Sampling theorem, Nyquist criterion and aliasing

Sampling is the process of taking values (samples) of a continuous-time signal at regular intervals Ts, to convert it into a discrete-time signal. The sampling frequency is fs = 1/Ts.

Sampling theorem

A band-limited continuous signal, with no frequency component above fm, can be completely recovered from its samples if it is sampled at a rate

fs ≥ 2 fm

The signal is then reconstructed by passing the samples through an ideal low-pass filter with cutoff fm.

Nyquist criterion

• The sampling frequency must be at least twice the highest frequency in the signal. The minimum rate fN = 2fm is called the Nyquist rate, and fs/2 is called the Nyquist frequency.
• Sampling at fs > 2fm: the copies of the spectrum, repeated around multiples of fs, do not overlap.
• Sampling at fs = 2fm: the limiting case, recoverable only in theory (with an ideal filter).
• In practice fs = 2.5 to 10 times fm is used, to allow for a practical (non-ideal) filter.

Example: a signal with a maximum frequency of 4 kHz needs fs ≥ 8 kHz (telephone speech is sampled at 8 kHz).

Aliasing

If fs < 2fm, the spectral copies at multiples of fs overlap. High-frequency components then appear as lower frequencies in the sampled signal and cannot be separated again. This distortion is called aliasing; a component at frequency f appears at |f − k·fs|.

 Spectrum, fs > 2fm        Spectrum, fs < 2fm (overlap)
 |\    /|\    /|\          |\  /|\  /|\
 | \  / | \  / | \         | \/ | \/ | \
 +--fm--fs-----> f         +--fm fs------> f

Example: a 7 kHz tone sampled at 10 kHz appears as 3 kHz.

Eliminating aliasing

1. Anti-aliasing filter: place a low-pass filter before the sampler, with its cutoff at (or below) fs/2, to remove all components above fs/2.
2. Increase the sampling rate so that fs ≥ 2 × the highest frequency component present (including noise).
3. Oversampling, followed by digital filtering and decimation.

Aliasing cannot be removed after sampling, so the filter must be placed before the sampler.