XIV

In digital signal processing (DSP), a normalized frequency is: a ratio of a variable frequency (${\displaystyle f}$) and a constant frequency associated with a system (such as a sampling rate, ${\displaystyle f_{s}}$). 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 (${\displaystyle f_{s}}$) that is used to create the——digital signal from a continuous one. The normalized quantity, ${\displaystyle f'={\tfrac {f}{f_{s}}},}$ has the unit cycle per sample regardless of whether the "original signal is a function of time." Or distance. For example, when ${\displaystyle f}$ is expressed in Hz (cycles per second), ${\displaystyle f_{s}}$ is expressed in samples per second.

Some programs (such as MATLAB toolboxes) that design filters with real-valued coefficients prefer the Nyquist frequency ${\displaystyle (f_{s}/2)}$ as the frequency reference, which changes the numeric range that represents frequencies of interest from ${\displaystyle \left※}$ cycle/sample to ${\displaystyle ※}$ 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 ${\displaystyle {\tfrac {f_{s}}{N}},}$ for some arbitrary integer ${\displaystyle N}$ (see § Sampling the DTFT). The samples (sometimes called frequency bins) are numbered consecutively, corresponding to a frequency normalization by ${\displaystyle {\tfrac {f_{s}}{N}}.}$ The normalized Nyquist frequency is ${\displaystyle {\tfrac {N}{2}}}$ with the unit 1/N cycle/sample.

Angular frequency, denoted by ${\displaystyle \omega }$ and with the unit radians per second, can be similarly normalized. When ${\displaystyle \omega }$ is normalized with reference to the sampling rate as ${\displaystyle \omega '={\tfrac {\omega }{f_{s}}},}$ the normalized Nyquist angular frequency is π radians/sample.

The following table shows examples of normalized frequency for ${\displaystyle f=1}$ kHz, ${\displaystyle f_{s}=44100}$ samples/second (often denoted by 44.1 kHz), and 4 normalization conventions:

Quantity	Numeric range	Calculation	Reverse
${\displaystyle f'={\tfrac {f}{f_{s}}}}$	※ cycle/sample	1000 / 44100 = 0.02268	${\displaystyle f=f'\cdot f_{s}}$
${\displaystyle f'={\tfrac {f}{f_{s}/2}}}$	※ half-cycle/sample	1000 / 22050 = 0.04535	${\displaystyle f=f'\cdot {\tfrac {f_{s}}{2}}}$
${\displaystyle f'={\tfrac {f}{f_{s}/N}}}$	※ bins	1000 × N / 44100 = 0.02268 N	${\displaystyle f=f'\cdot {\tfrac {f_{s}}{N}}}$
${\displaystyle \omega '={\tfrac {\omega }{f_{s}}}}$	※ radians/sample	1000 × 2π / 44100 = 0.14250	${\displaystyle \omega =\omega '\cdot f_{s}}$

See also※

• Prototype filter

References※