This document describes a linear system and the convolution integral used to represent it. It defines the convolution integral and shows that a linear system can be represented as the convolution of the input function with the impulse response function. It then provides examples of calculating the output of a linear system using the convolution integral.

3. )(tx
0
0
0
)(
1 T j
n dt
tn
etx
T
C
n
tnj
enCtx 0
)(
00 /1 Tf00 2 f
n
Cj
nn eCC ||
|| nC nC
|| nC
nC

4. n
nTt
tx 0
)(

5. 0
2
0 0
( ) sinc
t
jn
T
n
n
x t e
T T
0
2
0 0
( ) sinc
tM jn
T
n M
n
x t e
T T
5.0 2T

6. dtetxX tj
)()(
deXtx tj
)(
2
1
)(

7. )]([)( txFX
)]([)( 1
XFtx
)()( Xtx
)(X )(X
)(X
)(X
dxdfe
dfedextx
tfj
ftjfj
)(
)()(
)(2
22
dxttx )()()(
)(t
dfet tfj )(2
)(
dfet ftj2
)(
)(t

8. )( t
)
2
sinc
)
2
sin(2
1
)()]([
2/2/
2
2
(
ee
j
dte
dtettF
jj
tj
tj
)( t

9. )()( ttx
1
)()]([ 2
dtettF ftj
1)( 2
dtet ftj
)(]1[ fF
)(t

12. )()]([
)()]([
22
11
XtxF
XtxF
)()( 21 txtx
)()()]()([ 2121 XXtxtxF
)(1 tx )(2 tx
)]([)(and)]([)(then)]([)(If tXFxtXFxtxFX
0t
0t
)]([)]([ 0
0 txFettxF tj
0a
a
X
a
atxF
1
)]([

13. )(tx )(ty
)(*)(
2
1
)]([*)]([
2
1
)]().([
)()()]([)]([)]()([
YXtyFtxFtytxF
YXtyFtxFtytxF
tj
etx 0
)( )( 0X
)cos()( 0ttx
)(
2
1
)(
2
1
00 XX
)(tx )(ty
)( fX )( fY
dYXdttytx )()(
2
1
)()( **
dXdttx
22
)(
2
1
)(

14. )(xR
dttxtxRx )()()( *
2
)()]([ XRF x
)()()( Xjtx
dt
d
F
)()0(
2
1)(
)( X
j
X
dxF
)()]([ XtxF dttxtn
)(
0
)(
2
)(
f
n
nn
n
X
d
dj
dttxt

15. xE
dttxEx
2
)(
dttx
T
P
T
TT
x
2
2
2
)(
1
lim
xE
xP0
0xP xE
dttx
T
P
T
x
0 2
0
)(
1
0T

16. )(xR
dttxtx
dttxtx
xxRx
)()(
)()(
)()()(
*
*
*
0
x
x
E
dffX
dttxR
2
2
)(
)()(
2
)()]([)( XRFG xx

17. dttxtx
T
R
T
T
T
x
2
2
*
)()(
1
lim)(
)0(
)(
1
lim 2
2
2
x
T
T
T
x
R
dttx
T
P
)(xS
)]([)( xx RFS
)(xS
dS
RP
x
xx
)(
)0(

18. dthxty )()()(
*2
2
* *2
2
1
( ) lim ( ) ( )
1
lim ( ) ( ) ( ) ( )
T
Ty
T
T
T
T
R y t y t dt
T
h u x t u du h v x t v dv dt
T
)(tx )(ty)(th
utw
)()()(
)()]()([
)()()(
)()(
1
lim)()()(
*
*
*
2
2
**
hhR
dvvhvhvR
dudvvhuhuvR
dudvdwvwuxwx
T
vhuhR
x
b
x
b
x
a
u
T
u
T
T
y
xR

19. 2
*
)()(
)()()()(
HS
HHSS
x
xy
0T }{ nx
dttxtx
T
dttxtx
kT
k
dttxtx
kT
dttxtx
T
R
T
T
T
T
k
kT
kT
k
T
T
T
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

20. dteexx
T
R
T
T
n m
t
T
mn
j
T
m
j
mnx
2
2
22
*
0
0
0
00
1
)(
nm
T
T
t
T
mn
j
mn
mn
dte
T
,
2
2
2
0 0
11 0
0
0
n
T
n
j
nx exR 0
22
)(
n
nxx
T
n
fxRFfS
0
2
)]([)(
n
n
n
nxx
x
df
T
n
fxdffSP
2
0
2
)(

21. 0
2
0
2
0
22
)()(
T
n
f
T
n
Hx
T
n
fxfHfS
n
n
n
ny
n
ny
T
n
HxP
2
0
2
)(tx )(ty)( fH

22. ( ) ( )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

23. Convolution Integral :
Linear system
)(t )(th
Linear system
)(tu )()( thtu
)()()()(
)()()()(
txthdtxth
dtxhthtx
( ) ( ) ( ),x t h t p t
( )p t
tT0

24. 0
( ) ( ) ( ) , 0y t x h t d t
0t
( )x
T0
( )h t
t T t
( ) 0y t

25. 0 t T
( )x
T0
( )h t
t T t
0
( )
t
y t d t
( )x
T0
( )h t
t T t
( ) ( ) 2
T
t T
y t d T t T T t
0 2t T T T t T

26. ( ) 0y t
( )x
T0
( )h t
t T t
2T t T T t
( ) ( ) ( )y t x t h t
T0 t2T

27. 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
d
t
ty
2
)
2
4(2)(
(3) 63 t
2 3
)(u)(th
t8t
3
2
)
2
4(2)( d
t
ty

28. (4) 116 t
2 3
)(u )(th
t8t
3
8
)
2
4(2)(
t
d
t
ty
(5) t11
0)(ty
2 3
)(u
)(th
t8t
0)(ty2t
32 t
t
d
t
ty
2
)
2
4(2)(
63 t
3
2
)
2
4(2)( d
t
ty
116 t
3
8
)
2
4(2)(
t
d
t
ty
t11 0)(ty
Ans:

29. )(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

30. ( ) ( ) ( )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 t
x t v t v t
dt dt
2
2
( ) ( )
( ) ( )
d dx t dv t
x t v t
dt dt dt

31. 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

32. ( ) ( ) ( )
( ) ( )
dg t dh t du t
u t h t
dt dt dt
( )
( )
du t
t
dt
( ) ( ) ( )h t h t t
( )
( )
dg t
h t
dt
and
0
( ) ( )
t
g t h dor