Normalized frequency (signal processing)

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.

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.^[1]

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 ${\displaystyle \left[0,{\tfrac {1}{2}}\right]}$ cycle/sample to ${\displaystyle [0,1]}$ 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 ${\displaystyle {\tfrac {f_{s}}{N}}.}$^{[2]: p.56 eq.(16) [3]} The normalized Nyquist frequency is ${\displaystyle {\tfrac {N}{2}}}$ with the unit ⁠1/N⁠^th 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
${\displaystyle f'={\tfrac {f}{f_{s}}}}$	[0, ⁠1/2⁠] cycle/sample	1000 / 44100 = 0.02268	${\displaystyle f=f'\cdot f_{s}}$
${\displaystyle f'={\tfrac {f}{f_{s}/2}}}$	[0, 1] half-cycle/sample	1000 / 22050 = 0.04535	${\displaystyle f=f'\cdot {\tfrac {f_{s}}{2}}}$
${\displaystyle f'={\tfrac {f}{f_{s}/N}}}$	[0, ⁠N/2⁠] bins	1000 × N / 44100 = 0.02268 N	${\displaystyle f=f'\cdot {\tfrac {f_{s}}{N}}}$
${\displaystyle \omega '={\tfrac {\omega }{f_{s}}}}$	[0, π] radians/sample	1000 × 2π / 44100 = 0.14250	${\displaystyle \omega =\omega '\cdot f_{s}}$

• Prototype filter