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Unformatted text preview: Homework 7 Solution (ECE220  Fall 2007) 1. (40 points) Modulation of signals (Upconversion, Downconversion) Property to use: Duality between Fourier Transforms properties and properties in the time domain. Premise: One of the most basic ways of sharing a channel using wireless RF transmission/optical or audio waves is that of modulating the signals with a sinusoidal waveforms: literally this is a jargon term that means using either the amplitude (AM) or the phase (more properly its derivative, from which comes the name FM) to transmit a signal. More generally, the problem of sharing the spectrum among several signals is called the problem of multiplexing signals. Using a modulating sinusoid is a technique called frequency multiplexing. Assume the signal we want to transmit is m ( t ) and we have been designated a carrier frequency of f . That means that rather than having the signal vary with its own natural frequency con tent, we want to center its frequency content around the frequency f . How do we do that? AM signal: m ( t )cos(2 f t + ) FM signal: cos 2 f t + a R t m ( ) d (where a is called modulation index) How do we know that we achieved the desired effect? You will learn FM in a later com munication course. However, you already know enough to fully understand AM. The first thing to understand is what happens if you multiply to signals in time. More specifi cally: (a) (10 points) What is the Fourier Transform of x ( t ) y ( t )? You want to use 2 things here: i) Do you know what is the inverse Fourier transform of a product X(f)Y(f)? Please find out because you will need this. ii) Fourier and time domains are dual of each other, as it is clarified in section 4.3.6....
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This note was uploaded on 05/31/2008 for the course ECE 2200 taught by Professor Johnson during the Fall '05 term at Cornell University (Engineering School).
 Fall '05
 JOHNSON

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