ps5_08 - EE 350 PROBLEM SET 5 DUE: 13 October 2008 Reading...

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EE 350 PROBLEM SET 5 DUE: 13 October 2008 Reading assignment: Lathi Sections 3.1 through 3.3 Exam II is scheduled for Thursday, October 16 from 8:15 pm to 10:15 pm in 102 Forum (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. Recitations sections will meet during the week of September 29. Problem 24: (12 points) Using the relationship f ( t ) * δ ( t - T )= f ( t - T ) (1) derived in lecture and in Problem 21, part 1, evaluation of the convolution integral y ( t f ( t ) * h ( t ± -∞ f ( τ ) h ( t - τ ) is simple when either f ( t )or h ( t ) is a sum of weighted impulses. This problem extends this result to the case where either the derivative of either f ( t h ( t ) yields a sum of weighted impulses. 1. (3 points) As as an example of the utility of equation (1), suppose that y ( t f ( t ) * h ( t ) where f ( t - 2 δ ( t - 4) h ( t u ( t +1) - u ( t - 1) . Determine y ( t ), and sketch f ( t ), h ( t ), and y ( t ) on a single plot. 2. (9 points) Now suppose we apply the input f ( t u ( t ) - u ( t - 1) to a LTI system that has the impulse response function h ( t )=2 u ( t - 2) - 2 u ( t - 3) . Neither f ( t 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/dt in terms of impulse functionals and sketch df/dt . (3 points) Let g ( t ) denote the response of the system to the input df/dt , that is g ( t df dt * h ( t ) . Calculate and sketch g ( t ). (1 point) Using the result from Problem 21 part 4, show the zero-state response of the system to f ( t ) can be expressed as y ( t ± t -∞ g ( τ ) dτ. (3 points) Using the last two results, calculate and sketch y ( t ).
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Problem 25: (12 points) In lecture it was shown that when the impulse response function h ( t ) is a causal signal, then the system is causal. Conversely, if the system impulse response is noncausal then the system is noncausal. To illustrate this important concept, consider two LTI systems that are represented by the impulse response functions System 1: h 1 ( t )= u ( t +1) - u ( t - 1) System 2: h 2 ( t u ( t ) - u ( t - 2) . 1. (2 points) Sketch h 1 ( t ) and h 2 ( t ), and specify whether or not each impulse response function is a causal or noncausal signal. 2. (2 points) Specify whether or not each system is casual. 3. (8 points) Using convolution, Fnd and sketch the zero-state unit-step response of each system. By comparing a sketch of the zero-state unit-step response to a sketch of the input f ( t u ( t ), explain why which system, if any, is noncausal. Problem 26: (12 points) 1. (8 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 tu ( t ) Classify each system, corresponding to the impulse functions considered above, as either BIBO stable or not BIBO stable. In order to receive credit, justify your answer.
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ps5_08 - EE 350 PROBLEM SET 5 DUE: 13 October 2008 Reading...

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