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PROFESSOR: In this lecture we
begin a discussion of the
topic of modulation, which is,
among other things, a very
important topic in
practical terms.
For example, it forms the
cornerstone for many
communication systems.
And also, as we'll see as these
lectures go along, a
particular form of modulation
referred to as pulse amplitude
modulation, and eventually
impulse modulation or impulse
train modulation, forms a very
important bridge between
continuous time signals and
discrete time signals.
Now in general terms what
we mean when we refer to
modulation is the notion of
using one signal to vary a
parameter of another signal.
For example, a sinusoidal signal
has three parameters,
amplitude, frequency,
and phase.
And we could think, for example,
of using one signal
to vary, let's say,
the amplitude of
a sinusoidal signal.
And what that leads to is a
notion, which we'll develop in
some detail, referred to
as sinusoidal amplitude
modulation, and would correspond
to a sinusoidal
signal, referred to as the
carrier, and it's amplitude
being varied on the basis
of another signal.
Now alternatively we could think
of varying either the
frequency or the phase of a
sinusoidal signal, again with
another signal.
And what that leads to is
another very important notion,
which is referred to sinusoidal
frequency
modulation, where essentially
it's the frequency of a
sinusoid that's changing
depending on the signal that's
we're using to modulate
the sinusoid.
Now sinusoidal amplitude,
frequency, and phase
modulation are extremely
important topics and ideas in
the context of communication
systems.
One of the reasons is that if
you want to transmit a signal,
let's say for example a voice
signal, the voice signal that
you're listening to now.
If you try to transmit that over
long distances, because
of the frequencies involved
the medium that you use to
transmit it won't carry
it long distances.
The idea then is to essentially
take that signal,
like a voice signal, use
it to modulate a much
higher-frequency signal,
and then transmit that
higher-frequency signal over a
medium that essentially can
support long-distance
transmission at those
frequencies.
Then at the other end of course,
the voice information,
or whatever else the information
is, is taken off.
Now also, a notion that that
leads to, and we'll be
developing in some detail,
is the idea that you can
simultaneously transmit more
than one signal by in essence
taking several voice signals or
other signals, using them
to modulate either the frequency
or amplitude of
sinusoidal signals at different
frequencies, adding
all those together--
that's a process called
multiplexing--
and then at the other end of
the transmission system,
taking those sinusoidal
signals apart.
And then extracting the
envelope or frequency
modulation information to get
back to the voice signal or
other information-carrying
signal.
So that's one of the very
important ways in which
sinusoidal modulation is used
in communication systems.
And what we'll see, in
particular as we go through
today's lecture, is that
sinusoidal amplitude
modulation, follows in a fairly
straightforward way
from the properties of the
Fourier transform that we've
developed in some of the
earlier lectures.
So our focus in today's lecture
will be on sinusoidal
amplitude modulation
in continuous time.
In the next lecture we'll
consider the same set of
notions related to discrete
time, and also a concept
referred to as pulse amplitude
modulation.
And all of these follow, in a
very straightforward way, from
the modulation property for
the Fourier transform.
Issues of frequency and phase
modulation are a little more
difficult to analyze.
But many of the techniques that
we've developed in the
previous lectures also provide
important insights into
frequency and phase
modulation.
And some of this is developed
in more detail in the book.
So what I'd like to do is focus,
for now, on the concept
of amplitude modulation.
And as I indicated, there are
several kinds of carrier
signals on which the
modulation can be
superimposed.
The basic structure for an
amplitude modulation system is
one in which there is the
modulating signal, let's say
for example, voice, and a
carrier signal-- what's
referred to as the carrier.
And then of course the resulting
output is the
modulated output.
Now to analyze this, since we
have multiplication in the
time domain, we know from the
property of the Fourier
transform that we've developed
previously--
the modulation property--
that multiplication in the time
domain corresponds to
convolution in the
frequency domain.
And it's this basic property
or equation that lets us
analyze, in some detail in fact,
the notions of amplitude
modulation.
As we go through this lecture
and the next lecture, we'll be
talking, as I indicated, about
several different types of
carrier signals.
One is what's referred
to as pulse carriers.
And that leads to, among other
things, the concept of pulse
amplitude modulation.
That will be deferred until
the next lecture.
In today's lecture what I'll
focus on is first, the case of
a complex exponential carrier,
second, the case
of sinusoidal carrier.
And in fact the complex
exponential carrier and
sinusoidal carrier are obviously
very closely
related, since the complex
exponential carrier is, in
effect, two sinusoidal
carriers.
One for the real part and one
for the imaginary part.
So let's first begin the
discussion of amplitude
modulation by considering a
complex exponential carrier,
and then moving on to
a discussion of
a sinusoidal carrier.
So the issue then is that we
have a signal, x of t.
It's multiplied by a carrier.
And the carrier that we're
considering is a carrier
signal, c of t, of the form
e to the j omega c t
plus theta sub c.
That's the form of our
carrier signal.
And what we can first analyze
is what the resulting signal
or spectrum is at the output
of the modulator.
Well, we can do that by
concentrating on the
modulation property.
And let's consider, just as a
general form for a spectrum,
what I've indicated here for the
Fourier transform of the
input signal or modulating
signal, X of omega.
And so this is intended
to represent the
spectrum of x of t.
And then the carrier signal,
since it's a single complex
exponential, has a Fourier
transform which is an impulse
in the frequency domain.
And the amplitude of the impulse
is 2 pi e to the j
theta sub c, where we notice
that the complex amplitude
incorporates the phase
information.
So now if we multiply in the
time domain, we convolve in
the frequency domain.
And as you know, convolving a
signal with an impulse just
shifts that signal to the
location of the impulse.
And so as a consequence of
taking care of various
factors, what we end up with is
a spectrum that is centered
at the carrier frequency
omega sub c.
So what this says is that if we
have a signal, x of t, and
we use it to modulate a complex
exponential carrier in
the frequency domain, what we've
simply done is to take
the original spectrum and
shift it in frequency.
So that what was originally at
zero frequency is now centered
around the carrier frequency.
We've now modulated, in effect,
to a higher frequency.
Things are happening in a
higher-frequency band.
And the next question is, how do
we demodulate, or in other
words, how do we get the
original signal back?
Of course one way that we
can think of doing it,
particularly in the context of
this specific carrier, if we
look back at the top equation we
have, as the result of the
modulation, x of t times c of
t, where c of t is this.
And we could consider, for
example, just simply dividing
the modulator output by this.
Or equivalently, taking the
modulated output and
multiplying by e to the minus
j omega c t plus theta c.
Let's track that through in
terms of the spectra.
We have, again, the spectrum
of the output of the
modulator, which is the original
spectrum shifted up
to the carrier frequency.
We have, below that, the
spectrum of e to the minus j
omega c t plus theta c.
And if we now convolve this
with this, that results in
simply shifting this spectrum--
except for an issue
of a scale factor--
shifting this spectrum back
down to the origin.
So convolving these two
together, the spectrum that we
end up with is that.
So we can track this through
in the frequency domain.
In the frequency domain it says
shift the spectrum up.
When you want to demodulate,
shift the spectrum back down.
And alternatively, we can look
at it algebraically in the
time domain.
And what it says is, if you
multiply by e to the plus j
omega c t, then when you want to
get back, multiply by e to
the minus j omega c t.
Now one question that you could
conceivably be asking
is, if we're talking about
practical systems and not
simply mathematics, does it make
sense in the real world
to consider using a complex
exponential carrier?
And the answer to that,
in fact, is yes.
That very often in practical
systems one considers using a
carrier which in fact is
a complex exponential.
Well, a complex exponential
is complex.
There's a square root of
minus one in there.
And you could ask well,
how do we get a square
root of minus one?
And the answer is
fairly simple.
Let's look again at the
modulator, which we have here.
And in effect, what that says
is we want to multiply a
real-valued signal by e to the
j omega c t plus theta c.
Now, we can equivalently use
Euler's relationship to break
this down into a cosine
and sine term.
And so what that means in terms
of an implementation,
equivalently, is modulating x
of t onto a cosine carrier.
And that then gives
us the real part
of the complex output.
And modulating it onto a
sinusoidal carrier--
these two being 90 degrees
out of phase--
and that gives us the
imaginary part.
And so in effect, this is
the complex signal.
If we just simply think of
hanging a tag on here that
says square root of minus 1,
or j, and we appropriately
combine complex signals
following the rules of complex
arithmetic.
And indeed, that's exactly the
way things are done in the
real world.
A complex signal is simply a
set of two real signals.
And of course, if we look at the
spectra involved, we have
here the real part and
the imaginary part
of the complex output.
If we again refer back to the
original spectrum, X of omega,
and the modulated spectrum which
I show down here, the
original spectrum shifted up
to the carrier frequency.
In effect we're building
this out of two lines.
One line representing the
real part of that.
And the real part in the time
domain corresponds to the even
part in the frequency domain.
And so with the output of the
cosine modulator, we have a
spectrum that looks like this.
And the output along the
imaginary branch has a
spectrum that looks like this.
Recall in the top branch that
this, for positive frequencies
was positive, and was
positive here.
And so in effect when you add
them, this portion of the
spectrum will cancel out.
So in effect, what we're doing
is building the complex signal
out of two real signals.
Or we're building the spectrum
of the complex signal out of
separate lines that represent
the even and the odd parts.
Now, there are lots of
applications of amplitude
modulation.
And we'll be seeing a number of
these as we go through the
discussion.
What I'd like to do is just
indicate briefly one now,
which is an application that in
fact surfaces fairly often
in the context of a complex
exponential carrier.
And that is the notion of using
modulation to permit the
application of a very well
designed and implemented
low-pass filter to be used as
a band-pass filter and in
fact, as a set of band-pass
filters.
And here's the idea.
The idea is if we have
a fixed filter--
let's say we have a signal.
And we want to think of a
filter, which we want to move
along the signal, one way to
do it is to somehow have
filters that move along
the signal.
The other possibility is to keep
the filter fixed and let
the signal move in frequency
in front of the filter.
Let me be a little
more specific.
Suppose that we have
a signal, x of t.
And we modulate it with a
complex exponential carrier
with a carrier frequency,
omega c.
And the output of that is then
processed with a low-pass
filter and then we demodulate
the result.
Then what we've done is to take
the spectrum of the input
signal, shift it, pull out
what is now around low
frequencies, and then shift that
part of the spectrum back
to where it belongs.
So if we look at that in terms
of actually tracking through
the spectra, we would have
initially a spectrum for the
original signal, which I show
at the top as X of omega.
After modulating or shifting
that spectrum up to a center
frequency of omega c, we then
have what I indicate here.
And the dotted line corresponds
to the pass band
of the low-pass filter.
Well, the result of low-pass
filtering rejects all the
spectrum except the part
around low frequencies.
And the next step is then
to demodulate this.
And so in effect, demodulating
will shift this spectrum back
to where it originally
came from.
And so that result will be
what I show in the final
result, which is here.
And what we can see is that
this is equivalent.
If we can look back at the top
spectrum, this is equivalent
to having extracted, with a
band-pass filter, a section
out of this part of
the spectrum.
So in terms of tracking through
the spectrum and
looking at the equivalent
filtering operation, then what
we accomplished was to pull out
this part of the spectrum
using a low-pass filter
and modulation.
But equivalently what we
implemented was a band-pass
filter as I indicated here.
Now of course, a signal with
this spectrum, since the
spectrum is not conjugate
symmetric, we know that this
signal does not correspond
to a real-valued signal.
Equivalently this filter doesn't
correspond to a filter
whose impulse response
is real.
If we add another step to this,
which is to take the
real part of the output, then by
taking the real part of the
output we would be taking the
even part of the spectrum
associated with that
complex signal.
And the equivalent filter that
we would end up with then is
the filter that I indicate
at the bottom, which is a
band-pass filter.
Now just to reiterate a point
that I made earlier.
A question, of course, is why
would you go to this trouble?
Why not just build a
band-pass filter?
And one of the reasons is that
it's often much easier to
build a fixed filter, a filter
with a fixed-center frequency,
for example a low-pass filter,
than it is to build a filter
that has variable components
in it so that when you vary
them the filter's center
frequency shifts around.
Now, if you want to look at
the energy in a signal in
different frequency bands, then
you'd like to look at it
through different filters.
And so the idea here, which is
really the basis for many
spectrum analyzers, is to build
a really good quality
low-pass filter and then use
modulation, which is often
easier to implement.
Use modulation to shift the
signal essentially in front of
the filter.
So we've worked our way through
modulation with a
complex exponential carrier.
And what we saw, among other
things with a complex
exponential carrier,
is that what it
corresponds to is two branches.
One being modulation with
a cosine, and the other,
modulation with a sine.
And so in the real world, or
in a practical system,
modulation of the complex
exponential carrier really
would be accomplished with
modulation with a sinusoidal
carrier, and in particular with
sinusoidal carriers that
are in quadrature, as it's
referred to, or equivalently
90 degrees out of phase.
Well, in fact sinusoidal
modulation, in other words,
modulation using only a
sinusoidal carrier, very often
is used in its own right not
only for generating a complex
exponential carrier, but
as a carrier by itself.
Let's look at what the
consequences of modulation
with a sinusoidal carrier are.
And in particular work through,
again, what the
spectra are and how we get the
original signal back again.
So we are talking about a
carrier signal which is simply
a sinusoidal signal
with some phase.
And of course we can write that
as the sum of two complex
exponential signals.
And so now, when we apply the
modulation property we have
the original spectrum, which
I show here, X of omega.
And that's convolved with the
spectrum of the carrier.
And the spectrum of the
carrier, in this
case, is two impulses.
One at plus omega c, and
one at minus omega c.
And the amplitudes of these
incorporate the phase.
And later on in the lecture,
and in subsequent lectures,
I'll have a tendency to drop the
theta sub c, just to keep
the notation and algebra a
little cleaner, but for now
I've incorporated it.
And so now when we apply the
modulation property, what we
will do is convolve this
spectrum with this spectrum,
and the result is that the
spectrum of the original
signal gets replicated at
both omega sub c and at
minus omega sub c.
And the resulting spectrum at
the output of the modulator,
then, is the spectrum
that I show here.
Now the question,
of course, is--
so now what's happened is that
with a sinusoidal carrier,
we've moved the spectrum
to both plus omega c
and minus omega c.
And now if we want to get the
original signal back again,
what we would like to do somehow
is move that spectrum
back down to the origin.
Now in the case of a complex
exponential,
that was easy to do.
We'd shifted one up, we'd
just shift it back down.
Let's see what happens if we
attempt to demodulate by again
multiplying by the same
sinusoidal carrier.
So let's examine what happens
if we now take our modulated
signal and, again, modulate it
onto the same sinusoidal
carrier to generate
the output w of t.
If we look at the spectra, we
have the modulated spectrum
which we had initially.
And we now want to convolve
that, again, with the spectrum
of the carrier signal.
The spectrum of the carrier
signal, I indicate here.
And if you track through the
convolution, which is fairly
straightforward, then what
happens as you convolve this
with this is you end up with a
composite spectrum, which is
what I've indicated on the
bottom curve, and has the
spectrum of the original signal,
x of t, replicated in
three places.
One is at minus 2 omega sub c.
One is around the origin.
And one is shifted up to twice
the carrier frequency.
Well it's this piece
that we want.
If we could eliminate everything
else and keep this,
then that would correspond to
the spectrum of the original
signal, x of t.
How do we do that?
Well, we know how to eliminate
part of the spectrum and keep
another part of the spectrum.
That's called filtering.
So what we would do is put the
result of this through a
low-pass filter.
The low-pass filter route would
retain the part of the
spectrum around DC and eliminate
the remaining part
of the spectrum.
So we would keep this part and
eliminate the part of the
spectrum that we
have over here.
And let me just draw your
attention to the fact that,
because of the way the algebra
works out, the amplitude of
this replication of the spectrum
is half what the
original spectrum was.
And that means that ideally, to
keep scale factors correct,
we would choose the amplitude of
this to be 2, to scale this
back up to 1.
So what we have is the modulator
and demodulator.
And just to summarize, for the
case of a sinusoidal carrier
as opposed to a complex
exponential carrier, the
modulator is just as it is in
the complex exponential case.
It's multiplication with the
sinusoidal carrier, with
frequency, omega c, and
phase, theta sub c.
In the demodulator we would
take the modulated signal,
modulate it again with the
same carrier signal--
and as we'll see later, it's
important to keep the same
phase relationship.
This result is not yet quite
the demodulated signal.
We need to process that with a
low-pass filter that extracts
the part of the spectrum around
DC and throws away the
upper part of the spectrum that
gets generated in the
second modulation process.
And the resulting output is the
original signal, x of t.
What we've done then is
we've taken x of t.
We've modulated it
onto a carrier.
And then we've taken that
modulated signal and we've
figured out how to
get back x of t.
And of course one could ask,
well, if you start with x of t
and you want to get x of t back
again, why bother going
through all that?
Why not just use x of t at the
beginning and at the end?
And obviously there are
lots of reasons
as I indicated before.
And just to reiterate
what they are.
The notion, often, is that what
you'd like to do is shift
the signal into a different
frequency band for
transmission over some medium
that is more matched to that
frequency band than the
frequency range of the
original signal.
Also, as I alluded to, is the
notion that you can take lots
of signals and transmit them
simultaneously over one channel--
whether the channel is a wire,
a microwave link, a satellite
link, or whatever--
again, using the idea
of modulation.
And what that process
is referred to as is
multiplexing.
And let me just quickly
indicate what that
multiplexing process
corresponds to.
We could think, for example,
of taking one signal and
modulating in it onto one
carrier with one carrier
frequency, taking a second
signal, modulating it onto a
different carrier frequency,
taking a third signal and
modulating it onto a third
carrier frequency, et cetera.
And if we choose these carrier
frequencies appropriately,
then we can add all
those together--
and do it in such a way that
the spectra don't overlap--
and end up with one broader band
signal that incorporates
the information simultaneously
in all of those signals.
So just to illustrate that
in the frequency domain.
What we have are our three
spectra, Xa, Xb, and Xc.
And we would, for example,
take this spectrum and
modulate it to a carrier
frequency, omega sub a.
We can take this spectrum and
modulate it to a carrier
frequency, omega sub b, where
omega sub b is chosen so that
when we add these two together
they don't overlap, so that
they can eventually
be separated out.
And then we can do the same
thing with the third signal,
and put that in a frequency
range over here, being careful
that none of those overlap.
And when we add all those
together, the composite
spectrum is what I show here.
And as you can see, essentially,
by doing
appropriate band-pass filtering
we can pull out
whatever part of the spectrum
we choose to, and then
demodulate that in the
appropriate way.
And of course we can do this,
not just with three signals,
but perhaps with tens or
hundreds of signals.
So that's a process that is
typically referred to as
multiplexing.
And as I've described it here,
it's referred to as
frequency-division
multiplexing.
That is, dividing the frequency
band into cells and
plunking different signals
into each one of those.
And so if we want now to recover
one of those channels
in a frequency-division
multiplex system, as I
indicated, we would
first demultiplex.
Demultiplexing corresponding to
pulling out the appropriate
channel with a band-pass
filter.
And after demultiplexing, we
would then demodulate.
And we would demodulate with the
carrier appropriate to the
channel that we've pulled out.
And the demodulation, of course,
involves multiplying
by the carrier and doing
appropriate low-pass filtering
to finally get the
signal back.
And frequency-division
multiplexing is the type of
multiplexing that's used, for
example, in typical broadcast
AM radio systems, where
all the channels are
superimposed together.
And it's your home radio
receiver that does the
appropriate demultiplexing
and demodulating.
And of course, you can see that
not only is modulation an
important part of that, but as
I alluded to in the last
lecture, filtering also becomes
important part of
these practical systems.
Now, the kind of amplitude
modulation that I've talked
about so far is what's referred
to as synchronous
modulation.
And the reason for the term
synchronous is that what's
implied in these systems is a
synchronization between the
transmitter and receiver.
In particular, in the system as
we've talked about it, the
modulator and the demodulator
have a synchronization in both
frequency and phase.
The phase here is indicated
as theta sub c.
And if we take a look at the
demodulator, the demodulator
has phase of theta sub c.
And in general, there's the
issue of whether we can
maintain that synchronization
between the modulator and
demodulator.
And so what we want to examine
now, more generally, is what
the consequence might be, and
the solution to the resulting
problems, if we don't have
synchronization between the
modulator and demodulator.
Synchronization in
terms of phase.
And there also is another
problem, which is the issue of
synchronization in frequency.
That's examined more
in the text.
And what I'll focus on here
is just the issue of
synchronization in phase, to
give you some sense of what
the issue is.
So now what we want to look at
is what happens if we have a
modulator with phase, theta sub
c, and a demodulator where
the phase, instead of being
theta sub c, is some other
phase, phi sub c.
And if you track through the
details and the algebra, then
what you'll find is that the
output of the low-pass filter,
rather than being x of t, the
signal that we want, is x of t
multiplied by a scale factor.
And the scale factor is the
cosine of the phase
difference.
Now one could ask, OK well,
what's the big deal about
scale factor?
If it's too small we'll make it
big, it it's too big we'll
make it small.
But there are several points.
One is, notice, for example,
that if the phase difference
between the modulator and
demodulator is 90 degrees,
then the output of the
demodulator is zero.
Or if it isn't quite
90 degrees, the
amplitude might be small.
And the implication would be
that if there's other noise it
gets injected in the system,
the signal-to-noise
ratio is very low.
Now even worse is the issue
that if there's a phase
difference, but the exact
phase difference isn't
maintained, so that the
modulator and demodulator kind
of fade in and out of phase,
then the output of the
demodulator is x of t multiplied
by a time-varying
fading term, which is the
cosine of the phase
difference.
Well what that means,
essentially, is that if you
use this kind of system to do
the demodulation, then what
you need to be careful about is
maintaining synchronization
in phase, and also in frequency,
between the
modulator and the demodulator.
Now there are alternatives
to this.
And the alternative is what's
referred to as asynchronous
demodulation.
And let me indicate what the
idea behind asynchronous
demodulation is.
Now, recall that what we've done
in amplitude modulation
is to take the carrier signal
and vary its amplitude with
the signal that eventually
we want to get back.
So if we look at the
amplitude-modulated waveform,
it might typically look
as I indicate here.
And we're trying to get
back the envelope.
Well, one could imagine
building a circuit, or
designing a device, which
in some sense
will track the envelope.
And a common circuit to do
that is a fairly simple
circuit consisting of a diode
and a resistor and capacitor
in parallel.
The idea being that the
capacitor charges up as this
waveform moves up to its peak.
And then as the waveform drops
down, the capacitor discharges
through the resistor.
And it kind of tracks
the envelope.
In fact, the kind of output that
we would get is the type
of behavior that I've
indicated here.
And then that is a type
of demodulation.
It's a demodulation that
doesn't require
synchronization between the
modulator and demodulator.
And it's fairly inexpensive
to build.
But it has, obviously, some
tradeoffs associated with it.
Well, to indicate where the
tradeoff comes from, or where
the issue surfaces, notice
that what we're doing is
tracking the envelope of
the sinusoidal signal.
And we're calling that, or we're
assuming that that is
our original signal, x of t.
Well, suppose that x of t, the
original signal, is sometimes
positive and sometimes
negative.
What might we see as we look
at the output of the
demodulator?
Well, the output of the
demodulator would follow the
envelope down, and then
it would follow the
envelope back up again.
In other words, what it would
tend to generate is a
full-wave rectified version of
the signal that you were
really trying to get back.
Now, there's a simple
solution to this.
The simple solution is to make
sure that the signal that is
the modulating signal, x of
t, never goes negative.
So if it happens to-- a voice
signal tends to go negative.
If it happens to, we can simply
add a constant to it,
add a large enough constant,
so that it
always stays positive.
Well let's look at that.
What we want to do then, if
we're considering asynchronous
demodulation, is to take our
original signal, x of t, and
add to it a constant, where
the constant is made large
enough so that we're sure that
this is a positive signal.
And incidentally, let me just
draw your attention to the
fact that I'm now suppressing
the phase on the carrier
signal, since the phase is not
important to the argument and
it's just some additional
notation to carry around.
So the idea then, is add
a constant to x of t.
Notice that if we just take this
term and expand it out
into two terms, x of t cosine
omega c t plus a times cosine
omega c t, then in block diagram
terms we can represent
that as I've shown here.
And so it would correspond to
modulating the signal, x of t,
onto the carrier, omega sub c
t, and also injecting some
carrier with an amplitude,
A. And the output of the
modulator is then the
sum of those two.
And depending on exactly what
this value A is will influence
what the envelope
will look like.
And I indicate below,
two possibilities.
One is where I've made A fairly
large, and one is where
I've made A significantly
smaller.
And there are both positive and
negative issues associated
with whether A is too large
or A is too small.
For example, if A is large in
relation to the amplitude of
the signal, then this envelope
tends to be very flat.
And it tends to be easy to
track it with that simple
diode RC circuit, as compared
with the case down here.
On the other hand, there is a
price that you pay for this
kind of envelope.
And the price that you pay is
perhaps best seen in the
frequency domain.
If we look in the frequency
domain, here is
our original spectrum.
Here is the spectrum at the
output of the modulator.
And the impulse that occurs
here corresponds to the
carrier that's injected.
The larger A is, the more
carrier is injected.
The more carrier that's
injected, the easier it is for
the envelope detector
to demodulate.
So one can ask, why not
just put a lot in?
Well, the obvious answer
is that it's not an
information-carrying
part of the signal.
And so in some sense it
represents an inefficiency in
transmission, because what
you're transmitting is power,
energy, that doesn't have
any information
associated with it.
It's simply the injection
of a carrier to make the
demodulation for an asynchronous
demodulator--
to make the demodulation
easier.
And so there's this tradeoff.
And in fact, one represents
the tradeoff and the
associated parameters very
often in terms of percent
modulation, where the percent
modulation is essentially the
ratio of the maximum signal
level to the amplitude of the
injected carrier.
And depending on whether the
modulation's very high or very
low, the tradeoff is that
the transmission is more
inefficient and it takes
more energy, but the
demodulator is simpler.
Or the demodulator is more
complicated but the
transmission is simpler.
Now, there are situations where
you might very well want
to use one or the other.
For example, in home radio
you're often willing to
transmit a lot of power so that
you can have inexpensive
consumer-oriented receivers.
On the other hand, in satellite
communication you're
willing to pay a very high price
for the modulators and
demodulators, but it's the
amount of power that's
transmitted that's
at a premium.
And so in one case, satellite
communication, you would use
synchronous modulation
and demodulation.
Whereas in typical
consumer-oriented
broadcasting, you would use
an asynchronous system and
transmit more power, even if
it's inefficient, so that the
demodulator can be simpler.
Now, in the asynchronous system,
as we've indicated,
there's one source of
inefficiency, which is this
injection of the carrier.
There also is a somewhat
different issue, related to
inefficiency in sinusoidal
amplitude modulation.
And it's an inefficiency that is
separate from the issue of
synchronous versus asynchronous
systems.
In other words, it's not
associated with the injection
of the carrier, it's a
very different issue.
Let me indicate what that is.
Let's look again at the spectrum
of x of t, which I've
indicated here.
And in a sinusoidal amplitude
modulation system, we would
center it around plus and minus
the carrier frequency.
Now, notice that in the original
system we occupy a
frequency spectrum that's 2
times omega sub M. By the time
we've shifted it, thinking
of positive and negative
frequencies, we've used
up twice as much of
the frequency spectrum.
Well you could say, OK, let's
just shift this up this way
and get rid of this part.
That's of course what
the complex
exponential carrier did.
And the issue there is that
now you've got to transmit
both a real part and
an imaginary part.
So what you can think about, and
ask, is if you still want
to transmit a real-valued
signal, how can you somehow
remove the inefficiency or
redundancy in the spectrum?
Well, notice that what we have
is this spectrum moved here,
and moved here.
And we could imagine building
real-valued signal by
eliminating what I refer to here
as the lower sideband out
of the positive frequencies, and
the lower sideband out of
the negative frequencies.
And in effect, what we've done
is taken just the positive
frequencies here, shifted
them there, the negative
frequencies here, and
shifted them here.
And the resulting spectrum
is what I indicate below.
Well, this is what
is often done.
And what it's referred to
as is single sideband.
What we've done is kept the
upper sideband, in this
particular case.
We could alternatively think
of putting this system
together where we retain the
lower sideband instead of the
upper sideband.
And in either case, what we've
removed is an inefficiency in
transmission of the signal.
Namely we have a real-valued
signal, but it only requires
as much total bandwidth, in
terms of the frequencies in
which there's energy present,
as the original signal.
Well how do we do this?
There are a variety of ways.
And there's one procedure that
is discussed in more detail in
the text, which uses what's
referred to as a 90 degree
phase splitter.
The simplest way, at least
conceptually, is to think
about doing it with filtering.
And the idea simply is that if
we have our modulated signal--
here's the spectrum of
the modulated signal.
And if that modulated signal
is simply put through a
high-pass filter, then the
result will be to eliminate
the lower sideband, if we choose
the high-pass filter to
have a characteristic
as I indicate here.
So this, conceptually, is a
very sharp cutoff filter.
And what it eliminates are
the lower sidebands.
And the resulting spectrum
is what we have below.
And this in fact is really
the basic idea behind
single-sideband transmission.
Again, there's a tradeoff.
It's clearly more efficient
than double-sideband
transmission, but also has the
complication, or additional
issue, that the modulator
becomes a little more
complicated because you need
this filtering operation, or
some equivalent operation,
to get rid of
the unwanted sideband.
Well, this is a fairly quick
tour through a variety of
issues related to modulation.
And it really is just the tip
of the iceberg, obviously.
Modulation in the context of
sinusoidal modulation, as
we've talked about, has a
lot of detailed issues
associated with it.
It's important to recognize, and
to be somewhat pleased by
the fact, that not only with the
mathematical foundations
that we've developed can we
understand the basics of
sinusoidal amplitude
modulation.
But what you'll find if you dig
into this somewhat deeper
that the basic background that
we built up so far--
the mathematical tools--
are really pretty much what
you need for a much deeper
understanding of all of
the issues involved.
So from what might have seemed
like a fairly abstract
mathematical property associated
with the Fourier
transform, we've begun to
develop what should give you
the sense of some important
practical considerations.
And as we'll see the next
lecture, very much the same
kinds of notions apply for
discrete time, sinusoidal, and
complex exponential amplitude
modulation.
And also as I indicated at the
beginning of the lecture, in
the next lecture we'll also talk
about what's referred to
as pulse amplitude modulation.
It's a different kind
of carrier.
And what that will lead to,
among other things, is a very
important bridge between the
notions of continuous time and
the notions of discrete time.
Thank you.
