Table of Fourier Transforms

Definition of Fourier Transforms


If f(t) is a function of the real variable t, then the Fourier transform F(?) of f is given by the integral
F(?) = ?-?+? e - j ? t f(t) dt
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).

unit step function

Table of Fourier Transforms

f(t) F(?)
u(t) e -a t , a > 0 1 / (a + j ?)
1 for - a ? t ? a
and 0 otherwise 		2 sin (? a) / ?
A (constant) 2 ? A ? (?)
? (t) 1
? (t - a) e - j ? a
cos (a t) ? [ ?(? + a) + ?(? - a) ]
sin (a t) - j ? [ ? (? - a) - ? (? + a) ]
e j a t 2 ? [?(? - a) ]
f '(t) j ? F (?)
f ''(t) (j ?) 2 F(?)
t f(t) j d [F(?)] / d?
t 2 f(t) j 2 d 2[F(?)] / d? 2

More References and links

integrals and their applications in calculus.