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What is Shannon–Hartley theorem?

A theorem stating that for a channel with bandwidth B and signal-to-noise ratio S/N, the theoretical upper limit on the error-free transmission rate (the channel capacity) is C = B·log₂(1 + S/N). S/N is used as a linear ratio. Capacity is proportional to bandwidth but grows only logarithmically with S/N.

Japanese name: シャノンの定理

Shannon–Hartley theorem appears in 3 practice questions across 2 topics, including Information Theory and Coding Basics, Wireless Communication Basics.

Field: Wireless Communication Basics

Reading the formula

In C = B·log₂(1 + S/N), C is bits per second, B is bandwidth (Hz) and S/N is the linear power ratio of signal to noise. For example, with a 20 MHz bandwidth and an S/N of 30 dB (a power ratio of 1000), C = 20×10⁶ × log₂(1001) ≈ 200 Mbit/s. Capacity divided by bandwidth, C/B, is the upper limit on bits sent per hertz (spectral efficiency).

How bandwidth and S/N affect capacity

Doubling the bandwidth at the same S/N doubles the capacity. On the other hand, when S/N is large, doubling it (about +3 dB) adds only about 1 bit/s per Hz. If the bandwidth is widened while the transmit power stays fixed, however, the noise power also rises in proportion to bandwidth and S/N falls, so capacity grows less than in proportion to bandwidth. This is why millimeter-wave bands, where wide bandwidths are available, are used for higher speed.

Relation to practical communication

Shannon's theorem gives the limit below which the error rate can be made arbitrarily small with ideal codes; it does not specify a concrete scheme. Practical radio systems approach this limit by combining multilevel modulation with powerful error-correcting codes such as turbo codes, LDPC codes and polar codes. MIMO spatial multiplexing obtains rates beyond the limit of one channel by running several channels side by side.

Try it: questions about Shannon–Hartley theorem

In a binary symmetric channel, 0s and 1s are sent and each bit flips independently with probability 0.1. With the binary entropy function H(p) = −p log₂ p − (1 − p) log₂(1 − p), H(0.1) ≈ 0.469. Which of the following is closest to the channel capacity of this channel per transmission?

  1. AAbout 0.47 bits
  2. BAbout 0.53 bits
  3. C0.9 bits
Show answer and explanation

Answer: B. About 0.53 bits

The channel capacity of a binary symmetric channel is C = 1 − H(p), the value of the mutual information when 0 and 1 are sent with equal probability. For p = 0.1, C ≈ 1 − 0.469 = 0.531, about 0.53 bits per transmission. With suitable error-correcting codes, data can be sent at up to this rate with an arbitrarily small error rate. At p = 0.5, the received value is unrelated to the value sent, and the capacity is 0.

For a channel with a bandwidth of 20 MHz and an S/N (signal-to-noise ratio) of 30 dB, which is closest to the channel capacity (the theoretical upper limit on the error-free transmission rate) given by Shannon's theorem?

  1. A100 Mbit/s
  2. B200 Mbit/s
  3. C600 Mbit/s
Show answer and explanation

Answer: B. 200 Mbit/s

By Shannon's theorem, channel capacity is C = B·log₂(1 + S/N), with S/N as a linear (power) ratio. 30 dB is a power ratio of 10³ = 1000, so C = 20×10⁶ × log₂(1001) ≈ 20×10⁶ × 9.97 ≈ 200 Mbit/s. Doubling the bandwidth at the same S/N doubles the capacity, but when S/N is large, doubling S/N (about +3 dB) adds only about 1 bit/s per Hz. Practical systems combine error-correcting codes and multilevel modulation to approach this limit.

FAQ

Can S/N be put into the formula in dB?

No. S/N in the formula is a linear power ratio. Convert 30 dB to 10³ = 1000 and 20 dB to 100 before calculating. Using 30 for 30 dB underestimates the capacity by about half.

How does it differ from the Nyquist formula?

The Nyquist formula shows that a noiseless channel of bandwidth B can carry at most 2B·log₂ M bits per second with M-level signals. Increasing M seems to raise the rate without limit, but in reality noise limits how many levels can be distinguished. Shannon's theorem gives that limit with noise taken into account.

Shannon–Hartley theorem in the explanations

Questions that mention Shannon–Hartley theorem (3)

  1. In a binary symmetric channel, 0s and 1s are sent and each bit flips independently with probability 0.1. With the binary entropy function H(p) = −p log₂ p − (1 − p) log₂(1 − p), H(0.1) ≈ 0.469. Which of the following is closest to the channel capacity of this channel per transmission?
  2. For a channel with a bandwidth of 20 MHz and an S/N (signal-to-noise ratio) of 30 dB, which is closest to the channel capacity (the theoretical upper limit on the error-free transmission rate) given by Shannon's theorem?
  3. Which of the following correctly states Shannon's channel coding theorem?

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Related terms

Binary symmetric channel Information entropy Mutual information Beamforming Bluetooth Low Energy (BLE) Direct-conversion receiver Fading Free-space path loss (FSPL) Hidden node problem Link budget

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