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ASE 365 - Lecture 10

# ASE 365 - Lecture 10 - Complex Fourier series have we and...

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Unformatted text preview: Complex Fourier series have we and Since , 2 sin 2 cos i e e t p e e t p t ip t ip t ip t ip ϖ ϖ ϖ ϖ ϖ ϖ--- = + = t ip p p p t ip p p t ip p p p t ip t ip p t ip t ip p e C e ib a e ib a a i e e b e e a a t f 1 1 2 2 2 2 2 2 ) ( ϖ ϖ ϖ ϖ ϖ ϖ ϖ ∑ ∑ ∑ ∞-∞ = ∞ =- ∞ =-- = + + - + = - + + + = dt e t f T dt t p i t p t f T p ib a p a p ib a C T t t t ip T t t p p p p p ∫ ∫ +- +-- =- = < + =- = ˆ ˆ ˆ ˆ ) ( 1 ) sin )(cos ( 1 , 2 , 2 , 2 ϖ ϖ ϖ for for for where = ≠ = = ∫ +- q p q p dt e e T e pq T t t t iq t ip t ip if if : s ' of ity Orthogonal , 1 , 1 ˆ ˆ δ ϖ ϖ ϖ t ip p p p t p i p p p t ip p p e C G e C ip G t x e C t f p ) ( ) ( ) ( , ) ( ϖ φ ϖ ϖ ϖ ∑ ∑ ∑ ∞-∞ =- ∞-∞ = ∞-∞ = = = = then If Don’t like negative frequencies? + = ⇒ = = = = = + = ∑ ∫ ∫ ∑ ∞ = +- + ∞ = 1 ˆ ˆ ˆ ˆ 1 Re 2 ) ( 2 ) ( 2 ) ( 2 2 , Re 2 ) ( p t ip p p p p T t t t ip p T t t p t ip p e A G A t x C dt e t f T A dt t f T a C A e A A t f ϖ ϖ ϖ and where Use . ) ( ) ( t ip p p t ip p p p p p p t ip p p p p p e C G e C G C C G G e C G t x t x ϖ ϖ ϖ--- ∞-∞ = + = = = ∑ in cancels part imaginary so , and because , from even real, is that Note Analysis process: ) ( ) ( T t f t f + = s ' or s, ' or s, ' and s ' p p p p A C b a s ' p G ) ( ) ( T t x t x p p + = in Terms “Time domain” “Frequency domain” { } - = - = +-- = -- + - = = =------ ∞-∞ = ∫ ∑ 2 , / 2 , 1 ) 2 ( ) ( 1 ) ( 1 ) ( ) 2 ( 2 / 2 / p p ip ip ip T T t ip T t ip t ip T p p t ip p ib a p p iA p e e e p...
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ASE 365 - Lecture 10 - Complex Fourier series have we and...

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