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ps9-Problem1Only

# ps9-Problem1Only - 6.450 Principles of Digital...

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6.450 Principles of Digital Communication Wednesday, November 6, 2002 MIT, Fall 2002 Handout #32 Due: Wednesday, November 13, 2001 Problem Set 9 Problem 9.1 (Carrierless modulation) In Lecture 13, we saw how to modulate a base- band QAM waveform to passband and then demodulate by shifting down to passband, followed by filtering and sampling at baseband. This exercise explores the interesting con- cept of completely eliminating the baseband operations by modulating and demodulating directly at passband. (a) Let { a k } be a complex data sequence and let u ( t ) = k a k p ( t kT ) be the corre- sponding modulated output. Let p ˆ( f ) be equal to T over f [3 / (2 T ) , 5 / (2 T )] and be equal to 0 elsewhere. At the receiver, u ( t ) is filtered using p ( t ) and the output y ( t ) is then T-space sampled at time instants kT . Show that y ( kT ) = a k for all k Z . Don’t worry about the fact that the transmitted waveform u ( t ) is complex. (b) For what other brick-wall positive frequency filters P ( f ) would this scheme work? Could it work if P ( f ) was not brickwall? (c) We cannot send complex signals over real channels. Give a general method for sending a complex signal u ( t ) whose spectrum is bandlimited to a given positive-frequency band ˆ [ f 0 , f 1 ] through a real channel with a brick wall

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ps9-Problem1Only - 6.450 Principles of Digital...

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