20102ee113_1_HW 4 Solution Spring 2010

20102ee113_1_HW 4 Solution Spring 2010 - EE113: Digital...

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Unformatted text preview: EE113: Digital Signal Processing Spring 2010 Prof. Mihaela van der Schaar Homework #4 Solutions Prepared by Nick Mastronarde, Yu Zhang, and Shaolei Ren 8.7 Consider the difference equation y ( n ) = 3 4 y ( n- 1)- 1 8 y ( n- 2) + x ( n ) . 1. The modes of the system are the roots of the charateristic polynomial: 2- 3 4 + 1 8 = 0 1 = 1 2 2 = 1 4 . 2. The solutions to the homogeneous equation are: y h ( n ) = 1 n 1 + 2 n 2 = 1 1 2 n + 2 1 4 n , for all n. 3. Particular solutions: i. For x ( n ) = u ( n ): Letting y ( n ) = Ku ( n ) yields Ku ( n ) = 3 4 Ku ( n- 1)- 1 8 Ku ( n- 2) + u ( n ) and for n 2 we get K = 8 3 and y p ( n ) = 8 3 u ( n ) , n 2 . ii. For x ( n ) = nu ( n ): Letting y ( n ) = ( K 1 + K 2 n ) u ( n ) yields ( K 1 + K 2 n ) u ( n ) = 3 4 ( K 1 + K 2 ( n- 1)) u ( n- 1)- 1 8 ( K 1 + K 2 ( n- 2)) u ( n- 2)+ nu ( n ) and for n 2 we get K 1 + nK 2 = 3 4 ( K 1 + nK 2- K 2 )- 1 8 ( K 1 + nK 2- 2 K 2 ) + n or equivalently, n ( 3 4 K 2- 1 8 K 2- K 2 + 1) + 3 4 ( K 1- K 2 )- 1 8 ( K 1- 2 K 2 )- K 1 = 0 . Solving the equations 3 4 K 2- 1 8 K 2- K 2 + 1 = 0 , 3 4 ( K 1- K 2 )- 1 8 ( K 1- 2 K 2 )- K 1 = 0 yields K 2 = 8 3 and K 1 =- 32 9 . Thus, y p ( n ) = 8 3 n- 32 9 u ( n ) , n 2 . 1 iii. For x ( n ) = ( 1 3 ) n u ( n ): Letting y ( n ) = K ( 1 3 ) n u ( n ) yields K 1 3 n u ( n ) = 3 4 K 1 3 n- 1 u ( n- 1)- 1 8 K 1 3 n- 2 u ( n- 2) + 1 3 n u ( n ) and for n 2 we get K 1 3 n- K 3 4 1 3 n- 1 + K 1 8 1 3 n- 2 = 1 3 n or equivalently, K 1 3 n " 1- 3 4 1 3 - 1 + 1 8 1 3 - 2 # = 1 3 n Solving for K yields K = " 1- 3 4 1 3 - 1 + 1 8 1 3 - 2 #- 1 =- 8 and hence y p ( n ) =- 8 1 3 n u ( n ) , n 2 . 4. For the complete solutions we will assume that the system is relaxed since no initial conditions are given. i. For x ( n ) = u ( n ): y ( n ) = y h ( n ) + y p ( n ) = 1 1 2 n + 2 1 4 n + 8 3 u ( n ) , n ....
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This note was uploaded on 06/01/2010 for the course EE EE113 taught by Professor Mihaela during the Spring '10 term at UCLA.

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20102ee113_1_HW 4 Solution Spring 2010 - EE113: Digital...

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