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Nyquist Sampling Theorem Calculator

Calculate Nyquist sampling rate, oversampling ratio, and aliasing frequency. Verify your ADC sampling meets the Nyquist criterion and determine data rate. Free, instant results.

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Formula

fN=2fsig,OSR=fsfN,SNR=20log⁡10(2)⋅N+10log⁡10(3/2) dBf_N = 2 f_{sig},\quad OSR = \frac{f_s}{f_N},\quad SNR = 20\log_{10}(2)\cdot N + 10\log_{10}(3/2)\text{ dB}

Reference: Nyquist, H. (1928). "Certain Topics in Telegraph Transmission Theory". AIEE Transactions. Shannon-Nyquist sampling theorem.

f_N— Nyquist rate (minimum sampling rate) (Hz)
f_sig— Signal maximum frequency / bandwidth (Hz)
f_s— Actual sampling rate (Sa/s)
OSR— Oversampling ratio
N— ADC resolution (bits)
SNR— Signal-to-quantization-noise ratio (dB)

How It Works

The Nyquist Sampling Calculator computes minimum sampling frequency and aliasing-free bandwidth — essential for ADC selection, anti-aliasing filter design, and digital signal processing system architecture. DSP engineers, embedded developers, and audio professionals use this to ensure faithful signal reconstruction. The Nyquist-Shannon theorem (1949) states that sampling frequency must exceed 2x the highest signal frequency to prevent aliasing. Practical systems use 2.2-2.5x oversampling to accommodate real filter roll-off. Per Oppenheim "Signals and Systems" (2nd ed., Ch. 7), aliasing folds high-frequency content into the baseband, causing irreversible distortion. CD audio samples at 44.1 kHz for 20 kHz bandwidth (2.205x). Professional audio uses 96 kHz (4.8x oversampling) achieving -120 dB aliasing rejection with practical filters. Modern delta-sigma ADCs use 64-256x oversampling, trading speed for resolution — a 64x oversampled 1-bit converter achieves 16-bit equivalent resolution per Schreier.

Worked Example

Design sampling system for 5 kHz vibration sensor requiring 90 dB dynamic range. Step 1: Nyquist minimum = 2 * 5 kHz = 10 kHz. Step 2: Select 2.5x oversampling for practical anti-aliasing: fs = 25 kHz. Step 3: Anti-alias filter cutoff = 5 kHz, stopband at 12.5 kHz (fs/2) needs 90 dB attenuation. Step 4: Filter order: Butterworth needs log(10^9)/log(12.5/5) = 22.5 -> 23rd order (impractical). Step 5: Use 8th-order elliptic filter (90 dB stopband) or increase to 4x oversampling (fs = 20 kHz) allowing 4th-order Butterworth. Per Kester, 4x oversampling relaxes filter requirements by 40 dB. Step 6: Resolution: 90 dB requires (90-1.76)/6.02 = 14.7 bits -> select 16-bit ADC.

Practical Tips

  • ✓Per AES17-2015, use minimum 2.2x oversampling for audio; 4x enables simpler anti-alias filters
  • ✓Delta-sigma ADCs with 64x+ oversampling eliminate external anti-alias filter requirement per Analog Devices AN-283
  • ✓Budget 10-20% bandwidth margin above signal frequency for filter transition band per IEEE 1057
  • ✓For wideband signals, consider undersampling (bandpass sampling) when signal bandwidth << center frequency

Common Mistakes

  • ✗Sampling at exactly 2x Nyquist rate — requires infinite-order brick-wall filter; use 2.2-2.5x minimum per Oppenheim
  • ✗Neglecting anti-aliasing filter design — aliased signals cannot be recovered and corrupt all lower frequencies
  • ✗Overlooking ADC aperture bandwidth — sample-and-hold must track signals to fs/2 with < 0.1 dB droop

Frequently Asked Questions

Aliasing occurs: frequencies above fs/2 fold back into baseband. A 15 kHz tone sampled at 20 kHz appears at 5 kHz (fs - fsignal). Per Shannon, aliased information is permanently lost — no digital processing can recover it. Always use anti-aliasing lowpass filter before ADC.
Practical anti-alias filters have finite roll-off. Per Kester, achieving 80 dB rejection at fs/2 requires: 2x oversampling needs 26th-order filter, 2.5x needs 10th-order, 4x needs 5th-order. Higher oversampling trades sample rate for filter simplicity and reduced in-band phase distortion.
Required bits N = (dynamic_range_dB - 1.76) / 6.02. Voice: 8-12 bits (48-72 dB). Consumer audio: 16 bits (98 dB). Professional audio: 24 bits (144 dB theoretical, ~120 dB practical). Scientific: 18-24 bits. Per IEEE 1241, add 2-bit margin for processing headroom.
No — aliasing irreversibly mixes high and low frequency content per Shannon theorem. A 30 kHz signal sampled at 44.1 kHz aliases to 14.1 kHz and cannot be distinguished from a true 14.1 kHz tone. Prevention via anti-aliasing filter is the only solution.
No. Baseband signals (DC to fmax): sample at 2.2-4x fmax. Bandpass signals (narrowband around fc): use undersampling at 2.2x bandwidth. Per Proakis, a 100 MHz signal with 10 MHz bandwidth can be sampled at 25 MHz using bandpass sampling, reducing ADC speed requirements 4x.

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