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PROFESSOR: Last time we
introduced poles.

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And in particular, we introduced
how to move from

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the manipulation of feed-forward
and feedback

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systems and the geometric
sequence that fell out into

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using the base of that geometric
sequence to attempt

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to predict the long-term
behavior of the system.

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When we're solving for poles and
we're only interested in

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long term behavior, one of the
easiest ways to do so to solve

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for the roots of z, where z is a
substitution for 1 over R in

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the denominator of the
system function.

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Once we've done that, we
have a list of poles.

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From that list of poles, we
would like to select, the

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dominant pole, or the pole with
the greatest magnitude,

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and then based on the magnitude
and period of that

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pole we can determine what the
long term behavior of our

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system looks like.

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Today I'd like to mention
to you some notable

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things about poles.

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If you are interested in this
information or feedback and

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controls in the general sense,
I highly recommend 6.003.

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But here's some information you
should at least be aware

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of as a consequence of 6.01.

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The other thing I would like to
do is just walk through a

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couple of pole problems to
familiarize you, or get you

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more comfortable with the idea
of solving for the poles of a

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system function, or looking at
the unit sample response of a

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system function and then
graphing the poles.

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The first thing that I want
to mention is pole-zero

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cancellation.

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And what do I mean
when I say that?

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I mean that if both the
numerator and the denominator

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have a degree of R in them, then
you're going to have both

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a zero and a pole.

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If the zero and the pole have
the same value associated with

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them, you may be tempted
to cancel them out.

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Unless both the zero and the
pole are equal to 0 --

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don't do it.

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The reason why is that when you
get to a implementation of

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a real system, it is highly
unlikely that both the zero

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and the pole will be implemented
to a degree of

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accuracy that you will actually
see those two things

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cancel out.

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The only exception to this is
when both the pole and the

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zero are equal to 0,
or in this case.

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This you should feel free
to convert to this.

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In almost any other situation,
don't factor.

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The other thing I want to talk
about is repeated roots.

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If you have a repeated root,
you'll have repeated poles.

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This does get tricky when you're
talking about how to

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add the unit response
of those poles.

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But the long term behavior
of your system is

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going to look the same.

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So if both of these poles are
the dominant pole, then the

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characteristics of both, which
are the same, are going to

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determine what your long term
behavior looks like.

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If they're not, then the
dominant pole is going to

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determine what your long term
behavior looks like.

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The last thing I want to mention
is superposition.

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So far we've only talked about
the unit sample response of a

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system function and how we use
poles to determine what the

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long term behavior of our
system's going to be.

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We can look at the response to
more complicated inputs than

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the unit sample response,
or the delta.

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In fact, one of the things we'
probably end up looking at

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some point is the
step function.

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The thing that you need to know
to go from talking about

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unit sample response to any
other sort of response, is

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that we're still working
with an LTI system.

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What that means is if you take
the summation of your inputs,

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and apply the system function
to that summation, it is the

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same as the output that would
result from inputting all

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those values at once.

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The best way I would like to
explain it is by referring

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again to if your function was
a system function, the same

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property applies.

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Now let's walk through
a pole problem.

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Here I have a second order
system set up.

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We've got two degrees of R. I
have feedback, and I can solve

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for an expression of
y in terms of x.

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In fact, let's do
that right now.

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y is the result of the
summation or a linear

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combination of x plus a delayed
signal of y scaled by

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1.6 in a linear combination with
a delayed value of the

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delayed value of y scaled
by negative 0.63.

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There's my first degree.

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For consistency's sake, there's
my second degree.

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Let's first solve for
the system function.

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If you're confused, I recommend
doing the algebra

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from here to this expression.

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You should get this
fraction out.

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Our second step is to solve
for the roots of z.

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Remember that z is equal to 1
over R in the denominator of

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the system function.

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In this case, we'll
be working with--

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all I've done there is taken
every degree of R, substitute

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it in for 1/z.

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and then multiply it out, so
that I'm not working with z in

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the denominator anymore.

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I'm actually just working with
everything in the numerator.

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If I follow this back out,
I get this expression.

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And my poles are going
to be 0.7 and 0.9.

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All right, based on my poles,
what are the properties of the

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unit sample response
in the long term?

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First thing I'm going to do is
look for the dominant pole

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among the poles that I found.

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In this case, I don't even have
to worry about finding

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the length of the distance from
the origin for poles in

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the complex plane.

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All I have to worry about
is the magnitude of

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poles on the real axis.

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0.9 is my dominant pole, because
it's the largest pole.

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0.9 is less than 1.

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So I'm going to end up
with convergence.

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Eventually, my system is
going to converge, or

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tend towards 0.

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The other interesting property
of my system is what is its

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period, how does that relate
to what my function's

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going to look like.

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In this case, we're only working
on the positive real

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axis, so the angle associated
with graphing this pole on the

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complex plane is 0, so there is
no period for our system.

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This means that our system
is going to converge

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monotonically.

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Now let's walk through some unit
sample responses and then

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graph the poles that generated
those unit sample responses on

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the unit circle, where this
is the complex plane.

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Let's look at this
graph first.

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The first thing that I notice
about this graph is that, like

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in the previous example, we have
monotonic convergence.

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We're tending towards 0, and
we're not alternating or

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oscillating about the x-axis.

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So I know I'm going to be
working somewhere along this

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line before the edge
of the unit circle.

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Because at the edge of the unit
circle, the distance from

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the origin is equal to 1.

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If you made me guess, then I
would look at the distance

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here and compare it
to the distance at

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the next time step.

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I realize this is
a blackboard.

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It's not entirely to scale.

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But for the purposes of this
demonstration, I'd like to say

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that the signal at this time
step is 0.5 the signal from

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the previous times.

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Likewise at the next time step,
I would like to say that

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this signal is 0.5 the signal
from the previous time step,

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and so on and so forth.

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Therefore, I'm going to graph
my pole right here.

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Let's take a look
at this graph.

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I've drawn these squiggles to
indicate that the unit sample

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response exceeds the bounds
of the space that I

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gave for this graph.

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So just assume that these values
are much larger than

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I've drawn them.

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The first thing that I notice
about this unit sample

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response graph is the fact that
not only am I increasing

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in a way that does not seem
to change in any way--

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we're going to end
up diverging--

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is that I'm actually alternating
about the x-axis.

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And that particularly, that
if I were to call this an

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oscillation, then I
would say it's an

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oscillation with period 2.

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This means that I'm working
with a negative real pole.

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The fact that I'm diverging
means I'm working with a

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negative real pole that has
magnitude greater than 1.

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If you had to make me guess, I
would look at the distance

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associated with this time
step, compare it to the

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distance associated with
this time step.

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And if you had to ask me, I
would say this is about 1.3

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the value at the
previous step.

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Likewise, if I were to look at
the next time step, I would

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say that this increase
is about 30% of

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the previous value.

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I'm not even going
to try that one.

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But what I'm trying to get
at is that you can use

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comparisons of previous and
future time steps in order to

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attempt to determine the
magnitude of the pole if

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you're working with the
first order system.

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If you're working with the
second order system, then it's

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possible that you'll see
some really interesting

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initialization effects.

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And you should probably ask
one of us what's up.

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But for this example,
we're going to put

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our pole over here.

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Here's the last graph I
want to talk about.

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The first thing that I notice is
that it doesn't seem to be

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diverging, but it doesn't
really seem to

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be converging either.

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If this is the case, then
I'm going to put

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it on the unit circle.

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The second thing that I notice
is that it's not monotonic and

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it's not alternating.

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This is oscillating.

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So in order to determine what
angle I'm going to sign to my

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unit simple response, I'm going
to count out the time

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steps that it takes to cycle
through an entire period and

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then from there figure out what
the angle would have to

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be in order to determine a
period of that length.

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So I start here.

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I'm just going to count 1,
2, 3, 4, 5, 6, 7, 8 --

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to complete one full
oscillation.

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This means that my
period is 8.

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If I have to divide 2pi by a
particular angle in order to

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get out 8, I want to
divide by pi/4.

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So at this point, I'm working
with a magnitude of about 1,

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and I want this angle
to be about pi/4.

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This concludes my tutorial
on solving poles.

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Next time, we'll end up talking
about circuits.

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