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Electrical Engineering 20N - Spring 2003 - Varaiya - Final

Electrical Engineering 20N - Spring 2003 - Varaiya - Final...

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EECS 20. Final Exam Solution May 20, 2003. 1. 15 points. 5 points for (a), 10 points for (b) (a) The deterministic machine A is like the CodeRecognizer machine studied in the text and in the homework. start 1 11 {1} {1} {0} {0} {0} {1}/ recognize {1} {0} 111 A Let x denote an input signal and y the corresponding output signal. Complete the ex- pression for y ( n ) below, ignoring stuttering inputs, (i.e. replace the · · · by an expression involving x ) x InputSignals , n Naturals 0 , y ( n ) = recognize , if · · · absent , otherwise Answer ( x ( n 3) , x ( n 2) , x ( n 1) , x ( n )) = (1 , 1 , 1 , 1) (b) Determine whether the non-deterministic machine B simulates A and write down the relevant simulation relation if it does. start maybe {1} {0} {0} {1}/ recognize {1} B Answer Yes, B simulates A with the simulation relation { ( start, start ) , (1 , maybe ) , (11 , maybe ) , (111 , maybe ) } 1
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2. 20 points. 5 points for (a), (b), 10 points for (c) The input signal x and output signal y of an LTI system are related by the differential equation t Reals , ˙ y ( t ) + y ( t ) = x ( t ) . (a) The frequency response of this system is ω Reals , H ( ω ) = 1 1+ and the magnitude and phase response for ω = 0 , ± 1 , and ω → ±∞ are: | H (0) | = 1 , H (0) = 0; | H ( ± 1) | = 1 2 , H ( ± 1) = π 4 lim ω →±∞ | H ( ω ) | = 0 , lim ω →±∞ H ( ω ) = π 2 , (b) Using the hint, the impulse response of this system is t Reals h ( t ) = 0 , if t < 0 e t , if t > 0 (c) Now consider an LTI system whose impulse response g = h h , where h is as in ( 2b ).
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