EEL 3135: DiscreteTime Signals and Systems
Dr. Fred J. Taylor, Professor
Lesson Title: Modulation
Lesson Number: 33 [Section 12.2 – 12.3]
Background:
Communications and wireless communications is an enabling technology.
Over the years the
textures of communications and communication protocols have evolved from simple to complex.
The basic structure of a communications system is abstracted in Figure 1.
For commercial
radio applications, the two most commonly encountered protocols are amplitude modulation
(AM) and frequency modulation (FM).
Their environments are summarized in Table 1.
Figure 1:
Communications system.
Table 1:
Commercial Radio
Item
AM
FM
Carrier frequency
5401600 kHz
88.1107.9 MHz
Transmission Bandwidth
10kHz
200kHz
Audio bandwidths
35kHz
15kHz
AM Modulation
In Chapter 12 basic modulation and demodulation schemes, which are core to communications,
are presented and analyzed.
The basic building blocks are multipliers (sometime referred to
mixers), adders, and filters.
Even though these technologies have been previously studied, but
will be reviewed in depth when necessary.
This process will begin with a study of the mixer
(multiplier) shown in Figure 2 which defines a communications process called
doublesideband
amplitude modulation
(DSAM or AMDS).
Figure 2:
Basic mixer (modulator) circuit.
The information carried by the signal y(t) = x(t) cos(
ϖ
c
t), where
ϖ
c
is called the
carrier
frequency,
rides on the time varying envelope of the cosine ware and is correlated to the
information process x(t), thus giving rise to the name amplitude modulation.
The output of the
mixer can be expressed as:
y(t) = (1/2)x(t)e
j
ϖ
c
t
+
(1/2)x(t)e
j
ϖ
c
t
1.
Information
Transmitter
Channel
Receiver
Information
1
x(t)
⊗
cos(
ϖ
c
t)
y(t) = x(t) cos(
ϖ
c
t)
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View Full DocumentEEL 3135: DiscreteTime Signals and Systems
Dr. Fred J. Taylor, Professor
From previously studied applications of the Fourier transform, it then follows that the spectrum
of y(t) is given by:
Y(j
ϖ
) = (1/2)(X(j(
ϖ

ϖ
c
) + X(j(
ϖ
+
ϖ
c
))
2.
Since multipliers are nonlinear device, they have the ability to create new frequencies beyond
those externally supplied signal sources which is the case here.
Example
Consider the case where the modulating signal x(t) is a square pulse train with a DC level (bias)
and the carrier frequency is 5 times higher than the repetition frequency of x(t).
The outcome is
shown in Figure 3.
The envelope of the carrier signal c(t) is seen to be correlated to that of x(t).
In Figure 4 the magnitude frequency response of x(t), c(t), and y(t) are displayed.
It can be
seen that the spectrum of carrier is essentially a twosided line spectrum.
The modulated output
is also seen to have the spectrum of x(t) centered about each of the carrier sidebands (upper
and lower). A question to be engaged momentarily is how noe determined the location of these
subband frequencies.
» t=0:.001:1; x=(0.125*square(2*pi*5*t))+1;
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 Spring '08
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 Dr. Fred J. Taylor, Dr. Fred J., Fred J. Taylor

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