Computer Science & Data Science · Information theory

Shannon-Hartley channel capacity

The Shannon-Hartley limit gives the maximum reliably transmitted data rate of a band-limited AWGN channel.

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Formula

C = B lo g_{2} (1 + S / N)

C in bit/s; B in Hz; S/N is a dimensionless linear power ratio, not dB

Variables & units

Variables & units
Symbol	Meaning	Unit
$C$	Theoretical channel capacity	bit/s
$B$	Channel bandwidth	Hz
$S/N$	Received signal-to-noise power ratio	dimensionless

LaTeX

C=B\log_2(1+S/N)

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Derivation & background

The formula applies to a band-limited additive white Gaussian noise (AWGN) channel and a linear SNR. It is an asymptotic limit for reliable communication with suitable coding, not the rate every practical system achieves. If SNR is given in decibels, first calculate SNRlin=10^(SNRdB/10). Bandwidth raises C linearly, whereas increasing SNR has only a logarithmic effect.

Exam blueprint

Validity range

Derivation steps

The Shannon-Hartley limit gives the maximum reliably transmitted data rate of a band-limited AWGN channel.

1Read the relation from its definition: C=B\log_2(1+S/N).
2Identify the input, requested output and assumptions of the relation.
3For B=1 MHz and SNR=15 (about 11.76 dB), C=10⁶·log₂(16)=4 Mbit/s as a theoretical limit.

Rearrangements

SNR from capacity

SNR=2^(C/B)−1

SNR in the equation is linear, not in dB.

Task variant

Use the example for Shannon-Hartley channel capacity to explain which inputs determine the requested output.

The example connects concrete inputs to its result.
The Shannon-Hartley limit gives the maximum reliably transmitted data rate of a band-limited AWGN channel.
The relation applies only under its stated assumptions.

Common mistakes

Applying the relation to a different case without checking its stated assumptions.

Check the validity range before substituting values: The formula applies to a band-limited additive white Gaussian noise (AWGN) channel and a linear SNR. It is an asymptotic limit for reliable communication with suitable coding, not the rate.

Exam context

Interpret theoretical data rates and spectral efficiency of communication channels.

Formula cluster

Information theory

The Shannon-Hartley limit gives the maximum reliably transmitted data rate of a band-limited AWGN channel.

H(X) = −Σ p_i log₂(p_i)

Shannon entropy

dH(x,y) = Zahl unterschiedlicher Stellen

Hamming distance between bit strings

H(p,q) = −Σ p_i ln(q_i)

Cross-entropy (classification loss)

Worked example

For B=1 MHz and SNR=15 (about 11.76 dB), C=10⁶·log₂(16)=4 Mbit/s as a theoretical limit.

Applications

Interpret theoretical data rates and spectral efficiency of communication channels.

How do you calculate with Shannon-Hartley channel capacity?

Here is how to work through a typical Shannon-Hartley channel capacity (C = B·log₂(1+SNR)) task step by step:

1
Task
Use the example for Shannon-Hartley channel capacity to explain which inputs determine the requested output.
Solution path
1. The example connects concrete inputs to its result.
2. The Shannon-Hartley limit gives the maximum reliably transmitted data rate of a band-limited AWGN channel.
3. The relation applies only under its stated assumptions.

Frequently asked questions about Shannon-Hartley channel capacity

What does channel capacity mean?+

Channel capacity is the theoretical rate below which suitable coding can make communication arbitrarily reliable. It does not promise that a particular modem reaches that rate. For an AWGN channel, the Shannon-Hartley limit depends on bandwidth B and linear SNR. Real systems need margin and resources for protocols, synchronization and error correction. The bound assumes a band-limited AWGN channel and suitable coding. Practical protocols fall below it because of finite block length, hardware and overhead; interpret C as an upper limit. It is not a guaranteed payload rate for an arbitrary Wi-Fi link or modem. Also check the assumed noise model.

Why must SNR be entered in linear form?+

The logarithm takes the power ratio itself, not its decibel representation. At SNRdB=10 dB, SNRlin=10^(10/10)=10; treating a dB value as a ratio gives a mis-scaled result. Convert first using 10^(SNRdB/10). Voltage ratios use a different decibel definition and require extra care. The bound assumes a band-limited AWGN channel and suitable coding. Practical protocols fall below it because of finite block length, hardware and overhead; interpret C as an upper limit. It is not a guaranteed payload rate for an arbitrary Wi-Fi link or modem. Also check the assumed noise model.

How do I calculate a capacity example?+

Substitute B=1 MHz and SNR=15. Since 1+15=16 and log₂16=4, C=10⁶·4=4·10⁶ bit/s, or 4 Mbit/s. Check the unit: hertz times a dimensionless spectral-efficiency factor gives bit/s. The value is idealized; a practical finite-delay code does not reach it exactly. The bound assumes a band-limited AWGN channel and suitable coding. Practical protocols fall below it because of finite block length, hardware and overhead; interpret C as an upper limit. It is not a guaranteed payload rate for an arbitrary Wi-Fi link or modem. Also check the assumed noise model.

Does doubling bandwidth double capacity?+

At fixed linear SNR, C is proportional to B, so doubling bandwidth doubles the formula’s capacity. In real channels, extra bandwidth can also admit more noise, so SNR may not stay fixed. The simple scaling holds only when S/N is kept fixed for the bandwidth considered and the AWGN model applies. The bound assumes a band-limited AWGN channel and suitable coding. Practical protocols fall below it because of finite block length, hardware and overhead; interpret C as an upper limit. It is not a guaranteed payload rate for an arbitrary Wi-Fi link or modem. Also check the assumed noise model.

Is the limit practically attainable?+

Information theory says rates below capacity can be made arbitrarily reliable with suitable coding and sufficiently long blocks. This is an existence limit, not a recipe for a short code. Approaching it usually costs computation, latency and complexity. Account for bandwidth limits, channel variation and implementation losses before planning a real data rate. The bound assumes a band-limited AWGN channel and suitable coding. Practical protocols fall below it because of finite block length, hardware and overhead; interpret C as an upper limit. It is not a guaranteed payload rate for an arbitrary Wi-Fi link or modem. Also check the assumed noise model.

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Subject references and further reading

This list names subject references and further reading. A page or section number appears when it is part of the reference; these are not verbatim quotations.

[1]MIT OpenCourseWare 6.451, Principles of Digital Communication II, complete lecture notes, Chapter 3: Shannon capacity.

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Shannon-Hartley channel capacity

The Shannon-Hartley limit gives the maximum reliably transmitted data rate of a band-limited AWGN channel.

AdvancedUniversity/collegeBachelor foundationsSubject specialization

Free to browse · flashcards or a multiple-choice quiz · import from Essential

Symbol

Meaning

Unit

C

Theoretical channel capacity

bit/s

B

Channel bandwidth

S/N

Received signal-to-noise power ratio

dimensionless

Derivation & background

Exam blueprint

Validity range

Derivation steps

The Shannon-Hartley limit gives the maximum reliably transmitted data rate of a band-limited AWGN channel.

1Read the relation from its definition: C=B\log_2(1+S/N).
2Identify the input, requested output and assumptions of the relation.
3For B=1 MHz and SNR=15 (about 11.76 dB), C=10⁶·log₂(16)=4 Mbit/s as a theoretical limit.

How do you calculate with Shannon-Hartley channel capacity?

Here is how to work through a typical Shannon-Hartley channel capacity (C = B·log₂(1+SNR)) task step by step:

Task

Use the example for Shannon-Hartley channel capacity to explain which inputs determine the requested output.

Solution path

The example connects concrete inputs to its result.
The Shannon-Hartley limit gives the maximum reliably transmitted data rate of a band-limited AWGN channel.
The relation applies only under its stated assumptions.

Frequently asked questions about Shannon-Hartley channel capacity

What does channel capacity mean?+

Why must SNR be entered in linear form?+

How do I calculate a capacity example?+

Does doubling bandwidth double capacity?+

Is the limit practically attainable?+

Retain Shannon-Hartley channel capacity for exams

Create a curated FSRS exam set for C = B·log₂(1+SNR): formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.

Curated formula set · LaTeX · FSRS spaced repetition · import from Essential