MCEN90032 Final Exam 2024

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MCEN90032 Sensor Systems


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Note: There are 5 short questions (8+8+8+8+10) and 3 long questions (18+20+20)
Short Questions
Short Question 1 (8 Marks)
Let be a random variable with PDF given by
= 12 1 + cos − ≤ ≤ 0 otherwise
(1) (2 marks) compute
Note: ∫ = ! + cos + "
(2) (4 marks) Let a random signal be given by 3sin % + , determine whether this
random signal is stationary? Explain how whether the signal is stationary or not
would affect the outcome of frequency analysis.
(3) (2 marks) Let the signal & % = 3 sin % be sampled with a sampling period of '(s over
an interval from [0,2 seconds. Compute the 4-point Discrete Fourier Transform
(DFT) of &.
Short Question 2 (8 Marks)
Consider the following linear time-invariant (LTI) dynamic system:
+, = -1 2 02 1 20 0 −1. + + -
001. /
0 = [1 ( 23+
(1) (1 mark) Determine if this system is controllable.
(2) (2 marks) Is it possible to find a set of parameters {1, (, 2} such that the system is
observable? If it is possible, find such a set. If it is not possible, explain your result.
(3) (2 marks) Determine if an open-loop observer could work for the system above.
(4) (3 marks) Consider the following revised LTI dynamic system:
+, = -1 2 02 1 20 0 1. + + -
001. / 0 = [0 1 13+
design a Luenberger observer with the closed-loop poles located at {−1, −2, −3}
Short Question 3 (8 Marks)
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You are designing an accelerometer-based pedometer, after some investigation you have
identified that there is a bias in your signal such that you will need a high-pass filter to
process the raw acceleration data.
(1) (2 marks) Determine the transfer function and circuit (including a set of resistance
and capacitance values) for a continuous-time first-order RC high-pass filter with a
cut-off frequency of 2Hz. It is assumed that the amplitude of this filter is
approximately 1 when the input frequency is 15 rad/s.
(2) (2 marks) Sketch the approximate Bode plot (amplitude and phase spectrum) of the
filter that you designed, including labels of the slope and cut-off frequency. Using
this filter what would you expect the response to be to the input & % = sin 5%?
(3) (2 marks) Upon seeing the output of your filter, you observe that the filter is allowing
too much noise to pass. You therefore decide to create a band-pass filter using your
existing high pass filter and a low-pass filter. Design a low-pass filter (transfer
function only) such that your final band-pass filter has a pass band between 2 and 5
Hz and show in a block diagram how your overall band-pass filter is obtained.
(4) (2 marks) Your friend proposes to use second-order filters with the same cut-off
frequency. Comment on a pro and a con of using second-order filters for your band
pass filter design. Please note numerical calculations are not required for
evaluation.
Short Question 4 (8 Marks)
The following circuit is a voltage amplifier.

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(1) (1 mark) Compute the relationship between the output voltage :; and input voltage :< for the voltage amplifier.
(2) (2 marks) Compute the sensitivity of this amplifier with respect to = and => provided
that the initial voltage of the output is :? = 5@ and the resistances are given by = =5AΩ and => = 10AΩ.
(3) (2 marks) How should = and => be designed such that the absolute value of the
amplifier gain is 1000?
(4) (3 marks) The following figure shows a filter circuit using an amplifier:

a) (1 mark) Compute the transfer function of this filter.
b) (1 mark) What type of filter is this?
c) (1 mark) How would you measure the sensitivity of this filter?
Short Question 5 (10 Marks)
(1) (5 marks) The speed of an autonomous vehicle traveling on a highway is
represented by a continuous random variable .
a) (2 marks) Initially, the speed is assumed to be xinit = 50 km/h. The uncertainty in this
estimate is modelled by a Normal distribution with a variance of 9 (km/h)2.
Construct the prior probability density function p(x). Outline the steps to compute
the prior probability that the vehicle speed lies between 50 km/h to 60 km/h.
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b) (2 marks) A speed sensor is employed to determine the vehicle speed. The sensor
measurement uncertainty is modelled by a Normal distribution with a variance of 16
(km/h)2. Write the probability density function for sensor measurement model
p(z|x). Explain the steps to compute the probability that the speed measured by the
sensor is in the range of 70 km/h to 80 km/h if the actual speed 75 km / h.
c) (1 mark) Describe how the prior knowledge of vehicle speed can be updated using
information from the speed sensor.
1. (5 marks) Consider a simple example of the application of Bayes’ theorem to
estimate a discrete parameter based on sensor observation and some prior
information. Here, the environment of interest for a vehicle is modelled by a single
state x, which can take on one of three values:
 x1: x is a type 1 target.
 x2: x is a type 2 target.
 x3: x is a type 3 target.
Two sensors observe x independently and return three possible values:
 z1: The observation of a type 1 target.
 z2: The observation of a type 2 target.
 z3: The observation of a type 3 target.
The two sensors are described by the likelihood matrices C1 Z1|x and C( Z(|x,
respectively.

P1(Z1| x):


P2(Z2| x):

z1 z2 z3
x1 0.4 0.3 0.3
x2 0.3 0.3 0.4
x3 0.3 0.4 0.3
z1 z2 z3
x1 0.45 0.1 0.45
x2 0.15 0.7 0.15
x3 0.4 0.2 0.4
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The first sensor observes the first instance of the type 2 target and the second
sensor observes the first instance of the type 2 target. The prior probabilities for
type-1, type-2, and type-3 targets are 0.2,0.4, and 0.4, respectively. Use Bayes’
theorem to determine the posterior probability for this combination and
comment on identifying the target type.



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Long Questions
Long Question 1 (18 Marks)
In this question, we will estimate the angle G and its velocity G, of an inverted pendulum
using a Kalman filter. The inverted pendulum has a mass H = 1AI and a massless rod of
length J = 1H. Our inverted pendulum has been placed in a room that has a motion
tracking system such that the angle of the pendulum is the measured output.

(1) (2 marks) Linearise the inverted pendulum dynamics about the upright equilibrium
point.
(2) (1 mark) Determine if the linearised system is observable.
(3) (2 marks) Discretise the system such that the model is suitable for a Kalman Filter.
In your answer, discuss how to design an appropriate sampling rate for this model.
(4) (3 marks) Explain how you could determine the values of the initial state co-
variance matrix C;, the model uncertainty co-variance matrix @ and the
measurement noise uncertainty matrix K.
(5) (6 marks) Let the initial state co-variance matrix be given by C; = L3 00 3M, the model
uncertainty co-variance matrix be given @ = L0.0025 0.050.05 1 M and the measurement
noise uncertainty matrix be given by K = 1.21. Given the measurements &[03 = 1
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and &[13 = 1, determine the first two values of the estimated state O and the
estimated co-variance CP.
(6) (2 marks) If you decided to shorten the length of the rod, how might this affect the
choice of the model uncertainty co-variance matrix? How might different sampling
rates affect the model uncertainty co-variance matrix?
(7) (2 marks) The inverted pendulum is to be operated at the same time as a controller
tracks a moving reference trajectory. Explain (without numerical computation) how
you might change your Kalman Filter to make it suitable for this scenario.
Long Question 2 (20 marks)
A ground vehicle (GV) is equipped with a pinhole camera that has a focal length of f mm. This
camera enables the vehicle to visualize its surroundings and make navigation decisions
based on the visual information. The figure below shows the geometry of the imaging setup,
with the K-axes of the camera and the world aligned. The imaging plane is positioned at the
principal point (px,py) mm.

(1) (4 marks) Determine the image coordinates (x0, y0) for the world point (X0,Y0,Z0) and
construct the corresponding camera calibration matrix.
(2) (2 marks) Given a world coordinate of (10,20,10) m, a principal point of (10,20) mm,
and a focal length of 100 mm, what would be the corresponding image coordinate
values for x0 and y0?
(3) (6 marks) The GV now moves such that the image plane coordinates captured in (2)
are scaled by (3, 4) and rotated by 450 in counter clockwise direction. Construct a
unified homogeneous transformation matrix and determine the new image plane
coordinates.
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(4) (4 marks) The image below was captured by GV's camera. It shows a sudden jump in
image intensities. What kind of image filtering is typically used to minimize or blur
these abrupt transitions? Select an appropriate 3x3 filter kernel to perform the image
filtering and reduce these sharp changes. You can discard border pixels for filtering.
10 20 30 10
5 15 35 10
10 20 40 55
40 70 90 100

(5) (4 marks) An effective method to summarize image contents is by constructing a
histogram. It provides a visual representation of the distribution of pixel intensities.
Construct a histogram for the image in (4) using a bin width of 10. You can show your
histogram as a table.
Long Question 3 (20 marks)
(1) (4 marks) The GV mentioned in the previous question also employs a radar sensor
that uses radio signals to detect targets in short to mid ranges. When a radio signal
of 100mW is transmitted from the GV, it reaches a truck 200m ahead and returns to
the radar receiver with a power of 10mW. If the truck slows down and its distance to
the vehicle sensor is halved, what should the power of the transmitted signal be to
maintain the same the received signal power of 10mW?
(2) (6 marks) Radar is also used to localize the GV with respect to the two nearby
landmarks. The radio signals sent to landmarks 1 and 2 are received by the radar
antenna after roundtrip delays of 100 ms and 80 ms, respectively. If landmark 1 is
located at (0,0) m and landmark 2 is at (2,0) m, how many unique locations can you
determine for GV with respect to the landmarks. Determine the locations. Assume
that radio signals travel at 100 m/s.
(3) (4 marks) A LiDAR sensor has been added to GV's sensor system. It uses short pulses
to determine the distance to target objects. The light signal is transmitted at 20 ns,
with an azimuth of 450 and an elevation of 600. It reaches the target and returns to the
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receiver after a delay of 70 ns. Determine the location (x,y,z) for the target object,
assuming a signal propagation speed of 100 m/s.
(4) (4 marks) The LiDAR in (3) is now replaced by a continuous wave LiDAR that sends
amplitude-modulated continuous waves. If the target range computed in (3) is the
maximum distance that the LiDAR sensor can measure, what should the signal
modulation frequency be? Assume the signal propagation speed of 100 m/s.
(5) (2 marks) The amplitude modulated waves in (4) are replaced by linearly frequency
modulated (LFM) ramp signals to measure the distances to target objects. If a target
object located at 40 m from the sensor results in a beat frequency of 100 Hz,
determine the slope of the LFM ramp waves. Assume the signal propagation speed
of 100 m/s.







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