082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

24.3 Some properties of the Fourier transform 761
Table24.1
CommonFourier transforms.
f(t)
f(t) =Au(t)e −αt ,α >0
{ 1 −αtα
f(t)=
0 otherwise
F(ω)
A
α+jω
2sinωα
ω
f(t) =A constant 2πAδ(ω)
( )
f(t) =u(t)A A πδ(ω)− j ω
f(t) = δ(t) 1
f(t) = δ(t −a)
f(t) =cosat
f(t) =sinat
f(t) =e −α|t| ,α>0
{ 1 t>0
f(t) =sgn(t) =
−1 t<0
f(t)= 1 t
e −jωa
π(δ(ω +a)+δ(ω −a))
π
(δ(ω −a)−δ(ω +a))
j
2α
α 2 +ω 2
2
jω
−jπsgn(ω)
f(t) =e −at2 √ π
a e−ω2 /4a
Solutions
1 (a)
(d)
sin3ω
(b)
2ω
1+jω
(1+jω) 2 +1
1−cos2ω
ω 2
τ
(e)
1+jωτ
(c)
2α
α 2 +ω 2
2 (a)
3
1
α+jω
2sin2(1−ω)
1 − ω
(b)
1
s + α
24.3 SOMEPROPERTIESOFTHEFOURIERTRANSFORM
A number of the properties of Laplace transforms that we have already discussed hold
forFourier transforms. We considerlinearityand two shifttheorems.
24.3.1 Linearity
If f andgarefunctions oft andkisaconstant, then
F{f +g}=F{f}+F{g}
F{kf}=kF{f}