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HW7_ece220_2007_solution

# HW7_ece220_2007_solution - Homework 7 Solution(ECE220 Fall...

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Homework 7 Solution (ECE220 - Fall 2007) 1. (40 points) Modulation of signals (Up-conversion, Down-conversion) 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 0 . 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 0 . How do we do that? AM signal: m ( t ) cos (2 πf 0 t + φ ) FM signal: cos 2 πf 0 t + a R t -∞ m ( τ ) · (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.

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HW7_ece220_2007_solution - Homework 7 Solution(ECE220 Fall...

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