# What is the specific meaning of “Fourier frequency” (as opposed to simply “frequency”)?

I've noticed that many journal articles (in optics) use the phrase "Fourier frequency" to describe, well, the frequency of something.

To compare with previous sensitivity curves of a single LISA Michelson interferometer, we construct the SNRs as a function of Fourier frequency for sinusoidal waves from sources uniformly distributed on the celestial sphere.

This phrase seems redundant to me. Does the adjective "Fourier" add anything? Is it trying to make a distinction of audio frequencies from optical frequencies? Or temporal frequencies from spatial frequencies?

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In my opinion is completely redundant. It can be added to strengthen the importance of the Fourier transform but that's all there is to it. –  Marek Jul 29 '11 at 21:26
This article makes a distinction between "Fourier frequency" and "instantaneous frequency": rp-photonics.com/instantaneous_frequency.html –  nibot Jul 29 '11 at 21:46
I suppose that "frequency" is also used to describe the relative frequency of events (without implying periodicity), for instance in histograms. "Outcome B occurs with frequency 0.5". But one would never confuse this with [Fourier] frequency. –  nibot Jul 29 '11 at 21:52

A fourier series of a function can look quite different to the function itself. The Fourier series is a periodic function over a finite integral and a Fourier transform is a periodic function over an infinite integral.

Lets look at a periodic function over the finite integral for a bit (i.e. a fourier series example):

Let $f(x)=cosx$

We already know lots about $cosx$ so it is a nice example, since we know it has a fundamental period of $2\pi$ and it is even.

Now, what will this function when we are expanding it over the finite range $-\pi\le x \le\pi$. What will $cosx$ look like then? What will it's fundamental period be?:

We no longer see a function that we recognise as $cosx$ because in Fourier analysis we only care about what is in the range.

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The term Fourier frequency most usually denotes the frequency of one of several components of a function which may or may not be periodic. Take, for example, a gaussian pulse, which you can decompose as a "sum" (integral) of different periodic waves, $\cos(\omega t)$, with weights that vary with $\omega$: $$e^{-{t^2}/{2T^2}}=\int_{-\infty}^\infty \frac{e^{-T^2\omega^2/2}}{\sqrt{2\pi}}\cos(\omega t)d\omega.$$ In this context it's quite appropriate to call $\omega$ a Fourier frequency since it's not the frequency of the function under consideration, but only of part of it in a certain decomposition. The general techniques for doing this are Fourier transforms and series, depending on what type of function you are dealing with.

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See the standard text by Peter J. Brockwell & Richard A. Davis "Time Series:Theory and Methods" 2 ed, p.331, where authors define $\omega_j=2\pi j/n\in(-\pi,\pi]$ (integer multiples of the fundamental frequency $2\pi /n$) as Fourier frequencies. Later on p.335, they say "if $\omega$ is not a Fourier frequency, the analysis is a little more complicated..." This is still in reference to frequencies within interval $(-\pi,\pi]$, but those that are not integer multiples, for example "$\omega=\pi$" (also p.335).

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