The Sampling Theorem
Sampling replicates the spectrum. Keep the copies apart and reconstruction is exact — not approximate.
Skip to the animationSampling replicates a signal's spectrum at every multiple of the sampling rate, so if the signal is band-limited to f_m and sampled faster than 2f_m the copies stay apart and the original can be recovered exactly — which is a claim about spectra, not about samples.
Why samples alone are not enough
Infinitely many continuous curves pass through any finite set of samples, so the samples cannot single one out. The theorem's real content is the extra hypothesis that makes the choice unique — and it is a condition on the spectrum.
The mechanism
- 1Assume the signal is band-limited: its spectrum is exactly zero above
f_m. - 2Sampling multiplies by an impulse train, and multiplication in time is convolution in frequency.
- 3Convolving with an impulse train replicates the spectrum at every multiple of
f_s. - 4If
f_s > 2f_mthe copies do not overlap, so an ideal low-pass filter isolates the original.
The images are not an artefact to be removed — they are what sampling *is*. Everything else in the theorem is reading this picture.
Two terms that get confused
- Nyquist rate
2f_m— a property of the signal. The minimum acceptable sampling rate.- Nyquist frequency
f_s/2— a property of the sampler. The highest frequency it can represent.
The hypothesis is severe: a signal of finite duration cannot be exactly band-limited, so nothing you can actually record satisfies it. Real systems approximate it with an anti-aliasing filter, which is why that filter is not optional.
Reconstruction is not joining the dots
Perfect reconstruction places a sinc at every sample, scaled by that sample's value, and adds them — the sinc being the ideal low-pass filter's impulse response.
That sinc is infinitely long and non-causal, so ideal reconstruction is unbuildable. Real converters use a zero-order hold plus a reconstruction filter and accept a known, correctable error. Linear interpolation between samples is an approximation to this, not the thing itself.
The numbers you already know
| System | Rate | Reason |
|---|---|---|
| Audio CD | 44.1 kHz | Hearing to ~20 kHz, plus filter transition band |
| Telephony | 8 kHz | Speech band-limited to 3.4 kHz |
| Oversampling ADC | 4× and up | Widens the gap so a gentle analogue filter suffices |
| Undersampling | Below 2f_m, deliberately | Uses an image to shift a radio band down |
44.1 exceeds 40 because a real filter cannot cut off vertically — the theorem sets the floor and the filter sets the margin. Undersampling shows the images are a resource rather than merely a hazard.
The numbers you will be asked for
- Sampling theorem
f_s > 2·f_m
Strictly greater — the boundary case fails.
- Nyquist rate and frequency
2f_m (signal) · f_s/2 (sampler)
Different quantities, frequently confused.
- Sampled spectrum
X_s(f) = f_s · Σ X(f − k·f_s)
Replication at every multiple of f_s.
- Reconstruction
x(t) = Σ x[n]·sinc((t − nT)/T)
Exact, and unbuildable.
Advantages and disadvantages
Advantages
- Guarantees exact reconstruction, not an approximation.
- Explains every standard sampling rate in use.
- Its picture — replicated spectra — makes aliasing predictable.
- Undersampling turns the images into a deliberate tool.
Disadvantages
- Requires exact band-limiting, which no finite-duration signal has.
- Ideal reconstruction needs an infinite, non-causal sinc.
- Real anti-aliasing filters need a transition band, so practical rates exceed the theoretical minimum.
- Says nothing about quantisation, which is a separate error source.
Watch it work
Check yourself
question 1 / 4
One question at a time. Pick an answer to see why it is right or wrong, then move on — there is no score to keep and nothing is saved.