formulas

# Formulas - F o u r i e r T r a n s f o r m s X f = Z ∞ ∞ x t e j 2 π f t d t x t = Z ∞ ∞ X f e j 2 π f t d f Z ∞ ∞ | x t | 2 d t = Z

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Unformatted text preview: F o u r i e r T r a n s f o r m s X ( f ) = Z ∞- ∞ x ( t ) e- j 2 π f t d t x ( t ) = Z ∞- ∞ X ( f ) e j 2 π f t d f Z ∞- ∞ | x ( t ) | 2 d t = Z ∞- ∞ | X ( f ) | 2 d f T R A N S F O R M T H E O R E M S N a m e o f T h e o r e m S i g n a l F o u r i e r T r a n s f o r m S u p e r p o s i t i o n a 1 x 1 ( t ) + a 2 x 2 ( t ) a 1 X 1 ( f ) + a 2 X 2 ( f ) T i m e D e l a y x ( t- t ) X ( f ) e- j 2 π f t S c a l e C h a n g e x ( a t ) | a |- 1 X ( f / a ) F r e q u e n c y S h i f t x ( t ) e j 2 π f t X ( f- f ) M o d u l a t i o n x ( t ) c o s ( 2 π f t ) 1 2 X ( f- f ) + 1 2 X ( f + f ) D i ff e r e n t i a t i o n d n d t n x ( t ) ( j 2 π f ) n X ( f ) I n t e g r a t i o n Z t- ∞ x ( τ ) d τ ( j 2 π f )- 1 X ( f ) + 1 2 X ( ) δ ( f ) C o n v o l u t i o n Z ∞- ∞ x 1 ( t- τ ) x 2 ( τ ) d τ X 1 ( f ) X 2 ( f ) M u l t i p l i c a t i o n x 1 ( t ) x 2 ( t ) Z ∞- ∞ X 1 ( f- f ) X 2 ( f ) d f T R A N S F O R M P A I R S S i g n a l x ( t ) T r a n s f o r m X ( f ) A- τ / 2 τ / 2 t A τ s i n ( π f τ ) π f τ ≡ A τ s i n c ( f τ ) B t- t t B τ s i n 2 ( π f τ ) ( π f τ ) 2 ≡ B τ s i n c 2 ( f τ ) e- α t u ( t ) 1 α + j 2 π f e- | t | / τ 2 τ 1 + ( 2 π f τ ) 2 e- π ( t / τ ) 2 τ e- π ( f τ ) 2 s i n ( 2 π W t ) 2 π W t ≡ s i n c ( 2 W t ) 1 / ( 2 W ) f- W W e j ( 2 π f c t + φ ) e j φ δ ( f- f c ) c o s ( 2 π f c t + φ ) 1 2 δ ( f- f c ) e j φ + 1 2 δ ( f + f c ) e- j φ δ ( t- t ) e- j 2 π f t ∞ X m =- ∞ δ ( t- m T ) 1 T ∞ X n =- ∞ δ f- n T s g n ( t ) = + 1 t >- 1 t <- j π f u ( t ) = + 1 t > t < 1 2 δ ( f ) + 1 j 2 π f 1 D i s c r e t e F o u r i e r T r a n s f o r m s T h e D F T a n d t h e i n v e r s e...
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## This note was uploaded on 09/29/2009 for the course EC 505 taught by Professor Karl during the Fall '04 term at BU.

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Formulas - F o u r i e r T r a n s f o r m s X f = Z ∞ ∞ x t e j 2 π f t d t x t = Z ∞ ∞ X f e j 2 π f t d f Z ∞ ∞ | x t | 2 d t = Z

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