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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.