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DENNIS FREEMAN:
Hello, and welcome.
00:00:29.950 --> 00:00:32.320
So before starting
today, there's
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one of those kind of required
announcements for events
00:00:36.010 --> 00:00:37.640
coming up.
00:00:37.640 --> 00:00:40.690
So even though it seems
like we just did exam 1,
00:00:40.690 --> 00:00:42.700
next week is exam 2.
00:00:42.700 --> 00:00:47.440
So next Wednesday
evening, 7:30 to 9:30.
00:00:47.440 --> 00:00:51.940
Same as before other
than it's Walker.
00:00:51.940 --> 00:00:54.850
So don't go to Building 26.
00:00:54.850 --> 00:00:58.010
Go to Walker.
00:00:58.010 --> 00:00:58.690
OK.
00:00:58.690 --> 00:01:01.420
Other than that, the rules
are pretty much the same.
00:01:01.420 --> 00:01:04.900
No recitations on
the day of the exam.
00:01:04.900 --> 00:01:08.680
Coverage will be up until
the end of this week.
00:01:08.680 --> 00:01:11.860
That includes Lecture
12, Recitation 12.
00:01:11.860 --> 00:01:14.260
It includes Homework
7, but Homework 7
00:01:14.260 --> 00:01:18.640
will be handled the
way Homework 4 was.
00:01:18.640 --> 00:01:20.500
So Homework 7
won't be collected.
00:01:20.500 --> 00:01:21.520
It won't be graded.
00:01:21.520 --> 00:01:24.700
There will be solutions posted.
00:01:24.700 --> 00:01:27.560
We'll post previous exams.
00:01:27.560 --> 00:01:30.790
You'll be allowed to use two
pages of notes, presumably
00:01:30.790 --> 00:01:33.640
the page you used last
time and one more.
00:01:33.640 --> 00:01:36.280
Although, we are not going to
check whether it was the page
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you used last time.
00:01:37.250 --> 00:01:41.030
This is supposed to
be a convenience.
00:01:41.030 --> 00:01:42.250
No calculators.
00:01:42.250 --> 00:01:44.470
No electronic devices.
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Just like last time--
00:01:45.610 --> 00:01:47.440
I hope you all found
it to be this way--
00:01:47.440 --> 00:01:49.540
the exam was designed
to be done in one hour.
00:01:49.540 --> 00:01:51.550
And you have two hours,
so there's not supposed
00:01:51.550 --> 00:01:54.760
to be any time pressure.
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Review sessions during the
normal open office hours.
00:01:57.970 --> 00:01:59.440
And conflicts,
please let me know
00:01:59.440 --> 00:02:03.440
so that I can arrange to
get somebody to proctor.
00:02:03.440 --> 00:02:05.340
And so that I can get a room.
00:02:05.340 --> 00:02:07.780
So for those two
reasons, try to tell me
00:02:07.780 --> 00:02:09.384
about conflicts before Friday.
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I guess the other
important thing
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is remember that the theory
of the exams in this class
00:02:15.850 --> 00:02:17.590
is that they ramp up.
00:02:17.590 --> 00:02:20.050
The first exam only
counted 10%, with the idea
00:02:20.050 --> 00:02:22.210
that that's to let you
get acclimated to things.
00:02:22.210 --> 00:02:24.460
You're sort of walking in
the shallow end of the pool
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is the idea.
00:02:26.080 --> 00:02:28.310
This one will count 15%.
00:02:28.310 --> 00:02:29.410
So it's coming up.
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The next one will count 20.
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And the final one will count 40.
00:02:33.760 --> 00:02:35.920
So this one counts slightly
more than the last one,
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but you're still kind of in the
not-so-deep end of the pool.
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Questions?
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Comments?
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Questions on the exams?
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Good.
00:02:49.000 --> 00:02:52.120
So I want to start talking
about a new topic today, which
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is really an elaboration
of what we talked about
00:02:54.250 --> 00:02:55.776
with frequency response.
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But I want to start by
telling you the big picture,
00:02:57.900 --> 00:03:01.930
the sort of 30,000-foot view.
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We've been focusing
in this class
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on multiple representations
for thinking
00:03:05.560 --> 00:03:08.440
about how linear systems work.
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Differential equations, block
diagrams, impulse responses,
00:03:13.330 --> 00:03:15.430
frequency response-- all
kinds of different ways
00:03:15.430 --> 00:03:19.200
of thinking about it, with
the idea that if you know all
00:03:19.200 --> 00:03:22.240
of them, then you can use the
one that's the easiest to solve
00:03:22.240 --> 00:03:24.610
your particular problem.
00:03:24.610 --> 00:03:26.380
In almost all of
the cases, we've
00:03:26.380 --> 00:03:29.740
tried to find
representations that are
00:03:29.740 --> 00:03:32.465
small in conceptual complexity.
00:03:32.465 --> 00:03:34.090
So for example,
differential equations.
00:03:34.090 --> 00:03:37.010
In principle, differential
equations could be complicated.
00:03:37.010 --> 00:03:39.430
In fact, we're only interested
in linear time invariant
00:03:39.430 --> 00:03:41.860
differential equations,
which means that they always
00:03:41.860 --> 00:03:43.090
have a simple form.
00:03:43.090 --> 00:03:47.650
They always look something like
y plus some coefficient times y
00:03:47.650 --> 00:03:50.750
dot plus some other
coefficient times y dot.
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y double dot plus a whole bunch
of things turns into say b0
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x plus b1 x dot plus b2 x dot--
double dot plus a whole bunch
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of things.
00:04:03.400 --> 00:04:06.490
Point being that although the
differential equations could,
00:04:06.490 --> 00:04:08.140
in fact, be arbitrarily
complicated,
00:04:08.140 --> 00:04:09.640
we only focus on
the ones that have
00:04:09.640 --> 00:04:16.329
a simple representation in terms
of a small number of constants.
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There they are-- a1,
a2, a3, b0, b1, b2.
00:04:21.490 --> 00:04:23.960
Small number of constants.
00:04:23.960 --> 00:04:26.410
So that's the way of
managing the complexity.
00:04:26.410 --> 00:04:28.360
There's only a small
number of constants
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you need to worry about.
00:04:29.620 --> 00:04:32.320
Similarly, when we think about
things like poles and zeros,
00:04:32.320 --> 00:04:36.610
the whole point of a pole-zero
diagram is that you reduce.
00:04:36.610 --> 00:04:39.290
So say we're doing s-plane.
00:04:39.290 --> 00:04:42.190
So say we're doing some
continuous kind of a problem.
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We reduced the whole system to
thinking about a small number
00:04:48.190 --> 00:04:50.830
of singularities.
00:04:50.830 --> 00:04:53.740
Again, we're trying to think
about the system in terms
00:04:53.740 --> 00:04:56.650
of a small number of numbers.
00:04:56.650 --> 00:05:03.820
We like this one because the
numbers are in some sense more
00:05:03.820 --> 00:05:06.220
intuitive or help us to
solve problems better
00:05:06.220 --> 00:05:08.290
than perhaps these numbers did.
00:05:08.290 --> 00:05:10.330
Both of these representations
are characterized
00:05:10.330 --> 00:05:11.538
by a small number of numbers.
00:05:14.140 --> 00:05:17.070
But for certain purposes,
one set of numbers
00:05:17.070 --> 00:05:20.730
is easier to work with than
another set of numbers.
00:05:20.730 --> 00:05:25.230
Last week, we started to think
about frequency responses.
00:05:25.230 --> 00:05:28.040
Frequency responses
were really good
00:05:28.040 --> 00:05:30.210
for thinking about
systems, like audio.
00:05:30.210 --> 00:05:32.370
I did some examples with audio.
00:05:32.370 --> 00:05:35.726
Or just other kinds of systems.
00:05:35.726 --> 00:05:37.350
It's very good to
think about this kind
00:05:37.350 --> 00:05:39.030
of a system as a
frequency response
00:05:39.030 --> 00:05:42.150
because there's a certain
frequency at which
00:05:42.150 --> 00:05:44.010
a very small input--
00:05:44.010 --> 00:05:45.480
you can barely see
my hand moving.
00:05:45.480 --> 00:05:50.610
Or, if I weren't shaking,
it wouldn't move so much.
00:05:50.610 --> 00:05:54.510
And in fact, the
mass moves a lot.
00:05:54.510 --> 00:05:59.670
So that's some unique property
of a particular frequency
00:05:59.670 --> 00:06:02.550
that's interesting
for us to know about.
00:06:02.550 --> 00:06:04.140
The problem with
frequency responses
00:06:04.140 --> 00:06:09.840
is that when we start to think
about a frequency response,
00:06:09.840 --> 00:06:15.660
we're now thinking about,
say, the magnitude of a system
00:06:15.660 --> 00:06:18.660
function as a function
of omega, which
00:06:18.660 --> 00:06:24.030
might be horribly complicated,
and an angle which
00:06:24.030 --> 00:06:26.400
might be horribly complicated.
00:06:26.400 --> 00:06:28.045
So we've kind of
lost the modularity.
00:06:28.045 --> 00:06:29.670
We've kind of lost
the ability to think
00:06:29.670 --> 00:06:31.086
about the whole
frequency response
00:06:31.086 --> 00:06:32.340
by a handful of numbers.
00:06:32.340 --> 00:06:34.080
That's what Bode plots are.
00:06:34.080 --> 00:06:36.870
Bode plots are a way of thinking
about the entire frequency
00:06:36.870 --> 00:06:39.810
response in terms of
a handful of numbers.
00:06:39.810 --> 00:06:41.550
That's why we like it.
00:06:41.550 --> 00:06:44.820
We like frequency response
because it's a very good way
00:06:44.820 --> 00:06:47.790
to think about certain systems.
00:06:47.790 --> 00:06:50.850
Audio because we like to
think about bass differently
00:06:50.850 --> 00:06:54.040
from the way we
think about treble.
00:06:54.040 --> 00:06:57.240
Mass-spring dashpots because
there are certain frequencies
00:06:57.240 --> 00:07:00.150
at which the system goes crazy.
00:07:00.150 --> 00:07:04.290
Airplanes because you don't
want it to do that, right?
00:07:04.290 --> 00:07:08.250
So there are certain reasons why
frequency responses are good.
00:07:08.250 --> 00:07:11.100
And Bode plots are a way of
getting back to this idea
00:07:11.100 --> 00:07:13.530
that we can think about a
whole function of frequency
00:07:13.530 --> 00:07:15.390
with just a handful of numbers.
00:07:15.390 --> 00:07:16.720
That's what we're doing.
00:07:16.720 --> 00:07:17.220
OK.
00:07:17.220 --> 00:07:21.630
So just to get things
going, remember
00:07:21.630 --> 00:07:23.310
where we're coming from.
00:07:23.310 --> 00:07:26.970
We think about
frequency responses
00:07:26.970 --> 00:07:29.880
by thinking about
eigenfunctions and eigenvalues.
00:07:29.880 --> 00:07:32.640
We think about how if your
system is linear and time
00:07:32.640 --> 00:07:37.770
invariant, last week we showed
that complex exponentials are
00:07:37.770 --> 00:07:40.730
eigenfunctions for
all such systems.
00:07:40.730 --> 00:07:43.980
You put in any complex
exponential and what comes out
00:07:43.980 --> 00:07:49.020
is a weighted version of the
same complex exponential.
00:07:49.020 --> 00:07:52.740
Property of linear time
invariant systems Furthermore,
00:07:52.740 --> 00:07:57.210
the eigenvalues are really easy
to find from vector diagrams.
00:07:57.210 --> 00:08:01.620
The eigenvalue is the value of
the system function evaluated
00:08:01.620 --> 00:08:04.080
at the frequency of the
complex exponential.
00:08:07.360 --> 00:08:08.939
Really easy.
00:08:08.939 --> 00:08:11.230
And you can calculate that
from a vector diagram, which
00:08:11.230 --> 00:08:15.780
capitalizes on the small number
of numbers representation
00:08:15.780 --> 00:08:18.780
of the pole-zero plot.
00:08:18.780 --> 00:08:19.690
So it's all easy.
00:08:19.690 --> 00:08:21.540
We only do easy things.
00:08:21.540 --> 00:08:23.700
And then, we're
really interested
00:08:23.700 --> 00:08:27.280
in these sinusoidal responses.
00:08:27.280 --> 00:08:31.350
So we think about a sinusoid
by using Euler's formula
00:08:31.350 --> 00:08:34.380
as the sum of two
complex exponentials.
00:08:34.380 --> 00:08:37.020
And that lets us see
that all we need to do
00:08:37.020 --> 00:08:41.370
is evaluate the system
function at j omega 0,
00:08:41.370 --> 00:08:44.610
j times the frequency
of interest,
00:08:44.610 --> 00:08:46.560
the magnitude, and phase.
00:08:46.560 --> 00:08:48.870
And that lets us
compute, how much
00:08:48.870 --> 00:08:50.850
is the amplification
at that frequency
00:08:50.850 --> 00:08:57.150
and how much is the time delay
associated with that frequency?
00:08:57.150 --> 00:09:01.290
And so that motivated
us to look at, how
00:09:01.290 --> 00:09:04.560
does frequency response
map to pole-zero?
00:09:04.560 --> 00:09:07.140
There's a simple mapping,
the vector diagram.
00:09:07.140 --> 00:09:10.920
Think about for example, if
you have an isolated zero,
00:09:10.920 --> 00:09:12.675
a single zero in your system.
00:09:12.675 --> 00:09:17.030
A Single zero here at minus 2.
00:09:17.030 --> 00:09:19.640
The magnitude can be determined
by the magnitude of the--
00:09:19.640 --> 00:09:22.880
by the length of the
angle that goes from the 0
00:09:22.880 --> 00:09:24.380
to the frequency of interest.
00:09:24.380 --> 00:09:26.060
Frequencies are on
the j omega axis.
00:09:28.670 --> 00:09:30.320
Angle can be found--
00:09:30.320 --> 00:09:32.870
this angle can be
computed from the angle
00:09:32.870 --> 00:09:34.760
that this vector
makes with the x-axis.
00:09:38.250 --> 00:09:39.680
And that always works.
00:09:39.680 --> 00:09:42.101
Here it's illustrated
for a single zero.
00:09:42.101 --> 00:09:43.850
If you think about the
magnitude and angle
00:09:43.850 --> 00:09:47.090
as a function of
frequency, you map out
00:09:47.090 --> 00:09:49.190
features that look like so.
00:09:49.190 --> 00:09:52.760
As the frequency goes
from 0 to very big,
00:09:52.760 --> 00:09:55.610
the arrow goes from
short to very long.
00:09:55.610 --> 00:09:58.790
That's manifested by this.
00:09:58.790 --> 00:10:00.410
And the angle changes
systematically
00:10:00.410 --> 00:10:06.341
from being near 0 to
being up around pi over 2.
00:10:06.341 --> 00:10:06.840
OK.
00:10:06.840 --> 00:10:08.510
And the same sort of
thing works for a pole,
00:10:08.510 --> 00:10:09.968
except now a pole
is in the bottom.
00:10:13.310 --> 00:10:17.690
So when it used to get
big, now it gets small.
00:10:17.690 --> 00:10:20.510
The angle, which used to go
positive, now goes negative.
00:10:20.510 --> 00:10:22.580
But it's the same idea.
00:10:22.580 --> 00:10:25.970
And you can compose
more elaborate functions
00:10:25.970 --> 00:10:28.670
by thinking about the
individual arrows that
00:10:28.670 --> 00:10:29.885
correspond to factors.
00:10:29.885 --> 00:10:33.020
That's using the factor
theorem to reduce the system
00:10:33.020 --> 00:10:38.750
function, which is always a
polynomial in s, into factors.
00:10:38.750 --> 00:10:39.250
OK.
00:10:39.250 --> 00:10:41.722
So that was what we
talked about last time
00:10:41.722 --> 00:10:43.180
what I want to talk
about today now
00:10:43.180 --> 00:10:45.550
is how to reduce the
frequency response, which
00:10:45.550 --> 00:10:47.470
looks like a lot of numbers--
00:10:47.470 --> 00:10:50.410
one for every frequency-- to
thinking about it in terms
00:10:50.410 --> 00:10:52.840
of a small number of numbers.
00:10:52.840 --> 00:10:58.257
So the idea is that this
magnitude response is simple
00:10:58.257 --> 00:10:59.590
if you look at it the right way.
00:11:02.390 --> 00:11:07.310
In particular, it has a simple
value at low frequencies.
00:11:07.310 --> 00:11:09.860
If we think about the smallest
frequencies you can have,
00:11:09.860 --> 00:11:11.840
that's omega equals 0.
00:11:11.840 --> 00:11:14.570
And the smallest
frequencies, the magnitude
00:11:14.570 --> 00:11:19.700
response asymptotes becomes
arbitrarily close to the line
00:11:19.700 --> 00:11:21.590
that you would get
by substituting omega
00:11:21.590 --> 00:11:23.900
equals 0 here.
00:11:23.900 --> 00:11:29.070
If omega were 0, then h of
j omega would be minus z1.
00:11:29.070 --> 00:11:34.910
The magnitude of minus z1 is z1.
00:11:34.910 --> 00:11:37.820
So at very low frequencies,
the response-- so
00:11:37.820 --> 00:11:39.650
when you get to
frequencies near 0,
00:11:39.650 --> 00:11:41.690
the response asymptotes
to that line.
00:11:44.920 --> 00:11:49.080
And if you get to very high
frequencies, then the fact
00:11:49.080 --> 00:11:51.330
that you're adding a small
constant to a number that's
00:11:51.330 --> 00:11:55.260
already very big, if omega
were large, then that little
00:11:55.260 --> 00:11:58.600
constant, z1, wouldn't matter.
00:11:58.600 --> 00:12:01.900
So the magnitude becomes very
close to the magnitude of omega
00:12:01.900 --> 00:12:03.450
shown by the green dotted line.
00:12:05.980 --> 00:12:08.380
So you can think
about the blue line
00:12:08.380 --> 00:12:10.615
being some kind of an
interpolation between those two
00:12:10.615 --> 00:12:13.360
dash lines, the
low-frequency behavior
00:12:13.360 --> 00:12:14.920
to the high-frequency behavior.
00:12:14.920 --> 00:12:19.060
And the relation looks
even simpler if you plot it
00:12:19.060 --> 00:12:20.980
on a different kind of axis.
00:12:20.980 --> 00:12:24.280
If you make it
log-log, then you can
00:12:24.280 --> 00:12:27.840
think about frequencies going--
so the log of frequency.
00:12:27.840 --> 00:12:29.590
Well, there's some--
if the frequency
00:12:29.590 --> 00:12:31.320
were 1, whatever that means.
00:12:31.320 --> 00:12:33.370
Or in fact here, it's
important to note
00:12:33.370 --> 00:12:36.580
that I'm plotting here
versus a normalized version
00:12:36.580 --> 00:12:39.160
of frequency.
00:12:39.160 --> 00:12:40.750
I'm normalizing by z1.
00:12:40.750 --> 00:12:46.090
What That has the effect that
when omega is the same as z1,
00:12:46.090 --> 00:12:48.456
I get the log of 1, which is?
00:12:48.456 --> 00:12:49.170
AUDIENCE: 0.
00:12:49.170 --> 00:12:50.850
DENNIS FREEMAN: 0, right?
00:12:50.850 --> 00:12:54.030
That's a way of shifting
the axis to make
00:12:54.030 --> 00:12:55.770
the interesting
stuff happen at 0.
00:12:55.770 --> 00:12:56.700
That's all.
00:12:56.700 --> 00:13:02.130
So if you scale frequency by
the singularity-- in this case,
00:13:02.130 --> 00:13:05.740
a 0--
00:13:05.740 --> 00:13:09.490
then the critical frequency
becomes the frequency zero.
00:13:09.490 --> 00:13:12.580
We all like 0, right?
00:13:12.580 --> 00:13:14.980
So what you can see then
if you plot it this way--
00:13:14.980 --> 00:13:19.060
and I also scaled the
magnitude the same way.
00:13:19.060 --> 00:13:22.510
So if you realize that
there's a simple asymptote,
00:13:22.510 --> 00:13:24.970
you can see very clearly
on this log-log plot
00:13:24.970 --> 00:13:27.290
that you get a good
approximation to the function,
00:13:27.290 --> 00:13:29.110
which is showed in blue.
00:13:29.110 --> 00:13:32.320
That one's calculated exactly.
00:13:32.320 --> 00:13:36.280
And you get a good approximation
by the low- and high-frequency
00:13:36.280 --> 00:13:37.010
asymptotes.
00:13:37.010 --> 00:13:40.360
So instead of thinking about
the whole function of frequency,
00:13:40.360 --> 00:13:44.570
what we do is we think
about just the asymptotes.
00:13:44.570 --> 00:13:46.550
That's the simplification.
00:13:46.550 --> 00:13:49.340
We don't think about this
complicated function.
00:13:49.340 --> 00:13:51.855
We only think about what
happens at low frequencies
00:13:51.855 --> 00:13:52.730
and high frequencies.
00:13:52.730 --> 00:13:55.100
Of course, this was
too simple, right?
00:13:55.100 --> 00:14:00.110
This is what happens
with a single zero.
00:14:00.110 --> 00:14:01.580
The point is that
a similar thing
00:14:01.580 --> 00:14:05.010
happens with a single pole
or when you combine them.
00:14:05.010 --> 00:14:06.890
Let's look at the
case of a pole.
00:14:06.890 --> 00:14:09.680
Here, the pole idea is a
little bit more complicated.
00:14:09.680 --> 00:14:11.840
If you go to very
low frequencies,
00:14:11.840 --> 00:14:14.630
think about s being j omega.
00:14:14.630 --> 00:14:18.110
The frequency response
lives on the j omega axis.
00:14:18.110 --> 00:14:20.720
Always look at s equals
j omega if you're
00:14:20.720 --> 00:14:23.030
interested in the
frequency response.
00:14:23.030 --> 00:14:26.840
If you think about s
equals j omega, then
00:14:26.840 --> 00:14:31.940
if you go to small omega,
this term goes away.
00:14:31.940 --> 00:14:34.760
And we're left with,
again, a constant at low
00:14:34.760 --> 00:14:36.220
frequency showed
by the red line.
00:14:39.540 --> 00:14:41.820
If we look at high
frequencies, now the curve's
00:14:41.820 --> 00:14:44.880
a little more funky.
00:14:44.880 --> 00:14:48.960
The curve at high
frequencies-- the real answer
00:14:48.960 --> 00:14:52.290
is showed by this sort
of bell-shaped curve.
00:14:52.290 --> 00:14:57.312
If you plot for different values
of omega where s is j omega,
00:14:57.312 --> 00:14:59.020
plot out the magnitude,
you get this sort
00:14:59.020 --> 00:15:01.510
of funky, bell-shaped curve.
00:15:01.510 --> 00:15:04.940
The point is that--
00:15:04.940 --> 00:15:06.400
and the high-frequency
asymptote.
00:15:06.400 --> 00:15:09.190
If you just say, what
if j omega were so big
00:15:09.190 --> 00:15:11.800
that the p1 wouldn't matter?
00:15:11.800 --> 00:15:18.860
Then, you would get 9 over
omega being the asymptote.
00:15:18.860 --> 00:15:21.410
So that's this hyperbolic
sort of thing here.
00:15:21.410 --> 00:15:21.910
OK.
00:15:21.910 --> 00:15:23.740
Well, that's ugly.
00:15:23.740 --> 00:15:27.650
But if you go to log-log,
it's very simple.
00:15:27.650 --> 00:15:30.890
That's why we like log-log.
00:15:30.890 --> 00:15:35.990
If you think about the function
y' equals 1 over omega and plot
00:15:35.990 --> 00:15:40.190
that function on
log-log axes, reciprocal
00:15:40.190 --> 00:15:41.630
turns into a straight line.
00:15:41.630 --> 00:15:43.560
We like straight lines.
00:15:43.560 --> 00:15:45.590
They're easy.
00:15:45.590 --> 00:15:48.260
So what happens then-- again,
I've normalized things.
00:15:48.260 --> 00:15:52.100
I've normalized
frequency by the pole.
00:15:52.100 --> 00:15:56.360
That makes the critical
frequency be omega equals pole.
00:15:56.360 --> 00:15:58.850
The critical
frequency comes out 1.
00:15:58.850 --> 00:16:00.890
The log of 1 is 0.
00:16:00.890 --> 00:16:05.860
So the interesting behavior
happens at log equals 0.
00:16:08.710 --> 00:16:12.340
Below that frequency, the
frequency response magnitude
00:16:12.340 --> 00:16:17.440
is well approximated by the
constant whose log divided
00:16:17.440 --> 00:16:19.180
by some funny number
that was put there
00:16:19.180 --> 00:16:20.430
for normalization is 0.
00:16:24.320 --> 00:16:27.140
And at high frequencies,
it falls off
00:16:27.140 --> 00:16:30.300
with a slope of minus 1.
00:16:30.300 --> 00:16:32.090
So again, for the
case of the pole,
00:16:32.090 --> 00:16:37.610
we get this simple behavior
if we focus on the asymptotes.
00:16:37.610 --> 00:16:38.960
OK.
00:16:38.960 --> 00:16:42.140
So now, that's kind of the
theory behind everything.
00:16:42.140 --> 00:16:44.702
Make sure you're all up
to speed on the theory.
00:16:44.702 --> 00:16:47.380
So compare log-log
plots of the frequency
00:16:47.380 --> 00:16:49.630
response magnitudes of the
following system functions.
00:16:49.630 --> 00:16:52.810
H1, 1 over s plus 1.
00:16:52.810 --> 00:16:55.728
Where's the pole?
00:16:55.728 --> 00:16:57.100
AUDIENCE: [INAUDIBLE]
00:16:57.100 --> 00:16:58.860
DENNIS FREEMAN: Minus 1.
00:16:58.860 --> 00:17:01.080
And s2, 1 over s plus 10.
00:17:01.080 --> 00:17:02.750
Pole is?
00:17:02.750 --> 00:17:03.920
Minus 10.
00:17:03.920 --> 00:17:06.440
Compare the magnitude.
00:17:06.440 --> 00:17:07.880
Compare their
magnitude functions
00:17:07.880 --> 00:17:11.300
when plotted on log-log and
answer the following question,
00:17:11.300 --> 00:17:14.930
which of these best describes
the transformation between H1
00:17:14.930 --> 00:17:16.250
and H2?
00:17:16.250 --> 00:17:17.540
You should shift horizontally.
00:17:17.540 --> 00:17:19.730
You should shift and
scale horizontally.
00:17:19.730 --> 00:17:21.770
You should shift
horizontally and vertically.
00:17:21.770 --> 00:17:24.410
You should shift and scale
horizontally and vertically.
00:17:24.410 --> 00:17:28.050
Or, it's something
completely different.
00:17:28.050 --> 00:17:29.600
OK, turn to your neighbor.
00:17:29.600 --> 00:17:30.740
Say hi.
00:17:30.740 --> 00:17:33.530
Right now, say hi.
00:17:33.530 --> 00:17:36.080
And now, figure out
what answer best
00:17:36.080 --> 00:17:40.404
classifies that transformation.
00:17:40.404 --> 00:17:44.388
[SIDE CONVERSATIONS]
00:19:02.149 --> 00:19:02.940
DENNIS FREEMAN: OK.
00:19:02.940 --> 00:19:05.162
So which transformation
do you like the best?
00:19:05.162 --> 00:19:06.870
Everybody raise your
hand with the number
00:19:06.870 --> 00:19:08.845
of fingers between 1 and 5.
00:19:12.672 --> 00:19:14.130
Let me make sure
I know the answer.
00:19:14.130 --> 00:19:15.960
I think I know the answer.
00:19:15.960 --> 00:19:16.770
OK.
00:19:16.770 --> 00:19:19.590
Greater audience participation.
00:19:19.590 --> 00:19:20.730
Blame it on your neighbor.
00:19:20.730 --> 00:19:23.625
My stupid partner
thought it was--
00:19:27.935 --> 00:19:28.435
OK.
00:19:28.435 --> 00:19:31.000
It's about 90% correct.
00:19:33.970 --> 00:19:38.590
If I plot the magnitude
response for H1 on log-log axes,
00:19:38.590 --> 00:19:40.490
what will it look like?
00:19:40.490 --> 00:19:41.650
Sketch it in the air.
00:19:44.480 --> 00:19:47.190
Sketch it in the air is
like that sort of thing.
00:19:50.290 --> 00:19:50.790
Go ahead.
00:19:50.790 --> 00:19:51.610
Go ahead.
00:19:51.610 --> 00:19:52.110
Sketch.
00:19:54.710 --> 00:19:57.060
OK.
00:19:57.060 --> 00:19:58.730
Maybe that didn't work.
00:19:58.730 --> 00:19:59.230
OK.
00:19:59.230 --> 00:20:05.160
What I want to do is plot the
magnitude of H1 of j omega
00:20:05.160 --> 00:20:10.145
log versus the log of omega.
00:20:10.145 --> 00:20:13.590
But I only want to think
about the asymptotes.
00:20:13.590 --> 00:20:16.680
Where should I draw the
low-frequency asymptote
00:20:16.680 --> 00:20:18.600
for that function?
00:20:18.600 --> 00:20:20.550
What's the
low-frequency asymptote
00:20:20.550 --> 00:20:23.360
for the log magnitude of H1?
00:20:33.940 --> 00:20:36.270
What's the low-frequency
asymptote--
00:20:36.270 --> 00:20:37.300
forget the log part.
00:20:37.300 --> 00:20:39.330
What's the low-frequency
asymptote for H1?
00:20:41.720 --> 00:20:42.220
0.
00:20:46.290 --> 00:20:46.982
Yes.
00:20:46.982 --> 00:20:49.877
AUDIENCE: [INAUDIBLE]
00:20:49.877 --> 00:20:50.960
DENNIS FREEMAN: I'm sorry.
00:20:50.960 --> 00:20:51.380
I can't hear.
00:20:51.380 --> 00:20:52.796
AUDIENCE: Why is
it not symmetric?
00:20:52.796 --> 00:21:00.165
So if you look at the amplitude
of the system function, j
00:21:00.165 --> 00:21:01.516
omega, it's symmetric.
00:21:01.516 --> 00:21:05.432
But the Bode plots
are not symmetric.
00:21:05.432 --> 00:21:07.390
DENNIS FREEMAN: The Bode
plots are unsymmetric.
00:21:07.390 --> 00:21:09.556
Why are the Bode plots
unsymmetric and the frequency
00:21:09.556 --> 00:21:10.930
response was symmetric?
00:21:10.930 --> 00:21:11.770
Somebody--
00:21:11.770 --> 00:21:14.020
AUDIENCE: I get why, if you
have a negative frequency.
00:21:14.020 --> 00:21:16.210
DENNIS FREEMAN: Negative
frequency, exactly.
00:21:16.210 --> 00:21:18.875
Why do we have
negative frequencies?
00:21:18.875 --> 00:21:19.750
AUDIENCE: [INAUDIBLE]
00:21:19.750 --> 00:21:23.696
DENNIS FREEMAN: Because
time runs backwards.
00:21:23.696 --> 00:21:26.070
Negative frequency corresponds
to running time backwards,
00:21:26.070 --> 00:21:26.850
right?
00:21:26.850 --> 00:21:30.150
The argument, it gets bigger
when time gets smaller.
00:21:30.150 --> 00:21:30.780
OK.
00:21:30.780 --> 00:21:32.550
That's not the reason
we do it, right?
00:21:32.550 --> 00:21:36.798
Why do we use
negative frequencies?
00:21:36.798 --> 00:21:38.270
AUDIENCE: [INAUDIBLE]
00:21:38.270 --> 00:21:41.752
DENNIS FREEMAN: Why do we
consider negative frequencies?
00:21:41.752 --> 00:21:42.700
AUDIENCE: [INAUDIBLE]
00:21:42.700 --> 00:21:44.033
DENNIS FREEMAN: To make it real.
00:21:44.033 --> 00:21:45.166
To make what real?
00:21:45.166 --> 00:21:46.812
AUDIENCE: A system.
00:21:46.812 --> 00:21:49.270
DENNIS FREEMAN: Why do we think
about negative frequencies?
00:21:49.270 --> 00:21:49.770
Come on.
00:21:49.770 --> 00:21:50.348
Yeah.
00:21:50.348 --> 00:21:52.340
AUDIENCE: [INAUDIBLE]
00:21:52.340 --> 00:21:54.590
DENNIS FREEMAN: We
want sines and cosines.
00:21:54.590 --> 00:21:57.320
It's easy to find
complex exponentials.
00:21:57.320 --> 00:21:59.930
We like Euler, right?
00:21:59.930 --> 00:22:00.800
We like Euler.
00:22:00.800 --> 00:22:04.910
We like that cosine
of omega t can
00:22:04.910 --> 00:22:08.115
be written as 1/2 e to
the j omega t plus 1/2
00:22:08.115 --> 00:22:11.060
e to the minus j omega t.
00:22:11.060 --> 00:22:12.502
That's what we like.
00:22:12.502 --> 00:22:13.770
Whoops.
00:22:13.770 --> 00:22:17.050
I shouldn't put the
parentheses there.
00:22:17.050 --> 00:22:19.410
That's what we like.
00:22:19.410 --> 00:22:21.357
We invent the
negative frequencies.
00:22:21.357 --> 00:22:23.190
They are completely an
invention of our own.
00:22:23.190 --> 00:22:26.940
When we do the Bode
plots, we throw them away.
00:22:26.940 --> 00:22:30.610
We know that we can
always figure them out
00:22:30.610 --> 00:22:32.980
because it was an
invention of ours,
00:22:32.980 --> 00:22:34.480
so we don't need
to write them down.
00:22:34.480 --> 00:22:36.370
If we ever needed them,
we could figure them out.
00:22:36.370 --> 00:22:38.740
So we don't bother to write
them down on the Bode plot.
00:22:38.740 --> 00:22:40.850
What's the low-frequency
asymptote of H1?
00:22:43.610 --> 00:22:48.120
If omega 0 goes to 0,
what's the value of H1?
00:22:48.120 --> 00:22:50.130
1.
00:22:50.130 --> 00:22:53.130
s equals j omega,
omega equals 0.
00:22:53.130 --> 00:22:53.920
1.
00:22:53.920 --> 00:22:55.520
What's the log of 1?
00:22:55.520 --> 00:22:56.380
AUDIENCE: 0.
00:22:56.380 --> 00:22:57.280
DENNIS FREEMAN: 0.
00:22:57.280 --> 00:23:01.310
So the low-frequency
asymptote for H1 is 0.
00:23:01.310 --> 00:23:03.150
What's the high-frequency
asymptote of H1?
00:23:07.030 --> 00:23:11.168
If you put a high frequency-
00:23:11.168 --> 00:23:14.240
AUDIENCE: [INAUDIBLE]
00:23:14.240 --> 00:23:17.810
DENNIS FREEMAN: We're
thinking about H1 of j omega.
00:23:17.810 --> 00:23:22.790
H1 of j omega is 1
over j omega plus 1.
00:23:22.790 --> 00:23:24.800
What happens when
you make omega high?
00:23:24.800 --> 00:23:27.900
What happens to the magnitude?
00:23:27.900 --> 00:23:29.370
AUDIENCE: [INAUDIBLE]
00:23:29.370 --> 00:23:33.660
DENNIS FREEMAN: 1 over omega
for the amplitude, right?
00:23:33.660 --> 00:23:35.159
You could say it approaches 0.
00:23:35.159 --> 00:23:37.200
That's kind of right, but
we're interested in how
00:23:37.200 --> 00:23:39.120
it approaches 0.
00:23:39.120 --> 00:23:43.950
On a log plot, 0
is very far away.
00:23:43.950 --> 00:23:47.190
So we never think
about being there.
00:23:47.190 --> 00:23:49.480
We think about getting there.
00:23:49.480 --> 00:23:52.380
So we get there with
a slope of minus 1.
00:23:52.380 --> 00:23:53.250
So it's like this.
00:23:53.250 --> 00:23:55.770
What's the crossover point?
00:23:55.770 --> 00:23:56.400
Omega equals?
00:23:59.870 --> 00:24:02.960
What's the omega at
the crossover point?
00:24:02.960 --> 00:24:04.190
AUDIENCE: 0.
00:24:04.190 --> 00:24:04.940
DENNIS FREEMAN: 0.
00:24:08.405 --> 00:24:12.860
AUDIENCE: We're plotting
for negative [INAUDIBLE]..
00:24:12.860 --> 00:24:14.510
And I would think
that since we're
00:24:14.510 --> 00:24:17.315
taking the absolute
value [INAUDIBLE]
00:24:17.315 --> 00:24:20.780
and not having it just be 0.
00:24:20.780 --> 00:24:23.255
Because you're talking
about as omega goes to 0,
00:24:23.255 --> 00:24:25.740
but not necessarily as
omega goes to negative.
00:24:25.740 --> 00:24:28.190
DENNIS FREEMAN: We're
thinking about the limit
00:24:28.190 --> 00:24:30.200
as we go towards
0, because we're
00:24:30.200 --> 00:24:32.420
plotting on the log of omega.
00:24:32.420 --> 00:24:39.170
0, omega equals 0,
is way over there.
00:24:39.170 --> 00:24:41.442
Way, way, way off
the blackboard.
00:24:41.442 --> 00:24:42.650
So we don't think about that.
00:24:42.650 --> 00:24:45.380
We think about how it got there.
00:24:45.380 --> 00:24:47.360
And it got there
by being constant.
00:24:49.950 --> 00:24:52.770
The trend-- if you start
at the critical frequency
00:24:52.770 --> 00:24:57.990
in the middle, the value
becomes closer and closer to 1
00:24:57.990 --> 00:25:00.400
as you go to the left.
00:25:00.400 --> 00:25:05.280
That's why we say the
asymptote is at log equals 0.
00:25:05.280 --> 00:25:06.450
Log of 1 is 0.
00:25:06.450 --> 00:25:09.500
The answer goes to 1.
00:25:09.500 --> 00:25:10.490
The log of 1 is 0.
00:25:10.490 --> 00:25:12.850
Does everybody know
what I'm talking about?
00:25:12.850 --> 00:25:15.540
I'm getting a lot
of blank stares.
00:25:15.540 --> 00:25:19.020
Then, if you try to
go to a high frequency
00:25:19.020 --> 00:25:23.290
and think about
the magnitude, you
00:25:23.290 --> 00:25:24.670
can go to a high
enough frequency
00:25:24.670 --> 00:25:26.128
where the magnitude
of this doesn't
00:25:26.128 --> 00:25:29.020
matter that that's there.
00:25:29.020 --> 00:25:31.900
The magnitude never matters
that there's a j there.
00:25:31.900 --> 00:25:35.380
So the magnitude goes
like 1 over omega.
00:25:35.380 --> 00:25:37.480
If you plot the
function, 1 over omega,
00:25:37.480 --> 00:25:39.670
a reciprocal function
on log coordinates,
00:25:39.670 --> 00:25:43.510
it turns into slope of minus 1.
00:25:43.510 --> 00:25:45.280
So that gives us
this region where
00:25:45.280 --> 00:25:48.687
it has a slope of minus 1.
00:25:48.687 --> 00:25:50.395
What's the frequency
at which they cross?
00:25:55.012 --> 00:25:56.446
AUDIENCE: [INAUDIBLE].
00:25:56.446 --> 00:25:59.320
In log scale, it's
only [INAUDIBLE]..
00:25:59.320 --> 00:26:01.630
DENNIS FREEMAN: On a log
scale, it's omega equals 0.
00:26:01.630 --> 00:26:06.770
On a linear scale,
it's omega equals 1.
00:26:06.770 --> 00:26:10.970
So the crossover point
where 1 over omega--
00:26:10.970 --> 00:26:13.460
the high-frequency
asymptote is 1 ever omega.
00:26:13.460 --> 00:26:15.920
The low-frequency asymptote's 1.
00:26:15.920 --> 00:26:20.480
They are equal when 1
over omega equals 1.
00:26:20.480 --> 00:26:23.350
Omega equals 1, right?
00:26:23.350 --> 00:26:24.860
Log 0.
00:26:24.860 --> 00:26:29.600
So they cross at a point
where the log is 0.
00:26:29.600 --> 00:26:31.640
Now, how about H2?
00:26:31.640 --> 00:26:34.040
What if I plot H2
on top of this?
00:26:34.040 --> 00:26:36.180
What's the low-frequency
limit of H2?
00:26:39.568 --> 00:26:40.540
AUDIENCE: [INAUDIBLE]
00:26:40.540 --> 00:26:43.790
DENNIS FREEMAN: 1/10.
00:26:43.790 --> 00:26:46.760
What's the log of 1/10?
00:26:46.760 --> 00:26:48.000
Minus 1.
00:26:48.000 --> 00:26:57.160
So now, H2, if I do log
of H2, it's going to--
00:26:57.160 --> 00:26:59.800
so the low-frequency
asymptote is 1/10.
00:26:59.800 --> 00:27:02.290
It's minus 1 on a log plot.
00:27:02.290 --> 00:27:04.510
What's the high-frequency
asymptote for H2?
00:27:07.365 --> 00:27:10.850
AUDIENCE: [INAUDIBLE]
00:27:10.850 --> 00:27:11.870
DENNIS FREEMAN: 1/s.
00:27:11.870 --> 00:27:14.210
1 over omega.
00:27:14.210 --> 00:27:14.960
It's the same.
00:27:18.100 --> 00:27:20.710
If you go to a high
enough frequency,
00:27:20.710 --> 00:27:23.470
the fact that there
is divide by 10--
00:27:23.470 --> 00:27:27.220
the fact that you've added
10 or added 1 doesn't matter.
00:27:27.220 --> 00:27:29.770
If omega gets sufficiently
high, the thing you added to it
00:27:29.770 --> 00:27:31.810
has no relevance.
00:27:31.810 --> 00:27:33.880
You get the same
high-frequency asymptote.
00:27:38.310 --> 00:27:43.830
So to transform the asymptotic
view of the magnitude function
00:27:43.830 --> 00:27:48.375
from H1 to H2, what should
you do to the curves?
00:27:48.375 --> 00:27:49.770
[INTERPOSING VOICES]
00:27:49.770 --> 00:27:51.660
DENNIS FREEMAN: Shift and shift.
00:27:51.660 --> 00:27:53.640
No scaling.
00:27:53.640 --> 00:27:54.930
That's why we like Bode.
00:27:54.930 --> 00:27:57.850
That's why we like log-log.
00:27:57.850 --> 00:28:01.770
Every pole looks the same.
00:28:01.770 --> 00:28:04.050
Every pole is straight
across and down.
00:28:04.050 --> 00:28:09.382
Every zero looks the same,
straight across and up.
00:28:09.382 --> 00:28:10.590
There's a critical frequency.
00:28:10.590 --> 00:28:13.920
That frequency is the
frequency equal to the position
00:28:13.920 --> 00:28:15.780
of the pole or zero.
00:28:15.780 --> 00:28:18.941
That's all of the rules.
00:28:18.941 --> 00:28:19.440
OK.
00:28:19.440 --> 00:28:21.060
That's why we like this.
00:28:21.060 --> 00:28:21.940
Does that make sense?
00:28:21.940 --> 00:28:25.900
So the answer here
was that we want
00:28:25.900 --> 00:28:27.900
to be able to shift
horizontally and vertically.
00:28:27.900 --> 00:28:30.390
We don't need to scale.
00:28:30.390 --> 00:28:34.080
The shape of the curve is
invariant to the position
00:28:34.080 --> 00:28:35.670
of the pole and zero.
00:28:35.670 --> 00:28:36.600
That's what we like.
00:28:36.600 --> 00:28:37.780
It's an invariant.
00:28:37.780 --> 00:28:40.820
We like invariant things.
00:28:40.820 --> 00:28:42.550
OK.
00:28:42.550 --> 00:28:48.290
Then, if you wanted to construct
a more complicated system,
00:28:48.290 --> 00:28:50.205
it's also easy.
00:28:50.205 --> 00:28:52.330
If you want to construct
a more complicated system,
00:28:52.330 --> 00:28:56.920
you can use the factor theorem
to convert the system function,
00:28:56.920 --> 00:29:02.140
which for a system that's built
out of integrators, adders,
00:29:02.140 --> 00:29:04.210
and gains.
00:29:04.210 --> 00:29:05.730
If you build a
system out of that,
00:29:05.730 --> 00:29:10.600
the system function will always
be a ratio of polynomials in s.
00:29:10.600 --> 00:29:13.180
We proved that a while back.
00:29:13.180 --> 00:29:16.780
You can always
factor such things.
00:29:16.780 --> 00:29:19.030
We just found out a
rule for how you think
00:29:19.030 --> 00:29:21.350
about the individual factors.
00:29:21.350 --> 00:29:23.290
And so all you need
to worry about is
00:29:23.290 --> 00:29:25.870
the rule for combination.
00:29:25.870 --> 00:29:27.880
If you multiply a bunch
of zeros and divide
00:29:27.880 --> 00:29:34.240
by the product of a bunch of
poles, that's pretty easy.
00:29:34.240 --> 00:29:37.829
If you take the magnitude--
00:29:37.829 --> 00:29:40.370
the magnitude of a product is
the product of the magnitudes--
00:29:40.370 --> 00:29:44.590
that's also pretty easy.
00:29:44.590 --> 00:29:47.160
Everybody's with me?
00:29:47.160 --> 00:29:48.910
And if you take the
log, it's even easier.
00:29:51.650 --> 00:29:57.670
The log of a product
is the sum of the logs.
00:29:57.670 --> 00:29:58.490
OK.
00:29:58.490 --> 00:30:01.150
Sum's easier to multiply.
00:30:01.150 --> 00:30:04.210
That's another reason we
like this log-log thing.
00:30:04.210 --> 00:30:06.880
So all you need to do is
think about each singularity.
00:30:06.880 --> 00:30:09.730
Each one of them can be
represented by this and down
00:30:09.730 --> 00:30:11.770
or this and up.
00:30:11.770 --> 00:30:13.300
And compose them by adding.
00:30:13.300 --> 00:30:14.710
So it's really easy.
00:30:14.710 --> 00:30:18.840
So say I had this system
function, a 0 at 0,
00:30:18.840 --> 00:30:21.655
a pole at minus 1, and
a pole at minus 10.
00:30:25.430 --> 00:30:28.680
Think about the
Bode representation.
00:30:28.680 --> 00:30:31.951
Bode's just a word that means
think about the asymptotes.
00:30:31.951 --> 00:30:34.760
Think about the Bode
representation for the top.
00:30:34.760 --> 00:30:39.060
Well, that grows
linearly with omega.
00:30:39.060 --> 00:30:41.640
A function that grows
linearly with omega
00:30:41.640 --> 00:30:45.150
has a log that grows
with a slope of 1.
00:30:45.150 --> 00:30:48.440
So that's this.
00:30:48.440 --> 00:30:51.250
So that's the zero.
00:30:51.250 --> 00:30:55.240
That's the contribution to the
magnitude function of the zero.
00:30:55.240 --> 00:30:57.430
The contribution to
the magnitude function
00:30:57.430 --> 00:31:01.450
of the pole at minus 1 is this.
00:31:01.450 --> 00:31:04.090
The pole at minus 10 is this.
00:31:04.090 --> 00:31:07.770
All pole's look the same.
00:31:07.770 --> 00:31:09.880
So all you need to
do is add them up.
00:31:09.880 --> 00:31:12.750
So you add the first
two, you get the--
00:31:12.750 --> 00:31:15.060
so backing up.
00:31:15.060 --> 00:31:16.020
0.
00:31:16.020 --> 00:31:17.350
Pole.
00:31:17.350 --> 00:31:18.900
Add them.
00:31:18.900 --> 00:31:21.690
In this region, the
sum of a constant
00:31:21.690 --> 00:31:24.240
and a straight line sloping up
is a straight line sloping up.
00:31:27.590 --> 00:31:30.890
In this region, the sum of
a straight line sloping up
00:31:30.890 --> 00:31:35.576
with this sloping down is flat.
00:31:35.576 --> 00:31:36.700
So that's why you get that.
00:31:39.240 --> 00:31:41.880
Then, we add in
this contribution.
00:31:41.880 --> 00:31:43.211
Add a constant to this.
00:31:43.211 --> 00:31:44.460
It just shifts it up and down.
00:31:44.460 --> 00:31:44.959
Who cares?
00:31:47.380 --> 00:31:48.510
That's easy.
00:31:48.510 --> 00:31:50.760
The important thing is
that it breaks down.
00:31:50.760 --> 00:31:53.204
So the result of summing
that is that it breaks down.
00:31:53.204 --> 00:31:55.889
Easy.
00:31:55.889 --> 00:31:57.930
So instead of thinking
about a frequency response
00:31:57.930 --> 00:31:59.638
as something that's
horribly complicated,
00:31:59.638 --> 00:32:02.100
we think about it
as having parts that
00:32:02.100 --> 00:32:04.470
came from each pole and zero.
00:32:04.470 --> 00:32:06.990
So instead of thinking about
it as a collection of arbitrary
00:32:06.990 --> 00:32:08.364
numbers at different
frequencies,
00:32:08.364 --> 00:32:09.900
we think about it
as a little part
00:32:09.900 --> 00:32:12.150
that comes from the first
pole, another part that
00:32:12.150 --> 00:32:14.100
comes from the second
pole, another part that
00:32:14.100 --> 00:32:15.450
comes from the first zero--
00:32:15.450 --> 00:32:16.950
blah, blah, blah.
00:32:16.950 --> 00:32:19.290
It's a way of thinking
about it, reducing
00:32:19.290 --> 00:32:23.672
the complexity, the
conceptual complexity,
00:32:23.672 --> 00:32:24.755
of the frequency response.
00:32:28.560 --> 00:32:33.700
So the angles are the same.
00:32:33.700 --> 00:32:35.370
If we think about
the low-frequency and
00:32:35.370 --> 00:32:39.660
high-frequency behavior of
the angle starting with a 0.
00:32:39.660 --> 00:32:43.730
For low frequencies,
the angle is 0.
00:32:43.730 --> 00:32:46.520
So that's the dot.
00:32:46.520 --> 00:32:48.004
For high frequencies,
as frequency
00:32:48.004 --> 00:32:49.420
goes higher and
higher and higher,
00:32:49.420 --> 00:32:51.710
the angle stands straighter
and straighter up.
00:32:51.710 --> 00:32:53.060
The asymptotic value is?
00:32:59.310 --> 00:33:01.980
Pi over 2.
00:33:01.980 --> 00:33:05.040
So we have two asymptotes,
0 and pi over 2.
00:33:05.040 --> 00:33:07.710
If we plot that on
log-log axes, we
00:33:07.710 --> 00:33:11.710
get something slightly
more complicated.
00:33:11.710 --> 00:33:14.105
The blue line shows
the calculated value.
00:33:17.010 --> 00:33:19.110
The red line shows
a way of thinking
00:33:19.110 --> 00:33:22.560
about that from a straight line
approximation point of view.
00:33:22.560 --> 00:33:26.100
Very nice construction
is put a straight line
00:33:26.100 --> 00:33:29.700
starting one decade--
00:33:29.700 --> 00:33:33.870
a factor of 10, minus
1 on a log scale.
00:33:33.870 --> 00:33:39.520
Draw a line from minus 1
to plus 1, a straight line.
00:33:39.520 --> 00:33:42.000
And you actually get a
very good approximation
00:33:42.000 --> 00:33:45.004
to the angle function
over the whole range.
00:33:45.004 --> 00:33:46.920
Notice that there are
two critical frequencies
00:33:46.920 --> 00:33:48.572
associated with the phase.
00:33:48.572 --> 00:33:51.030
There was one critical frequency
associated with magnitude.
00:33:54.270 --> 00:33:57.120
What was the crossing point
of the low- and high-frequency
00:33:57.120 --> 00:33:59.000
asymptotes?
00:33:59.000 --> 00:34:01.130
In phase, we think
about that same thing,
00:34:01.130 --> 00:34:02.830
but then we bump
it up and down 1.
00:34:02.830 --> 00:34:05.400
So there are two
critical frequencies.
00:34:05.400 --> 00:34:10.170
The same critical frequency
we use with the magnitude
00:34:10.170 --> 00:34:13.380
plus or minus 1.
00:34:13.380 --> 00:34:15.560
The same sort of thing
happens with a pole,
00:34:15.560 --> 00:34:18.260
but now it's upside down.
00:34:18.260 --> 00:34:19.850
Other than that, it's identical.
00:34:23.400 --> 00:34:25.139
Same thing.
00:34:25.139 --> 00:34:31.400
Having calculated the phase
for a single pole or zero,
00:34:31.400 --> 00:34:33.679
it's easy to think about
how you would generalize
00:34:33.679 --> 00:34:36.440
to multiple poles and zeros.
00:34:36.440 --> 00:34:41.420
The angle of a product
is the sum of the angles.
00:34:41.420 --> 00:34:43.806
Don't need to take
the log this way.
00:34:43.806 --> 00:34:45.389
This time, when we're
doing the phase,
00:34:45.389 --> 00:34:46.330
we don't need to take the log.
00:34:46.330 --> 00:34:48.000
There's a way of
thinking about phase.
00:34:48.000 --> 00:34:51.960
Because it's in the
exponent, e to the j angle
00:34:51.960 --> 00:34:53.639
by Euler's equation.
00:34:53.639 --> 00:34:56.219
There's a way of thinking
about the logs already
00:34:56.219 --> 00:34:58.170
in the angle function, right?
00:34:58.170 --> 00:35:02.650
The angle was in the
exponent, e to the j omega.
00:35:02.650 --> 00:35:05.040
So since the angle is
already up in the e,
00:35:05.040 --> 00:35:06.320
it's kind of like an--
00:35:06.320 --> 00:35:09.650
it's already a
logarithmic function.
00:35:09.650 --> 00:35:11.530
And that shows up here.
00:35:11.530 --> 00:35:16.080
The angle of a product
is the sum of the angles.
00:35:16.080 --> 00:35:21.739
So same sort of thing happens if
we had a complicated function.
00:35:21.739 --> 00:35:24.280
Just think about the angle that
results for each of the poles
00:35:24.280 --> 00:35:26.830
and zeros and add them.
00:35:26.830 --> 00:35:37.180
The angle associated with
the zero at 0 is pi over 2.
00:35:37.180 --> 00:35:39.900
j is the same as e
to the j pi over 2.
00:35:39.900 --> 00:35:41.100
e to the j pi over 2.
00:35:41.100 --> 00:35:44.310
The angle's pi over 2.
00:35:44.310 --> 00:35:50.040
So the angle associated with
this is always pi over 2.
00:35:50.040 --> 00:35:54.780
The angle associated
with the pole at minus 1.
00:35:54.780 --> 00:35:58.980
Poles in the left-half plane
cause the angle to start out
00:35:58.980 --> 00:36:01.560
at 0 and go negative.
00:36:01.560 --> 00:36:04.690
So we start out at
0 and go negative.
00:36:04.690 --> 00:36:07.845
We find the critical frequency
labeled in this axis by 0.
00:36:07.845 --> 00:36:11.740
s equals 1 is the
critical frequency.
00:36:11.740 --> 00:36:14.880
So we go up and
down 1 unit and we
00:36:14.880 --> 00:36:17.950
draw the straight-line
approximation.
00:36:17.950 --> 00:36:20.360
s equals 10 is the same
thing, except now it's
00:36:20.360 --> 00:36:21.860
shifted to a higher frequency.
00:36:21.860 --> 00:36:26.910
Factor of 10, units
shift on a log plot.
00:36:26.910 --> 00:36:30.240
And now all we do is sum them,
add the first to the second,
00:36:30.240 --> 00:36:31.380
add the third.
00:36:31.380 --> 00:36:35.070
That's our angle approximation.
00:36:35.070 --> 00:36:36.780
Again, the idea is
to take something
00:36:36.780 --> 00:36:38.647
that's conceptually hard--
00:36:38.647 --> 00:36:40.230
what's the value of
the angle function
00:36:40.230 --> 00:36:44.090
as a function of frequency-- and
turn it into something simple.
00:36:44.090 --> 00:36:45.830
A few straight line
segments associated
00:36:45.830 --> 00:36:46.940
with every pole and zero.
00:36:50.020 --> 00:36:53.820
So this is just a summary.
00:36:53.820 --> 00:36:57.090
Because we can represent
a system that's
00:36:57.090 --> 00:37:03.430
composed of integrators,
summers, and gains
00:37:03.430 --> 00:37:05.957
by a linear
differential equation
00:37:05.957 --> 00:37:07.540
with constant
coefficients, it follows
00:37:07.540 --> 00:37:18.490
that the system function is a
quotient of polynomials in s.
00:37:18.490 --> 00:37:20.450
Because of the fundamental
theorem in algebra,
00:37:20.450 --> 00:37:21.930
there's n roots.
00:37:21.930 --> 00:37:24.614
Because of factor theorem, you
can break it up into factors.
00:37:24.614 --> 00:37:27.030
Because of all that, we can
think about them one at a time
00:37:27.030 --> 00:37:28.950
and glue them together.
00:37:28.950 --> 00:37:31.101
Gluing them together in
the case of the magnitude
00:37:31.101 --> 00:37:33.600
works best if you use the log
because then the product turns
00:37:33.600 --> 00:37:36.330
into a sum.
00:37:36.330 --> 00:37:38.100
In the case of the
angle, it's for free
00:37:38.100 --> 00:37:39.930
because the angle
is, in some sense,
00:37:39.930 --> 00:37:43.672
already a logarithmic function.
00:37:43.672 --> 00:37:45.670
OK.
00:37:45.670 --> 00:37:47.710
OK, we'll see if you got it.
00:37:47.710 --> 00:37:49.544
So here is a complicated
frequency response.
00:37:49.544 --> 00:37:51.876
This is the straight-line
approximation to the frequency
00:37:51.876 --> 00:37:53.020
response of a system.
00:37:53.020 --> 00:37:53.850
Which system?
00:39:20.160 --> 00:39:21.660
So which of the
system functions--
00:39:21.660 --> 00:39:24.780
1, 2, 3, 4, or
none of the above--
00:39:24.780 --> 00:39:29.160
is represented by the
Bode straight-line plot
00:39:29.160 --> 00:39:30.360
showed above?
00:39:30.360 --> 00:39:32.010
Raise your hands,
which one is better--
00:39:32.010 --> 00:39:33.755
1, 2, 3, 4, or 5?
00:39:37.395 --> 00:39:38.510
Ah, 100%.
00:39:38.510 --> 00:39:40.070
That's exactly the right answer.
00:39:40.070 --> 00:39:41.240
Wonderful.
00:39:41.240 --> 00:39:43.620
The idea is that Bode is easy.
00:39:43.620 --> 00:39:45.690
That's why we do it.
00:39:45.690 --> 00:39:46.280
It's easy.
00:39:46.280 --> 00:39:47.270
So tell me a rule.
00:39:47.270 --> 00:39:48.830
How do I think about this one?
00:39:48.830 --> 00:39:49.820
What would happen here?
00:39:49.820 --> 00:39:51.804
What's this saying?
00:39:51.804 --> 00:39:53.470
What would be the
Bode plot of this one?
00:39:53.470 --> 00:39:54.344
Sketch it in the air.
00:39:57.660 --> 00:39:58.610
Exactly.
00:39:58.610 --> 00:40:02.841
You start at 0, right?
00:40:02.841 --> 00:40:03.340
OK.
00:40:03.340 --> 00:40:04.420
So start at 0.
00:40:04.420 --> 00:40:07.210
Then what?
00:40:07.210 --> 00:40:10.480
What do you run into first?
00:40:10.480 --> 00:40:11.590
You start at 0.
00:40:11.590 --> 00:40:13.840
What do you run into first
when you're doing this guy?
00:40:18.730 --> 00:40:21.820
You run into this factor, or
that factor, or that factor
00:40:21.820 --> 00:40:23.090
first?
00:40:23.090 --> 00:40:24.360
AUDIENCE: [INAUDIBLE]
00:40:24.360 --> 00:40:26.220
DENNIS FREEMAN:
Yeah, the left one.
00:40:26.220 --> 00:40:28.440
So think about the order.
00:40:28.440 --> 00:40:32.070
By how close were
they to the origin?
00:40:32.070 --> 00:40:34.710
Think about them in
order from the origin.
00:40:34.710 --> 00:40:37.010
So you hit the first one
first, the one at 10,
00:40:37.010 --> 00:40:38.350
then the one at 100.
00:40:38.350 --> 00:40:41.600
This would start
flat, break down at 1,
00:40:41.600 --> 00:40:43.995
break down again at 10,
break down again at 100.
00:40:43.995 --> 00:40:45.120
That's not the right shape.
00:40:48.150 --> 00:40:51.880
This one starts with a 0.
00:40:51.880 --> 00:40:54.600
That's a break up.
00:40:54.600 --> 00:40:59.870
The 0 happens at frequency 1.
00:40:59.870 --> 00:41:02.060
So that's log frequency 0.
00:41:02.060 --> 00:41:04.610
That's right.
00:41:04.610 --> 00:41:07.920
Then, you break
down, break down.
00:41:07.920 --> 00:41:09.230
OK.
00:41:09.230 --> 00:41:11.840
So break up, break
up, break down.
00:41:11.840 --> 00:41:14.850
Except that it would
actually break down first.
00:41:14.850 --> 00:41:16.310
So the way to think
about this one
00:41:16.310 --> 00:41:20.090
is break down,
break up, break up
00:41:20.090 --> 00:41:23.090
ordered from going
away from the origin.
00:41:23.090 --> 00:41:24.050
OK.
00:41:24.050 --> 00:41:26.900
So the point is it's very
easy to take a system function
00:41:26.900 --> 00:41:31.060
and immediately draw
the frequency response.
00:41:31.060 --> 00:41:36.010
That lets you take a pole-zero
representation, which
00:41:36.010 --> 00:41:39.070
can be very concise in terms
of the number of numbers
00:41:39.070 --> 00:41:41.890
you need to know, and
quickly map out the frequency
00:41:41.890 --> 00:41:44.200
response, which can be
very intuitive for thinking
00:41:44.200 --> 00:41:45.449
about how the system responds.
00:41:47.820 --> 00:41:49.260
OK.
00:41:49.260 --> 00:41:54.650
So one more issue.
00:41:54.650 --> 00:41:58.220
We don't really like log
plots when it gets down
00:41:58.220 --> 00:42:03.560
to talking about the frequency
at which the log was 7.129.
00:42:03.560 --> 00:42:06.590
We just don't like that because
nobody measures frequency
00:42:06.590 --> 00:42:10.010
in the frequency whose log is x.
00:42:10.010 --> 00:42:12.920
So what we usually do,
instead of plotting
00:42:12.920 --> 00:42:18.901
versus the log of omega, we plot
versus omega on a log scale.
00:42:18.901 --> 00:42:19.400
OK.
00:42:19.400 --> 00:42:20.600
Very reasonable.
00:42:20.600 --> 00:42:25.710
We just put tick marks and we
just label them exponentially.
00:42:25.710 --> 00:42:29.660
So the tick mark associated
with omega is labeled 1, not 0.
00:42:32.270 --> 00:42:34.920
That's frequency 1.
00:42:34.920 --> 00:42:38.960
The tick mark associated with
frequency 10 is labeled 10.
00:42:38.960 --> 00:42:42.350
What that means, though, is that
the growth inside that interval
00:42:42.350 --> 00:42:45.074
is on a log scale.
00:42:45.074 --> 00:42:46.490
That's why we write
log scale here
00:42:46.490 --> 00:42:50.030
to keep reminding ourselves that
the ticks between 1 and 10--
00:42:50.030 --> 00:42:53.330
1, 2, 3, 4, 5--
00:42:53.330 --> 00:42:54.710
are not uniformly spaced.
00:42:54.710 --> 00:42:56.427
They're log spaced.
00:42:56.427 --> 00:42:58.760
The distance between 1 and 2
is bigger than the distance
00:42:58.760 --> 00:43:00.710
between 9 and 10.
00:43:00.710 --> 00:43:03.350
That's the way logs work.
00:43:03.350 --> 00:43:04.080
OK.
00:43:04.080 --> 00:43:07.230
So we think about frequency
on a log scale rather than
00:43:07.230 --> 00:43:08.460
log frequency.
00:43:08.460 --> 00:43:12.510
Similarly, we think about
amplitude on a dB scale
00:43:12.510 --> 00:43:14.330
rather than log amplitude.
00:43:14.330 --> 00:43:17.220
dB is for Alexander Graham Bell.
00:43:17.220 --> 00:43:19.970
It's decibel.
00:43:19.970 --> 00:43:24.510
A bell is a factor of 10.
00:43:24.510 --> 00:43:26.560
It's a little unintuitive,
a factor of 10
00:43:26.560 --> 00:43:32.110
is labeled 20, because Bell
was really thinking energy.
00:43:32.110 --> 00:43:33.520
And you have to square energy.
00:43:33.520 --> 00:43:36.260
You have to square
voltage to get energy.
00:43:36.260 --> 00:43:39.000
Bell was thinking a
factor of 10 in energy.
00:43:39.000 --> 00:43:40.990
We like to think amplitude.
00:43:40.990 --> 00:43:46.471
Therefore, a factor of 10
for us would be 20 decibels.
00:43:46.471 --> 00:43:46.970
20.
00:43:46.970 --> 00:43:48.720
OK, so that's a little weird.
00:43:48.720 --> 00:43:52.190
So we will think about
this axis in decibels
00:43:52.190 --> 00:43:56.940
and we'll think about
that one in decades.
00:43:56.940 --> 00:43:59.390
And so what that means
is that the slopes are
00:43:59.390 --> 00:44:01.880
no longer minus 1 and 1.
00:44:01.880 --> 00:44:06.270
The slopes are 20
decibels per decade,
00:44:06.270 --> 00:44:08.344
or minus 20 decibels per decade.
00:44:08.344 --> 00:44:10.010
Now, this is all
completely meaningless.
00:44:10.010 --> 00:44:13.170
It's a slope of 1.
00:44:13.170 --> 00:44:17.190
And you could equally have
labeled this axis in dB
00:44:17.190 --> 00:44:19.110
and this axis on a log
scale, and then you
00:44:19.110 --> 00:44:24.600
would have the slope
of 1 decade per 20 dB.
00:44:24.600 --> 00:44:26.550
And that, too, would be 1.
00:44:26.550 --> 00:44:29.350
We just don't do it that way.
00:44:29.350 --> 00:44:29.850
OK.
00:44:29.850 --> 00:44:30.433
What do we do?
00:44:30.433 --> 00:44:34.680
What we do do is
frequency on a log scale.
00:44:34.680 --> 00:44:40.440
So therefore, the unit of
frequency is the decade.
00:44:40.440 --> 00:44:42.300
You have a frequency
and a decade
00:44:42.300 --> 00:44:45.750
higher and a decade higher
and a decade higher.
00:44:45.750 --> 00:44:49.050
The unit of frequency
is decade or octave.
00:44:49.050 --> 00:44:51.780
Octave is a factor of 2.
00:44:51.780 --> 00:44:53.130
Octave higher, factor of 2.
00:44:53.130 --> 00:44:54.180
Octave higher.
00:44:54.180 --> 00:44:56.790
Octave makes more sense if
you're a musician, right?
00:44:56.790 --> 00:45:01.220
The distance between
two C's is an octave.
00:45:01.220 --> 00:45:02.300
So that's a dB scale.
00:45:05.150 --> 00:45:07.265
And that results in a
little bit of funny math.
00:45:10.610 --> 00:45:13.520
So if you convert linear
measures of amplitude
00:45:13.520 --> 00:45:19.390
to decibel measures, a
factor of 1 in amplitude
00:45:19.390 --> 00:45:22.000
is a factor of 0 dB.
00:45:22.000 --> 00:45:24.602
A factor of 10 is 20 dB.
00:45:24.602 --> 00:45:25.810
We already talked about that.
00:45:25.810 --> 00:45:28.150
There are some convenient
middle grounds.
00:45:28.150 --> 00:45:30.424
2 is 6 dB.
00:45:30.424 --> 00:45:31.880
That's a little weird.
00:45:31.880 --> 00:45:35.112
So we'll go around saying
it's 6 dB and we mean 2.
00:45:35.112 --> 00:45:39.240
But every engineer in the
world will call it 6 dB.
00:45:39.240 --> 00:45:43.670
Half of that is the square
root of 2, which is 3 dB.
00:45:43.670 --> 00:45:46.280
It's just convenient,
at least it
00:45:46.280 --> 00:45:48.710
is after you spend
30 years doing that.
00:45:48.710 --> 00:45:50.420
The other point
from the slide is
00:45:50.420 --> 00:45:54.550
that the asymptotic responses
are really quite good.
00:45:54.550 --> 00:45:57.490
The magnitude for a
single pole deviates
00:45:57.490 --> 00:46:01.750
from the straight-line
approximation by only 3 dB.
00:46:01.750 --> 00:46:05.260
I can hear sounds
that range 120 dB.
00:46:05.260 --> 00:46:08.290
That's the sense in
which 3 is small.
00:46:08.290 --> 00:46:11.080
The kinds of signals
we work with every day
00:46:11.080 --> 00:46:14.110
have ranges that are
big compared to 3.
00:46:14.110 --> 00:46:16.540
So we think of 3
as a small thing.
00:46:16.540 --> 00:46:18.890
The phases are within 6 degrees.
00:46:18.890 --> 00:46:24.130
And for a lot of applications,
6 degrees is a small number.
00:46:24.130 --> 00:46:26.950
Again, we're thinking
6 out of 180.
00:46:26.950 --> 00:46:30.240
So it's a small
fraction of a cycle.
00:46:30.240 --> 00:46:33.320
OK, this is a good thing--
00:46:33.320 --> 00:46:35.320
especially the fact that
it's a trick question--
00:46:35.320 --> 00:46:37.140
for you to practice
for the exam.
00:46:37.140 --> 00:46:38.890
So let me skip it in
the interest of time,
00:46:38.890 --> 00:46:41.730
because there's one
more important thing.
00:46:41.730 --> 00:46:44.480
Don't look at the answer.
00:46:44.480 --> 00:46:47.470
Use this as practice.
00:46:47.470 --> 00:46:50.890
There's one more
important thing in trying
00:46:50.890 --> 00:46:56.820
to reduce poles and zeros
to a frequency response.
00:46:56.820 --> 00:46:59.610
And that is that when you
use the fundamental theorem
00:46:59.610 --> 00:47:04.590
of algebra, even though
the polynomial can
00:47:04.590 --> 00:47:08.980
have real-valued coefficients,
that does not mean the roots
00:47:08.980 --> 00:47:11.250
are real.
00:47:11.250 --> 00:47:16.090
Right
00:47:16.090 --> 00:47:19.210
The roots to a polynomial
with real value coefficients
00:47:19.210 --> 00:47:23.460
can have complex parts.
00:47:23.460 --> 00:47:25.650
So the remaining thing
I want to talk about
00:47:25.650 --> 00:47:30.910
is, what do you do with these
poles that have complex parts?
00:47:33.610 --> 00:47:37.740
The imaginary part has
to do with oscillations.
00:47:37.740 --> 00:47:41.110
And so we want to think about,
what's a Bode plot looks like?
00:47:41.110 --> 00:47:44.040
What's the asymptotes
look like when
00:47:44.040 --> 00:47:46.000
you have a system like this?
00:47:46.000 --> 00:47:50.041
This was a system that has
a mass spring, and dashpot.
00:47:50.041 --> 00:47:51.915
You should have done
this by homework by now.
00:47:55.700 --> 00:47:58.160
The differential
equation is second order.
00:47:58.160 --> 00:48:03.500
It has real-valued coefficients,
but the holes are complex.
00:48:03.500 --> 00:48:06.290
What happens with complex roots?
00:48:06.290 --> 00:48:10.460
Well, if the com--
00:48:10.460 --> 00:48:14.840
if the polynomial had
real-valued coefficients,
00:48:14.840 --> 00:48:18.530
complex roots come in pairs.
00:48:18.530 --> 00:48:21.560
That's the only way to
take a complex number
00:48:21.560 --> 00:48:23.870
and end up with a
product that's all real.
00:48:23.870 --> 00:48:25.190
You take the complex number.
00:48:25.190 --> 00:48:27.410
It has to be paired with
its complex conjugate.
00:48:27.410 --> 00:48:34.550
So that when you pair them, the
result has real coefficients.
00:48:34.550 --> 00:48:36.920
So we only need to
worry about the case
00:48:36.920 --> 00:48:42.830
when the poles or zeros
come in complex pairs.
00:48:42.830 --> 00:48:45.800
And for that purpose, it's
convenient to think about--
00:48:45.800 --> 00:48:50.045
you might think that if you
had mass-spring dashpot system,
00:48:50.045 --> 00:48:52.580
you might be expecting
something like s
00:48:52.580 --> 00:48:57.350
squared m plus sb plus
k, something like that.
00:48:57.350 --> 00:48:59.090
Mass-spring and dashpot.
00:48:59.090 --> 00:49:00.980
Three parameters.
00:49:00.980 --> 00:49:03.594
Well, you don't
really need three.
00:49:03.594 --> 00:49:05.760
Three is nice because it
has an association with how
00:49:05.760 --> 00:49:07.460
stiff is this thing
and how massive
00:49:07.460 --> 00:49:08.862
is that thing, et cetera.
00:49:08.862 --> 00:49:10.820
But in terms of thinking
about poles and zeros,
00:49:10.820 --> 00:49:12.650
you don't need to
think about all three.
00:49:12.650 --> 00:49:18.140
First off, you could divide
the top and bottom by k
00:49:18.140 --> 00:49:21.770
and the 1/k then is
just a gain factor.
00:49:21.770 --> 00:49:22.830
Gains are easy.
00:49:22.830 --> 00:49:24.420
They don't affect shape.
00:49:24.420 --> 00:49:24.920
OK.
00:49:24.920 --> 00:49:26.370
We don't care about that one.
00:49:26.370 --> 00:49:29.330
So we went from 3 to 2.
00:49:29.330 --> 00:49:31.970
Now, there's another
simplification
00:49:31.970 --> 00:49:35.970
because all of these are
going to be oscillatory.
00:49:35.970 --> 00:49:39.330
Pole pairs work in an
oscillatory fashion.
00:49:39.330 --> 00:49:43.170
This has a natural frequency
that if I didn't shake it,
00:49:43.170 --> 00:49:45.240
it has a preferred frequency.
00:49:45.240 --> 00:49:50.250
If I divide by that
preferred frequency, omega 0,
00:49:50.250 --> 00:49:52.500
I can get rid of
another parameter.
00:49:52.500 --> 00:49:53.610
So don't think about s.
00:49:53.610 --> 00:49:56.940
Think about s over omega 0.
00:49:56.940 --> 00:50:00.591
Now, frequencies
are normalized to 1.
00:50:00.591 --> 00:50:01.090
OK.
00:50:01.090 --> 00:50:03.010
So by dividing by
omega 0, I turn
00:50:03.010 --> 00:50:06.220
every frequency-dependent
system into something
00:50:06.220 --> 00:50:10.360
whose best frequency
is near omega equals 1.
00:50:10.360 --> 00:50:13.990
Then finally, for
my third parameter,
00:50:13.990 --> 00:50:16.406
if I write my third
parameter as 1 over q,
00:50:16.406 --> 00:50:18.280
something very magical
and wonderful happens.
00:50:21.460 --> 00:50:25.280
The roots fall on a circle.
00:50:25.280 --> 00:50:30.140
So what I want to show
here then is as I change q,
00:50:30.140 --> 00:50:31.790
I don't need to
think about omega.
00:50:31.790 --> 00:50:34.266
I can plot this on an
s over omega 0 plane
00:50:34.266 --> 00:50:35.390
and it works for all omega.
00:50:39.110 --> 00:50:40.300
I do need to worry about q.
00:50:40.300 --> 00:50:47.630
So if I change q, here q is 1/2.
00:50:47.630 --> 00:50:48.470
So there's q of 1/2.
00:50:51.790 --> 00:50:53.970
1/4.
00:50:53.970 --> 00:50:54.936
1/8.
00:50:54.936 --> 00:50:58.570
Excuse me, I'm
doing it backwards.
00:50:58.570 --> 00:51:02.190
I want 1/q to equal--
00:51:02.190 --> 00:51:05.070
I want 1 over 2q to equal 1/2.
00:51:05.070 --> 00:51:06.750
I need q equals 1.
00:51:06.750 --> 00:51:07.600
OK, that's better.
00:51:07.600 --> 00:51:10.585
That's q equals 1.
00:51:10.585 --> 00:51:15.900
q equals 2 in order to half
the distance to the origin--
00:51:15.900 --> 00:51:17.445
4, 8, 16.
00:51:20.310 --> 00:51:23.390
Notice that as I change--
now watch this side.
00:51:23.390 --> 00:51:26.294
So q equals 1.
00:51:26.294 --> 00:51:28.660
So q equals 1, q equals 1.
00:51:31.630 --> 00:51:33.380
Low-frequency magnitude is flat.
00:51:33.380 --> 00:51:36.110
High-frequency is slope of 2.
00:51:36.110 --> 00:51:37.590
Sloping down with minus 2.
00:51:40.990 --> 00:51:51.060
As I change q from 1 to 2, 4,
8, 16, the peak gets bigger.
00:51:51.060 --> 00:51:53.280
In fact, if you do a
little bit of math,
00:51:53.280 --> 00:51:59.310
you can show that the
peak gets bigger with q.
00:51:59.310 --> 00:52:02.800
The peak value, if
you measure how big
00:52:02.800 --> 00:52:06.430
is the peak compared to
where is the crossover,
00:52:06.430 --> 00:52:10.620
that distance is a factor of q.
00:52:10.620 --> 00:52:13.480
Similarly-- and you
can reason about that
00:52:13.480 --> 00:52:14.470
with vector diagrams.
00:52:14.470 --> 00:52:17.230
And we'll do homework
problems to practice that.
00:52:17.230 --> 00:52:19.900
Similarly, it got peakier.
00:52:19.900 --> 00:52:22.120
It got sharper.
00:52:22.120 --> 00:52:24.040
If you do the
vector story to try
00:52:24.040 --> 00:52:26.160
to figure out why
it got peakier,
00:52:26.160 --> 00:52:28.030
the width turns out to be 1/q.
00:52:30.590 --> 00:52:32.130
So that's kind of impulse-y.
00:52:32.130 --> 00:52:34.100
The height got bigger with q.
00:52:34.100 --> 00:52:37.280
The width got bigger with 1/q.
00:52:37.280 --> 00:52:39.599
The product always 1.
00:52:39.599 --> 00:52:41.140
So that's a way of
thinking about why
00:52:41.140 --> 00:52:42.760
it got peaky that way.
00:52:42.760 --> 00:52:45.940
And finally, if you think about
the angle, the angle changes.
00:52:45.940 --> 00:52:51.750
As you make q bigger and bigger,
the angle changes very quickly.
00:52:51.750 --> 00:52:55.300
And it turns out that the
angle changes abruptly
00:52:55.300 --> 00:52:56.950
over the same bandwidth.
00:52:56.950 --> 00:52:59.080
Bandwidth is how many
frequencies are there
00:52:59.080 --> 00:53:03.430
between the low-frequency part
and the high-frequency part.
00:53:03.430 --> 00:53:07.480
The phase change over the
bandwidth is always pi over 2.
00:53:07.480 --> 00:53:08.020
OK.
00:53:08.020 --> 00:53:09.460
So that's the whole story then.
00:53:09.460 --> 00:53:12.070
Think about isolating the
poles on the real axes.
00:53:12.070 --> 00:53:13.090
They're just this.
00:53:13.090 --> 00:53:16.000
Isolated zeros,
they're just this.
00:53:16.000 --> 00:53:19.400
Pairs can be more complicated
because they can be peaky.
00:53:19.400 --> 00:53:19.900
OK.
00:53:19.900 --> 00:53:21.450
Thanks.