ECE 466 Homework 5

ECE 466 Homework 5


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import numpy as np
import matplotlib.pyplot as plt
from scipy import signal
1:
[20] 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.
Answer for 1.1:

# Your code for 1.1 should be written and executed in this cell.
Answer for 1.2:

# 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
.
Answer for 2:

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
]
.
Answer for 3.1:

Answer for 3.2:

Answer for 3.3:

Answer for 3.4:

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.
Answer for 4.1:

Answer for 4.2:

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

Answer for 5:

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
)
}
.