---
title: "Nyquist channel capacity theorem"
subject: "Communication Systems"
chapter: 6
key: "semester_6/communication_systems:nyquist-channel-capacity"
source: https://www.bctengineer.com/q/semester_6/communication_systems/nyquist-channel-capacity
site: BCT Engineering
---

# Nyquist channel capacity theorem

Communication Systems · Chapter 6: Baseband Digital Data Transmission

- Asked in: 2080 Bhadra (CS II), 8 marks
- 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_6/communication_systems/nyquist-channel-capacity
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-6

## Question

State and explain Nyquist Channel Capacity Theorem.

## Short answer

Nyquist theorem states a noiseless channel of bandwidth B Hz can transmit up to 2B symbols per second without ISI. Using M signal levels, the maximum bit rate is C = 2B log2 M bits/s. This assumes ideal sinc pulses; practical systems use raised cosine filtering which reduces the rate to 2B/(1+α) symbols/s.

## Answer

**Nyquist channel capacity theorem:** for a noiseless channel of bandwidth $B$ Hz, the maximum rate at which independent pulses (symbols) can be sent without ISI is $2B$ symbols per second. If each symbol takes one of $M$ distinct levels, the maximum bit rate (capacity) is

$$
C = 2B\log_2 M\ \text{bits/s}
$$

#### Explanation

1. **Signalling rate limit.** A channel of bandwidth $B$ can pass the sinc pulse $p(t) = \text{sinc}(2Bt)$, whose zeros occur every $\frac{1}{2B}$ s. Pulses spaced $T_s = \frac{1}{2B}$ apart therefore do not interfere at the sampling instants (Nyquist zero-ISI criterion). Hence the maximum symbol rate is

$$
r_{max} = \frac{1}{T_s} = 2B\ \text{symbols/s (bauds)}
$$

   This rate $2B$ is called the **Nyquist rate** and $B = r/2$ the **Nyquist bandwidth**. Sending faster than $2B$ causes ISI that cannot be removed.

2. **Bits per symbol.** With $M$ equally likely levels, each symbol carries $\log_2 M$ bits. With binary signalling ($M = 2$), $C = 2B$ bits/s.

3. **Capacity.** Multiplying, $C = 2B\log_2 M$.

#### Examples

For a telephone channel with $B = 3$ kHz:

$$
\begin{aligned}
M = 4: \quad C &= 2(3000)\log_2 4 = 12\ 000\ \text{bits/s} \\
M = 16: \quad C &= 2(3000)\log_2 16 = 24\ 000\ \text{bits/s}
\end{aligned}
$$

#### Points to note

- The theorem assumes a **noiseless** channel. In theory, $C$ grows without limit as $M$ increases.
- In practice, noise limits $M$: closely spaced levels cannot be told apart. The noise-limited capacity is given by Shannon–Hartley, $C = B\log_2(1 + S/N)$. Comparing the two, the useful number of levels is about $M \approx \sqrt{1 + S/N}$.
- The ideal pulse needed for $2B$ symbols/s (brick-wall filter) is not realizable; with a raised cosine pulse of roll-off $\alpha$, the rate is $\frac{2B}{1+\alpha}$ symbols/s.

| Nyquist capacity | Shannon capacity |
|---|---|
| $C = 2B\log_2 M$ | $C = B\log_2(1+S/N)$ |
| Noiseless channel | Channel with AWGN |
| Limited by ISI (bandwidth) | Limited by noise |
| Depends on number of levels | Depends on SNR |

## Check yourself

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

1. A noiseless channel of bandwidth B can carry at most:
   A. B/2 symbols/s
   B. 2B symbols/s
   C. B^2 symbols/s
   D. B symbols/s
   Answer: B. Sinc pulses with zeros every 1/2B s give a max symbol rate of 2B (Nyquist rate). (Key point: Maximum symbol rate)

2. With M levels, the Nyquist capacity is:
   A. B log2(1+M)
   B. B log2 M
   C. 2B M
   D. 2B log2 M
   Answer: D. Each symbol carries log2 M bits at 2B symbols/s. (Key point: Capacity formula)

3. For B = 3 kHz and M = 16, C is:
   A. 24 kb/s
   B. 12 kb/s
   C. 6 kb/s
   D. 48 kb/s
   Answer: A. 2(3000) log2 16 = 6000 x 4 = 24000 b/s. (Key point: Example)

4. What limits M in practice?
   A. Noise
   B. Nothing
   C. Carrier frequency
   D. Filter length only
   Answer: A. Noise makes closely spaced levels indistinguishable; Shannon gives M about sqrt(1+S/N). (Key point: Real limit)

## Related in Communication Systems

- [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
- [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×
- [State and prove Nyquist sampling theorem](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anyquist-sampling-prove) — Communication Systems, chapter 4
- [Nyquist sampling theorem; flat top sampling](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anyquist-sampling-flat-top) — Communication Systems, chapter 4
- [Nyquist sampling theorem; natural sampling](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anyquist-natural-sampling) — Communication Systems, chapter 4

## Related in other subjects

- [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×
- [State Nyquist and Shannon capacity formulae](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Anyquist-shannon-capacity-formula) — Data Communication, chapter 1, asked 2×
- [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