ecen314-fall11-hw3

# ecen314-fall11-hw3 - x 2 n = u n[4 pt(c Determine the...

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ECEN 314: Signals and Systems, Fall ‘11 Homework #3 Homework Assignment #3 Due date – Sep. 29, 2011 (Thu), 3:55PM in class. Problem 1. Convolution (9 points) Consider an input x [ n ] to an LTI system with an impulse response h [ n ] as shown below. Find the output y [ n ] of the system for each case. [3 pt] (a) x [ n ] = δ [ n ] + 2 δ [ n - 1] - δ [ n - 3] , h [ n ] = 2 δ [ n + 1] + 2 δ [ n - 1] . [3 pt] (b) x [ n ] = (1 / 2) n - 2 u [ n - 2] , h [ n ] = u [ n + 2] . [3 pt] (c) x [ n ] = (1 / 3) - n u [ - n - 1] , h [ n ] = u [ n - 1] . Problem 2. Convolution (7 points) Let x ( t ) = u ( t - 3) - u ( t - 5) and h ( t ) = e - 3 t u ( t ) . [3 pt] (a) Compute y ( t ) = x ( t ) * h ( t ) . [2 pt] (b) Compute g ( t ) = ( dx ( t ) /dt ) * h ( t ) . [2 pt] (c) How is g ( t ) related to y ( t ) ?

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Problem 3. Difference Equation (12 points) Consider the difference equation y [ n ] - 1 2 y [ n - 1] = x [ n ] . Suppose the systems is initially at rest. [4 pt] (a) Determine the output y 1 [ n ] when the input is x 1 [ n ] = δ [ n ] . [4 pt] (b) Determine the output
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Unformatted text preview: x 2 [ n ] = u [ n ] . [4 pt] (c) Determine the output y 3 [ n ] when the input is x 3 [ n ] = (1 / 3) n u [ n ] . Problem 4. Differntial Equation (12 points) Consider a system whose input x ( t ) and output y ( t ) satisfy the following differential equation dy ( t ) dt + 3 y ( t ) = x ( t ) . Assume that the system is initially at rest. [4 pt] (a) Determine the output y 1 ( t ) when the input is x 1 ( t ) = e 2 t u ( t ) [4 pt] (b) Determine the output y 2 ( t ) when the input is x 2 ( t ) = e t u ( t ) [4 pt] (c) Determine the output y 3 ( t ) when the input is x 3 ( t ) = 6 e 2 t u ( t )-12 e t u ( t )...
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