Sampling, Aliasing and Filtering

Nyquist and sampling

Sampling reads a signal at regular intervals. If the sample rate is f_s, the highest frequency that can be represented is f_s ÷ 2, the Nyquist frequency. Components above it do not vanish: they are folded down into the range below it and look like slower signals. Once sampled, an alias cannot be told apart from a real low-frequency signal, so the only cure is to remove the high frequencies before sampling (see the filter below).

f_alias = | f − k · f_s |, with k the whole number that makes the result lie between 0 and f_s ÷ 2

How aliasing appears

Signal frequency	Status at fs = 50 Hz	Appears at
10 Hz	below Nyquist: correct	10 Hz
20 Hz	below Nyquist: correct	20 Hz
25 Hz	above Nyquist: aliased	25 Hz
40 Hz	above Nyquist: aliased	10 Hz
50 Hz	above Nyquist: aliased	0 Hz
60 Hz	above Nyquist: aliased	10 Hz
90 Hz	above Nyquist: aliased	10 Hz
110 Hz	above Nyquist: aliased	10 Hz

• Aliases look real. A slow oscillation in a log that nobody expected may be fast noise folded down.
• A suspicious frequency that changes when you change the sample rate is an alias; a real signal stays at the same frequency.
• Sampling at an exact multiple of the interference (for example 50 Hz mains at 100 Hz) maps it to a constant offset instead of an oscillation: harmless-looking, still wrong.

The anti-alias filter

A low-pass filter ahead of the ADC removes frequencies above those you want. The simplest is a resistor and capacitor with a cut-off f_c = 1 ÷ (2π R C), where the signal is down 3 dB (to 71 %). A single pole falls only 20 dB per decade, so it does not stop much just above the cut-off.

f_c = 1 ÷ (2π R C) |H(f)| = 1 ÷ √( 1 + (f ÷ f_c)² )

• Filter first, then sample. A digital filter after sampling cannot remove an alias.
• Set the corner by the signal, not the sample rate: the filter must pass everything you care about, and stop everything that would fold into it. Aim for the stop-band to start at or below f_s − f_max.
• Slow signals still need a filter if the environment is noisy: a temperature signal sampled once a minute can still carry 50 Hz pick-up that folds into the data.
• Integrating converters (dual-slope, delta-sigma at a low rate) average over a window, which acts as a filter: integrating over 20 ms nulls 50 Hz and its harmonics.

Choosing the sample rate

Goal	Rule of thumb	Example
Detect that a frequency exists	Just over 2 × the frequency (Nyquist)	A 10 Hz signal needs more than 20 Hz
See the waveform shape	5 to 10 × the highest frequency	About 100 Hz for the same 10 Hz signal
Measure peaks and rise times	10 to 20 × the highest frequency, or 5 samples across the fastest edge	A 1 ms edge needs a sample every 0.2 ms
Trend of a slow process	Interval about τ ÷ 5	A 5 min time constant: every 60 s (see Planning a Logging System)
Power and energy	Many samples per mains cycle (at least 20 to 64); more for harmonics	At 50 Hz, at least 1 kHz for harmonics to the 9th
Vibration and sound	2.56 × the analysis frequency in analyzers, higher for time waveforms	A 5 kHz band: 12.8 kHz

• Sampling faster is cheap in the ADC and expensive in storage. Often the best design samples fast, filters or averages digitally, and stores a lower rate (see below).
• Consistency matters: a rate from a timer interrupt is regular; one from a software loop with sleep() is not (see Time Stamps and Clocks).
• Multi-channel scanning reads the channels one after another, so they are not simultaneous. For phase or power calculations use simultaneous-sampling converters or correct for the delay.

Oversampling and decimation

Oversampling samples much faster than needed, then decimates: low-pass filters digitally and keeps every M-th sample. The analog filter can then be gentle (the aliases are far above the signal), and the averaging reduces noise.

# Decimate by M with a simple block average (a basic low-pass), then keep one value per block
import numpy as np
M = 40
x = np.asarray(samples[: len(samples) // M * M], dtype=float)
y = x.reshape(-1, M).mean(axis=1)          # 4000 Hz  ->  100 Hz
# For a sharper filter use scipy.signal.decimate(x, M, ftype="fir") in stages of 10 or less

Record length and resolution

The record length sets the frequency resolution of any spectrum you compute: resolution Δf = f_s ÷ N = 1 ÷ T. A short record cannot separate close frequencies, and a record that does not hold a whole number of cycles “leaks” energy across the spectrum unless a window (Hann, for example) is applied.

Digital filters for logged data

Filter	What it does	Notes
Moving average	Mean of the last N samples	Simple; cut-off about 0.443 fs ÷ N (4.4 Hz for N = 10 at 100 Hz); poor stop-band; delays by (N−1) ÷ 2 samples
Exponential (first order IIR)	y = y + α (x − y)	One line of code, one variable; cut-off set by α; a long tail for step changes
Median	Middle value of the last N	Removes single-sample spikes without smearing steps (see Numerical Methods)
Butterworth / FIR	Designed low-pass with a defined slope	Use scipy.signal; forward and backward filtering (filtfilt) gives zero phase lag offline, which a real-time filter cannot

• Every filter adds delay. Compare the filtered time against events; do not use a filtered signal to decide when something happened without allowing for the lag.
• Filter the data you use, store the raw data you can. Raw data can be re-filtered with better ideas; filtered data cannot be un-filtered.
• A filter cannot create information: it cannot repair an aliased signal, and a smoothing filter turns a short spike into a wide bump (check the peak is not underestimated).

Checking for aliasing

Test	What it shows
Change the sample rate by 10 to 30 % and compare	A real frequency stays; an alias moves
Look at the signal with a fast scope or a burst at a high rate	The true spectrum, including high-frequency noise you did not know about
Add the filter and see what disappears	Whatever vanishes was out of band (or an alias of it)
Plot a spectrum	Lines at suspicious frequencies, and a raised noise floor near the Nyquist frequency
Check the numbers: the alias formula for known interferers (mains, a PWM drive, a switching supply)	Whether a mystery frequency is explained

Timing jitter also limits fast signals: a sampling-time uncertainty of 1 µs on a 1 000 Hz sine gives an amplitude error up to 2π f t = 0.63 % of the amplitude. Use hardware timing for fast signals (see Time Stamps and Clocks).

Related

• Data Logging: the overview of this guide.
• Sensors, Signal Conditioning and ADCs: the converter and its noise.
• Storage, Formats and Data Rates: what the sample rate costs in storage.
• Numerical Methods and Embedded Systems: filters and ADCs in code.

FAQ

What sample rate should I use?

At least twice the highest frequency present, and 5 to 10 times the highest frequency you want to see, after an anti-alias filter. Check by changing the rate: a real signal does not move.

Why does 60 Hz appear as 10 Hz?

At a 50 Hz sample rate, a 60 Hz input folds to | 60 − 50 | = 10 Hz. Filter it out before the converter or sample faster.

Can I filter the aliasing out afterwards?

No. Once sampled, an alias is indistinguishable from a real low-frequency signal. The filter must act before the converter.

Is averaging the same as an anti-alias filter?

Only if the averaging is done on much faster samples (oversampling). Averaging a few slow samples does not stop high-frequency noise folding in.

References

• Oppenheim, A. V. and Schafer, R. W., Discrete-Time Signal Processing.
• Smith, S. W., The Scientist and Engineer’s Guide to Digital Signal Processing (free online).
• Analog Devices, Data Conversion Handbook and Analog Filter Design application notes; the SciPy scipy.signal documentation.

Last reviewed: October 2026. Filter values are examples: compute your own corner frequency and check the stop-band attenuation.