Higher Diploma in Mechanical
Engineering
Higher Diploma in Engineering
Technology
A202ET
Engineering Mathematics 2
by: Alfred Tan {BEng(Civil), MSc(WES), BSc(Maths)}
email: alfredtan.ct@auston.edu.sg
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TOPIC 3
FOURIER SERIES &
TRANSFORM
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Syllabus
Topic 1a -- Laplace Transform & Inverse Laplace
Topic 1b – z Transform & Inverse z
Topic 2 – Systems of Linear Differential Equations
Topic 2b -- Intro to Control Theory
Topic 3a – Fourier Series
Topic 3b -- Fourier Transforms
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TOPIC 3
FOURIER SERIES
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Fourier Series(FS)- Intro
• Invented by Joseph Fourier(1768-1830)
• While considering T(temperature)=T(x,y,z,t),over space and
(t)ime, he encountered partial differential equations(pdes)
• Solving these pdes, he found it necessary to express
temperature(T) as an infinite series of sines and cosines of
the form: Hence the birth of FS. But first we need to
understand periodic functions.
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Periodic Functions
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Periodic Functions-sinusoidal periodic functions
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Periodic Functions: Examples of non-sinusoidal
periodic functions- square wave
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Periodic Functions: Examples of non-sinuisodal
periodic functions- saw-tooth wave
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Periodic Functions: Examples of non-sinusoidal
periodic functions- Triangular wave
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Fourier Series for f(x) with a period 𝑃 = 2𝜋
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Periodic Functions: Odd and Even functions
• A function is even if:
𝑓 −𝑡 = 𝑓 𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑡.
eg 𝐹𝑜𝑟 𝑎𝑛𝑦 𝑘, cos 𝑘𝑡 𝑖𝑠 𝑒𝑣𝑒𝑛
• A function is odd if:
𝑓 −𝑡 = −𝑓 𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑡.
eg 𝐹𝑜𝑟 𝑎𝑛𝑦 𝑘 ≠ 0 , sin 𝑘𝑡 𝑖𝑠 𝑜𝑑𝑑
For periodic functions, we consider over the period ie [
−𝑟 +𝑟
𝟐
,
𝟐
]
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Example 1: Half rectified square wave
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Example 1: Half rectified square wave (contd)
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Example 1: Half rectified square wave (contd)
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Example 1: Half rectified square wave (contd)
n=10
n=100
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Fourier Series for f(x) with a period 𝑃 = 2𝐿 ie
replace 𝜋 by L
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Example 2 – Period 𝑟 = 2𝑡0(Step function)
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Example 2 – Period 𝑟 = 2𝑡0 (contd)
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Example 2 – Period 𝑟 = 2𝑡0 (contd)
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Recap Fourier Series
The FS is given by:
where
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• break
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More Examples: Example 3:
𝑥, [−, 𝑙)
f(x)={
𝑓 𝑥 + 2𝑙 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
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More Examples: Example 3 (contd)
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More Examples: Example 3 (contd)
n=10
n=100
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• break2
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Sine and Cosine Functions- intro
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Sine and Cosine Functions- intro (contd)
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Sine and Cosine Functions- intro (contd)
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Complex Fourier series: intro for info
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TOPIC 3b
FOURIER TRANSFORM
(intro)
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Fourier Transform –intro for info
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Fourier Transform(2) –intro for info
Compared with Laplace Transform
end Topic 6
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Example e – Fourier Transform example
end Topic 3
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