FT_Properties

FT_Properties - 1 Review FS: ( ) ( ) 2 1 jk t K K jK t K T...

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Unformatted text preview: 1 Review FS: ( ) ( ) 2 1 jk t K K jK t K T x t c e T c x t e d t T + = = = = FT: ( ) ( ) ( ) ( ) 1 2 j t j t x t X e d X x t e d t + = = Relationship: ( ) If 0, 2 T x t t = > ( ) ( ) ( ) ( ) ( ) is periodic with period define , FS of ( ) : k T jk t k K t T x t x t kT x x t Tc x t e dt X = = = = = Example ( ) x t 1 1 1 ) s i n c sin( ) ( ( 2 2 ) T X T T = = t 2T T T 1 T/2-T 1-T/2-T 1 1 1 sinc sin 2 ( ( ) 2 ) k k Tc k T T k T = = ( ) x t t T 1 T/2-T 1-T/2-T 2 Properties of FT 1. Linearity: If x t ( ) FT X ( ) and y t ( ) FT Y ( ) 2. Time shift then ax t ( ) + by t ( ) FT aX ( ) + bY ( ) If x t ( ) FT X ( ) then x t t ( ) FT e j t X ( ) Proof: ( ) ( ) ( ) ( ) ( ) FT u t t j u t j t j t x t t x t t e dt x u e du e X = + = = 3. Time and frequency scaling 3....
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This note was uploaded on 10/15/2010 for the course BIM BIM108 taught by Professor Qiu during the Winter '09 term at UC Davis.

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FT_Properties - 1 Review FS: ( ) ( ) 2 1 jk t K K jK t K T...

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