lecture 8 topics fourier transforms –as the limit of fourier series –spectra –convergence of...
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Lecture 8 Topics
• Fourier Transforms – As the limit of Fourier Series– Spectra– Convergence of Fourier Transforms– Fourier Transform:
• Synthesis equation• Analysis equation
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( ) ( ) kskH s h e d
We defined:
Note that:
( ) ( ) stH s h t e dt
is the Laplace Transform
( ) ( ) j tH j h t e dt
is the Fourier Transform
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Relate this to Fourier Series for x(t) periodic, period
0( ) jk tk
k
x t c e
00( ) ( ) jk t
kk
y t c H jk e
0
2T
Where( ) ( ) j tH j h t e dt
so each term has a gain and phase due to 0( )H jk
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Recall from last lecture:
x(t) y(t)
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Recall from last lecture:
x(t) y(t)
0j te
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Recall from last lecture:
x(t) y(t)
0j te 00( ) j tH j e
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Recall from last lecture:
x(t) y(t)
0j te 00( ) j tH j e
( ) ( ) j tH j h t e dt
where
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Fourier Transform – Limit of Fourier Series
0
2
T
0 0 0
/ 2/ 2/ 2
/ 20/ 2 / 2
1 1 1( ) 1
p
p
p
p
TTjn t Tjn t jn t
n T
T T
c x t e dt e dt eT T jn T
0 02 20
1 1sin( )
2 2
p pT Tjn jn pT
e e njn n
Fourier Series:
…..…..
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Recall (Euler)
cos( ) sin( )
cos( )2
sin( )2
2 sin( )
jx
jx jx
jx jx
jx jx
e x j x
e ex
e ex
j
e e j x
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