ECE 466 Homework 3

ECE 466 Homework 3


You have 10 extra points. I will carry your points over the other homeworks as well. So, if you score 105 in this homework, I will add 0.5 points to your total homework score for the whole semester.
You can solve the question by paper and pen and embed an image in the markdown cell. Make sure that the image is legible in the output .html file.
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Include your codes and answers within the cells that are requested. Do not create additional cells.
Upload the .ipynb document and a generated .html in a zip:
I will import the necessary modules for you. Please only use these modules for now.

Your Name: [Name Surname]

import numpy as np
import matplotlib.pyplot as plt
from scipy import signal
1:
[20] Determine the z-transform of the following signals and define the ROC.

x
[
n
]
=
{
3
,
0
,
0
,
0
,
6
–
 
,
1
,
−
4
}
x
[
n
]
=
−
n
(
−
1
)
n
u
(
n
−
1
)
x
[
n
]
=
(
−
1
)
n
cos
(
π
3
n
)
x
[
n
]
=
(
1
2
)
n
(
u
(
n
)
−
u
(
n
−
10
)
)
Your answers for 1. should be written below in this cell.

2:
[20] Consider the recursive discrete-time system described by the difference equation:
y
[
n
]
=
−
a
1
y
[
n
−
1
]
−
a
2
y
[
n
−
2
]
+
b
0
x
[
n
]
,
where 
a
1
=
−
0.8
, 
a
2
=
0.64
, and 
b
0
=
0.866
. (Hint: Check out lfilter, impulse, and convolve)

Write a code to compute and plot the impulse response 
h
[
n
]
 of the system for 
0
≤
n
≤
49
.
Write a code to compute and plot the zero-state step response 
s
[
n
]
 of the system for 
0
≤
n
≤
99
.
Define an FIR system with impulse response 
h
F
I
R
[
n
]
 given by

h
F
I
R
[
n
]
=
{
h
[
n
]
,
0
≤
n
≤
19
0
,
otherwise.
where 
h
[
n
]
 is the impulse response computed in part (1). Write a code to compute and plot its step response.

Compare the results obtained in parts (2) and (3) and explain their similarities and differences.
# Your code for 2.1 should be written and executed in this cell.
# Your code for 2.2 should be written and executed in this cell.
# Your code for 2.3 should be written and executed in this cell.
Your answers for 2.4 should be written below in this cell.

3:
[20] Determine the z-transforms of the following signals and describe the pole-zero patterns and ROC. You can use the properties of z-transform.

x
[
n
]
=
(
n
+
1
)
u
[
n
]
.
Your answers for 3.1 should be written below in this cell.

x
[
n
]
=
{
(
1
3
)
n
,
n
≥
0
(
1
2
)
−
n
,
n
<
0
.
Your answers for 3.2 should be written below in this cell.

x
[
n
]
=
(
1
4
)
n
(
u
[
n
]
−
u
[
n
−
5
]
)
.
Your answers for 3.3 should be written below in this cell.

x
[
n
]
=
(
1
2
)
n
u
[
n
]
∗
2
n
u
[
n
−
1
]
.
Your answers for 3.4 should be written below in this cell.

x
[
n
]
=
n
sin
(
π
2
n
)
u
[
−
n
]
.
Your answers for 3.5 should be written below in this cell.

4:
[20] Given the z-transform pair 
n
2
3
n
u
[
n
]
↔
X
(
z
)
, use the z-transform properties to determine the time domain signals corresponding to the following z transforms:

Y
(
z
)
=
X
(
2
z
)
Your answers for 4.1. should be written below in this cell.

Y
(
z
)
=
z
2
−
z
−
2
2
X
(
z
)
Your answers for 4.2. should be written below in this cell.

Y
(
z
)
=
δ
X
(
z
)
δ
z
Your answers for 4.3. should be written below in this cell.

Y
(
z
)
=
X
(
z
−
1
)
Your answers for 4.4. should be written below in this cell.

5:
[10] Compute the time domain signals with the z-transform described by:

X
(
z
)
=
1
1
−
z
−
1
+
1
2
z
−
2
.
Your answers for 5.1 should be written below in this cell.

X
(
z
)
=
5
z
−
1
(
1
−
2
z
−
1
)
(
3
−
z
−
1
)
.
Your answers for 5.2 should be written below in this cell.

6:
[20] A causal discrete-time system is implemented using the difference equation:
y
[
n
]
=
0.2
y
[
n
−
1
]
+
x
[
n
]
+
0.5
x
[
n
−
1
]
Find the system function 
H
(
z
)
.
Your answers for question 6.1. should be written below in this cell.

Determine the pole-zero patterns using scipy.signal. Comment on the stability of the system.
# Your code for 6.2 should be written and executed in this cell.
Your answers for question 6.2. should be written below in this cell.

What is the impulse response of this system, 
h
[
n
]
? Verify 
h
[
n
]
 using impulse in scipy.signal for the first ten time points.
# Your code for 6.3 should be written and executed in this cell.
Your answers for question 6.3. should be written below in this cell.

Find the zero-state response of the system to the input signal 
x
[
n
]
=
(
1
3
)
n
u
[
n
]
. What is the steady-state response of the system? Verify your answer using lfilter. (Hint: Use z-transforms to solve the problem.)
# Your code for 6.4 should be written and executed in this cell.
Your answers for question 6.4. should be written below in this cell.