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

24.8 Convolution and correlation 775
Theconvolution theoremstatesthatconvolutioninthetimedomaincorresponds to
multiplication inthe frequency domain:
The convolution theorem:
If F{f(t)} =F(ω)and F{g(t)} =G(ω)then
F{f ∗g} =F(ω)G(ω)
Thistheoremgivesusatechniqueforcalculatingtheconvolutionoftwofunctionsusing
the Fourier transform,since
f∗g=F −1 {F(ω)G(ω)}
So,itis possible tofind the convolution of f (t)andg(t), by
(1) finding the corresponding Fourier transforms,F(ω)andG(ω),
(2) multiplying these together toformF(ω)G(ω),
(3) finding the inverse Fourier transform which then yields f ∗g.
This is a process often used for finding convolutions using a computer as will be explained
inSection 24.15.
Agraphicalrepresentationofconvolutionisusefulasitcanthrowlightontheunderlyingprocessandhelpustodetermineappropriatelimitsofintegration.Wewillillustrate
thisinthe following example.
Example24.16 (a) Usingthe definition ofconvolution, calculate the convolution f ∗gwhen
f (t) =u(t)e −t andg(t) =u(t)e −2t , whereu(t) isthe unit stepfunction.
(b) Verifythe convolution theorem for these functions.
Solution (a) The convolution of f andgisgiven by
1
f(t)=u(t)e –t
g(t) = u(t)e –2t t
Figure24.12
Graphsof
f(t) =u(t)e −t and
g(t) =u(t)e −2t .
f∗g=
∫ ∞
−∞
f(λ)g(t − λ)dλ
Graphs of f (t)andg(t) areshown inFigure 24.12.
Evaluatingaconvolutionintegralcanbedifficult,sowewilldevelopthesolutioninstages.Firstofallitisnecessarytobeclearaboutthemeaningofthedifferent
termsintheintegrand.Notethatif f (t) =u(t)e −t then f (λ) =u(λ)e −λ .Similarly
g(λ) =u(λ)e −2λ .Thefunctiong(−λ)isfoundbyreflectingg(λ)intheverticalaxis
asshowninFigure24.13(a).Insignalprocessingterminologythisisalsoknownas
folding.Thefoldedgraphcanbetranslatedadistancet totheleftortotherightby
changingtheargumentofgtog(t −λ).Ift isnegativethegraphmovestotheleftas
shown in Figure 24.13(b) whereas ift is positive it moves to the right as shown in
Figure 24.13(c). In Figure 24.13 we have superimposed the graphs of f (λ), shown
dashed, andg(t − λ), fort being negative, zero and positive. Where the graphs do
notoverlap,theproduct f (λ)g(t −λ),andhencetheconvolution,mustbezero.We
examineseparatelythedomainst < 0andt 0correspondingtowherethegraphs
do notoverlap and where they do overlap respectively.