Fourier Series Pt 1 - University of Waterloo CivE 331...

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Unformatted text preview: University of Waterloo CivE 331 Compute the Fourier Series expansion of the following periodic function. Plot the expansion for =3, 5, and 10. By inspection, the period of the function is =16 and 8 2 / = = T l . Substitute l into the general expression for the Fourier Series of ( ): ( 29 ∑ ∞ = + + = 1 8 sin 8 cos n n n x n b x n a a x f π π The coefficients (i.e. , , and ) are: ( 29 ( 29 ( 29 ∫ ∫ ∫ +- +- +- = = = = = 8 8 8 8 8 8 ... 3 , 2 , 1 8 sin 8 1 ... 3 , 2 , 1 8 cos 8 1 16 1 n dx x n x f b n dx x n x f a dx x f a n n π π By inspection of the graph, ( ) is piecewise continuous and odd: ( 29 < < < <- < <--- < <- +- < <- = 8 6 ; 6 4 ; 6 4 4 ; 2 4 6 ; 6 6 8 ; x x x x x x x x x f-3-2-1 1 2 3-20-18-16-14-12-10-8-6-4-2 2 4 6 8 10 12 14 16 University of Waterloo CivE 331 Because ( ) is odd, the even components of the Fourier Series (i.e. and ) are zero: = = b a a Now evaluate . The following integral will be useful (and would be provided in an exam or quiz context): ( 29 ( 29 ( 29 a ax x a ax dx ax x cos sin sin 2- = ∫ Begin the evaluation of by substituting for ( ): ( 29 ( 29 + - + - + + + = ∫ ∫ ∫ ∫ ∫ + + + + +----- 8 6 6 4 4 4 4 6 6 8 8 sin 8 sin 6 8 sin 2 8 sin 6 8 sin 8 1 dx x n dx x n x dx x n x dx x n x dx x n b n π π π π π ( 29 ( 29...
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Fourier Series Pt 1 - University of Waterloo CivE 331...

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