PROFESSOR: Hi.
Today I'd like to talk to about
a new module called
Signals and Systems.
Our previous module on
programming, we focused on
object-oriented programming
and state machines as
different methods by which
we could model
the physical world.
In this module we're going to
talk about a new way to model
the physical world called
the discrete linear
time-invariant system.
You might say, Kendra, why do
I need to do that, or why
would I move away from
state machines.
I seem to be able to model
everything using state machines.
The short answer is, you want
to be able to move away from
state machines, because
you want to be able
to predict the future.
But I'm not really going to be
able to get to how you're
going to be able to predict
the future until
two videos from now.
So for now we'll introduce
discrete linear time-invariant
systems and also talk about
different knowledge
representations you might
encounter them in, so that you
recognize them when you see them
and can manipulate them
as needed to talk to other
people about them.
Let's go to the board.
Last time we talked about
state machines.
We're able to use them to model
pretty much any system.
We can model the evolution of a
particular process over time
using a record of all previous
inputs and outputs and
possibly previous states, if we
included that in our output.
But the problem with state
machines is that they, like
most programming, encounter
the halting problem.
At some point your program can
reach a little complexity at
which you cannot determine what
it's going to do without
actually just running it and
seeing what happens.
This is great for researchers.
This is the method by which
people go out and have to find
something or discover something
instead of being
able to like, simulate
it using a machine.
But if you want to be able to
promise things to other people
or predict what's going to
happen, then it helps if you
can model something that has a
high level of complexity by
abstracting away the complexity
and interacting
with it as though it has
features that indicate that
it's going to behave in
a particular way.
So if you want to make a control
system that does a
particular thing, or look at
an existing system and say,
well, I know that your power
plant's going to blow up in
five years, because of this,
it helps if you have some
understanding of particular
classes of systems and what
kind of long term behaviors
those particular classes of
systems generate.
And that's why we look
at discrete LTI.
It's a particular class
of system --
in particular discrete LTI is
what we're going to look at in
6.01 because it's
fairly simple.
But once you've learned how to
deal with the system in that
manner, you can use the skills
that you've learned there to
approach more complex feedback
problems or more complex
control problems.
First, an incredibly brief
review of linear
time-invariant systems.
When you're talking about a
linear time-invariant system,
both the inputs and the outputs
are going to be real.
We're not going to deal with
the complex plane at all.
If you were to model your LTI
system using a state machine,
that state machine would be
dependent on a fixed amount of
previous inputs, outputs
or states.
You give some amount of leeway
for start state or the fixed
number of states before you get
to your fixed amount of
data structure that represents
your current state.
But if you're running a linear
time-invariant system, the
amount of information that you
need to figure out your state
in a long term sense is always
finite and fixed.
In terms of functional
expressions, you've probably
seen these associated with
linear time-invariant systems.
Know that this also means that
you can represent linear
time-invariant systems as
additions or scalar
multiplications of your
existing function.
So that's out of the way.
These are all true of LTI.
We're going to focus
on discrete LTI.
One, because we can use
a discrete state
machine to model it.
And two, because it
makes it easier to
represent using computers.
Once you've got that digital
abstraction, you don't have to
worry about having access to a
truly continuous function.
All right.
Now that you know what we're
going to talk about, how do
you talk about it
to other people?
Here are the different
representations that you might
see a discrete linear
time-invariant system in.
One of them is called a
difference equation.
And you've probably encountered
this in one of
your math classes.
It says that you're going to
take a sample from a signal y
at some sort of given time
step, typically n.
And when you're writing out
a difference equation that
represents an entire system, you
usually see y of n on one
side on the left, and then
everything that represents the
functional component on the
system on the right, which in
this case determines what
determines the output
at every time step.
In this very particular case,
we're talking about an
accumulator, which means that at
every time step the output
is determined by the input
plus the previous output.
So at every time step we take
the input and whatever it is
we were outputting before
and output it again.
Pretty straightforward.
If somebody described to you a
difference equation in words,
you could probably turn it into
a equation at this point.
Note that even though there are
variables associated with
these samples, they are still
very particular samples.
And the thing that you're
interested in is probably not
even the samples at all.
It's the relationship between
the samples, when they were
sampled, and relative to one
another, and how that relates
to the output at a
given time step.
If you then want to talk about
an entire signal, then you can
use an operator equation.
And you'll probably see these
things when you do any sort of
research on control theory
or feedback.
Instead of representing the
sample of the signal y at a
given time n, we're just going
to talk about the overall
signal capital Y. Likewise,
instead of talking about x at
a given time n, we're going to
talk about the signal capital
X.
Remember that I said that the
most important part of this
equation is the fact that
there's a different
relationship between these two
signals and when you were
sampling from them.
When we want to represent this
relative delay, we represent
it using an R. And in
particular, I want to note the
fact that the degree of R
represents the amount of delay
associated with the sample
from a particular signal.
So if I wanted to make this n
minus 2, I would reflect that
change in my operator equation
by changing the degree of R.
And because we're working with
a linear time-invariant
system, if we wanted to scale
this by 2, it'd be the same as
scaling this by 2, et cetera.
Now I can talk about signals.
What if I want to talk about a
physical manifestation of the
relationship between these
signals, and/or what if I want
to use something kind of like
circuit diagrams to talk about
what kind of signal manipulation
I'm going to do?
This is where block
diagrams come in.
Block diagrams are incredibly
PowerPoint-friendly.
Block diagrams mean that
everybody is on the same page,
because all you have to do
is trace out the arrows.
Block diagrams mean you're going
to do a combination of
gains, delays and adders to
visually represent the
relationship between your output
signal and any input
signals that you have.
Up here I have actually
drawn an accumulator.
And it's got a box around
it, which will be
relevant in two minutes.
But right now you
can ignore it.
I've got my input signal,
or my input signals, are
typically on the left.
And the progression of gains,
delays and adders associated
with the block diagram represent
the relationship
between the input signals
and the output signal.
And then I'll have my output
signal on the right.
Most people indicate flow with
arrows, in part to make things
easier to read.
But typically, you only need
the arrow indicating where
you're sampling from and
what your output's
going to look like.
In this case in order to get Y,
and in the general sense, I
can backtrace from the signal
that I'm interested in through
my diagram and figure
out what values I'm
actually interested in.
So in this particular case, Y
is a linear combination of X
and whatever this represents,
which is RY.
If I did want to put a 2 here,
I've been talking about gains,
delays and adders, but
this diagram doesn't
contain a gain yet.
So let me show you how you
would include a gain.
Gains are typically represented
by the value of
the gain inside an arrow.
It looks like an op-amp from
schematic diagrams.
We'll get to those later
in circuits.
It's not essential that the
arrow point in the direction
of flow, but people might look
at you funny if you don't put
it in the direction of flow.
And the other thing that I think
I want to note here is
that if you see a minus sign
here, it means X minus 2RY,
which is the same as putting a
negative value on your gain.
So right now, we're back
to X plus 2RY.
OK.
We've got to block diagrams.
You might be saying at this
point, Kendra, what are block
diagrams good for other than
PowerPoint presentations and
possibly arguing with my friends
over what's going on.
And I say, they're really
good for abstraction.
I could draw a box around this
and abstract it away into a
function and say, if I put
something into this box, and I
want to get something out where
the action that happens
in here is identical to this
schematic, then I can find an
equation that actually
represents that operation.
And the way I do that is I'm
going to take my operator
equation and solve in particular
for Y over X. In
this case, with the 2, Y
over X is going to be 1
over (1 minus 2R).
Note that if H is equal to the
expression Y over X, and I
multiply H by X, I get out Y.
This concludes my coverage of
motivations for learning about
discrete LTI systems and also
the different representations
that we will want to use when
talking to other people about
discrete LTI.
At this point we can also start
talking about how we get
to the point where we can start
predicting the future.
And then in the video after
that I'll actually start
talking about poles.
