Use MATHLAB to solve the questions in the doc.

UNIVERSITY COLLEGE DUBLIN
SCHOOL OF ELECTRICAL & ELECTRONIC ENG.
EEEN30110 SIGNALS AND SYSTEMS
LAB 3 – FIRST GRADED LAB FOR OFFERING 1
Instructions and suggestions
See the document “2019_Regulations_and_grading” on Brightspace for additional information
about grading and submission. Here a few key instructions are reported for your convenience:
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•
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The report you are going to produce on this lab activity is individual;
The deadline for the submission of his/her report for this lab is 18/OCT/19 (no later
than noon on that day);
Each student will have to submit the hard copy of your report to the School Office;
The soft copy (.pdf files only) needs to be sent (by the above deadline) to
giovanni.russo1@ucd.ie;
The front copy of your report needs to include a signed declaration that the work is your
own and that you have read and understood the University policy on plagiarism;
See below for screenshots from where you can download the “Ownership of Work”
form you will need for the front page:
•
Each problem contains some parameters that depend on your student number. Your
student number consists of 8 digits. “Digit A” from your student number is the most
rightward digit, while “Digit H” is the most leftward. See the example below:
Student Number Example
Digit ID
•
•
•
•
8
H
7
G
6
F
5
E
4
D
3
C
2
B
1
A
So, for example, where you see the statement “the parameter a in the above equation is
digit A of your student number”, you will have to replace a with the right digit. In the
example above, digit A is 1;
The grade steps associated to each problem are stated in bold in the problem text;
Remember to use the MATLAB command stem for discrete-time signals and plot for
continuous time signals;
It is up to the student to determine what is the best way for presenting the results. The
grades for each problem will depend on how you explained your conclusions and on
the level of details you included. For example, all the steps that you followed to obtain
a given result need to be properly documented in the report.
Problem sets
Problem 1. Plot, using MATLAB, the following signals (one grade step):
1. 𝑥[𝑛] = 𝛿 [𝑛] − 𝛿 [𝑛 − 𝑎]
2. 𝑥(𝑡) = 𝑢(𝑡 + 𝑎) − 𝑢(𝑡 − 𝑎)
Where: the parameter “a” in the above equation is Digit A of your student number (if this is 0,
then a = 5).
Problem 2. Assume that the impulse response of a discrete-time system is given by:
0 3
ℎ[𝑛] = /1 2 𝑢[𝑛] + 𝑥[𝑛].
Where the parameter “a” in the above equation is Digit A of your student number (if this is 0,
then a = 5) and where x[n] is the signal in item 1 of the previous problem. Answer the following
(two grade steps):
1. Is the system causal? Motivate your question and plot h[n] using MATLAB;
2. Assume that 𝑥[𝑛]=0 and find, using MATLAB, the output of the system (i.e. y[n])
when the input is a step function. Carefully describe the procedure you followed in
MATLAB and how you came to the result.
Problem 3. Consider the signal:
𝑥(𝑡) = 𝑒 516 𝑢(𝑡),
where the parameter “a” in the above equation is Digit A of your student number (if this is 0,
then a = 5). Answer the following (two grade steps):
1. Analytically compute the energy, i.e. 𝐸9 of the signal;
2. Analytically compute the power of the signal.
Problem 4. Consider the system model (as usual, y(t) is the system output and x(t) the input):
𝑑
𝑦(𝑡) + sin(𝑎)𝑦 ? (𝑡) = 𝑥(𝑡),
𝑑𝑡
where the parameter “a” in the above equation is Digit A of your student number (if this is 0,
then a = 5). Answer the following (two grade steps):
1. Is the system linear? Carefully describe how you reached your conclusion.
2. Is the system time-invariant? Carefully describe how you reached your conclusion.
Problem 5. Consider the circuit below:
A simple high-pass filter
Signals and Systems
2
Chap. 1
R
c
Figure 1.2 An automobile responding to an
applied force t from the engine and to a retarding
frictional
force pv
proportionalof
to the
output
is the
voltage
the
automobile’s velocity v.
A simple RC circuit with source
Figure 1. 1
If we consider the outputvoltage
beingVs the
voltagevoltage
on R:Vc.
and capacitor
that: (i) the input is Vs(t) and 𝑉A (𝑡) = 𝑒 BC6 ;
Assume
(ii) the
dVr (t)
dVfor
s (t))the system is given by:
resistance, Vr(t). In this case, it is known
that
the
model
RC
+ Vr (t) = RC
dt
AAACH3icbVDLSgMxFM34rPVVdekmWISKUGZE1I1Q7MZlFfuAtgyZTKYNzWSG5I5QhvkTN/6KGxeKiLv+jekD0dYDgZNzziW5x4sF12DbI2tpeWV1bT23kd/c2t7ZLeztN3SUKMrqNBKRanlEM8ElqwMHwVqxYiT0BGt6g+rYbz4ypXkkH2AYs25IepIHnBIwklu4uK92AkVo6jdcVYKTLPUhw6d4esPX+Mc3kjbSNOEWinbZngAvEmdGimiGmlv46vgRTUImgQqidduxY+imRAGngmX5TqJZTOiA9FjbUElCprvpZL8MHxvFx0GkzJGAJ+rviZSEWg9DzyRDAn09743F/7x2AsFVN+UyToBJOn0oSASGCI/Lwj5XjIIYGkKo4uavmPaJqQNMpXlTgjO/8iJpnJUdu+zcnRcrN7M6cugQHaESctAlqqBbVEN1RNETekFv6N16tl6tD+tzGl2yZjMH6A+s0TcrQKEv
dt
F
F
𝑅𝐶 F6 𝑉G (𝑡) + 𝑉G (𝑡) = 𝑅𝐶 F6 𝑉A (𝑡).
Then:
j!RC
1 + j!RC
G(j!) =
I
I
I
.!_ _____ 1_ _ _ _ _ !._ _____1 _ _ _ _ _ _ _ _ _ _
1_ _ _ _ _
I
_ _ _ _ _ I_ _ _ _ _ J
Moreover: (i) the parameter
“R”
is Digit
A of your student number (if
j
sh in the above equation
oul
d
this is 0, then R = 1); (ii) “C” is equal to 1. Answer the following (three grade steps):
AAACGHicbZDLSgMxFIYz9VbrbdSlm2ARKkKdEUE3QrELXVaxF+iUkkkzbdpkMiQZoQx9DDe+ihsXirjtzrcxbQfU1h8CP985h5Pz+xGjSjvOl5VZWl5ZXcuu5zY2t7Z37N29mhKxxKSKBROy4SNFGA1JVVPNSCOSBHGfkbo/KE/q9UciFRXhgx5GpMVRN6QBxUgb1LZPbwp9T3DSRcfwCnqBRDhJAbwvjxIXnsA+/AFtO+8UnangonFTkwepKm177HUEjjkJNWZIqabrRLqVIKkpZmSU82JFIoQHqEuaxoaIE9VKpoeN4JEhHRgIaV6o4ZT+nkgQV2rIfdPJke6p+doE/ldrxjq4bCU0jGJNQjxbFMQMagEnKcEOlQRrNjQGYUnNXyHuIROONlnmTAju/MmLpnZWdJ2ie3eeL12ncWTBATgEBeCCC1ACt6ACqgCDJ/AC3sC79Wy9Wh/W56w1Y6Uz++CPrPE3VUqeEQ==
G. Russo
Hamilton Institute
r – – – I
EEEN 30110 – Signals and Systems
– – -Sec.
– I -4.3
– – – I -Properties
– – – – of
– -the
– I Continuous-Time
– – – – – – – IFourier
– – – –
1. Compute the frequency response of the system;
2. Plot the magnitude
and phase of the frequency response.
Sec. 4.3
Properties of the Continuous-Time Fourier Transform
I
I
w
Example
I
I
I
I
I
I
303
Example of a recording of speech. [Adapted from Apof
I
I
1 I 2 plications
3
4 Digital Signal Processing, A.V. Oppenheim, ed. (Englewood
(a)
𝑥(𝑡) = 𝑥0 (𝑡 −1 𝑎)2 +3 𝑥?4 (𝑡 − 2𝑎),
Cliffs, N.J.: Prentice-Hall, Inc., 1978),
I
(a)
p. 121.] The signal represents acous.!_ _____
tic pressure variations as a function
ch
a
of timenumber
for the spoken
above signal
be seen
as: equation is Digit A of your student
the The
parameter
“a” can
in the
above
(ifwords
this”should
is 0,
we chase.” The top line of the figure
= 5). The signals x1(t) and x12(t) 2are
given
below:
corresponds
to
the
word
“should,”
_1
1
1_1 3 1 4
2
x(t) = x2 1(b)(t2 2.5) + x2 (t 2.5)2
the second line to the word “we,”
(a)
2
(b)
and the last two lines to the word
“chase.” (We have indicated the apI
I
I
I
proximate beginnings and endings
1_ _ _ _ _
_ _ _ _ _ 1_ _ _ _ _
_ _ _ _ _1 _ _ _ _ _ I_ _ _ _ _
_ _ _ _ _ 1_ _ _ _ _ J
I
of each successive sound in each
-1.
1.
a2
se
2
I
word.)
(c)
_ _ _ _ _1 _ _ _ _ _
____
Figure 1.3
303
– – – – I – – – – I – – – – –
r – – – –
where
then a
Lecture 8
303
e
Problem
6. Consider
the signal:Fourier Transform
Sec. 4.3
Properties
of the Continuous-Time
Transform
– – – – I – – – –
– – – – I – – – –
_ _ _ _ _ 1_ _ _ _ _
I _____
_____I
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I
_1
I
I
I
I
-1.
1
1.
2
2
2Figureusing
24. 1 5 aDecomposing
a signaltointosense
the linear
combination of
simmicrophone
variations
intwoacoustic
pressure, which are then converted into
(c)
pler(b)
signals. (a) The signal x(t) for Example 4.9; (b) and (c) the two compoan electrical
signal.
nent signals
used to represent
x(t). As can be seen in the figure,
Answer the
different sounds correspond to different
patterns2 in
the variations
of1acoustic
human
vocal
system produces
intelFigure 4.
5 Decomposing
a and
signaltheinto
the linear
combination
of two simsin(!/2)
2 pressure,
sin(3!/2)
X1grade
(j!)
= steps):
, signals.
X2 (j!)
= signalsequences
ligible
speech bypler
generating
particular
of these
Alternatively,
for the
(a)
The
x(t) for Example
4.9;patterns.
(b) and (c)
the two compofollowing4.3.3
(three
! picture,
! 1.4,
Conjugation
and Conjugate
Symmetry
nent
signals
usedintoFigure
represent
x(t).
monochromatic
shown
it is the pattern of variations in brightness
The conjugation
property
states
thatimage,
if
-1.
1.the
across
that is important.
I
I
I
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2
1. Analytically compute the
Fourier
transform
X(jw) for x(t);
(c)
x(t)
X(jw ),
G. Russo
EEEN 30110 – Signals and Systems
Institute
2. Plot, inHamilton
MATLAB,
the signal
and the modulus
of X(jw).
4.3.3 x(t)
Conjugation
and Conjugate
Symmetry
then
2
Decomposing a signal into the linear combination of two simpler signals. (a) The signal x(t) for
4.9; (b)property
and (c) the
two that
compoTheExample
conjugation
states
if
x * (t)
X * (- jw ).
nent signals used to represent x(t).
Lecture 10
Figure 4. 1 5
Problem 7. Consider a system characterized by the transfer function:
(4.28)
X(jw ),
This property follows from the evaluation of the complex conjugate of eq.x(t)
(4.25):
[r:
then
X'(jw) =
4.3.3 Conjugation and Conjugate Symmetry
𝐻(𝑗𝑤) =
The conjugation property states that if
(𝑗𝑤)?
=
f
+oo
-oo
(𝑗𝑤) + 2
(𝑎 + 𝑏)(𝑗𝑤)* + 𝑎 ∙ 𝑏*
+
x*(t)ejwt dt.
x(t)e-iwt dtr
X (- jw ).
x (t)
(4.28)
X(jw
Replacing wx(t)
by -w, we see
that ),
Where: (i) the parameter “a” in the
isthe
Digit
A ofofyour
student
number
this is
Thisabove
propertyequation
follows from
evaluation
the complex
conjugate
of eq.(if
(4.25):
X*(- jw) =
x*(t)e- jwt dt.
(4.29)
then
f
0, then a = 5); (ii) the parameter “b” in the above equation is Digit B of your student number
x(t)e-iwt dtr
(if this is 0, then b = 3). Answer
the* following (three X'(jw)
grade=steps):
*
+oo
-oo
X (- jw ).
x (t)
[r:
(4.28)
+oo
=
x*(t)ejwt dt.
Compute
the partial-fraction
expansion
for of𝐻(𝑗𝑤)
analytically;
This1.property
follows from
the evaluation of the complex
conjugate
eq. (4.25):
2. Use MATLAB to verify your answer, describing which command you used;
Replacing w by -w, we see that
3. Plot, using MATLAB,
X'(jw) = the response
x(t)e-iwt dtrto the system to the impulse;
X*(- to
jw)the
= input
x*(t)ejwt dt. 𝑥(𝑡) = sin (2𝜋𝑡).
(4.29)
4. Plot, using MATLAB, the response to the system
signal
f
-oo
[r:
=
f
-oo
f
+oo
-oo
+oo
x*(t)ejwt dt.
Replacing w by -w, we see that
X*(- jw) =
f
+oo
-oo
x*(t)e- jwt dt.
(4.29)
Problem 8. Consider the periodic signal having period of 2𝜋:
𝑓(𝑡) = Q
0
𝑎𝑡
−𝜋 < 𝑡 < 0 0