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MIT6_003S10_assn06

# MIT6_003S10_assn06 - 6.003 Homework 6 Due at the beginning...

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6.003 Homework 6 Due at the beginning of recitation on Wednesday, March 17, 2010 . Problems 1. Second-order systems The impulse response of a second-order CT system has the form h ( t ) = e σt cos( ω d t + φ ) u ( t ) where the parameters σ , ω d , and φ are related to the parameters of the characteristic polynomial for the system: s 2 + Bs + C . a. Determine expressions for σ and ω d (not φ ) in terms of B and C . b. Determine the time required for the envelope e σt of h ( t ) to diminish by a factor of e , the period of the oscillations in h ( t ) , and the number of periods of oscillation before h ( t ) diminishes by a factor of e . Express your results as functions of B and C only. c. Estimate the parameters in part b for a CT system with the following poles: 10 100 100 s -plane The unit-sample response of a second-order DT system has the form n h [ n ] = r 0 cos(Ω 0 n + Φ) u [ n ] where the parameters r 0 , 0 , and Φ are related to the parameters of the characteristic polynomial for the system: z 2 + Dz + E . d. Determine expressions for r 0 and 0 (not Φ ) in terms of D and E . e. Determine n the length of time required for the envelope r 0 of h [ n ] to diminish by a factor of e . the period of the oscillations in h [ n ] , and the number of periods of oscillation in h [ n ] before it diminishes by a factor of e . Express your results as functions of D and E only.

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2 6.003 Homework 6 / Spring 2010 0 . 938 0 . 149 z -plane f. Estimate the parameters in part e for a DT system with the following poles: 2. Maximum gain For each of the following systems, find the frequency ω m for which the magnitude of the gain is greatest. 1 a. 2 1 + s + s b. 1 + s s + s 2 2 s c. 2 1 + s + s Compare the ω m for these systems and explain qualitatively any similarities or differences.
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MIT6_003S10_assn06 - 6.003 Homework 6 Due at the beginning...

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