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:

• Prototype filter