tut1_problems_17_01_09

tut1_problems_17_01_09 - h ( t ), show that Im { H ( f ) }...

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Department of Electrical Engineering EE 308: Communication Systems (Spring 2009) Course Instructor: Prof. Abhay Karandikar Problem Set 1 1. What is the Fourier transform of a function that is (a) Real and even? (b) Real and odd? (c) Imaginary and even? (d) Imaginary and odd? 2. For k > 0, ±nd the Fourier transform of g ( t ) = 2 k k 2 +(2 πt ) 2 and use it to evaluate i -∞ dx (1+ x 2 ) 2 . 3. Let g ( t ) = e - a | t | sgn( t ) ,a > 0. (a) Find G ( f ), the Fourier transform of g ( t ). (b) What happens to G ( f ) as a 0? (c) Can you now evaluate the Fourier transform of the step function u ( t )? 4. Let J n ( x ) be the Fourier coe²cients of the function f ( θ ) = cos( x sin θ ). Show that sinc( x ) * J 0 ( πx ) = J 0 ( πx ). 5. Let x ( t ) be a periodic square wave with amplitude A and period T . Find the power in x ( t ). What happens when T → ∞ ? 6. Given a causal signal
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Unformatted text preview: h ( t ), show that Im { H ( f ) } is the Hilbert transform of Re { H ( f ) } . What is the Hilbert transform of Im { H ( f ) } ? 7. The signal x ( t ) = sinc 2 40 t is to be transmitted using AM with μ < 1. (a) Find the Fourier transform of x ( t ). (b) Sketch the spectrum of x c ( t ). (c) What is the bandwidth of the transmitted signal? 8. The signal x ( t ) = 4 sin π 2 t is transmitted using DSB. Is there any limitation on the range of carrier frequencies that can be used? Explain. 9. The signal x ( t ) = cos2 π 40 t + 0 . 5 cos2 π 90 t is transmitted using DSB. Sketch the positive-frequency ‘line’ (what is this?) spectrum. 1...
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This note was uploaded on 02/05/2012 for the course EE EE308 taught by Professor B.k.dey during the Spring '09 term at IIT Bombay.

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