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Solutions_Midterm_F09

# Solutions_Midterm_F09 - UNIVERSITY OF CALIFORNIA SAN DIEGO...

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Unformatted text preview: UNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical & Computer Engineering Department ECE 101 - Fall 2009 Linear Systems Fundamentals MIDTERM EXAM You are allowed one 2-sided sheet of notes. No books, no other notes, no calculators. PRINT YOUR NAME Marc-Antoine Parseval des Chˆ enes Signature 1 T i T | x ( t ) | 2 dt = ∑ ∞ k =-∞ | a k | 2 Student ID Number . . . , a-2 , a-1 , a , a 1 , a 2 , . . . Problem Weight Score 1 30 pts 30 2 36 pts 36 3 24 pts 24 4 10 pts 10 Total 100 pts 100 Please do not begin until told. Write your name on all pages. Show your work. Use back of previous page and attached scratch sheets as needed. Tables 3.1 and 3.2 from the textbook are attached to the back of the exam. Good luck! 1 Name/Student ID: Problem 1 (30 points) For each part, check the appropriate boxes. Justify your answers. Each answer is worth 6 points. (a) Let x [ n ] = δ [ n ] + 2 δ [ n-1] + 3 δ [ n-2] . Let x e [ n ] be the even part of x [ n ] and let x o [ n ] be the odd part of x [ n ]. Check the box next to the correct statement: a x e [0] = 1 2 and x o [0] = 1 2 a x e [1] = 3 2 and x o [1] = 1 2 a X x e [-2] = 3 2 and x o [-2] =-3 2 a x e [2] = 2 and x o [2] = 1 Since x e [ n ] = x [ n ]+ x [-n ] 2 , we see that x e [0] = 1 , x e [1] = 1 , x e [-2] = 3 2 , x e [2] = 3 2 . So, the only possible answer is the third one. Recalling that x o [ n ] = x [ n ]-x [-n ] 2 , we con±rm that x o [-2] =-3 2 . This con±rms the choice of the third answer. 2 Name/Student ID: Problem 1 (cont.) (b) Consider the signal x [ n ] = e j 2 π 9 ( n-1) + ∞ s k =-∞ δ [ n-6 k + 2] . The smallest positive integer N such that x [ n + N ] = x [ n ] for all integer values of n is: a 9 a 12 a X 18 a 54 a No such N Note that the Frst term in x [ n ], namely x 1 [ n ] = e j 2 π 9 ( n-1) , has fundamental period N 1 = 9 because it is simply a time-shifted version of the periodic discrete-time exponential signal e j 2 π 9 n with fundamental period 9. The second part of x [ n ], namely x 2 [ n ] = ∑ ∞ k =-∞ δ [ n-6 k + 2], has fundamental period N 2 = 6 because is simply a time shift of the periodic discrete-time impulse train ∑ ∞ k =-∞ δ [ n-6 k ] with fundamental period 6....
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Solutions_Midterm_F09 - UNIVERSITY OF CALIFORNIA SAN DIEGO...

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