Final 2010

Final 2010 - Name:______________________________

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Unformatted text preview: Name:______________________________ ID:________________________ EL6113 Final Exam  ­ Spring 2010 Directions: Answer all questions completely in the blue exam book. Clearly identify your results. Time: 2 Hours, 30 Minutes 1. (20 pts) Consider the following causal system: ! ! − 6! + 8 !! = 3 1! !+4 !+2 ! a. Find an Hm(z) that has the same magnitude squared response as above, but is minimum phase. Find the all pass factor that corresponds to the minimum phase Hm(z). b. Sketch the pole ­zero diagrams for H(z), Hm(z), and Hap(z). c. Is Hm(z) causal and stable? Is the all ­pass factored out causal and stable? Explain. d. Write a difference equation describing Hm(z). 2. (15 pts) Consider the following LTI system: !! = ! ! ! + .09 a. Sketch the Pole ­Zero diagram for this system. b. Find the stable inversion of H(z). c. Is this a real system? d. Find a difference equation describing H(z). 3. (15pts) Consider the following causal LTI system: !! = !− 1 2 ! !+1 ! Find y(t) for the following inputs: a. x(t)=u(t) b. x(t)=sin(2t) 4. (15 pts) Consider the following causal first order LTI system: ! ! ! + 4! ! = ! ! a. Find the impulse response h(t). b. Find y(t) if x(t) = t u(t). (Note : ! = ! ! !" , ! = 0) c. Roughly sketch |H(jw)| as a function of w. 5. (20 pts) Consider the following magnitude squared response: 5 − cos !  4 ! (cos ! ) = ! ! + 45 cos ! + 125 cos 16 64 Find a stable, causal, minimum phase H(z) that will satisfy the above requirement. 6. (15 pts) Assume a bandlimited signal has a frequency response such that F(f)=0, |f|> 5KHz, and we wish to sample this signal. The maximum sampling frequency is 8000 samples/sec. Design a sampling system that best recovers f(t) assuming we have imperfect filters that require 200Hz guard banding on each side. Draw the frequency response after the sampler, after all filters and at the output. ...
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