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Shannon-Hartley Theorem Calculator

Calculates the maximum rate at which information can be transmitted over a communications channel with a specific bandwidth and noise.

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Channel Capacity

Formula first

Overview

The Shannon-Hartley theorem defines the maximum rate at which information can be transmitted over a communications channel of a specified bandwidth in the presence of noise. It provides the fundamental theoretical limit for data transmission, establishing that capacity is constrained by both physical frequency range and signal power relative to interference.

Symbols

Variables

C = Channel Capacity, B = Bandwidth, S/N = Signal-to-Noise Ratio

Channel Capacity
bps
Bandwidth
Hz
S/N
Signal-to-Noise Ratio
ratio

Apply it well

When To Use

When to use: Apply this theorem when calculating the maximum possible throughput of a digital communication link such as Wi-Fi, LTE, or satellite systems. It assumes the presence of Additive White Gaussian Noise (AWGN) and serves as a benchmark for evaluating the efficiency of different modulation schemes.

Why it matters: It transformed telecommunications by proving that increasing signal power has diminishing returns compared to increasing bandwidth. This insight guides modern network engineering, driving the move toward higher frequency bands and sophisticated error-correction codes to approach the 'Shannon Limit'.

Avoid these traps

Common Mistakes

  • Using SNR in dB directly without converting to a linear ratio.
  • Confusing log2 with ln or log10.

One free problem

Practice Problem

A communication channel has a bandwidth of 3,000 Hz and a linear signal-to-noise ratio (SN) of 31. Calculate the maximum theoretical channel capacity in bits per second.

Bandwidth3000 Hz
Signal-to-Noise Ratio31 ratio

Solve for:

Hint: Add 1 to the signal-to-noise ratio before calculating the base-2 logarithm.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. A Mathematical Theory of Communication by Claude E. Shannon (1948)
  2. Communication Systems by Simon Haykin
  3. Digital Communications by John G. Proakis
  4. Wikipedia: Shannon-Hartley theorem
  5. Claude E. Shannon, 'A Mathematical Theory of Communication', Bell System Technical Journal, Vol. 27, No. 3, pp. 379-423, 1948
  6. Herbert Taub, Donald L. Schilling, Goutam Saha, 'Principles of Communication Systems', 4th ed., McGraw-Hill, 2013
  7. Shannon, C. E. (1948). A Mathematical Theory of Communication. The Bell System Technical Journal, 27(3), 379-423.
  8. Cover, T. M., & Thomas, J. A. (2006). Elements of Information Theory (2nd ed.). Wiley-Interscience.