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ECE111_2007SPRING_EXAM2__[0]

# ECE111_2007SPRING_EXAM2__[0] - The Cooper Union Department...

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The Cooper Union Department of Electrical Engineering ECEl11 Signal Processing & Systems Analysis Exam II April, Friday the 13'" 2007 Time: 2 hours. Closed book, closed notes. No calculators. SHOW ALL WORK!! L 14 pts.] A digital filter is given by, y(n) = 3x(n) - 2x(n-l) +5x(n- 2) +6x(n - 3) Specify the impulse response, transfer function and frequency response of the filter. Sketch a transversal filter realization. Is the filter FIR or 1IR? 2. [4 pts.) Sketch a direct fonn II transposed realization of: ,I H (z) = 2z' + 5z + 4 4z 2 - z+2 3. [6 pt's.] Use the method of partial fractions to find the inverse Laplace transform of: 2s 2 +s+1 H (5) = (5 + 5) (5 H) 4. [6 ptis.) Use the method. of partial fractions to find the inverse z-transfonn of: Hz _ z2+ z +1 ( ) - (2z - 1) (3z + 2) 5. [6 ptls.] The unit step response of a stable analog system is: Y (t) = 5u (t) + [4t 3 - t + 7J .-"u (t) + te-' cos (3t) u (t) - e-' sin (3t) u (t) , (a) Specify the term or terms associated that comprise the forced response. I (b) he remaining terms are denoted the response. (c) Specify the system poles, with multiplicity. [ (d) If we wanted to approximate this system with a second order model, should we J se:~e:~d:~~:~:~ :e~:l~:~ ,?~~:iticallY damped system? m~:~i~f ~~: I ----------- transient response does (does not) exhibit " 1 . t

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6. [9 pts.] Recall that, for a discrete-time system, the time-con.stant (in seconds) asso- ciated with a mode with pole p is given by: T T~-- lin Ipll where T is the sample period. A real discrete-time system is operated at a sampling rate Is = 1M H z. The system is represented in state-space form as {A, B, C, D}, and the associated transfer function matrix is H (z). The eigenvalues of A are: simple at -2, simple at -1/3 ± j1/3 and triple multiplicity at +0.20. (a) The system is not internally stable. Why not? Be BRIEF. (b) Suppose H (z) is a stable transfer function for this system. What could be the poles of H? Give the most general possibility, including restrictions if any for the multiplicities.
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ECE111_2007SPRING_EXAM2__[0] - The Cooper Union Department...

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