Website owner: James Miller
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 = reiθ, 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)
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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) iω
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
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