)]([)( txFX
)]([)( 1 XFtx
)()( Xtx
)(X )(X
)(X
)(X
dxdfe
dfedextx
tfj
ftjfj
)(
)()(
)(2
22
dxttx )()()(
)(t
dfet tfj )(2)(
dfet ftj2)(
)(t
)()]([
)()]([
22
11
XtxF
XtxF
)()( 21 txtx
)()()]()([ 2121 XXtxtxF
)(1 tx )(2 tx
)]([)( and )]([)( then )]([)( If tXFxtXFxtxFX
0t
0t
)]([)]([ 0
0 txFettxF tj
0a
aX
aatxF
1)]([
)(tx )(ty
)(*)(2
1)]([*)]([
2
1)]().([
)()()]([)]([)]()([
YXtyFtxFtytxF
YXtyFtxFtytxF
tjetx 0)( )( 0X
)cos()( 0ttx
)(2
1)(
2
100 XX
)(tx )(ty
)( fX )( fY
dYXdttytx )()(2
1)()( **
dXdttx22
)(2
1)(
)(xR
dttxtxRx )()()( *
2)()]([ XRF x
)()()( Xjtxdt
dF
)()0(2
1)()( X
j
XdxF
)()]([ XtxF dttxtn )(
0
)(2
)(
f
n
nnn X
d
djdttxt
dttxtxT
RT
TT
x2
2
* )()(1
lim)(
)0(
)(1
lim 2
2
2
x
T
TT
x
R
dttxT
P
)(xS
)]([)( xx RFS
)(xS
dS
RP
x
xx
)(
)0(
dthxty )()()(
*2
2
* *2
2
1( ) lim ( ) ( )
1lim ( ) ( ) ( ) ( )
T
TyT
T
TT
R y t y t dtT
h u x t u du h v x t v dv dtT
)(tx )(ty)(th
utw
)()()(
)()]()([
)()()(
)()(1
lim)()()(
*
*
*
2
2
**
hhR
dvvhvhvR
dudvvhuhuvR
dudvdwvwuxwxT
vhuhR
x
b
x
b
x
a
uT
uT
Ty
xR
2
*
)()(
)()()()(
HS
HHSS
x
xy
0T }{ nx
dttxtxT
dttxtxkT
k
dttxtxkT
dttxtxT
R
T
T
T
Tk
kT
kTk
T
TT
x
2
2
*
0
2
2
*
0
2
2
*
0
2
2
*
0
0
0
0
0
0
)()(1
)()(lim
)()(1
lim
)()(1
lim)(
integrated over one period
dteexxT
RT
T
n m
tT
mnj
T
mj
mnx2
2
22*
0
0
0
001
)(
nm
T
T
tT
mnj
mn
mndte
T
,
2
2
2
0 0
11 0
0
0
n
T
nj
nx exR 0
22
)(
n
nxxT
nfxRFfS
0
2)]([)(
n
n
n
nxx
x
dfT
nfxdffSP
2
0
2)(
( ) ( )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 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
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