hw5_sol

hw5_sol - EE113 Digital Signal Processing Prof Mihaela van...

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EE113: Digital Signal Processing Spring 2008 Prof. Mihaela van der Schaar Homework #5 Solutions Prepared by Martin Andersen and Hyunggon Park 10.4. The input sequence can be expressed as x ( n )= u ( n 1) 2 δ ( n 3) . Note that the bilateral z -transform can be used to Fnd the zero-state solution only. The zero-input solution is found by using the method described in Chapter 8. The characteristic equation is λ 2 5 2 λ +1=0 , or, ± λ 1 2 ² ( λ 2) = 0 . Then the zero-input solution has the form is given by y zi ( n )= C 1 ± 1 2 ² n + C 2 (2) n . Using the initial conditions y ( 2) = 0 and y ( 1) = 1, we can Fnd that C 1 = 1 / 6and C 2 =8 / 3. Therefore, the zero-input solution is then given by y zi ( n )= ³ 1 6 ± 1 2 ² n + 8 3 (2) n ´ u ( n ) . By assuming that the system is relaxed, the zero-state solution can be found in the following ways: (a) Using z -transform: Take the z -transform of the di±erence equation Y zs ( z ) 5 2 z - 1 Y zs ( z )+ z - 2 Y zs ( z )= X ( z ) , or Y zs ( z ) ³ 1 5 2 z - 1 + z - 2 ´ = 1 z 1 2 z - 3 . Then, we have a partial fraction expression for Y zs ( z ) as follows: Y zs ( z )= z 3 2 z +2 z ( z 1)( z 1 / 2)( z 2) = 2 z 2 z 1 + 3 z 1 / 2 + 2 z 2 .
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Therefore, the inverse z -transform gives y zs ( n )= 2 δ ( n 1) + " 2+3 ± 1 2 ² n - 1 +2 n # u ( n 1) . (1) (b) Without using z -transform: Let us Fnd the impulse response sequence of the relaxed system and convolve it with the input sequence x ( n ). The di±erence equation with x ( n )= δ ( n )isg ivenby h ( n ) 5 2 h ( n 1) + h ( n 2) = δ ( n ) . ²or n 1, this equation becomes homogeneous, and thus, h ( n )= C 1 ± 1 2 ² n + C 2 (2) n ,n 1 . We use the relaxed initial conditions h ( 1) = h ( 2) = 0, which gives that h (0) = 1. Using h (0) and h (1), we Fnd that C 1 = 1 / 3and C 2 =4 / 3. Therefore, h ( n )= ³ 1 3 ± 1 2 ² n + 4 3 (2) n ´ u ( n ) . The zero-state solution is thus given by
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This note was uploaded on 10/03/2011 for the course EE 113 taught by Professor Walker during the Spring '08 term at UCLA.

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hw5_sol - EE113 Digital Signal Processing Prof Mihaela van...

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