20115ee102_1_hw3

# 20115ee102_1_hw3 - x ( t ) = ( x * )( t ) , where ( t ) is...

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1 EE102 Systems and Signals Fall Quarter 2011 Jin Hyung Lee Homework #3 Due: Wednesday, October 19, 2011 at 5 PM. 1. Analytically compute the convolution ( f * g )( t ) , where f ( t ) and g ( t ) are f ( t ) = u ( t ) e - t g ( t ) = rect( t ) and sketch a plot of the result. 2. Show the somewhat surprising result that the convolution of two impulse functions, y ( t ) = Z -∞ δ ( τ ) δ ( t - τ ) is itself an impulse function. Hint: To make sense of the integral, ﬁrst replace one of the impulses with lim ± 0 g ± ( t ) , where g ± ( t ) is g ± ( t ) = ( 1 | t | < ±/ 2 0 otherwise This is a model for an impulse function, as we discussed in class. After the convolution, show you get the same model function back. 3. The derivative of a function x ( t ) may be written as the convolution

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Unformatted text preview: x ( t ) = ( x * )( t ) , where ( t ) is the derivative of ( t ) . If f * g = y , show that f * g = y 00 . (hint: try to use properties of convolution) 4. The output of this system y ( t ) is a convolution of the input x ( t ) and an impulse response h ( t ) . Find a simple expression for the impulse response h ( t ) . Simplify the expression so that it doesnt contain any explicit convolutions. d dt * g ( t ) x ( t ) y ( t ) * ( t-1 ) 2 5. Graphically compute the convolution of these two functions: 3-1 1 2 1 2 f ( t ) t 3-1 1 2 1 2 t g ( t )...
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## This note was uploaded on 03/30/2012 for the course ELEC ENGR 102 taught by Professor Lee during the Fall '11 term at UCLA.

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20115ee102_1_hw3 - x ( t ) = ( x * )( t ) , where ( t ) is...

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