Move across the waveform to inspect the nearest acquired sample.
Seven setups worth seeing. Each one is a way a real acquisition goes wrong.
The record represents the input.
- Input
- 1.00 kHz
- Nyquist
- 4.00 kHz
- Predicted alias
- 1.00 kHz
- Measured from record
- 1.00 kHz
- Samples per cycle
- 8.00
Where the false frequency comes from
Sampling copies the spectrum arriving at the converter around every multiple of the sample rate. Anything the copies drop below fs/2 is indistinguishable from a real signal down there. The pale trace is the actual spectrum of the record above, computed from the samples themselves.
arriving at the sampler folded copies spectrum of the record fs/2
Every combination at once
Input frequency across, sample rate up. Colour is the frequency the sampler reports. The dark wedge on the left is the only region where it reports the truth; everything to the right of it is a fold. The fan of ridges is what makes aliasing so easy to walk into by accident.
Notes from the bench
Short answers to the questions this instrument tends to raise.
- Nyquist frequency, fs/2
- The highest frequency a sampler can represent without ambiguity. Above it, a sinusoid produces exactly the same numbers as a lower one, so no amount of processing downstream can separate them.
- The folding rule
- Pick the integer k that brings
|f − k·fs|into 0 … fs/2. That is what you will measure. Even-numbered Nyquist zones also come back mirrored, so sweeping the input upward makes the reported frequency walk downward. - Two samples per cycle is a limit, not a target
- The theorem needs
fs > 2Bwith a strict inequality, a band-limited signal, and an ideal interpolator. At exactly two samples per cycle the amplitude you recover depends on sample phase, which is why the clock offset control can null the waveform completely. - A square wave is not band-limited
- Its edges are built from odd harmonics that run far above the fundamental, so “two samples per period” says nothing useful about it. Sample a 1 kHz square at 3 kS/s and the fundamental is legal while the third harmonic onwards folds back on top of it. Rate the acquisition against the edge, not the repetition rate.
- A real square wave is band-limited, by its edge rate
- Nothing physical carries infinite harmonics. A trapezoid with transition time
trrolls its harmonics off past roughly1/(π·tr), which is what makes the bandwidth question answerable at all. The edge rate control sets that, and every stage downstream inherits it. - Why a clean trace proves nothing
- An alias is a genuine sinusoid in the data, not noise. It is stable, it triggers, it measures cleanly. The record is not corrupted; it is honestly reporting a frequency that was never at the input.
- What the anti-alias filter actually buys
- It throws away information you were never going to represent correctly, before the sampler turns it into information you cannot identify. Losing a harmonic is recoverable by design; a fold is not. A realisable filter has skirts, though — set its response to anything but brick wall and some energy still gets past.
- Bandwidth and sample rate are separate specifications
- Bandwidth decides what reaches the sampler; sample rate decides what the sampler can represent. A fast converter behind a wide front end with no filtering is the classic way to fold noise and harmonics into a measurement band. A fast converter behind a narrow front end is a lot of points describing a signal that already lost its detail.
- Rise time, bandwidth, and the constant between them
- The familiar
0.35/BWbelongs to a single-pole or Gaussian response. This page measures the 10–90% transition off each modelled response rather than assuming it: 0.35 for a single pole, 0.34 for a Gaussian, 0.39 for a four-pole maximally flat filter and 0.45 for a brick wall — the same spread instrument vendors quote as “0.35 to 0.45, depending on response”. - Reconstruction modes
- Hold is what a plain DAC does. Linear is what most plotting code does, and it exaggerates how bad sparse sampling looks. Band-limited interpolation is the one the theorem is about, and it is the only one that shows the alias for what it is: a smooth sinusoid at the wrong frequency.
- Memory depth is the third specification
- Record length divided by sample rate is the longest window an instrument can hold at full speed. Ask for a longer one and it quietly drops the sample rate to fit, which is how a setup that was safe at full rate starts aliasing halfway through a zoom-out. The record length control does exactly that.
- Where the instrument figures come from
- On the Instruments page, bandwidth, sample rate, resolution and memory depth are published specifications for the named models. Response shape, effective bits and jitter are modelled per class to sit in a plausible place, not copied from a datasheet, and they are quoted at each instrument’s full bandwidth.
Keyboard
- 1–7load a preset
- Ffront end in or out
- Aanti-alias filter in or out
- Ssampler in or out
- Rreset the bench
- ←→nudge whichever control has focus
Every control is in the address bar, so a link carries the exact setup with it.