lecture 8 topics fourier transforms –as the limit of fourier series –spectra –convergence of...

Lecture 8 Topics

• Fourier Transforms – As the limit of Fourier Series– Spectra– Convergence of Fourier Transforms– Fourier Transform:

• Synthesis equation• Analysis equation

( ) ( ) 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

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

Recall from last lecture:

x(t) y(t)

Recall from last lecture:

x(t) y(t)

0j te

Recall from last lecture:

x(t) y(t)

0j te 00( ) j tH j e

Recall from last lecture:

x(t) y(t)

0j te 00( ) j tH j e

( ) ( ) j tH j h t e dt

where

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:

…..…..

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