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ECE633F09_HW5solutions - ECE633 Signals and Systems I Fall...

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ECE633 Signals and Systems I, Fall 2009 – Homework 5 Solutions 1 From: Linear Systems and Signals, 2 nd ed . B. P. Lathi, Oxford University Press, 2005. 2.4-7 The unit impulse response of an LTIC system is ( ) ( ) t h t e u t = . Find this system’s (zero- state) response ( ) y t if the input ( ) x t is: (a) ( ) u t (c) ( ) 2 t e u t (b) ( ) t e u t (d) ( ) ( ) sin 3 t u t Use the convolution table (Table 2.1) to find your answers. (a) ( ) ( ) ( ) ( ) ( ) 1 1 t t e y t e u t u t u t = = − − ( ) ( ) 1 t y t e u t = (b) ( ) ( ) ( ) t t y t e u t e u t = ( ) ( ) t y t te u t = (c) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 1 2 t t t t e e y t e u t e u t u t = = − − ( ) ( ) 2 t t y t e e u t = (d) ( ) ( ) ( ) ( ) ( ) ( ) 3 3 sin 3 2 j t j t t t e e y t e u t t u t e u t u t j = = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 3 3 3 1 1 2 2 1 3 1 3 t j t t j t t j t t j t e e e e y t e u t e u t e u t e u t u t j j j j = = − − ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) 3 3 1 3 1 3 1 2 1 3 1 3 t j t t j t j e e j e e y t u t j j j − + − − − = − + − − ( ) ( ) ( ) ( ) 3 3 3 3 3 2 1 2 10 j t j t t j t j t e e j e e e y t u t j + + = ( ) ( ) ( ) 3 3 3 3 1 3 3 10 2 2 j t j t j t j t t e e e e y t e u t j + = + ( ) ( ) ( ) ( ) 1 3 3 sin 3 cos 3 10 10 10 t y t t e t u t = + OR
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ECE633 Signals and Systems I, Fall 2009 – Homework 5 Solutions 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 sin 3 cos 3 90 t t t y t e u t t u t e u t e t u t = = ° ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 0 1 2 2 3 3 cos 90 tan cos 3 90 tan 0 1 0 1 0 1 3 t t e e t y t u t °− °− + − + − = + − + ( ) ( ) ( ) ( ) ( ) ( ) 1 1 cos 90 tan 3/ 1 cos 3 90 tan 3/ 1 10 t e t y t u t °− °− = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 3/ 10 cos 3 18.43 cos tan 1/ 3 cos 3 tan 1/ 3 10 10 t t e t e t y t u t u t + ° + = ( ) ( ) ( ) ( ) ( ) cos 3 18.43 3 0.3 0.3162cos 3 18.43 10 10 t t t y t e u t e t u t
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