EECE 301 Discussion 05 - Fourier Series Examples

# EECE 301 Discussion 05 - Fourier Series Examples - EECE 301...

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EECE 301 Signals & Systems Prof. Mark Fowler Discussion #5 • Fourier Series Examples

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Fourier Series Examples
Example #1 t e ) ( t x 12 1 -2 -1 …… sec / rad 2 2 2 0 π ω = = = T choose 1 0 ) 1 ( 1 0 ) 1 ( 2 1 1 0 2 0 1 1 2 1 2 1 0 2 1 ) ( 2 1 ) ( 1 0 0 0 + = = × + = = = + + + t jk t jk t jk t jk t t jk T t t t jk k e jk dt e dt e dt e e dt e t x dt e t x T c [ ] ) 1 ( 2 1 1 ) 1 ( 2 1 1 1 0 ) 1 ( jk e e e jk jk jk + = + = + = k odd k even e jk , 1 , 1 : Note k k j jk e e ) 1 ( ) ( ly equivalent or = = ) 1 ( 2 ) 1 ( 1 1 jk e c k k + = So…

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Now we can use Matlab to plot | c k | & c k (rad/sec) (rad/sec) (rad)
Example #2 ) ( t x 1 12 -2 -1 -3 3 T π ω= = 0 2 T 1 0 = t pick [] ) 1 ( ) 1 ( ) 1 ( 0 ) 1 ( ) ( 2 2 1 1 2 1 1 = = = jk jk k k t jk jk e dt te c k t jk t jk k 1 ) ( ) 1 ( : Note 2 2 = = = ± j j e k jk cancel 0 ) 1 ( Table Integral From 2 = a ax a e dx xe ax ax

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0 ) 1 ( 2 ) 1 ( 2 2 2 + = + = k k j k jk c k k k π 0 0 2 0 = = c k k c k When k < 0: c k is j (neg)(-1) k When k > 0: c k is j (pos)(-1) k Since j (neg) = - /2 Since j (pos) = /2 we get this = = even negative k odd positive k odd negative k even

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EECE 301 Discussion 05 - Fourier Series Examples - EECE 301...

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