ELEC 212: Digital Signal Processing (200
9
Fall Semester)
Homework 3
Question 1:
An FIR filter digital filter is required to meet the following specification:
(If you use Matlab to design the filter, please also submit the Matlab code)
Stopband attenua
tion
>40dB
Passband edge frequency
100Hz
Passband ripple
<0.05dB
Transition width
10Hz
Sampling frequency
1024Hz
a.
Select one of the following windows for designing your filter: ‘rectangular’,
‘Bartlett’, ‘Hanning’, ‘Hamming’ and ‘Blackman’. Plot
the log magnitude response
and explain your reasoning for your window selection.
b.
Use ‘Kaiser’ window method to design the filter and plot the log magnitude response.
c.
Use ‘Optimal’ method to design the filter and plot the log magnitude response.
Compare t
he filters designed in part a,b and c.
Question 2:
The figure below shows a finite

length sequence
x
[
n
]. Sketch the sequences:
4
1
))
2
((
]
[
n
x
n
x
3
0
n
and
4
2
))
((
]
[
n
x
n
x
3
0
n
3
4
5
6
0 1 2 3
n
x
[
n
]
Question 3:
Consider the
finite

length sequence:
]
3
[
]
1
[
]
[
2
]
[
n
n
n
n
x
We perform the following operation on this sequence:
(i)
We compute the five

point DFT
X
[
k
].
(ii)
We compute a five

point inverse DFT of
2
]
[
]
[
k
X
k
Y
to obtain a sequence
y
[
n
].
a.
Determine the sequence
y
[
n
] for
n
= 0, 1, 2, 3, 4.
b.
If
N

point DFTs are used in the two

step procedure, how should we choose
N
so
that
]
[
]
[
]
[
n
x
n
x
n
y
for
1
0
N
n
?
Question 4:
A continuous

time
filter with
impulse
response
c
h t
and frequency

response
magnitude
,10
0,10
c
H j
is to be used as the prototype for the design of a discrete

time filter.
T
he resulting discrete

time
system is to be used in the configuration
of Figure
1
to filter the
continuous

time signal
c
x t
.
Figure
1
a)
A
discrete time system with
impulse
response
1
h n
and the system function
1
H z
is
obtained from the prototype
continuous

time system by impulse invariance
with
Td=0.01; i.e.
1
0.01 0.01n
c
h n h
.
P
lot the magnitude of
the
overall effective frequency response
/
eff c c
H j Y j X j
when this discret
e

time system is used in Figure
1
.
b)
Alternatively, suppose that a disc
rete

time system with impulse response
2
h n
and system
function
2
H z
is obtained from the prototype
continuous

time
system
by the bilinear
transformation
with Td=2; i.e.,
1 1
2
(1 )/(1 )

c
s z z
H z H s
Plot th
e magnitude of the overall effective frequency response
eff
H j
when this
discrete

time system is used in Figure
1
.
Question 5:
Consider a
continuous

time lowpass filter
c
H s
with
passband and stopband
specificat
ions
1 1
1 1
c
H j
,
p
,
2
1
c
H j
,
s
.
This filter is transformed to a lowpass discrete

time filter
1
H z
by the transformation
1 1
1
(1 )/(1 )

c
s z z
H z H s
,
A
nd the s
ame
continuous

time filter is transformed to a highpass discrete

time filter by the
transformation
1 1
2
(1 )/(1 )

c
s z z
H z H s
.
a)
D
etermine a relationship between the passband cutoff
frequenc
y
p
of
continuous

time
lowpass filter and
the passband cutoff frequency
1
p
of the discrete

time lowpass filter.
b)
Determine a relationship between the passpand cutoff frequency
p
of
continuous

time
lowpass filter and the passband cutoff frequency
2
p
of the discrete

time highpass filter.
c)
Determine a relationship between the passpand cutoff frequency
1
p
of the discrete

time
lowpass filter and the passband cutoff frequency
2
p
of the
discrete

time highpass filter.
d)
T
he network in Figure
2
depicts an implementation of the discrete

time lowpass filter with
system
function
1
H z
.
The
coefficients A, B, C, and D are real. How should these
coefficients be modified to ob
tain a network that implements the discrete

time high pass filter
with system
function
2
H z
?
Figure
2
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