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PROFESSOR: In the last
lecture, we began the
discussion of modulation.
And in particular, what
we focused on, was the
continuous time case.
And in talking about continuous
time modulation,
we've covered a number
of topics.
We talked about the properties
and analysis of modulation
when we had a complex
exponential carrier signal.
We talked about the properties
and analysis in the case of a
sinusoidal carrier.
And in that context and related
to the application
associated with communications,
we talked
about synchronous modulation,
asynchronous modulation and
also the notion of single
side band modulation.
In the lecture today, there are
two issues that I'd like
to address, broad topics.
One is a parallel discussion,
particularly, as associated
with complex exponential and
sinusoidal modulation for
discrete time signals.
And the second is the
introduction and analysis of
another kind of carriers,
specifically a pulse kind of
carrier in continuous time
leading to the notions of
pulse amplitude modulation
and, eventually, a very
powerful theorem and result
called the sampling theorem.
Well, let me begin the lecture,
though, focusing on
the discrete time modulation
to essentially draw your
attention to the fact that the
analysis in discrete time very
much parallels the analysis
in continuous time.
Well, let's consider, in the
discrete time case, just as we
had in continuous time, a signal
modulating a carrier
signal and the resulting
modulated signal is y of n.
And it was in continuous time
the modulation property
associated with the Fourier
Transform that provided the
basis for the analysis.
And exactly the same thing
is true in the
discrete time case.
In particular, what we have
in discrete time, is the
modulation property as it
relates to the Fourier
Transform, which tells us that
the Fourier Transform of the
modulated signal is the
convolution of the Fourier
Transform of the carrier and the
Fourier Transform of the
modulated signal.
And the only real difference at
issue here is that, in the
discrete time case, what we're
talking about is a periodic
convolution because
the specter,
of course is periodic.
Whereas, in the continuous time
case, it was an aperiodic
convolution.
So let's parallel the
discussion, and in particular,
what we'll focus on is, first,
a complex exponential carrier
and second a sinusoidal
carrier.
And we'll see how this parallels
our discussion in
continuous time, and we'll make
fairly brief reference as
we introduce the pulse carrier
for continuous time.
We'll make very brief reference
to the pulse carrier
for discrete time indicating
that, again, the analysis and
discrete time and continuous
time is very parallel.
So let's, first, consider
complex exponential and
sinusoidal carriers for the
discrete time case,
emphasizing the very strong
parallel and similarity
between discrete time
and continuous time.
Well we have, once, again
the modulation property.
And the modulation property
tells us that the spectrum of
the modulated signal is the
periodic convolution of the
two spectra.
And let's consider, for
example, an input, or
modulating spectrum, as
I've indicated here.
And since we want to consider,
first of all, a complex
exponential carrier, we'll
consider the case of c of n
equal to e to the
j omega sub cn.
And let me stress, by the way,
as I did in the continuous
time case, that I'll tend to
suppress the phase angle
which, of course, can
be associated
with the carrier also.
All right, so we have,
then, the spectrum of
the modulated signal.
The spectra, the carrier
signal, if this is the
carrier, then it's spectrum is
an impulse train, and that
impulse train, I've
indicated here.
And let me stress, also, that in
the discrete time case, of
course, these spectra and all
of the spectra involved, are
periodic with a period
of 2 pi.
So this then is the spectrum
of the carrier signal.
This is the spectrum of
the input signal.
The periodic convolution of
these two is the spectrum the
modulated signal.
And the result is, then, this
spectrum shifted to a center
frequency, which is the carrier
frequency omega sub c.
So the result of modulation with
a complex exponential is
a straightforward shift of the
spectrum so that what occurred
around zero frequency now occurs
around the frequency
omega sub c.
Now, in the continuous time
case, we demodulated, when we
had a complex exponential
carrier, we demodulated by,
essentially, just shifting
the spectrum back.
And in fact, in the discrete
time case, were able to do
exactly the same thing.
So if we were to replace c of n
which is either j omega sub
cn by c of n equals e to the
minus j omega sub cn, the
resulting spectra would be an
impulse train, as I indicate
here, and the result of multiply
y of n by that new
carrier, in the frequency domain
as a convolution of
these two, and it's relatively
straightforward to verify that
if you convolve these with a
periodic convolution, then
that will get us back to the
original spectrum that we
started with.
So what's happened in the
discrete time case, with the
complex exponential, is
exactly the same as in
continuous time.
Namely, we modulate that
corresponds to
shifting the spectrum.
We demodulate by multiplying
by the complex conjugate of
the original modulated carrier
and that shifts the spectrum
back to where it
was originally.
OK.
Now let's consider the case
of a sinusoidal carrier in
discrete time.
And again, things very much
parallel what we saw in
continuous time.
And again, as we look at the
spectra, I will choose a phase
angle of zero, mainly for
notational and analytical
convenience.
So in this case, now, rather
than a carrier signal, which
is a single complex exponential,
it's now a
sinusoidal carrier and the
sinusoidal carrier is the sum
of two complex exponential.
And so if we consider a
modulated spectrum, that is
the spectrum of x of n,
something of the type that I
indicate here, and the spectrum
of the carrier, now,
since the carrier is sinusoidal
rather than a
complex exponential consists of
two impulses, one at plus
omega sub c and one at minus
omega sub c, convolving this
spectrum with this spectrum
gives us a replication of x of
omega around plus and
minus omega sub c.
And incidentally, with an
amplitude change of a half.
So again, things have
worked as they did
in continuous time.
In continuous time or in
discrete time, modulating with
a sinusoidal carrier would
correspond to a replication of
the spectrum around, plus the
carrier frequency and a
replication of the spectrum
around minus the carrier
frequency, in both cases, as
long as the carrier frequency
is large enough compared with
the bandwidth of the signal so
that those two replication
don't overlap, then it's
reasonable to suppose that we
should be able to recover the
original signal.
Well, in fact, to demodulate in
the discrete time case, we
would again follow very much
the strategy that we did in
continuous time.
In particular, let's consider
demodulating by taking the
modulated signal and, again,
putting that through a
modulator, again, with
the carrier which is
cosine omega sub cn.
If we do that, we have a
demodulator or what will turn
out to be part of a demodulator,
as I indicate
here, the spectrum of the input
signal is, as I had just
developed, a replication of the
original spectrum around
plus and minus omega sub c with
an amplitude of a half.
When this is, again, convolved
with the spectrum of the
carrier, then we get a
replication of the original
spectrum, first around zero
frequency, as I indicate here,
and then around twice the
carrier frequency and minus
twice the carrier frequency.
And as long as the carrier
frequency is large enough
compared with the width of the
original signal, then, as you
can see, by extracting this part
of the spectrum with a
low pass filter, we can, in
principle, recover the
spectrum associated with
the original signal.
And again, just as in continuous
time, because this
amplitude is a half, we would
want to choose, for scaling
purposes, a low pass filter
amplitude which is 2 to
compensate for this
factor of a half.
So once again things work out
basically the same way as they
had in continuous time.
We have sinusoidal modulation
which consists of using a
sinusoidal carrier.
And we have the demodulator
which consists of taking a
modulated signal, multiplying
by the carrier, and then
processing that with a low pass
filter to extract the
portion of the spectrum, which
is around zero frequency, as I
indicate in the spectrum below
and the result, then, that
this low pass filter having
a gain of 2 is that we've
recovered the original spectrum,
x of omega, which is
the spectrum that
we started with.
Now several other things
to stress.
This is a fairly quick tour
through sinusoidal modulation
for discrete time.
There are very similar issues
that arise in the discrete
time case in terms of having
phase synchronization and
frequency synchronization
between the modulator and
demodulator.
And we had discussed that in a
fair amount of detail for the
continuous time case.
In some sense, in practical
terms, that becomes much more
of an issue in continuous time
than it does in discrete time,
in part, because synchronization
between a
modulator and demodulator is
often much harder in a
continuous time system, which
is essentially an analog
system as compared with
a digital system.
Another very important reason
and it's important to stress
this at the outset is that,
whereas the theory involving
the use of complex exponential
and sinusoidal modulation
parallels very strongly in the
continuous time and the
discrete time case.
In practical terms, it has
much more significance in
continuous time than it
does in discrete time.
That is, the notion called
sinusidal modulation, in the
context of communication
systems, is extremely
important for continuous time
systems, and less so in
discrete time systems.
Now as a preview of a point to
be raised later on, I should
modify that slightly with the
statement that one very
important place in which
sinusoidal modulation in a
discrete time context arises,
is in a class of systems
called transmultiplexers or
transmodulation systems.
And this surface is basically
because so many communication
systems are now becoming digital
and, specifically,
discrete time, although the
actual transmission is
continuous time, the signal
processing manipulation and
switching is discrete time.
And so, in fact, it turns out
to be very important and
useful to take a discrete time
representation of the analog
signals or continuous time
signals and, in a discrete
time, or digital representation,
to convert
them from one modulation scheme
or one multiplexing
scheme to another.
And although I said a lot there
that really requires
much more detail to develop in
any sense at all, you should
get the notion that discrete
time modulation systems become
very important, in
part, because of
implementational issues.
OK, now, there is another
application that we have
discussed for both continuous
time and actually, previously,
for discrete time, amplitude
modulation with sinusoidal
complex exponential carriers.
And let me just remind you of
that, because, in fact, it
becomes a very important
one in the case of
discrete time systems.
And that is the notion of using
modulation together with
fixed filtering to implement a
filter, which either has a
variable cut off or converts,
let's say, a low pass filter
to a high pass filter.
We had originally talked about
this when we introduce the
modulation property in the
context of converting a low
pass filter to a high
pass filter.
And the notion was that, if we
modulate the signal with a
carrier which is minus 1 to the
n, and that's just simply
a complex exponential or
sinusoidal carrier with a
carrier frequency of pi, then
that, in effect, interchanges
the low frequencies and
the high frequencies.
And if, after modulation, that
is processed with a low pass
filter, and then the result is
demodulated, using exactly the
same carrier, namely a carrier
which is minus 1 to the n,
then the effect of that is
equivalent to high pass
filtering on the original
signal.
And a generalization of that
would involve, instead of this
specific choice of minus 1 to
the n, would involve a choice,
in general, of e to the j omega
sub cn, that is a more
general carrier frequency, and
a demodulator which is e to
the minus j omega sub cn.
And as I've represented it here,
and as we had talked
about it when we talked about
the modulation property for
discrete time signals, we had
specifically chosen the
conversion of a low pass
to a high pass filter.
Well, let me continue the
review of that just by
reminding you of the details
of what happens with the
spectra, and, specifically, the
notion, if we take this
particular case of omega sub
c is equal to pi, or
equivalently, a carrier signal
which is minus 1 to the n,
then if we have the original
spectra and the spectrum of
the carrier signal, the spectrum
of the carrier signal
convolved with this spectrum
will then, in effect, shift
this by pi.
And so, after modulating, the
result that we have is a shift
of that spectrum so that what
happened in low frequencies
now happens at high frequencies,
namely around pi,
and what happened at high
frequencies now happens at low
frequencies.
Well, if that's processed now,
with a low pass filter, and
this dashed line indicates the
low pass filter, then the
result that we get is shown
here, where we've extracted
the low frequency portion
of the modulated signal.
And now when we modulate or
demodulate back, then this
spectrum is shifted back
to where it belongs.
Namely, it's shifted back to
be centered around minus pi
and around plus pi.
So if we just compare this
spectrum with the original
spectrum at the top, what we
can see is that, in effect,
what we've done is to extract
a portion of the spectrum
equivalent to processing with
a high pass filter.
And, again, this is very similar
to what we did in
continuous time and all of the
analytical processes and
convolution involved are
very much the same.
Really, the biggest difference
between continuous time
discrete time has to do, not
so much with the details of
the analysis, but perhaps has
more to do with issues of
practical applications.
OK, well, so what we've done,
so far, for continuous time
and discrete time, is to talk
about modulation, amplitude
modulation with complex
exponential
and sinusoidal carriers.
We saw that the analysis is
very similar, although
applications are slightly
different.
And now what I'd like to turn
to is a different choice of
carrier signal.
And the carrier signal, in this
particular case, is a
pulse train rather than
a sinusoidal signal.
Now the idea is the following.
In general, of course, the
modulator consists of all of
multiplying x of t by whatever
the carrier signal is.
And previously, we've talked
about a carrier signal which
is sinusoidal signal.
The carrier signal that we
want to consider now is a
carrier signal which, in
fact, is a pulse train.
And so, in fact, what we want
to do is multiply the input
signal by a pulse train and, in
effect, then, the modulated
signal consists of the original
signal, simply with
time slices pulled out of it,
as I've indicated in the
bottom curve.
So what we have now is a
modulated signal that is a
chopped or sliced version of the
original waveform and that
is what's referred to as pulse
amplitude modulation.
Now it seems like it's
kind of a crazy idea.
The idea is to chop out slices
of the wave form and hope that
you could put things back
together again.
And the amazing thing about it,
as we'll see, is that, in
fact, under fairly broad
general and applicable
conditions, you really can put
the waveform back together
again if you just have
these time slices.
Not only that, but that basic
notion, as we'll see, is
independent, in fact,
of what the width of
those time slices are.
In fact the width
can go to zero.
And, in fact, we're going to
make it go to zero, and really
only dependent on what
the frequency of
the pulse train is.
So let's explore that
in some detail.
And what we want to look at is
the analysis, but let me,
first, just comment, very
briefly, that all of the
analysis we go through, as has
been true in the case of
sinusoidal modulation, all of
the analysis then we go
through holds just as well
with, essentially minor
analytical modifications, to
discrete time pulse amplitude
modulation as it does to
continuous time pulse
amplitude modulation.
And so we'll really only go
through this in terms of
tracking the wave forms
and spectra for the
continuous time case.
But bear in mind that the
results are basically similar
for discrete time.
OK, well, let's see how so we
get the basic result that we
want to get.
What we have is modulated signal
which is a pulse train,
basically a square wave, and
as we've seen in previous
lectures, the spectra or Fourier
transform associated
with that is an impulse train.
And the envelope of that impulse
train is on the form
of a sine x over x function.
The Fourier transform
is impulses.
And the spacing of the impulses
is associated with
the fundamental frequency of
the pulse train and that's
omega sub p.
So omega sub p is pi divided
by the period
of the pulse train.
And the amplitude and shape of
this envelope is dictated by
the parameter delta, which
has to do with how
wide the pulses are.
OK, so we have a
time function.
It's multiplied by
this pulse train.
Now we're talking
continuous time.
So, in the frequency domain, we
have the Fourier transform
of the time function convolved
with this Fourier transform
for the pulse train.
And let's see what
that looks like.
If we were to consider, let's
say, a Fourier transform,
which I have chosen as more or
less a general one, then in
fact, when we convert all of
this with this impulse train,
what we end up with is a
replication of this spectrum
at the places in the frequency
domain where the individual
impulses occurred.
So we can see that this spectrum
is replicated at each
of these locations.
And as long as the frequency
of the pulse train is large
enough, compared with the
maximum frequency in the
original signal, x of t, so
that there's no overlap
between these triangles, then
what you can see, in fact,
somewhat amazingly is that,
simply by low pass filtering
the result, we can get back,
except for amplitude factor,
we can get back to the
original signal.
Now it's amazing.
It really is amazing that all
that this depends on is the
original signal being band
limited and the frequency of
the pulse train being high
enough so that when you
replicate the spectrum the
frequency domain, there's no
overlap between these individual
replications.
And we'll have address that a
little more in a few minutes.
But let me, first of all, point
out that this has a
whole variety of very important
implications.
One is, in the context of
communications, it leads to
another very important
multiplexing scheme for
communications.
We had talked last time about
frequency division
multiplexing, where individual
signals were put into
individual frequencies slots by
choosing different carrier
frequencies for a sinusoidal
modulating signal.
What this suggests is that
what we can put different
signals into, non-overlapping
time slots and, in fact, be
able to recover the original
signals back again.
So in particular, suppose that
I had a signal which I
modulated with a pulse train
and I chose another signal,
modulated with another pulse
train, where the time slot was
different, and I continued
this process.
And after I'd done this with
some number of channels,
simply added all those together
as I indicate here.
Then as long as I knew what time
slots to associate with
what signal, I could get the
original modulated signals
back again.
And then as long as the
frequency of the impulse train
was such that I was able to do
this reconstruction by simply
low pass filtering, then I would
be able to demodulate.
So it's a very different very
important modulation scheme
called time division
multiplexing in contrast to
frequency division multiplexing
as we had talked
about last time.
I had made reference earlier
to the concept of
trans-multiplexing.
And in fact, what happens in
many communication systems is
that the signals are
represented, in fact, in
discrete time.
The analog and continuous time
signals are represented in
discrete time.
And very often the conversion
from frequency division
multiplexing to time division
multiplexing and back is done,
in fact, in the discrete
time domain.
OK, so what we have then, is
the notion that we can
multiply a time function
by a pulse train,
as I indicate here.
And from the output I can, if
the frequency of this pulse
train is high enough in relation
to this bandwidth,
from the output, which consists
of time slices, from
those time slices I can recover
the original signal.
Stressing again the reason it
relates to the spectra, and
the reason is that the original
spectra is simply
replicated at multiples of the
fundamental frequency of the
pulse train.
Now there's a very important
thing to observe here, which
is that the ability to do the
reconstruction is associated
with the notion of whether
we can extract
that central triangle.
I happened to choose a
triangular shape but obviously
I could be talking about any
shape, as long as it's band
limited, the ability
to extract that.
And notice that, in this
modulated output spectrum, the
ability to recover this is
totally independent called
what the value of delta is.
In other words, if we look back
at the modulator, then,
in fact, we can make delta,
the width of these pulses,
arbitrarily small.
And, in theory, that doesn't
affect our ability to do the
reconstruction.
Now in practical
terms it might.
Looking back once more at the
spectrum of the output, notice
that this amplitude is
proportional to delta.
And what that suggests is that,
as we make delta smaller
and smaller, which we might,
in fact, want to do, if you
want to time division multiplex
lots of channels, in
principle, in theory, you could
make it an infinite
number of channels just
by making that
infinitesimally small.
The smaller it is, in
some sense, the less
energy there is.
And again, in practical terms,
this one of those things if
you push down here pops up
there, namely, you eventually
run into issues such
as noise problems.
So, more typically what's done
is to, in fact, eliminate this
scale factor of delta.
And the way that that's
done is very simply.
It's done by choosing the width
of the pulses, and the
height of the pulses, in such
a way that the area is
constant, even as we make delta
get arbitrarily small.
So we can just modify our
argument so that what we're
referring to is a modulated
pulse train, which is a pulse
train with pulses of
width delta and
height, 1 over delta.
In that case, as delta gets
arbitrarily small, then, in
fact, what these rectangles
become are impulses, in which
case, what we're talking about
is a carrier signal which, in
fact, is an impulse train.
And the resulting modulated
signal is an impulse train for
which the amplitudes of the
impulses are proportional to
the original input waveform at
the times at which these
impulses occur.
OK well, let's look at
the analysis of that.
And so now, what we're talking
about, is a spectrum that
consists of the result of the
spectrum we talked about
before with the sine x over x
envelope, except that, now, as
delta goes to zero that
becomes flat.
In other words, the modulated
signal is an impulse train.
And so as we look at the
spectrum of the modulated
signal, that is, then,
an impulse train in
the frequency domain.
The height is proportional to
the frequency of the impulse
train and omega sub s now
denotes the frequency of the
impulse train.
And the resulting output of the
modulator has a spectrum
which is this original spectrum,
again, replicated
around each of these impulses,
in other words, replicated in
multiples of the sampling
frequency
Now this is very much
identical to the
more general case.
We have this replication
of the spectra.
And as long as the frequency of
the impulse train is large
enough, compared with the
bandwidth of the signal so
that these triangles don't
overlap, I can extract this
portion of the spectrum by low
pass filtering, in fact, would
then give us back the
original signal.
Now if, instead, this frequency
omega sub m is
greater than omega sub s minus
omega sub m, we would have a
spectrum that looked something
more like this.
And what's happened, in this
case, is that, because we have
an overlap here, we've destroyed
the ability to
recover the original signal
from the impulse train.
And that would be true, also
in a more general case, of
pulse amplitude modulation with
pulses of non-zero width.
This effect by the way, is one
that we'll be exploring in
considerably more detail
in the next lecture.
And it's a phenomenon or
distortion refer to as
aliasing which, in fact,
is an important
and interesting topic.
But going back to the case
in which we've chosen the
frequency of the impulse train
high enough, then we would
recover the original signal by
processing it through a low
pass filter.
And in that case, what this
says is, that if we have a
signal, and we modulate it with
an impulse train, if we
then process that impulse train
through an idea low pass
filter, given the right
conditions on the frequency of
impulse train and the bandwidth
of the signal, we
can recover the original
signal back again.
Now let me stress, just going
back to the picture in which
we had done this modulation,
that this process, where the
modulation, where the carrier
signal involves an impulse
train, is often referred
to as sampling.
And what that means,
specifically, is that, if we
notice, this resulting impulse
train is, in fact, a sequence
of samples of the original
continuous time signal.
In other words, what we've
done, in effect, is taken
instantaneous sample
of this wave form.
And the implication is that,
if we do that at a rapid
enough rate in relation to the
bandwidth of the signal, then
we can, in fact, recover the
original signal back again.
And, finally to remind you of
the argument once more, we
have an original signal and
we have its spectrum.
When we've sampled it, and
this is now the sampled
signal, it's an impulse train
whose instantaneous values are
samples of the original
waveform, the spectrum of that
is the original one
replicated.
And when that is processed,
through a low pass filter, to
extract this part of the
spectrum, then, after the low
pass filter, we can recover the
original signal back again.
OK well, in fact, although if
you follow through the spectra
and the wave forms, this all
seems fairly straightforward
and, perhaps or perhaps not,
obvious, it's really worth
reflecting on how amazing
the result really is.
We began this discussion by
talking about modulation.
And in fact modulation and
sinusoidal of modulation is
important in its own right.
We ended the discussion by
talking about first pulse
amplitude modulation, and then
showing how, under the right
set of conditions, you can, in
fact, take a wave form and
sample it with a set of
instantaneous samples.
And that set of instantaneous
samples, in fact, are
sufficient to totally
represent and
reconstruct the signal.
What in fact, the formal
statement that is, is refer to
as the sampling theorem, a very
powerful theorem that
says, if we're given equally
spaced samples of a time
function, and if that time
function is band limited, and
if the bandwidth and if the
sampling frequency is chosen
in the right way, in relation
to the bandwidth, then, in
fact, the original time
function is uniquely
recoverable with a
low pass filter.
Now the sampling theorem is,
I would say, a watershed or
cornerstone of a lot of the
discussion that we've been
having for a whole variety
of reasons.
It, first of all, drops out
almost as a straightforward
obvious statement.
But more importantly what
it says is, if I have a
continuous time signal which
satisfies the right set of
conditions, I could represent it
by what it does at sampling
instance or, equivalently, at
discrete instance of time.
Now what that leads to is
a whole host of things.
One of which is this statement
that says, if we have a
continuous time signal, I could
in fact, represent it as
a discrete time signal.
And I could even think of
processing a continuous time
signal using discrete
time concepts.
And when I'm all done converting
back, through the
power of the sampling theorem,
converting back to a
continuous time signal.
So the sampling theorem provides
us with a very major
important bridge between
continuous time and discrete
time implementations
and ideas.
In the next several lectures,
we will be exploring some of
this in considerable detail.
First, to focus in more, next
time, on some of the specific
issues and distortions
associated with sampling.
And following that, a discussion
of what is referred
to discrete time processing of
continuous time signals.
Thank you.
