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

and theirproduct is
F(ω)G(ω) =
1
(1+jω)(2+jω)
24.8 Convolution and correlation 777
The Fourier transform of the convolution is, usinglinearityand Table 24.1,
F{u(t)(e −t −e −2t )} = 1
1+jω − 1
2+jω
= (2+jω)−(1+jω)
(1+jω)(2+jω)
1
=
(1+jω)(2+jω)
(24.11)
whichisthesameasEquation(24.11).Wehaveshownthat F{f ∗g} =F(ω)G(ω)
and sothe convolution theorem has been verified.
Example24.17 (a) Usingthe definition ofconvolution find the convolution, f ∗g, ofthe ‘top-hat’
function
{ 1 −1t1
f(t)=
0 otherwise
and the functiong(t) =u(t)e −t , whereu(t) isthe unit stepfunction.
(b) Verifythe convolution theoremforthesefunctions.
Solution (a) Thefunctions f (t)andg(t) are shown inFigure 24.14.
Theconvolution of f (t)andg(t) isgiven by
f∗g=
∫ ∞
−∞
f(λ)g(t − λ)dλ
Note that since g(t) = u(t)e −t , it follows that g(λ) = u(λ)e −λ as shown in Figure
24.15(a). The function g(−λ) is found by reflecting, or folding, g(λ) in the
vertical axis. This folding is shown in Figure 24.15(b). The folded graph can be
translated a distancet to the left or to the right by changing the argument of g to
g(t − λ).Ift isnegativethegraphinFigure24.15(b)movestotheleft,whereasift
ispositive itmoves tothe right.Study Figures24.15(c--g)toobserve this.
Convolution is the integral of the product of f (λ) andg(t − λ). We have superimposed
f (λ) on the graphs in Figure 24.15. For values of λ where the graphs do
notoverlap, thisproduct mustbezero.
Inspectionofthegraphsshowsthatwhent islessthan −1(Figure24.15(c))there
isno overlap and hence f (λ)g(t − λ) = 0.So:
ift<−1
f∗g=0
f(t)
1
g(t) =
u(t) e – t
–1 1
t
t
Figure24.14
The‘top-hat’ function f (t),
andg(t) =u(t)e −t .