# Table of Fourier Transforms



## Definition of Fourier Transforms

If $$f(t)$$ is a function of the real variable $$t$$, then the Fourier transform $$F(\omega)$$ of $$f$$ is given by the integral $F(\omega) = \int_{-\infty}^{+\infty} e^{-j \omega t} f(t) \, dt$ where $$j = \sqrt{-1}$$, the imaginary unit.
In what follows, $$u(t)$$ is the unit step function defined by
$$u(t) = 1$$ for $$t \geq 0$$ and $$u(t) = 0$$ for $$t < 0$$. (see figure below).

## Table of Fourier Transforms

 $$f(t)$$ $$F(\omega)$$ $$u(t) e^{-a t}$$, $$a > 0$$ $$\dfrac{1}{a + j \omega}$$ $$f(t) = 1$$ for $$-a \leq t \leq a$$ and $$0$$ otherwise $$\dfrac{2 \sin (\omega a)}{\omega}$$ $$f(t) = A$$ (constant) $$2 \pi A \delta (\omega)$$ $$\delta (t)$$ 1 $$\delta (t - a)$$ $$e^{-j \omega a}$$ $$\cos (a t)$$ $$\pi [\delta (\omega + a) + \delta (\omega - a)]$$ $$\sin (a t)$$ $$-j \pi [\delta (\omega - a) - \delta (\omega + a)]$$ $$e^{j a t}$$ $$2 \pi [\delta (\omega - a)]$$ $$f'(t)$$ $$j \omega F(\omega)$$ $$f''(t)$$ $$(j \omega)^2 F(\omega)$$ $$t f(t)$$ $$j \dfrac{d F(\omega)}{d \omega}$$ $$t^2 f(t)$$ $$j^2 \dfrac{d^2 F(\omega)}{d \omega^2}$$