Nyquist-Shannon Sampling Theorem
Nyquist theorem states sampling rate must be at least 2× the signal frequency.
Explains why CD audio uses 44,100 Hz to capture up to 22,050 Hz accurately.
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
Key Notes
- Sampling below the Nyquist rate causes aliasing — high-frequency components fold back and appear as false lower-frequency artifacts (e.g., a spinning wheel that appears to spin backward on video)
- In practice, sampling rates are set above the strict minimum — CD audio uses 44.1 kHz (not 40 kHz) to allow for a non-ideal low-pass filter that rolls off gradually before 20 kHz
- The theorem requires the signal to be band-limited (no frequencies above f_max); real-world signals with sharp transients are not perfectly band-limited and may still exhibit aliasing
Key Notes
- Nyquist-Shannon theorem: f_sample ≥ 2 × f_max: To perfectly reconstruct a band-limited signal, the sampling rate must be at least twice the highest frequency in the signal. The minimum sufficient sampling rate is called the Nyquist rate; sampling at exactly twice f_max is called sampling at the Nyquist frequency.
- Aliasing — the consequence of undersampling: When a signal is sampled below the Nyquist rate, high-frequency components fold back and appear as spurious low-frequency components. A 15 kHz tone sampled at 22 kHz appears as a 7 kHz alias. This is why anti-aliasing filters are applied before sampling.
- Audio CD: 44,100 Hz is not arbitrary: Human hearing extends to ~20 kHz. Nyquist requires at least 40 kHz; 44,100 Hz provides margin above the hearing limit and matches earlier video frame rate standards. The anti-aliasing filter cuts off above ~20 kHz.
- Oversampling for practical filters: Real anti-aliasing filters cannot cut off sharply at exactly f_max/2. Practical systems oversample (2× to 8×) so the filter can transition gradually. The extra samples are later decimated back to the target rate.
- Applications: The Nyquist theorem governs digital audio (CDs, streaming), analog-to-digital converter (ADC) design, digital oscilloscopes, MRI signal acquisition, radar and sonar signal processing, and any system that converts continuous signals to discrete digital representations.