[MUSIC PLAYING]
PROFESSOR: I'm Al Oppenheim, and I'd like to welcome you to
this videotape course on signals and systems.
Signals, at least as an informal definition, are
functions of one or more independent variables that
typically carry some type of information.
Systems, in our setting, would typically be
used to process signals.
One very common example of a signal might be, let's say, a
speech signal.
And you might think of the air pressure as a function of
time, or perhaps the electrical signal after it
goes through the microphone transducer as a function of
time, as representing the speech signal.
And we might see a typical speech signal looking
something like I've indicated here.
It's a function of time, in this particular case.
And the independent variable, being time, is, in fact,
continuous.
And so a signal like this, we will typically be referring to
as a continuous time signal.
Now, it also, for this particular example, is a
function of one independent variable.
And that will be referred to as a one-dimensional signal
corresponding to the fact that there's only one independent
variable instead of several independent variables.
So the speech signal is an example of a continuous time,
one-dimensional signal.
Now, signals can, of course, be multi-dimensional.
And they may not have, as their independent variables,
time variables.
One very common example are the examples
represented by images.
Images, as signals, we might think of as representing
brightness, as it varies in a horizontal
and vertical direction.
And so the brightness as a function of these two spatial
variables is then a two-dimensional signal.
And the independent variables would typically be continuous,
but of course they're not time variables.
And incidentally, it's worth just commenting that very
often, simply for convenience, we'll have a tendency to refer
to the independent variables when we talk about signals as
time variables, whether or not they really do represent time.
Well, let me illustrate one example of an image.
And this is a picture of J. B. J. Fourier, who, perhaps, more
than anyone else, is responsible for the elegance
and beauty of a lot of the concepts that we'll be talking
about throughout this course.
And when you look at this, in addition to seeing Fourier
himself, you should recognize that what you're looking at is
basically a signal which is brightness as a function of
the horizontal and vertical spatial variables.
As another example of an image as a signal, let's look at an
aerial photograph.
This is an aerial photograph taken over a set of roads,
which you can more or less recognize in the picture.
And one of the difficulties with this signal is that the
road system is somewhat obscured by cloud cover.
And what I'll want to show later as an example of what a
system might do to such a signal, in terms of processing
it, is an attempt to at least compensate somewhat for the
cloud cover that's represented in the photograph.
Although in terms of the detailed analysis that we go
through during the course, our focus of attention is pretty
much restricted to one-dimensional signals.
In fact, we will be using two-dimensional signals, more
specifically images, very often to illustrate a variety
of concepts.
Now, speech and images are examples of what we've
referred to as continuous-time signals in that they are
functions of continuous variables.
An equally important class of signals that we will be
concentrating on in the course are signals that are
discrete-time signals, where by discrete-time, what we mean
is that the signal is a function of an integer
variable, and so specifically only takes on values at
integer values of the argument.
So here is a graphical illustration of a
discrete-time signal.
And discrete-time signals arise in a variety of ways.
One very common example that is seen fairly often is
discrete-time signals in the context of economic time
series, for example, stock market analysis.
So what I show here is one very commonly occurring
example of a discrete-time signal.
It represents the weekly stock market index.
The independent variable in this case is the week number.
And we see what the stock market is doing over this
particular period as a function of the
number of the week.
And, of course, along the vertical axis
is the weekly index.
Incidentally, this particular period was
not chosen at random.
It In fact captures a very interesting aspect of stock
market history, namely the stock market crash in 1929,
which, in fact, is represented by the behavior of this
discrete-time signal, or sequence, in
this particular area.
So this dramatic dip, in fact, is the stock
market crash of 1929.
Well, the Dow Jones weekly average is an example of a
one-dimensional discrete-time signal.
And just as with continuous time, we had not just
one-dimensional but multi-dimensional signals,
likewise we have multi-dimensional signals in
the discrete-time case where, in that case, then, the
discrete-time signal that we're talking about, or
sequence, is a function of two integer variables.
And as one example, this might, let's say, represent a
spatial antenna array where this is array number in, let's
say, the horizontal direction, and this is array number in
the vertical direction.
Both classes of signals, continuous-time and
discrete-time, as I've indicated, are very important.
And it should be emphasized that the importance of
discrete-time signals and associated processing
continues to grow in large part because of the current
and emerging technologies that permit, basically, the
processing of continuous-time signals by first converting
them to discrete-time signals and processing them with
discrete-time systems.
And that, in fact, is a topic that we will discuss in a fair
amount of detail later on in the course.
Let's now our attention to systems.
And as I indicated, a system basically processes signals.
And they have, of course, inputs and outputs.
And depending on whether we're talking about continuous time
or discrete time, the system may be a continuous-time
system or a discrete-time system.
So in the continuous-time case, I indicate here an input
x(t) and an output y(t) If we were talking about a
discrete-time system, I would represent the input in terms
of a discrete-time variable, and, of course, the output in
terms of a discrete-time variable also.
Now, in very general terms, systems are hard to deal with
because they are defined very broadly and very generally.
And in dealing with systems and analyzing them, what we
will do is attempt to exploit some very specific, and as
we'll see, very useful system properties.
To indicate what I mean and how things might be split up,
we could talk about systems, and will talk about systems,
that are linear.
And we could divide systems, basically, into systems that
are either linear or nonlinear, and we will, and
also divide systems into systems that are what we'll
refer to as time-invariant or time-varying systems.
And these aren't terms that we've defined yet, of course,
but we will be defining in the course very shortly.
And while, in some sense, this division represents all
systems, and this does, too, the focus of the course is
really going to be principally on linear,
time-invariant systems.
So it's basically these systems that we will be
focusing on.
And we'll be referring to those systems as linear,
time-invariant systems.
Well, as a brief glimpse at some of the kinds of things
that systems can do, let me illustrate, first in a
one-dimensional continuous-time context, and
then later with a discrete-time example, one
example of some processing of signals with
an appropriate system.
The particular example that I want to illustrate relates to
the restoration of old recordings.
And this is some work that was done by Professor Thomas
Stockham, who is at the University of Utah, and work
that he had done a number of years ago relating to the fact
that in old recordings, for example in Caruso recordings,
the recording was done through a mechanical horn, and the
characteristics of the horn tended to
vary from day to day.
And because of the characteristics of the horn,
the recording tended to have a muffled quality, something
like this, sort of the sense that you would get if you were
speaking through a megaphone.
What Professor Stockham did was develop a system to
process these old recordings in such a way that a lot of
the characteristics and distortion due to that
recording system was removed.
So I'd like to illustrate that as one example of some signal
processing with an appropriate continuous-time system.
And what you'll hear is a two-track recording.
On the first track is the original, unrestored Caruso
recording, and on the second track is the result of the
restoration.
And so as I switch back and forth from channel one to
channel two, we'll be switching from the original to
the restored.
We'll begin the tape by playing the original.
And then, as it proceeds, we'll switch.
So we'll begin on channel one.
[MUSIC PLAYING, MUFFLED]
That's the original recording.
And switch now to the processed.
[MUSIC PLAYING, CLEARER]
Now let's switch back, back to the original.
Back to the restoration.
And once again, back to the original.
And presumably and hopefully, what you heard was that in the
restoration, in fact, a lot of the muffled characteristics of
the original recording were compensated for or removed.
Now one of the interesting things that happened, in fact,
in the work that Professor Stockham did is that in the
process of the restoration--
and perhaps you heard this--
in the process of the restoration, in fact, some of
the background noise on the recording was emphasized.
And so he processed the signal further in an attempt to
remove that background noise.
And with that particular processing, the processing was
very highly nonlinear.
A very interesting thing happened, which was that not
only in that processing was the background noise removed,
but somewhat surprisingly, also the
orchestra was removed.
And let me just play that now as an example of some very
sophisticated processing with a nonlinear system.
What you'll hear on channel one is the restoration as we
had just played it.
When I switch to channel two, it will be after the
processing with an attempt to remove the orchestra and the
background noise.
Channel one now.
And now the noise and orchestra removed.
Back to channel one.
And finally, once again, with the orchestra removed.
So that's an example of processing of a
continuous-time signal with a corresponding
continuous-time system.
Now I'd like to illustrate an example of some processing on
a discrete-time signal.
And I'd like to do that in the context of the example that I
showed before of a discrete-time signal, which
was the Dow Jones Industrial weekly stock market index.
I had shown it before, as I've shown it here again, over a
period of slightly more than a year, where this is the number
of weeks and this is the weekly index.
And to illustrate some of the processing, what I'd like to
do is show the stock market index, the weekly index, over
a much longer time period, in particular, the weekly index
over a 10 year period.
And that's what I show here.
So what this covers is roughly 1927 to 1937.
And in this case, although this is still a discrete-time
signal, just simply for the purposes of display, what
we've done is to essentially connect the dots and draw a
continuous curve through the points so that this picture
isn't filled up with a lot of vertical lines.
So this is the discrete-time sequence that represents the
weekly Dow Jones Index over a 10 year period.
And here, by the way, again, is the crash of 1929.
It's interesting to note, by the way, that actually the
disaster in the stock market wasn't so much the 1929 crash
but the long downward trend that followed that.
And you can see that here by filtering through, by eye, the
rapid variations in the index.
And what you see is this smooth downward trend
followed, eventually, by an upward trend.
Now, this issue of looking at something like this, looking
at a sequence, and following the smoother parts of it,
namely the long term trends, is, in fact, something that is
done quite typically with economic
time series like this.
And in particular, what's done is to smooth it, or average
over some time period, to emphasize the slow variations
and de-emphasize the rapid variations.
And that, in fact, is processing that is done with a
discrete-time system.
So when you hear referred to, let's say, in stock market
reports, a 51-day moving average, that, in fact, is
processing the stock market index with a particular
discrete-time system.
The result of doing that on this particular example
generates a smooth version of the curve,
which I overlay here.
And the overlay, then, is really attempting to track the
smoother variations and de-emphasize the more rapid
variations.
Let me just slightly offset that so that the difference
stands out a little more.
And so here you see what is the original weekly index.
And this is the result of processing that sequence with
an appropriate system to apply smoothing.
And in fact, what it is is a moving average.
And so here again, you can see, in the smoother curve,
this general downward trend up until this time period,
followed by, eventually, a recovery.
Well, we've seen an example with a continuous-time signal,
the Caruso recording, an example of the discrete-time
signal, this stock market index.
And what I'd also like to show is a third example, which is
the result of some processing on an image, in particular the
image that we talked about before, which was the aerial
photograph that had the problem of some cloud cover.
So once again, what we see here is the original aerial
photograph with the cloud cover.
And some processing was applied to this using a system
which, in fact, was both nonlinear and quote
"time-varying," or, in the case of these independent
variables, we would refer to it as spatially-varying.
And the result of applying that processing is shown in
the adjoining picture.
And what we see there is hopefully a reasonable attempt
to compensate for the cloud cover.
And this, by the way, was some work that was done by
Professor Lim at MIT, and has been very successful type of
processing for aerial photographs.
I should say, also, that this particular example is one
where, although the original signal was a signal that is
continuous-time, that is, the independent variables are
continuous, as they are in a spatial, aerial photograph, in
fact, for the processing, that picture was first converted to
a sequence through a process called sampling, which we'll
be talking about later.
And then the processing, in fact, was done
on a digital computer.
Well, these then are some examples of the use of some
systems to process some signals, both in continuous
time and discrete time, for one-dimensional signals and
for multi-dimensional signals.
And as I've referred to systems, we've thought of them
as one big block with an appropriate, or associated,
input and output.
And as we'll be getting into in the first part of the
course, very often, systems are interconnected together
for a variety of reasons.
Some of the kinds of interconnections that we'll
talk about are connecting systems in what are called
series, or cascade interconnections, parallel
interconnections, feedback interconnections.
And feedback interconnections, in fact, are very interesting,
very important, and very useful, and will be a major
topic toward the end of the course.
Feedback, as you may or may not know, comes into play in a
variety of situations, for example, in amplifier design,
as we'll talk about, feedback plays an important role.
In a situation where you have a basically unstable system,
feedback is often used to stabilize the system.
And feedback interconnections of systems in that sense are
very often used in high performance aircraft, which
are inherently unstable, and are stabilized through this
kind of interconnection.
Just to give you a little sense of this without going
into any of the details, what I'd like to show you is an
excerpt from a lecture that we'll be seeing toward the end
of the course relating to the analysis of feedback systems
and the uses of feedback.
And this is in the context of what's referred to as the
inverted pendulum, which is a system that's basically
unstable, and feedback interconnections are used to
stabilize it.
The idea, as you'll see in this brief clip, is that there
is a cart that's moving on a track with a rod that's
pivoted at the base.
And so that system, in the absence of anything, is
unstable in that the rod would tend to fall.
And as we go into in detail in the lecture later, we use
feedback to position the cart under the
pendulum to balance it.
And in fact, that balancing can be done even when we
modify the system in a variety of ways, as
you'll see in this clip.
So let's take a look at that, remembering that this is just
a brief excerpt from a longer discussion.
[VIDEO PLAYBACK]
I can change the overall system even further by, let's
say, for example, pouring a liquid in.
And now, let me also comment that I've changed the physics
of it a little bit because the liquid can slosh around a
little bit.
It becomes a little more complicated a system, but as
you can see, it still remains balanced.
[END VIDEO PLAYBACK]
As you'll see when we get to it, by the way, that
demonstration was a lot of fun to do.
Now, in talking about signals and systems as we go through
the course, there are several domains, two in particular,
that we will find convenient for the analysis and
representation of signals and systems.
One is the time domain, which is what we tend to think of,
and which we have kind of been focusing on in the discussion
so far in this lecture.
But equally important is what's referred to as the
frequency domain as a representation for signals,
and as a means for analysis for systems.
And in the context of the frequency domain
representation, some of the kinds of ideas and topics that
we'll be exploring are the Fourier Transform, and the
Laplace Transform, and a discrete-time counterpart of
the Laplace Transform, which is the z-Transform.
The Fourier Transform discussion we'll get into
fairly early in the course.
And the Laplace Transform and z-Transform represent
extensions of the Fourier transform, and we'll be
getting into that later in the course.
Just initially to think about the time domain and frequency
domain, you might think, for example, of
a note being played.
And the time-domain representation would be how
the sound pressure, as a function
of time, would change.
And the frequency-domain representation would
correspond to a representation of the frequency
content of the note.
And, in fact, what I'd like to do is illustrate that and
those two domains simultaneously by playing a
glockenspiel note.
What you'll hear is the note repeated several times over.
And at the same time, you'll see two displays, one on the
left representing the time domain display a
representation of the signal, and the one on the right
representing the frequency domain.
So let's look at that.
And you'll hear the note and simultaneously
see these two displays.
So there's the note on the left, the time waveform.
And on the right, what we see is the frequency content, in
particular, indicating the fact that there are several
harmonic lines in the tone.
Well, what I've gone through in this lecture represents a
brief overview of signals and systems.
And beginning with the next lecture, we will be much more
specific and precise, first discussing some basic signals,
and then talking about systems, and system
properties, and how to exploit them.
As one final comment that I'd like to make in this lecture,
I'd like to emphasize at the outset that the taped lectures
represent only one component of the course.
And equally important will be both the textbook and the
video course manual.
In particular, it's important not only to be viewing the
tapes, but simultaneously, or in conjunction with that,
doing the appropriate reading in the textbook and also
working through the problems carefully in the
video course manual.
In a course like this, you basically only get out of it
as much as you put into it.
The hope is that if you put the right amount of time and
effort into it, you'll find the course to be educational
and interesting.
And I certainly hope that that will be the case.
Thank you.