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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 difference 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 , y ( 2) = 0 . 1
a. Find the homogeneous solution for this system. Solution. λ 2 7 12 λ + 1 12 = λ 1 3 λ 1 4 = 0 so y h ( n ) = c 1 1 3 n + c 2 1 4 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 ku ( n 1) + 4 12 ku ( n 2) = u ( n ) . Evaluating this at n = 2 yields 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 n + c 2 1 4 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 find c 1 = 6 . 67 , c 2 = 2 . 25 . y ( n ) = 6 . 67 1 3 n + 2 . 25 1 4 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 , y (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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113_1_EE113_Midterm_Solutions_Spring_2007 - EE 113 Midterm...

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