Normalized frequency (signal processing)

In digital signal processing (DSP), a normalized frequency is a ratio of a variable frequency (f) and a constant frequency associated with a system (such as a sampling rate, f_s). 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 (f_s) that is used to create the digital signal from a continuous one. The normalized quantity, f′ = f / f_s, has the unit cycle per sample regardless of whether the original signal is a function of time or distance. For example, when f is expressed in Hz (cycles per second), f_s is expressed in samples per second.

Some programs (such as MATLAB toolboxes) that design filters with real-valued coefficients prefer the Nyquist frequency (f_s/2) as the frequency reference, which changes the numeric range that represents frequencies of interest from [0, 1/2] cycle/sample to [0, 1] half-cycle/sample. Therefore, the normalized frequency unit is obviously important when converting normalized results into physical units.

Example of plotting samples of a frequency distribution in the unit "bins", which are integer values. A scale factor of 0.7812 converts a bin number into the corresponding physical unit (hertz).

A common practice is to sample the frequency spectrum of the sampled data at frequency intervals of f_s/N, for some arbitrary integer N (see § Sampling the DTFT). The samples (sometimes called frequency bins) are numbered consecutively, corresponding to a frequency normalization by f_s/N. The normalized Nyquist frequency is N/2 with the unit ⁠1/N⁠^th cycle/sample.

Angular frequency, denoted by ω and with the unit radians per second, can be similarly normalized. When ω is normalized with reference to the sampling rate as ω′ = ω / f_s, the normalized Nyquist angular frequency is π radians/sample.

The following table shows examples of normalized frequency for f = 1 kHz,  f_s = 44100 samples/second (often denoted by 44.1 kHz), and 4 normalization conventions:

Quantity	Numeric range	Calculation	Reverse
f′ = f / fs	[0, ⁠1/2⁠] cycle/sample	1000 / 44100 = 0.02268	f = f′ × fs
f′ = f / (fs/2)	[0, 1] half-cycle/sample	1000 / 22050 = 0.04535	f = f′ × fs / 2
f′ = f / (fs/N)	[0, ⁠N/2⁠] bins	1000 × N / 44100 = 0.02268 N	f = f′ × fs / N
ω′ = ω / fs	[0, π] radians/sample	1000 × 2π / 44100 = 0.14250	ω = ω′ × fs

See also

• Prototype filter