ECE 466 Homework 6

ECE 466 Homework 6


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import numpy as np
import matplotlib.pyplot as plt
from scipy import signal
1:
[20] Determine if the Fourier Transform 
X
(
ω
)
 of a signal 
x
[
n
]
 is a) even or odd, and b) imarinary or real when 
x
[
n
]
 is:

Real and even.
Imaginary and even.
Real and odd.
Imaginary and odd.
Show your work.

Answer for 1.1:

Answer for 1.2:

Answer for 1.3:

Answer for 1.4:

2:
[25 points] Determine and sketch the Fourier transforms 
X
1
(
ω
)
, 
X
2
(
ω
)
, and 
X
3
(
ω
)
 of the following signals:

x
l
[
n
]
=
{
1
,
1
,
1
–
 
,
1
,
1
}
.
x
2
[
n
]
=
{
1
,
0
,
1
,
0
,
1
–
 
,
0
,
1
,
0
,
1
}
.
x
3
[
n
]
=
{
1
,
0
,
0
,
1
,
0
,
0
,
1
–
 
,
0
,
0
,
1
,
0
,
0
,
1
}
.
Is there any relation between 
X
1
(
ω
)
, 
X
2
(
ω
)
, and 
X
3
(
ω
)
? What is the physical meaning behind the relationship?
Show that if
x
k
[
n
]
=
{
x
[
n
k
]
,
if 
n
/
k
 is integer
,
0
,
otherwise
then
X
k
(
ω
)
=
X
(
k
ω
)
.
Answer for 2.1:

Answer for 2.2:

Answer for 2.3:

Answer for 2.4:

Answer for 2.5:

3:
[15 points] A signal 
x
[
n
]
 has the following Fourier transform:
X
(
ω
)
=
1
1
−
a
e
−
j
ω
Determine the Fourier transforms of the following signals:

x
[
2
n
+
1
]
e
π
n
/
2
x
[
n
+
2
]
x
[
n
]
c
o
s
(
0.3
π
n
)
Answer for 3.1:

Answer for 3.2:

Answer for 3.3:

4:
[40 points] Write a python function to compute the DTFT of a finite duration sequence. The format of the function should be as following:

def dtft(x, n, w):
""" DTFT computed at w

Parameters:
---
    x : finite duration sequence over n
    n : sample position vector
    w : frequency location vector

Returns:
---
    X : DTFT values computed at frequencies w
"""

    return X
Compute the DTFT for the following sequences by using your code. Choose 
ω
 as an array sampled between 
−
π
 and 
π
 uniformly (You can use np.arange for this). Plot the magnitude and angle of 
X
(
ω
)
:

x
[
n
]
=
{
4
–
 
,
3
,
2
,
1
,
2
,
3
,
4
}
. Comment on the angle plot.
x
[
n
]
=
{
1
,
−
N
≤
n
≤
N
,
0
,
otherwise
.
Determine the DTFT for 
N
=
5
,
25
,
100
. Scale the DTFT so that 
|
X
(
0
)
|
=
1.
 Comment on the behavior of the magnitude plot as a function of N.
# Code for question 4
Answer for 4.1:

Answer for 4.2: