{{Short description|Frequency divided by a characteristic frequency}} <!-- Hide the merge tag to avoid a loop, because Digital frequency now redirects here. {{mergewith|Digital frequency|date=September 2012}} -->
In [[digital signal processing]] (DSP), a '''normalized frequency''' is a ratio of a variable [[frequency]] (<math>f</math>) and a constant frequency associated with a system (such as a ''[[sampling rate]]'', <math>f_s</math>). 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]]'' (<math>f_s</math>) that is used to create the digital signal from a continuous one. The normalized quantity, <math>f' = \tfrac{f}{f_s},</math> has the unit ''cycle per sample'' regardless of whether the original signal is a function of time or distance. For example, when <math>f</math> is expressed in [[hertz|Hz]] (''cycles per second''), <math>f_s</math> is expressed in ''samples per second''.<ref name=Carlson/>
Some programs (such as [[MATLAB]] toolboxes) that design filters with real-valued coefficients prefer the [[Nyquist frequency]] <math>(f_s/2)</math> as the frequency reference, which changes the numeric range that represents frequencies of interest from <math>\left[0, \tfrac{1}{2}\right]</math> ''cycle/sample'' to <math>[0, 1]</math> ''half-cycle/sample''. Therefore, the normalized frequency unit is important when converting normalized results into physical units.
[[File:Normalized_frequency_example.svg|thumb|350px|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 <math>\tfrac{f_s}{N},</math> for some arbitrary integer <math>N</math> (see {{slink|Discrete-time_Fourier_transform|Sampling_the_DTFT|nopage=y}}). The samples (sometimes called frequency ''bins'') are numbered consecutively, corresponding to a frequency normalization by <math>\tfrac{f_s}{N}.</math><ref name=Harris/>{{rp|p.56 eq.(16)}}<ref name=Taboga/> The normalized Nyquist frequency is <math>\tfrac{N}{2}</math> with the unit {{sfrac|1|N}}<sup>th</sup> ''cycle/sample''.
[[Angular frequency]], denoted by <math>\omega</math> and with the unit ''[[radian per second|radians per second]]'', can be similarly normalized. When <math>\omega</math> is normalized with reference to the sampling rate as <math>\omega' = \tfrac{\omega}{f_s},</math> the normalized Nyquist angular frequency is {{nowrap|''π radians/sample''}}.
The following table shows examples of normalized frequency for <math>f = 1</math> ''kHz'', <math>f_s = 44100</math> ''samples/second'' (often denoted by [[44.1 kHz]]), and 4 normalization conventions:
{| class="wikitable" |+ !'''Quantity''' !'''Numeric range''' !'''Calculation''' !'''Reverse''' |- |<math>f' = \tfrac{f}{f_s}</math> | {{math|[|size=150%}}0, {{sfrac|1|2}}{{math|]|size=150%}} ''cycle/sample'' |1000 / 44100 = 0.02268 |<math>f = f' \cdot f_s</math> |- |<math>f' = \tfrac{f}{f_s / 2}</math> | [0, 1] ''half-cycle/sample'' |1000 / 22050 = 0.04535 |<math>f = f' \cdot \tfrac{f_s}{2}</math> |- |<math>f' = \tfrac{f}{f_s / N}</math> | {{math|[|size=150%}}0, {{sfrac|''N''|2}}{{math|]|size=150%}} ''bins'' |1000 × {{mvar|N}} / 44100 = 0.02268 {{mvar|N}} |<math>f = f ' \cdot \tfrac{f_s}{N}</math> |- |<math>\omega' = \tfrac{\omega}{f_s}</math> | [0, ''π''] ''radians/sample'' |1000 × 2π / 44100 = 0.14250 |<math>\omega = \omega' \cdot f_s</math> |}
==See also== *[[Prototype filter]]
==References== {{reflist|1|refs= <ref name=Carlson> {{cite book |last=Carlson |first=Gordon E. |title=Signal and Linear System Analysis |publisher=©Houghton Mifflin Co |year=1992 |isbn=8170232384 |location=Boston, MA |pages=469, 490 }}</ref>
<ref name=Harris> {{cite journal |doi=10.1109/PROC.1978.10837 |last=Harris |first=Fredric J. |title=On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform |journal=Proceedings of the IEEE |volume=66 |issue=1 |pages=51–83 |date=Jan 1978 |url=http://web.mit.edu/xiphmont/Public/windows.pdf|citeseerx=10.1.1.649.9880 |bibcode=1978IEEEP..66...51H |s2cid=426548 }}</ref>
<ref name=Taboga> Taboga, Marco (2021). "Discrete Fourier Transform - Frequencies", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/discrete-Fourier-transform-frequencies. </ref> }} [[Category:Digital signal processing]] [[Category:Frequency]]