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2.161 Signal Processing: Continuous and Discrete
Fall 2008
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Massachusetts
Institute
of
Technology
Department
of
Mechanical
Engineering
2.161
Signal
Processing
 Continuous
and
Discrete
Fall
Term
2008
Lecture
17
1
Reading:
•
Class
Handout:
FrequencySampling
Filters
.
•
Proakis
and
Manolakis:
Secs.
10.2.3,
10.2.4
•
Oppenheim,
Schafer
and
Buck:
7.4
•
Cartinhour:
Ch.
9
FrequencySampling
Filters
In
the
frequencysampling
ﬁlters
the
parameters
that
characterize
the
the
ﬁlter
are
the
values
of
the
desired
frequency
response
H
(
e
jω
)
at
a
discrete
set
of
equally
spaced
sampling
frequencies.
In
particular,
let
2
π
ω
k
=
k
k
=0
,...,N
−
1
(1)
N
as
shown
below
for
the
cases
of
N
even,
and
N
odd.
Note
that
when
N
is
odd
there
is
no
sample
at
the
Nyquist
frequency,
ω
=
π
.
The
frequencysampling
method
guarantees
that
the
resulting
ﬁlter
design
will
meet
the
given
design
speciﬁcation
at
each
of
the
sample
frequencies.
1

1
1

1
I
m
(
z
)
R
e
(
z
)
M
±
²
2
F
2
1
0
1
1
z

p
l
a
n
e
R
e
(
z
)
z

p
l
a
n
e
I
m
(
z
)
N
=
1
0
(
e
v
e
n
)
N
=
1
1
(
o
d
d
)
(
a
)
(
b
)
1
copyright
c
±
D.Rowell
2008
17–1
±
²
For
convenience
denote
the
complete
sample
set
{
H
k
}
as
H
k
=
H
(
e
jω
k
)
k
=1
,...,N
−
1
.
a
ﬁlter
with
a
real
impulse
response
{
h
n
}
we
require
conjugate
symmetry,
that
is
¯
H
N
−
k
=
H
k
and
further,
for
a
ﬁlter
with
a
real,
even
impulse
response
we
require
{
H
k
}
to
be
real
and
even,
that
is
H
N
−
k
=
H
k
.
Within
these
constraints,
it
is
suﬃcient
to
specify
frequency
samples
for
the
upper
half
of
the
z
plane,
that
is
for
2
π
k
=0
,...,
N
2
−
1
N
odd
ω
k
=
k
N
N
k
2
N
even
.
and
use
the
symmetry
constraints
to
determine
the
other
samples.
If
we
assume
that
H
(
e
)
may
be
recovered
from
the
complete
sample
set
{
H
k
}
by
the
cardinal
sinc
interpolation
method,
that
is
N
−
1
H
(
e
)=
²
H
k
sin (
ω
−
2
πk/N
)
ω
−
2
πk/N
k
=0
then
H
(
e
)
is
completely
speciﬁed
by
its
sample
set,
and
the
impulse
response,
of
length
N
,
may
be
found
directly
from
the
inverse
DFT,
{
h
n
}
=
IDFT
{
H
k
}
where
N
−
1
1
j
2
πkn
h
n
=
H
k
e
N
n
−
1
N
k
=0
As
mentioned
above,
this
method
guarantees
that
the
resulting
FIR
ﬁlter,
represented
by
{
h
n
}
,
will
meet
the
speciﬁcation
H
(
e
H
k
at
ω
=
ω
k
=2
kπ/N
.
Between
the
given
sampling
frequencies
the
response
H
(
e
)
will
be
described
by
the
cardinal
interpolation.
1.1
LinearPhase
FrequencySampling
Filter
The
ﬁlter
described
above
is
ﬁnite,
with
length
N
,
but
is
noncausal.
To
create
a
causal
ﬁlter
with
a
linear
phase
characteristic
we
require
an
impulse
response
that
is
real
and
symmetric
about
its
midpoint.
This
can
be
done
by
shifting
the
computed
impulse
response
to
the
right
by
(
N
−
1)
/
2
samples
to
form
H
±
(
z
z
−
(
N
−
1)
/
2
H
(
z
)
but
this
involves
a
noninteger
shift
for
even
N
.
Instead,
it
is
more
convenient
to
add
the
appropriate
phase
taper
to
the
frequency
domain
samples
H
k
before
taking
the
IDFT.
The
noninteger
delay
then
poses
no
problems:
17–2
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±
±
•
Apply
a
phase
shift
of
πk
(
N
−
1)
φ
k
=
−
(2)
N
to
each
of
the
samples
in
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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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