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PROFESSOR: Last time, which is now two weeks ago, we started
the talk about the signals and systems approach, by which I
mean, think about a system by the way it transforms its
input signal into an output signal.
That's kind of a bizarre way of thinking about systems.
I demonstrated that last time by thinking about a mass and a
spring, something you have a lot of experience with, but
you probably didn't use this kind of approach for
thinking about it.
Over this lecture and over the next lecture, what I'd like to
do is show you the advantages of this kind of approach.
And so for today, what I'd like to do is talk about how
to think about feedback within this structure, and I'd like
to also think about how you can use this structure to
characterize the performance of a system in a
quantitative fashion.
So first off, I want to just think about feedback.
Feedback is so pervasive that you don't notice
it most of the time.
You use feedback in virtually everything that you do.
Here is a very simple example of driving a car.
If you want to keep the car in the center of your lane,
something that many people outside of Boston, at least,
think is a good idea, then you are mentally doing feedback.
You're constantly comparing where you are to where you'd
like to be and making small adjustments to the system
based on that.
When you think of even the most simple of systems, like
the thermostat in a house--
I'm not talking about a cheap motel.
The cheap motels don't do this.
But in a real house, there's a thermostat, which regulates
the temperature.
That's important, because if the temperature suddenly
drops, it would never do that of course, but if the
temperature ever dropped, it could compensate for it.
Here's one of my favorite examples.
Feedback is enormously pervasive in biology.
There's no general rules in biology, but to a first cut,
everything is regulated.
And in many cases, the regulation is amazing.
Here's an example from 6.021, where it's illustrating the
system that your body uses to regulate glucose delivery from
food sources to every cell in your body, which is crucial to
your being cognizant and mobile.
And the idea is that it does that with amazing precision
despite the fact that eating and exercise
are enormously episodic.
In order for you to remain healthy and functional, you
need to have something between approximately two and ten
millimoles per liter of glucose in your
blood at all times.
Were it to go higher than that, systematically, you
would develop cardiac problems and could lead to even
congestive heart failure.
If you were to have lower than two millimoles per liter, you
would go comatose.
That's a very narrow range, two to five, especially
because what we eat is so episodic, when we exercise is
so episodic.
It's amazing, and just to dramatize how amazing it is,
how much sugar do you think circulates in
your blood right now?
Well, if you convert five millimoles per liter, and if
you assume the average person has about three liters of
blood, which is true, and if you calculate it, that comes
out to 2.7 grams.
That's this much.
This is 2.7 grams of table sugar.
So this is how much sugar is in your blood, now.
This is what's keeping you healthy.
This is what's keeping you from becoming comatose.
Or not.
How much sugar do you think is in this?
AUDIENCE: The amount in that other cup.
PROFESSOR: Exactly.
We call that the Theory of Lectures.
I don't want to look like an idiot, so my
problems make sense.
This is the amount of sugar in a can of soda.
Numerically, this is 39 grams.
That's more than 13 times this much.
So when you down one of these, it's very important that the
sugar gets taken out of the blood quickly, and it does.
And that happens by way of a feedback system, and the
feedback system is illustrated here.
Basically, the feedback involves the hormone insulin,
and that's why insulin deficiency is such a
devastating disease.
Finally, everything we do has feedback in it.
Think about how your life would be
different if it didn't.
Even a simple task, like removing a light bulb, would
be virtually impossible except for the fact that you get
feedback from everything.
Your hand's amazing.
You have touch sensors, you have proprioceptive sensors,
you have stress sensors on the muscles and ligaments, and
they all coordinate to tell you when to stop squeezing on
the light bulb so you don't break it.
That's all completely amazing, and what we'd like to do,
then, is think about that kind of a system, a feedback
system, within the signals and systems construct.
As an example, I want to think through the WallFinder problem
that you did last week in Design Lab.
I'm sure you all remember that, in that problem, we were
trying to move the robot to a fixed distance away from the
wall, and we thought about that as a feedback system
comprised of three parts, a controller, a
plant, and a sensor.
We wrote difference equations to characterize each of those
parts, and then we figured out how to solve those difference
equations to make some meaningful prediction about
how the robot would work.
So just to make sure you're all on board with me, now,
here's a question for you.
These are the equations that describe
the WallFinder problem.
How many equations and how many unknowns are there?
Take 20 seconds, talk to your neighbor, figure out an answer
between (1) and (5).
Unlike during the exam next week, you are allowed to talk.
[CLASSROOM SIDE CONVERSATIONS]
PROFESSOR: T and K are no.
[CLASSROOM SIDE CONVERSATIONS]
PROFESSOR: OK, so what's the answer, number (1), (2), (3),
(4), or (5)?
Everybody raise your hands.
Let me see if I got the answer.
OK.
Come on, raise your hands.
Come on, everybody vote.
If you're wrong, just blame it on your neighbor.
You had a poor partner.
That's the idea.
Right?
So you can all vote, and you don't need to worry about
being wrong.
And you're all wrong, so that worked out well.
OK, so I don't like the-- so the predominant answer is
number (2).
I don't like number (2).
Can somebody think of a reason, now that you know the
answer by the Theory of Lectures, the
answer is not (2)?
Why isn't the answer (2)?
Yeah?
AUDIENCE: Oh, I was saying (5).
PROFESSOR: You were saying (5).
Why did you say (5)?
AUDIENCE: I don't remember for sure, but can you substitute n
for different things?
Like Df of n, you could substitute n for, and if you
did that you would have two equations, which is
[INAUDIBLE].
PROFESSOR: [UNINTELLIGIBLE] kind of thing, you count
before or after doing substitution and
simplification.
And I mean to count before you do simplification.
Any other issues?
Yeah?
AUDIENCE: Is D0 [UNINTELLIGIBLE]
PROFESSOR: Say again?
AUDIENCE: D0 [UNINTELLIGIBLE]
PROFESSOR: Is D0_n different or the same from D0_n-1?
That's the key question.
So I want to think about this as a system of algebraic
equations, and if I do that, then there's a lot of them.
So it looks like there's three equations.
The problem with that approach is that there's actually three
equations for every value of n.
If you think about a system of equations that you could solve
with an algebra solver, you would have to treat all the
n's separately.
That's what we call the samples approach.
So here's a way you could solve them.
You could think about what if k and t are parameters, so
they're known, they're given?
What if my input signal is known, say it's a unit sample
signal, for example?
What would I need to solve this system?
Well, I'd need to tell you the initial conditions.
So in some sense, I want to consider those to be known.
So my knowns kind of comprise t and k, the initial
conditions for the output of the robot, and the sensor, all
of the input signals because I'm telling you the input and
asking you to calculate the output.
My unknowns are all of the different velocities for all
values of n bigger than or equal to 0 because I didn't
tell you those.
All of the values of robot's output at samples
n bigger than 0.
All the values of the sensor output for values
n bigger than 0.
So I get a lot of unknowns, infinitely many, and I get a
lot of equations, also infinitely many.
So the thing I want you think about is, if you're thinking
about solving difference equations using algebra,
that's a big system of equations.
By contrast, what if you were to try to solve the system
using operators?
Now, how many equations and unknowns do you see?
By the Theory of Lectures.
OK, raise your hand.
Or talk to your neighbor so you can blame your neighbor.
Talk to your neighbor.
Get a good alibi.
[CLASSROOM SIDE CONVERSATIONS]
PROFESSOR: So the idea here is that if you think about
operators instead, so that you look at a whole signal at a
time, then each equation only specifies one relationship
among signals, and there's a small number of signals.
So if I think about the knowns being k, t the parameters and
the signal d(i), and if I think about the unknowns being
the velocity, the output, and the sensor signal, then I get
three equations and three unknowns.
So one of the values of thinking about the operator
approach is that it just simply reduces the amount of
things you need to think about.
It reduces complexity.
That's what we're trying to do in this course.
We're trying to think of methods that allow you to
solve problems by reducing complexity.
We would like, ultimately, to solve very complicated
problems, and this operator approach is an approach that
lets you do that.
It does a lot more than that, too.
It also generates new kinds of insights.
So it lets you focus on the relations, but the relation is
now not quite the same as we would have
expected from algebra.
Now, the relation between the input signal and the output
signal is an operator.
We're going to represent the operation that transforms the
input to the output by this symbol, h.
We'll call that the system functional.
It's an operator.
It's a thing that, when operated on x, gives you y.
And this is one of the main purposes of today's lecture,
it's also convenient to think about h as a ratio.
We like to think of it that way because, as we'll see,
there's a way of thinking about h as a ratio of
polynomials in R. So two ways of thinking about it.
We're trying to develop a signals and systems approach
for thinking about feedback.
We want to think about the input goes into a box.
The box represents an operation.
We will characterize that by a functional.
We'll call the functional h.
The functional, when applied to the input, produces the
output, and what we'd like to do is infer what is the nature
of that functional, and what are the properties of the
system that functional represents.
OK, so I see that you're all with me.
Think about the WallFinder system.
Think about the equations for the various components of that
system when expressed an operator form.
And figure out the system functional for that system.
figure out the ratio of polynomials that can be in r
that is represented by h.
Take 30 seconds, talk to your neighbor, figure out whether
the answer is (1), (2), (3), (4), or (5).
So what's the answer -- (1), (2), (3), (4), or (5)?
Come on, more voter participation.
All right?
Blame it on your partner.
OK, virtually 100% correct.
So the idea is algebra.
You solve the operator equations exactly as though
they were algebraic.
Here, I've started with the second equation and just done
substitutions until I got rid of everything other
than d(0) and d(i).
So I express v in terms of ke, then I express e in terms of
d(i) minus rd(0).
Then I'm left with one equation that relates d(0) and
d(i), which I can solve for the ratio.
And the answer comes out there, which was number (3).
Point is that you can treat the operator just as though it
were algebra, so that results in enormous implications.
But what we want to understand is what's the relationship
between that functional, that thing that we just calculated,
and the behaviors.
These are the kinds of behaviors that you observed
with the WallFinder system.
When you built the WallFinder system, depending on what you
made k, you could get behaviors that were monotonic
and slow, faster and oscillatory, or even faster
and even more oscillatory.
And what we'd like to know is, before we build it, how should
we have constructed the system so that it
has a desirable behavior?
And, incidentally, are these the best you can do?
Or is there some other set of parameters that's lurking
behind some door that we just don't know about, and if we
could discover it, it would work a lot better?
So the question is, given the structure of our problem,
what's the most general kind of answer that we can expect?
And how do we choose the best behavior out of that set of
possible behaviors?
So that's what I want to think about for the rest of the hour
and a half, and I want to begin by taking a step
backwards and look at something simpler.
The idea's going to be the same as the idea that we used
when we studied Python.
I want to find simple behaviors, think about that as
a primitive, and combine primitives to get a more
complicated behavior, so I want to use an approach that's
very much PCAP.
Find the most simple behavior, and then see if I can leverage
that simple behavior to somehow understand more
complicated things.
So let's think about this very simple system that has a
feedback loop that has a delay in it and a gain of P0.
What I want to do is think about what would be the
response of that very simple system if the input were a
unit sample.
So find y, given that the input x is delta.
In order to do that, I have to start the system somehow.
I will start it at rest.
You've all seen already, I'm sure, that rest is the
simplest assumption I can make.
I'll say something at the end of the hour about how you deal
with things that are not at rest.
For the time being, we'll just assume rest
because that's simple.
Assume that the system is at rest, that means that the
output of every delay box is 0.
That's what rest means.
If the output of this starts at 0, then the output of the
scale by P0 is also 0.
And if that's 0, and if the input is at 0 because I'm at
time before 0, then the output is 0, indicated here.
So now if I step, then the input becomes 1 because
delta of 0 is 1.
The output of the delay box is still 0, so the first answer
is 1, the 1 just propagates straight through
[UNINTELLIGIBLE] box.
Then I step, and the 1 that was here goes through the r
and becomes 1, the 1 goes through P0 and becomes P0, but
at the same time, the 1 that was at the input goes to 0
because the input is 1 only at time equals 0.
So the result, then, is that after one step, the
output has become P0.
This propagated to 1, that became P0, add it to 0, and it
became P0, so now the answer, which had been 1, is P0.
On the next step, a very similar thing happens.
The P0 that was here becomes the output of the delay, gets
multiplied by P0 to give you P0 squared, gets added to 0 to
give you P0 squared, et cetera.
The thing that I want you to see is that the output was in
some sense simple.
The value simply increased as P0 to the n, geometrically.
There's another way we can think about that.
I just did the sample by sample approach, but the whole
theme of this part of the course
is the signals approach.
If I think about the whole signal in one fell swoop, then
I can develop an operator expression to
characterize the system.
The operator expression says the signal y is constructed by
adding the signal x to the signal P0RY.
If I solve that for the ratio of the output to input, I get
1 over (1 minus P0R).
Again, going back to the idea that I started with, that
we're going to get ratios of polynomials in R now the R is
in the bottom, and now I can expand that just as though it
were an algebraic expression.
I can expand 1 over P0R in a power series by using
synthetic division.
The result is very similar in structure to the result we saw
in sample by sample.
It consists of an ascending series in R, which means an
ascending number of delays.
Every time you increase the number of delays by 1, you
also multiply the amplitude by P0, so this is, in fact, the
same kind of result, but viewed from a
signal point of view.
Finally, I want to think about it in terms of block diagrams.
Same idea, I've got the same feedback system, but now I
want to take advantage of this ascending series expansion
that I did and think about each of the terms in that
series as a signal flow path through the feedback system.
So one, the first term in the ascending series, represents
the path that goes directly from the input to the output,
passing through no delays.
The second term in the series, P0R, represents the path that
goes to the output, loops around, comes back through the
adder, and then comes out.
In traversing that more complicated path, you picked
up 1 delay and 1 multiply by P0.
Second term, two loops.
Third term, three loops.
Fourth term, four loops.
The idea is that the block diagram gives us a way to
visualize how the answer came about.
It came about by all the possible paths that lead from
the input to the output.
Those possible paths all differed by a delay, and
that's why the decomposition was so simple, each path
corresponding to a different number of
delays through the system.
That won't always be true from more complicated systems, but
it is true for this one.
This flow diagram also lets you see something that's
extremely interesting.
Cyclical flow paths, which are characteristic of feedback--
feedback means the signal comes back.
Cyclical flow paths require that transient inputs generate
persistent outputs.
They generate persistent outputs because the output at
time n is not triggered by the input at time n.
It's triggered by the output at time n minus 1.
It keeps going on itself.
That's fundamental to feedback.
There's no way of getting around that.
That's what feedback is.
And it also shows why you got that funny oscillatory
behavior in WallFinder.
There wasn't any way around that.
Feedback meant that you were looping back.
That meant that there was a cycle in
the signal flow paths.
That means that even transient signals, signals that go away
very quickly like the [INAUDIBLE] sample, generate
responses that go on forever.
So that's a fundamental way of thinking about systems.
Systems are either feedforward or feedback.
Feedforward means that there are no cyclic
paths in the system.
No path in the system that take you from the input to the
output has a cycle in it.
That's what acyclic means.
That's what feedforward means.
Acyclic, feedforward, those all have responses to
transient inputs that are transient.
That contrasts with cyclic systems.
A cyclic system has feedback and will have the property
that transient signals can generate
outputs that go on forever.
OK, how many of these systems are cyclic?
Easy questions.
15 seconds, talk to your neighbor.
OK, so what's the answer?
How many?
OK, virtually 100%.
Correct, the answer's (3).
I've illustrated the cycles in red, so there's a cycle here,
there's two cycles in this one, and there's a cycle here.
So the idea is that, when you see a block diagram, one of
the first things you want to characterize, because it's
such a big difference between systems, is whether or not
there's a cycle in it.
If there's a cycle, then you know there's feedback.
If there's feedback, then you know you have the potential to
have a persistent response to even a transient signal.
OK, so if you only have one loop of the type that I
started with, where we had just one loop with an R and a
P0, then the question is, as you go around the loop, do the
samples get bigger, or smaller, or do
they stay the same?
That's a fundamental characterization of how the
simple feedback system works.
So here, if on every cycle the amplitude of the signal
diminishes by multiplication by half, that means that the
response ultimately decays.
Mathematically, it goes on forever, just like I said
previously, but the amplitude is decaying, so practically it
stops after a while.
It becomes small enough that you lose track of it.
By contrast, if every time you go around the loop, you pick
up amplitude, if the amplitude here were multiplied by 1.2,
then it gets bigger.
So the idea, then, is that you can characterize this kind of
a feedback by one number.
We call that number the pole.
Very mysterious word.
I won't go into the origins of the word.
For our purposes, it just simply means the base of the
geometric sequence that characterizes the response of
a system to the unit sample signal.
So here, I've showed an illustration of what can
happen if p is 1/2, p is one, p is 1.2, which you can see
decay, persistence, divergence.
Can you characterize this system by P0?
And if so, what is P0?
Yes?
No?
[CLASSROOM SIDE CONVERSATIONS]
PROFESSOR: Yeah, and virtually everybody's
getting the right answer.
The right answer's (2).
So we like algebra.
We like negative numbers, so we're allowed to think about
poles being negative.
In fact, by the end of the hour, we'll even think about
poles having imaginary parts, but for the time
being, this is fine.
If the pole were negative, what that means is the
consecutive terms in the unit sample response, the response
of the system to a unit sample signal, the unit sample
response, the unit sample response can
alternate in sine.
OK, so this then represents all the possible behaviors
that you could get from a feedback system
with a single pole.
If a feedback system has a single pole, the only
behaviors that you can get are represented
by these three cartoons.
So here, this z-axis contains all possible values of P0.
If P0 is bigger than 1, then the magnitude diverges, and
the signal grows monotonically.
If the pole is between 0 and 1, the response is also
monotonic, but now it converges towards 0.
If you flip the sign, the relations are still the same,
except that you now get sign alternation.
So if the P0 is between 0 and minus 1, which is here, the
output still converges because the magnitude of the pole is
less than 1.
But now the sign flips.
And if the pole is below minus 1, then you get alternation,
but you also get divergence.
The important thing is we started with a simple system,
and we ended up with an absolutely complete
characterization of it.
This is everything that can happen.
That's a powerful statement.
When I can analyze a system, even if it's simple, and find
all the possible behaviors, I have something.
If you have a simple system with a single pole, this is
all that can happen.
There might be offsets.
There might be delays.
The signal may not start until the fifth sample, but the
persistent signal will either grow without bounds, the k to
0, or do one of those two with alternating sign.
That's the only things that can happen, which of course,
begs the question, well, what if the system's more
complicated.
OK, so here's a more complicated system.
This system cannot be represented by just one pole.
In fact, the system's complicated enough you should
think through how you would solve it.
You should all be very comfortable with
this sort of thing.
So if you were to think about what if I had a system like
so, and I want it to be 1.6 minus 0.63.
What would be the output signal at time 2 if the input
were a unit sample signal?
OK, as with all systems we're going to think about, we have
to specify initial conditions.
The simplest kind of initial conditions we could think
about would be rest.
If I thought about this system at rest, then the initial
outputs of the R's would be 0.
That's at rest.
For times less than 0, the input would be 0.
0 times 1.6 plus 0 times -0.63 plus 0 would give me 0.
Now, the clock ticks.
When the clock ticks, it becomes times 0.
At times 0, the input is 1.
This 0 just propagated down to here, but this was 0, so
nothing interesting happens at the R's.
But now my output is 1.
Now, the clock ticks.
What happens?
When the clock ticks, this 1 propagates down here.
This 0 propagates down here, but that was 0.
This 1 goes to 0 because the input's only 1 at times 0.
So what's the output?
1.6.
Now, the clock ticks.
What happens?
Well, this 1 comes down here.
This 1.6 comes down here.
This 0 becomes another 0 because the input has an
infinite stream of 0's after the initial time.
So what's the output?
Well, it's 1.6 times 1.6 plus 1 times -0.63, so the answer
is number (3).
Yeah?
OK.
I forgot to write it up there, so the
answer's in red down here.
1.6 squared minus 0.63.
OK?
The point is that it's slightly more complicated to
think about than the case with a single pole, and in fact, if
you use that logic to simply step through all the
responses, you get a response that doesn't look geometric.
The geometric sequences that we looked at previously either
monotonically increased, monotonically decreased
towards 0, or give one of those two things and
alternated.
This does none of those behaviors.
So the point is Freeman's an idiot.
He spent all that time telling us what one pole does, and now
two poles does something completely different.
Right?
So the response is not geometric.
The response grows and then decays.
It never changes sign.
It does something completely different from what we would
have expected from a single pole system.
As you might expect from the Theory of Lectures, that's not
the end of the story.
So the idea is to now capitalize on this notion that
we can think about operators as algebra.
If our expressions behaved like I told you they did last
lecture, if they behaved as entities upon which--
if they are isomorphic with polynomials, as I said, then
there's a very cute thing we can do with this system to
make it a lot simpler.
The thing we can do is factor.
If we think about the operator expression to characterize
this system, the thing that's different is that
there's an R squared.
But if R operators work just like polynomials--
you can factor polynomials.
That's the factor theorem from algebra.
And if I factor it, I get two things that look like
first-order systems.
Well, that's good.
The factored form means that I can think about this more
complicated system as the cascade of
two first-order systems.
Well, that's pretty good.
In fact, it doesn't even matter what order I put them
in because, as we've seen previously, if the system
started at initial rest, then you can swap things because
they obey all the principles of polynomials, which include
commutation.
So what we've done, then, is transform this more
complicated system into the cascade of two simple systems,
and that's very good.
Even better, we can think about the complicated system
as the sum of simpler parts, and that uses more intuition
from polynomials.
If we have one over a second-order polynomial, we
can write it in a factored form here, but we can expand
it in what we call partial fractions.
We can expand this thing in this sum, and if you think
about putting this over a common denominator and working
out the relationship, this difference, 4.5 over 1 minus
0.9R minus 3.5 over 1 minus 0.7R.
That's precisely the same using the normal rules for
polynomials.
That's precisely the same as that expression.
But the difference, from the point of view of thinking
about systems, is enormous.
We know the answer to that one.
That's the sum of the responses to two first-order
systems, so we can write that symbolically this way.
We can think about having a sum system that
generates this term.
This term is a simple system of the type of that we looked
at previously that, then, gets multiplied by 4.5.
I'm just factoring again.
I'm saying I've got something over something, which means
that I can put something in each of two different parts of
two things that I multiply together.
And I can think about this as having been generated by this
system, and you just add them together.
The amazing thing is that that says that, despite the fact
that the response looked complicated, it was in fact
the sum of two geometrics.
So it wasn't very different from the answer
for a single pole.
What I've just done is amazing.
I've just taken something that, had you studied the
difference equations and had you studied the block
diagrams, it would have been very hard for you to conclude
that something this complicated has a response
that can be written as the sum of two geometrics.
By thinking about the system as a polynomial in R, it's
completely trivial.
It's a simple application of the rules for polynomials that
you all know.
So what we've shown, then, is that this complicated system
has a way of thinking about as just two
of the simpler systems.
The complicated response that grew and decayed, that's just
the difference, really, 4.5 minus 3.5.
It's the weighted difference of a part that goes 0.7 to the
n, then a different part that goes 0.9 to the n.
So far, we've got to results, the n equals 1 case, the
first-order polynomial in our case, the one pole case,
that's trivial.
It's just a geometric sequence.
The response is just a geometric sequence.
If it happens to be second-order, this is
second-order because when you write the operator expression,
the polynomial in the bottom is second-order, second-order
polynomial in R.
This second-order system has a response that
looks like two pieces.
Each piece looks like a piece that was from
a first-order system.
And in fact, that idea generalizes.
If we have a system that can be represented by linear
difference equation with constant coefficients that
will always be true if the system was constructed out of
the parts that we talked about, adders, gains, delays.
If the system is constructed out of adders, gains, and
delays, then it will be possible to express the system
in terms of one difference equation.
General form is showed here.
Y then can be constructed out of parts that are delayed
versions of Y and delayed versions of X. If you do that,
then you can always write the operator that expresses the
ratio between the output and the input as the ratio of two
polynomials.
That will always be true.
So this, now, is the generalization step.
We did the n equals 1 case, we did the n equals 2 case, and
now we're generalizing.
We will always get, for any system that can be represented
by a linear difference equation with constant
coefficients, we can always represent the system
functional in this form.
Then just like we did in the second-order case, we can use
the factor theorem to break this polynomial in the
denominator into factors.
That comes from the factor theorem in algebra.
Then we can re-express that in terms of partial fractions.
And what I've just showed is that, in the general case,
regardless of how many delays are in the system, if the
system only has adders, gains, and delays, I can always
express the answer as a sum of geometrics.
That's interesting.
That means that if I knew the bases for all of those
geometric sequences, I know something about the response.
The bases are things we call poles.
If you knew all the poles, you'd know something very
powerful about the system.
So every one of the factors corresponds to a pole, and by
partial fractions, you'll get one response for each pole.
The response for each pole goes like pole to the n.
You know the basic shape.
You don't know the constants, but you know the basic shape
of the response just by knowing the poles.
We can go one more step, which makes the
computation somewhat simpler.
I used the factor theorem.
Here, I'm using the fundamental theorem of
algebra, which says that if I have a polynomial of order n,
I have n roots.
The poles are related to the roots of the R polynomial.
The relationship is take the functional,
substitute for R --
1 over Z. Re-express the functional as a ratio of
polynomials in Z. The poles are the roots of the
denominator.
So recapping, I started with a first-order system.
I showed you how to get a second-order system.
I showed that, in general, you can use the factor theorem to
break down the response of a higher-order system into a sum
of responses of first-order systems.
Now, I've shown that you can use the fundamental theorem of
algebra to find the poles directly, and then by knowing
the poles, you know each of the behaviors, monotonic
divergence, monotonic convergence,
or alternating signs.
And so here is this same example that I started with,
worked out by thinking about what are the poles.
The poles are 0.7 and 0.9, which we see by a simple
application of the fundamental theorem of algebra.
OK, we got a long way by just thinking about operators as
polynomials.
We haven't done anything that you haven't
done in high school.
Polynomials are very familiar and we've made an isomorphism
between systems and polynomials.
OK, make sure you're all with me.
Here's a higher order system.
How many of these statements are true --
0, 1, 2, 3, 4, or 5?
Talk to your neighbor, get an answer.
So how many are true --
0, 1, 2, 3, 4, or 5?
Oh, come on.
Blame it on your neighbor.
You weren't talking, but I didn't hear you not talking.
How many are true --
0, 1, 2, 3, 4, or 5?
Raise your hands.
AUDIENCE: They can't all be true.
PROFESSOR: They can't all be true.
Are they mutually contradictory?
AUDIENCE: Well, yeah.
5 --
PROFESSOR: N1 of the above, that sounds like--
OK, so you've eliminated 1.
Which one's true?
How many statements are true?
Looks like about 75%.
Correct?
What should I do?
How do I figure it out?
What's my first step?
What do I do?
AUDIENCE: Operators.
PROFESSOR: Operators, absolutely.
So turn it into operators.
So take the difference equation,
turn it into operators.
The important thing to see is that there are three Y terms.
Take them all to the same side, and I get an operator
expression like that.
The ones that depend on X, there are two of them, that's
represented here.
The thing this is critical for determining poles is figuring
out the denominator.
The poles are going to come from this one.
After I get the ratio of two polynomials in R, I substitute
1 over Z for each R. So for this R, I get 1 over Z. For
this R squared, I get 1 over Z squared.
Then I want to turn it back into a ratio of polynomials in
Z, so I have to multiply top and bottom by Z squared.
And when I do that, I get this ratio of polynomials in Z.
The poles are the roots of the denominator polynomial in Z.
The poles are minus 1/2 and plus 1/4.
So the unit sample response converges to 0.
What would be the condition that that represents?
AUDIENCE: Take the polynomial on the bottom.
PROFESSOR: Something about the polynomial on the bottom.
Would all second-order systems have that property that the
unit sample response would converge to 0?
AUDIENCE: [INAUDIBLE]
PROFESSOR: Louder?
AUDIENCE: Absolute value of the poles?
PROFESSOR: Absolute value of the poles has to be?
AUDIENCE: Less than 1.
PROFESSOR: Less than 1.
If the magnitude of the poles is less than 1, then the
response magnitude will decay with time.
So that's true here, and it would be true so long as none
of the poles have a magnitude exceeding 1.
There are poles at 1/2 and 1/4.
No, that's not right.
It's -1/2 or 1/4.
There's a pole at 1/2.
No, that's not right.
There's a pole at -1/2.
There are two poles.
Yes, that's true.
None of the above.
No, that's not true.
So the answer was (2).
Everybody's comfortable?
We've done something very astonishing.
We took an arbitrary system, and we've figured out a rule
that let's us break it into the sum
of geometric sequences.
We can always write the response to a unit sample
signal, we can always write, as a weighted sum of geometric
sequences, and the number of geometric sequences in the sum
is the number of poles, which is the order of the operator
that operates on Y. OK, so we've done
something very powerful.
There's one more thing that we have to think about, and then
we have a complete picture of what's going on.
Think about when you learned polynomials.
One of the big shocks was that roots can be complex.
What would that mean?
What would it mean if we had a system whose poles were
complex valued?
So first off, does such a system exist?
Well, here's one.
I just pulled that out of the air.
If I think about the functional, 1 over (1 minus R
plus R squared) --
if I convert that into our ratio of polynomials in Z and
then find the roots, I find that the roots
have a complex part.
The roots are 1/2 plus or minus root 3 over 2j.
There's an imaginary part.
So the question is what would that mean?
Or is perhaps that system just meaningless?
Well, complex numbers work in algebra, and complex numbers
work here, too.
So the fact that a pole has a complex value in the context
of signals and systems simply means that the pole is
complex, that the base of the geometric sequence, that base
is complex.
So that means that we can still rewrite the denominator,
which was 1 minus R plus R squared, we can rewrite that
denominator in terms of a product of two first-order R
polynomials.
The coefficients are now complex, but it still works.
The algebra still works right.
That has to work because that's just polynomials.
That's the way polynomials behave.
Now, we can still factor it.
We can still use the factor theorem.
In fact, we can still use the fundamental theorem of algebra
to find the poles by the Z trick.
That's fine.
We can still use partial fractions.
All of these numbers are complex, but
the math still works.
The funny thing is that it implies that
the fundamental modes--
by fundamental mode, I mean the simple geometric for the
case of a first-order system, more complex behaviors for
higher order systems.
The mode is the time response associated with a pole.
So the modes are, in this case, complexes sequences.
So in general, the modes look like P0 to the n.
Here, my modes are simply have a complex value.
So what did I say they were?
The poles were 1/2 plus or minus--
so my modes simply look that.
Same thing.
The strange thing that happened was that those modes,
those geometric sequences, are now have complex values.
The first one up here, if I just look at the denominator,
these coefficients mean that it's proportionate to the mode
associated with this, which is that, which has a real part,
which is the blue part, and the imaginary part,
which is a red part.
There were two poles, plus and minus.
The other pole just flips the imaginary part.
So if I have imaginary poles, all I get is complex modes,
complex geometric sequences.
An easier way of thinking about that is thinking about--
so when we had a simple, real pole, we just had P0 to the n.
That's easy to visualize because we just think about
each time you go from 0 to 1 to 2 to 3, it goes from 1 to
P0 to P0 squared to P0 cubed.
Here, when you're multiplying complex numbers, it's easier
to imagine that on the complex plane.
Think about the location of the point 1, think about the
location of the point P0, think about the location of
the point P0 squared, and in this particular case, where
the pole was 1/2 plus or minus the square root of 3 over 2
times j, this would be pole to the 0.
This is pole to the 1.
This is pole squared, pole cubed.
As you can see when you have a complex number, the trajectory
in complex space can be complicated.
In this case, it's circular.
The circular trajectory in the complex plane corresponds to
the sinusoidal behavior in time.
So there's a correlation between the way you think
about the modes evolving on the complex plane and the way
you think about the real and imaginary
parts evolving in time.
It seems a little weird that the
response should be complex.
We're studying this kind of system theory primarily
because we're trying to gain insight into real systems.
We want to know how things like robots work.
How does the WallFinder work?
What would it mean if the WallFinder went to position
one plus the square root of 3 over 2j?
That doesn't make sense.
So there's a little bit of a strange thing going on here.
How is it that we need complex numbers to model real things?
That doesn't seem to sound right.
But the answer is that, if the difference equation had real
coefficients, as they will for a real system--
if you think about a real system, like a bank account,
the coefficients in the difference equation are real
numbers, not complex numbers.
If you think about the WallFinder system, the
coefficients in the WallFinder system, the coefficients of
the difference equations--
the coefficients of the different equations describe
the WallFinder behavior were all real numbers.
Here, I'm thinking about the denominator polynomial.
If we try to find the roots of a polynomial, and if we find a
complex root, if the coefficients were all real, it
follows that the complex conjugate of the original root
is also a root.
That's pretty simple, if you think about what it means to
be a polynomial.
If you think about a polynomial is whatever--
so I've got 1 plus Z plus Z squared plus blah, blah, blah.
2, 3 minus 16, if all of those coefficients real, then the
only way the--
If P were a root of this polynomial, then P* would have
to be a root, too, because if you complex conjugate each of
the Z's, it's the same thing as complex conjugating the
whole thing because the coefficients are real valued.
So the idea then is that if I happen to get a complex root
for my system that can be described by real value
coefficients, it must also be true that it's complex
conjugate is a root.
If that happens, the two roots co-conspire so that the modes
have canceling imaginary parts.
You can prove that.
I'm not worried about you being able to prove that.
I just want you to understand that if you have two roots
that are complex conjugates, they can conspire to have
their imaginary parts cancel, and that's
exactly what happens.
Here, in this example, the example that I started with,
where the system was 1 over (1 minus R plus R squared), the
unit sample responses showed here.
You can write as the sum of sinusoidal and cosinusoidal
signals, and the sum that falls out has the property
that the imaginary parts cancel.
It's still useful to look at the imaginary parts the same
as it is when you're trying to solve polynomials.
It's useful because the period of all of these
signals is the same.
If I think about the period of the individual modes, if I
think about the period of P0 to the n, 1/2 plus or minus
root 3 over 2j to the n, the period of that signal --
I can see in the complex plane, this is the n equals 0,
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 --
the period is 6.
If I were to take the minus 1, I would go around the circle
the other way.
This would be zero term 1, 2, 3, 4, 5, 6, et cetera.
Both of the modes, the geometric sequence associated
with the two complex conjugate poles, both of those modes had
the same period, they're both 6, as does the
response to the real part.
So you can deduce the period of the real signal by looking
at the periods of the two complex signals.
So think about this system.
Here, I've got a system whose responses showed here.
I'm going to tell you that the response was generated by a
second-order system, that is to say a system whose
polynomial was second-order, whose polynomial in R was
second-order.
Which of these statements is true?
I want to think about the pole as being a complex number.
Here, I'm showing you the complex number in polar form.
It's got a magnitude and an angle, and I'd like you to
figure how what must the magnitude have been, and what
must the angle have been.
So what's the utility?
Why did I tell you the pole in terms of it's magnitude and
angle rather than telling it to you in it's Cartesian form
as a real and imaginary part?
What's good about thinking about magnitude and angle?
Or if I break the pole into magnitude and angle, and if I
think about the mode--
the modes are always of the form P0 to the n.
If I think about modes as having a magnitude and an
angle, when I raise it to the n, something
very special happens.
What happens?
You can separate it.
What is R e to the j omega raised to the n?
That's the same as R to the n, e to the j n omega.
It's the product of a real thing times a very simple
complex thing.
What's simple about the complex thing?
The magnitude is everywhere 1 e to the j, the
magnitude of that term.
So all of the magnitude is here, none of the
magnitude is here.
All of the angle is here, none of the angle is here.
I've separated the magnitude and the angle.
So it's very insightful to think about poles in terms of
magnitude and angle because it decouples the
parts of the mode.
So I can think, then, of this complicated signal that I gave
you as being the product of a magnitude part and a pure
angle part.
From the magnitude part, I can infer something about R. R is
the ratio of the nth 1 to the n minus 1-th one, so R is a
lot bigger than 1/2.
In fact, R in this case is 0.97.
And this is pure angle.
This lets me infer something about the oscillations.
In fact, I can say something about the period.
The period is, here's a peak, here's a peak, 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12.
The period's 12.
If the period is 12, what is omega?
Omega is a number, such that by the time I got up to 12
times it, I got up to 2 pi.
What's omega?
AUDIENCE: Pi over 6.
PROFESSOR: 2 pi over 12.
So it's about 1/2.
So that's a way that I can infer the form of the answer.
I ask you what pole corresponds to this behavior.
Well, the decay that comes from the real part, that comes
from R to the n.
The oscillation, that comes from the imaginary part, and I
can figure that out by thinking about the period and
thinking about that relationship.
So the answer, then, is this one, R is between 1/2 and 1,
and omega is about 1/2.
OK, one last-- whoops, wrong button.
One last example.
So what we've seen is a very powerful way of decomposing
systems, so that we can always think about them in terms of
poles and complex geometrics, which we will call modes,
poles and modes.
And that behavior works for any system in an enormous
class of systems, so I want to think about one last one,
which is Fibonacci sequence.
You all know the Fibonacci sequence.
You've all programmed it.
We started with that when we were doing Python, and we made
some illustrations about the recursion and all that sort of
thing by thinking about it.
Now, I'm going to use signals and
systems do the same problem.
So Fibonacci was interested in population growth.
How many pairs of rabbits can be produced from a single pair
in a year if it is suppose that every month each pair
begets a new pair, from which the second month it becomes
productive?
OK, it's not quite the same English I would have used.
From this statement, you can infer a difference equation.
I've written it in terms of X. What do you think X is?
X is the input signal, and here, I'm thinking about X as
something that's specifies the initial condition.
This is the thing I alluded to earlier.
One trick that we use to make it easy to think about initial
conditions is that we embed them in the input.
So in this particular case, I'll think about the initial
condition arising from a delta function.
If I think about X as a delta, then the sequence of results
Y0, Y1, Y2, Y3 from this difference equation, is the
conventional Fibonacci sequence.
It would correspond to what if you had a baby rabbit, one
baby rabbit, that's the one, in generation
0, that's the delta.
So the input is a way that I can specify initial
conditions, and that's a very powerful way of thinking about
initial conditions.
So here's the problem.
I've got one set of baby rabbits at times 0.
They grow up.
They have baby rabbits, which grow up at the same time the
parents had more baby rabbits, at which point more babies
grow into bigger rabbits.
And big rabbits have more babies, et cetera, et cetera,
et cetera, et cetera, et cetera.
So you all know that.
You all know about Fibonacci sequence.
It blows up very quickly.
What are the poles of the Fibonacci sequence?
The difference equation looks just like the difference
equations we've looked at throughout
this hour and a half.
We can do it just like we did all the other problems.
We write the difference equation in terms of R. We
rewrite the system functional in terms of a ratio of two
polynomials in R. We substitute R goes to 1 over Z,
deduce a ratio of polynomials in Z, factor the denominator,
find the roots of the denominator, and find that
there are two poles.
The poles for the Fibonacci sequence are plus or minus the
root of 5 over 2.
That's curious.
There's no recursion there.
It's a different way of thinking about things.
There are two poles.
The first pole, the plus one --
1 plus the root of 5 over 1, corresponds to a pole whose
magnitude is bigger than 1.
It explodes.
There it is.
The second one is a negative number.
So the first one is the golden ratio, 1.618...
The second one is the negative reciprocal of the golden
ratio, which is -0.618...
And those two numbers, amazingly, conspire so that
their sum, horrendous as they are, is an integer.
And in fact, that's the integer that we computed here.
So we've used Fibonacci before to think about the way you
structure programs, iteration, recursion, that sort of thing.
Here, by thinking about signals and systems, we can
think about exactly the same problem as poles.
There's no complexity in this problem.
It doesn't take N or N squared or N log N
or anything to compute.
It's closed form.
The answer is (pole one) to the N plus (pole two) to the
N. That's it.
That's the answer.
So what we've done is we found a whole new way of thinking
about the Fibonacci sequence in terms of poles, and more
than that, we found that that way of thinking about poles
works for any difference equation of this type.
And we found that poles, this way of thinking about systems
in terms of polynomials, is a powerful abstraction that's
exactly the same kind of PCAP abstraction
that we used for Python.