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Unformatted text preview: UNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical & Computer Engineering Department ECE 101  Fall 2008 Linear Systems Fundamentals FINAL EXAM WITH SOLUTIONS You are allowed one 2sided sheet of notes. No books, no other notes, no calculators. PRINT YOUR NAME Johann Peter Gustav Lejeune Dirichlet Signature J . P . G . L . DIRICHLET Student ID Number A10101010101010101010101010101 Problem Weight Score 1 20 pts 20 2 24 pts 24 3 20 pts 20 4 20 pts 20 5 16 pts 16 Total 100 pts 100 Please do not begin until told. Write your name on all pages. Show your work. Tables from the textbook are at the back of the exam. Good luck! 1 Name/Student ID: Problem 1 Consider the continuoustime signal x ( t ) = sin(1000 t ) t . (a) Determine and sketch X ( j ), the CTFT of x ( t ). Referring to Table 4.2, we get X ( j ) = braceleftbigg 1 ,   < 1000 ,   > 1000 . 1 1000 S1000 S Z For each of the signals, denoted y ( t ), defined on the following pages, you are asked to: (i) Determine its CTFT Y ( j ) and sketch precisely its magnitude; (ii) Determine the bandwidth W of the signal; (iii) Assume that signal undergoes impulsetrain sampling with sampling in terval T , and find the largest value of T such that no aliasing occurs. 2 Name/Student ID: Problem 1 (cont.) (b) y ( t ) = x ( t ) + x ( t ) (i) Determine Y ( j ) and sketch precisely its magnitude. From Table 4.1, Time Reversal Property, we get x ( t ) X ( j ) So, Y ( j ) = X ( j ) + X ( j ) . Since X ( j ) is even, we know X ( j ) = X ( j ). Therefore, Y ( j ) = 2 X ( j ) . 2 Z1000 S 1000 S (ii) Determine the bandwidth W of the signal. W = 1000 (iii) Find the largest sampling interval T such that no aliasing occurs. By the sampling theorem, we want s = 2 T > 2 W = 2000 . This requires that T < 2 2 W = W = 10 3 sec. 3 Name/Student ID: Problem 1 (cont.) (c) y ( t ) = x ( t ) x ( t ) (i) Determine Y ( j ) and sketch precisely its magnitude. By the Multiplication Property, Y ( j ) = 1 2 X ( j ) X ( j ) . 1000 Z2000 S 2000 S (ii) Determine the bandwidth W of the signal. W = 2000 (iii) Find the largest sampling interval T such that no aliasing occurs. T < W = 1 2000 sec. 4 Name/Student ID: Problem 1 (cont.) (d) y ( t ) = dx ( t ) dt (i) Determine Y ( j ) and sketch precisely its magnitude. By the Differentiation in Time property, dx ( t ) xt jX ( j ) Therefore,  Y ( j )  = braceleftbigg   ,   < 1000 ,   > 1000 . 1000 S Z1000 S 1000 S (ii) Determine the bandwidth W of the signal. W = 1000 (iii) Find the largest sampling interval T such that no aliasing occurs....
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 Fall '08
 Siegel

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