---
title: "Sampling theorem, Nyquist criterion and aliasing"
subject: "Instrumentation I"
chapter: 6
key: "semester_4/instrumentation_i:sampling-nyquist-aliasing"
source: https://www.bctengineer.com/q/semester_4/instrumentation_i/sampling-nyquist-aliasing
site: BCT Engineering
---

# Sampling theorem, Nyquist criterion and aliasing

Instrumentation I · Chapter 6: Digital Instrumentation

- Asked in: year not given, 8 marks
- Video: [SAMPLING THEOREM in digital communication - sampling rate and Nyquist rate](https://www.youtube.com/watch?v=wgS5OglyB80) — Electronics Subjectified
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/instrumentation_i/sampling-nyquist-aliasing
- Whole chapter: https://www.bctengineer.com/md/semester_4/instrumentation_i/chapter-6

## Question

What is sampling and sampling theorem? State and explain Nyquist criterion. Also explain the phenomenon of aliasing and the way to eliminate it.

## Answer

**Sampling** is the process of taking values (samples) of a continuous-time signal at regular intervals Ts, to convert it into a discrete-time signal. The sampling frequency is fs = 1/Ts.

#### Sampling theorem
A band-limited continuous signal, with no frequency component above fm, can be completely recovered from its samples if it is sampled at a rate

**fs ≥ 2 fm**

The signal is then reconstructed by passing the samples through an ideal low-pass filter with cutoff fm.

#### Nyquist criterion
- The sampling frequency must be at least twice the highest frequency in the signal. The minimum rate **fN = 2fm** is called the **Nyquist rate**, and fs/2 is called the **Nyquist frequency**.
- Sampling at fs > 2fm: the copies of the spectrum, repeated around multiples of fs, do not overlap.
- Sampling at fs = 2fm: the limiting case, recoverable only in theory (with an ideal filter).
- In practice fs = 2.5 to 10 times fm is used, to allow for a practical (non-ideal) filter.

Example: a signal with a maximum frequency of 4 kHz needs fs ≥ 8 kHz (telephone speech is sampled at 8 kHz).

#### Aliasing
If fs < 2fm, the spectral copies at multiples of fs overlap. High-frequency components then appear as lower frequencies in the sampled signal and cannot be separated again. This distortion is called **aliasing**; a component at frequency f appears at |f − k·fs|.

```text
 Spectrum, fs > 2fm        Spectrum, fs < 2fm (overlap)
 |\    /|\    /|\          |\  /|\  /|\
 | \  / | \  / | \         | \/ | \/ | \
 +--fm--fs-----> f         +--fm fs------> f
```
Example: a 7 kHz tone sampled at 10 kHz appears as 3 kHz.

#### Eliminating aliasing
1. **Anti-aliasing filter:** place a low-pass filter before the sampler, with its cutoff at (or below) fs/2, to remove all components above fs/2.
2. **Increase the sampling rate** so that fs ≥ 2 × the highest frequency component present (including noise).
3. **Oversampling**, followed by digital filtering and decimation.

Aliasing cannot be removed after sampling, so the filter must be placed before the sampler.

## Check yourself

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

1. According to the sampling theorem, a signal band-limited to fm can be fully recovered if:
   A. fs ≤ 2 fm
   B. fs = fm
   C. fs ≥ 2 fm
   D. fs ≥ fm/2
   Answer: C. The sampling rate must be at least twice the highest frequency component. (Key point: Sampling theorem)

2. What is the Nyquist rate for a signal with maximum frequency 4 kHz?
   A. 4 kHz
   B. 8 kHz
   C. 2 kHz
   D. 16 kHz
   Answer: B. Nyquist rate = 2 fm = 8 kHz. (Key point: Nyquist rate)

3. What happens when fs < 2 fm?
   A. The spectral copies separate widely
   B. The signal is perfectly reconstructed
   C. The signal amplitude doubles
   D. The spectral copies overlap and high frequencies appear as lower ones
   Answer: D. Overlap causes aliasing; a 7 kHz tone sampled at 10 kHz appears at 3 kHz. (Key point: Aliasing)

4. Where must the anti-aliasing filter be placed and why?
   A. Before the sampler, because aliasing cannot be removed after sampling
   B. After the sampler, to remove aliases
   C. After the DAC only
   D. Anywhere, it makes no difference
   Answer: A. A low-pass filter before the sampler removes components above fs/2; after sampling aliased components cannot be separated. (Key point: Anti-aliasing)

## Related in other subjects

- [Nyquist-Kotelnikov sampling theorem; aliasing and aperture](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anyquist-kotelnikov-aliasing-aperture) — Communication Systems, chapter 4
- [Sampling theorem; aliasing cause; Nyquist](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Asampling-aliasing-cause-nyquist) — Signals and Systems, chapter 4
- [How Nyquist criterion prevents aliasing](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Anyquist-prevents-aliasing) — Signals and Systems, chapter 4
- [Sampling theorem; aliasing and aperture effect remedies](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asampling-theorem-aliasing-aperture-remedy) — Communication Systems, chapter 4