dsp_foehu - lec 02 - frequency domain analysis of signals and systems
TRANSCRIPT
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0
0
0
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n dttn
etxT
C
n
tnjenCtx 0)(
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nn eCC ||
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n
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0
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T
n
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n M
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5.0 2T
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dfet ftj2)(
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2/2/
2
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T
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R y t y t dtT
h u x t u du h v x t v dv dtT
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integrated over one period
dteexxT
RT
T
n m
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)(tx )(ty)( fH
( ) ( )x t t
( ) ( )y t h t
( )y t( )x t System
( )h t( )t System
0t
( ) ( )x t h t
( ) ( ) ( )y t x t h t
( )y t( )x t ( )h t
Convolution Integral :
Linear system
)(t )(th
Linear system
)(tu )()( thtu
)()()()(
)()()()(
txthdtxth
dtxhthtx
( ) ( ) ( ),x t h t p t
( )p t
tT0
0
( ) ( ) ( ) , 0y t x h t d t
0t
( )x
T0
( )h t
t T t
( ) 0y t
0 t T
( )x
T0
( )h t
t T t
0
( )t
y t d t
( )x
T0
( )h t
t T t
( ) ( ) 2T
t T
y t d T t T T t
0 2t T T T t T
( ) 0y t
( )x
T0
( )h t
t T t
2T t T T t
( ) ( ) ( )y t x t h t
T0 t2T
2/4)( tth)(tu
2 3 8
(1) 2t
2 3
)(u)(th
tt
t8t
0)(ty
(2) 32 t
2 3
)(u)(th
t8t
t
dt
ty2
)2
4(2)(
(3) 63 t
2 3
)(u)(th
t8t
3
2)
24(2)( d
tty
(4) 116 t
2 3
)(u )(th
t8t3
8)
24(2)(
td
tty
(5) t11
0)(ty2 3
)(u)(th
t8t
0)(ty2t
32 tt
dt
ty2
)2
4(2)(
63 t3
2)
24(2)( d
tty
116 t3
8)
24(2)(
td
tty
t11 0)(ty
Ans:
)(th)(tu )()()( thtuty
)(sH)(sU )()()( sHsUsY
integral
Algebra operator
( ) ( ( ) ( )) ( ( ) ( )) ( )x t v t w t x t v t w t
( ) ( ) ( ) ( )x t v t v t x t
( ) ( ( ) ( )) ( ) ( ) ( ) ( )x t v t w t x t v t x t w t
( ) ( ) ( )w t x t v t
( ) ( ) ( ) ( ) ( )q qw t q x t v t x t v t
( ) ( )qx t x t q
( ) ( )qv t v t q
( ) ( ) ( )x t t x t
( ) ( ) ( )qx t t x t q
( )( ) ( ) ( )
d dx tx t v t v t
dt dt
2
2
( ) ( )( ) ( )
d dx t dv tx t v t
dt dt dt
then ( 1) ( 1) ( 1)( ) ( ) ( ) ( ) ( ) ( )x v t x t v t x t v t
( ) ( )x t u t 0t
( ) ( ) ( )g t h t u t
( )g t( )u t ( )h t
( ) ( ) ( )( ) ( )
dg t dh t du tu t h t
dt dt dt
( )( )
du tt
dt( ) ( ) ( )h t h t t
( )( )
dg th t
dt
and
0
( ) ( )t
g t h dor