Mathematical Methods for Physics and Engineering : A Comprehensive Guide

INTEGRAL TRANSFORMS

13.10 In many applications in which the frequency spectrum of an analogue signal is
required, the best that can be done is to sample the signalf(t) a finite number of
times at fixed intervals, and then use adiscrete Fourier transformFkto estimate
discrete points on the (true) frequency spectrum ̃f(ω).

(a) By an argument that is essentially the converse of that given in section 13.1,

show that, ifNsamplesfn, beginning att= 0 and spacedτapart, are taken,

thenf ̃(2πk/(Nτ))≈Fkτwhere

Fk=

1

√

2 π

N∑− 1

n=0

fne−^2 πnki/N.

(b) For the functionf(t) defined by

f(t)=

{

1for0≤t< 1 ,

0otherwise,

from which eight samples are drawn at intervals ofτ=0.25, find a formula

for|Fk|and evaluate it fork=0, 1 ,..., 7.

(c) Find the exact frequency spectrum off(t) and compare the actual and

estimated values of

√

2 π| ̃f(ω)|atω=kπfork=0, 1 ,..., 7 .Note the

relatively good agreement fork<4 and the lack of agreement for larger

values ofk.

13.11 For a functionf(t) that is non-zero only in the range|t|<T/2, the full frequency
spectrum ̃f(ω) can be constructed, in principle exactly, from values at discrete
sample pointsω=n(2π/T).Provethisasfollows.

(a) Show that the coefficients of a complex Fourierseriesrepresentation off(t)

with periodTcan be written as

cn=

√

2 π

T

̃f

(

2 πn

T

)

.

(b) Use this result to representf(t) as an infinite sum in the defining integral for

̃f(ω), and hence show that

̃f(ω)=

∑∞

n=−∞

f ̃

(

2 πn

T

)

sinc

(

nπ−

ωT

2

)

,

where sincxis defined as (sinx)/x.

13.12 A signal obtained by sampling a functionx(t) at regular intervalsTis passed
through an electronic filter, whose responseg(t) to a unitδ-function input is
represented in atg-plot by straight lines joining (0,0) to (T, 1 /T)to(2T,0) and
is zero for all other values oft. The output of the filter is the convolution of the
input,

∑∞

−∞x(t)δ(t−nT), withg(t).

Using the convolution theorem, and the result given in exercise 13.4, show that

the output of the filter can be written

y(t)=

1

2 π

∑∞

n=−∞

x(nT)

∫∞

−∞

sinc^2

(

ωT

2

)

e−iω[(n+1)T−t]dω.

13.13 Find the Fourier transform specified in part (a) and then use it to answer part
(b).