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

Review exercises 24 821
REVIEWEXERCISES24
1 Find the Fourier transformsof
{
1−t 2 |t|<1
(a)f(t) =
0 otherwise
{ sint |t|<π
(b)f(t) =
0 otherwise
{ 1 0<t<τ
(c)f(t) =
0 otherwise
{ e −αt t>0
(d)f(t) =
−e αt α >0
t<0
2 Find the Fourier integral representations of
{ 3t |t|<2
(a) f(t) =
0 otherwise
⎧
⎨0 t<0
(b) f(t) = 6 0<t<2
⎩
0 t>2
3 (a) IfF(ω) = F{f(t)},showthat
F{f ′ (t)} = jωF(ω).
(b) IfF(ω) = F{f(t)},showthat
F{f (n) (t)} = (jω) n F(ω).Theseresultsenableus
to calculate the Fourier transformsofderivatives
offunctions.
4 IfF(ω) = F{f(t)},showthat
( )
F{f(at)} = 1 a F ω
.
a
5 IfF(ω)is the Fourier transform of f (t)show that
(a) F(0) = ∫ ∞
−∞ f(t)dt.
(b) f(0) = 1 ∫ ∞
F(ω)dω.
2π −∞
(c) Show that if f (t)isan even function,
F(ω) =2 ∫ ∞
0 f(t)cosωtdt.
6 Theconvolution theoremgiven in Section24.8.1
representsconvolution in the time domain.
Convolution can alsobe performedin the frequency
domain,in whichcasethe equivalent convolution
theoremis
F{f(t)g(t)} = 1
2π [F(ω)∗G(ω)]
Provethe convolution theorem in thisform.
7 (a) Given that the Fourier transform of f (t) = e −α|t|
2α
is
α 2 use thet--ω duality principleto find
+ω2 the Fourier transformof
1
α 2 +t 2.
(b) Fromatable oftransformswrite down the
Fourier transformofcosbt.
(c) Use the convolution theoremobtained in
Question 6 to find the Fourier transformof
cosbt
α 2 +t 2,forb>0.
(d) Using the result in part(c)evaluate the integral
∫ ∞ cosbt
−∞ α 2 +t 2dt
8 Find the d.f.t. ofthe sequence f[n] = {3,3,0,3}.
Verify Rayleigh’stheoremforthissequence.
9 Findthelinearconvolutionofthetwofinitesequences
f[n] = 3, −1, −7 andg[n] = 4,0, 1 2 .
10 The signum functionis defined to be
⎧
⎨ 1 t>0
sgn(t) = −1 t<0
⎩
0 t=0
This functioncan be representedasthe exponential
functione −ǫt ,ift > 0, andas −e ǫt ,ift < 0,in the
limitas ǫ → 0.
(a) Show that
∫ ∞
sgn(t)e −jωt dt = 2
−∞ jω
(b) Usethet--ω duality principle to showthat
∫ ∞
−∞
1
πt e−jωt dt = −jsgn(ω)
(c) Usethe second resultin part (b), the convolution
theorem and the integral propertiesofthe delta
function to showthat
1
∗cos(αt) = sin(αt)
πt
11 Show that f[n] ○⋆ g[n] = f[n] ○∗ g[−n] andhence
deduce thatacorrelationcanbe expressed in termsof
a convolution.