ECE111_2011SPRING_EXAM2__[0]

ECE111_2011SPRING_EXAM2__[0] - The Cooper lJnion Department...

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The Cooper lJnion Department of Electrical Engineering ECE11l Signal Processing & Systems Analysis Exam II Mav 6, 2011 Time: 2 hours. Closed book, closed notes. No calculators. 1. [6 pts.] Given the discrete-time transfer function: H(z\:22-Bz+! I\z):+rr_"+1 4 rfu?-Z+l (a) Write a difference equation with r (n) as input and y(n) as output. (b) Draw a direct form II realization. (c) Draw a direct form II transposed realization. 2. [5 pts.] The unit step response of a stable LTI discrete-tirae system with transfer function H (z) has the form: Ku (n) * (an2 * bn * c) 10.27" u (n) +d (0.8)" cos (ntr f S + g) u (n) (a) Identify the natural and forced responses. (b) Identify the dominant mode in the transient response (i.e., not counting the Ku(n) term). (c) Specify the locations and multiplicities of the poles of H ("). 3. [3 pts.] The impulse response of a continuous-time system is characterized by several modes as follows: one mode decays with a time constant of 0.25 sec without oscillation; another mode decays with a time constant of 0.5 sec and a frequency of oscillation of l)H z. (a) Specify the locations of the poles in the s-plane. (b) can you determine anything about the multiplicities of the poles?

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a' [9 pts'] A continuous-time system with transfer function matrix 11(s) and state-space representation {,4, B,C,D} has 5 state variables, 3 inputs and 2 outputs. (a) Specify the dimensions of the A, B, C, D matrices. (b) Specify the dimension of II ("). (c) The H"(t) entryinrl(s) isthetransferfunctionfromthe (Iul2.o)inputtothe (t",12"a7 output. (d) suppose the system is externally stable and all eigenvalues of A arein the LHp. Is this enough information to determine if this is a minimal realization? (yes or no, no justification). (e) Every eigenvalue of -4 must be a system pole- true or false? If false, what are eigenvalues that are NOT system poles called? (f)
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ECE111_2011SPRING_EXAM2__[0] - The Cooper lJnion Department...

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