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
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
)
dτ
·
(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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 Fall '05
 JOHNSON
 2J

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