102_1_hw03 - 1 EE102 Spring 2009-10 Systems and Signals Lee...

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1 EE102 Spring 2009-10 Lee Systems and Signals Homework #3 Due: Tuesday April 27, 2010 at 5PM. 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 ) = t -∞ x ( τ ) d τ (b) y ( t ) = t t - T x ( τ ) d 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 ) g ( t ) = rect( t - 1) and plot the result. 4. Show the somewhat surprising result that the convolution of two impulse functions, y ( t ) = -∞ δ ( τ ) δ ( t - τ ) d 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 ( t ) = ( x * δ )( t ) , where δ ( t ) is the derivative of δ ( t ) , described in Lecture 3. If f * g = y , show that f * g = y . 6. Cross-correlation and convolution are closely related. Show that if x ( t ) and y ( t ) are two signals, and (note the “ ” for cross correlation!) ( x y )( t ) = z ( t ) then ( y x )( t ) = z ( - t ) meaning if we change the order of the signals in the cross correlation, we time reverse the resulting signal. 7. An exponential Fourier series can be simplified into the following form: f ( t ) = cos(3 π t ) + 1 2 sin(4 π t ) . (a) What is the fundamental ( i.e. shortest) period T 0 for f ( t ) ? (b) Which Fourier coefficients are non-zero, and what are their values? 8. Suppose that f ( t ) is a periodic signal with period T 0 , and that f ( t ) has a Fourier series f ( t ) = n = -∞ D n e jn ω 0 t If a is a positive real number, find a Fourier series for f ( at ) in terms of the Fourier series coefficients for f ( t ) . Note that if the period of f ( t ) is T 0 , the period of f ( at ) is T 0 /a . 9. Suppose that f ( t ) is a periodic signal with period T 0 , and that f ( t ) has a Fourier series. If τ is a real number, show that f ( t - τ ) can be expressed as a Fourier series identical to that for f ( t ) except for the multiplication by a complex constant, which you must find.
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3 MATLAB Homework 3 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.
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