+S/N).

The square root effectively converts the power ratio back to a voltage ratio, so the number of levels is approximately proportional to the ratio of signal RMS amplitude to noise standard deviation.

This similarity in form between Shannon's capacity and Hartley's law should not be interpreted to mean that M pulse levels can be literally sent without any confusion.

Frequency-dependent (colored noise) case

In the simple version above, the signal and noise are fully uncorrelated, in which case S+N is the total power of the received signal and noise together. A generalization of the above equation for the case where the additive noise is not white (or that the ⁠ S/N ⁠ is not constant with frequency over the bandwidth) is obtained by treating the channel as many narrow, independent Gaussian channels in parallel: C=∫ ₀^(B)log ₂(1+(S(f))/(N(f)))df

Approximations

Bandwidth-limited case

When the SNR is large ( （公式略）), the logarithm is approximated by log ₂(1+S/N)≈ log ₂S/N=(ln 10)/(ln 2)· log ₁₀S/N≈ 3.32· log ₁₀S/N,

in which case the capacity is logarithmic in power and approximately linear in bandwidth (not quite linear, since N increases with bandwidth, imparting a logarithmic effect). This is called the bandwidth-limited regime. C≈ 0.332· B· SNR (in dB)

where （公式略）

Power-limited case

Similarly, when the SNR is small (if ⁠ （公式略） ⁠), applying the approximation to the logarithm: （公式略）

then the capacity is linear in power. This is called the power-limited regime. （公式略）

Examples

At a SNR of 0 dB (Signal power = Noise power) the Capacity in bits/s is equal to the bandwidth in hertz.

If the SNR is 20 dB, and the bandwidth available is 4 kHz, which is appropriate for telephone communications, then C = 4000 log2(1 + 100) = 4000 log2 (101) = 26.63 kbit/s. Note that the value of S/N = 100 is equivalent to the SNR of 20 dB.

If the requirement is to transmit at 50 kbit/s, and a bandwidth of 10 kHz is used, then the minimum S/N required is given by 50000 = 10000 log2(1+S/N) so C/B = 5 then S/N = 25 − 1 = 31, corresponding to an SNR of 14.91 dB (10 x log10(31)).