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# HW13 - ECE 301 Homework#13 due date...

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ECE 301, Homework #13, due date: 11/24/2010 http://cobweb.ecn.purdue.edu/ chihw/10ECE301F/10ECE301F.html Question 1: Consider an AM-DSB signal y ( t ) = ( x ( t ) + 0 . 5) cos(3 πt ) is sent by a trans- mitter. 1. In what condition can we use asynchronous demodulation for the above system so that x ( t ) can be reconstructed perfectly? 2. Suppose the receiver receives y ( t ) as follows. What is the end result if we use asynchronous demodulation to construct the original signal x ( t ). -3 -2 -1 0 1 2 3 -1.5 -1 -0.5 0 0.5 1 1.5 y(t) t 3. The receiver realizes that the synchronous carrier at the receiver side is actually sin(3 πt ) instead of cos(3 πt ). Explain why it changes from cos(3 πt ) to sin(3 πt ). 4. How to construct x ( t ) from y ( t ) using synchronous demodulation. Please describe it by a diagram and explicitly specify all the frequencies involved. 5. What is the end result of using synchronous demodulation to reconstruct x ( t )? Question 2: [Basic] Find and plot the DTFT of x [ n ] = k = -∞ δ [ n - 4 k ]. Find and plot the CTFT of y ( t ) = k = -∞ δ ( t - πk ). Observe the similarity between both your answers. Hint: you should use the generalized DTFT and CTFT formulas. Namely, find the

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