---
title: "State Nyquist and Shannon capacity formulae"
subject: "Data Communication"
chapter: 1
key: "semester_5/data_communication:nyquist-shannon-capacity-formula"
source: https://www.bctengineer.com/q/semester_5/data_communication/nyquist-shannon-capacity-formula
site: BCT Engineering
---

# State Nyquist and Shannon capacity formulae

Data Communication · Chapter 1: Introduction

- Asked in: 2074 Asoj, 2 marks; 2072 Kartik, 2 marks (asked 2 times)
- Video: [L17: Nyquist Limit Theorem, Shannon's Channel Capacity, Maximum Bit Rate with Example | DCN Lectures](https://www.youtube.com/watch?v=0O6jxk5i_Lc) — Easy Engineering Classes
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/data_communication/nyquist-shannon-capacity-formula
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-1

## Question

State Nyquist's and Shannon's channel capacity formulae.

## Short answer

Nyquist (noiseless): C = 2B log2 L bps. Shannon (noisy): C = B log2(1 + S/N) bps.

## Answer

**Nyquist formula (noiseless channel)**

C = 2B log₂L bps

where B = bandwidth in Hz and L = number of discrete signal levels. For a binary signal (L = 2), C = 2B.

**Shannon formula (noisy channel)**

C = B log₂(1 + S/N) bps

where S/N is the signal-to-noise power ratio (SNR in dB = 10 log₁₀(S/N)).

Nyquist gives the rate for an ideal channel; Shannon gives the upper limit when noise is present.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which is Nyquist's formula for a noiseless channel?
   A. C = 2B log2 L
   B. C = B log2(1 + S/N)
   C. C = B log2 L
   D. C = 2B(1 + S/N)
   Answer: A. L is the number of signal levels; Nyquist ignores noise. (Key point: Nyquist formula)

2. For a binary signal (L = 2), Nyquist capacity is
   A. B
   B. 2B
   C. 4B
   D. B/2
   Answer: B. log2 2 = 1, so C = 2B. (Key point: Binary Nyquist)

3. When does Shannon's formula apply?
   A. Only for noiseless channels
   B. Only for binary signals
   C. When noise is present, giving an upper limit
   D. Only in optical fibre
   Answer: C. Shannon includes S/N and gives the limit for a noisy channel. (Key point: Shannon applicability)

## Related in Data Communication

- [Shannon-Hartley channel capacity theorem](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Ashannon-hartley-theorem) — Data Communication, chapter 1, asked 3×
- [Capacity and Nyquist levels, 3 to 4 MHz, 24 dB](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Anyquist-shannon-bandwidth-3-4mhz) — Data Communication, chapter 1
- [Nyquist levels for 3 to 4 MHz channel](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Anyquist-noiseless-24db) — Data Communication, chapter 1
- [Channel capacity for 6 kHz, 6 dB](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Acapacity-6khz-6db) — Data Communication, chapter 1
- [Entropy, information and channel capacity](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Ainformation-entropy-capacity) — Data Communication, chapter 8

## Related in other subjects

- [Shannon channel capacity theorem and its limitations](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashannon-capacity-limitations) — Communication Systems, chapter 6, asked 2×
- [Shannon channel capacity theorem and its implication](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashannon-capacity-implication) — Communication Systems, chapter 6
- [Nyquist channel capacity theorem](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anyquist-channel-capacity) — Communication Systems, chapter 6
- [Shannon-Hartley theorem; capacity limit 1.44S/N₀](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashannon-hartley-1-44) — Communication Systems, chapter 6, asked 2×