Fourier transform pairs

x(t)	$Δ t = 1 / Δ f x (t) \circ - • X (f) Δ f = 1 / Δ t$		X(f)
$1$			$δ (f)$
$δ (t)$			$1$
$\cos (2 π f_{0} t)$			$\frac{1}{2} [δ (f - f_{0}) + δ (f + f_{0})]$
$\frac{1}{2} [δ (t - t_{0}) + δ (t + t_{0})]$			$\cos (2 π t_{0} f)$
$\begin{matrix} 1 f ü r - \frac{Δ t}{2} < t < \frac{Δ t}{2} \\ 0 s o n s t \end{matrix}} = \prod (\frac{t}{Δ t})$			$Δ t \cdot \frac{\sin (π Δ t f)}{π Δ t f} = Δ t s i (π Δ t f)$
$Δ f \cdot \frac{\sin (π Δ f t)}{π Δ f t}$			$\begin{matrix} 1 f ü r - \frac{Δ f}{2} < f < \frac{Δ f}{2} \\ 0 s o n s t \end{matrix}} = \prod (\frac{f}{Δ f})$
$\begin{matrix} \frac{t}{Δ t} + 1 f ü r - Δ t \leq t \leq 0 \\ \frac{- t}{Δ t} + 1 f ü r 0 \leq t \leq Δ t \\ 0 s o n s t \end{matrix}} = Λ (\frac{t}{Δ t})$			$Δ t {[\frac{\sin (π Δ t f)}{π Δ t f}]}^{2}$
$Δ f {[\frac{\sin (π Δ f t)}{π Δ f t}]}^{2}$			$\begin{matrix} \frac{f}{Δ f} + 1 f ü r - Δ f \leq f \leq 0 \\ \frac{- f}{Δ f} + 1 f ü r 0 \leq f \leq Δ f \\ 0 s o n s t \end{matrix}} = Λ (\frac{f}{Δ f})$
$e^{- \| \frac{2 t}{Δ t} \|}$			$Δ t \cdot \frac{1}{1 + {(π Δ t f)}^{2}}$
$Δ f \cdot \frac{1}{1 + {(π Δ f t)}^{2}}$			$e^{- \| \frac{2 f}{Δ f} \|}$
$e^{- π {(\frac{t}{Δ t})}^{2}}$			$Δ t \cdot e^{- π {(\frac{f}{Δ f})}^{2}}; Δ t = \frac{1}{Δ f}$
$\sum_{n = - \infty}^{\infty} δ (t - n T_{0})$			$f_{0} \cdot \sum_{n = - \infty}^{\infty} δ (f - n f_{0})$

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 domain	Frequency 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