102_1_hw5 - 1 EE102 Spring 2009-10 Systems and Signals Lee...

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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 π t ) 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 π t ) . Let the convolution of the two signals be f ( t ) = ( f 1 * f 2 )( t ) (a) Find the Fourier transform F [ f ( t )] = F ( j ω ) . (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 ( j ω ) and F 2 ( j ω ) , show that -∞ f 1 ( t ) f * 2 ( t ) dt = 1 2 π -∞ F 1 ( j ω ) F * 2 ( j ω ) d ω 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 -∞ f 1 ( t ) f 2 ( t ) dt = 1 2 π -∞ F 1 ( j ω ) F 2 ( - j ω ) d ω
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2 4. Evaluate the integral: -∞ 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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