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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, 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, t0, and ω0 =2π/T are real constants and n = 1, 2, .... Furthermore, fe(t) is an even function, fo(t) is an odd function, and u(t) is the step function.

f(t)                                                                             F(ω)

___________________________________________________________________________

a1f1(t) + a2f2(t)                                                                   a1F1(ω)+ a2F2(ω)

f(at)

f(-t)                                                                                        F(-ω)

f(t-t0)

F(ω - ω0)

f(t) cos ω0t

f(t) sin ω0t

fe(t) = ½[f(t) + f(-t)]                                                           R(ω)

fo(t) = ½[f(t) - f(-t)]                                                            iX(ω)

f(t) = fe(t) + fo(t)                                                     F(ω) = R(ω) + iX(ω)

F(t)                                                                                        2π f(-ω)

iω F(ω)

f(n)(t)                                                                                      (iω)nF(ω)

-itf(t)

(-it)nf(t)                                                                                 F(n)(ω)

F1(ω)F2(ω)

f1(t)f2(t)

e-atu(t)

e-a |t|

P2a (ω)

te-at u(t)

e-atsin bt u(t)

e-atcos bt u(t)

δ(t)                                                                                                     1

δ(t - t0)

δʹ(t)

δ(n)(t)                                                                                                 (iω)n

u(t)                                                                                                     πδ(ω) + 1/(iω)

u(t - t0)

1                                                                                                         2π δ(ω)

t                                                                                                          2πi δʹ(ω)

tn                                                                                                        2πin δ(n)(ω)

2π δ(ω - ω0)

cos ω0t                                                                                   π [δ(ω - ω0) + δ(ω + ω0)]

sin ω0t                                                                                    -iπ [δ(ω - ω0) - δ(ω + ω0)]

sin ω0t u(t)

cos ω0t u(t)

t u(t)                                                                                      iπ δʹ(ω) - 1/ω2

1/t                                                                                           πi - 2πi u(ω)

1/tn

sgn t                                                                                                                2/iω

Other properties:

References

Hsu. Fourier Analysis