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20115ee102_1_hw5 - 1 EE102 Fall Quarter 2011 Systems and...

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1 EE102 Systems and Signals Fall Quarter 2011 Jin Hyung Lee Homework #5 Due: Wednesday, November 9, 2011 at 5 PM. 1. Each of these signals can be written as a sum of scaled and shifted unit rectangles and triangles, 2 -1 1 0 -2 1 2 -2 -1 x(t) a) 2 -1 1 0 -2 1 2 -2 -1 t x(t) b) Find a simple expression for each signal, and then compute the Fourier transform. 2. Determine whether the assertions are true or false, and provide a supporting argument. (a) If ( f * g )( t ) = f ( t ) , then g ( t ) must be an impulse, δ ( t ) . (b) If the convolution of two functions f 1 ( t ) and f 2 ( t ) is identically zero, ( f 1 * f 2 )( t ) = 0 then either f 1 ( t ) or f 2 ( t ) is identically zero, or both are identically zero. 3. 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 ) . 4. Generalized Parseval’s Theorem
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2 (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 )
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