113_1_EE113_Midterm_Solutions_Spring_2007

113_1_EE113_Midterm_Solutions_Spring_2007 - EE 113 Midterm...

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EE 113 Midterm Solution Spring 2007 Inst: Dr. C.W. Walker Problem Points Score 1 9 2 11 3 15 4 15 5 10 6 10 7 10 8 10 9 10 Total 100
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Problem 1. For parts (a), (b) and (c) determine whether or not the system is i. linear ii. time-invariant iii. BIBO stable, i.e., bounded input-bounded output stable a. y ( n )=2 x ( n ) . Solution. linear, time-invariant, BIBO stable b. y ( n )=log( n +1) x ( n ) , log is the natural logarithm. Solution. linear, not time-invariant, not BIBO stable c. y ( n )=[ x ( n )] n . Solution. not linear, not time-invariant, not BIBO stable Note: You do not need to show any work on this problem if you can quickly recognize the answer. Problem 2. Consider the system described by the following di±erence equa- tion: y ( n ) 7 12 y ( n 1) + 1 12 y ( n 2) = x ( n ) , where, x ( n )=(1 / 2) n u ( n ) ,y ( 1) = 1 ( 2) = 0 . 1
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a. Find the homogeneous solution for this system. Solution. λ 2 7 12 λ + 1 12 = ± λ 1 3 ²± λ 1 4 ² =0 so y h ( n )= c 1 ± 1 3 ² n + c 2 ± 1 4 ² n . b. Find the particular solution for this system. Solution. k (1 / 2) n u ( n ) 7 12 k (1 / 2) n 1 u ( n 1)+ 1 12 k (1 / 2) n 2 u ( n 2) = (1 / 2) n u ( n ) or ku ( n ) 14 12 ( n 1) + 4 12 ( n 2) = u ( n ) . Evaluating this at n =2y ie lds k 14 12 k + 4 12 k =1 so k =6 . Thus , y p ( n )=6(1 / 2) n ,n 2 . c. Find the complete solution for this system. Solution. y (0) = 7 12 y ( 1) 1 12 y ( 2) + x (0) = 7 12 +1= 19 12 y (1) = 7 12 y (0) 1 12 y ( 1) + x (1) = 133 144 1 12 + 1 2 = 193 144 y ( n y h ( n )+ y p ( n ) or y ( n c 1 ± 1 3 ² n + c 2 ± 1 4 ² n +6(1 / 2) n . 2
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y (0) = c 1 + c 2 +6= 19 12 . y (1) = c 1 3 + c 2 4 +3= 193 144 . We fnd c 1 = 6 . 67 ,c 2 =2 . 25 . y ( n )= 6 . 67 ± 1 3 ² n +2 . 25 ± 1 4 ² n +6(1 / 2) n ,n 0 . d. Evaluate your y ( n )For n =0 , 1 , 2. Solution. y (0) = 1 . 58 ,y (1) = 1 . 34 (2) = 0 . 90 . Problem 3. Compute X ( z ), the Forward z-transForm, (iF it exists) For each oF the Following. Remember to speciFy the region oF convergence in each case.
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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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113_1_EE113_Midterm_Solutions_Spring_2007 - EE 113 Midterm...

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