Nyquist-Shannon Sampling Theorem
Determine the minimum sampling rate to accurately capture a signal.
Essential for digital audio and signal processing.
The Formula
f_s ≥ 2 × f_max
The Nyquist theorem states that to perfectly reconstruct a continuous signal from digital samples, you must sample at least twice the highest frequency present in the signal.
Variables
| Symbol | Meaning |
|---|---|
| f_s | Sampling frequency (samples per second, Hz) |
| f_max | Highest frequency in the signal (Hz) |
| 2 × f_max | The Nyquist rate — minimum sampling frequency |
Example 1
Human hearing goes up to about 20,000 Hz. What sampling rate is needed?
f_max = 20,000 Hz
f_s ≥ 2 × 20,000
f_s ≥ 40,000 Hz (CD audio uses 44,100 Hz for a safety margin)
Example 2
A radio signal has a maximum frequency of 108 MHz. What sampling rate is needed?
f_max = 108 × 10⁶ Hz
f_s ≥ 2 × 108 × 10⁶
f_s ≥ 216 MHz (216 million samples per second)
When to Use It
Use the Nyquist theorem when:
- Choosing sampling rates for audio recording
- Designing analog-to-digital converters
- Avoiding aliasing artifacts in digital signals
- Determining bandwidth requirements for digital communication