core24 min studyLesson 957 in Physics

Sampling, aliasing, and antialias filters

Identify indistinguishable sampled frequencies and allocate analog filtering and clock performance before digitization.

Before you begin

On this page

The question for today

What you’ll understand

Calculate the baseband alias of a sinusoidal input.
Explain why analog rejection needs a transition band before sampling.
Estimate a declared small-jitter signal-to-noise limit.

Sampling replaces a continuous waveform by values at selected times. Even an ideal converter with infinitely fine voltage resolution cannot distinguish all possible waveforms from those values alone. Frequencies separated by integer multiples of the sampling rate produce the same discrete complex phase progression. An analog bandwidth assumption is therefore part of the measurement model. The antialias filter enforces that assumption approximately before the information is folded, while the sampling clock determines how faithfully the intended sample times are realized.

Uniform samples identify frequency only modulo the clock rate

Assume ideal instantaneous samples at times n/fs with a fixed, uniform sampling frequency fs. Adding an integer multiple of fs to an input frequency changes each sampled complex phase by an integer number of full rotations and leaves the samples unchanged. For real sinusoids, negative-frequency aliases can be represented at positive frequency with the appropriate phase convention. A convenient reported alias lies between zero and fs/2.

The baseband reconstruction theorem assumes that the input spectrum is strictly confined below half the sampling rate and that the ideal sampling and reconstruction conditions hold. At exactly the Nyquist frequency, particular phases can give degenerate samples, so equality should not be used as a comfortable design margin. Bandpass sampling can deliberately use a higher Nyquist zone, but then band placement and filtering require their own analysis. A single sample-rate number does not establish the accepted analog bandwidth.

e^{i2\pi(f+kf_s)n/f_s}=e^{i2\pi fn/f_s}\quad(k\in\mathbb Z),\qquad f_{\mathrm{alias}}=|f-kf_s|\in[0,f_s/2]

f and fs are in hertz, n is an integer sample index, and k is chosen to fold the tone into the stated baseband interval. Real cosine phase must be adjusted when using a negative-frequency representative. The identity assumes exactly uniform ideal samples.↔ Scroll wide equations horizontally.

Filtering after aliasing cannot identify the original frequency

An antialias filter reduces unwanted signals and broadband noise before they reach the sampler. A practical filter has finite rolloff, so its passband must end early enough to provide a transition region and the required stopband attenuation. Merely choosing a cutoff equal to fs/2 does not give strong suppression immediately above that frequency. Strong disturbances may require more rejection than ordinary signal-flatness considerations suggest.

Once two contributions have folded onto the same digital frequency, a digital filter cannot identify their analog origins. It can reject that frequency only by rejecting all content there, including any wanted component. Noise over several analog Nyquist zones can also fold into the same digital band. Input protection, amplifier linearity, and prefilter placement must be considered together, because a disturbance that clips an earlier stage can create new frequencies before the sampler.

\delta v\simeq\frac{dv}{dt}\,\delta t,\qquad \mathrm{SNR}_{\mathrm{jitter}}\simeq-20\log_{10}(2\pi f_{\mathrm{in}}\sigma_t)\ \mathrm{dB}

The SNR relation applies to a sinusoid with small random RMS sampling-time error σt, uncorrelated with the signal, and excludes all other errors. fin is the analog input frequency in hertz, not its lower aliased frequency. The approximation needs 2πfinσt≪1.↔ Scroll wide equations horizontally.

Timing and acquisition introduce additional response limits

Clock jitter produces voltage errors in proportion to waveform slope, making high-frequency inputs sensitive even when their digital aliases are low. A fixed aperture delay instead shifts timing systematically and can often be calibrated; random aperture uncertainty is different. A frequency error in the clock changes the inferred time scale and should not be described as independent white jitter. Synchronization and trigger uncertainty likewise need a stated relationship to the measurand.

A real acquisition system also has finite analog input bandwidth, sample-and-hold aperture, settling, and possibly channel multiplexing. An integrating aperture introduces its own frequency weighting. A multiplexer can leave a channel contaminated by the previous channel if the input does not settle before conversion. These are separate from the ideal alias identity and from voltage quantization. End-to-end acquisition tests should therefore check analog response, timing, and channel conditions rather than relying solely on the nominal number of samples per second.

Work it through

Follow an interferer through a filter and sampler

A hypothetical uniform sampler runs at fs=1000 samples/s. A 700 Hz cosine interferer with peak amplitude 70 mV first passes through an ideal one-pole analog low-pass filter with fc=100 Hz and unit DC gain. Find its sampled alias and filtered amplitude. Also state the filter's amplitude transmission for a desired 100 Hz tone.

The 700 Hz tone has representative 700−1000=−300 Hz, so its real-signal baseband alias is 300 Hz with the corresponding phase reversal.
The analog magnitude at 700 Hz is $1/\sqrt{1+(700/100)^2}=1/\sqrt{50}$ . Its peak amplitude becomes $70/\sqrt{50}\simeq9.89949$ mV before aliasing.
At 100 Hz the magnitude transmission is 1/√2≈0.707107. Thus this filter attenuates the wanted tone too and leaves a sizable 300 Hz alias; calling 100 Hz its cutoff does not prove adequate stopband rejection.

The result

The filtered interferer appears at 300 Hz with about 9.899 mV peak amplitude; the desired 100 Hz amplitude is multiplied by about 0.7071.

The living notebook

Explore the model

Interactive lab

Interactive lab · modular

Discrete phase progresses modulo a full turn

For each fresh comparison, Reset and set BOTH Modulus m=10 and Additive step a. Compare a=7 and a=3 over the first three Steps. Predict residues (7,4,1) and (3,6,9). Interpreting each residue r as phase 2πr/10 gives equal zero-initial-phase cosine samples in the two runs. This additive orbit illustrates frequency folding; it does not sample a waveform, implement a filter, or simulate clock jitter.

Additive orbitCurrent residue / next edge

Residue r0: 0
Orbit period · m/gcd(m,a): 12
Distinct residues visited: 1 of 12

Modulus m12Additive step a5

t = 0 s

Addition modulo m starts at zero: rₙ = na mod m. Playback advances two additions per second. The orbit has period m/gcd(m,a), including period one for a = 0. This is not a cryptographic security test or a primality algorithm. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forDifferent phase progressions can give the same real samples when one is the negative-frequency representative of the other.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

Timing error couples to the actual analog waveform slope. A low digital alias does not reduce the high-frequency input slope at the sampling instant.

Your turn

Try it yourself

Under the stated small, uncorrelated random-jitter model, let σt=10 ps. Compute the jitter-only SNR for analog input frequencies 10 MHz and 100 MHz. Assume adequate analog bandwidth and an otherwise ideal acquisition. Use log10(2π)≈0.798180.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Convert picoseconds to seconds. Raising the analog frequency by a factor ten changes the logarithmic estimate by twenty decibels.

Compare with the solution

At 10 MHz, $2\pi f\sigma_t=2\pi\times10^{-4}$ , giving SNR≈64.0364 dB. At 100 MHz the factor is ten times larger and SNR≈44.0364 dB. The results describe a jitter-only ceiling within the small-error model, not a measured converter specification. A lower alias frequency would not change either number, and a systematic clock-rate error would require a separate timing calibration rather than this random-jitter formula.

Keep the idea

Take it with you

Uniform sampling creates frequency equivalence modulo the sample rate.
Analog antialias rejection and a practical transition band precede irreversible folding.
Clock jitter, aperture response, settling, and voltage resolution impose different limits.

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