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# ECE 466 Homework 5

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ECE 466 Homework 5

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import numpy as np
import matplotlib.pyplot as plt
from scipy import signal
1:

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ECE 466 Homework 5

You can collaborate on homeworks and turn in a homework for 2 people. Make sure both of you submit.
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:
Your Name: [Name Surname]

import numpy as np
import matplotlib.pyplot as plt
from scipy import signal
1:
 Use the one-sided z-transform to determine
y
[
n
]
,
n

0
for the following difference equations. Verify your answers scipy.signal.lfilter function with the initial conditions. Compare your results for the first 3 time points.

y
[
n
]

1.5
y
[
n

1
]
+
0.5
y
[
n

2
]
=
0
,

y
[

1
]
=
1
,

y
[

2
]
=
0.
y
[
n
]
=
0.5
y
[
n

1
]
+
x
[
n
]
,

x
[
n
]
=
(
1
/
3
)
n
u
[
n
]
,

y
[

1
]
=
1.

# Your code for 1.1 should be written and executed in this cell.

# Your code for 1.2 should be written and executed in this cell.
2:
[10 points] Let
x
[
n
]
be a causal sequence.

What conclusion can you draw about the value of its
z
-transform
X
(
z
)
at
z
=

?
Use the result in the first part to check which of the following transforms cannot be associated with a causal sequence.

a)
X
(
z
)
=
(
z

0.5
)
4
(
z

1
3
)
3
.
b)
X
(
z
)
=
(
1

1
2
z

1
)
2
(
1

1
3
z

1
)
.
c)
X
(
z
)
=
(
z

1
3
)
2
(
z

1
2
)
3
.

3:
[20 points] Compute the
z
-transform of the following signals and specify their ROC. Determine whether the Fourier transform exists and if it does determine the Fourier transform.

x
[
n
]
=
2
n
u
[
n
]
.
x
[
n
]
=
u
[
n
]

u
[
n

6
]
.
x
[
n
]
=
a
n
sin
(
ω
n
)
u
[
n
]
,

|
a
|
<
1.
x
[
n
]
=

2
δ
[
n
+
2
]

δ
[
n
+
1
]
+
δ
[
n

1
]
+
2
δ
[
n

2
]
.

4:
[20 points] Consider the following periodic signal with period
N
=
6
:
x
[
n
]
=
{

,
1
,
0
,
1
,
2
,
3

,
2
,
1
,
0
,
1
,

}
Sketch the signal
x
[
n
]
and its magnitude and phase spectra by pen and paper (You can use your tablet if you’d like).
Using the results in part 1, verify Parseval’s relation by computing the power in the time and frequency domains.

5:
[10 points] Determine the signal
x
[
n
]
if its Fourier transform is as given in the figure below: (There is no phase component, the frequency response does not contain any imaginary parts.)

Figure 1

6:
Signal \$x[n]\$

[20 points] Let
X
(
ω
)
denote the Fourier transform of the signal
x
[
n
]
shown above. Perform the following calculations without explicitly evaluating
X
(
ω
)
.

Evaluate ܺ
X
(
0
)
.
Find

X
(
ω
)
.
Evaluate

π
π
X
(
ω
)
δ
ω
.
Determine and sketch the signal whose Fourier transform is
R
e
{
X
(
ω
)
}
.

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