ECE 466 Homework 1
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I will import the necessary modules for you. Please only use these modules for now.
Your Name: [Name Surname]
(Double Click Here to edit. Delete this expression after edit.)
import numpy as np
import matplotlib.pyplot as plt
1:
[18 points] For each
x
[
n
]
given below:
Plot the signal using np.stem() for
−
20
≤
n
≤
20
.
Is the signal an energy signal or a power signal?
Is it periodic or aperiodic? What’s the period if it is periodic?
Is it even or odd?
a.
x
[
n
]
=
(
1.2
)
−
n
sin
(
π
n
4
)
u
[
n
]
.
## Your code for plotting the signal in 1.a. should be in this cell.
Your answers for signal 1.a. should be written in this cell. (Double click on this text)
b.
x
[
n
]
=
sin
(
π
n
)
sin
(
π
n
10
)
.
## Your code for plotting the signal in 1.b. should be in this cell.
Your answers for signal 1.b. should be written in this cell.
c.
x
[
n
]
=
cos
(
π
n
4
)
sin
(
π
n
3
)
.
## Your code for plotting the signal in 1.c. should be in this cell.
Your answers for signal 1.c. should be written below in this cell.
2:
[15] Determine whether the following systems are linear, shift-invariant, memoryless, causal, and BIBO stable:
a.
y
[
n
]
=
x
[
−
n
]
.
b.
y
[
n
]
=
e
x
[
n
]
.
c.
y
[
n
]
=
g
[
n
]
x
[
n
]
with any given
g
[
n
]
.
Your answers for question 2. should be written and executed below in this cell.
3:
[15] The continuous time signal
x
(
t
)
=
sin
(
10
π
t
)
10
π
t
is is the input to an ideal A/D converter with sampling period ܶ
T
, to obtain the discrete-time signal
x
[
n
]
=
sin
(
π
n
)
2
π
n
2
.
a. What is the ideal
T
for such a conversion?
b. Is your choice of
T
unique? If not, what other
T
s can be used?
Your answers for question 3. should be written and executed in this cell.
4:
[20] Consider the following continuous-time sinusoidal signal:
x
a
(
t
)
=
sin
(
2
π
F
0
t
)
.
The sampled version of this signal can be described by it’s values every
T
s
seconds. Namely,
x
[
n
]
=
x
a
(
n
T
s
)
= \sin(2\pi\frac{F}{F_s} n)$.
a. Plot the signals
x
[
n
]
with given
F
0
for
0
≤
n
≤
99
.
F
s
=
5000
=
5
kHz. Explain the similarities and differences between these plots.
F
0
=
500
=
0.5
kHz.
F
0
=
2000
=
2
kHz.
F
0
=
3000
=
3
kHz.
F
0
=
4500
=
4.5
kHz.
# Give the code of your answer for signal with F_0=0.5kHz.
# Give the code of your answer for signal with F_0=2kHz.
# Give the code of your answer for signal with F_0=3kHz.
# Give the code of your answer for signal with F_0=4.5kHz.
What similarities do you observe? What differences?:
b. Suppose that
F
0
=
2
kHz and
F
s
=
50
kHz.
i. Plot
x
[
n
]
. What is the frequency
f
0
of
x
[
n
]
?
ii. Plot the signal
y
[
n
]
=
x
[
2
n
]
. Is this a sinusoidal? What is the frequency?
# Code for 4.b.i.
Type the frequency of
x
[
n
]
here.
# Code for 4.b.ii.
Type the frequency of
y
[
n
]
here.
5:
[20] An analog signal
x
a
(
t
)
=
sin
(
480
π
t
)
+
3
sin
(
720
π
t
)
is sampled 600 times per second.
(a) Determine the Nyquist sampling rate for
x
a
(
t
)
.
Answer
(b) Determine the folding frequency.
Answer
(c) What are the frequencies, in radians, in the resulting discrete time signal
x
[
n
]
?
Answer
(d) If
x
[
n
]
is passed through an ideal D/A converter, what is the reconstructed signal
Y
a
(
t
)
?
Answer
