Shannon Channel Capacity
C = B × log₂(1 + SNR)
Quick answer
This calculator solves C = B × log₂(1 + SNR) for Channel Capacity. With Bandwidth 2.2 kHz, SNR (linear) 0.5, the result is 1.287 kHz. It solves in 2 directions, so you can enter the values you have and get the one you need.
Calculator
Formula
Description
The Shannon-Hartley theorem establishes the theoretical maximum rate at which information can be reliably transmitted over a communication channel with a given bandwidth and signal-to-noise ratio. Published by Claude Shannon in 1948, it is the foundation of information theory and sets an absolute upper bound that no practical modulation or coding scheme can exceed. Real-world systems like Wi-Fi, LTE, and cable modems approach this limit using advanced techniques like OFDM, turbo codes, and LDPC codes.
Variables
- C — Channel capacity in bits per second (bps)
- B — Channel bandwidth in hertz (Hz)
- SNR — Signal-to-noise ratio (linear, not dB)
Practical Notes
To convert SNR from dB to linear: SNR_linear = 10^(SNR_dB/10). Doubling the bandwidth doubles the capacity, but doubling the SNR gives diminishing returns due to the logarithmic relationship. Modern 5G systems achieve near-Shannon capacity through massive MIMO and advanced coding.
Symbols and units
| Symbol | Meaning | Default unit | Accepted units |
|---|---|---|---|
| C | Channel Capacity | Hz | Hz, kHz, MHz, GHz |
| B | Bandwidth | Hz | Hz, kHz, MHz, GHz |
| SNR | SNR (linear) | — | — |
Step by step
- 1. What you know: Bandwidth 2.2 kHz, SNR (linear) 0.5
- 2. The relation: C = B × log₂(1 + SNR)
- 3. Substitute: B = 2.2 kHz, SNR = 0.5
- 4. Result: Channel Capacity = 1.287 kHz
Worked examples
Minimum range
Given: Bandwidth 11 Hz, SNR (linear) 0.025
Result: Channel Capacity = 0.3919 Hz
Low range
Given: Bandwidth 110 Hz, SNR (linear) 0.25
Result: Channel Capacity = 35.41 Hz
Typical
Given: Bandwidth 2.2 kHz, SNR (linear) 0.5
Result: Channel Capacity = 1.287 kHz
High range
Given: Bandwidth 44 kHz, SNR (linear) 0.8
Result: Channel Capacity = 37.31 kHz
Maximum range
Given: Bandwidth 0.44 MHz, SNR (linear) 8
Result: Channel Capacity = 1.395 MHz
Solving for each variable
Solve for Channel Capacity
Bandwidth 11 Hz, SNR (linear) 0.025 → 0.3919 Hz
Solve for Bandwidth
Channel Capacity 5 Hz, SNR (linear) 0.025 → 140.4 Hz
How Channel Capacity changes with Bandwidth
| Bandwidth (input) | Channel Capacity (result) |
|---|---|
| 0.22 kHz | 128.7 Hz |
| 0.55 kHz | 0.3217 kHz |
| 1.1 kHz | 0.6435 kHz |
| 1.65 kHz | 0.9652 kHz |
| 2.2 kHz | 1.287 kHz |
| 3.3 kHz | 1.93 kHz |
| 4.4 kHz | 2.574 kHz |
| 8.8 kHz | 5.148 kHz |
| 15.4 kHz | 9.008 kHz |
| 22 kHz | 12.87 kHz |
The same result in other units
| Unit | Value |
|---|---|
| Hz | 1287 |
| kHz | 1.287 |
| MHz | 0.001287 |
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<p><a href="https://vbatpower.com/formulas/shannon-capacity/">Shannon Channel Capacity</a> Calculator by vbatpower.com</p>Frequently asked questions
How do you calculate Shannon Channel Capacity?
Use C = B × log₂(1 + SNR). With Bandwidth 2.2 kHz, SNR (linear) 0.5, Channel Capacity works out to 1.287 kHz.
What is the formula for Shannon Channel Capacity?
The formula is C = B × log₂(1 + SNR). The Shannon-Hartley theorem establishes the theoretical maximum rate at which information can be reliably transmitted over a communication channel with a given bandwidth and signal-to-noise ratio.
How do you find Bandwidth when you know the other values?
Rearrange for Bandwidth: with Channel Capacity 5 Hz, SNR (linear) 0.025, Bandwidth is 140.4 Hz.
What units does Shannon Channel Capacity use?
Channel Capacity is shown in Hz, kHz, MHz. Values are held in SI base units internally, so you can mix prefixes without converting by hand.
Related Concepts
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