Digital Signal Processing Reference
In-Depth Information
5.2
Problem Solving
Exercise 1: Solve the following problems, briefly outlining the important
steps
a.
The frequency response of a certain class of digital filters called
binomial filters is written as
H
r
(
e
j
ω
) = 2
N
[sin(
ω
/2)]
r
[cos(
ω
/2)]
N
-
r
in the range -
.
Selecting
N
= 2, approximately sketch the
response of the filters in the range
0
π ≤
ω
≤
π
≤
ω
≤
π
for the following cases:
i.
r
= 0
ii.
r
= 1
iii.
r
= 2
b. A 3-point symmetric moving average discrete-time filter is of the form:
y
(
n
) =
b
[
a
x
(
n
- 1) +
x
(
n
) +
a
x
(
n
+ 1)]
where
a
and
b
are constants. Determine, as a function of
a
and
b
, the
frequency response
H
(
e
j
ω
) of the system.
c.
We wish to use the Kaiser window method to design a real-valued
FIR filter that meets the following specifications:
()
<
09
.
<
He
j
ω
11
. ,
0
≤
ωπ
≤
0.2
()
<
j
ω
−<
006
.
He
006
.
,
0.3
πω
≤≤
0 475
.
π
()
<
He
j
ω
19
.
<
21
. ,
0.525
πωπ
≤
≤
The ideal frequency response
H
d
(
e
j
ω
) is given by
⎧
1, 0
≤≤
ω
0.25
π
⎪
⎪
j
()
ω
He
d
=
0, 0.25
πω
≤≤
05
.
π
⎨
⎪
⎪
2, 0.
5
πωπ
≤≤
⎩
i.
What is the maximum value of
that can be used to meet this
specification? What is the corresponding value of
δ
β
?
ii.
What is the maximum value of
that can be used to meet this
specification? What is the corresponding value of
M
?
∆ω
