# Fourier transform pairs

x(t)$\Delta t=1/\Delta f\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\left(t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\circ -•\text{\hspace{0.17em}}\text{\hspace{0.17em}}X\left(f\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\Delta f=1/\Delta t$ X(f)
$1$  $\delta \left(f\right)$
$\delta \left(t\right)$  $1$
$\mathrm{cos}\left(2\pi \text{\hspace{0.17em}}{f}_{0}t\right)$  $\frac{1}{2}\left[\delta \left(f-{f}_{0}\right)+\delta \left(f+{f}_{0}\right)\right]$
$\frac{1}{2}\left[\delta \left(t-{t}_{0}\right)+\delta \left(t+{t}_{0}\right)\right]$  $\mathrm{cos}\left(2\pi \text{\hspace{0.17em}}{t}_{0}f\right)$
$\begin{array}{c}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}für\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{\Delta t}{2}  $\Delta t\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\frac{\mathrm{sin}\left(\pi \text{\hspace{0.17em}}\Delta t\text{\hspace{0.17em}}f\right)}{\pi \text{\hspace{0.17em}}\Delta t\text{\hspace{0.17em}}f}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\Delta t\text{\hspace{0.17em}}\text{\hspace{0.17em}}si\left(\pi \text{\hspace{0.17em}}\Delta t\text{\hspace{0.17em}}f\right)$
$\Delta f\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\frac{\mathrm{sin}\left(\pi \text{\hspace{0.17em}}\Delta f\text{\hspace{0.17em}}t\right)}{\pi \text{\hspace{0.17em}}\Delta f\text{\hspace{0.17em}}t}\text{\hspace{0.17em}}\text{\hspace{0.17em}}$  $\begin{array}{c}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}für\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{\Delta f}{2}
$\begin{array}{c}\frac{t}{\Delta t}+1\text{\hspace{0.17em}}\text{\hspace{0.17em}}für\text{\hspace{0.17em}}-\Delta t\le t\le 0\\ \frac{-t}{\Delta t}+1\text{\hspace{0.17em}}\text{\hspace{0.17em}}für\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le t\le \Delta t\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}sonst\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}\right\}=\Lambda \left(\frac{t}{\Delta t}\right)$  $\Delta t{\left[\frac{\mathrm{sin}\left(\pi \text{\hspace{0.17em}}\Delta t\text{\hspace{0.17em}}f\right)}{\pi \text{\hspace{0.17em}}\Delta t\text{\hspace{0.17em}}f}\right]}^{2}$
$\Delta f{\left[\frac{\mathrm{sin}\left(\pi \text{\hspace{0.17em}}\Delta f\text{\hspace{0.17em}}t\right)}{\pi \text{\hspace{0.17em}}\Delta f\text{\hspace{0.17em}}t}\right]}^{2}$  $\begin{array}{c}\frac{f}{\Delta f}+1\text{\hspace{0.17em}}\text{\hspace{0.17em}}für\text{\hspace{0.17em}}-\Delta f\le f\le 0\\ \frac{-f}{\Delta f}+1\text{\hspace{0.17em}}\text{\hspace{0.17em}}für\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le f\le \Delta f\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}sonst\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}\right\}=\Lambda \left(\frac{f}{\Delta f}\right)$
${e}^{-|\frac{2t}{\Delta t}|}$  $\Delta t\cdot \frac{1}{1+{\left(\pi \text{\hspace{0.17em}}\Delta t\text{\hspace{0.17em}}f\right)}^{2}}$
$\Delta f\cdot \frac{1}{1+{\left(\pi \text{\hspace{0.17em}}\Delta f\text{\hspace{0.17em}}t\right)}^{2}}$  $\text{\hspace{0.17em}}{e}^{-|\frac{2f}{\Delta f}|}$
${e}^{-\pi {\left(\frac{t}{\Delta t}\right)}^{2}}$  $\Delta t\cdot {e}^{-\pi {\left(\frac{f}{\Delta f}\right)}^{2}}\text{\hspace{0.17em}};\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\Delta t=\frac{1}{\Delta f}$
$\sum _{n=-\infty }^{\infty }\delta \left(t-n{T}_{0}\right)$  ${f}_{0}\cdot \sum _{n=-\infty }^{\infty }\delta \left(f-n{f}_{0}\right)$
Table of Fourier transform pairs

This experiment illustrates the Fourier transform. Demo examples of time signals and corresponding spectra are simulated. Click the play button to open an oscilloscope and spectrum analyzer. Vary the time signal (amplitude, frequency or pulse witdth) and watch the impact on the spectrum!

Time domainFrequency domain Start DC component Spectral line at frequency f=0  Dirac delta impulse Constant spectrum  Cosine Spectral lines  Rectangular pulse Sinc Function  Sinc pulse Rectangular spectrum (approximation)  Sinc^2 pulse Triangle spectrum (approximation)  Exponential pulse 1 / (1+f) spectrum  Gaussian pulse Gaussian spectrum Example demonstrations of time signals and corresponding frequency signals.

1. Fourier transforms. Time signals are shown on the left. Assign the corresponding spectra on the right per drag & drop!

Time domainFequency domain • • • •    