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PROFESSOR: Hi.

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Today I'd like to talk about
signals and systems again.

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At this point, you're probably
familiar with the motivation

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for why we're talking about
discrete linear time and

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variant systems, and also with
a few of the representations

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that we're going to end up
using in this course.

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But you're still not sure what
it is that we're trying to

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accomplish.

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Or where's the part where we
get to predict the future

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based on the fact that
we are capable of

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manipulating these systems?

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Well, we actually have
to be capable of

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manipulating these systems.

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And at this point, we can
describe this system as we see

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it, but we can't also manipulate
its representation

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in ways that make sense to us.

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So the thing that I'm going
to do today is talk about

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different system equivalences
and how to take a system and

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solve for an expression that
represents a complex system

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and also how if you know that
some things in your system are

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equivalent, how you can
convert between them.

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At that point, we should be
able to talk about poles,

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which is how we're going to
actually predict the future.

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So different equivalences that
I'd like to talk about.

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I'm first going to briefly
review the facts that last

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time we discovered the notion
of system function, right?

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We can take a representation
of a system and abstract it

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away into some sort of function,
where we take the

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input as it's given to us and
then multiply it by this

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function and then get
the output that

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we're interested in.

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How do we deal with something
more complex?

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I mean, y is all the
way over here.

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And we've got multiple
system functions.

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And I don't even know what
happens here, but it doesn't

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have to be that scary.

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Let's break it down.

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One of the easiest ways to
approach something like this

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is to identify each position
where you have a new signal,

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or if you were to sample here,
you would have a new signal,

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and label those values
appropriately.

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You can then start with your
final output and then back

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solve for the values that you're
interested in as a

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consequence of that
final output.

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In this particular example, y
is going to be y2 plus y3.

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y2 is going to be y1
times H2, where H2

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is some system function.

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And it probably is abstracting
away some combination of

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gains, delays, and adders
like this one here.

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y1 is going to be x times H1.

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And Y3 is going to
be x times H3.

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Now I've got all my expressions
in terms of either

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y or something for which
I have an equivalent

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expression for x.

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So I can do my substitutions,
come up for an expression for

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y over x, in terms of
H1, H2, and H3.

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Here I've just made the
substitutions of the equations

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above and factored out the x.

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If I wanted the system function,
I would then just

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divide by x.

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And then I would have y over x
is equal to this expression.

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The thing I wanted to indicate
is that if I wanted to

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abstract this away into its own
box-- maybe I wanted like

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a big H or an H0 or something
like that--

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and it represented what was
happening in this top line,

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cascading two system functions
is the functional equivalent

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of multiplying them together.

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So if I have an expression for
H1 and I have an expression

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for H2, and I want the
expression that is equal to

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cascading H1 and H2, I just
multiply them together.

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Likewise, if I want an
expression for the linear

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combination of two system
functions applied to an input

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individually, like the
combination H1 and H2, and H3,

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it's a summation of those
two values which

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is expressed here.

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This is the same as the
relationship that we reviewed

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in a very basic sense when we
were originally doing the

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accumulator.

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The only thing I'm attempting
to indicate is that, that

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relationship scales to an
arbitrary level of complexity.

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So if you need to, you could
shift around these values, if

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you can find some sort
of equivalence.

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Let's see what happens when
H2 is equal to H3.

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I'm going to take my
operator equation.

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What this means is that if I
wanted to rewrite this block

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diagram, I could do
so by doing--

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This is really similar
to bubble pushing.

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If you've done 6.004 or 6.002
and have experience with logic

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gates, I just wanted to indicate
that it's also a

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thing that you can do for block

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diagrams and system functions.

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There's one more type of
equivalence that I want to

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talk about.

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I call it feedback
equivalence.

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Here's our normal accumulator
rate.

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If I wanted to represent this
feedback system as a feed

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forward system, what
would I have to do?

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Well, the first time that
I sampled from x, it

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would just be y.

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So right now this diagram
matches for

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the first time step.

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On the second time step, if I
had an input from x from the

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previous time step, I would also
want to account for it by

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putting in a delay and then
summing it with the current

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value of x in order to get y.

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At the second time step, I
would want access to the

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starting value, the value from
the previous time step, and

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the value from the current
time step.

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And one more time, to exhaust
the example, at the third time

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step, my output would be a
linear combination of the

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starting value, the value from
the first time step, the value

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from the second time step, and
the value from the current

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third time step.

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We'd end up doing this
ad nauseum to model

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our feedback system.

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So it's difficult to do on
paper, but it turns out

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there's a great relationship
between these two equivalences

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and things that we already
know from--

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I want to say high school
calculus or

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possibly 18.01, 18.02.

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Geometric sequences.

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When we solved for the system
function, we found an

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expression for our
feedback system.

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If I wanted to find an
equivalent expression using

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this feed forward system, I
would look at this infinite

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summation of x terms.

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So if I wanted to know something
about the long-term

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behavior of the system, in terms
of this system function,

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I would solve for this
expression and then using my

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knowledge of geometric
sequences, in order to express

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the long-term behavior.

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In the general sense, in this
course, we're going to be

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looking at the unit sample
response of a system.

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What that means is, if the only
thing I ever do for input

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is a single value of 1 at
time 0, then what does

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my output look like?

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The reason we're looking at the
unit sample response is

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because it's (a) the simplest
way to look at the long-term

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behavior of a discrete linear
time invariant system.

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But the other reason (b) is --
once we have this, we can also

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use it to do things like to
make predictions about the

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long-term step response
and other more

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complicated input signals.

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In the case of the accumulator,
if I input 1 at

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time 0, my output is going
to be 1 forever more.

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That's reflected in
the coefficient of

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my geometric sequence.

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If I want to know what my
long-term response is going to

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look like, I can look at the
coefficient of R and make a

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decision about whether or not
I'm going to diverge or

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converge or do neither.

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So if I put a coefficient on R,
whatever p0 converges to is

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what my system is going
to converge to.

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So using my knowledge of p0, I
can make long-term predictions

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about the behavior
of the system.

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Next time I'm going to go over
some general classifications

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of those behaviors for the
system and how to more

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effectively use our knowledge
of p0 and how to deal with

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things like second
order systems.