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

780 Chapter 24 The Fourier transform
( )
= e−jω
−jω + ejω e
−(1+jω)
jω −e−1 −(1+jω)
( ) ( )
e
+e −1 (1+jω) e
+(e 1 −e −1 −(1+jω)
)
−(1+jω) 1+jω
Thefirsttwotermssimplifyto 2sinω
ω
.Bysimplifyingandrearrangingtheremainder
becomes − ejω −e −jω
. Putting all thistogether we find
1+jω
F{(f ∗g)(t)} = 2sinω
ω
= 2sinω
ω
= 2sinω
ω(1+jω)
− ejω −e −jω
1+jω
− 2jsin ω
1+jω
whichisthesameasEquation(24.12).Wehaveshownthat F{f ∗g} =F(ω)G(ω)
and so the convolution theorem has been verified.
24.8.2 Correlationandthecorrelationtheorem
If f (t) and g(t) are two piecewise continuous functions their correlation (also called
their cross-correlation), whichwedenoteby f ⋆g, isdefined asfollows:
Thecorrelationof f (t)andg(t):
f⋆g=
∫ ∞
−∞
f(λ)g(λ −t)dλ
Thisformulaisthesameasthatforconvolutionexceptfortheimportantdifferencethat
the functiongisnotfolded; thatis, wehaveg(λ −t) hererather thang(t − λ).
It is possible to show that the correlation of f (t) andg(t) can be written in the alternative
form
f⋆g=
∫ ∞
−∞
f(t + λ)g(λ)dλ
Sometextsusethisform,butnotethatitistheargumentof f whichist + λ(seeQuestion
8 inExercises 24.8).
We can now statethecorrelation theorem.
Thecorrelationtheorem:
If F{f(t)} =F(ω)and F{g(t)} =G(ω)then
F{f ⋆g} =F(ω)G(−ω)