MCEN90032 Final Exam 2024

Page 1 of 11

MCEN90032 Sensor Systems

MCEN90032 Final Exam 2024

Page 2 of 11

MCEN90032 Final Exam 2024

Page 3 of 11

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)

MCEN90032 Final Exam 2024

Page 4 of 11

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.

MCEN90032 Final Exam 2024

Page 5 of 11

(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.

MCEN90032 Final Exam 2024

Page 6 of 11

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

MCEN90032 Final Exam 2024

Page 7 of 11

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.

MCEN90032 Final Exam 2024

Page 8 of 11

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

MCEN90032 Final Exam 2024

Page 9 of 11

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.

MCEN90032 Final Exam 2024

Page 10 of 11

(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

MCEN90032 Final Exam 2024

Page 11 of 11

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.

51作业君版权所有