MCEN90032 Final Exam 2021
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MCEN90032 Sensor Systems
End of Semester 2 2021Exam Commencement Time:
• This exam starts on 8/6/2021 15:00 and ends on 8/6/2021 18:45 (Melbourne time)
Exam Duration:
• Recommended Reading Time: [15] minutes
• Recommended Writing Time: [180] minutes
Submission Time: [30] minutes to scan and upload
Exam Submission:
• You must upload your examination answer scripts by 8/6/2021 18:45 (Melbourne time) . No
late submission is accepted.
All students must connect to the Zoom meeting.
All exam-related announcements will be made via zoom.
If you experience internet problems, you must inform the examiner via zoom. If you have
Zoom meeting detail:
Topic: Sensor Systems (MCEN90032_2021_SM1)
Time: Jun 8, 2021 03:00 PM Canberra, Melbourne, Sydney
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exam.
Issues & Concerns:
If you run into any issues uploading your responses, inform the lecturer / exam supervising team
immediately within the first 10 minutes of uploading.
Mechanical Engineering
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Exam format and instructions:
• Total marks for this exam are 100
• Attempt all questions.
• All answers must be hand-written and show your own working for each question.
• Write legibly, preferably in blue or black pen
• Ensure your student number is written on each answer page that you upload
• Number each page prior to submission to indicate the order of the pages.
• Responses should be compiled into a single PDF document(NOT AS IMAGES)
Authorised Materials
This is an Open Book exam, which means that you will be allowed to use notes or textbook during
the exam. However, interaction with other persons outside of the teaching staff supervising the exam
is strictly prohibited during the examination and is a breach of the University’s Academic Integrity
Policy. Any evidence of collusion will be investigated as potential academic misconduct. The
Academic Integrity policy can be found at: https://academicintegrity.unimelb.edu.au/
The following materials are permitted:
• Any material loaded onto Canvas as part of the subject content
• Your own notes (printed, hand-written, and digital/electronic)
• Textbooks
• Online books and materials
• Language dictionaries
• Calculators (any model),
• Computers, electronic tablets, blank paper, pens, rulers, etc.
Academic Integrity
Collusion is not allowed under any circumstances. Collusion includes, but is not limited to, talking
to, phoning, emailing, texting or using the internet to communicate with other students. Similarly,
you cannot communicate with any other person via any means about the content of this exam during
the examination time. If another student contacts you during the examination period, please inform
the subject coordinator immediately.
Plagiarism/copying is not allowed under any circumstances. Your answers to the exam must be in
your own words and not directly copied from lecture notes, tutorial materials, the internet or study
notes you have prepared with your friends. You may refer to sources, but answers should be written
in your own words. This also applies to programming (code) related answers. Code must be written
on your own displaying originality in the content.
Any similarity detected between your answers, the answers from other students and/or from the
internet or other sources will be investigated and may result in severe penalties
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Academic Integrity Declaration
By commencing and/or submitting this assessment I agree that I have read and understood the University’s
policy on academic integrity. Links to an external site.
I also agree that:
1. The work I submit will be original and solely my own work (cheating);
2. I will not use any sources without proper acknowledgment or referencing (plagiarism).
3. Where the work I submit is a computer program or code, I will ensure that:
1. any code I have copied is clearly noted by identifying the source of that code at the start of
the program or in a header file or, that comments inline identify the start and end of the
copied code; and
2. any modifications to code sourced from elsewhere will be commented upon to show the
nature of the modification.
There are 5 short questions (8+8+8+8+10) and three long questions (18 +20+20)
Short Questions
1. This question tests your understanding of signals.
1) (4 marks) A continuous-time periodic signal has the following form
() = {
1 − < < 0
0 0 < <
.
Show that the Fourier series for this signal is () =
1
2
−
2
[sin() +
1
3
sin(3) +
1
5
sin(5) +
1
7
sin(7) + ⋯ ]
2) (1 mark) We sample this signal () from 1) with the sampling rate , which can
be found in your excel dataset. Check whether [] is a periodic signal or not. If
[] is periodic signal, compute the periodicity of the signal. If [] is not a
periodic signal, explain the reason.
3) (3 marks) The discrete-time Fourier transform (DTFT) of a signal [] has the
following form
(Ω) = cos2(Ω).
Using inverse DTFT to find the signal []. Hint: you might want to use the
properties of DTFT. Hint: You might need to use the following properties of
DTFT:
[] ⟷ 1
[ − ] ⟷ (Ω)−Ω, if [] ⟷ (Ω)
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You might also need to use the following relations:
cos(2) = 2 cos2() − 1.
cos(Ω) =
1
2
(−Ω + Ω).
2. A filter circuit is given in Figure 1.
Figure 1 A filter circuit
.
1) (2 marks) Obtain the transfer function () =
()
()
of this filter circuit.
2) (2 marks) The values of R, L, and C can be found in your excel dataset. Sketch the
Bode plot of this transfer function in terms of amplitude spectrum and phase
spectrum. You need to label some characteristics: the cut-off frequency and the
slop. (.
3) (1 mark) Determine the type of the filter (lowpass filter, highpass filter, bandpass
filter, and bandstop filter) and provide your explanation.
4) (3 marks). Assume that the input signal () = sin(10), after a sufficiently long
time, compute the output signal ().
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3. Consider a simple mechanical system consisting of a mass, a spring, and a damper as in
Figure 2. The values of M, B, and k can be found in the excel. Here is the mass of the
mechanical system. is the damping coefficient of the damper and is the spring
constant. Let the x is the position of system when there is no extra force.
1) (3 marks) Assume that the initial position is 0, and the initial velocity (0) = ̇(0)
is a random variable satisfying normal distribution (0)~(0, 2), where the
value of can be found in your excel dataset. Compute that (()) and
(()) when () = 0.
2) (2 marks) Compute (( + )()). Is the random signal () stationary?
3) (3 marks) Assume that (0) = 0 , (0) = 0 , and () is a unit white noise.
Compute (2(0)).
4. A potentiometer circuit is shown in Figure 3. The output is voltage from the load
resistor, which can be denoted as
1) (3 marks) Derive the corresponding displacement–output voltage relation. Here
is corresponding to the position of slider and the length of overall coil is = .
2) (3 marks) Let
= / and
= / . We
can treat the
displacement–
output voltage
relationship as a
nonlinear
mapping
between input
Figure 2 A mechanical system
Figure 3 A potentiometer
()
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=
and the output =
. That is: = (). How to select and such
that this nonlinear function is almost linear in the form as ≈ .
3) (2 marks) The values =
, and =
can be found in your excel dataset.
Linearize this nonlinear mapping at = 0.5.
5. An environment of interest for a vehicle is modelled by a single state x, which can take
on one of four values:
• x1: x is a type 1 target.
• x2: x is a type 2 target.
• x3: x is a type 3 target.
• x4: x is a type 4 target.
Two sensors observe x independently and return four possible values:
• z1: Observation of a type 1 target.
• z2: Observation of a type 2 target.
• z3: Observation of a type 3 target.
• z4: Observation of a type 4 target.
The two sensors are described by the likelihood matrices P1(z1 | x) and P2(z2 | x),
respectively.
P1(z1 | x):
P2(z2 | x):
1) (6 marks) Compute the posterior probability for the combination of sensors if the
first sensor observes first instance of type-1 target and second sensor observes first
instance of type-2 target. What does this posterior probability indicate on
identifying the target type? Assume uniform priors.
2) (4 marks) Determine the conditional entropies for sensor 1 observing first instance
of type-1 target and sensor 2 observing first instance of type-2 target. Which has
higher information content and why? Assume uniform priors.
Long Questions
Long Question 1 (18 marks)
z1 z2 z3 z4
x1 0.3 0.1 0.3 0.3
x2 0.3 0.3 0.3 0.1
x3 0.3 0.3 0.4 0.0
x4 0.1 0.3 0.0 0.6
z1 z2 z3 z4
x1 0.4 0.2 0.2 0.2
x2 0.2 0.6 0.2 0.0
x3 0.3 0.1 0.4 0.2
x4 0.1 0.1 0.2 0.6
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In order to obtain a position of a ship = (, ) in a 2-D plane, four sensors are used to
measure its distances from 4 lighthouses at positions.
1 = (1,, 1,) = (10,0), 2 = (2,, 2,) = (−10,0),
3 = (3,, 3,) = (0,10), 4 = (4,, 4,) = (10,10).
Figure 4 The movement of a ship
The ship was stop at the harbor at (0,0). Then it heads to northeast direction with 45°with a
nominal velocity 2km/hour. After one hour, the measurements from four lighthouses are 1 =
(, 1), 2 = (, 2), 3 = (, 3), 4 = (, 4) can be found in your excel dataset.
The distance measure (1, 2) = √(1, − 2,)
2
+ (1, − 2,)
2
computes the distance
between any two points in 2-D plane with 1 = (1,, 1,) and 2 = (2,, 2,). Here, the unit
of the distance is in kilometer.
We want to estimate the position of this ship by using Kalman filter techniques or extended
Kalman filter techniques. Following steps are used.
1) (4 marks) Using the nominal velocity, represent the system as a discrete-time state
space model, assuming a sampling period of 1 hour. The state of this system is the
position (, ) of this ship. The output of this system is the distance measurement
from four lighthouses. The input of this system can be a constant value. You have
to consider modelling uncertainties and measurement noises in your
discrete-time model. You could fist build the model in continuous time. Then you
can discretize the model using Euler discretization.
2) (2 marks) In order to apply the Kalman filter, a linearization technique is needed.
We linearize the model of the system obtained in 1) around the harbor location.
Obtain the linearized model.
3) (4 marks) Let 1 = (11
) , 1 = (11
) and 0 = ((0 − ̂0)(0 − ̂0)
),
where 0 is a random initial condition and ̂0 is its estimation. Select appropriate
45°
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values of 1, 1, 0 for he linearized model obtained in 2). Justify your choice by
providing detailed explanations.
Hint: Usually, the choice of and depends on the confidence of your model
and your measurements. You might know the relatively accurate position (1, 1).
With this position, you can estimate the distance of this position to four lighthouses.
From the measurements (1, 2, 3, 4)you got, you might roughly know how
accurate your sensors (from each lighthouse) are. This information can provide
some insight on how to choose 1.
4) (5 marks) Implement the Kalman filter using the model obtained in 2) by the
following steps:
Step 1: predict the state ̂1
− and the error covariance matrix1
− . You can use your
own choice of 1 1.
Step 2: Compute the Kalman filter gain 1
Step 3: Compute the estimate ̂1 and 1
5) (3 marks) If an extended Kalman filter (EKF) is used, discuss the difference
between the Kalman filter and the extended Kalman filter for this question. For
example, you might want to discuss how Step 1 in Question 4) will be affected
when an EKF is used.
Long Question 2 (20 marks)
A vehicular ad hoc network consists of groups of stationary vehicles connected by a
wireless network. Vehicle A in this vehicular network can localise itself in two ways: (a) by
using an on-board camera sensor, and (b) by communicating with other vehicles in the network.
1) (4 marks) The camera on-board the vehicle A has a focal length of 0.02 mm. It
images a building by the road, which has its left top corner at (200,100,500) m. Find
the corresponding image plane coordinates for this location assuming that the
camera axis is aligned with the world’s z-axis.
2) (16 marks) Vehicle A communicates with two other vehicles, B and C, in its
network. Vehicle B and C are located at (1, 2) km and (5, 4) km, respectively.
Vehicle A communicates with vehicle B by issuing a beacon signal at 50 second,
which is received by Vehicle B at 60 second (on its local clock).
Vehicle B responds with a beacon issued at 70 second, subsequently received by
vehicle A at 75 second. Similarly, vehicle C issues a beacon signal at 6 second,
received by Vehicle A at 12 second (on its local clock). Assume the beacon
propagation time is 300 m/s. Determine possible locations for vehicle A. Show
detailed steps.
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Long Question 3 (20 marks)
A vehicle plans its travel from Melbourne Cricket Ground (MCG) to the Melbourne
University campus at Parkville. The road network connecting the two locations is sampled
and is represented in the form of a weighted graph given in Figure 5. Vertex A corresponds to
MCG and the university campus corresponds to vertex J. Other vertices represent different
intermediate landmarks in the route and edge weights represent associated cost in traversing a
road segment from one landmark to another. The (x,y) tuple next to a vertex represents its
corresponding approximate physical world location with respect to a fixed coordinate system.
1) (8 marks) Employ Dijkstra’s search algorithm to plan the shortest route for the
vehicle in getting from source A to destination J. Show the shortest route as a
sequence of vertices.
2) (10 marks) Employ A* search algorithm to plan the shortest route for the vehicle in
getting from source A to destination J. Use Manhattan distance as heuristic function
h to estimate the cost of the cheapest path from a vertex (x1, y1) to the goal (x2, y2):
|x1- x2| + |y1- y2|. Show the shortest route as a sequence of vertices.
3) (2 marks) Comment on the number of vertices explored in arriving at the final
shortest route solution in both the cases in (a) and (b).
END OF EXAM
Figure 5 a weighted graph of a given road network