Science:Math Exam Resources/Courses/MATH307/April 2012/Question 06 (a)

MATH307 April 2012

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Question 06 (a)
A continuous measurement of the temperature in Vancouver, y(t), for $0\leq t\leq T$, is decomposed as a Fourier series of the form $y(t)=\sum _{n=-\infty }^{\infty }c_{n}e^{2\pi i\omega _{n}t}$ where $\omega _{n}=n/T$. What important property do the set of functions $\displaystyle \{e^{2\pi i\omega t}\}$ have? What is the expression for the Fourier coefficients cn in terms of y(t)?

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Hint

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. For the first half of the question, the set of functions $\{e^{2\pi i\omega _{n}t}\}$ form an orthonormal basis for the space $L^{2}([0,T])$. Using Euler’s formula we can express $e^{2\pi iw_{n}t}$ as $\displaystyle \cos(2\pi w_{n}t)+i\sin(2\pi w_{n}t)$ So the temperature function y(t) can be viewed as a wave-like function with period T decomposed into a combination of sinusoidal waves. The Fourier series represents it as a purely oscillatory components with frequency $\omega _{n}$. Here the phase is 0. For the Fourier coefficients $c_{n}$, since we have y(t) = $\sum _{n=-\propto }^{\propto }c_{n}e^{2\pi iw_{n}t}$ then we an wirte $c_{n}$ as $c_{n}={\frac {1}{T-0}}\int _{0}^{T}e^{-2\pi i\omega _{n}t}y(t)dt={\frac {1}{T}}\int _{0}^{T}e^{-2\pi \ i{\frac {n}{T}}t}y(t)dt$

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