# 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?

• 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
• In many cases this is redundant as you perceived. However, in cases where data is sampled the phrase "Fourier frequency" has an important meaning. Please refer to Oleg's answer. – DanielSank Aug 19 '15 at 6:18

## 4 Answers

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.

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 $f(x) = \cos x$ We already know lots about $\cos x$ so it is a nice example, since we know it has a fundamental period of $2 \pi$ and it is even.

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

We no longer see a function that we recognize as $\cos x$ because in Fourier analysis we only care about what is in the range.

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).

• This is the only correct answer. Upvoted. – DanielSank Aug 19 '15 at 6:17
• Ok now it's one of two correct answers :) – DanielSank Aug 19 '15 at 6:50

In many cases people do seem to say "Fourier frequency" when they mean "frequency". However, when dealing with data defined only on discrete time points the phrase "Fourier frequency" has important meaning.

Consider a sequence of $N$ values $\{ x_n \}$ where $n \in \{1, 2, \ldots N \}$. This situation comes up all the time if we have a physical signal $x(t)$ which we sample at times $t_n = n \Delta t$, in which case of course $x_n = x(t=n\Delta t)$. We can analyze the frequency content of this signal by computing its discrete Fourier transform (DFT)$^{a}$

$$X_k \equiv \sum_{n=1}^N x_n \exp \left( -i 2 \pi n k / N \right) \, .$$

This sum expresses the sequence $\{ x_n \}$ as a sum of sinusoidal signals $\exp(i 2 \pi n k / N)$. The frequencies of these sinusoids (i.e. the number of radians of oscillation per time step) are

$$\omega_k \equiv 2 \pi k / N \, .$$

These frequencies are some times called "Fourier frequencies".

As noted in Emilio's and Magpie's answers, it can be important to distinguish the notion of a frequency component of a signal from that signal's true underlying frequency. However, in my experience as an experimental physicist who does a fair amount of signal processing, the phrase "Fourier frequency" is not used in that way. In fact the usage quoted in the original question is pretty foreign to my ears/eyes, and it seems awkward since the word "frequency" would serve just fine. I have only seen and used "Fourier frequency" in the way described here and in Oleg's answer.

$[a]$: Many people refer to the discrete Fourier transform as the "FFT". In fact, FFT stands for "fast Fourier transform", which is a particular algorithm for computing the discrete Fourier transform.

## protected by Qmechanic♦Aug 19 '15 at 7:30

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