ELEC3004: Signals, Systems and Control Sem 1, 2024
Problem Set 1: An Introduction to Signals & Systems
Total marks: 100 Due Date: March 28, 2024 at 16:00 AEST
Note: This assignment is worth 20% of the final course mark. Please submit answers
via Gradescope on Blackboard, including your name and student number. Solutions, in-
cluding equations, should be typed. Explain your solutions as if you are trying to teach
a peer. Demonstrate your insight and understanding. Answering an entire question with
bare equations, lone numbers or without any explanation is not acceptable. Marks may be
reduced if an answer is of poor quality, demonstrates little effort or significant misunder-
standing.
Questions
Question 1. Not So Complex Exponentials (4 marks)
Consider the following complex exponential signal:
x(t) = −4ej( π10 t−π6 )
(a) Determine the even and odd components of the signal. (2 marks)
(b) Is the signal periodic? If periodic, determine the fundamental period and frequency.
(2 marks)
Question 2. System Classification (16 marks)
Consider the following systems where x is the input signal and y is the output signal.
Determine and justify if the following systems are (i) Linear (ii) Causal (iii) Time invariant,
specifying key conditions/assumptions as needed.
(a) y(t) = 3d
2x
dt2
− 2dx
dt
+ x(t) (4 marks)
(b) y(t) = t2x(t) + sin (x (t)) (4 marks)
(c) y[n] = x[n− 2] + 0.5x[n]− x[n+ 2] (4 marks)
Now consider the following continuous-time system with input and output specified sepa-
rately, where u(t) is the unit step function and δ(t) is the Dirac delta function:
x(t) = u(t) + 2u(t− 1)
y(t) = 2u(t) + e−tu(t) + 4u(t− 1) + 2e−tu(t− 1)− 3δ(t)− 6δ(t− 1)
(d) Is this system linear, and why? (4 marks)
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ELEC3004: Signals, Systems and Control Sem 1, 2024
Question 3. Signal Transforms (18 marks)
(a) Find the Fourier transform by integration of a unit gate function with width T , given
by:
rect
(
t
T
)
= Π
(
t
T
)
=
{
1 −T
2
≤ t ≤ T
2
0 otherwise
Plot the magnitude and phase of its spectrum when T = 1. (4 marks)
(b) Write down expressions for the frequency domain representation of the unit impulse,
δ(t), the unit gate function, rect (t), and a constant signal, i.e., x(t) = 1. Plot and
compare the magnitude spectra.
(6 marks)
(c) Now consider a signal x(t) = cos(2πt).
Find the Fourier domain representation when x(t) is multiplied by a gate function,
i.e. the Fourier transform of y(t) = x(t) · rect (t).
Compare the resulting magnitude spectrum, Y (ω), to the un-gated magnitude spec-
trum, X(ω). What is the relation between the spectrum of Y (ω) and your answer
from part (a)?
(8 marks)
Question 4. Sampling and Aliasing (17 marks)
Prof. Gineur has built a flashing Light Emitting Diode (LED) circuit powered by a function
generator with the following current waveform:
I(t) = 2 cos(50πt)
The light intensity of the LED is linearly proportional to the current (you can assume a
linear constant of 1), except that the LED does not turn on when the current is negative.
To demonstrate the effect of sampling and aliasing, Prof. Gineur records the LED with
his mobile phone recording at a frame rate of 30 frames per second (fps), and the students
compare the LED light signal visible to their eyes versus that in the resultant video.
(a) Plot the continuous current signal being supplied to the LED, I(t), for t = 0 to 1
second. What is the frequency (Hz) of this waveform, and at what frequency is the
LED flashing? (3 marks)
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ELEC3004: Signals, Systems and Control Sem 1, 2024
(b) Plot the sampled LED light intensity signal as captured by the mobile phone and
compare to the continuous LED light intensity signal, for t = 0 to 1 second, noting
that the LED does not light up for negative values. (4 marks)
(c) What is the perceived frequency (fundamental frequency) of the sampled LED signal?
Justify your answer with plots and/or equations. If using Matlab, include your code
in your answer, and you may make use of the fft and fftshift functions. What
should the minimal sampling rate have been to avoid aliasing (include justification)?
(10 marks)
Question 5. Nyquist Rate (15 marks)
Determine the Nyquist rate and Nyquist interval of the following signals:
(a) a(t) = 2 cos(10πt) + sin(30πt) + 4 sin(20πt) (3 marks)
(b) b(t) = 2 cos(10πt)× sin(20πt) (3 marks)
(c) c(t) = sinc(10πt) (3 marks)
(d) d(t) = sinc2(10πt) (3 marks)
(e) e(t) = 0.1 sinc2(10πt) (3 marks)
Question 6. Sampling and Fourier Series (14 marks)
Consider a real, even, and periodic signal x(t) with Fourier series representation given by:
x(t) =
2∑
k=0
(
1
2
)k
cos(4kπt).
Let xs(t) represent the signal obtained by performing impulse-train sampling on x(t) using
a sampling period of Ts = 1/6 seconds.
(a) Does aliasing occur when this impulse-train sampling is performed on x(t) and why?
(6 marks)
(b) If the sampled signal, xs(t), is reconstructed by passing it through an ideal low-pass
filter with cutoff frequency ωc = 6π, determine the Fourier series representation of
the (reconstructed) output signal. (8 marks)
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ELEC3004: Signals, Systems and Control Sem 1, 2024
Question 7. Transfer Function and Zero State Response of an LTIC system
(16 marks)
Consider a zero-state LTIC system described by the equation
d2y
dt2
+ 6
dy
dt
+ 8y(t) = 2
dx
dt
+ x(t)
(a) Determine the transfer function of the LTIC system. (5 marks)
(b) Using your answer from part (a), determine the impulse response of the LTIC system.
(5 marks)
(c) Using your answer from part (a), find the response of the system to the input signal
x0(t) = 5e
−3tu(t) . (6 marks)