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, $f'={\frac {f}{f_{s}}}$ , 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.^[1]

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.

A common practice is to sample the frequency spectrum of the sampled data at frequency intervals of ${\frac {f_{s}}{N}}$ , for some arbitrary integer $N$ (see § Sampling the DTFT). The samples (sometimes called frequency bins) are numbered consecutively, corresponding to a frequency normalization by ${\frac {f_{s}}{N}}$ .^[2]^{: p.56 eq.(16)} The normalized Nyquist frequency is ${\frac {N}{2}}$ 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 $\omega '={\frac {\omega }{f_{s}}}$ , the normalized Nyquist angular frequency is π radians/sample.

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


Quantity	Numeric range	Calculation	Reverse
$f'={\frac {f}{f_{s}}}$	$[$ 0, ⁠1/2⁠ $]$ cycle/sample	1000 / 44100 = 0.02268	$f=f'\cdot f_{s}$
$f'={\frac {f}{f_{s}/2}}$	[0, 1] half-cycle/sample	1000 / 22050 = 0.04535	$f=f'\cdot {\frac {f_{s}}{2}}$
$f'={\frac {f}{f_{s}/N}}$	$[$ 0, ⁠N/2⁠ $]$ bins	1000 × $N$ / 44100 = 0.02268 $N$	$f=f'\cdot {\frac {f_{s}}{N}}$
$\omega '={\frac {\omega }{f_{s}}}$	[0, π] radians/sample	1000 × 2π / 44100 = 0.14250	$\omega =\omega '\cdot f_{s}$