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2.161 Signal Processing: Continuous and Discrete
Fall 2008
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Institute
of
Technology
Department
of
Mechanical
Engineering
2.161
Signal
Processing
 Continuous
and
Discrete
Fall
Term
2008
Lecture
15
1
Reading:
Proakis
&
Manolakis,
Ch.
7
•
•
Oppenheim,
Schafer
&
Buck.
Ch.
10
Cartinhour,
Chs.
6
&
9
•
1
Frequency
Response
and
Poles
and
Zeros
As
we
did
for
the
continuous
case,
factor
the
discretetime
transfer
functions
into
as
set
of
poles
and
zeros;
b
0
z
0
+
b
1
z
−
2
+
+
b
M
z
−
M
M
i
=1
(
z
−
z
i
)
H
(
z
) =
···
=
K
a
0
z
0
+
a
1
z
−
2
+
+
a
N
z
−
N
N
i
=1
(
z
−
p
i
)
where
Z
i
are
the
system
zeros
,
the
p
i
are
the
system
poles
,
and
K
=
b
0
/a
0
is
the
overall
gain
.
We
note,
as
in
the
continuous
case
that
the
polse
and
zeros
must
be
either
real,
or
appear
in
complex
conjugate
pairs.
As
in
the
continuous
case,
we
can
draw
a
set
of
vectors
from
the
poles
and
zeros
to
a
test
point
in
the
z
plane,
and
evaluate
H
(
z
)
in
terms
of
the
lengths
and
angles
of
these
vectors.
In
particular,
we
choose
to
evaluate
H
(e
j
ω
)
on
the
unit
circle,
M
±
e
²
H
j
ω
) =
K
i
=1
j
ω
−
z
i
N
i
=1
j
ω
−
p
i
)
and
M
i
=1
³
³
e
−
z
i
³
³
M
i
=1
q
i
³
³
H
j
ω
)
³
³
=
K
M
N
i
=1

e
j
j
ω
ω
−
p
i

=
K
N
i
=1
r
i
M
N
N
j
ω
j
ω
H
j
ω
)
=
´
±
e
−
z
i
²
−
´
±
e
−
p
i
²
=
´
θ
i
−
´
φ
i
i
=1
i
=1
i
=1
i
=1
where
the
q
i
and
θ
i
are
the
lengths
and
angles
of
the
vectors
from
the
zeros
to
the
point
z
= e
j
ω
,
the
r
i
and
φ
i
are
the
lengths
and
angles
of
the
vectors
from
the
poles
to
the
point
z
= e
j
ω
,
as
shown
below:
1
copyright
c
D.Rowell
2008
15–1
X
X
X
O
O
1
 1
 j 1
j 1
Â
{
z
}
Á
{
z
}
f
f
f
q
q
p
p
p
z
z
1
1
1
1
2
2
2
2
3
3
r
r
r
q
q
1
1
2
2
3
z
= e
j
w
A t f r e q u e n c y
w
:
 H
( e
)  =
K
q
q
r
r
j
w
1
1
2
2
3
H
) = (
+ q
)
 ( f
+ f
)
1
2
j
w
1
2
3
We
can
interpret
the
eﬀect
of
pole
and
zero
locations
on
the
frequency
response
as
follows:
(a)
A
pole
(or
conjugate
pole
pair)
on
the
unit
circle
will
cause
H
( e
j
ω
) to
become
inﬁnite
at
frequency
ω
.
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.161 taught by Professor Derekrowell during the Fall '08 term at MIT.
 Fall '08
 DerekRowell

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