Normalized frequency (signal processing)

In digital signal processing, the reference value is usually the sampling frequency, denoted $f_{s},\,$  in samples per second, because the frequency content of a sampled signal is completely defined by the content within a span of $f_{s}\,$ hertz, at most. In other words, the frequency distribution is periodic with period $f_{s}.\,$  When the actual frequency$,f,\,$ has units of hertz (SI units), the normalized frequencies, also denoted by $f,\,$  have units of cycles per sample, and the periodicity of the normalized distribution is 1.  And when the actual frequency$,\omega ,\,$ has units of radians per second (angular frequency), the normalized frequencies have units of radians per sample, and the periodicity of the distribution is 2п.

If a sampled waveform is real-valued, such as a typical filter impulse response, the periodicity of the frequency distribution is still $f_{s}.\,$  But due to symmetry, it is completely defined by the content within a span of just $f_{s}/2.\,$  Accordingly, some filter design procedures/applications use that as the normalization reference (and the resulting units are half-cycles per sample). A filter design can be used at different sample-rates, resulting in different frequency responses. Normalization produces a distribution that is independent of the sample-rate. Thus one plot is sufficient for all possible sample-rates.

The following table shows examples of normalized frequencies for a 1 kHz signal, and a sample rate $f_{s}$ = 44.1 kHz.

Type	Computation	Value
Radians/sample	2 pi 1000 / 44100	0.1425
w.r.t. fs	1000 / 44100	0.02676
w.r.t. Nyquist	1000 / 22050	0.04535