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BME 343_Fall 2009_HW 7_solution

# BME 343_Fall 2009_HW 7_solution - 7.14(a T T 0 0 X = e at e...

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7.1 4 (a) ( ) ( ) 0 0 1 ( ) ( ) T T j a T at j t j a t e X e e dt e dt j a ω ω ω ω ω + + = = = + 7.2 4 (a) 0 0 0 0 0 0 ( ) 1 1 ( ) 2 2 j t j t t j t x t e e d e d ω ω ω ω ω ω ω ω ω π π = = 0 0 0 0 ( ) ( ) 0 0 0 0 2 sin[ ( )] 2 ( ) 2 ( ) j t t j t t j t t e e j t t j t t ω ω ω π π = = 0 0 0 0 0 0 sin[ ( )] sin [ ( )] ( ) t t c t t t t ω ω ω π π = = 7.3 3 (c) ( ) cos( )[ ( ) ( )] cos( ) ( ) cos( ) ( ) 2 2 x t t u t u t t u t t u t π π = = Since sin( ) cos( ) 2 t t π = − Î ( ) cos( ) ( ) sin( ) ( ) 2 2 x t t u t t u t π π = + /2 2 2 1 ( ) [ ( 1) ( 1)] { [ ( 1) ( 1)] } 2 1 2 1 j j X e j πω π ω π ω δ ω δ ω δ ω δ ω ω ω = + + + + + + /2 /2 ( 1) ( 1) ( 1) j j e e j πω π δ ω δ ω δ ω + = + = + ; /2 /2 ( 1) ( 1) ( 1) j j e e j πω π δ ω δ ω δ ω = = − + Î /2 [ ( 1) ( 1)] [ ( 1) ( 1)] 0 2 2 j e j π π π δ ω δ ω δ ω δ ω + + + + = /2 2 2 1 ( ) 1 1 j j X e πω ω ω ω ω = + 7.3 6 (a) The signal x(t) in this case is a triangle pulse ( ) 2 t π Δ multiplied by cos( ) t ( ) ( )cos(10 ) 2 t x t t π = Δ From Table 7.1, pair 19 (page 702) 2 ( ) sin ( ) 2 2 t c πω π π Δ

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BME 343_Fall 2009_HW 7_solution - 7.14(a T T 0 0 X = e at e...

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