Normalized frequency

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

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.

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 cycles per half-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.

References