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

24.3 Some properties of the Fourier transform 765
Example24.7 Giventhatwhen f (t) =
{ 1 |t|1
0 |t|>1
tofind the Fourier transform of
g(t) =
{ 1 1t3
0 otherwise
Verifyyour resultdirectly.
2sinω
,F(ω) =
ω
,applythesecondshifttheorem
g(t)
1
1 2 3
t
Figure24.3
The function
{ 1 1t3
g(t) =
0 otherwise.
Solution The functiong(t) is depicted in Figure 24.3. Clearlyg(t) is the function f (t)translated
2unitstotheright,thatisg(t) = f (t−2).NowF(ω) = 2sinω istheFouriertransform
ω
of f (t).Therefore, by the second shift theorem
F{g(t)} = F{f(t −2)} =e −2jω F(ω) = 2e−2jω sinω
ω
To verifythis resultdirectly wemustevaluate
∫ ∞ ∫ 3
[ ] e
F{g(t)} = g(t)e −jωt dt = e −jωt −jωt 3
dt =
−∞ 1 −jω
1
as required.
= e−3jω −e −jω
−jω
= 2e−2jω sinω
ω
= e −2jω ( e −jω −e jω
−jω
)
EXERCISES24.3
1 Provethe first shift theorem.
2 Find the Fourier transform of
{
1−t 2 |t|1
f(t)=
0 |t|>1
Usethe first shift theoremto deduce the Fourier
transformsof
{
e
(a)g(t) =
3jt (1−t 2 ) |t|1
0 |t|>1
{
e
(b)h(t) =
−t (1−t 2 ) |t|1
0 |t|>1
3 Find the inverse Fourier transformsof
(a)
(b)
1
(ω+7)j+1
2
1+2(ω−1)j
4 Prove the second shift theorem.