HW02s09_soln

HW02s09_soln - ECE-2025 Homework #2 Solutions Spring-2009...

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Unformatted text preview: ECE-2025 Homework #2 Solutions Spring-2009 Problem 2.1: (a) ) 6 . 999 cos( 5 ) 2 . 999 cos( 3 ) ( 1 + + = t t t x . First define t j j t j j j t j j t j j e Ae e e e e e e e t z 999 999 6 . 2 . 999 6 . 999 2 . 1 ) 5 3 ( 5 3 ) ( = + = + = , and )] ( Re[ ) ( 1 1 t z t x = , then we can get the amplitude and phase with a calculator, or get the amplitude via the following mathematical operations. The amplitude is 1192 . 3 )] 2 . 6 . cos( 30 25 9 [ ] ) 6 . sin 5 2 . sin 3 ( ) 6 . cos 5 2 . cos 3 [( 2 / 1 2 / 1 2 2 = + + + = + + + = A ; and phase 4088 . )] 6 . cos 5 2 . cos 3 /( ) 6 . sin 5 2 . sin 3 [( tan 1 = + + = . Finally, we get ) 4088 . 999 cos( 1192 . 3 ) 999 cos( ) ( 1 + = + = t t A t x . (b) ) 8 / 3 cos( 8 ) 8 / 9 cos( 8 ) 8 / 5 cos( 7 ) 8 / cos( 7 ) ( 2 + + + + = t t t t t x ; First define t j j t j j t i j j j t j j j e e e e e e e e e e e t z 8 / 5 8 / 3 4 / 4 / 8 / 3 8 / 5 8 / 4 2 7 2 7 ) ( 7 ] [ 7 ) ( = = + = + = and t j j t i j j j t j j j e e e e e e e e e t z 8 / 5 4 / 4 / 8 / 5 8 / 3 8 / 9 5 2 8 ) ( 8 ] [ 8 ) ( = + = + = . Then define t j j t j j e e A e e t z t z t z 2 2 8 / 5 5 4 2 2 ) ( ) ( ) ( = = + = Therefore, we have ) 8 / 5 cos( 2 )] ( Re[ ) ( 2 2 + =...
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HW02s09_soln - ECE-2025 Homework #2 Solutions Spring-2009...

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