102_1_hw03

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, ±nd a function h ( t ) such that y ( t )=( x * h )( t ) . (a) y ( t )= ± t -∞ x ( τ ) (b) y ( t ± 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 ) 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 ) is itself an impulse function. Hint: To make sense of the integral, ±rst replace one of the impulses with lim ± 0 g ± ( t ) , where g ± ( t ) is g ± ( t ² 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 simpli±ed into the following form: f ( t ) = cos(3 πt )+ 1 2 sin(4 ) . (a) What is the fundamental ( i.e. shortest) period T 0 for f ( t ) ? (b) Which Fourier coef±cients 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 jnω 0 t If a is a positive real number, ±nd a Fourier series for f ( at ) in terms of the Fourier series coef±cients 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 ±nd.
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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 deFne new functions in matlab.
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102_1_hw03 - 1 EE102 Spring 2009-10 Systems and Signals Lee...

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