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

760 Chapter 24 The Fourier transform
where
F(ω) =
=
∫ ∞
−∞
∫ 1
−1
f (t)e −jωt dt
1e −jωt dt
[ ] e
−jωt 1
=
−jω
−1
= e−jω −e jω
−jω
= ejω −e −jω
jω
Using Euler’s relation (Section 9.6)
sinθ = ejθ −e −jθ
2j
we find
F(ω) = 2sinω
ω
so that
f(t)= 1 ∫ ∞
2sinω
2π ω
−∞
since f (t)iszero outside [−1,1]
ejωt dω
istherequiredintegralrepresentation.NotethatF(ω) = 2sinω istheFouriertransform
ω
of f (t).Thefunction sin ω
ω
occursfrequentlyandisoftenreferredtoasthesincfunction
(see Section 3.5).
As with Laplace transforms, tables have been compiled for reference. Such a table of
common transforms appearsinTable 24.1.
EXERCISES24.2
1 Find the Fourier transformsof
{ 1/4 |t| 3
(a)f(t) =
0 |t|>3
⎧
1 − t 0t2
⎪⎨ 2
(b)f(t) =
1 + t −2t0
⎪⎩ 2
0 otherwise
{ e −αt t0 α>0
(c)f(t) =
e αt t<0
{ e
(d)f(t) =
−t cost t 0
0 t<0
(e) f (t) =u(t)e −t/τ
where τ isaconstant
2 Find
(a) the Fourier transform, and
(b) the Laplace transformof
f(t) =u(t)e −αt α >0
Show thatmakingthe substitutions = jω in the
Laplacetransform of f resultsin the Fourier
transform.
{ 1 |t|2
3 Iff(t)=
0 otherwise
andg(t) = e jt ,find F{f (t)g(t)}.