ece11112final - The Cooper Union Department of Electrical...

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The Cooper Union Department of Electrical Engineering ECE111 Signal Processing & Systems Analysis Final May 4, 2012 Time: 3 hours. Close book, closed notes. No calculators. Part I: ANSWER ALL PARTS. WRITE YOUR ANSWERS IN THE SPACE PROVIDED HERE, NOT IN YOUR EXAM BOOKS. TOTAL: 30 points. Any errors will cause disproportionate penalty. IA: FORMULAS 1. Laplace transform. 2. z -transform. 3. Continuous-time convolution. 4. Discrete-time convolution. 5. CTFT. 1
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6. Inverse CTFT. 7. DTFT. 8. Inverse DTFT. 9. DFT. 10. Inverse DFT. 2
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IB: Basic Concepts 1. An analog transistor ampli°er has simple poles at ° 2 , ° 3 ± 4 j and a double pole at ° 1 ; also zeros at ° 4 ± j and at +3 . True or False (no justi°cation): The system is stable. _________________ 2. A digital °lter realized as a cascade of several direct form II transposed sections has poles at 1 2 e ° j°= 3 , ± 2 3 j , and double poles at 1 3 e ° j 2 °= 3 . It also has zeros at 2 3 e ° j°= 4 , ° 2 , ° 3 , and double zeros at 1 2 e ° j°= 2 . True or False (no justi°cation): The system is stable. ___________ 3. BIBO stands for (do not de°ne it; write what the acronym stands for): 4. The Laplace transform of 3 ° ( t ) + 4 e ± 3 t u ( t ) is (no need to simplify): 5. The discrete-time unit step u [ n ] is de°ned as: 6. True or False (no justi°cation needed): A transversal °lter is always FIR (can not be IIR). _________________ 7. An LTI discrete-time system is FIR if its ___________ response has °nite ______________, and is IIR if its _______________ response has in- °nite ________________. 3
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IC: Digital Filter Structures Draw the °lters on THIS PAGE in the space provided below. 1. Draw a transversal °lter realization for: H ( z ) = 4 + 3 z ± 1 + 2 z ± 2 ° 6 z ± 3 2. Draw a direct-form II realization for: H ( z ) = 4 z 2 + 2 z + 1 2 z 2 ° 4 z ° 3 4
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Part II: Answer all parts. Points per problem indicated. Partial credit will be granted ±proportionately² when warranted. 1. [8 pts.] Consider the following analog transfer function: H ( s ) = K s + 2 ( s + 0 : 5) ( s + 10) 2 First select K so that the DC gain is 20dB. Using that value for K : graph the Bode magnitude plot. Speci°cally, graph the straight-line asymptotes, determine corrections and sketch the actual curve. Be sure to label your diagram carefully.
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