Sampling theorem

Uniform sampling records x[n]=x(nT), with sampling rate fₛ=1/T. Arbitrary continuous signals can vary between the recorded points, so samples alone do not determine an unrestricted waveform. Band limitation supplies the missing constraint: when a baseband signal has no frequency content above B in absolute value, fₛ>2B permits unique ideal reconstruction from its complete sample sequence.

The quantity 2B is commonly called the signal's Nyquist rate; fₛ/2 is the sampling system's Nyquist frequency. They are not interchangeable definitions. Using a strict inequality avoids boundary-frequency ambiguities. The slogan “two points per cycle” omits assumptions needed to use the result correctly.

References: [1]

Sampling produces spectral copies separated by the sampling frequency. When these copies remain separated, appropriate ideal low-pass reconstruction can retain the desired spectrum. Overlap can instead make distinct continuous signals produce identical samples: this is aliasing.

For example, sample zero-phase cosine waves at 1000 samples per second. A 100-hertz wave and a 900-hertz wave give the same values because cos(2π·900n/1000)=cos(2π·100n/1000) for integer n. Without additional frequency-band information, those samples cannot identify which waveform was present. Drawing a smoother curve or adding interpolated points does not supply the missing distinction.

References: [1] [2]

A conventional baseband acquisition system places an analog anti-alias low-pass filter before analog-to-digital conversion. It attenuates higher-frequency signals and noise that would fold into the band of interest. Digital filtering after sampling cannot generally separate a desired component from interference that has already aliased onto it.

Real filters have transition bands and finite attenuation rather than an infinitely sharp cutoff. Sampling above the theoretical minimum provides room for practical filtering. The required margin depends on the desired band, out-of-band interference and acceptable error, not a universal multiplier.

Deliberate undersampling of a suitably restricted bandpass signal is also possible. This uses controlled spectral relocation rather than defeating the theorem; the occupied bands and their sampled images must still be arranged to avoid destructive overlap.

References: [2]

Ideal reconstruction assumes appropriate band limitation, exact samples and the complete sample sequence. Actual recordings cover finite intervals and use approximate reconstruction filters. A smooth curve drawn through a finite file's samples is therefore not automatically an exact recovery of the original signal. Sample density alone does not specify endpoint handling or reconstruction accuracy.

Sampling and quantization are separate operations. Sampling discretizes the positions in time, while quantization maps amplitudes into a finite set of levels. Increasing sample rate does not automatically remove quantization error, and increasing amplitude resolution cannot undo aliasing. These distinctions matter when applying the theorem to digital audio, images and measurements rather than treating it as an unconditional guarantee about hardware.

References: [1] [3]