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# 10.3: Fourier Transforms for Time-Domain Functions


So far, we have been dealing with functions of a spatial coordinate $$x$$. Of course, mathematical relations don’t care about what kind of physical variable we are dealing with, so the same equations could be applied to functions of time $$t$$. However, there is a important difference in convention. When dealing with functions of the time coordinate $$t$$, it is customary to use a different sign convention in the Fourier relations!

The Fourier relations for a function of time, $$f(t)$$, are:

Definition: Fourier relations

\begin{align}\left\{\;\,\begin{aligned}F(\omega) &= \;\int_{-\infty}^\infty dt\; e^{i\omega t}\, f(t) \\ f(t) &= \int_{-\infty}^\infty \frac{d\omega}{2\pi}\; e^{-i\omega t}\, F(\omega).\end{aligned}\;\,\right.\end{align}

Compared to the Fourier relations previously given in Eq. (10.2.9), the signs of the $$\pm i \omega t$$ exponents are flipped.

There’s a good reason for this difference in sign convention: it arises from the need to describe propagating waves, which vary with both space and time. As discussed in Chapter 5, a propagating plane wave can be described by a wavefunction of the form $f(x,t) = A e^{i(kx - \omega t)},$ where $$k$$ is the wave-number and $$\omega$$ is the angular frequency. We write the plane wave function this way so that positive $$k$$ indicates forward propagation in space (i.e., in the $$+x$$ direction), and positive $$\omega$$ indicates forward propagation in time (i.e., in the $$+t$$ direction). This requires the $$kx$$ and $$\omega t$$ terms in the exponent to have opposite signs. Thus, when $$t$$ increases by some amount, a corresponding increase in $$x$$ leaves the exponent unchanged.

As we have seen, the inverse Fourier transform relation describes how a wave-form is broken up into a superposition of elementary waves. For a wavefunction $$f(x,t)$$, the superposition is given in terms of plane waves: $f(x,t) = \int_{-\infty}^\infty \frac{dk}{2\pi} \int_{-\infty}^\infty \frac{d\omega}{2\pi}\;\; e^{i(kx-\omega t)}\, F(k,\omega).$ To be consistent with this, we need to treat space and time variables with oppositely-signed exponents: \begin{align} f(x) &= \int_{-\infty}^\infty \frac{dk}{2\pi}\; e^{ikx}\, F(k) \\ f(t) &= \int_{-\infty}^\infty \frac{d\omega}{2\pi}\; e^{-i\omega t}\, F(\omega).\end{align} The other equations follow similarly.

This page titled 10.3: Fourier Transforms for Time-Domain Functions is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Y. D. Chong via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.