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16.362 Signal and System I • Continuous-time filters described by differential equations
)(tx)(
2
1
2
1)( tx
dt
dyty
)()(2 txtydt
dy)(th
)(ty
2
1
+
dt
d
2
1
Recall in Ch. 2
• Continuous time Fourier transform.
• LTI system response properties, Ch. 2
Two different ways:
16.362 Signal and System I
)(ty)(tx )(th
)(tx )(2
1
2
1)( tx
dt
dyty
)()(2 txtydt
dy
)(th
)(ty
2
1
+
dt
d
2
1
)()(2 txtydt
dy
Time domain Frequency domain
16.362 Signal and System I
)(ty)(tx )(th
)(tx )(2
1
2
1)( tx
dt
dyty
)()(2 txtydt
dy
)(th
)(ty
2
1
+
dt
d
2
1
)()(2 txtydt
dy
tjk
kkeatx 0)(
tjk
kk ejkHaty 0)()( 0
dehjkH jk 0)()( 0
16.362 Signal and System I
)(ty)(tx )(th
)()(2 txtydt
dy
tjk
kkeatx 0)(
tjk
kk ejkHaty 0)()( 0
tjk
kk ejkHjka
dt
tdy0)(
)(00
)()(2 txtydt
dy tjk
kk
tjk
kk
tjk
kk eaejkHaejkHjka 000 )(2)( 000
Valid for any k
1)(2)( 000 jkHjkHjk0
0 2
1)(
jkjkH
16.362 Signal and System I
)(ty)(tx )(th
)()(2 txtydt
dy
00 2
1)(
jkjkH
jjH
2
1)(
)()( 2 tueth t
)2(
1
)()(
0
)2(
0
2
2
j
de
dee
detuejH
j
j
j
16.362 Signal and System I
)(ty)(tx )(th
)()(2 txtydt
dy
A simpler way
Time domain Frequency domain
)()()(2)()( jXjXjHjXjHj
)2(
1)(
jjH
)(tx )( jX
)(ty )()()( jXjHjY
dt
dy )( jYj
16.362 Signal and System I
)(ty)(tx )(th
)()(2 txtydt
dy
A simpler way
)2(
1)(
jjH
jejH
2/12 )4(
1)(
2tan 1
Low-pass
16.362 Signal and System I A simple RC low pass filter
+-)(tvs
R
C )(tvc)(ti
)(
)(
tv
tQC
c
c )()( tCvtQ cc
dt
tdvC
dt
tdQti cC )()()(
)(tvR
dt
tdvRCtRitv c
R
)()()(
)()()(
tvtvdt
tdvRC sc
c
16.362 Signal and System I
)(tvc)(tvs )(th
A simple RC low pass filter
+-)(tvs
R
C )(tvc)(ti
)(tvR)()(
)(tvtv
dt
tdvRC sc
c
)()()(
tvtvdt
tdvRC sc
c )()()()()( jVjVjHjVjHRCj sss
RCjjH
1
1)(
2121
1)(
RC
jH
RC 1tan
)(1
)( tueRC
th RC
t
16.362 Signal and System I
)(tvc)(tvs )(th
+-)(tvs
R
C )(tvc)(ti
)(tvR)()(
)(tvtv
dt
tdvRC sc
c
2121
1)(
RC
jH
RC 1tan
)(1
)( tueRC
th RC
t
sRC 1.0
sRC 1
)( jH
16.362 Signal and System I A simple RC high pass filter
+-)(tvs
R
C
)(tvc
)(ti
)(
)(
tv
tQC
c
c )()( tCvtQ cc
dt
tdvC
dt
tdQti cC )()()(
)(tvR
)()()( tvtvtv CsR
dt
tdv
dt
tdv
dt
tdv CsR )()()(
dt
tdv
dt
tdv
dt
tdv RsC )()()(
dt
tdvRC
dt
tdvRCtv Rs
R
)()()( RtitvR )()(
16.362 Signal and System I
+-)(tvs
R
C
)(tvc
)(ti
)(tvR
dt
tdvRCtv
dt
tdvRC s
RR )(
)()(
Time domain Frequency domain
)()()( jVRCjjVjVRCj sRR
dt
tdvRCtv
dt
tdvRC s
RR )(
)()(
)()()()()( jVRCjjVjHjVjHRCj sss
RCj
RCjjH
1
)(
2
1
2
2
1)(
RC
RCjH
sRC 1.0
sRC 1
)( jH
16.362 Signal and System I • discrete-time filters described by difference equations
Recall in Ch. 2
• discrete-time Fourier transform.
• LTI system response properties, Ch. 2
Two different ways:
][nx
][]1[2
1][ nxnyny
?][ ny ?][ nh][nh
][ny
2
1
+
delay
16.362 Signal and System I
][nx
][]1[2
1][ nxnyny
][nh
][ny
2
1
+
delay
][ny][nx ][nh
Time domain Frequency domain
][]1[2
1][ nxnyny
16.362 Signal and System I
][ny][nx ][nh
Time domain Frequency domain
][]1[2
1][ nxnyny
][nx )( 0jkX
][ny )( 0jkY
)()()( 000 jkXjkHjkY
]1[ nx )( 00 jkXe jk
][]1[2
1][ nxnyny )()(
2
1)( 000
0 jkXjkYejkY jk
)()()(2
1)()( 00000
0 jkXjkXjkHejkXjkH jk
16.362 Signal and System I
][ny][nx ][nh ][]1[2
1][ nxnyny
][2
1][ nunh
n
0
21
1
1)( 0
jkejkH
0
0
0
21
1
1
2
1
][)(
0
0
jk
n
njk
n
njk
e
e
enhjkH
16.362 Signal and System I
][ny][nx ][nh
M
Nmm mnxbny ][][
M
Nm
mjkkebjkH 0)( 0
Nonrecursive discrete-time filter
FIR filter Finite Impulse Response filter
Example
]1[][]1[3
1][ nxnxnxny 00 1
3
1)( 0
jkjk eejkH
jj eejH 13
1)(
Moving average filter
