Small Group Activity: Fourier Transform of a Shifted Function

Periodic Systems 2022

Suppose you have a definite function \(f(x)\) in mind and you already know its Fourier transform, i.e. you know how to do the integral \begin{equation} \tilde{f}(k)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty}e^{-ikx}\, f(x)\, dx \end{equation} Find the Fourier transform of the shifted function \(f(x-x_0)\).

Instructor's Guide


Students will need a short lecture giving the definition of the Fourier Transform \begin{equation} {\cal{F}}(f) =\tilde{f} (k)= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-ikx}\, f(x)\, dx \end{equation}

Student Conversations

This example will feel very abstract to some students. It may be difficult for them to understand that the conditions of the problem state that the know both \(f(x)\) and \(\tilde{f}(k)\). This problem is about changing \(f\) slightly (by shifting its argument by \(x_0\)) and then asking how \(\tilde{f}\) changes, in response.


The result from this calculation underlies why it is possible to factor out the time dependence in the Fourier transform of a plane wave, Fourier Transform of a Plane Wave. Even though the problem is somewhat abstract, it is super important in applications for this reason.

Learning Outcomes