# Normalized frequency (digital signal processing)

﻿
Normalized frequency (digital signal processing)

In digital signal processing, the normalized frequency of a periodic signal is its frequency expressed in units of cycles (or radians) per sample, rather than in the usual SI units of hertz (cycles per second). The cycles-per-sample frequency is computed by dividing the cycles-per-second frequency by the sampling rate (samples per second); symbolically, the "/second" (per second) units cancel:

(cycles/second) / (samples/second) = cycles/sample.

## Applications

The abstract reason for using normalized frequency is that, from the point of view of signal processing, a second is an arbitrary unit of time, while the sampling interval is a meaningful quantity (formally, a characteristic unit for the system): the frequency of a signal with respect to 1 second does not tell you about the behavior of the signal, but the frequency of a signal with respect to the sampling interval tells you the effect of sampling on the signal, via the sampling theorem. Stated alternatively, this process is called "normalization", and the sampling frequency is a normalizing constant.

In filter design, a given design can be used at different sample-rates, resulting in different frequency responses. Normalization produces a distribution that is independent of the sample rate, and thus one plot is sufficient for all possible sample rates.

## Alternative normalizations

The reference value is usually the sampling frequency, denoted $f_s,\,$  in samples per second, because the frequency spectrum of a sampled signal (with real or complex values) is periodic with period $f_s.\,$  When the actual frequency$, f,\,$ has units of hertz (SI units), the normalized frequencies, also denoted by $f,\,$  have units of cycles per sample, and the periodicity of the normalized spectrum is 1.

Alternatively, if the actual frequency$, \omega,\,$ is written with units of radians per second (angular frequency), the normalized frequencies have units of radians per sample, and the periodicity of the distribution is 2π.

If a sampled waveform is real-valued, such as a typical filter impulse response, the periodicity of the frequency distribution is still $f_s.\,$  But due to symmetry, it is completely defined by the content within a span of just $f_s/2,\,$ half the sampling frequency – the Nyquist frequency. Accordingly, some filter design procedures/applications use that as the normalization reference[citation needed] (and the resulting units are half-cycles per sample).

### Example

The following table shows examples of normalized frequencies for a 1 kHz signal, a sample rate fs = 44.1 kHz, and these 3 different choices of normalization constant

 Type Computation Value Radians/sample 2 π 1000 / 44100 0.1425 cycles/sample (w.r.t. fs, sampling frequency) 1000 / 44100 0.02268 half-cycles/sample (w.r.t. fs/2, Nyquist frequency) 1000 / 22050 0.04535

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Normalized frequency — can refer to: Normalized frequency (digital signal processing) Normalized frequency (fiber optics), also known as V number This disambiguation page lists articles associated with the same title. If an …   Wikipedia

• Digital frequency — is the analogue for discrete signals as frequency is to continuous signals. Since a discrete signal is a sequence (merely a series of symbols; typically, numbers) it contains no direct information as to determine the frequency of the… …   Wikipedia

• Frequency — For other uses, see Frequency (disambiguation). Three cyclically flashing lights, from lowest frequency (top) to highest frequency (bottom). f is the frequency in hertz (Hz), meaning the number of cycles per second. T is the period in seconds (s) …   Wikipedia

• Signal conditioning — In electronics, signal conditioning means manipulating an analogue signal in such a way that it meets the requirements of the next stage for further processing. For example, the output of an electronic temperature sensor, which is probably in the …   Wikipedia

• Angular frequency — Not to be confused with angular velocity. Angular frequency ω (in radians per second), is larger than frequency ν (in cycles per second, also called Hz), by a factor of 2π …   Wikipedia

• Normalization — may refer to: Contents 1 Mathematics and statistics 2 Science 3 Technology …   Wikipedia

• Discrete-time Fourier transform — In mathematics, the discrete time Fourier transform (DTFT) is one of the specific forms of Fourier analysis. As such, it transforms one function into another, which is called the frequency domain representation, or simply the DTFT , of the… …   Wikipedia

• Discrete Fourier transform — Fourier transforms Continuous Fourier transform Fourier series Discrete Fourier transform Discrete time Fourier transform Related transforms In mathematics, the discrete Fourier transform (DFT) is a specific kind of discrete transform, used in… …   Wikipedia

• Nyquist–Shannon sampling theorem — Fig.1: Hypothetical spectrum of a bandlimited signal as a function of frequency The Nyquist–Shannon sampling theorem, after Harry Nyquist and Claude Shannon, is a fundamental result in the field of information theory, in particular… …   Wikipedia

• Window function — For the term used in SQL statements, see Window function (SQL) In signal processing, a window function (also known as an apodization function or tapering function[1]) is a mathematical function that is zero valued outside of some chosen interval …   Wikipedia