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Homework 1 Sol

The document contains instructions for Homework 1 that is due on March 17th. It includes 3 questions - the first asks about analyzing the periods and pitches of two signals, the second asks about a system and filter transfer function, and the third asks about plotting signal harmonics in Microsoft Excel. The solutions provide the analysis for questions 1 and 2, and guidelines for plotting the signals as requested in question 3.

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Alison Lee
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41 views4 pages

Homework 1 Sol

The document contains instructions for Homework 1 that is due on March 17th. It includes 3 questions - the first asks about analyzing the periods and pitches of two signals, the second asks about a system and filter transfer function, and the third asks about plotting signal harmonics in Microsoft Excel. The solutions provide the analysis for questions 1 and 2, and guidelines for plotting the signals as requested in question 3.

Uploaded by

Alison Lee
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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CORE1240 Homework 1 [20] Deadline: 10 pm on 17 March (Fri)

Please submit ONE soft copy (pdf only) of your assignment to Canvas (Assignments)
before the deadline. NO LATE Submission! (Zero mark!)

1. Consider two signals: [6]

𝑥 = 5 sin(π416𝑡) + sin(π1144𝑡) + 10sin(π1560𝑡)

𝑥 = 20sin(π784t) + 5 sin(π1568t) + 30sin(π2156𝑡)

a) Find the period of x1 and x2.


b) Do they have the same pitch? Explain your answer with justifications.

Solution:
a) The period of x1 is 0.01923 second or 19.23 ms
The period of x2 is 0.01020 second or 10.20 ms

b) No.
Since the fundamental frequency of the two signals is different.
The fundamental frequency of x1 is 52 Hz and x2 is 98 Hz.

2. Consider the following system. [10]

1 of 4
S(f) Y(f)
7 4

3.5
6

3
5

2.5
4
2
3
1.5

2
1

1
0.5

0
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
frequency (kHz) frequency (kHz)

a) Write the mathematical expression of 𝑠(𝑡).


b) What is the fundamental frequency of 𝑠(𝑡).
c) What are the harmonics of S(f) shown in 𝑌(𝑓).
d) If we use a filter H(f) to obtain 𝑌(𝑓), what kind of filter will H(f) be?
e) Draw the filter H(f) and briefly describe the filter with appropriate values.
Solution:
a) 𝑠(𝑡) = 7 sin(2𝜋10000𝑡) + 4 sin(2𝜋15000𝑡) + 5 sin(2𝜋20000𝑡) +
3 sin(2𝜋30000𝑡) + 2sin(2𝜋35000𝑡)

b) 5000 𝐻𝑧 𝑜𝑟 5𝑘𝐻𝑧

c) 3rd and 4th harmonics

d) A bandpass filter

e) The low cutoff frequency must be > 10𝑘𝐻𝑧 and < 15𝑘𝐻𝑧. The high
cutoff must be > 20 𝑘𝐻𝑧 and < 30 𝑘𝐻𝑧. The amplitude (response) at 15 kHz
must be 1 and at 20 kHz must be 0.5. There are many solutions.

2 of 4
3. Microsoft Excel Question: [4]

A signal is a sum of its first 4 harmonics (first to fourth harmonic). The graph below is
showing its 4th harmonic of this signal. The amplitudes of the first harmonic, 2nd harmonic
and the 3rd harmonic are 4, 3 and 2 respectively (Note: The amplitude of the 4th harmonic
is given in the plot below).
4th harmonic
2

1.5

0.5
Amplitude

-0.5

-1

-1.5

-2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
t(second)

a) Plot the fundamental frequency of this signal ONLY in the time domain and show
TWO completed cycles (periods) only of that signal using Microsoft Excel.
b) Plot the entire signal in the time domain and show TWO completed cycles (periods)
only using Microsoft Excel.

[*Note: guidelines on using EXCEL for plotting the sine function are provided on Canvas]

3 of 4
Solution:

Part a
5
4
3
2
1
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
-1
-2
-3
-4
-5

Part b
10
8
6
4
2
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
-2
-4
-6
-8
-10

4 of 4

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