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20105ee102_1_hw3

# 20105ee102_1_hw3 - 1 EE102 Fall Quarter 2010 Systems and...

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1 EE102 Systems and Signals Fall Quarter 2010 Jin Hyung Lee Homework #3 Due: Wednesday, October 20, 2010 at 5 PM. 1. It is often useful to represent operations on signals as convolutions. For each of the follow- ing, ﬁnd a function h ( t ) such that y ( t ) = ( x * h )( t ) . (a) y ( t ) = Z t -∞ x ( τ ) (b) y ( t ) = Z t t - T x ( τ ) (c) y ( t ) = x ( t ) (d) y ( t ) = x ( t - 1) 2. Graphically compute the convolution of these two functions: 3 -1 0 1 2 1 2 f ( t ) t 3 -1 0 1 2 1 2 t g ( t ) 3. 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. 4. 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.

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2 5. The derivative of a function x ( t ) may be written as the convolution x 0 ( t ) = ( x * δ 0 )( t ) , where δ 0 ( t ) is the derivative of δ ( t ) . If f * g = y , show that f 0 * g 0 = y 00 . (hint: try to use properties of convolution) 6. 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 doesn’t contain any explicit convolutions. d dt * g ( t ) x ( t ) y ( t ) * δ ( t - 1 )
3 Matlab Assignment #2 This laboratory will be concerned with numerically evaluating continuous time convolution integrals. Matlab provides a function conv() that performs a discrete-time convolution of two discrete-time sequences. We will add a new function to matlab that uses conv() to numerically integrate the continuous time convolution. To do this, we’ll need to learn about how to deﬁne new functions in matlab. Matlab provides two ways of executing commands that you have programmed in a ﬁle. Both are stored in a ”.m” ﬁle or m-ﬁle. This ﬁrst is a script . Invoking the script at the command line causes the ﬁle to be treated as terminal input in your current matlab environment. Most of the examples in the book are matlab scripts. Variables created in the script are visible when it

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20105ee102_1_hw3 - 1 EE102 Fall Quarter 2010 Systems and...

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