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

24.9 The discrete Fourier transform 783
EXERCISES24.8
1 Find the convolution of
{ 2
f(t)= 3 t 0t3
0 otherwise
and
{ 4 −1t3
g(t) =
0 otherwise
2 Theconvolution ofafunctionwith itselfis known as
autoconvolution.Find the autoconvolution f ∗ f
when
{ 1 −1t1
f(t)=
0 otherwise
3 Find the correlationof f (t) = 1 for −1 t 1 and
zerootherwise, andg(t) =u(t)e −t .Verify the
correlationtheoremforthese functions.
4 Prove thatconvolution iscommutative, thatis
f ∗g=g ∗ f.Note thatcorrelationis not.
5 Show thatif f (t)andg(t) are both zero,fort < 0,
then (f ∗g)(t) = ∫ t
0 f(λ)g(t − λ)dλ.
6 Prove the convolution theorem.
7 Show that f (t) ⋆g(t) = f (t) ∗g(−t).Deduce thata
correlationcanbeexpressedintermsofaconvolution.
Showalsothat f(t) ∗g(t) = f(t) ⋆g(−t).
8 Prove thatthe correlationintegral
f⋆g= ∫ ∞
−∞ f (λ)g(λ −t)dλcanalsobe writtenin
the form ∫ ∞
−∞g(λ)f(t + λ)dλ.
9 Prove the correlationtheorem.
Solutions
⎧
0 t−1
4
⎪⎨ 3 (t+1)2 −1<t2
1 f∗g= 12 2<t3
12 − 4 ⎪⎩ 3 (t−3)2 3<t6
0 t>6
⎧
⎪⎨
t+2 −2t0
2 f∗f= 2−t 0<t2
⎪⎩
0 otherwise
⎧
⎪⎨
e t (e 1 −e −1 ) t <−1
3 f⋆g= 1−e
⎪⎩
t−1 −1t1
0 t>1
24.9 THEDISCRETEFOURIERTRANSFORM
FormostpracticalengineeringproblemsrequiringtheevaluationofaFouriertransform
it is necessary to use a computer and so some form of approximation is needed. In this
section we show how a function f (t), with Fourier transformF(ω), can be sampled at
intervalsofT togiveasequenceofvalues f (0), f (T), f (2T)....ThediscreteFourier
transform (d.f.t.) takes such a sequence and processes it to produce a new sequence
whichcanbethoughtofasasampledversionofF(ω).Thed.f.t.isimportantasitisthe
basis of most signal and image processing methods. The use of a computer is essential
because ofthe enormousnumberofcalculationsrequired.
We start by stating the transform and provide a simple example to show how it is
calculated. The interested reader should refer to Section 24.10 for a derivation which
shows the relationshipbetween the Fourier transform and the d.f.t.