Normalized frequency
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213016705
2008-05-17T10:49:23Z
219.98.34.1
'''Normalized frequency''' is the [[ratio]] of an actual frequency and a reference value.
==Digital signal processing==
In [[digital signal processing]], the reference value is usually the sampling frequency, denoted <math>f_s,\,</math> in '''samples per second''', because the frequency content of a sampled signal is completely defined by the content within a span of <math>f_s\,</math> [[hertz]], at most. In other words, the frequency distribution is periodic with period <math>f_s.\,</math> When the actual frequency<math>, f,\,</math> has units of [[hertz]] ([[SI]] units), the normalized frequencies, also denoted by <math>f,\,</math> have units of '''cycles per sample''', and the periodicity of the normalized distribution is 1. And when the actual frequency<math>, \omega,\,</math> has units of radians per second ([[angular frequency]]), the normalized frequencies have units of '''radians per sample''', and the periodicity of the distribution is 2п.
If a sampled waveform is real-valued, such as a typical filter impulse response, the periodicity of the frequency distribution is still <math>f_s.\,</math> But due to symmetry, it is completely defined by the content within a span of just <math>f_s/2.\,</math> Accordingly, <u>some</u> filter design procedures/applications use that as the normalization reference (and the resulting units are ''half-cycles per sample''). A filter design can be used at different sample-rates, resulting in different frequency responses. Normalization produces a distribution that is independent of the sample-rate. Thus one plot is sufficient for all possible sample-rates.
==Fiber optics==
In an [[optical fiber]], the normalized frequency, ''V'' (also called the '''V number'''), is given by
:<math>V = {2 \pi a \over \lambda} \sqrt{{n_1}^2 - {n_2}^2}\quad = {2 \pi a \over \lambda} \mathrm{NA},</math>
where ''a'' is the [[Fiber_optics#Principle_of_operation|core]] radius, λ is the [[wavelength]] in vacuum, ''n''<sub>1</sub> is the maximum [[refractive index]] of the core, ''n''<sub>2</sub> is the refractive index of the homogeneous cladding, and applying the usual definition of the [[numerical aperture]] ''NA''.
In multimode operation of an optical fiber having a [[power-law refractive index profile]], the approximate number of bound modes (the [[mode volume]]), is given by
:<math>{V^2 \over 2} \left( {g \over g + 2} \right)\quad,</math>
where ''g'' is the profile parameter, and ''V'' is the normalized frequency, which must be greater than 5 for the approximation to be valid.
For a [[step index fiber]], the mode volume is given by ''V''<sup>2</sup>/2. For single-mode operation is required that ''V'' < 2.405, which is the first root of the [[Bessel function]] ''J''<sub>0</sub>.
==References==
* [[Federal Standard 1037C]]
* [[MIL-STD-188]]
[[Category:Fiber optics]]
[[Category:Digital signal processing]]