# Normalized frequency (signal processing)

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Normalized_frequency_(signal_processing)
> Markdown URL: https://mediated.wiki/source/Normalized_frequency_(signal_processing).md
> Source: https://en.wikipedia.org/wiki/Normalized_frequency_(signal_processing)
> Source revision: 1214351093
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

Frequency divided by a characteristic frequency

In [digital signal processing](/source/Digital_signal_processing) (DSP), a **normalized frequency** is a ratio of a variable [frequency](/source/Frequency) ( f {\displaystyle f} ) and a constant frequency associated with a system (such as a *[sampling rate](/source/Sampling_rate)*, f s {\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](/source/Sampling_rate)* ( f s {\displaystyle f_{s}} ) that is used to create the digital signal from a continuous one. The normalized quantity, f ′ = f f s , {\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 f {\displaystyle f} is expressed in [Hz](/source/Hertz) (*cycles per second*), f s {\displaystyle f_{s}} is expressed in *samples per second*.[1]

Some programs (such as [MATLAB](/source/MATLAB) toolboxes) that design filters with real-valued coefficients prefer the [Nyquist frequency](/source/Nyquist_frequency) ( f s / 2 ) {\displaystyle (f_{s}/2)} as the frequency reference, which changes the numeric range that represents frequencies of interest from [ 0 , 1 2 ] {\displaystyle \left[0,{\tfrac {1}{2}}\right]} *cycle/sample* to [ 0 , 1 ] {\displaystyle [0,1]} *half-cycle/sample*. Therefore, the normalized frequency unit is important when converting normalized results into physical units.

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 f s N , {\displaystyle {\tfrac {f_{s}}{N}},} for some arbitrary integer N {\displaystyle N} (see [§ Sampling the DTFT](/source/Discrete-time_Fourier_transform#Sampling_the_DTFT)). The samples (sometimes called frequency *bins*) are numbered consecutively, corresponding to a frequency normalization by f s N . {\displaystyle {\tfrac {f_{s}}{N}}.} [2]: p.56 eq.(16)[3] The normalized Nyquist frequency is N 2 {\displaystyle {\tfrac {N}{2}}} with the unit ⁠1/N⁠th *cycle/sample*.

[Angular frequency](/source/Angular_frequency), denoted by ω {\displaystyle \omega } and with the unit *[radians per second](/source/Radian_per_second)*, can be similarly normalized. When ω {\displaystyle \omega } is normalized with reference to the sampling rate as ω ′ = ω f s , {\displaystyle \omega '={\tfrac {\omega }{f_{s}}},} the normalized Nyquist angular frequency is *π radians/sample*.

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

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

## See also

- [Prototype filter](/source/Prototype_filter)

## References

1. **[^](#cite_ref-Carlson_1-0)** Carlson, Gordon E. (1992). *Signal and Linear System Analysis*. Boston, MA: ©Houghton Mifflin Co. pp. 469, 490. [ISBN](/source/ISBN_(identifier)) [8170232384](https://en.wikipedia.org/wiki/Special:BookSources/8170232384).

1. **[^](#cite_ref-Harris_2-0)** Harris, Fredric J. (Jan 1978). ["On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform"](http://web.mit.edu/xiphmont/Public/windows.pdf) (PDF). *Proceedings of the IEEE*. **66** (1): 51–83. [Bibcode](/source/Bibcode_(identifier)):[1978IEEEP..66...51H](https://ui.adsabs.harvard.edu/abs/1978IEEEP..66...51H). [CiteSeerX](/source/CiteSeerX_(identifier)) [10.1.1.649.9880](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.649.9880). [doi](/source/Doi_(identifier)):[10.1109/PROC.1978.10837](https://doi.org/10.1109%2FPROC.1978.10837). [S2CID](/source/S2CID_(identifier)) [426548](https://api.semanticscholar.org/CorpusID:426548).

1. **[^](#cite_ref-Taboga_3-0)** Taboga, Marco (2021). "Discrete Fourier Transform - Frequencies", Lectures on matrix algebra. [https://www.statlect.com/matrix-algebra/discrete-Fourier-transform-frequencies](https://www.statlect.com/matrix-algebra/discrete-Fourier-transform-frequencies).

---
Adapted from the Wikipedia article [Normalized frequency (signal processing)](https://en.wikipedia.org/wiki/Normalized_frequency_(signal_processing)) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Normalized_frequency_(signal_processing)?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.