ELEC 242 - Midterm 2.0 Solution (2016-W).pdf - ELEC242 Continuous-Time Signals and Systems Question 1 Consider a continuous-time LTI system which has

# ELEC 242 - Midterm 2.0 Solution (2016-W).pdf - ELEC242...

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ELEC242 Continuous-Time Signals and Systems Concordia University 1 Sample Midterm Exam II Question 1: Consider a continuous-time LTI system which has impulse response of ) 1 ( ) ( ) ( - - = t u t u t h . If { } ) 1 ( ) ( ) 1 ( ) ( - - - = t u t u t t x is applied at the input of the system, evaluate the output ) ( t y of the system using convolution integral - - = τ τ τ d t h x t y ) ( ) ( ) ( as follows: a) Draw ) ( τ x and ) ( τ - t h for different intervals of “ t ”. b) Write the expressions for calculating the output ) ( t y for the intervals of “ t indicated in part (a). Do not calculate the result of integrals. Solution 1a: - - = τ τ τ d t h x t y ) ( ) ( ) ( Therefore, we draw ) ( τ x and ) ( τ - t h and perform the convolution: 1 ) ( τ - h τ -1 c) 1 ) ( τ h τ 1 b) 1 ) ( τ x τ 1 a)

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ELEC242 Continuous-Time Signals and Systems Concordia University 2 Sample Midterm Exam II Solution 1b: Based on the overlap of ) ( τ x in figure “a” and ) ( τ - t h in figures “d”, “e”, “f” and “g”, ) ( t y can be calculated for various regions as shown below: < < < + - = - = - < < - = - = - < = - - ) . 2 0 ) . 2 1 2 2 2 2 ) 1 ( ) . 1 0 2 2 ) 1 ( ) . 0 0 ) ( 1 1 2 1 1 2 0 2 0 2 g Fig t f Fig t t t d e Fig t t t d d Fig t t y t t t t τ τ τ τ τ τ τ τ 1 t t h < - 2 ) ( τ τ g) 1 t 1 - t 1 1 2 1 ) ( < < - t t h τ τ f) t 1 - t 1 1 1 0 ) ( < < - t t h τ τ e) t 1 - t 1 1 0 ) ( < - t t h τ τ d) t 1 - t
ELEC242 Continuous-Time Signals and Systems

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