Nyquist-Shannon Sampling Theorem
Minimum sampling rate to perfectly reconstruct a signal.
Nyquist frequency and aliasing explained with examples.
The Theorem
The Nyquist-Shannon sampling theorem states that to perfectly reconstruct a continuous signal from its digital samples, you must sample at a rate at least twice the highest frequency present in the signal.
This minimum rate (2 × fmax) is called the Nyquist rate. Half the sampling rate (fs / 2) is called the Nyquist frequency — it is the highest frequency that can be faithfully represented.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| fs | Sampling rate (samples per second) | Hz |
| fmax | Maximum frequency in the signal | Hz |
| fN | Nyquist frequency = fs / 2 | Hz |
Aliasing
When the sampling rate is too low (fs < 2 × fmax), high-frequency components are misrepresented as lower frequencies. This distortion is called aliasing and cannot be corrected after sampling. Anti-aliasing filters remove frequencies above fN before the signal is sampled.
Example 1
CD audio captures frequencies up to 22,050 Hz. What sampling rate is needed?
fmax = 22,050 Hz
fs ≥ 2 × 22,050
fs ≥ 44,100 Hz — this is exactly the standard CD sampling rate (44.1 kHz)
Example 2
A sensor records vibrations up to 500 Hz. You sample at 800 Hz. Is this enough?
Nyquist rate = 2 × 500 = 1,000 Hz
Your sampling rate: 800 Hz < 1,000 Hz
Not enough — aliasing will occur. You need at least 1,000 Hz sampling rate.
Example 3
A digital phone system samples at 8,000 Hz. What is the maximum frequency it can capture?
fN = fs / 2 = 8,000 / 2
fN = 4,000 Hz — this covers the essential range of human speech (300–3,400 Hz)
When to Use It
- Choosing the right sampling rate for audio recording and music production
- Designing analog-to-digital converters (ADCs)
- Setting up data acquisition systems for sensors and instruments
- Understanding why digital images can show moire patterns (spatial aliasing)
- Telecommunications system design and bandwidth planning