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Properties of the Fourier transform

Fourier transform. The Fourier transform of the function *f(t) *is *F(ω). We can write
F[f(t)] = F(ω) and F ^{-1}[F(ω)]= f(t).*

Magnitude spectrum and phase spectrum of a function *f(t*). Let F(ω) be
the Fourier transform of a function *f *(t). The function F(ω) is, in general, complex. Let us write it
as

F(ω) = R(ω) + i X(ω) = |F(ω)| e ^{i φ(ω)}

where |F(ω)| is called the magnitude spectrum of *f *(t) and φ(ω) is called the phase spectrum
of *f *(t). Since for any complex number x + iy, x + iy = re^{iθ}, the magnitude spectrum is given by

and the phase spectrum is given by

In the table below the functions are periodic with period T, a > 0. Moreover, b, t_{0}, and ω_{0} =2π/T
are real constants and n = 1, 2, .... Furthermore, *f _{e}(t)* is an even function,

*f(t)* *F(ω)*

___________________________________________________________________________

*a _{1}f_{1}(t) + a_{2}f_{2}(t)*

*f(at)*

*f(-t)* *F(-ω)*

*f(t-t _{0})*

*F(ω - ω _{0})*

*f(t) cos ω _{0}t*

*f(t) sin ω _{0}t*

*f _{e}(t) = ½[f(t) + f(-t)]*

*f _{o}(t) = ½[f(t) - f(-t)]*

*f(t) = **f _{e}(t) + f_{o}(t) *

*F(t)* *2π f(-ω)*

*iω F(ω)*

*f ^{(n)}(t)*

*-itf(t)*

*(-it) ^{n}f(t)* F

F_{1}(ω)F_{2}(ω)

*f _{1}(t)f_{2}(t)*

*e ^{-at}u(t)*

*e ^{-a }^{|t|}*

*P _{2a} (ω)*

*te ^{-at} u(t)*

*e ^{-at}sin bt u(t)*

*e ^{-at}cos bt u(t)*

*δ(t)* 1

*δ(t - t _{0})*

*δʹ(t)* *iω*

*δ ^{(n)}(t)*

*u(t)* *πδ(ω) + 1/(iω)*

*u(t - t _{0})*

1 *2π δ(ω)*

*t* *2πi δʹ(ω)*

*t ^{n}*

*2π δ(ω - ω _{0})*

*cos ω _{0}t* π [δ(ω - ω

*sin ω _{0}t* -iπ [δ(ω - ω

*sin ω _{0}t* u(t)

*cos ω _{0}t* u(t)

*t u(t)* *iπ δʹ(ω) - 1/ω ^{2}*

1/t *πi - 2πi u(ω)*

1/t^{n}

sgn t 2/iω

Other properties:

References

Hsu. Fourier Analysis

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