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Unformatted text preview: 6.003 Homework 6 Due at the beginning of recitation on Wednesday, October 21, 2009 . Problems 1. DT convolution Let y represent the DT signal that results when f is convolved with g , i.e., y [ n ] = ( f ∗ g )[ n ] which is sometimes written as y [ n ] = f [ n ] ∗ g [ n ]. Determine closed-form expressions for each of the following: y [ n ] f [ n ] g [ n ] y a [ n ] u [ n ] u [ n ] y b [ n ] u [ n ] ( 1 2 ) n u [ n ] y c [ n ] ( 1 2 ) n u [ n ] ( 1 3 ) n u [ n ] y d [ n ] ( 1 2 ) n u [ n ] ( 1 2 ) n u [ n ] 2. CT convolution Let y represent the CT signal that results when f is convolved with g , i.e., y ( t ) = ( f ∗ g )( t ) which is sometimes written as y ( t ) = f ( t ) ∗ g ( t ). Determine closed-form expressions for each of the following: y ( t ) f ( t ) g ( t ) y a ( t ) u ( t ) u ( t ) y b ( t ) u ( t ) e − t u ( t ) y c ( t ) e − t u ( t ) e − 2 t u ( t ) y d ( t ) e − t u ( t ) e − t u ( t ) 6.003 Homework 6 / Fall 2009 2 3. 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 h [ n ] = r n cos(Ω n + Φ)...
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- Spring '11
- Impulse response, unit-sample response, Determine closed-form expressions