# ps5 - EE 350 PROBLEM SET 5 DUE: 16 October 2007 Laboratory...

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EE 350 PROBLEM SET 5 DUE: 16 October 2007 Laboratory sections will meet during the week of October 8. Reading assignment: Lathi Sections 3.1 and 3.2 Exam II is scheduled for Thursday, October 18 from 8:15 pm to 10:15 pm in 121 Sparks (all sections). The second exam covers material from Problem Sets 4 and 5, and Laboratory #2. The date and location of a review session will be announced on the EE 350 web page. Problem 22: (16 points) In Problem Set 4, Problem 19, you showed that f ( t ) δ ( t T )= f ( t T ) . Using this relationship you will now investigate the relationship between causal signals and causal systems, and simplify the task of calculating the zero-state response in situations where either the derivative of either f ( t )o r h ( t ) yields a sum of weighted impulses. 1. (6 points) Consider a LTI system with input f ( t ), impulse response h ( t ), and zero-state response y ( t ). (a) (1 point) Using the convolution integral, show that the input f ( t δ ( t ) results in the zero-state response y ( t h ( t ). (b) (3 points) Suppose that h ( t u ( t +1) u ( t 1) and f ( t δ ( t ). Is h ( t ) a causal or noncausal signal? Find the zero-state response y ( t ). Based on the relationship between the given input f ( t ) and the resulting zero-state response y ( t ), is the system causal or noncausal? (c) (2 points) Consider an arbitrary LTI system with input f ( t ), impulse response function h ( t ), and the zero-state response y ( t Z −∞ f ( τ ) h ( t τ ) dτ. What restriction must be placed on h ( t ) so that the system is causal? Justify your answer. 2. (10 points) Suppose we apply the input f ( t )=2 u ( t 2 u ( t ) to a LTI system that has the impulse response function h ( t u ( t 1) u ( t 3) . Neither f ( t )or h ( t ) is expressed directly as a sum of weighted impulse functionals. However, careful inspection of f ( t ) reveals that its derivative with respect to time can be expressed as the sum of two impulse functionals. (2 points) Find an expression for df / d t in terms of impulse functionals and sketch . (2 points) Let g ( t ) denote the response of the system to the input ,thatis g ( t df dt h ( t ) . Calculate and sketch g ( t ). (3 points) Using the result from Problem 19 part 4, show the zero-state response of the system to f ( t )canbe expressed as y ( t Z t −∞ g ( τ ) dτ. (3 points) Using the last two results, calculate and sketch y ( t ).

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Problem 23: (14 points) 1. (9 points) Consider three linear time-invariant systems whose impulse responses h ( t ) are specifed as h 1 ( t )= e t u ( t ) h 2 ( t u ( t +1) h 3 ( t )=[ e 2 t 5 e 10 t ] u ( t ) (a) (3 points) ClassiFy each impulse response as either a causal or noncausal signal . In order to receive credit, justiFy your answer. (b) (3 points) ClassiFy each system , corresponding to the impulse Functions considered above, as either causal or noncausal. In order to receive credit, justiFy your answer.
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## ps5 - EE 350 PROBLEM SET 5 DUE: 16 October 2007 Laboratory...

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