[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.
