Variables

Symbol	Quantity	Unit
C	Channel capacity	bps
B	Bandwidth	Hz
S/N	Signal-to-noise ratio (linear)

What it means

The Shannon–Hartley theorem establishes the theoretical upper bound on the rate of error‑free data transmission over a noisy channel. It states that the channel capacity C (in bits per second) is C = B log₂(1 + S/N), where B is the bandwidth in hertz, and S/N is the signal‑to‑noise ratio (power ratio, not dB). This formula is foundational in information theory and telecommunications, guiding the design of modems, wireless systems, and data networks. It shows that capacity increases linearly with bandwidth and logarithmically with S/N. For example, to double the data rate, either the bandwidth must double or the S/N must be squared. In practice, achieving capacity requires sophisticated coding and modulation schemes. The theorem also implies that, given enough bandwidth, even a weak signal can transmit data reliably, as long as the rate is below capacity. It is used to evaluate the efficiency of communication systems and to set realistic performance targets. The Shannon limit is a benchmark against which actual systems are measured.

Worked example

Shannon-Hartley Channel Capacity – Two Examples

Real‑World

Scenario 1 – Wi-Fi Network: A Wi-Fi channel has bandwidth 20 MHz (20,000,000 Hz) and SNR = 30 dB (ratio 1000). What is the maximum data rate?

Parameter	Value
B	20,000,000 Hz
SNR	1000

1log₂(1 + 1000) = log₂(1001) ≈ 9.967

2C = 20,000,000 × 9.967 ≈ 199,340,000 bps ≈ 199.3 Mbps

Result ≈ 199 Mbps ✓ Theoretical max for 20 MHz, 30 dB SNR

Scenario 2 – Fiber Optic Link: A fibre link has bandwidth 10 GHz (10,000,000,000 Hz) and SNR = 20 dB (ratio 100). Find maximum bit rate.

Parameter	Value
B	10,000,000,000 Hz
SNR	100

1log₂(1 + 100) = log₂(101) ≈ 6.658

2C = 10,000,000,000 × 6.658 ≈ 66.58 Gbps

Result ≈ 66.6 Gbps ✓ High‑capacity link

Key insight: Shannon capacity depends on both bandwidth and SNR – increasing either improves data rate, but with diminishing returns.

Common mistakes

SNR units: S/N must be a linear ratio, not dB – convert dB to linear using 10^(dB/10) before using the formula.
Bandwidth B: In hertz (Hz) – not kHz or MHz without conversion.
Log base: The formula uses log₂ (base 2) – not natural log or log₁₀.
Channel capacity: This is the maximum theoretical limit; real channels have lower rates due to implementation losses.
Additive white Gaussian noise: The theorem assumes AWGN; for other noise types, the capacity may differ.

Applications

The Shannon‑Hartley theorem defines the maximum error‑free data rate (channel capacity) of a communication channel given its bandwidth and signal‑to‑noise ratio. This fundamental law of information theory guides the design of all modern communication systems, from Wi‑Fi and cellular networks to satellite and optical fibre communications. Engineers use the formula to determine the theoretical upper bound on data throughput, helping to select modulation schemes, coding rates, and bandwidth allocations. In practice, it is applied in spectrum planning, network capacity analysis, and the evaluation of new transmission technologies. Understanding this capacity helps in designing systems that approach the limit while managing interference and noise, ultimately enabling higher data rates and more reliable connections.

Design of wireless and wired communication systems
Network capacity planning and performance analysis
Selection of modulation and coding schemes
Spectrum management and interference mitigation
Development of 5G, Wi‑Fi, and fibre‑optic technologies

Shannon-Hartley Channel Capacity

Shannon-Hartley Channel Capacity Calculator C = B · log₂(1 + S/N)