Calculate the Fourier series approximation for f(t)

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Mathematics

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Question 1: A function f(t) is defined as, f(t) = 𝜋 − 𝑡 for 0 < t < 𝜋 Write down the even extension of f(t) for −𝜋 < 𝑡 < 0. Determine the Fourier cosine series, and hence, calculate the Fourier series approximation for f(t) up to the 5th harmonics when t = 1.08. Use 𝜋 = 3.42. Give your answers to 3 decimal places Question 2: A function f(t) is defined as, f(t) = 𝜋 − 𝑡 for 0 < t < 𝜋 Write down the odd extension of f(t) for −𝜋 < 𝑡 < 0. Determine the Fourier sine series, and hence, calculate the Fourier series approximation for f(t) up to the 3rd harmonics when t = 0.6. Use 𝜋 = 3.42. Give your answers to 3 decimal places
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Explanation & Answer

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The Fourier series




𝑛=1

𝑛=1

𝑎0
+ ∑ 𝑎𝑛 cos 𝑛𝑡 + ∑ 𝑏𝑛 sin 𝑛𝑡
2
of a function 𝑔 defined on [−𝜋, 𝜋] has the coefficients
1 𝜋
1 𝜋
1 𝜋
𝑎0 = ∫ 𝑓(𝑡)𝑑𝑡 , 𝑎𝑛 = ∫ 𝑓(𝑡) cos 𝑛𝑡 𝑑𝑡 , 𝑏𝑛 = ∫ 𝑓(𝑡) sin 𝑛𝑡 𝑑𝑡 , 𝑛 ≥ 1.
𝜋 −𝜋
𝜋 −𝜋
𝜋 −𝜋

We are given the function 𝑓(𝑡) = 𝜋 − 𝑡, 0 < 𝑡 < 𝜋 defined on (0. 𝜋).

1. If we perform the even extension of 𝑓, the resulting function will be even and defined on (−𝜋, 𝜋).
When computing the coefficients for that even function, all 𝑏𝑛 become zeros while 𝑎𝑛 will be
𝑎0 =
2

𝜋

2 𝜋
2 𝜋
∫ 𝑓(𝑡)𝑑𝑡 , 𝑎𝑛 = ∫ 𝑓(𝑡) cos 𝑛𝑡 𝑑𝑡.
𝜋 0
𝜋 0
2

𝜋

1

Compute them: 𝑎0 = 𝜋 ∫0 (𝜋 − 𝑡)𝑑𝑡 = 𝜋 (𝜋𝑡 − 2 𝑡 2 )

2

𝑡=0

1

= 𝜋 (𝜋 2 − 2 𝜋 2 ) = 𝜋.

𝜋

𝜋
2
2 𝜋
(𝜋
𝑎𝑛 = ∫
− 𝑡) cos 𝑛𝑡 𝑑𝑡 = 2 ∫ cos 𝑛𝑡 𝑑𝑡 ...

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