Digital Signal Processing Reference
In-Depth Information
We will now explore some filters which can be obtained from
H
(
z
)
:
(d) Sketch the magnitude response of a filter
g
[
n
]
whose taps are as fol-
lows:
l
g
r
,
y
i
d
.
,
©
,
L
s
]=
0.005, 0,
0.11, 0, 0.005
g
[
n
−
0.11, 0, 0.27, 0,
−
(i.e.
g
[
0
]=
0.005,
g
[
1
]=
0,
g
[
2
]=
−
0.11, etc.)
We now want to obtain a linear phase highpass filter
f
[
n
]
from
h
[
n
]
and the
following design is proposed:
−
y
[
n
]
x
[
n
]
H
(
z
)
+
+
However the design is faulty:
(e) From the impulse response of the above system, show that the result-
ing filter is
not
linear phase.
(f ) Clearly, the designer's idea was to obtain a system with magnitude
F
e
j
ω
)
=
1
e
j
ω
)
|
(notethemagnitudesignsaround
H
e
j
ω
)
);
this, however, is not the magnitude response of the above system.
Write out the actual magnitude response.
(
Hint:
it is easier to consider the squared magnitude response and,
since
H
(
e
j
ω
)
is linear phase, to express
H
(
e
j
ω
)
as a real term
A
(
e
j
ω
)
∈
(
−|
H
(
(
, multiplied by a pure phase term.)
Now it is your turn to design a highpass filter:
(g) Howwould youmodify the above design to obtain a linear phase high-
pass filter?
(h) Sketch the magnitude response of the resulting filter.
Exercise 7.8: FIR filters analysis - II.
Consider a generic
N
-tapType I FIR
filter. Since the filter is linear phase, its frequency response can be expressed
as
e
j
ω
)=
e
j
ω
)
e
j
ω
)
H
(
A
(
H
r
(
e
j
ω
)
e
j
ω
)
where
H
r
(
is a real function of
ω
and
A
(
is a pure phase term.
