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[MUSIC PLAYING]

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PROFESSOR: I'm Al Oppenheim, and
I'd like to welcome you to

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this videotape course on
signals and systems.

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Signals, at least as an informal
definition, are

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functions of one or more
independent variables that

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typically carry some type
of information.

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Systems, in our setting,
would typically be

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used to process signals.

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One very common example of a
signal might be, let's say, a

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speech signal.

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And you might think of the air
pressure as a function of

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time, or perhaps the electrical
signal after it

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goes through the microphone
transducer as a function of

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time, as representing
the speech signal.

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And we might see a typical
speech signal looking

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something like I've
indicated here.

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It's a function of time, in
this particular case.

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And the independent variable,
being time, is, in fact,

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continuous.

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And so a signal like this, we
will typically be referring to

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as a continuous time signal.

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Now, it also, for this
particular example, is a

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function of one independent
variable.

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And that will be referred to
as a one-dimensional signal

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corresponding to the fact that
there's only one independent

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variable instead of several
independent variables.

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So the speech signal is an
example of a continuous time,

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one-dimensional signal.

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Now, signals can, of course,
be multi-dimensional.

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And they may not have, as their
independent variables,

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time variables.

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One very common example
are the examples

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represented by images.

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Images, as signals, we might
think of as representing

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brightness, as it varies
in a horizontal

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and vertical direction.

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And so the brightness as a
function of these two spatial

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variables is then a
two-dimensional signal.

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And the independent variables
would typically be continuous,

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but of course they're
not time variables.

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And incidentally, it's worth
just commenting that very

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often, simply for convenience,
we'll have a tendency to refer

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to the independent variables
when we talk about signals as

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time variables, whether or not
they really do represent time.

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Well, let me illustrate one
example of an image.

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And this is a picture of J. B.
J. Fourier, who, perhaps, more

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than anyone else, is responsible
for the elegance

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and beauty of a lot of the
concepts that we'll be talking

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about throughout this course.

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And when you look at this, in
addition to seeing Fourier

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himself, you should recognize
that what you're looking at is

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basically a signal which is
brightness as a function of

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the horizontal and vertical
spatial variables.

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As another example of an image
as a signal, let's look at an

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aerial photograph.

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This is an aerial photograph
taken over a set of roads,

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which you can more or less
recognize in the picture.

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And one of the difficulties with
this signal is that the

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road system is somewhat obscured
by cloud cover.

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And what I'll want to show later
as an example of what a

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system might do to such a
signal, in terms of processing

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it, is an attempt to at least
compensate somewhat for the

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cloud cover that's represented
in the photograph.

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Although in terms of the
detailed analysis that we go

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through during the course, our
focus of attention is pretty

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much restricted to
one-dimensional signals.

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In fact, we will be using
two-dimensional signals, more

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specifically images, very often
to illustrate a variety

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of concepts.

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Now, speech and images are
examples of what we've

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referred to as continuous-time
signals in that they are

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functions of continuous
variables.

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An equally important class of
signals that we will be

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concentrating on in the course
are signals that are

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discrete-time signals, where by
discrete-time, what we mean

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is that the signal is a
function of an integer

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variable, and so specifically
only takes on values at

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integer values of
the argument.

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So here is a graphical
illustration of a

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discrete-time signal.

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And discrete-time signals arise
in a variety of ways.

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One very common example that
is seen fairly often is

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discrete-time signals in the
context of economic time

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series, for example, stock
market analysis.

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So what I show here is one
very commonly occurring

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example of a discrete-time
signal.

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It represents the weekly
stock market index.

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The independent variable in this
case is the week number.

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And we see what the stock market
is doing over this

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particular period as
a function of the

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number of the week.

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And, of course, along
the vertical axis

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is the weekly index.

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Incidentally, this particular
period was

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not chosen at random.

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It In fact captures a very
interesting aspect of stock

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market history, namely the stock
market crash in 1929,

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which, in fact, is represented
by the behavior of this

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discrete-time signal,
or sequence, in

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this particular area.

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So this dramatic dip, in
fact, is the stock

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market crash of 1929.

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Well, the Dow Jones weekly
average is an example of a

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one-dimensional discrete-time
signal.

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And just as with continuous
time, we had not just

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one-dimensional but
multi-dimensional signals,

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likewise we have
multi-dimensional signals in

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the discrete-time case where,
in that case, then, the

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discrete-time signal that
we're talking about, or

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sequence, is a function of
two integer variables.

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And as one example, this might,
let's say, represent a

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spatial antenna array where this
is array number in, let's

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say, the horizontal direction,
and this is array number in

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the vertical direction.

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Both classes of signals,
continuous-time and

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discrete-time, as I've
indicated, are very important.

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And it should be emphasized
that the importance of

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discrete-time signals and
associated processing

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continues to grow in large part
because of the current

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and emerging technologies that
permit, basically, the

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processing of continuous-time
signals by first converting

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them to discrete-time signals
and processing them with

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discrete-time systems.

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And that, in fact, is a topic
that we will discuss in a fair

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amount of detail later
on in the course.

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Let's now our attention
to systems.

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And as I indicated, a system
basically processes signals.

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And they have, of course,
inputs and outputs.

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And depending on whether we're
talking about continuous time

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or discrete time, the system
may be a continuous-time

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system or a discrete-time
system.

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So in the continuous-time case,
I indicate here an input

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x(t) and an output y(t) If
we were talking about a

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discrete-time system, I would
represent the input in terms

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of a discrete-time variable,
and, of course, the output in

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terms of a discrete-time
variable also.

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Now, in very general terms,
systems are hard to deal with

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because they are defined very
broadly and very generally.

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And in dealing with systems and
analyzing them, what we

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will do is attempt to exploit
some very specific, and as

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we'll see, very useful
system properties.

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To indicate what I mean and how
things might be split up,

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we could talk about systems, and
will talk about systems,

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that are linear.

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And we could divide systems,
basically, into systems that

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are either linear or nonlinear,
and we will, and

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also divide systems into systems
that are what we'll

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refer to as time-invariant
or time-varying systems.

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And these aren't terms that
we've defined yet, of course,

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but we will be defining in
the course very shortly.

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And while, in some sense, this
division represents all

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systems, and this does, too,
the focus of the course is

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really going to be principally
on linear,

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time-invariant systems.

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So it's basically these systems
that we will be

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focusing on.

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And we'll be referring to
those systems as linear,

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time-invariant systems.

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Well, as a brief glimpse at some
of the kinds of things

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that systems can do, let me
illustrate, first in a

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one-dimensional continuous-time
context, and

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then later with a discrete-time
example, one

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example of some processing
of signals with

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an appropriate system.

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The particular example that I
want to illustrate relates to

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the restoration of
old recordings.

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And this is some work that was
done by Professor Thomas

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Stockham, who is at the
University of Utah, and work

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that he had done a number of
years ago relating to the fact

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that in old recordings, for
example in Caruso recordings,

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the recording was done through
a mechanical horn, and the

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characteristics of the
horn tended to

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vary from day to day.

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And because of the
characteristics of the horn,

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the recording tended to have a
muffled quality, something

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like this, sort of the sense
that you would get if you were

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speaking through a megaphone.

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What Professor Stockham did
was develop a system to

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process these old recordings
in such a way that a lot of

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the characteristics and
distortion due to that

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recording system was removed.

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So I'd like to illustrate that
as one example of some signal

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processing with an appropriate
continuous-time system.

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And what you'll hear is
a two-track recording.

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On the first track is the
original, unrestored Caruso

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recording, and on the second
track is the result of the

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restoration.

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And so as I switch back and
forth from channel one to

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channel two, we'll be switching
from the original to

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the restored.

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We'll begin the tape by
playing the original.

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And then, as it proceeds,
we'll switch.

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So we'll begin on channel one.

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[MUSIC PLAYING, MUFFLED]

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That's the original recording.

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And switch now to
the processed.

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[MUSIC PLAYING, CLEARER]

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Now let's switch back,
back to the original.

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Back to the restoration.

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And once again, back
to the original.

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And presumably and hopefully,
what you heard was that in the

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restoration, in fact, a lot of
the muffled characteristics of

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the original recording were
compensated for or removed.

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Now one of the interesting
things that happened, in fact,

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in the work that Professor
Stockham did is that in the

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process of the restoration--

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and perhaps you heard this--

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in the process of the
restoration, in fact, some of

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the background noise on the
recording was emphasized.

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And so he processed the signal
further in an attempt to

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remove that background noise.

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And with that particular
processing, the processing was

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very highly nonlinear.

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A very interesting thing
happened, which was that not

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only in that processing was the
background noise removed,

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but somewhat surprisingly,
also the

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orchestra was removed.

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And let me just play that now
as an example of some very

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sophisticated processing with
a nonlinear system.

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What you'll hear on channel one
is the restoration as we

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had just played it.

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When I switch to channel two,
it will be after the

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processing with an attempt to
remove the orchestra and the

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background noise.

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Channel one now.

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And now the noise and
orchestra removed.

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Back to channel one.

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And finally, once again, with
the orchestra removed.

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So that's an example
of processing of a

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continuous-time signal
with a corresponding

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continuous-time system.

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Now I'd like to illustrate an
example of some processing on

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a discrete-time signal.

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And I'd like to do that in the
context of the example that I

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showed before of a discrete-time
signal, which

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was the Dow Jones Industrial
weekly stock market index.

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I had shown it before, as I've
shown it here again, over a

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period of slightly more than a
year, where this is the number

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of weeks and this is
the weekly index.

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And to illustrate some of the
processing, what I'd like to

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do is show the stock market
index, the weekly index, over

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a much longer time period, in
particular, the weekly index

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over a 10 year period.

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And that's what I show here.

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00:17:12,290 --> 00:17:19,329
So what this covers is
roughly 1927 to 1937.

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00:17:19,329 --> 00:17:22,950
And in this case, although this
is still a discrete-time

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00:17:22,950 --> 00:17:26,319
signal, just simply for the
purposes of display, what

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00:17:26,319 --> 00:17:31,010
we've done is to essentially
connect the dots and draw a

249
00:17:31,010 --> 00:17:34,550
continuous curve through the
points so that this picture

250
00:17:34,550 --> 00:17:37,110
isn't filled up with a lot
of vertical lines.

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00:17:37,110 --> 00:17:42,050
So this is the discrete-time
sequence that represents the

252
00:17:42,050 --> 00:17:46,970
weekly Dow Jones Index over
a 10 year period.

253
00:17:46,970 --> 00:17:53,250
And here, by the way, again,
is the crash of 1929.

254
00:17:53,250 --> 00:17:56,990
It's interesting to note, by
the way, that actually the

255
00:17:56,990 --> 00:18:02,050
disaster in the stock market
wasn't so much the 1929 crash

256
00:18:02,050 --> 00:18:04,780
but the long downward trend
that followed that.

257
00:18:04,780 --> 00:18:10,430
And you can see that here by
filtering through, by eye, the

258
00:18:10,430 --> 00:18:13,220
rapid variations in the index.

259
00:18:13,220 --> 00:18:17,160
And what you see is this
smooth downward trend

260
00:18:17,160 --> 00:18:23,000
followed, eventually,
by an upward trend.

261
00:18:23,000 --> 00:18:27,800
Now, this issue of looking at
something like this, looking

262
00:18:27,800 --> 00:18:33,180
at a sequence, and following
the smoother parts of it,

263
00:18:33,180 --> 00:18:37,950
namely the long term trends, is,
in fact, something that is

264
00:18:37,950 --> 00:18:40,470
done quite typically
with economic

265
00:18:40,470 --> 00:18:42,325
time series like this.

266
00:18:42,325 --> 00:18:47,560
And in particular, what's done
is to smooth it, or average

267
00:18:47,560 --> 00:18:53,820
over some time period, to
emphasize the slow variations

268
00:18:53,820 --> 00:18:56,750
and de-emphasize the
rapid variations.

269
00:18:56,750 --> 00:19:01,070
And that, in fact, is processing
that is done with a

270
00:19:01,070 --> 00:19:02,560
discrete-time system.

271
00:19:02,560 --> 00:19:06,660
So when you hear referred to,
let's say, in stock market

272
00:19:06,660 --> 00:19:11,070
reports, a 51-day moving
average, that, in fact, is

273
00:19:11,070 --> 00:19:15,630
processing the stock market
index with a particular

274
00:19:15,630 --> 00:19:17,460
discrete-time system.

275
00:19:17,460 --> 00:19:21,960
The result of doing that on
this particular example

276
00:19:21,960 --> 00:19:25,090
generates a smooth version
of the curve,

277
00:19:25,090 --> 00:19:27,040
which I overlay here.

278
00:19:27,040 --> 00:19:34,330
And the overlay, then, is really
attempting to track the

279
00:19:34,330 --> 00:19:37,600
smoother variations and
de-emphasize the more rapid

280
00:19:37,600 --> 00:19:39,370
variations.

281
00:19:39,370 --> 00:19:42,930
Let me just slightly offset that
so that the difference

282
00:19:42,930 --> 00:19:44,680
stands out a little more.

283
00:19:44,680 --> 00:19:51,230
And so here you see what is
the original weekly index.

284
00:19:51,230 --> 00:19:56,280
And this is the result of
processing that sequence with

285
00:19:56,280 --> 00:20:00,990
an appropriate system
to apply smoothing.

286
00:20:00,990 --> 00:20:04,110
And in fact, what it is
is a moving average.

287
00:20:04,110 --> 00:20:08,330
And so here again, you can see,
in the smoother curve,

288
00:20:08,330 --> 00:20:12,800
this general downward trend
up until this time period,

289
00:20:12,800 --> 00:20:17,620
followed by, eventually,
a recovery.

290
00:20:17,620 --> 00:20:23,560
Well, we've seen an example with
a continuous-time signal,

291
00:20:23,560 --> 00:20:26,070
the Caruso recording, an example
of the discrete-time

292
00:20:26,070 --> 00:20:28,900
signal, this stock
market index.

293
00:20:28,900 --> 00:20:33,810
And what I'd also like to show
is a third example, which is

294
00:20:33,810 --> 00:20:38,320
the result of some processing on
an image, in particular the

295
00:20:38,320 --> 00:20:40,720
image that we talked about
before, which was the aerial

296
00:20:40,720 --> 00:20:45,000
photograph that had the problem
of some cloud cover.

297
00:20:45,000 --> 00:20:50,570
So once again, what we see here
is the original aerial

298
00:20:50,570 --> 00:20:53,770
photograph with the
cloud cover.

299
00:20:53,770 --> 00:20:58,260
And some processing was applied
to this using a system

300
00:20:58,260 --> 00:21:02,520
which, in fact, was both
nonlinear and quote

301
00:21:02,520 --> 00:21:06,280
"time-varying," or, in the
case of these independent

302
00:21:06,280 --> 00:21:10,520
variables, we would refer to
it as spatially-varying.

303
00:21:10,520 --> 00:21:16,100
And the result of applying that
processing is shown in

304
00:21:16,100 --> 00:21:18,590
the adjoining picture.

305
00:21:18,590 --> 00:21:24,350
And what we see there is
hopefully a reasonable attempt

306
00:21:24,350 --> 00:21:26,890
to compensate for
the cloud cover.

307
00:21:26,890 --> 00:21:31,080
And this, by the way, was some
work that was done by

308
00:21:31,080 --> 00:21:37,150
Professor Lim at MIT, and has
been very successful type of

309
00:21:37,150 --> 00:21:40,160
processing for aerial
photographs.

310
00:21:40,160 --> 00:21:44,480
I should say, also, that this
particular example is one

311
00:21:44,480 --> 00:21:49,670
where, although the original
signal was a signal that is

312
00:21:49,670 --> 00:21:52,790
continuous-time, that is, the
independent variables are

313
00:21:52,790 --> 00:21:57,080
continuous, as they are in a
spatial, aerial photograph, in

314
00:21:57,080 --> 00:22:01,180
fact, for the processing, that
picture was first converted to

315
00:22:01,180 --> 00:22:05,050
a sequence through a process
called sampling, which we'll

316
00:22:05,050 --> 00:22:06,620
be talking about later.

317
00:22:06,620 --> 00:22:09,180
And then the processing,
in fact, was done

318
00:22:09,180 --> 00:22:10,430
on a digital computer.

319
00:22:12,890 --> 00:22:19,340
Well, these then are some
examples of the use of some

320
00:22:19,340 --> 00:22:25,490
systems to process some signals,
both in continuous

321
00:22:25,490 --> 00:22:30,290
time and discrete time, for
one-dimensional signals and

322
00:22:30,290 --> 00:22:34,190
for multi-dimensional signals.

323
00:22:34,190 --> 00:22:38,580
And as I've referred to systems,
we've thought of them

324
00:22:38,580 --> 00:22:42,310
as one big block with an
appropriate, or associated,

325
00:22:42,310 --> 00:22:43,990
input and output.

326
00:22:43,990 --> 00:22:47,530
And as we'll be getting into
in the first part of the

327
00:22:47,530 --> 00:22:52,320
course, very often, systems are
interconnected together

328
00:22:52,320 --> 00:22:54,100
for a variety of reasons.

329
00:22:54,100 --> 00:22:56,010
Some of the kinds of
interconnections that we'll

330
00:22:56,010 --> 00:23:02,300
talk about are connecting
systems in what are called

331
00:23:02,300 --> 00:23:06,080
series, or cascade
interconnections, parallel

332
00:23:06,080 --> 00:23:09,930
interconnections, feedback
interconnections.

333
00:23:09,930 --> 00:23:14,590
And feedback interconnections,
in fact, are very interesting,

334
00:23:14,590 --> 00:23:18,160
very important, and very useful,
and will be a major

335
00:23:18,160 --> 00:23:21,340
topic toward the end
of the course.

336
00:23:21,340 --> 00:23:27,280
Feedback, as you may or may not
know, comes into play in a

337
00:23:27,280 --> 00:23:31,250
variety of situations, for
example, in amplifier design,

338
00:23:31,250 --> 00:23:36,770
as we'll talk about, feedback
plays an important role.

339
00:23:36,770 --> 00:23:42,310
In a situation where you have
a basically unstable system,

340
00:23:42,310 --> 00:23:45,320
feedback is often used to
stabilize the system.

341
00:23:45,320 --> 00:23:49,520
And feedback interconnections
of systems in that sense are

342
00:23:49,520 --> 00:23:53,020
very often used in high
performance aircraft, which

343
00:23:53,020 --> 00:23:56,830
are inherently unstable, and
are stabilized through this

344
00:23:56,830 --> 00:23:59,770
kind of interconnection.

345
00:23:59,770 --> 00:24:03,760
Just to give you a little sense
of this without going

346
00:24:03,760 --> 00:24:08,970
into any of the details, what
I'd like to show you is an

347
00:24:08,970 --> 00:24:15,490
excerpt from a lecture that
we'll be seeing toward the end

348
00:24:15,490 --> 00:24:20,340
of the course relating to the
analysis of feedback systems

349
00:24:20,340 --> 00:24:22,100
and the uses of feedback.

350
00:24:22,100 --> 00:24:25,880
And this is in the context of
what's referred to as the

351
00:24:25,880 --> 00:24:29,670
inverted pendulum, which is
a system that's basically

352
00:24:29,670 --> 00:24:32,790
unstable, and feedback
interconnections are used to

353
00:24:32,790 --> 00:24:34,340
stabilize it.

354
00:24:34,340 --> 00:24:39,720
The idea, as you'll see in this
brief clip, is that there

355
00:24:39,720 --> 00:24:44,220
is a cart that's moving on
a track with a rod that's

356
00:24:44,220 --> 00:24:45,880
pivoted at the base.

357
00:24:45,880 --> 00:24:49,500
And so that system, in the
absence of anything, is

358
00:24:49,500 --> 00:24:52,390
unstable in that the rod
would tend to fall.

359
00:24:52,390 --> 00:24:57,510
And as we go into in detail in
the lecture later, we use

360
00:24:57,510 --> 00:25:01,000
feedback to position
the cart under the

361
00:25:01,000 --> 00:25:03,680
pendulum to balance it.

362
00:25:03,680 --> 00:25:07,730
And in fact, that balancing
can be done even when we

363
00:25:07,730 --> 00:25:10,690
modify the system in a
variety of ways, as

364
00:25:10,690 --> 00:25:11,970
you'll see in this clip.

365
00:25:11,970 --> 00:25:15,850
So let's take a look at that,
remembering that this is just

366
00:25:15,850 --> 00:25:19,340
a brief excerpt from a
longer discussion.

367
00:25:19,340 --> 00:25:19,700
[VIDEO PLAYBACK]

368
00:25:19,700 --> 00:25:25,460
I can change the overall system
even further by, let's

369
00:25:25,460 --> 00:25:28,810
say, for example, pouring
a liquid in.

370
00:25:28,810 --> 00:25:32,870
And now, let me also comment
that I've changed the physics

371
00:25:32,870 --> 00:25:36,800
of it a little bit because the
liquid can slosh around a

372
00:25:36,800 --> 00:25:37,160
little bit.

373
00:25:37,160 --> 00:25:40,170
It becomes a little more
complicated a system, but as

374
00:25:40,170 --> 00:25:43,080
you can see, it still
remains balanced.

375
00:25:43,080 --> 00:25:45,560
[END VIDEO PLAYBACK]

376
00:25:45,560 --> 00:25:48,090
As you'll see when we get
to it, by the way, that

377
00:25:48,090 --> 00:25:51,290
demonstration was a
lot of fun to do.

378
00:25:51,290 --> 00:25:56,090
Now, in talking about signals
and systems as we go through

379
00:25:56,090 --> 00:26:01,360
the course, there are several
domains, two in particular,

380
00:26:01,360 --> 00:26:05,850
that we will find convenient
for the analysis and

381
00:26:05,850 --> 00:26:10,070
representation of signals
and systems.

382
00:26:10,070 --> 00:26:17,100
One is the time domain, which
is what we tend to think of,

383
00:26:17,100 --> 00:26:21,910
and which we have kind of been
focusing on in the discussion

384
00:26:21,910 --> 00:26:23,950
so far in this lecture.

385
00:26:23,950 --> 00:26:28,890
But equally important is what's
referred to as the

386
00:26:28,890 --> 00:26:34,220
frequency domain as a
representation for signals,

387
00:26:34,220 --> 00:26:37,630
and as a means for analysis
for systems.

388
00:26:37,630 --> 00:26:40,600
And in the context of
the frequency domain

389
00:26:40,600 --> 00:26:44,750
representation, some of the
kinds of ideas and topics that

390
00:26:44,750 --> 00:26:50,270
we'll be exploring are the
Fourier Transform, and the

391
00:26:50,270 --> 00:26:55,930
Laplace Transform, and a
discrete-time counterpart of

392
00:26:55,930 --> 00:26:59,140
the Laplace Transform, which
is the z-Transform.

393
00:26:59,140 --> 00:27:02,080
The Fourier Transform discussion
we'll get into

394
00:27:02,080 --> 00:27:03,990
fairly early in the course.

395
00:27:03,990 --> 00:27:08,110
And the Laplace Transform and
z-Transform represent

396
00:27:08,110 --> 00:27:11,200
extensions of the Fourier
transform, and we'll be

397
00:27:11,200 --> 00:27:14,520
getting into that later
in the course.

398
00:27:14,520 --> 00:27:18,550
Just initially to think about
the time domain and frequency

399
00:27:18,550 --> 00:27:21,610
domain, you might think,
for example, of

400
00:27:21,610 --> 00:27:23,490
a note being played.

401
00:27:23,490 --> 00:27:28,800
And the time-domain
representation would be how

402
00:27:28,800 --> 00:27:32,250
the sound pressure,
as a function

403
00:27:32,250 --> 00:27:34,330
of time, would change.

404
00:27:34,330 --> 00:27:37,560
And the frequency-domain
representation would

405
00:27:37,560 --> 00:27:40,960
correspond to a representation
of the frequency

406
00:27:40,960 --> 00:27:43,440
content of the note.

407
00:27:43,440 --> 00:27:46,720
And, in fact, what I'd like to
do is illustrate that and

408
00:27:46,720 --> 00:27:51,720
those two domains simultaneously
by playing a

409
00:27:51,720 --> 00:27:53,370
glockenspiel note.

410
00:27:53,370 --> 00:27:58,220
What you'll hear is the note
repeated several times over.

411
00:27:58,220 --> 00:28:02,490
And at the same time, you'll see
two displays, one on the

412
00:28:02,490 --> 00:28:05,950
left representing the time
domain display a

413
00:28:05,950 --> 00:28:08,820
representation of the signal,
and the one on the right

414
00:28:08,820 --> 00:28:11,650
representing the frequency
domain.

415
00:28:11,650 --> 00:28:13,350
So let's look at that.

416
00:28:13,350 --> 00:28:16,130
And you'll hear the note
and simultaneously

417
00:28:16,130 --> 00:28:17,380
see these two displays.

418
00:28:21,080 --> 00:28:25,850
So there's the note on the
left, the time waveform.

419
00:28:25,850 --> 00:28:30,285
And on the right, what we see
is the frequency content, in

420
00:28:30,285 --> 00:28:34,900
particular, indicating the fact
that there are several

421
00:28:34,900 --> 00:28:36,420
harmonic lines in the tone.

422
00:28:54,540 --> 00:29:00,410
Well, what I've gone through in
this lecture represents a

423
00:29:00,410 --> 00:29:04,450
brief overview of signals
and systems.

424
00:29:04,450 --> 00:29:08,340
And beginning with the next
lecture, we will be much more

425
00:29:08,340 --> 00:29:13,950
specific and precise, first
discussing some basic signals,

426
00:29:13,950 --> 00:29:16,880
and then talking about
systems, and system

427
00:29:16,880 --> 00:29:20,860
properties, and how
to exploit them.

428
00:29:20,860 --> 00:29:26,890
As one final comment that I'd
like to make in this lecture,

429
00:29:26,890 --> 00:29:34,640
I'd like to emphasize at the
outset that the taped lectures

430
00:29:34,640 --> 00:29:38,180
represent only one component
of the course.

431
00:29:38,180 --> 00:29:42,690
And equally important will be
both the textbook and the

432
00:29:42,690 --> 00:29:44,670
video course manual.

433
00:29:44,670 --> 00:29:49,140
In particular, it's important
not only to be viewing the

434
00:29:49,140 --> 00:29:55,470
tapes, but simultaneously, or
in conjunction with that,

435
00:29:55,470 --> 00:29:59,510
doing the appropriate reading
in the textbook and also

436
00:29:59,510 --> 00:30:02,500
working through the problems
carefully in the

437
00:30:02,500 --> 00:30:04,940
video course manual.

438
00:30:04,940 --> 00:30:08,830
In a course like this, you
basically only get out of it

439
00:30:08,830 --> 00:30:10,630
as much as you put into it.

440
00:30:10,630 --> 00:30:14,900
The hope is that if you put the
right amount of time and

441
00:30:14,900 --> 00:30:18,910
effort into it, you'll find the
course to be educational

442
00:30:18,910 --> 00:30:19,720
and interesting.

443
00:30:19,720 --> 00:30:22,250
And I certainly hope that
that will be the case.

444
00:30:22,250 --> 00:30:23,500
Thank you.