# hw 5 - ( t-n ) , for integer n ? Shifted sincs will play an...

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1 EE102 Spring 2009-10 Lee Systems and Signals Homework #5 Due: Tuesday, May 18, 2010 1. Given the signal f ( t ) = sinc ( t ) , evaluate the Fourier transforms of the following signals. Provide a labeled sketch for each function and its Fourier transform. (a) f ( t ) (b) f ( t - 1) (c) 1 2 ( f ( t - 1) - f ( t + 1)) (d) t f ( t ) (e) f ( t ) cos(10 πt ) 2. Two signals f 1 ( t ) and f 2 ( t ) are deﬁned as f 1 ( t ) = sinc (2 t ) f 2 ( t ) = sinc ( t ) cos(2 πt ) . Let the convolution of the two signals be f ( t ) = ( f 1 * f 2 )( t ) (a) Find the Fourier transform F [ f ( t )] = F ( ) . (b) Find a simple expression for f ( t ) . 3. Generalized Parseval’s Theorem (a) Given two possibly complex signals f 1 ( t ) and f 2 ( t ) with Fourier transforms F 1 ( ) and F 2 ( ) , show that Z -∞ f 1 ( t ) f * 2 ( t ) dt = 1 2 π Z -∞ F 1 ( ) F * 2 ( ) This is another form of Parseval’s theorem, which reduces to the form we discussed in class if f 1 ( t ) = f 2 ( t ) . (b) If f 1 ( t ) and f 2 ( t ) are real, show Z -∞ f 1 ( t ) f 2 ( t ) dt = 1 2 π Z -∞ F 1 ( ) F 2 ( - )

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2 4. Evaluate the integral: Z -∞ sinc ( t - n ) sinc ( t - k ) dt where n and k are integers. What property describes the family of signals sinc
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Unformatted text preview: ( t-n ) , for integer n ? Shifted sincs will play an very important roll when we discuss reconstructing a continuous signal from its samples. Hint: Use the form of Parseval’s theorem from the previous question. Recall that the inte-gral of a complex exponential over an integer number of periods is zero! 5. Consider the signal y ( t ) that is deﬁned as y ( t ) = x ( t ) * x (-t ) where x ( t ) is a real signal. (a) Find an expression for y ( at ) in terms of x ( t ) , where a is a positive constant. (b) Is y ( t ) even, odd, or neither? (c) Is Y ( jω ) even or odd? Is it real or imaginary? (d) If we know y ( t ) , can we uniquely determine x ( t ) ?...
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## This note was uploaded on 10/21/2010 for the course EE ee102 taught by Professor Levan during the Spring '09 term at UCLA.

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hw 5 - ( t-n ) , for integer n ? Shifted sincs will play an...

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