Normalized frequency (signal processing)
Frequency divided by a characteristic frequency
title: "Normalized frequency (signal processing)" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["digital-signal-processing", "frequency"] description: "Frequency divided by a characteristic frequency" topic_path: "engineering" source: "https://en.wikipedia.org/wiki/Normalized_frequency_(signal_processing)" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Frequency divided by a characteristic frequency ::
--
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' = \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 is expressed in Hz (cycles per second), f_s is expressed in samples per second.
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 \left[0, \tfrac{1}{2}\right] cycle/sample to [0, 1] half-cycle/sample. Therefore, the normalized frequency unit is important when converting normalized results into physical units.
::figure[src="https://upload.wikimedia.org/wikipedia/commons/b/b2/Normalized_frequency_example.svg" caption="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 \tfrac{f_s}{N}, for some arbitrary integer N (see ). The samples (sometimes called frequency bins) are numbered consecutively, corresponding to a frequency normalization by \tfrac{f_s}{N}. The normalized Nyquist frequency is \tfrac{N}{2} with the unit th cycle/sample.
Angular frequency, denoted by \omega and with the unit radians per second, can be similarly normalized. When \omega is normalized with reference to the sampling rate as \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 kHz, f_s = 44100 samples/second (often denoted by 44.1 kHz), and 4 normalization conventions:
::data[format=table title=""]
| Quantity | Numeric range | Calculation | Reverse |
|---|---|---|---|
| f' = \tfrac{f}{f_s} | [0, ] cycle/sample | 1000 / 44100 = 0.02268 | f = f' \cdot f_s |
| f' = \tfrac{f}{f_s / 2} | [0, 1] half-cycle/sample | 1000 / 22050 = 0.04535 | f = f' \cdot \tfrac{f_s}{2} |
| f' = \tfrac{f}{f_s / N} | [0, ] bins | 1000 × N / 44100 = 0.02268 N | f = f ' \cdot \tfrac{f_s}{N} |
| \omega' = \tfrac{\omega}{f_s} | [0, π] radians/sample | 1000 × 2π / 44100 = 0.14250 | \omega = \omega' \cdot f_s |
| :: |
References
::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::