ELEC 212: Digital Signal Processing (2009 Fall Semester)

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Nov 24, 2013 (4 years and 7 months ago)

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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

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