EE53053 – Signals, Systems and Signal Processing Problem 1: Consider an RLC circuit whose differential equation from voltage input to charge output can be written as: 𝑦̈(𝑡) + 10𝑦̇(𝑡) + 1500𝑦(𝑡) = 1000𝑥(𝑡). Provide a well-commented MATLAB code and complete the following tasks:

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

EE53053 – Signals, Systems and Signal Processing

Assignment 2 (30% of the overall course grade)

Due 22/11/2021 before 12 noon

• This assignment is worth 30% of your overall course grade.

• Use the MS WORD equivalent of justified text format, single-line spacing, 12 pt

Calibri font and 2 cm margin all over.

• Solutions must not exceed 15 A4 sides excluding the Appendix.

• Submit typed answers. Handwritten / scanned solutions will not be graded.

• For full credit, show all the intermediate steps and relevant work. Only

reporting the answers will result in an automatic penalty of 80% of the overall

grade.

• Include the full, well-commented MATLAB code for each of the three problems

in the Appendix under headings: Code for Problem 1, Code for Problem 2 and

Code for Problem 3. Do not embed pieces of code in the main document.

• Generate all figures using MATLAB. Use appropriate legends and labels for each

figure. To ensure that the figures are sized properly, follow the procedure given

below:

o On the figure window go File Export Setup. Then set:

Size = 14 (width), 12 (height), units centimetres. And tick Expand

axes to fill figure.

Fonts Use fixed font size 12, Custom name Times New

Roman

Lines Use fixed line width 1.5

o Then hit Apply to Figure

o Then Export and save as .fig file.

o Then insert this figure (not the file) into your WORD doc. Once included

in the Word document, do not resize. If you are using LaTeX, export as

eps or pdf and then include in your document.

• Attach a signed Plagiarism Sheet with your submission. Failing to do so or

adhere to plagiarism rules as set by the University will lead to appropriate

penalties.

• A single PDF file must be uploaded on MyAberdeen via the relevant link. The

file should be named FirstNameInitialLastName_sub2.pdf. Thus, John Doe’s

submission will be JDoe_sub2.pdf.

Problem 1: Consider an RLC circuit whose differential equation from voltage input to

charge output can be written as: 𝑦̈(𝑡) + 10𝑦̇(𝑡) + 1500𝑦(𝑡) = 1000𝑥(𝑡). Provide a

well-commented MATLAB code and complete the following tasks:

i. Generate a Bode plot of the system from 1 Hz to 100 Hz with a resolution of

0.1 Hz. Magnitude responses are plotted in dB vs Hz and Phase responses in

degrees vs Hz. Clearly annotate the resonant frequency on the magnitude

response plot and the phase at this frequency in the phase response plot

using MATLAB Data tips Figure 1.i.

ii. Compute the poles and zeros of this system by computing the appropriate

roots and plot the Pole-Zero Map of the system in MATLAB to verify

Figure 1.ii.

iii. If this circuit is given the following inputs:

a. 1 V step at 1 second for a period of 9 seconds (subfigure 1.iii.a)

b. 10 rad/s, 2 𝑉pk-pk sine wave input for 10 seconds (subfigure 1.iii.b)

c. 38.4 rad/s, 2 𝑉pk-pk sine wave input for 10 seconds (subfigure 1.iii.c)

d. 100 rad/s, 2 𝑉pk-pk sine wave input for 10 seconds (subfigure 1.iii.d)

Plot the four input-output pairs in the same figure Figure 1.iii. Explain the

output obtained based on the system’s frequency response.

iv. Briefly explain what you see in the figures and how it aligns with the

underlying theory. Finally, comment on what you learned via this exercise.

[10 marks each = 40 marks total]

Problem 2: Write a well-commented MATLAB code to generate:

i. A discrete-time version of the continuous-time signal

𝑥(𝑡) = 2 sin(50𝜋𝑡) + 1.5sin(2500𝜋𝑡) − 3cos(6000𝜋𝑡)

sampled at 20 kHz, for a duration of 1 second. Call it 𝑥[𝑛].

ii. A plot of the signal for two full periods Figure 2.ii.

iii. A plot of the amplitude spectrum of the signal (dB vs Hz) Figure 2.iii.

iv. The design of a Butterworth filter that:

a. Reduces the amplitude of the 3 kHz component in 𝑥(𝑡) by at least 50 dB.

while not reducing the amplitude of the 1.25 kHz component by more

than 2 dB.

b. Plot the magnitude response of the designed filter clearly showing the

design parameters Figure 2.iv.c

c. Plot the pole-zero map of the designed filter Figure 2.iv.d

d. Give details of your filter (order, poles, zeros, stability etc)

v. A plot that superimposes the amplitude spectrum of the filtered signal

(name it 𝑥𝑓

(𝑡) or 𝑥𝑓[𝑛]), over the one plotted in Figure 2.iii

vi. From everything done in Problem 2 (i – v), derive the mathematical

expression for the filtered signal (𝑥𝑓

(𝑡) or 𝑥𝑓[𝑛]) as a combination of sine

waves with appropriate amplitudes and phases.

[10 marks each = 60 marks total]

Built-in MATLAB functions / commands that you might find useful:

abs, real, imag, filt, lsim, filter, tf, tfdata, freqresp, fft, c2d, bode, bodeplot, pole, zero,

pzmap, axis, xlabel, ylabel, legend, figure, subplot, log10, max, min, grid, sin, cos, step,

impulse, linspace.

Good Luck!

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