Normalized frequency

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Normalized frequency is the ratio of an actual frequency and a reference value.

[edit] Digital signal processing

In digital signal processing, the reference value is usually the sampling frequency, denoted f_s,\,  in samples per second, because the frequency content of a sampled signal is completely defined by the content within a span of f_s\, hertz, at most. In other words, the frequency distribution 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 distribution is 1.  And when the actual frequency, \omega,\, 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 f_s.\,  But due to symmetry, it is completely defined by the content within a span of just f_s/2.\,  Accordingly, some 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.

[edit] Fiber optics

In an optical fiber, the normalized frequency, V (also called the V number), is given by

V = {2 \pi a \over \lambda} \sqrt{{n_1}^2 - {n_2}^2}\quad = {2 \pi a \over \lambda} \mathrm{NA},

where a is the core radius, λ is the wavelength in vacuum, n1 is the maximum refractive index of the core, n2 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

{V^2 \over 2} \left( {g \over g + 2} \right)\quad,

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 V2/2. For single-mode operation is required that V < 2.405, which is the first root of the Bessel function J0.


[edit] References