PROFESSOR: Hi.
Today I'd like to talk about
signals and systems again.
At this point, you're probably
familiar with the motivation
for why we're talking about
discrete linear time and
variant systems, and also with
a few of the representations
that we're going to end up
using in this course.
But you're still not sure what
it is that we're trying to
accomplish.
Or where's the part where we
get to predict the future
based on the fact that
we are capable of
manipulating these systems?
Well, we actually have
to be capable of
manipulating these systems.
And at this point, we can
describe this system as we see
it, but we can't also manipulate
its representation
in ways that make sense to us.
So the thing that I'm going
to do today is talk about
different system equivalences
and how to take a system and
solve for an expression that
represents a complex system
and also how if you know that
some things in your system are
equivalent, how you can
convert between them.
At that point, we should be
able to talk about poles,
which is how we're going to
actually predict the future.
So different equivalences that
I'd like to talk about.
I'm first going to briefly
review the facts that last
time we discovered the notion
of system function, right?
We can take a representation
of a system and abstract it
away into some sort of function,
where we take the
input as it's given to us and
then multiply it by this
function and then get
the output that
we're interested in.
How do we deal with something
more complex?
I mean, y is all the
way over here.
And we've got multiple
system functions.
And I don't even know what
happens here, but it doesn't
have to be that scary.
Let's break it down.
One of the easiest ways to
approach something like this
is to identify each position
where you have a new signal,
or if you were to sample here,
you would have a new signal,
and label those values
appropriately.
You can then start with your
final output and then back
solve for the values that you're
interested in as a
consequence of that
final output.
In this particular example, y
is going to be y2 plus y3.
y2 is going to be y1
times H2, where H2
is some system function.
And it probably is abstracting
away some combination of
gains, delays, and adders
like this one here.
y1 is going to be x times H1.
And Y3 is going to
be x times H3.
Now I've got all my expressions
in terms of either
y or something for which
I have an equivalent
expression for x.
So I can do my substitutions,
come up for an expression for
y over x, in terms of
H1, H2, and H3.
Here I've just made the
substitutions of the equations
above and factored out the x.
If I wanted the system function,
I would then just
divide by x.
And then I would have y over x
is equal to this expression.
The thing I wanted to indicate
is that if I wanted to
abstract this away into its own
box-- maybe I wanted like
a big H or an H0 or something
like that--
and it represented what was
happening in this top line,
cascading two system functions
is the functional equivalent
of multiplying them together.
So if I have an expression for
H1 and I have an expression
for H2, and I want the
expression that is equal to
cascading H1 and H2, I just
multiply them together.
Likewise, if I want an
expression for the linear
combination of two system
functions applied to an input
individually, like the
combination H1 and H2, and H3,
it's a summation of those
two values which
is expressed here.
This is the same as the
relationship that we reviewed
in a very basic sense when we
were originally doing the
accumulator.
The only thing I'm attempting
to indicate is that, that
relationship scales to an
arbitrary level of complexity.
So if you need to, you could
shift around these values, if
you can find some sort
of equivalence.
Let's see what happens when
H2 is equal to H3.
I'm going to take my
operator equation.
What this means is that if I
wanted to rewrite this block
diagram, I could do
so by doing--
This is really similar
to bubble pushing.
If you've done 6.004 or 6.002
and have experience with logic
gates, I just wanted to indicate
that it's also a
thing that you can do for block
diagrams and system functions.
There's one more type of
equivalence that I want to
talk about.
I call it feedback
equivalence.
Here's our normal accumulator
rate.
If I wanted to represent this
feedback system as a feed
forward system, what
would I have to do?
Well, the first time that
I sampled from x, it
would just be y.
So right now this diagram
matches for
the first time step.
On the second time step, if I
had an input from x from the
previous time step, I would also
want to account for it by
putting in a delay and then
summing it with the current
value of x in order to get y.
At the second time step, I
would want access to the
starting value, the value from
the previous time step, and
the value from the current
time step.
And one more time, to exhaust
the example, at the third time
step, my output would be a
linear combination of the
starting value, the value from
the first time step, the value
from the second time step, and
the value from the current
third time step.
We'd end up doing this
ad nauseum to model
our feedback system.
So it's difficult to do on
paper, but it turns out
there's a great relationship
between these two equivalences
and things that we already
know from--
I want to say high school
calculus or
possibly 18.01, 18.02.
Geometric sequences.
When we solved for the system
function, we found an
expression for our
feedback system.
If I wanted to find an
equivalent expression using
this feed forward system, I
would look at this infinite
summation of x terms.
So if I wanted to know something
about the long-term
behavior of the system, in terms
of this system function,
I would solve for this
expression and then using my
knowledge of geometric
sequences, in order to express
the long-term behavior.
In the general sense, in this
course, we're going to be
looking at the unit sample
response of a system.
What that means is, if the only
thing I ever do for input
is a single value of 1 at
time 0, then what does
my output look like?
The reason we're looking at the
unit sample response is
because it's (a) the simplest
way to look at the long-term
behavior of a discrete linear
time invariant system.
But the other reason (b) is --
once we have this, we can also
use it to do things like to
make predictions about the
long-term step response
and other more
complicated input signals.
In the case of the accumulator,
if I input 1 at
time 0, my output is going
to be 1 forever more.
That's reflected in
the coefficient of
my geometric sequence.
If I want to know what my
long-term response is going to
look like, I can look at the
coefficient of R and make a
decision about whether or not
I'm going to diverge or
converge or do neither.
So if I put a coefficient on R,
whatever p0 converges to is
what my system is going
to converge to.
So using my knowledge of p0, I
can make long-term predictions
about the behavior
of the system.
Next time I'm going to go over
some general classifications
of those behaviors for the
system and how to more
effectively use our knowledge
of p0 and how to deal with
things like second
order systems.
