Normalized frequency (signal processing)

In digital signal processing, a normalized frequency is a ratio of a variable frequency and a constant frequency associated with a system. Some software applications require normalized inputs and produce normalized outputs, which can be re-scaled to physical units when necessary. Mathematical derivations are usually done in normalized units, relevant to a wide range of applications.

Examples of normalization

A typical choice of characteristic frequency is the sampling rate that is used to create the digital signal from a continuous one. The normalized quantity, has the unit cycle per sample regardless of whether the original signal is a function of time or distance. For example, when is expressed in Hz, is expressed in samples per second.
Some programs that design filters with real-valued coefficients prefer the Nyquist frequency as the frequency reference, which changes the numeric range that represents frequencies of interest from cycle/sample to half-cycle/sample. Therefore, the normalized frequency unit is important when converting normalized results into physical units.
A common practice is to sample the frequency spectrum of the sampled data at frequency intervals of for some arbitrary integer . The samples are numbered consecutively, corresponding to a frequency normalization by The normalized Nyquist frequency is with the unit ^th cycle/sample.
Angular frequency, denoted by and with the unit radians per second, can be similarly normalized. When is normalized with reference to the sampling rate as the normalized Nyquist angular frequency is.
The following table shows examples of normalized frequency for kHz, samples/second, and 4 normalization conventions:

Quantity	Numeric range	Calculation	Reverse
	0, cycle/sample	1000 / 44100 = 0.02268
	half-cycle/sample	1000 / 22050 = 0.04535
	0, bins	1000 × / 44100 = 0.02268
	radians/sample	1000 × 2π / 44100 = 0.14250