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.