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In digital signal processing, the reference value is usually the sampling frequency, denoted ${\displaystyle f_{s},\,}$  in samples per second, because the frequency content of a sampled signal is completely defined by the content within a span of ${\displaystyle f_{s}\,}$ hertz, at most. In other words, the frequency distribution is periodic with period ${\displaystyle f_{s}.\,}$  When the actual frequency${\displaystyle ,f,\,}$ has units of hertz (SI units), the normalized frequencies, also denoted by ${\displaystyle f,\,}$  have units of cycles per sample, and the periodicity of the normalized distribution is 1.  And when the actual frequency${\displaystyle ,\omega ,\,}$ 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 ${\displaystyle f_{s}.\,}$  But due to symmetry, it is completely defined by the content within a span of just ${\displaystyle f_{s}/2.\,}$  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 ${\displaystyle f_{s}}$ = 44.1 kHz.

• Digital signal processing