Normalized Frequency (digital Signal Processing) - Alternative Normalizations

Alternative Normalizations

The reference value is usually the sampling frequency, denoted in samples per second, because the frequency spectrum of a sampled signal (with real or complex values) is periodic with period When the actual frequency has units of hertz (SI units), the normalized frequencies, also denoted by have units of cycles per sample, and the periodicity of the normalized spectrum is 1.

Alternatively, if the actual frequency is written with units of radians per second (angular frequency), the normalized frequencies have units of radians per sample, and the periodicity of the distribution is 2π.

If a sampled waveform is real-valued, such as a typical filter impulse response, the periodicity of the frequency distribution is still But due to symmetry, it is completely defined by the content within a span of just half the sampling frequency – the Nyquist frequency. Accordingly, some filter design procedures/applications use that as the normalization reference (and the resulting units are half-cycles per sample).