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PROFESSOR: Last time, which is
now two weeks ago, we started
00:00:29.520 --> 00:00:33.050
the talk about the signals and
systems approach, by which I
00:00:33.050 --> 00:00:36.890
mean, think about a system by
the way it transforms its
00:00:36.890 --> 00:00:41.090
input signal into an
output signal.
00:00:41.090 --> 00:00:43.520
That's kind of a bizarre way
of thinking about systems.
00:00:43.520 --> 00:00:45.690
I demonstrated that last time by
thinking about a mass and a
00:00:45.690 --> 00:00:49.760
spring, something you have a
lot of experience with, but
00:00:49.760 --> 00:00:53.680
you probably didn't use this
kind of approach for
00:00:53.680 --> 00:00:55.060
thinking about it.
00:00:55.060 --> 00:00:57.020
Over this lecture and over the
next lecture, what I'd like to
00:00:57.020 --> 00:01:00.455
do is show you the advantages
of this kind of approach.
00:01:00.455 --> 00:01:04.920
And so for today, what I'd like
to do is talk about how
00:01:04.920 --> 00:01:09.900
to think about feedback within
this structure, and I'd like
00:01:09.900 --> 00:01:13.260
to also think about how you
can use this structure to
00:01:13.260 --> 00:01:15.760
characterize the performance
of a system in a
00:01:15.760 --> 00:01:18.900
quantitative fashion.
00:01:18.900 --> 00:01:21.400
So first off, I want to just
think about feedback.
00:01:21.400 --> 00:01:25.370
Feedback is so pervasive
that you don't notice
00:01:25.370 --> 00:01:27.470
it most of the time.
00:01:27.470 --> 00:01:31.110
You use feedback in virtually
everything that you do.
00:01:31.110 --> 00:01:33.680
Here is a very simple example
of driving a car.
00:01:33.680 --> 00:01:36.050
If you want to keep the car in
the center of your lane,
00:01:36.050 --> 00:01:38.630
something that many people
outside of Boston, at least,
00:01:38.630 --> 00:01:43.880
think is a good idea, then you
are mentally doing feedback.
00:01:43.880 --> 00:01:46.370
You're constantly comparing
where you are to where you'd
00:01:46.370 --> 00:01:49.950
like to be and making small
adjustments to the system
00:01:49.950 --> 00:01:52.760
based on that.
00:01:52.760 --> 00:01:54.900
When you think of even the most
simple of systems, like
00:01:54.900 --> 00:01:55.920
the thermostat in a house--
00:01:55.920 --> 00:01:58.110
I'm not talking about
a cheap motel.
00:01:58.110 --> 00:01:59.550
The cheap motels
don't do this.
00:01:59.550 --> 00:02:03.540
But in a real house, there's a
thermostat, which regulates
00:02:03.540 --> 00:02:06.060
the temperature.
00:02:06.060 --> 00:02:08.160
That's important, because if
the temperature suddenly
00:02:08.160 --> 00:02:11.550
drops, it would never do that
of course, but if the
00:02:11.550 --> 00:02:16.790
temperature ever dropped, it
could compensate for it.
00:02:16.790 --> 00:02:20.070
Here's one of my favorite
examples.
00:02:20.070 --> 00:02:23.280
Feedback is enormously
pervasive in biology.
00:02:23.280 --> 00:02:27.770
There's no general rules in
biology, but to a first cut,
00:02:27.770 --> 00:02:31.250
everything is regulated.
00:02:31.250 --> 00:02:35.550
And in many cases, the
regulation is amazing.
00:02:35.550 --> 00:02:39.930
Here's an example from 6.021,
where it's illustrating the
00:02:39.930 --> 00:02:44.540
system that your body uses to
regulate glucose delivery from
00:02:44.540 --> 00:02:48.330
food sources to every cell in
your body, which is crucial to
00:02:48.330 --> 00:02:50.950
your being cognizant
and mobile.
00:02:53.600 --> 00:03:00.580
And the idea is that it does
that with amazing precision
00:03:00.580 --> 00:03:04.690
despite the fact that
eating and exercise
00:03:04.690 --> 00:03:07.880
are enormously episodic.
00:03:07.880 --> 00:03:10.970
In order for you to remain
healthy and functional, you
00:03:10.970 --> 00:03:14.220
need to have something between
approximately two and ten
00:03:14.220 --> 00:03:16.510
millimoles per liter
of glucose in your
00:03:16.510 --> 00:03:18.260
blood at all times.
00:03:18.260 --> 00:03:20.380
Were it to go higher than that,
systematically, you
00:03:20.380 --> 00:03:25.050
would develop cardiac problems
and could lead to even
00:03:25.050 --> 00:03:28.170
congestive heart failure.
00:03:28.170 --> 00:03:32.310
If you were to have lower than
two millimoles per liter, you
00:03:32.310 --> 00:03:34.420
would go comatose.
00:03:34.420 --> 00:03:37.470
That's a very narrow range,
two to five, especially
00:03:37.470 --> 00:03:43.260
because what we eat is so
episodic, when we exercise is
00:03:43.260 --> 00:03:44.430
so episodic.
00:03:44.430 --> 00:03:49.440
It's amazing, and just to
dramatize how amazing it is,
00:03:49.440 --> 00:03:52.140
how much sugar do you
think circulates in
00:03:52.140 --> 00:03:53.390
your blood right now?
00:03:56.970 --> 00:03:59.430
Well, if you convert five
millimoles per liter, and if
00:03:59.430 --> 00:04:01.540
you assume the average person
has about three liters of
00:04:01.540 --> 00:04:05.290
blood, which is true, and if you
calculate it, that comes
00:04:05.290 --> 00:04:08.920
out to 2.7 grams.
00:04:08.920 --> 00:04:12.150
That's this much.
00:04:12.150 --> 00:04:14.690
This is 2.7 grams
of table sugar.
00:04:19.800 --> 00:04:21.740
So this is how much sugar
is in your blood, now.
00:04:21.740 --> 00:04:22.800
This is what's keeping
you healthy.
00:04:22.800 --> 00:04:26.730
This is what's keeping you
from becoming comatose.
00:04:26.730 --> 00:04:27.980
Or not.
00:04:30.310 --> 00:04:34.288
How much sugar do you
think is in this?
00:04:34.288 --> 00:04:35.650
AUDIENCE: The amount
in that other cup.
00:04:35.650 --> 00:04:38.110
PROFESSOR: Exactly.
00:04:38.110 --> 00:04:40.690
We call that the Theory
of Lectures.
00:04:40.690 --> 00:04:43.800
I don't want to look like
an idiot, so my
00:04:43.800 --> 00:04:45.220
problems make sense.
00:04:45.220 --> 00:04:47.550
This is the amount of sugar
in a can of soda.
00:04:50.160 --> 00:04:51.970
Numerically, this is 39 grams.
00:04:57.140 --> 00:05:00.350
That's more than 13
times this much.
00:05:00.350 --> 00:05:03.080
So when you down one of these,
it's very important that the
00:05:03.080 --> 00:05:07.400
sugar gets taken out of the
blood quickly, and it does.
00:05:07.400 --> 00:05:10.710
And that happens by way of a
feedback system, and the
00:05:10.710 --> 00:05:12.850
feedback system is
illustrated here.
00:05:12.850 --> 00:05:16.510
Basically, the feedback involves
the hormone insulin,
00:05:16.510 --> 00:05:19.370
and that's why insulin
deficiency is such a
00:05:19.370 --> 00:05:20.620
devastating disease.
00:05:22.890 --> 00:05:26.690
Finally, everything we do
has feedback in it.
00:05:26.690 --> 00:05:28.400
Think about how your
life would be
00:05:28.400 --> 00:05:30.520
different if it didn't.
00:05:30.520 --> 00:05:33.150
Even a simple task, like
removing a light bulb, would
00:05:33.150 --> 00:05:36.300
be virtually impossible except
for the fact that you get
00:05:36.300 --> 00:05:37.610
feedback from everything.
00:05:37.610 --> 00:05:39.170
Your hand's amazing.
00:05:39.170 --> 00:05:42.240
You have touch sensors, you have
proprioceptive sensors,
00:05:42.240 --> 00:05:45.160
you have stress sensors on the
muscles and ligaments, and
00:05:45.160 --> 00:05:48.520
they all coordinate to tell you
when to stop squeezing on
00:05:48.520 --> 00:05:51.310
the light bulb so you
don't break it.
00:05:51.310 --> 00:05:53.800
That's all completely amazing,
and what we'd like to do,
00:05:53.800 --> 00:05:58.070
then, is think about that kind
of a system, a feedback
00:05:58.070 --> 00:06:03.450
system, within the signals
and systems construct.
00:06:03.450 --> 00:06:06.330
As an example, I want to think
through the WallFinder problem
00:06:06.330 --> 00:06:09.820
that you did last week
in Design Lab.
00:06:09.820 --> 00:06:12.670
I'm sure you all remember that,
in that problem, we were
00:06:12.670 --> 00:06:15.300
trying to move the robot to a
fixed distance away from the
00:06:15.300 --> 00:06:19.220
wall, and we thought about
that as a feedback system
00:06:19.220 --> 00:06:21.820
comprised of three parts,
a controller, a
00:06:21.820 --> 00:06:24.180
plant, and a sensor.
00:06:24.180 --> 00:06:26.800
We wrote difference equations to
characterize each of those
00:06:26.800 --> 00:06:32.130
parts, and then we figured out
how to solve those difference
00:06:32.130 --> 00:06:35.100
equations to make some
meaningful prediction about
00:06:35.100 --> 00:06:36.350
how the robot would work.
00:06:38.690 --> 00:06:41.905
So just to make sure you're
all on board with me, now,
00:06:41.905 --> 00:06:44.210
here's a question for you.
00:06:44.210 --> 00:06:45.990
These are the equations
that describe
00:06:45.990 --> 00:06:48.760
the WallFinder problem.
00:06:48.760 --> 00:06:50.560
How many equations and how
many unknowns are there?
00:06:54.860 --> 00:06:56.870
Take 20 seconds, talk to your
neighbor, figure out an answer
00:06:56.870 --> 00:06:58.120
between (1) and (5).
00:07:09.920 --> 00:07:12.490
Unlike during the exam next
week, you are allowed to talk.
00:07:19.915 --> 00:07:39.220
[CLASSROOM SIDE CONVERSATIONS]
00:07:39.220 --> 00:07:40.705
PROFESSOR: T and K are no.
00:07:40.705 --> 00:08:09.010
[CLASSROOM SIDE CONVERSATIONS]
00:08:09.010 --> 00:08:10.614
PROFESSOR: OK, so what's the
answer, number (1), (2), (3),
00:08:10.614 --> 00:08:12.060
(4), or (5)?
00:08:12.060 --> 00:08:13.160
Everybody raise your hands.
00:08:13.160 --> 00:08:15.010
Let me see if I got the answer.
00:08:15.010 --> 00:08:15.920
OK.
00:08:15.920 --> 00:08:16.890
Come on, raise your hands.
00:08:16.890 --> 00:08:18.050
Come on, everybody vote.
00:08:18.050 --> 00:08:20.010
If you're wrong, just blame
it on your neighbor.
00:08:20.010 --> 00:08:21.240
You had a poor partner.
00:08:21.240 --> 00:08:22.020
That's the idea.
00:08:22.020 --> 00:08:23.730
Right?
00:08:23.730 --> 00:08:25.530
So you can all vote, and you
don't need to worry about
00:08:25.530 --> 00:08:26.540
being wrong.
00:08:26.540 --> 00:08:31.520
And you're all wrong, so
that worked out well.
00:08:31.520 --> 00:08:34.960
OK, so I don't like the-- so
the predominant answer is
00:08:34.960 --> 00:08:37.190
number (2).
00:08:37.190 --> 00:08:38.200
I don't like number (2).
00:08:38.200 --> 00:08:40.740
Can somebody think of a reason,
now that you know the
00:08:40.740 --> 00:08:43.909
answer by the Theory
of Lectures, the
00:08:43.909 --> 00:08:45.159
answer is not (2)?
00:08:49.060 --> 00:08:50.170
Why isn't the answer (2)?
00:08:50.170 --> 00:08:50.840
Yeah?
00:08:50.840 --> 00:08:53.110
AUDIENCE: Oh, I was
saying (5).
00:08:53.110 --> 00:08:53.590
PROFESSOR: You were
saying (5).
00:08:53.590 --> 00:08:55.682
Why did you say (5)?
00:08:55.682 --> 00:08:59.852
AUDIENCE: I don't remember for
sure, but can you substitute n
00:08:59.852 --> 00:09:01.092
for different things?
00:09:01.092 --> 00:09:05.804
Like Df of n, you could
substitute n for, and if you
00:09:05.804 --> 00:09:07.788
did that you would have two
equations, which is
00:09:07.788 --> 00:09:08.780
[INAUDIBLE].
00:09:08.780 --> 00:09:11.010
PROFESSOR: [UNINTELLIGIBLE]
kind of thing, you count
00:09:11.010 --> 00:09:13.502
before or after doing
substitution and
00:09:13.502 --> 00:09:15.240
simplification.
00:09:15.240 --> 00:09:18.940
And I mean to count before
you do simplification.
00:09:18.940 --> 00:09:22.065
Any other issues?
00:09:22.065 --> 00:09:22.540
Yeah?
00:09:22.540 --> 00:09:25.865
AUDIENCE: Is D0 [UNINTELLIGIBLE]
00:09:28.730 --> 00:09:29.422
PROFESSOR: Say again?
00:09:29.422 --> 00:09:32.185
AUDIENCE: D0 [UNINTELLIGIBLE]
00:09:32.185 --> 00:09:36.000
PROFESSOR: Is D0_n different
or the same from D0_n-1?
00:09:36.000 --> 00:09:38.310
That's the key question.
00:09:38.310 --> 00:09:41.330
So I want to think about this
as a system of algebraic
00:09:41.330 --> 00:09:46.615
equations, and if I do that,
then there's a lot of them.
00:09:50.545 --> 00:09:52.790
So it looks like there's
three equations.
00:09:52.790 --> 00:09:55.620
The problem with that approach
is that there's actually three
00:09:55.620 --> 00:09:59.670
equations for every
value of n.
00:09:59.670 --> 00:10:02.000
If you think about a system of
equations that you could solve
00:10:02.000 --> 00:10:05.940
with an algebra solver, you
would have to treat all the
00:10:05.940 --> 00:10:07.050
n's separately.
00:10:07.050 --> 00:10:10.600
That's what we call the
samples approach.
00:10:10.600 --> 00:10:13.660
So here's a way you
could solve them.
00:10:13.660 --> 00:10:18.660
You could think about what if
k and t are parameters, so
00:10:18.660 --> 00:10:21.730
they're known, they're given?
00:10:21.730 --> 00:10:25.500
What if my input signal is
known, say it's a unit sample
00:10:25.500 --> 00:10:28.220
signal, for example?
00:10:28.220 --> 00:10:30.650
What would I need to
solve this system?
00:10:30.650 --> 00:10:35.400
Well, I'd need to tell you
the initial conditions.
00:10:35.400 --> 00:10:39.810
So in some sense, I want to
consider those to be known.
00:10:39.810 --> 00:10:46.290
So my knowns kind of comprise
t and k, the initial
00:10:46.290 --> 00:10:52.680
conditions for the output of the
robot, and the sensor, all
00:10:52.680 --> 00:10:55.650
of the input signals because I'm
telling you the input and
00:10:55.650 --> 00:10:58.230
asking you to calculate
the output.
00:10:58.230 --> 00:11:01.700
My unknowns are all of the
different velocities for all
00:11:01.700 --> 00:11:06.260
values of n bigger than or equal
to 0 because I didn't
00:11:06.260 --> 00:11:08.360
tell you those.
00:11:08.360 --> 00:11:10.990
All of the values of robot's
output at samples
00:11:10.990 --> 00:11:13.100
n bigger than 0.
00:11:13.100 --> 00:11:16.510
All the values of the sensor
output for values
00:11:16.510 --> 00:11:18.750
n bigger than 0.
00:11:18.750 --> 00:11:23.030
So I get a lot of unknowns,
infinitely many, and I get a
00:11:23.030 --> 00:11:26.380
lot of equations, also
infinitely many.
00:11:26.380 --> 00:11:28.670
So the thing I want you think
about is, if you're thinking
00:11:28.670 --> 00:11:31.050
about solving difference
equations using algebra,
00:11:31.050 --> 00:11:32.565
that's a big system
of equations.
00:11:35.320 --> 00:11:40.320
By contrast, what if you were
to try to solve the system
00:11:40.320 --> 00:11:41.570
using operators?
00:11:43.990 --> 00:11:46.470
Now, how many equations and
unknowns do you see?
00:11:50.990 --> 00:11:52.515
By the Theory of Lectures.
00:11:56.610 --> 00:11:59.430
OK, raise your hand.
00:11:59.430 --> 00:12:02.736
Or talk to your neighbor so you
can blame your neighbor.
00:12:02.736 --> 00:12:05.196
Talk to your neighbor.
00:12:05.196 --> 00:12:06.180
Get a good alibi.
00:12:06.180 --> 00:12:25.880
[CLASSROOM SIDE CONVERSATIONS]
00:12:25.880 --> 00:12:28.830
PROFESSOR: So the idea here
is that if you think about
00:12:28.830 --> 00:12:32.030
operators instead, so that you
look at a whole signal at a
00:12:32.030 --> 00:12:38.620
time, then each equation only
specifies one relationship
00:12:38.620 --> 00:12:45.070
among signals, and there's a
small number of signals.
00:12:45.070 --> 00:12:48.970
So if I think about the knowns
being k, t the parameters and
00:12:48.970 --> 00:12:54.040
the signal d(i), and if I think
about the unknowns being
00:12:54.040 --> 00:12:57.520
the velocity, the output, and
the sensor signal, then I get
00:12:57.520 --> 00:13:00.160
three equations and
three unknowns.
00:13:00.160 --> 00:13:03.190
So one of the values of thinking
about the operator
00:13:03.190 --> 00:13:05.480
approach is that it just simply
reduces the amount of
00:13:05.480 --> 00:13:06.530
things you need to
think about.
00:13:06.530 --> 00:13:08.020
It reduces complexity.
00:13:08.020 --> 00:13:10.030
That's what we're trying
to do in this course.
00:13:10.030 --> 00:13:12.460
We're trying to think of methods
that allow you to
00:13:12.460 --> 00:13:16.180
solve problems by reducing
complexity.
00:13:16.180 --> 00:13:18.470
We would like, ultimately,
to solve very complicated
00:13:18.470 --> 00:13:21.740
problems, and this operator
approach is an approach that
00:13:21.740 --> 00:13:24.380
lets you do that.
00:13:24.380 --> 00:13:26.360
It does a lot more
than that, too.
00:13:26.360 --> 00:13:30.900
It also generates new
kinds of insights.
00:13:30.900 --> 00:13:36.840
So it lets you focus on the
relations, but the relation is
00:13:36.840 --> 00:13:38.800
now not quite the same
as we would have
00:13:38.800 --> 00:13:40.140
expected from algebra.
00:13:40.140 --> 00:13:43.290
Now, the relation between the
input signal and the output
00:13:43.290 --> 00:13:45.390
signal is an operator.
00:13:45.390 --> 00:13:49.580
We're going to represent the
operation that transforms the
00:13:49.580 --> 00:13:52.080
input to the output
by this symbol, h.
00:13:52.080 --> 00:13:54.300
We'll call that the
system functional.
00:13:54.300 --> 00:13:56.690
It's an operator.
00:13:56.690 --> 00:14:00.255
It's a thing that, when operated
on x, gives you y.
00:14:06.160 --> 00:14:08.890
And this is one of the main
purposes of today's lecture,
00:14:08.890 --> 00:14:12.860
it's also convenient to think
about h as a ratio.
00:14:12.860 --> 00:14:16.250
We like to think of it that
way because, as we'll see,
00:14:16.250 --> 00:14:20.310
there's a way of thinking
about h as a ratio of
00:14:20.310 --> 00:14:24.790
polynomials in R. So two ways
of thinking about it.
00:14:24.790 --> 00:14:27.190
We're trying to develop a
signals and systems approach
00:14:27.190 --> 00:14:28.900
for thinking about feedback.
00:14:28.900 --> 00:14:32.530
We want to think about the
input goes into a box.
00:14:32.530 --> 00:14:35.210
The box represents
an operation.
00:14:35.210 --> 00:14:36.950
We will characterize that
by a functional.
00:14:36.950 --> 00:14:39.610
We'll call the functional h.
00:14:39.610 --> 00:14:41.890
The functional, when applied
to the input, produces the
00:14:41.890 --> 00:14:45.660
output, and what we'd like to do
is infer what is the nature
00:14:45.660 --> 00:14:48.400
of that functional, and what
are the properties of the
00:14:48.400 --> 00:14:49.920
system that functional
represents.
00:14:52.970 --> 00:14:56.450
OK, so I see that you're
all with me.
00:14:56.450 --> 00:14:59.530
Think about the WallFinder
system.
00:14:59.530 --> 00:15:02.790
Think about the equations for
the various components of that
00:15:02.790 --> 00:15:06.000
system when expressed
an operator form.
00:15:06.000 --> 00:15:09.930
And figure out the system
functional for that system.
00:15:09.930 --> 00:15:14.280
figure out the ratio of
polynomials that can be in r
00:15:14.280 --> 00:15:17.420
that is represented by h.
00:15:17.420 --> 00:15:19.500
Take 30 seconds, talk to your
neighbor, figure out whether
00:15:19.500 --> 00:15:20.750
the answer is (1), (2),
(3), (4), or (5).
00:17:45.980 --> 00:17:47.230
So what's the answer -- (1),
(2), (3), (4), or (5)?
00:17:51.670 --> 00:17:53.910
Come on, more voter
participation.
00:17:53.910 --> 00:17:54.600
All right?
00:17:54.600 --> 00:17:55.850
Blame it on your partner.
00:17:58.630 --> 00:18:01.990
OK, virtually 100% correct.
00:18:01.990 --> 00:18:05.690
So the idea is algebra.
00:18:05.690 --> 00:18:09.180
You solve the operator equations
exactly as though
00:18:09.180 --> 00:18:10.670
they were algebraic.
00:18:10.670 --> 00:18:13.640
Here, I've started with the
second equation and just done
00:18:13.640 --> 00:18:15.970
substitutions until I got
rid of everything other
00:18:15.970 --> 00:18:17.220
than d(0) and d(i).
00:18:20.280 --> 00:18:25.730
So I express v in terms of ke,
then I express e in terms of
00:18:25.730 --> 00:18:28.640
d(i) minus rd(0).
00:18:28.640 --> 00:18:32.110
Then I'm left with one equation
that relates d(0) and
00:18:32.110 --> 00:18:34.140
d(i), which I can solve
for the ratio.
00:18:36.830 --> 00:18:41.830
And the answer comes out there,
which was number (3).
00:18:41.830 --> 00:18:47.240
Point is that you can treat the
operator just as though it
00:18:47.240 --> 00:18:52.103
were algebra, so that results
in enormous implications.
00:18:54.760 --> 00:18:57.660
But what we want to understand
is what's the relationship
00:18:57.660 --> 00:19:00.230
between that functional, that
thing that we just calculated,
00:19:00.230 --> 00:19:01.040
and the behaviors.
00:19:01.040 --> 00:19:03.710
These are the kinds of behaviors
that you observed
00:19:03.710 --> 00:19:05.850
with the WallFinder system.
00:19:05.850 --> 00:19:08.328
When you built the WallFinder
system, depending on what you
00:19:08.328 --> 00:19:13.280
made k, you could get behaviors
that were monotonic
00:19:13.280 --> 00:19:20.360
and slow, faster and
oscillatory, or even faster
00:19:20.360 --> 00:19:22.250
and even more oscillatory.
00:19:22.250 --> 00:19:27.120
And what we'd like to know is,
before we build it, how should
00:19:27.120 --> 00:19:29.640
we have constructed the
system so that it
00:19:29.640 --> 00:19:31.770
has a desirable behavior?
00:19:31.770 --> 00:19:36.010
And, incidentally, are these
the best you can do?
00:19:36.010 --> 00:19:38.790
Or is there some other set of
parameters that's lurking
00:19:38.790 --> 00:19:42.700
behind some door that we just
don't know about, and if we
00:19:42.700 --> 00:19:45.150
could discover it, it would
work a lot better?
00:19:45.150 --> 00:19:48.300
So the question is, given the
structure of our problem,
00:19:48.300 --> 00:19:51.860
what's the most general kind of
answer that we can expect?
00:19:51.860 --> 00:19:55.670
And how do we choose the best
behavior out of that set of
00:19:55.670 --> 00:19:58.280
possible behaviors?
00:19:58.280 --> 00:20:00.596
So that's what I want to think
about for the rest of the hour
00:20:00.596 --> 00:20:05.020
and a half, and I want to
begin by taking a step
00:20:05.020 --> 00:20:08.180
backwards and look at
something simpler.
00:20:08.180 --> 00:20:10.440
The idea's going to be the same
as the idea that we used
00:20:10.440 --> 00:20:11.400
when we studied Python.
00:20:11.400 --> 00:20:16.760
I want to find simple behaviors,
think about that as
00:20:16.760 --> 00:20:22.070
a primitive, and combine
primitives to get a more
00:20:22.070 --> 00:20:24.710
complicated behavior, so I want
to use an approach that's
00:20:24.710 --> 00:20:26.240
very much PCAP.
00:20:26.240 --> 00:20:29.770
Find the most simple behavior,
and then see if I can leverage
00:20:29.770 --> 00:20:32.310
that simple behavior to somehow
understand more
00:20:32.310 --> 00:20:34.270
complicated things.
00:20:34.270 --> 00:20:36.940
So let's think about this very
simple system that has a
00:20:36.940 --> 00:20:43.170
feedback loop that has a delay
in it and a gain of P0.
00:20:43.170 --> 00:20:45.250
What I want to do is think
about what would be the
00:20:45.250 --> 00:20:50.130
response of that very simple
system if the input were a
00:20:50.130 --> 00:20:51.550
unit sample.
00:20:51.550 --> 00:20:55.355
So find y, given that the
input x is delta.
00:20:58.060 --> 00:21:01.630
In order to do that, I have to
start the system somehow.
00:21:01.630 --> 00:21:03.160
I will start it at rest.
00:21:03.160 --> 00:21:05.890
You've all seen already, I'm
sure, that rest is the
00:21:05.890 --> 00:21:07.340
simplest assumption
I can make.
00:21:07.340 --> 00:21:10.520
I'll say something at the end of
the hour about how you deal
00:21:10.520 --> 00:21:13.950
with things that are
not at rest.
00:21:13.950 --> 00:21:16.140
For the time being, we'll
just assume rest
00:21:16.140 --> 00:21:17.600
because that's simple.
00:21:17.600 --> 00:21:19.950
Assume that the system is at
rest, that means that the
00:21:19.950 --> 00:21:23.280
output of every delay
box is 0.
00:21:23.280 --> 00:21:25.360
That's what rest means.
00:21:25.360 --> 00:21:27.820
If the output of this starts at
0, then the output of the
00:21:27.820 --> 00:21:31.970
scale by P0 is also 0.
00:21:31.970 --> 00:21:34.530
And if that's 0, and if the
input is at 0 because I'm at
00:21:34.530 --> 00:21:40.070
time before 0, then the output
is 0, indicated here.
00:21:40.070 --> 00:21:47.320
So now if I step, then the
input becomes 1 because
00:21:47.320 --> 00:21:49.920
delta of 0 is 1.
00:21:49.920 --> 00:21:54.390
The output of the delay box is
still 0, so the first answer
00:21:54.390 --> 00:21:57.010
is 1, the 1 just propagates
straight through
00:21:57.010 --> 00:21:58.260
[UNINTELLIGIBLE] box.
00:22:00.490 --> 00:22:05.990
Then I step, and the 1 that was
here goes through the r
00:22:05.990 --> 00:22:12.150
and becomes 1, the 1 goes
through P0 and becomes P0, but
00:22:12.150 --> 00:22:15.210
at the same time, the 1 that
was at the input goes to 0
00:22:15.210 --> 00:22:19.870
because the input is 1 only
at time equals 0.
00:22:19.870 --> 00:22:24.030
So the result, then, is that
after one step, the
00:22:24.030 --> 00:22:25.950
output has become P0.
00:22:25.950 --> 00:22:29.180
This propagated to 1, that
became P0, add it to 0, and it
00:22:29.180 --> 00:22:33.250
became P0, so now the answer,
which had been 1, is P0.
00:22:35.940 --> 00:22:38.250
On the next step, a very
similar thing happens.
00:22:38.250 --> 00:22:42.190
The P0 that was here becomes the
output of the delay, gets
00:22:42.190 --> 00:22:46.370
multiplied by P0 to give you P0
squared, gets added to 0 to
00:22:46.370 --> 00:22:49.140
give you P0 squared,
et cetera.
00:22:55.560 --> 00:22:59.460
The thing that I want you to see
is that the output was in
00:22:59.460 --> 00:23:00.710
some sense simple.
00:23:03.530 --> 00:23:08.930
The value simply increased as
P0 to the n, geometrically.
00:23:08.930 --> 00:23:11.370
There's another way we
can think about that.
00:23:11.370 --> 00:23:13.510
I just did the sample by sample
approach, but the whole
00:23:13.510 --> 00:23:15.270
theme of this part
of the course
00:23:15.270 --> 00:23:17.810
is the signals approach.
00:23:17.810 --> 00:23:21.630
If I think about the whole
signal in one fell swoop, then
00:23:21.630 --> 00:23:23.570
I can develop an operator
expression to
00:23:23.570 --> 00:23:25.400
characterize the system.
00:23:25.400 --> 00:23:28.760
The operator expression says the
signal y is constructed by
00:23:28.760 --> 00:23:32.090
adding the signal x to
the signal P0RY.
00:23:35.610 --> 00:23:39.660
If I solve that for the ratio of
the output to input, I get
00:23:39.660 --> 00:23:42.300
1 over (1 minus P0R).
00:23:42.300 --> 00:23:44.702
Again, going back to the idea
that I started with, that
00:23:44.702 --> 00:23:49.060
we're going to get ratios of
polynomials in R now the R is
00:23:49.060 --> 00:23:55.390
in the bottom, and now I can
expand that just as though it
00:23:55.390 --> 00:23:57.760
were an algebraic expression.
00:23:57.760 --> 00:24:04.730
I can expand 1 over P0R in
a power series by using
00:24:04.730 --> 00:24:07.630
synthetic division.
00:24:07.630 --> 00:24:11.820
The result is very similar in
structure to the result we saw
00:24:11.820 --> 00:24:14.230
in sample by sample.
00:24:14.230 --> 00:24:17.410
It consists of an ascending
series in R, which means an
00:24:17.410 --> 00:24:21.320
ascending number of delays.
00:24:21.320 --> 00:24:24.300
Every time you increase the
number of delays by 1, you
00:24:24.300 --> 00:24:29.770
also multiply the amplitude by
P0, so this is, in fact, the
00:24:29.770 --> 00:24:32.320
same kind of result,
but viewed from a
00:24:32.320 --> 00:24:35.490
signal point of view.
00:24:35.490 --> 00:24:39.000
Finally, I want to think about
it in terms of block diagrams.
00:24:39.000 --> 00:24:42.570
Same idea, I've got the same
feedback system, but now I
00:24:42.570 --> 00:24:45.870
want to take advantage of this
ascending series expansion
00:24:45.870 --> 00:24:49.870
that I did and think about
each of the terms in that
00:24:49.870 --> 00:24:55.720
series as a signal flow path
through the feedback system.
00:24:55.720 --> 00:24:59.970
So one, the first term in the
ascending series, represents
00:24:59.970 --> 00:25:02.730
the path that goes directly from
the input to the output,
00:25:02.730 --> 00:25:03.980
passing through no delays.
00:25:06.840 --> 00:25:10.370
The second term in the series,
P0R, represents the path that
00:25:10.370 --> 00:25:13.900
goes to the output, loops
around, comes back through the
00:25:13.900 --> 00:25:16.890
adder, and then comes out.
00:25:16.890 --> 00:25:19.490
In traversing that more
complicated path, you picked
00:25:19.490 --> 00:25:25.120
up 1 delay and 1
multiply by P0.
00:25:25.120 --> 00:25:27.990
Second term, two loops.
00:25:27.990 --> 00:25:29.300
Third term, three loops.
00:25:29.300 --> 00:25:32.560
Fourth term, four loops.
00:25:32.560 --> 00:25:37.440
The idea is that the block
diagram gives us a way to
00:25:37.440 --> 00:25:39.930
visualize how the answer
came about.
00:25:39.930 --> 00:25:43.620
It came about by all the
possible paths that lead from
00:25:43.620 --> 00:25:45.820
the input to the output.
00:25:45.820 --> 00:25:48.660
Those possible paths all
differed by a delay, and
00:25:48.660 --> 00:25:53.090
that's why the decomposition
was so simple, each path
00:25:53.090 --> 00:25:54.610
corresponding to a different
number of
00:25:54.610 --> 00:25:55.620
delays through the system.
00:25:55.620 --> 00:25:58.660
That won't always be true from
more complicated systems, but
00:25:58.660 --> 00:26:01.760
it is true for this one.
00:26:01.760 --> 00:26:04.170
This flow diagram also lets
you see something that's
00:26:04.170 --> 00:26:05.420
extremely interesting.
00:26:08.170 --> 00:26:12.540
Cyclical flow paths, which are
characteristic of feedback--
00:26:12.540 --> 00:26:15.780
feedback means the signal
comes back.
00:26:15.780 --> 00:26:24.720
Cyclical flow paths require that
transient inputs generate
00:26:24.720 --> 00:26:28.240
persistent outputs.
00:26:28.240 --> 00:26:30.670
They generate persistent outputs
because the output at
00:26:30.670 --> 00:26:34.450
time n is not triggered by
the input at time n.
00:26:34.450 --> 00:26:37.830
It's triggered by the output
at time n minus 1.
00:26:37.830 --> 00:26:41.880
It keeps going on itself.
00:26:41.880 --> 00:26:43.550
That's fundamental
to feedback.
00:26:43.550 --> 00:26:45.720
There's no way of getting
around that.
00:26:45.720 --> 00:26:49.690
That's what feedback is.
00:26:49.690 --> 00:26:53.150
And it also shows why you got
that funny oscillatory
00:26:53.150 --> 00:26:55.000
behavior in WallFinder.
00:26:55.000 --> 00:26:58.670
There wasn't any way
around that.
00:26:58.670 --> 00:27:01.760
Feedback meant that you
were looping back.
00:27:01.760 --> 00:27:04.790
That meant that there
was a cycle in
00:27:04.790 --> 00:27:07.230
the signal flow paths.
00:27:07.230 --> 00:27:11.060
That means that even transient
signals, signals that go away
00:27:11.060 --> 00:27:13.650
very quickly like the
[INAUDIBLE] sample, generate
00:27:13.650 --> 00:27:15.070
responses that go on forever.
00:27:18.540 --> 00:27:21.680
So that's a fundamental way
of thinking about systems.
00:27:21.680 --> 00:27:26.460
Systems are either feedforward
or feedback.
00:27:26.460 --> 00:27:30.230
Feedforward means that
there are no cyclic
00:27:30.230 --> 00:27:33.320
paths in the system.
00:27:36.830 --> 00:27:39.390
No path in the system that take
you from the input to the
00:27:39.390 --> 00:27:41.650
output has a cycle in it.
00:27:41.650 --> 00:27:42.820
That's what acyclic means.
00:27:42.820 --> 00:27:45.760
That's what feedforward means.
00:27:45.760 --> 00:27:51.350
Acyclic, feedforward, those
all have responses to
00:27:51.350 --> 00:27:55.880
transient inputs that
are transient.
00:27:55.880 --> 00:27:59.600
That contrasts with
cyclic systems.
00:27:59.600 --> 00:28:04.160
A cyclic system has feedback
and will have the property
00:28:04.160 --> 00:28:06.590
that transient signals
can generate
00:28:06.590 --> 00:28:10.200
outputs that go on forever.
00:28:10.200 --> 00:28:14.830
OK, how many of these
systems are cyclic?
00:28:14.830 --> 00:28:15.560
Easy questions.
00:28:15.560 --> 00:28:16.815
15 seconds, talk to
your neighbor.
00:29:46.420 --> 00:29:48.080
OK, so what's the answer?
00:29:48.080 --> 00:29:50.900
How many?
00:29:50.900 --> 00:29:52.310
OK, virtually 100%.
00:29:52.310 --> 00:29:53.720
Correct, the answer's (3).
00:29:53.720 --> 00:29:57.610
I've illustrated the cycles in
red, so there's a cycle here,
00:29:57.610 --> 00:30:00.730
there's two cycles in this one,
and there's a cycle here.
00:30:00.730 --> 00:30:05.340
So the idea is that, when you
see a block diagram, one of
00:30:05.340 --> 00:30:07.120
the first things you want to
characterize, because it's
00:30:07.120 --> 00:30:10.790
such a big difference between
systems, is whether or not
00:30:10.790 --> 00:30:11.580
there's a cycle in it.
00:30:11.580 --> 00:30:13.710
If there's a cycle, then you
know there's feedback.
00:30:13.710 --> 00:30:16.460
If there's feedback, then you
know you have the potential to
00:30:16.460 --> 00:30:19.715
have a persistent response to
even a transient signal.
00:30:22.870 --> 00:30:26.860
OK, so if you only have one
loop of the type that I
00:30:26.860 --> 00:30:29.340
started with, where we had just
one loop with an R and a
00:30:29.340 --> 00:30:36.680
P0, then the question is, as you
go around the loop, do the
00:30:36.680 --> 00:30:38.750
samples get bigger,
or smaller, or do
00:30:38.750 --> 00:30:40.570
they stay the same?
00:30:40.570 --> 00:30:43.130
That's a fundamental
characterization of how the
00:30:43.130 --> 00:30:45.030
simple feedback system works.
00:30:45.030 --> 00:30:48.810
So here, if on every cycle the
amplitude of the signal
00:30:48.810 --> 00:30:54.650
diminishes by multiplication by
half, that means that the
00:30:54.650 --> 00:30:57.730
response ultimately decays.
00:30:57.730 --> 00:31:00.370
Mathematically, it goes on
forever, just like I said
00:31:00.370 --> 00:31:05.440
previously, but the amplitude is
decaying, so practically it
00:31:05.440 --> 00:31:07.410
stops after a while.
00:31:07.410 --> 00:31:11.090
It becomes small enough that
you lose track of it.
00:31:11.090 --> 00:31:14.410
By contrast, if every time you
go around the loop, you pick
00:31:14.410 --> 00:31:18.990
up amplitude, if the amplitude
here were multiplied by 1.2,
00:31:18.990 --> 00:31:21.330
then it gets bigger.
00:31:21.330 --> 00:31:24.900
So the idea, then, is that you
can characterize this kind of
00:31:24.900 --> 00:31:28.490
a feedback by one number.
00:31:28.490 --> 00:31:31.460
We call that number the pole.
00:31:31.460 --> 00:31:33.350
Very mysterious word.
00:31:33.350 --> 00:31:36.270
I won't go into the origins
of the word.
00:31:36.270 --> 00:31:41.120
For our purposes, it just simply
means the base of the
00:31:41.120 --> 00:31:45.270
geometric sequence that
characterizes the response of
00:31:45.270 --> 00:31:49.440
a system to the unit
sample signal.
00:31:49.440 --> 00:31:52.140
So here, I've showed an
illustration of what can
00:31:52.140 --> 00:31:57.270
happen if p is 1/2, p is one,
p is 1.2, which you can see
00:31:57.270 --> 00:32:00.485
decay, persistence,
divergence.
00:32:04.780 --> 00:32:07.840
Can you characterize
this system by P0?
00:32:07.840 --> 00:32:09.231
And if so, what is P0?
00:32:21.950 --> 00:32:22.210
Yes?
00:32:22.210 --> 00:32:23.317
No?
00:32:23.317 --> 00:32:30.772
[CLASSROOM SIDE CONVERSATIONS]
00:32:30.772 --> 00:32:33.008
PROFESSOR: Yeah, and virtually
everybody's
00:32:33.008 --> 00:32:33.754
getting the right answer.
00:32:33.754 --> 00:32:35.710
The right answer's (2).
00:32:35.710 --> 00:32:37.960
So we like algebra.
00:32:37.960 --> 00:32:43.170
We like negative numbers, so
we're allowed to think about
00:32:43.170 --> 00:32:44.310
poles being negative.
00:32:44.310 --> 00:32:46.000
In fact, by the end of the hour,
we'll even think about
00:32:46.000 --> 00:32:48.820
poles having imaginary parts,
but for the time
00:32:48.820 --> 00:32:50.440
being, this is fine.
00:32:50.440 --> 00:32:53.400
If the pole were negative,
what that means is the
00:32:53.400 --> 00:32:57.370
consecutive terms in the unit
sample response, the response
00:32:57.370 --> 00:33:00.960
of the system to a unit sample
signal, the unit sample
00:33:00.960 --> 00:33:03.800
response, the unit sample
response can
00:33:03.800 --> 00:33:05.050
alternate in sine.
00:33:07.420 --> 00:33:12.320
OK, so this then represents
all the possible behaviors
00:33:12.320 --> 00:33:17.000
that you could get from
a feedback system
00:33:17.000 --> 00:33:20.020
with a single pole.
00:33:20.020 --> 00:33:24.360
If a feedback system has
a single pole, the only
00:33:24.360 --> 00:33:27.110
behaviors that you can
get are represented
00:33:27.110 --> 00:33:28.480
by these three cartoons.
00:33:34.310 --> 00:33:38.895
So here, this z-axis contains
all possible values of P0.
00:33:38.895 --> 00:33:44.540
If P0 is bigger than 1, then
the magnitude diverges, and
00:33:44.540 --> 00:33:48.010
the signal grows
monotonically.
00:33:48.010 --> 00:33:53.340
If the pole is between 0 and
1, the response is also
00:33:53.340 --> 00:34:00.870
monotonic, but now it
converges towards 0.
00:34:00.870 --> 00:34:04.350
If you flip the sign, the
relations are still the same,
00:34:04.350 --> 00:34:08.360
except that you now get
sign alternation.
00:34:08.360 --> 00:34:14.159
So if the P0 is between 0 and
minus 1, which is here, the
00:34:14.159 --> 00:34:17.989
output still converges because
the magnitude of the pole is
00:34:17.989 --> 00:34:20.429
less than 1.
00:34:20.429 --> 00:34:22.659
But now the sign flips.
00:34:22.659 --> 00:34:28.139
And if the pole is below minus
1, then you get alternation,
00:34:28.139 --> 00:34:30.460
but you also get divergence.
00:34:30.460 --> 00:34:33.710
The important thing is we
started with a simple system,
00:34:33.710 --> 00:34:35.699
and we ended up with an
absolutely complete
00:34:35.699 --> 00:34:36.610
characterization of it.
00:34:36.610 --> 00:34:40.090
This is everything
that can happen.
00:34:40.090 --> 00:34:41.860
That's a powerful statement.
00:34:41.860 --> 00:34:45.050
When I can analyze a system,
even if it's simple, and find
00:34:45.050 --> 00:34:46.974
all the possible behaviors,
I have something.
00:34:50.360 --> 00:34:52.820
If you have a simple system with
a single pole, this is
00:34:52.820 --> 00:34:55.460
all that can happen.
00:34:55.460 --> 00:34:56.350
There might be offsets.
00:34:56.350 --> 00:34:57.770
There might be delays.
00:34:57.770 --> 00:35:02.810
The signal may not start until
the fifth sample, but the
00:35:02.810 --> 00:35:06.660
persistent signal will either
grow without bounds, the k to
00:35:06.660 --> 00:35:11.250
0, or do one of those two
with alternating sign.
00:35:11.250 --> 00:35:16.970
That's the only things that can
happen, which of course,
00:35:16.970 --> 00:35:19.130
begs the question, well, what
if the system's more
00:35:19.130 --> 00:35:21.640
complicated.
00:35:21.640 --> 00:35:23.730
OK, so here's a more
complicated system.
00:35:23.730 --> 00:35:27.260
This system cannot be
represented by just one pole.
00:35:27.260 --> 00:35:31.280
In fact, the system's
complicated enough you should
00:35:31.280 --> 00:35:34.400
think through how you
would solve it.
00:35:34.400 --> 00:35:35.570
You should all be very
comfortable with
00:35:35.570 --> 00:35:36.590
this sort of thing.
00:35:36.590 --> 00:35:47.210
So if you were to think about
what if I had a system like
00:35:47.210 --> 00:36:02.670
so, and I want it to
be 1.6 minus 0.63.
00:36:02.670 --> 00:36:09.080
What would be the output signal
at time 2 if the input
00:36:09.080 --> 00:36:12.830
were a unit sample signal?
00:36:12.830 --> 00:36:17.050
OK, as with all systems we're
going to think about, we have
00:36:17.050 --> 00:36:18.430
to specify initial conditions.
00:36:18.430 --> 00:36:20.240
The simplest kind of initial
conditions we could think
00:36:20.240 --> 00:36:22.280
about would be rest.
00:36:22.280 --> 00:36:25.070
If I thought about this system
at rest, then the initial
00:36:25.070 --> 00:36:27.770
outputs of the R's would be 0.
00:36:40.650 --> 00:36:42.560
That's at rest.
00:36:42.560 --> 00:36:47.880
For times less than 0,
the input would be 0.
00:36:47.880 --> 00:36:56.530
0 times 1.6 plus 0 times -0.63
plus 0 would give me 0.
00:36:56.530 --> 00:36:59.240
Now, the clock ticks.
00:36:59.240 --> 00:37:04.840
When the clock ticks,
it becomes times 0.
00:37:04.840 --> 00:37:06.495
At times 0, the input is 1.
00:37:09.820 --> 00:37:12.770
This 0 just propagated down to
here, but this was 0, so
00:37:12.770 --> 00:37:14.325
nothing interesting happens
at the R's.
00:37:17.420 --> 00:37:20.560
But now my output is 1.
00:37:23.790 --> 00:37:26.100
Now, the clock ticks.
00:37:26.100 --> 00:37:27.350
What happens?
00:37:31.390 --> 00:37:34.315
When the clock ticks, this
1 propagates down here.
00:37:38.120 --> 00:37:40.300
This 0 propagates down
here, but that was 0.
00:37:43.360 --> 00:37:49.650
This 1 goes to 0 because the
input's only 1 at times 0.
00:37:49.650 --> 00:37:50.900
So what's the output?
00:37:53.540 --> 00:37:58.490
1.6.
00:37:58.490 --> 00:38:01.660
Now, the clock ticks.
00:38:01.660 --> 00:38:02.910
What happens?
00:38:08.170 --> 00:38:12.690
Well, this 1 comes down here.
00:38:12.690 --> 00:38:14.440
This 1.6 comes down here.
00:38:19.170 --> 00:38:23.240
This 0 becomes another 0 because
the input has an
00:38:23.240 --> 00:38:27.010
infinite stream of 0's after
the initial time.
00:38:27.010 --> 00:38:28.260
So what's the output?
00:38:30.880 --> 00:38:37.650
Well, it's 1.6 times 1.6 plus
1 times -0.63, so the answer
00:38:37.650 --> 00:38:38.900
is number (3).
00:38:42.700 --> 00:38:43.020
Yeah?
00:38:43.020 --> 00:38:44.270
OK.
00:38:47.350 --> 00:38:48.750
I forgot to write it
up there, so the
00:38:48.750 --> 00:38:50.720
answer's in red down here.
00:38:50.720 --> 00:38:53.260
1.6 squared minus 0.63.
00:38:53.260 --> 00:38:53.940
OK?
00:38:53.940 --> 00:38:55.850
The point is that it's slightly
more complicated to
00:38:55.850 --> 00:39:00.610
think about than the case with a
single pole, and in fact, if
00:39:00.610 --> 00:39:02.910
you use that logic to simply
step through all the
00:39:02.910 --> 00:39:08.740
responses, you get a response
that doesn't look geometric.
00:39:12.080 --> 00:39:15.060
The geometric sequences that we
looked at previously either
00:39:15.060 --> 00:39:17.240
monotonically increased,
monotonically decreased
00:39:17.240 --> 00:39:19.640
towards 0, or give one of
those two things and
00:39:19.640 --> 00:39:20.410
alternated.
00:39:20.410 --> 00:39:22.030
This does none of
those behaviors.
00:39:22.030 --> 00:39:25.430
So the point is Freeman's
an idiot.
00:39:25.430 --> 00:39:28.010
He spent all that time telling
us what one pole does, and now
00:39:28.010 --> 00:39:30.100
two poles does something
completely different.
00:39:30.100 --> 00:39:31.900
Right?
00:39:31.900 --> 00:39:34.750
So the response is
not geometric.
00:39:34.750 --> 00:39:37.110
The response grows
and then decays.
00:39:37.110 --> 00:39:38.350
It never changes sign.
00:39:38.350 --> 00:39:41.430
It does something completely
different from what we would
00:39:41.430 --> 00:39:45.020
have expected from a
single pole system.
00:39:45.020 --> 00:39:47.750
As you might expect from the
Theory of Lectures, that's not
00:39:47.750 --> 00:39:49.980
the end of the story.
00:39:49.980 --> 00:39:57.610
So the idea is to now capitalize
on this notion that
00:39:57.610 --> 00:40:02.720
we can think about operators
as algebra.
00:40:02.720 --> 00:40:06.950
If our expressions behaved like
I told you they did last
00:40:06.950 --> 00:40:12.280
lecture, if they behaved as
entities upon which--
00:40:12.280 --> 00:40:17.300
if they are isomorphic with
polynomials, as I said, then
00:40:17.300 --> 00:40:20.760
there's a very cute thing we
can do with this system to
00:40:20.760 --> 00:40:23.800
make it a lot simpler.
00:40:23.800 --> 00:40:27.070
The thing we can do is factor.
00:40:27.070 --> 00:40:30.510
If we think about the operator
expression to characterize
00:40:30.510 --> 00:40:34.690
this system, the thing that's
different is that
00:40:34.690 --> 00:40:37.850
there's an R squared.
00:40:37.850 --> 00:40:40.990
But if R operators work just
like polynomials--
00:40:40.990 --> 00:40:42.690
you can factor polynomials.
00:40:42.690 --> 00:40:47.410
That's the factor theorem
from algebra.
00:40:47.410 --> 00:40:51.120
And if I factor it, I get two
things that look like
00:40:51.120 --> 00:40:52.770
first-order systems.
00:40:52.770 --> 00:40:54.020
Well, that's good.
00:40:56.260 --> 00:41:01.080
The factored form means that
I can think about this more
00:41:01.080 --> 00:41:04.590
complicated system
as the cascade of
00:41:04.590 --> 00:41:07.140
two first-order systems.
00:41:07.140 --> 00:41:09.860
Well, that's pretty good.
00:41:09.860 --> 00:41:11.910
In fact, it doesn't even matter
what order I put them
00:41:11.910 --> 00:41:16.210
in because, as we've seen
previously, if the system
00:41:16.210 --> 00:41:20.350
started at initial rest, then
you can swap things because
00:41:20.350 --> 00:41:24.390
they obey all the principles of
polynomials, which include
00:41:24.390 --> 00:41:25.640
commutation.
00:41:27.830 --> 00:41:30.520
So what we've done, then,
is transform this more
00:41:30.520 --> 00:41:35.280
complicated system into the
cascade of two simple systems,
00:41:35.280 --> 00:41:36.530
and that's very good.
00:41:39.270 --> 00:41:44.880
Even better, we can think about
the complicated system
00:41:44.880 --> 00:41:51.940
as the sum of simpler parts, and
that uses more intuition
00:41:51.940 --> 00:41:53.970
from polynomials.
00:41:53.970 --> 00:41:57.750
If we have one over a
second-order polynomial, we
00:41:57.750 --> 00:42:01.680
can write it in a factored form
here, but we can expand
00:42:01.680 --> 00:42:04.870
it in what we call partial
fractions.
00:42:04.870 --> 00:42:09.110
We can expand this thing in
this sum, and if you think
00:42:09.110 --> 00:42:12.140
about putting this over a common
denominator and working
00:42:12.140 --> 00:42:17.100
out the relationship, this
difference, 4.5 over 1 minus
00:42:17.100 --> 00:42:22.280
0.9R minus 3.5 over
1 minus 0.7R.
00:42:22.280 --> 00:42:25.020
That's precisely the same using
the normal rules for
00:42:25.020 --> 00:42:26.110
polynomials.
00:42:26.110 --> 00:42:29.650
That's precisely the same
as that expression.
00:42:29.650 --> 00:42:31.400
But the difference, from the
point of view of thinking
00:42:31.400 --> 00:42:33.550
about systems, is enormous.
00:42:33.550 --> 00:42:37.310
We know the answer
to that one.
00:42:37.310 --> 00:42:39.670
That's the sum of the responses
to two first-order
00:42:39.670 --> 00:42:45.350
systems, so we can write that
symbolically this way.
00:42:45.350 --> 00:42:47.830
We can think about having
a sum system that
00:42:47.830 --> 00:42:51.190
generates this term.
00:42:51.190 --> 00:42:55.110
This term is a simple system of
the type of that we looked
00:42:55.110 --> 00:43:01.670
at previously that, then,
gets multiplied by 4.5.
00:43:01.670 --> 00:43:03.540
I'm just factoring again.
00:43:03.540 --> 00:43:07.690
I'm saying I've got something
over something, which means
00:43:07.690 --> 00:43:10.950
that I can put something in each
of two different parts of
00:43:10.950 --> 00:43:13.220
two things that I multiply
together.
00:43:13.220 --> 00:43:16.230
And I can think about this as
having been generated by this
00:43:16.230 --> 00:43:18.040
system, and you just
add them together.
00:43:21.430 --> 00:43:24.790
The amazing thing is that that
says that, despite the fact
00:43:24.790 --> 00:43:27.630
that the response looked
complicated, it was in fact
00:43:27.630 --> 00:43:30.720
the sum of two geometrics.
00:43:30.720 --> 00:43:33.850
So it wasn't very different
from the answer
00:43:33.850 --> 00:43:36.550
for a single pole.
00:43:36.550 --> 00:43:40.050
What I've just done
is amazing.
00:43:40.050 --> 00:43:43.210
I've just taken something that,
had you studied the
00:43:43.210 --> 00:43:45.410
difference equations and had
you studied the block
00:43:45.410 --> 00:43:49.880
diagrams, it would have been
very hard for you to conclude
00:43:49.880 --> 00:43:52.200
that something this complicated
has a response
00:43:52.200 --> 00:43:55.930
that can be written as the
sum of two geometrics.
00:43:55.930 --> 00:43:59.470
By thinking about the system
as a polynomial in R, it's
00:43:59.470 --> 00:44:00.650
completely trivial.
00:44:00.650 --> 00:44:05.290
It's a simple application of the
rules for polynomials that
00:44:05.290 --> 00:44:07.950
you all know.
00:44:07.950 --> 00:44:11.190
So what we've shown, then, is
that this complicated system
00:44:11.190 --> 00:44:15.305
has a way of thinking
about as just two
00:44:15.305 --> 00:44:18.190
of the simpler systems.
00:44:18.190 --> 00:44:22.860
The complicated response that
grew and decayed, that's just
00:44:22.860 --> 00:44:28.750
the difference, really,
4.5 minus 3.5.
00:44:28.750 --> 00:44:32.560
It's the weighted difference of
a part that goes 0.7 to the
00:44:32.560 --> 00:44:34.963
n, then a different part
that goes 0.9 to the n.
00:44:42.640 --> 00:44:46.250
So far, we've got to results,
the n equals 1 case, the
00:44:46.250 --> 00:44:48.620
first-order polynomial in our
case, the one pole case,
00:44:48.620 --> 00:44:49.550
that's trivial.
00:44:49.550 --> 00:44:51.490
It's just a geometric
sequence.
00:44:51.490 --> 00:44:54.010
The response is just a
geometric sequence.
00:44:54.010 --> 00:44:56.160
If it happens to be
second-order, this is
00:44:56.160 --> 00:45:03.240
second-order because when you
write the operator expression,
00:45:03.240 --> 00:45:07.260
the polynomial in the bottom is
second-order, second-order
00:45:07.260 --> 00:45:08.740
polynomial in R.
00:45:08.740 --> 00:45:12.270
This second-order system
has a response that
00:45:12.270 --> 00:45:15.300
looks like two pieces.
00:45:15.300 --> 00:45:17.750
Each piece looks like a
piece that was from
00:45:17.750 --> 00:45:19.410
a first-order system.
00:45:19.410 --> 00:45:21.675
And in fact, that idea
generalizes.
00:45:25.470 --> 00:45:29.790
If we have a system that can
be represented by linear
00:45:29.790 --> 00:45:34.500
difference equation with
constant coefficients that
00:45:34.500 --> 00:45:37.410
will always be true if the
system was constructed out of
00:45:37.410 --> 00:45:43.270
the parts that we talked about,
adders, gains, delays.
00:45:43.270 --> 00:45:46.060
If the system is constructed
out of adders, gains, and
00:45:46.060 --> 00:45:49.290
delays, then it will be possible
to express the system
00:45:49.290 --> 00:45:51.830
in terms of one difference
equation.
00:45:51.830 --> 00:45:53.850
General form is showed here.
00:45:53.850 --> 00:45:58.200
Y then can be constructed out
of parts that are delayed
00:45:58.200 --> 00:46:04.210
versions of Y and delayed
versions of X. If you do that,
00:46:04.210 --> 00:46:08.120
then you can always write the
operator that expresses the
00:46:08.120 --> 00:46:11.630
ratio between the output and the
input as the ratio of two
00:46:11.630 --> 00:46:13.000
polynomials.
00:46:13.000 --> 00:46:15.880
That will always be true.
00:46:15.880 --> 00:46:18.590
So this, now, is the
generalization step.
00:46:18.590 --> 00:46:23.310
We did the n equals 1 case, we
did the n equals 2 case, and
00:46:23.310 --> 00:46:24.730
now we're generalizing.
00:46:24.730 --> 00:46:28.220
We will always get, for any
system that can be represented
00:46:28.220 --> 00:46:31.080
by a linear difference equation
with constant
00:46:31.080 --> 00:46:34.170
coefficients, we can always
represent the system
00:46:34.170 --> 00:46:37.750
functional in this form.
00:46:37.750 --> 00:46:42.060
Then just like we did in the
second-order case, we can use
00:46:42.060 --> 00:46:46.420
the factor theorem to break
this polynomial in the
00:46:46.420 --> 00:46:49.940
denominator into factors.
00:46:49.940 --> 00:46:51.945
That comes from the factor
theorem in algebra.
00:46:54.640 --> 00:46:59.480
Then we can re-express that in
terms of partial fractions.
00:46:59.480 --> 00:47:02.820
And what I've just showed is
that, in the general case,
00:47:02.820 --> 00:47:05.910
regardless of how many delays
are in the system, if the
00:47:05.910 --> 00:47:09.430
system only has adders, gains,
and delays, I can always
00:47:09.430 --> 00:47:12.020
express the answer as
a sum of geometrics.
00:47:12.020 --> 00:47:14.710
That's interesting.
00:47:14.710 --> 00:47:19.130
That means that if I knew the
bases for all of those
00:47:19.130 --> 00:47:24.100
geometric sequences, I know
something about the response.
00:47:24.100 --> 00:47:26.050
The bases are things
we call poles.
00:47:26.050 --> 00:47:31.120
If you knew all the poles,
you'd know something very
00:47:31.120 --> 00:47:32.370
powerful about the system.
00:47:36.330 --> 00:47:40.590
So every one of the factors
corresponds to a pole, and by
00:47:40.590 --> 00:47:44.770
partial fractions, you'll get
one response for each pole.
00:47:44.770 --> 00:47:48.411
The response for each pole
goes like pole to the n.
00:47:48.411 --> 00:47:49.750
You know the basic shape.
00:47:49.750 --> 00:47:53.470
You don't know the constants,
but you know the basic shape
00:47:53.470 --> 00:47:58.880
of the response just by
knowing the poles.
00:47:58.880 --> 00:48:01.110
We can go one more step,
which makes the
00:48:01.110 --> 00:48:03.790
computation somewhat simpler.
00:48:03.790 --> 00:48:06.600
I used the factor theorem.
00:48:06.600 --> 00:48:08.160
Here, I'm using the fundamental
theorem of
00:48:08.160 --> 00:48:12.960
algebra, which says that if I
have a polynomial of order n,
00:48:12.960 --> 00:48:15.740
I have n roots.
00:48:15.740 --> 00:48:19.530
The poles are related to the
roots of the R polynomial.
00:48:22.510 --> 00:48:28.020
The relationship is take
the functional,
00:48:28.020 --> 00:48:29.250
substitute for R --
00:48:29.250 --> 00:48:38.500
1 over Z. Re-express the
functional as a ratio of
00:48:38.500 --> 00:48:43.240
polynomials in Z. The poles
are the roots of the
00:48:43.240 --> 00:48:44.490
denominator.
00:48:47.780 --> 00:48:50.350
So recapping, I started with
a first-order system.
00:48:50.350 --> 00:48:52.440
I showed you how to get
a second-order system.
00:48:52.440 --> 00:48:56.760
I showed that, in general, you
can use the factor theorem to
00:48:56.760 --> 00:49:00.730
break down the response of a
higher-order system into a sum
00:49:00.730 --> 00:49:03.880
of responses of first-order
systems.
00:49:03.880 --> 00:49:06.720
Now, I've shown that you can use
the fundamental theorem of
00:49:06.720 --> 00:49:11.590
algebra to find the poles
directly, and then by knowing
00:49:11.590 --> 00:49:16.030
the poles, you know each of
the behaviors, monotonic
00:49:16.030 --> 00:49:18.280
divergence, monotonic
convergence,
00:49:18.280 --> 00:49:20.870
or alternating signs.
00:49:20.870 --> 00:49:23.770
And so here is this same example
that I started with,
00:49:23.770 --> 00:49:26.260
worked out by thinking about
what are the poles.
00:49:26.260 --> 00:49:29.280
The poles are 0.7 and 0.9,
which we see by a simple
00:49:29.280 --> 00:49:32.300
application of the fundamental
theorem of algebra.
00:49:32.300 --> 00:49:38.210
OK, we got a long way by just
thinking about operators as
00:49:38.210 --> 00:49:39.080
polynomials.
00:49:39.080 --> 00:49:40.880
We haven't done anything
that you haven't
00:49:40.880 --> 00:49:42.300
done in high school.
00:49:42.300 --> 00:49:46.090
Polynomials are very familiar
and we've made an isomorphism
00:49:46.090 --> 00:49:50.100
between systems and
polynomials.
00:49:53.210 --> 00:49:55.920
OK, make sure you're
all with me.
00:49:55.920 --> 00:49:57.880
Here's a higher order system.
00:49:57.880 --> 00:50:00.140
How many of these statements
are true --
00:50:00.140 --> 00:50:02.030
0, 1, 2, 3, 4, or 5?
00:50:04.740 --> 00:50:06.060
Talk to your neighbor,
get an answer.
00:52:42.920 --> 00:52:44.650
So how many are true --
00:52:44.650 --> 00:52:46.660
0, 1, 2, 3, 4, or 5?
00:52:49.180 --> 00:52:50.290
Oh, come on.
00:52:50.290 --> 00:52:51.900
Blame it on your neighbor.
00:52:51.900 --> 00:52:53.595
You weren't talking, but I
didn't hear you not talking.
00:52:56.490 --> 00:52:57.385
How many are true --
00:52:57.385 --> 00:52:59.080
0, 1, 2, 3, 4, or 5?
00:52:59.080 --> 00:53:00.511
Raise your hands.
00:53:00.511 --> 00:53:02.420
AUDIENCE: They can't
all be true.
00:53:02.420 --> 00:53:03.555
PROFESSOR: They can't
all be true.
00:53:03.555 --> 00:53:05.270
Are they mutually
contradictory?
00:53:05.270 --> 00:53:06.180
AUDIENCE: Well, yeah.
00:53:06.180 --> 00:53:07.090
5 --
00:53:07.090 --> 00:53:08.390
PROFESSOR: N1 of the above,
that sounds like--
00:53:08.390 --> 00:53:10.650
OK, so you've eliminated 1.
00:53:13.310 --> 00:53:14.786
Which one's true?
00:53:14.786 --> 00:53:18.132
How many statements are true?
00:53:18.132 --> 00:53:21.700
Looks like about 75%.
00:53:21.700 --> 00:53:23.130
Correct?
00:53:23.130 --> 00:53:25.261
What should I do?
00:53:25.261 --> 00:53:26.390
How do I figure it out?
00:53:26.390 --> 00:53:27.120
What's my first step?
00:53:27.120 --> 00:53:27.878
What do I do?
00:53:27.878 --> 00:53:30.270
AUDIENCE: Operators.
00:53:30.270 --> 00:53:32.080
PROFESSOR: Operators,
absolutely.
00:53:32.080 --> 00:53:34.690
So turn it into operators.
00:53:34.690 --> 00:53:36.060
So take the difference
equation,
00:53:36.060 --> 00:53:39.110
turn it into operators.
00:53:39.110 --> 00:53:44.500
The important thing to see is
that there are three Y terms.
00:53:44.500 --> 00:53:47.950
Take them all to the same side,
and I get an operator
00:53:47.950 --> 00:53:51.420
expression like that.
00:53:51.420 --> 00:53:54.200
The ones that depend on X, there
are two of them, that's
00:53:54.200 --> 00:53:57.410
represented here.
00:53:57.410 --> 00:54:01.210
The thing this is critical for
determining poles is figuring
00:54:01.210 --> 00:54:02.770
out the denominator.
00:54:02.770 --> 00:54:04.225
The poles are going to
come from this one.
00:54:06.920 --> 00:54:16.080
After I get the ratio of two
polynomials in R, I substitute
00:54:16.080 --> 00:54:21.790
1 over Z for each R. So for this
R, I get 1 over Z. For
00:54:21.790 --> 00:54:25.350
this R squared, I get
1 over Z squared.
00:54:25.350 --> 00:54:29.020
Then I want to turn it back into
a ratio of polynomials in
00:54:29.020 --> 00:54:34.710
Z, so I have to multiply top
and bottom by Z squared.
00:54:34.710 --> 00:54:45.700
And when I do that, I get this
ratio of polynomials in Z.
00:54:45.700 --> 00:54:51.400
The poles are the roots of the
denominator polynomial in Z.
00:54:51.400 --> 00:54:57.820
The poles are minus
1/2 and plus 1/4.
00:54:57.820 --> 00:55:00.095
So the unit sample response
converges to 0.
00:55:04.680 --> 00:55:06.690
What would be the condition
that that represents?
00:55:10.050 --> 00:55:12.990
AUDIENCE: Take the polynomial
on the bottom.
00:55:12.990 --> 00:55:15.780
PROFESSOR: Something about the
polynomial on the bottom.
00:55:15.780 --> 00:55:18.030
Would all second-order systems
have that property that the
00:55:18.030 --> 00:55:20.774
unit sample response would
converge to 0?
00:55:20.774 --> 00:55:22.024
AUDIENCE: [INAUDIBLE]
00:55:24.150 --> 00:55:25.520
PROFESSOR: Louder?
00:55:25.520 --> 00:55:27.820
AUDIENCE: Absolute value
of the poles?
00:55:27.820 --> 00:55:31.526
PROFESSOR: Absolute value
of the poles has to be?
00:55:31.526 --> 00:55:32.990
AUDIENCE: Less than 1.
00:55:32.990 --> 00:55:35.110
PROFESSOR: Less than 1.
00:55:35.110 --> 00:55:37.380
If the magnitude of the poles
is less than 1, then the
00:55:37.380 --> 00:55:40.370
response magnitude will
decay with time.
00:55:40.370 --> 00:55:44.080
So that's true here, and it
would be true so long as none
00:55:44.080 --> 00:55:47.930
of the poles have a magnitude
exceeding 1.
00:55:47.930 --> 00:55:50.030
There are poles at
1/2 and 1/4.
00:55:50.030 --> 00:55:51.050
No, that's not right.
00:55:51.050 --> 00:55:53.640
It's -1/2 or 1/4.
00:55:53.640 --> 00:55:54.830
There's a pole at 1/2.
00:55:54.830 --> 00:55:55.640
No, that's not right.
00:55:55.640 --> 00:55:57.450
There's a pole at -1/2.
00:55:57.450 --> 00:55:58.260
There are two poles.
00:55:58.260 --> 00:56:00.870
Yes, that's true.
00:56:00.870 --> 00:56:01.520
None of the above.
00:56:01.520 --> 00:56:02.340
No, that's not true.
00:56:02.340 --> 00:56:05.210
So the answer was (2).
00:56:05.210 --> 00:56:07.410
Everybody's comfortable?
00:56:07.410 --> 00:56:08.970
We've done something
very astonishing.
00:56:08.970 --> 00:56:13.410
We took an arbitrary system, and
we've figured out a rule
00:56:13.410 --> 00:56:15.820
that let's us break
it into the sum
00:56:15.820 --> 00:56:17.820
of geometric sequences.
00:56:17.820 --> 00:56:20.750
We can always write the response
to a unit sample
00:56:20.750 --> 00:56:25.250
signal, we can always write, as
a weighted sum of geometric
00:56:25.250 --> 00:56:30.440
sequences, and the number of
geometric sequences in the sum
00:56:30.440 --> 00:56:35.030
is the number of poles, which
is the order of the operator
00:56:35.030 --> 00:56:38.920
that operates on Y.
OK, so we've done
00:56:38.920 --> 00:56:41.720
something very powerful.
00:56:41.720 --> 00:56:44.490
There's one more thing that we
have to think about, and then
00:56:44.490 --> 00:56:48.340
we have a complete picture
of what's going on.
00:56:48.340 --> 00:56:49.800
Think about when you learned
polynomials.
00:56:52.450 --> 00:56:57.180
One of the big shocks was that
roots can be complex.
00:56:57.180 --> 00:56:58.940
What would that mean?
00:56:58.940 --> 00:57:05.660
What would it mean if we had
a system whose poles were
00:57:05.660 --> 00:57:07.510
complex valued?
00:57:07.510 --> 00:57:09.480
So first off, does such
a system exist?
00:57:09.480 --> 00:57:11.050
Well, here's one.
00:57:11.050 --> 00:57:12.720
I just pulled that
out of the air.
00:57:12.720 --> 00:57:16.230
If I think about the functional,
1 over (1 minus R
00:57:16.230 --> 00:57:18.490
plus R squared) --
00:57:18.490 --> 00:57:23.480
if I convert that into our ratio
of polynomials in Z and
00:57:23.480 --> 00:57:25.640
then find the roots, I
find that the roots
00:57:25.640 --> 00:57:27.070
have a complex part.
00:57:27.070 --> 00:57:30.195
The roots are 1/2 plus or
minus root 3 over 2j.
00:57:33.020 --> 00:57:35.970
There's an imaginary part.
00:57:35.970 --> 00:57:40.330
So the question is what
would that mean?
00:57:40.330 --> 00:57:42.370
Or is perhaps that system
just meaningless?
00:57:45.970 --> 00:57:51.530
Well, complex numbers work in
algebra, and complex numbers
00:57:51.530 --> 00:57:54.000
work here, too.
00:57:54.000 --> 00:57:59.280
So the fact that a pole has a
complex value in the context
00:57:59.280 --> 00:58:02.770
of signals and systems simply
means that the pole is
00:58:02.770 --> 00:58:10.520
complex, that the base of the
geometric sequence, that base
00:58:10.520 --> 00:58:12.200
is complex.
00:58:12.200 --> 00:58:17.500
So that means that we can still
rewrite the denominator,
00:58:17.500 --> 00:58:20.960
which was 1 minus R plus R
squared, we can rewrite that
00:58:20.960 --> 00:58:25.830
denominator in terms of a
product of two first-order R
00:58:25.830 --> 00:58:28.460
polynomials.
00:58:28.460 --> 00:58:33.580
The coefficients are now
complex, but it still works.
00:58:33.580 --> 00:58:34.950
The algebra still works right.
00:58:34.950 --> 00:58:36.560
That has to work because that's
just polynomials.
00:58:36.560 --> 00:58:39.020
That's the way polynomials
behave.
00:58:39.020 --> 00:58:40.610
Now, we can still factor it.
00:58:40.610 --> 00:58:41.930
We can still use the
factor theorem.
00:58:41.930 --> 00:58:44.760
In fact, we can still use the
fundamental theorem of algebra
00:58:44.760 --> 00:58:47.820
to find the poles
by the Z trick.
00:58:47.820 --> 00:58:48.950
That's fine.
00:58:48.950 --> 00:58:50.890
We can still use partial
fractions.
00:58:50.890 --> 00:58:54.050
All of these numbers
are complex, but
00:58:54.050 --> 00:58:56.250
the math still works.
00:58:56.250 --> 00:59:00.120
The funny thing is that
it implies that
00:59:00.120 --> 00:59:02.620
the fundamental modes--
00:59:02.620 --> 00:59:06.440
by fundamental mode, I mean the
simple geometric for the
00:59:06.440 --> 00:59:09.380
case of a first-order system,
more complex behaviors for
00:59:09.380 --> 00:59:10.630
higher order systems.
00:59:10.630 --> 00:59:15.130
The mode is the time response
associated with a pole.
00:59:15.130 --> 00:59:20.006
So the modes are, in this case,
complexes sequences.
00:59:24.140 --> 00:59:32.040
So in general, the modes
look like P0 to the n.
00:59:32.040 --> 00:59:35.260
Here, my modes are simply
have a complex value.
00:59:35.260 --> 00:59:36.740
So what did I say they were?
00:59:36.740 --> 00:59:40.210
The poles were 1/2
plus or minus--
00:59:47.320 --> 00:59:48.940
so my modes simply look that.
00:59:48.940 --> 00:59:51.010
Same thing.
00:59:51.010 --> 00:59:55.625
The strange thing that happened
was that those modes,
00:59:55.625 --> 01:00:00.860
those geometric sequences, are
now have complex values.
01:00:00.860 --> 01:00:05.680
The first one up here, if I just
look at the denominator,
01:00:05.680 --> 01:00:08.680
these coefficients mean that
it's proportionate to the mode
01:00:08.680 --> 01:00:13.190
associated with this, which is
that, which has a real part,
01:00:13.190 --> 01:00:15.490
which is the blue part, and
the imaginary part,
01:00:15.490 --> 01:00:16.740
which is a red part.
01:00:19.260 --> 01:00:22.330
There were two poles,
plus and minus.
01:00:22.330 --> 01:00:25.420
The other pole just flips
the imaginary part.
01:00:28.480 --> 01:00:34.740
So if I have imaginary poles,
all I get is complex modes,
01:00:34.740 --> 01:00:36.660
complex geometric sequences.
01:00:42.620 --> 01:00:45.840
An easier way of thinking about
that is thinking about--
01:00:45.840 --> 01:00:51.020
so when we had a simple, real
pole, we just had P0 to the n.
01:00:51.020 --> 01:00:53.900
That's easy to visualize because
we just think about
01:00:53.900 --> 01:00:57.960
each time you go from 0 to 1 to
2 to 3, it goes from 1 to
01:00:57.960 --> 01:01:00.220
P0 to P0 squared to P0 cubed.
01:01:00.220 --> 01:01:02.990
Here, when you're multiplying
complex numbers, it's easier
01:01:02.990 --> 01:01:06.800
to imagine that on the
complex plane.
01:01:06.800 --> 01:01:10.020
Think about the location of the
point 1, think about the
01:01:10.020 --> 01:01:14.270
location of the point P0, think
about the location of
01:01:14.270 --> 01:01:18.180
the point P0 squared, and in
this particular case, where
01:01:18.180 --> 01:01:22.400
the pole was 1/2 plus or minus
the square root of 3 over 2
01:01:22.400 --> 01:01:28.700
times j, this would
be pole to the 0.
01:01:28.700 --> 01:01:30.510
This is pole to the 1.
01:01:30.510 --> 01:01:33.330
This is pole squared,
pole cubed.
01:01:33.330 --> 01:01:36.870
As you can see when you have a
complex number, the trajectory
01:01:36.870 --> 01:01:39.280
in complex space can
be complicated.
01:01:39.280 --> 01:01:42.560
In this case, it's circular.
01:01:42.560 --> 01:01:46.680
The circular trajectory in the
complex plane corresponds to
01:01:46.680 --> 01:01:50.070
the sinusoidal behavior
in time.
01:01:50.070 --> 01:01:51.850
So there's a correlation between
the way you think
01:01:51.850 --> 01:01:55.300
about the modes evolving on the
complex plane and the way
01:01:55.300 --> 01:01:56.720
you think about the
real and imaginary
01:01:56.720 --> 01:01:58.210
parts evolving in time.
01:02:02.560 --> 01:02:03.950
It seems a little
weird that the
01:02:03.950 --> 01:02:08.570
response should be complex.
01:02:08.570 --> 01:02:12.040
We're studying this kind of
system theory primarily
01:02:12.040 --> 01:02:14.290
because we're trying to gain
insight into real systems.
01:02:14.290 --> 01:02:16.690
We want to know how things
like robots work.
01:02:16.690 --> 01:02:17.900
How does the WallFinder work?
01:02:17.900 --> 01:02:21.190
What would it mean if the
WallFinder went to position
01:02:21.190 --> 01:02:26.040
one plus the square
root of 3 over 2j?
01:02:26.040 --> 01:02:29.100
That doesn't make sense.
01:02:29.100 --> 01:02:31.930
So there's a little bit of a
strange thing going on here.
01:02:31.930 --> 01:02:35.550
How is it that we need complex
numbers to model real things?
01:02:35.550 --> 01:02:37.690
That doesn't seem
to sound right.
01:02:37.690 --> 01:02:42.710
But the answer is that, if the
difference equation had real
01:02:42.710 --> 01:02:48.270
coefficients, as they will
for a real system--
01:02:48.270 --> 01:02:50.770
if you think about a real
system, like a bank account,
01:02:50.770 --> 01:02:53.850
the coefficients in the
difference equation are real
01:02:53.850 --> 01:02:55.940
numbers, not complex numbers.
01:02:55.940 --> 01:02:58.320
If you think about the
WallFinder system, the
01:02:58.320 --> 01:03:03.140
coefficients in the WallFinder
system, the coefficients of
01:03:03.140 --> 01:03:05.940
the difference equations--
01:03:05.940 --> 01:03:07.720
the coefficients of the
different equations describe
01:03:07.720 --> 01:03:11.080
the WallFinder behavior
were all real numbers.
01:03:16.050 --> 01:03:18.620
Here, I'm thinking about the
denominator polynomial.
01:03:18.620 --> 01:03:23.230
If we try to find the roots of a
polynomial, and if we find a
01:03:23.230 --> 01:03:28.430
complex root, if the
coefficients were all real, it
01:03:28.430 --> 01:03:33.150
follows that the complex
conjugate of the original root
01:03:33.150 --> 01:03:36.390
is also a root.
01:03:36.390 --> 01:03:38.210
That's pretty simple, if you
think about what it means to
01:03:38.210 --> 01:03:41.460
be a polynomial.
01:03:41.460 --> 01:03:45.020
If you think about a polynomial
is whatever--
01:03:45.020 --> 01:03:51.380
so I've got 1 plus Z plus Z
squared plus blah, blah, blah.
01:03:54.850 --> 01:04:04.230
2, 3 minus 16, if all of those
coefficients real, then the
01:04:04.230 --> 01:04:05.480
only way the--
01:04:10.530 --> 01:04:16.880
If P were a root of this
polynomial, then P* would have
01:04:16.880 --> 01:04:23.340
to be a root, too, because if
you complex conjugate each of
01:04:23.340 --> 01:04:25.790
the Z's, it's the same thing
as complex conjugating the
01:04:25.790 --> 01:04:30.970
whole thing because the
coefficients are real valued.
01:04:30.970 --> 01:04:36.780
So the idea then is that if I
happen to get a complex root
01:04:36.780 --> 01:04:39.530
for my system that can be
described by real value
01:04:39.530 --> 01:04:43.380
coefficients, it must also
be true that it's complex
01:04:43.380 --> 01:04:45.790
conjugate is a root.
01:04:45.790 --> 01:04:51.720
If that happens, the two roots
co-conspire so that the modes
01:04:51.720 --> 01:04:54.240
have canceling imaginary
parts.
01:04:54.240 --> 01:04:54.920
You can prove that.
01:04:54.920 --> 01:04:57.000
I'm not worried about you being
able to prove that.
01:04:57.000 --> 01:05:00.800
I just want you to understand
that if you have two roots
01:05:00.800 --> 01:05:06.080
that are complex conjugates,
they can conspire to have
01:05:06.080 --> 01:05:08.980
their imaginary parts
cancel, and that's
01:05:08.980 --> 01:05:10.120
exactly what happens.
01:05:10.120 --> 01:05:15.190
Here, in this example, the
example that I started with,
01:05:15.190 --> 01:05:18.860
where the system was 1 over (1
minus R plus R squared), the
01:05:18.860 --> 01:05:20.510
unit sample responses
showed here.
01:05:23.170 --> 01:05:29.350
You can write as the sum of
sinusoidal and cosinusoidal
01:05:29.350 --> 01:05:35.780
signals, and the sum that falls
out has the property
01:05:35.780 --> 01:05:39.490
that the imaginary
parts cancel.
01:05:39.490 --> 01:05:42.560
It's still useful to look at the
imaginary parts the same
01:05:42.560 --> 01:05:45.560
as it is when you're trying
to solve polynomials.
01:05:45.560 --> 01:05:49.230
It's useful because the
period of all of these
01:05:49.230 --> 01:05:50.490
signals is the same.
01:05:53.500 --> 01:05:56.900
If I think about the period of
the individual modes, if I
01:05:56.900 --> 01:06:00.380
think about the period of P0
to the n, 1/2 plus or minus
01:06:00.380 --> 01:06:04.740
root 3 over 2j to the n, the
period of that signal --
01:06:04.740 --> 01:06:07.810
I can see in the complex plane,
this is the n equals 0,
01:06:07.810 --> 01:06:13.000
1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12 --
01:06:13.000 --> 01:06:17.310
the period is 6.
01:06:17.310 --> 01:06:20.940
If I were to take the minus 1,
I would go around the circle
01:06:20.940 --> 01:06:22.220
the other way.
01:06:22.220 --> 01:06:26.850
This would be zero term 1,
2, 3, 4, 5, 6, et cetera.
01:06:26.850 --> 01:06:33.620
Both of the modes, the geometric
sequence associated
01:06:33.620 --> 01:06:37.670
with the two complex conjugate
poles, both of those modes had
01:06:37.670 --> 01:06:40.590
the same period, they're
both 6, as does the
01:06:40.590 --> 01:06:43.670
response to the real part.
01:06:43.670 --> 01:06:50.690
So you can deduce the period of
the real signal by looking
01:06:50.690 --> 01:06:54.055
at the periods of the
two complex signals.
01:06:57.440 --> 01:06:58.980
So think about this system.
01:06:58.980 --> 01:07:04.420
Here, I've got a system whose
responses showed here.
01:07:04.420 --> 01:07:07.740
I'm going to tell you that the
response was generated by a
01:07:07.740 --> 01:07:10.420
second-order system, that
is to say a system whose
01:07:10.420 --> 01:07:13.720
polynomial was second-order,
whose polynomial in R was
01:07:13.720 --> 01:07:15.440
second-order.
01:07:15.440 --> 01:07:17.375
Which of these statements
is true?
01:07:21.830 --> 01:07:25.730
I want to think about the pole
as being a complex number.
01:07:25.730 --> 01:07:29.640
Here, I'm showing you the
complex number in polar form.
01:07:29.640 --> 01:07:34.370
It's got a magnitude and an
angle, and I'd like you to
01:07:34.370 --> 01:07:38.210
figure how what must the
magnitude have been, and what
01:07:38.210 --> 01:07:39.997
must the angle have been.
01:08:40.140 --> 01:08:43.290
So what's the utility?
01:08:43.290 --> 01:08:46.553
Why did I tell you the pole in
terms of it's magnitude and
01:08:46.553 --> 01:08:51.310
angle rather than telling it to
you in it's Cartesian form
01:08:51.310 --> 01:08:54.100
as a real and imaginary part?
01:08:54.100 --> 01:08:58.266
What's good about thinking about
magnitude and angle?
01:09:02.577 --> 01:09:06.140
Or if I break the pole into
magnitude and angle, and if I
01:09:06.140 --> 01:09:07.390
think about the mode--
01:09:10.370 --> 01:09:14.870
the modes are always of
the form P0 to the n.
01:09:14.870 --> 01:09:20.010
If I think about modes as having
a magnitude and an
01:09:20.010 --> 01:09:25.680
angle, when I raise it
to the n, something
01:09:25.680 --> 01:09:29.090
very special happens.
01:09:29.090 --> 01:09:30.340
What happens?
01:09:32.830 --> 01:09:35.420
You can separate it.
01:09:35.420 --> 01:09:39.250
What is R e to the j omega
raised to the n?
01:09:39.250 --> 01:09:49.260
That's the same as R to the
n, e to the j n omega.
01:09:49.260 --> 01:09:55.040
It's the product of a real
thing times a very simple
01:09:55.040 --> 01:09:56.950
complex thing.
01:09:56.950 --> 01:09:58.660
What's simple about
the complex thing?
01:10:01.330 --> 01:10:05.050
The magnitude is everywhere
1 e to the j, the
01:10:05.050 --> 01:10:08.730
magnitude of that term.
01:10:08.730 --> 01:10:11.180
So all of the magnitude
is here, none of the
01:10:11.180 --> 01:10:12.210
magnitude is here.
01:10:12.210 --> 01:10:14.920
All of the angle is here, none
of the angle is here.
01:10:14.920 --> 01:10:18.270
I've separated the magnitude
and the angle.
01:10:18.270 --> 01:10:23.300
So it's very insightful to think
about poles in terms of
01:10:23.300 --> 01:10:28.910
magnitude and angle because
it decouples the
01:10:28.910 --> 01:10:31.080
parts of the mode.
01:10:31.080 --> 01:10:34.050
So I can think, then, of this
complicated signal that I gave
01:10:34.050 --> 01:10:39.350
you as being the product of a
magnitude part and a pure
01:10:39.350 --> 01:10:41.180
angle part.
01:10:41.180 --> 01:10:48.180
From the magnitude part, I can
infer something about R. R is
01:10:48.180 --> 01:10:54.480
the ratio of the nth 1 to the
n minus 1-th one, so R is a
01:10:54.480 --> 01:10:57.120
lot bigger than 1/2.
01:10:57.120 --> 01:11:04.390
In fact, R in this
case is 0.97.
01:11:04.390 --> 01:11:06.930
And this is pure angle.
01:11:06.930 --> 01:11:11.600
This lets me infer something
about the oscillations.
01:11:11.600 --> 01:11:13.530
In fact, I can say something
about the period.
01:11:13.530 --> 01:11:17.130
The period is, here's a peak,
here's a peak, 1, 2, 3, 4, 5,
01:11:17.130 --> 01:11:18.380
6, 7, 8, 9, 10, 11, 12.
01:11:21.460 --> 01:11:24.270
The period's 12.
01:11:24.270 --> 01:11:26.410
If the period is 12,
what is omega?
01:11:42.330 --> 01:11:47.350
Omega is a number, such that
by the time I got up to 12
01:11:47.350 --> 01:11:51.860
times it, I got up to 2 pi.
01:11:55.420 --> 01:11:56.070
What's omega?
01:11:56.070 --> 01:11:58.290
AUDIENCE: Pi over 6.
01:11:58.290 --> 01:11:59.540
PROFESSOR: 2 pi over 12.
01:12:02.180 --> 01:12:03.430
So it's about 1/2.
01:12:05.510 --> 01:12:10.710
So that's a way that I can infer
the form of the answer.
01:12:10.710 --> 01:12:15.660
I ask you what pole corresponds
to this behavior.
01:12:15.660 --> 01:12:18.570
Well, the decay that comes from
the real part, that comes
01:12:18.570 --> 01:12:21.580
from R to the n.
01:12:21.580 --> 01:12:26.100
The oscillation, that comes from
the imaginary part, and I
01:12:26.100 --> 01:12:28.400
can figure that out by thinking
about the period and
01:12:28.400 --> 01:12:29.650
thinking about that
relationship.
01:12:40.260 --> 01:12:43.720
So the answer, then, is this
one, R is between 1/2 and 1,
01:12:43.720 --> 01:12:47.040
and omega is about 1/2.
01:12:47.040 --> 01:12:50.150
OK, one last-- whoops,
wrong button.
01:12:50.150 --> 01:12:51.400
One last example.
01:12:54.330 --> 01:12:57.110
So what we've seen is a very
powerful way of decomposing
01:12:57.110 --> 01:13:01.650
systems, so that we can always
think about them in terms of
01:13:01.650 --> 01:13:06.420
poles and complex geometrics,
which we will call modes,
01:13:06.420 --> 01:13:09.070
poles and modes.
01:13:09.070 --> 01:13:12.530
And that behavior works for
any system in an enormous
01:13:12.530 --> 01:13:16.080
class of systems, so I want to
think about one last one,
01:13:16.080 --> 01:13:17.620
which is Fibonacci sequence.
01:13:17.620 --> 01:13:19.080
You all know the Fibonacci
sequence.
01:13:19.080 --> 01:13:20.260
You've all programmed it.
01:13:20.260 --> 01:13:22.640
We started with that when we
were doing Python, and we made
01:13:22.640 --> 01:13:24.800
some illustrations about the
recursion and all that sort of
01:13:24.800 --> 01:13:26.120
thing by thinking about it.
01:13:26.120 --> 01:13:27.190
Now, I'm going to
use signals and
01:13:27.190 --> 01:13:30.270
systems do the same problem.
01:13:30.270 --> 01:13:33.370
So Fibonacci was interested
in population growth.
01:13:33.370 --> 01:13:36.720
How many pairs of rabbits can be
produced from a single pair
01:13:36.720 --> 01:13:40.140
in a year if it is suppose that
every month each pair
01:13:40.140 --> 01:13:44.400
begets a new pair, from which
the second month it becomes
01:13:44.400 --> 01:13:45.040
productive?
01:13:45.040 --> 01:13:49.500
OK, it's not quite the same
English I would have used.
01:13:49.500 --> 01:13:54.470
From this statement, you can
infer a difference equation.
01:13:54.470 --> 01:13:56.910
I've written it in terms of
X. What do you think X is?
01:14:00.700 --> 01:14:05.740
X is the input signal, and here,
I'm thinking about X as
01:14:05.740 --> 01:14:09.960
something that's specifies
the initial condition.
01:14:09.960 --> 01:14:12.170
This is the thing I alluded
to earlier.
01:14:12.170 --> 01:14:16.780
One trick that we use to make it
easy to think about initial
01:14:16.780 --> 01:14:21.660
conditions is that we embed
them in the input.
01:14:21.660 --> 01:14:24.540
So in this particular case, I'll
think about the initial
01:14:24.540 --> 01:14:28.060
condition arising from
a delta function.
01:14:28.060 --> 01:14:33.630
If I think about X as a delta,
then the sequence of results
01:14:33.630 --> 01:14:37.590
Y0, Y1, Y2, Y3 from this
difference equation, is the
01:14:37.590 --> 01:14:39.860
conventional Fibonacci
sequence.
01:14:39.860 --> 01:14:44.590
It would correspond to what if
you had a baby rabbit, one
01:14:44.590 --> 01:14:48.330
baby rabbit, that's the
one, in generation
01:14:48.330 --> 01:14:52.030
0, that's the delta.
01:14:52.030 --> 01:14:54.960
So the input is a way that
I can specify initial
01:14:54.960 --> 01:14:57.530
conditions, and that's a very
powerful way of thinking about
01:14:57.530 --> 01:14:59.100
initial conditions.
01:14:59.100 --> 01:15:00.350
So here's the problem.
01:15:00.350 --> 01:15:04.370
I've got one set of baby
rabbits at times 0.
01:15:04.370 --> 01:15:07.370
They grow up.
01:15:07.370 --> 01:15:13.750
They have baby rabbits, which
grow up at the same time the
01:15:13.750 --> 01:15:19.980
parents had more baby rabbits,
at which point more babies
01:15:19.980 --> 01:15:22.110
grow into bigger rabbits.
01:15:22.110 --> 01:15:26.460
And big rabbits have more
babies, et cetera, et cetera,
01:15:26.460 --> 01:15:28.640
et cetera, et cetera,
et cetera.
01:15:33.520 --> 01:15:34.850
So you all know that.
01:15:34.850 --> 01:15:36.710
You all know about Fibonacci
sequence.
01:15:36.710 --> 01:15:38.390
It blows up very quickly.
01:15:38.390 --> 01:15:40.945
What are the poles of the
Fibonacci sequence?
01:15:49.920 --> 01:15:51.970
The difference equation looks
just like the difference
01:15:51.970 --> 01:15:53.560
equations we've looked
at throughout
01:15:53.560 --> 01:15:56.470
this hour and a half.
01:15:56.470 --> 01:16:00.320
We can do it just like we did
all the other problems.
01:16:00.320 --> 01:16:04.260
We write the difference equation
in terms of R. We
01:16:04.260 --> 01:16:08.340
rewrite the system functional
in terms of a ratio of two
01:16:08.340 --> 01:16:13.180
polynomials in R. We substitute
R goes to 1 over Z,
01:16:13.180 --> 01:16:18.340
deduce a ratio of polynomials in
Z, factor the denominator,
01:16:18.340 --> 01:16:20.810
find the roots of the
denominator, and find that
01:16:20.810 --> 01:16:24.050
there are two poles.
01:16:24.050 --> 01:16:27.830
The poles for the Fibonacci
sequence are plus or minus the
01:16:27.830 --> 01:16:30.400
root of 5 over 2.
01:16:30.400 --> 01:16:31.290
That's curious.
01:16:31.290 --> 01:16:33.330
There's no recursion there.
01:16:33.330 --> 01:16:36.310
It's a different way of
thinking about things.
01:16:36.310 --> 01:16:38.920
There are two poles.
01:16:38.920 --> 01:16:41.300
The first pole, the
plus one --
01:16:41.300 --> 01:16:46.540
1 plus the root of 5 over 1,
corresponds to a pole whose
01:16:46.540 --> 01:16:48.180
magnitude is bigger than 1.
01:16:48.180 --> 01:16:49.710
It explodes.
01:16:49.710 --> 01:16:51.970
There it is.
01:16:51.970 --> 01:16:55.170
The second one is a
negative number.
01:16:55.170 --> 01:17:00.770
So the first one is the
golden ratio, 1.618...
01:17:00.770 --> 01:17:03.070
The second one is the negative
reciprocal of the golden
01:17:03.070 --> 01:17:08.590
ratio, which is -0.618...
01:17:08.590 --> 01:17:14.710
And those two numbers,
amazingly, conspire so that
01:17:14.710 --> 01:17:21.240
their sum, horrendous as they
are, is an integer.
01:17:21.240 --> 01:17:25.330
And in fact, that's the integer
that we computed here.
01:17:25.330 --> 01:17:28.210
So we've used Fibonacci before
to think about the way you
01:17:28.210 --> 01:17:33.190
structure programs, iteration,
recursion, that sort of thing.
01:17:33.190 --> 01:17:35.030
Here, by thinking about signals
and systems, we can
01:17:35.030 --> 01:17:40.370
think about exactly the
same problem as poles.
01:17:40.370 --> 01:17:42.190
There's no complexity
in this problem.
01:17:42.190 --> 01:17:44.780
It doesn't take N or N
squared or N log N
01:17:44.780 --> 01:17:46.190
or anything to compute.
01:17:46.190 --> 01:17:48.470
It's closed form.
01:17:48.470 --> 01:17:53.100
The answer is (pole one) to the
N plus (pole two) to the
01:17:53.100 --> 01:17:54.400
N. That's it.
01:17:54.400 --> 01:17:55.500
That's the answer.
01:17:55.500 --> 01:17:57.810
So what we've done is we found
a whole new way of thinking
01:17:57.810 --> 01:18:02.160
about the Fibonacci sequence
in terms of poles, and more
01:18:02.160 --> 01:18:05.380
than that, we found that that
way of thinking about poles
01:18:05.380 --> 01:18:10.150
works for any difference
equation of this type.
01:18:10.150 --> 01:18:14.160
And we found that poles, this
way of thinking about systems
01:18:14.160 --> 01:18:18.190
in terms of polynomials, is a
powerful abstraction that's
01:18:18.190 --> 01:18:21.110
exactly the same kind
of PCAP abstraction
01:18:21.110 --> 01:18:22.360
that we used for Python.