Table of Fourier Transforms
Definition of Fourier Transforms
If f(t) is a function of the real variable t, then the Fourier transform F(ω) of f is given by the integral
where j = √(-1), the imaginary unit.
In what follows, u(t) is the unit step function defined by
u(t) = 1 for t ≥ 0 and u(t) = 0 for t < 0. (see figure below).
Table of Fourier Transforms
| f(t) | F(ω) |
| u(t) e -a t , a > 0 | 1 / (a + j ω) |
|
1 for - a ≤ t ≤ a
and 0 otherwise |
2 sin (ω a) / ω |
| A (constant) | 2 π A δ (ω) |
| δ (t) | 1 |
| δ (t - a) | e - j ω a |
| cos (a t) | π [ δ(ω + a) + δ(ω - a) ] |
| sin (a t) | - j π [ δ (ω - a) - δ (ω + a) ] |
| e j a t | 2 π [δ(ω - a) ] |
| f '(t) | j ω F (ω) |
| f ''(t) | (j ω) 2 F(ω) |
| t f(t) | j d [F(ω)] / dω |
| t 2 f(t) | j 2 d 2[F(ω)] / dω 2 |
