Field Guide · term

Also known as: Nyquist theorem, sampling theorem

The Nyquist–Shannon sampling theorem states that to represent a signal without loss you must sample at least twice its bandwidth.1 The minimum rate is the Nyquist rate; half the sample rate is the Nyquist frequency, the highest frequency a given rate can faithfully carry.

too few samples → a false (aliased) low-frequency wave
Nyquist: sample at least twice the bandwidth, or fast signals fold back as false low-frequency aliases.

How it works

The theorem’s guarantee is exact and constructive: if a signal contains no energy at or above the Nyquist frequency, its samples contain all the information in the original, and the continuous waveform can be reconstructed perfectly by sinc interpolation. Nothing is lost to sampling itself. The catch is the precondition — the signal must be band-limited. Sample too slowly for the bandwidth actually present and information does not merely vanish; it corrupts the surviving data through aliasing, folding out-of-band energy down onto real signals where it can never be separated again. That is why every practical sampler is preceded by an anti-alias filter.

For IQ (complex) sampling the useful restatement is that usable bandwidth ≈ sample rate, because the two quadrature channels resolve positive and negative frequencies separately and so double the span a real-only stream of the same rate would cover. In SDR terms, a rate of Fs lets you watch a band Fs wide centred on the tuned frequency.

Variants

  • Baseband sampling — the usual case, assuming energy from 0 up to the Nyquist frequency.
  • Bandpass sampling (undersampling) — the deeper form of the theorem: a signal of bandwidth B centred at a high carrier can be sampled at a rate just above 2B, deliberately aliasing it down to baseband, provided the whole band lands inside a single Nyquist zone.2 Direct-sampling and some HF receivers exploit this.
  • Oversampling — sampling well above the Nyquist rate to relax filter requirements and spread quantization noise.

In practice

Real filters are not brick walls, so designers leave a guard band and sample a little above the theoretical minimum — sampling at exactly 2× a component sitting right at the edge is fragile. The theorem also sets the floor for storage and processing: halving the bandwidth you need to capture halves the sample rate, the file size, and the CPU load. GopherTrunk relies on this floor to size its channel rates — 48 kHz comfortably carries a 4800-baud C4FM channel whose occupied bandwidth is well under 24 kHz.

Relevance to SDR

It is named for Harry Nyquist, who framed the sampling limit in 1928, and Claude Shannon, who proved it rigorously in 1949; it sets the floor on the sample rate needed to capture a given channel, and every choice of SDR rate and decimation factor is an application of it.

Sources

  1. Nyquist–Shannon sampling theorem — Wikipedia, on the minimum sampling rate for lossless representation. ↩

  2. Undersampling — Wikipedia, on the bandpass form of the theorem that samples below the carrier by controlled aliasing. ↩

See also

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