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PROFESSOR: Last time,
we talked about the
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representation of linear
time-invariant systems through
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the convolution sum in the
discrete-time case and the
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convolution integral in the
continuous-time case.
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Now, although the derivation
was relatively
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straightforward, in fact, the
result is really kind of
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amazing because what it tells
us is that for linear
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time-invariant systems, if we
know the response of the
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system to a single impulse at
t = 0, or in fact, at any
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other time, then we can
determine from that its
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response to an arbitrary input
through the use of
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convolution.
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Furthermore, what we'll see as
the course develops is that,
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in fact, the class of linear
time-invariant systems is a
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very rich class.
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There are lots of systems
that have that property.
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And in addition, there are
lots of very interesting
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things that you can do with
linear time-invariant systems.
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In today's lecture, what I'd
like to begin with is focusing
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on convolution as an algebraic
operation.
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And we'll see that it has a
number of algebraic properties
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that in turn have important
implications for linear
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time-invariant systems.
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Then we'll turn to a discussion
of what the
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characterization of linear
time-invariant systems through
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convolution implies, in terms
of the relationship, of
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various other system properties
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to the impulse response.
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Let me begin by reminding you
of the basic result that we
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developed last time, which
is the convolution sum in
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discrete time, as I indicate
here, and the convolution
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integral in continuous time.
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And what the convolution sum,
or the convolution integral,
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tells us is how to relate the
output to the input and to the
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system impulse response.
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And the expression looks
basically the same in
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continuous time and
discrete time.
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And I remind you also that we
talked about a graphical
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interpretation, where
essentially, to graphically
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interpret convolution required,
or was developed, in
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terms of taking the system
impulse response, flipping it,
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sliding it past the input, and
positioned appropriately,
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depending on the value of the
independent variable for which
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we're computing the convolution,
and then
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multiplying and summing in the
discrete-time case, or
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integrating in the
continuous-time case.
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Now convolution, as an algebraic
operation, has a
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number of important
properties.
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One of the properties of
convolution is that it is what
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is referred to as commutative.
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Commutative means that we can
think either of convolving x
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with h, or h with x, and the
order in which that's done
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doesn't affect the
output result.
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So summarized here is what the
commutative operation is in
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discrete time, or in
continuous time.
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And it says, as I just
indicated, that x[n]
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convolved with h[n]
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is equal to h[n]
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convolved with x[n].
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Or the same, of course,
in continuous time.
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And in fact, in the lecture
last time, we worked an
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example where we had, in
discrete time, an impulse
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response, which was an
exponential, and an input,
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which is a unit step.
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And in the text, what you'll
find is the same example
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worked, except in that case, the
input is the exponential.
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And the system impulse
response is the step.
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And that corresponds to the
example in the text, which is
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example 3.1.
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And what happens in that example
is that, in fact, what
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you'll see is that the same
result occurs in example 3.1
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as we generated in
the lecture.
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And there's a similar comparison
in continuous time.
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This example was worked
in lecture.
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And this example is worked
in the text.
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In other words, the text works
the example where the system
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impulse response
is a unit step.
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And the input is
an exponential.
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All right, now the commutative
property, as I said, tells us
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that the order in which we do
convolution doesn't affect the
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result of the convolution.
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The same is true for
continuous time
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and discrete time.
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And in fact, for the other
algebraic properties that I'll
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talk about, the results are
exactly the same for
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continuous time and
discrete time.
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So in fact, what we can do is
drop the independent variable
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as an argument so that we
suppress any kind of
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difference between continuous
and discrete time.
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And suppressing the independent
variable, we then
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state the commutative property
as I've rewritten it here.
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Just x convolved with h equals
h convolved with x.
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The same in continuous time
and discrete time.
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Now, the derivation of the
commutative property is, more
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or less, some algebra
which you can follow
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through in the book.
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It involves some changes
of variables and some
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things of that sort.
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What I'd like to focus on with
that and other properties is
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not the derivation, which you
can see in the text, but
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rather the interpretation.
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So we have the commutative
property, and now there are
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two other important algebraic
properties.
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One being what is referred to
as the associative property,
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which tells us that if we have x
convolved with the result of
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convolving h1 with h2, that's
exactly the same as x
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convolved with h1, and that
result convolved with h2.
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And what this permits is for
us to write, for example, x
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convolved with h1 convolved with
h2 without any ambiguity
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because it doesn't matter from
the associative property how
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we group the terms together.
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The third important property is
what is referred to as the
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distributive property, namely
the fact that convolution
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distributes over addition.
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And what I mean by that is what
I've indicated here on
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the slide: that if I think of x
convolved with the sum of h1
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and h2, that's identical to
first convolving x with h1,
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also convolving x with h2, and
then adding the two together.
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And that result will be the
same as this result.
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So convolution is commutative,
associative, and it
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distributes over addition.
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Three very important algebraic
properties.
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And by the way, there are other
algebraic operations
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that have that same property.
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For example, multiplication
of numbers is likewise
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commutative, associative,
and distributive.
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Now let's look at what these
three properties imply
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specifically for linear
time-invariant systems.
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And as we'll see, the
implications are both very
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interesting and very
important.
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Let's begin with the commutative
property.
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And consider, in particular,
a system with an impulse
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response h.
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And I represent that by simply
writing the h inside the box.
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An input x and an output,
then, which is x * h.
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Now, since this operation is
commutative, I can write
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instead of x * h, I
can write h * x.
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And that would correspond to a
system with impulse response
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x, and input h, and
output then h * x.
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So the commutative property
tells us that for a linear
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time-invariant system, the
system output is independent
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of which function we call the
input and which function we
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call the impulse response.
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Kind of amazing actually.
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We can interchange the role of
input and impulse response.
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And from an output point of
view, the output or the system
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doesn't care.
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Now furthermore, if we combine
the commutative property with
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the associative property,
we get another
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very interesting result.
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Namely that if we have two
linear time-invariant systems
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in cascade, the overall system
is independent of the order in
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which they're cascaded.
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And in fact, in either case, the
cascade can be collapsed
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into a single system.
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To see this, let's first
consider the cascade of two
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systems, one with impulse
response h1, the other with
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impulse response h2.
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And the output of the first
system is then x * h1.
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And then that is the input
to the second system.
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And so the output of that is
that result convolved with h2.
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So this is the result of
cascading the two.
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And now we can use the
associative property to
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rewrite this as x * (h1
* h2), where we group
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these two terms together.
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And so using the associative
property, we now can collapse
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that into a single system with
an input, which is x, and
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impulse response, which
is h1 * h2.
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And the output is then x
convolved with the result of
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those two convolved.
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Next, we can apply the
commutative property.
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And the commutative property
says we could write this
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impulse response that way, or
we could write it this way.
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And since convolution is
commutative, the resulting
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output will be exactly
the same.
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And so these resulting outputs
will be exactly the same.
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And now, once again we can use
the associative property to
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group these two terms
together.
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And x * h2 corresponds to
putting x first through the
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system h2 and then that output
through the system h1.
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And so finally applying the
associative property again, as
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I just outlined, we can expand
that system back into two
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systems in cascade with h2
first and h1 second,
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OK, well that involves a certain
amount of algebraic
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manipulation.
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And that is not the algebraic
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manipulation that is important.
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It's the result that
it's important.
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And what the result says, to
reiterate, is if I have two
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linear time-invariant systems in
cascade, I can cascade them
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in any order, and the
result is the same.
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Now you might think, well gee,
maybe that actually applies to
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systems in general, whether
you put them
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this way or that way.
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But in fact, as we talked
about last time, and I
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illustrated with an example, in
general, if the systems are
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not linear and time-invariant,
then the order in which
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they're cascaded is important
to the interpretation of the
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overall system.
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For example, if one system took
the square root and the
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other system doubled the input,
taking the square root
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and then doubling gives us a
different answer than doubling
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first and then taking
the square root.
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And of course, one can construct
much more elaborate
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examples than that.
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So it's a property very
particular to linear
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time-invariant systems.
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And also one that we will
exploit many, many times as we
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go through this material.
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The final property related to an
interconnection of systems
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that I want to just indicate
develops out of the
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distributive property.
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And what it applies to is
an interpretation of the
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interconnection of systems
in parallel.
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Recall that the parallel
combination of systems
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corresponds, as I indicate here,
to a system in which we
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simultaneously feed the input
into h1 and h2, these
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representing the impulse
responses.
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And then, the outputs
are summed to
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form the overall output.
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And using the fact that
convolution distributes over
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addition, we can rewrite
this as x * (h1 + h2).
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And when we do that then, we
can recognize this as the
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output of a system with input x
and impulse response, which
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is the sum of these two
impulse responses.
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So for linear time-invariant
systems in parallel, we can,
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if we choose, replace that
interconnection by a single
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system whose impulse response
is simply the sum of those
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impulse responses.
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OK, now we have this very
powerful representation for
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linear time-invariant systems
in terms of convolution.
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And we've seen so far in this
lecture how convolution and
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the representation through the
impulse response leads to some
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important implications for
system interconnections.
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What I'd like to turn to now
are other system properties
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and see how, for linear
time-invariant systems in
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particular, other system
properties can be associated
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with particular properties or
characteristics of the system
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impulse response.
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And what we'll talk about are
a variety of properties.
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We'll talk about the issue of
memory, we'll talk about the
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issue of invertibility, and
we'll talk about the issue of
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causality and also stability.
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00:17:10,210 --> 00:17:13,230
Well, let's begin with
the issue of memory.
245
00:17:13,230 --> 00:17:18,660
And the question now is what are
the implications for the
246
00:17:18,660 --> 00:17:22,490
system impulse response for a
linear time-invariant system?
247
00:17:22,490 --> 00:17:25,440
Remember that we're always
imposing that on the system.
248
00:17:25,440 --> 00:17:28,319
What are the implications on
the impulse response if the
249
00:17:28,319 --> 00:17:32,350
system does or does
not have memory?
250
00:17:32,350 --> 00:17:36,440
Well, we can answer
that by looking at
251
00:17:36,440 --> 00:17:39,860
the convolution property.
252
00:17:39,860 --> 00:17:43,660
And we have here, as a reminder,
the convolution
253
00:17:43,660 --> 00:17:51,230
integral, which tells us how
x(tau) and h(t - tau) are
254
00:17:51,230 --> 00:17:56,020
combined to give us y(t).
255
00:17:56,020 --> 00:18:00,940
And what I've illustrated
above is a
256
00:18:00,940 --> 00:18:02,730
general kind of example.
257
00:18:02,730 --> 00:18:04,890
Here is x(tau).
258
00:18:04,890 --> 00:18:08,670
Here is h(t - tau).
259
00:18:08,670 --> 00:18:13,450
And to compute the output at
any time t, we would take
260
00:18:13,450 --> 00:18:18,160
these two, multiply them
together, and integrate from
261
00:18:18,160 --> 00:18:19,730
-infinity to +infinity.
262
00:18:22,740 --> 00:18:27,050
So the question then is what
can we say about h(t), the
263
00:18:27,050 --> 00:18:33,020
impulse response in order to
guarantee, let's say, that the
264
00:18:33,020 --> 00:18:37,090
output depends only on
the input at time t.
265
00:18:37,090 --> 00:18:43,900
Well, it's pretty much obvious
from looking at the graphs.
266
00:18:43,900 --> 00:18:51,740
If we only want the output to
depend on x(tau) at tau = t,
267
00:18:51,740 --> 00:18:59,360
then h(t - tau) better be
non-zero only at tau = t.
268
00:18:59,360 --> 00:19:04,990
And so the implication then is
that for the system to be
269
00:19:04,990 --> 00:19:12,100
memoryless, what we require is
that h(t - tau) be non-zero
270
00:19:12,100 --> 00:19:16,500
only at tau = t.
271
00:19:16,500 --> 00:19:19,960
So we want the impulse response
to be non-zero at
272
00:19:19,960 --> 00:19:22,230
only one point.
273
00:19:22,230 --> 00:19:24,670
We want it to contribute
something after we multiply
274
00:19:24,670 --> 00:19:26,490
and go through an integral.
275
00:19:26,490 --> 00:19:29,410
And in effect, what that says is
the only thing that it can
276
00:19:29,410 --> 00:19:30,680
be and meet all those
277
00:19:30,680 --> 00:19:33,400
conditions is a scaled impulse.
278
00:19:33,400 --> 00:19:39,310
So if the system is to be
memoryless, then that requires
279
00:19:39,310 --> 00:19:43,020
that the impulse response
be a scaled impulse.
280
00:19:43,020 --> 00:19:49,190
Any other impulse response then,
in essence, requires
281
00:19:49,190 --> 00:19:52,010
that the system have memory,
or implies that the system
282
00:19:52,010 --> 00:19:54,210
have memory.
283
00:19:54,210 --> 00:19:59,950
So for the continuous-time case
then, memoryless would
284
00:19:59,950 --> 00:20:03,760
correspond only to the impulse
response being proportional to
285
00:20:03,760 --> 00:20:05,060
an impulse.
286
00:20:05,060 --> 00:20:09,720
And in the discrete-time case,
a similar statement, in which
287
00:20:09,720 --> 00:20:14,590
case, the output is just
proportional to the input,
288
00:20:14,590 --> 00:20:17,730
again either in the
continuous-time or in the
289
00:20:17,730 --> 00:20:18,980
discrete-time case.
290
00:20:22,430 --> 00:20:22,670
All right.
291
00:20:22,670 --> 00:20:27,650
Now we can turn our attention
to the issue of system
292
00:20:27,650 --> 00:20:29,410
invertibility.
293
00:20:29,410 --> 00:20:35,460
And recall that what is meant by
invertibility of a system,
294
00:20:35,460 --> 00:20:37,690
or the inverse of a system.
295
00:20:37,690 --> 00:20:41,080
The inverse of a system is a
system, which when we cascade
296
00:20:41,080 --> 00:20:45,360
it with the one that we're
inquiring about, the overall
297
00:20:45,360 --> 00:20:48,110
cascade is the identity
system.
298
00:20:48,110 --> 00:20:53,370
In other words, the output
is equal to the input.
299
00:20:53,370 --> 00:20:56,980
So let's consider a system
with impulse
300
00:20:56,980 --> 00:21:00,440
response h, input is x.
301
00:21:00,440 --> 00:21:04,480
And let's say that the impulse
response of the inverse system
302
00:21:04,480 --> 00:21:07,930
is h_i, and the output is y.
303
00:21:07,930 --> 00:21:16,850
Then, the output of this system
is x * (h * h_i).
304
00:21:16,850 --> 00:21:21,830
And we want this to come
out equal to x.
305
00:21:21,830 --> 00:21:28,740
And what that requires than is
that this convolution just
306
00:21:28,740 --> 00:21:34,590
simply be equal to an impulse,
either in the discrete-time
307
00:21:34,590 --> 00:21:38,270
case or in the continuous-time
case.
308
00:21:38,270 --> 00:21:42,380
And under those conditions
then, h_i is equal to the
309
00:21:42,380 --> 00:21:44,630
inverse of h.
310
00:21:44,630 --> 00:21:49,760
Notationally, by the way, it's
often convenient to write
311
00:21:49,760 --> 00:21:53,620
instead of h_i as the impulse
response of the inverse,
312
00:21:53,620 --> 00:21:57,330
you'll find it convenient often
and more typical to
313
00:21:57,330 --> 00:22:01,970
write as the inverse, instead
of h_i, h^(-1).
314
00:22:05,500 --> 00:22:10,670
And we mean by that the inverse
impulse response.
315
00:22:10,670 --> 00:22:14,120
And one has to be careful not
to misinterpret this as the
316
00:22:14,120 --> 00:22:16,610
reciprocal of h(t) or h(n).
317
00:22:16,610 --> 00:22:20,260
What's meant in this notation
is the inverse system.
318
00:22:24,250 --> 00:22:33,930
Now, if h_i is the inverse of
h, is h the inverse of h_i?
319
00:22:33,930 --> 00:22:38,710
Well, it seems like that ought
to be plausible or perhaps
320
00:22:38,710 --> 00:22:40,060
make sense.
321
00:22:40,060 --> 00:22:43,470
The question, if you believe
that the answer is yes, is
322
00:22:43,470 --> 00:22:47,300
how, in fact, do you
verify that?
323
00:22:47,300 --> 00:22:49,860
And I'll leave it to you
to think about it.
324
00:22:49,860 --> 00:22:51,020
The answer is yes,
325
00:22:51,020 --> 00:22:54,320
that if h_i is the inverse
of h, then h is
326
00:22:54,320 --> 00:22:56,140
the inverse of h_i.
327
00:22:56,140 --> 00:23:00,800
And the key to showing that is
to exploit the fact that when
328
00:23:00,800 --> 00:23:03,800
we take these systems and
cascade them, we can cascade
329
00:23:03,800 --> 00:23:05,050
them in either order.
330
00:23:07,920 --> 00:23:12,200
All right now let's turn to
another system property, the
331
00:23:12,200 --> 00:23:14,760
property of stability.
332
00:23:14,760 --> 00:23:21,160
And again, we can tie that
property directly to issues
333
00:23:21,160 --> 00:23:27,010
related, in particular, to the
system impulse response.
334
00:23:27,010 --> 00:23:30,845
Now, stability is defined as
we've chosen to define it and
335
00:23:30,845 --> 00:23:34,530
as I've defined it previously,
as bounded-input
336
00:23:34,530 --> 00:23:35,940
bounded-output stability.
337
00:23:35,940 --> 00:23:38,140
In other words, for every
bounded input
338
00:23:38,140 --> 00:23:40,750
is a bounded output.
339
00:23:40,750 --> 00:23:41,850
What you can show--
340
00:23:41,850 --> 00:23:45,690
and I won't go through the
algebra here; it's gone
341
00:23:45,690 --> 00:23:47,700
through in the book--
342
00:23:47,700 --> 00:23:53,160
is that a necessary and
sufficient condition for a
343
00:23:53,160 --> 00:23:57,550
linear time-invariant system
to be stable in the
344
00:23:57,550 --> 00:24:03,800
bounded-input bounded-output
sense is that the impulse
345
00:24:03,800 --> 00:24:08,360
response be what is referred
to as absolutely summable.
346
00:24:08,360 --> 00:24:12,410
In other words, if you take the
absolute values and sum
347
00:24:12,410 --> 00:24:16,300
them over infinite limits,
that's finite.
348
00:24:16,300 --> 00:24:21,300
Or in the continuous-time
case, that the impulse
349
00:24:21,300 --> 00:24:23,530
response is absolutely
integrable.
350
00:24:23,530 --> 00:24:27,450
In other words, if you take the
absolute values of h(t)
351
00:24:27,450 --> 00:24:29,690
and integrate, that's finite.
352
00:24:29,690 --> 00:24:34,370
And under those conditions,
the system is stable.
353
00:24:34,370 --> 00:24:39,070
If those conditions are
violated, then for sure, as
354
00:24:39,070 --> 00:24:42,950
you'll see in the text, the
system is unstable.
355
00:24:42,950 --> 00:24:46,010
So stability can also
be tied to the
356
00:24:46,010 --> 00:24:47,390
system impulse response.
357
00:24:50,470 --> 00:24:54,130
Now, the next property that I
want to talk about is the
358
00:24:54,130 --> 00:24:56,670
property of causality.
359
00:24:56,670 --> 00:25:01,890
And before I do, what I'd like
to do is introduce a
360
00:25:01,890 --> 00:25:04,690
peripheral result that
we'll then use--
361
00:25:04,690 --> 00:25:06,870
when we talked about
causality--
362
00:25:06,870 --> 00:25:10,690
namely what's referred to as the
zero input response of a
363
00:25:10,690 --> 00:25:11,940
linear system.
364
00:25:14,820 --> 00:25:18,570
The basic result, which is a
very interesting and useful
365
00:25:18,570 --> 00:25:21,180
one, is that for a
linear system--
366
00:25:21,180 --> 00:25:24,350
and in fact, it's whether it's
time-invariant or not, that
367
00:25:24,350 --> 00:25:26,600
this applies--
368
00:25:26,600 --> 00:25:31,640
if you put nothing into it,
you get nothing out of it.
369
00:25:31,640 --> 00:25:40,460
So if we have an input x(t)
which is 0 for all t, and if
370
00:25:40,460 --> 00:25:49,040
the output of that system is
y(t), if the input is 0 for
371
00:25:49,040 --> 00:25:54,440
all time, then the output
likewise is 0 for all time.
372
00:25:54,440 --> 00:26:00,380
That's true for continuous time,
and it's also true for
373
00:26:00,380 --> 00:26:01,630
discrete time.
374
00:26:04,750 --> 00:26:10,380
And in fact, to show that
result is pretty much
375
00:26:10,380 --> 00:26:11,280
straightforward.
376
00:26:11,280 --> 00:26:15,570
We could do it either by using
convolution, which would, of
377
00:26:15,570 --> 00:26:18,780
course, be associated with
linearity and time invariance.
378
00:26:18,780 --> 00:26:22,960
But in fact, we can show that
property relatively easily by
379
00:26:22,960 --> 00:26:27,350
simply using the fact that, for
a linear system, what we
380
00:26:27,350 --> 00:26:35,180
know is that if we have an
input x(t) with an output
381
00:26:35,180 --> 00:26:42,390
y(t), then if we scale the
input, then the output scales
382
00:26:42,390 --> 00:26:43,940
accordingly.
383
00:26:43,940 --> 00:26:51,290
Well, we can simply choose, as
the scale factor, a = 0.
384
00:26:51,290 --> 00:26:54,950
And if we do that, it
says put nothing in,
385
00:26:54,950 --> 00:26:57,370
you get nothing out.
386
00:26:57,370 --> 00:27:02,380
And what we'll see is that has
some important implications in
387
00:27:02,380 --> 00:27:05,420
terms of causality.
388
00:27:05,420 --> 00:27:09,400
It's important, though, while
we're on it, to stress that
389
00:27:09,400 --> 00:27:11,750
not every system, obviously,
has that property.
390
00:27:11,750 --> 00:27:14,380
That if you put nothing in,
you get nothing out.
391
00:27:14,380 --> 00:27:19,980
A simple example is, let's say,
a battery, let's say not
392
00:27:19,980 --> 00:27:21,330
connected to anything.
393
00:27:21,330 --> 00:27:25,250
The output is six volts no
matter what the input is.
394
00:27:25,250 --> 00:27:29,970
And it of course then doesn't
have this zero response to a
395
00:27:29,970 --> 00:27:31,150
zero input.
396
00:27:31,150 --> 00:27:35,170
It's very particular
to linear systems.
397
00:27:35,170 --> 00:27:40,640
All right, well now let's see
what this means for causality.
398
00:27:40,640 --> 00:27:45,500
To remind you, causality says,
in effect, that the system
399
00:27:45,500 --> 00:27:47,060
can't anticipate the input.
400
00:27:47,060 --> 00:27:50,260
That's what, basically,
causality means.
401
00:27:50,260 --> 00:27:54,470
When we talked about it
previously, we defined it in a
402
00:27:54,470 --> 00:27:59,100
variety of ways, one of which
was the statement that if two
403
00:27:59,100 --> 00:28:05,130
inputs are identical up until
some time, then the outputs
404
00:28:05,130 --> 00:28:08,140
must be identical up until
the same time.
405
00:28:08,140 --> 00:28:11,880
The reason, kind of intuitively,
is that if the
406
00:28:11,880 --> 00:28:13,190
system is causal--
407
00:28:13,190 --> 00:28:15,290
so it can't anticipate
the future--
408
00:28:15,290 --> 00:28:18,970
it can't anticipate whether
these two identical inputs are
409
00:28:18,970 --> 00:28:22,530
sometime later going to change
from each other or not.
410
00:28:25,670 --> 00:28:30,410
So causality, in general, is
simply this statement, either
411
00:28:30,410 --> 00:28:34,030
continuous-time or
discrete-time.
412
00:28:34,030 --> 00:28:37,200
And now, so let's look
at what that means
413
00:28:37,200 --> 00:28:39,400
for a linear system.
414
00:28:39,400 --> 00:28:44,020
For a linear system, what that
corresponds to or could be
415
00:28:44,020 --> 00:28:51,500
translated to is a statement
that says that if x(t) is 0,
416
00:28:51,500 --> 00:28:58,050
for t less than t_0, then
y(t) must be 0 for t
417
00:28:58,050 --> 00:29:00,580
less than t_0 also.
418
00:29:00,580 --> 00:29:07,790
And so what that, in effect,
says, is that the system--
419
00:29:07,790 --> 00:29:14,010
for a linear system to be
causal, it must have the
420
00:29:14,010 --> 00:29:19,200
property sometimes referred to
as initial rest, meaning it
421
00:29:19,200 --> 00:29:23,850
doesn't respond until there's
some input that happens.
422
00:29:23,850 --> 00:29:26,250
That it's initially
at rest until the
423
00:29:26,250 --> 00:29:29,680
input becomes non-zero.
424
00:29:29,680 --> 00:29:31,050
Now, why is this true?
425
00:29:31,050 --> 00:29:37,690
Why is this a consequence of
causality for linear systems?
426
00:29:37,690 --> 00:29:40,980
Well, the reason is we know that
if we put nothing in, we
427
00:29:40,980 --> 00:29:43,630
get nothing out.
428
00:29:43,630 --> 00:29:48,320
If we have an input that's 0 for
t less than t_0, and the
429
00:29:48,320 --> 00:29:51,830
system can't anticipate whether
that input is going to
430
00:29:51,830 --> 00:29:58,950
change from 0 or not, then the
system must generate an output
431
00:29:58,950 --> 00:30:03,940
that's 0 up until that time,
following the principle that
432
00:30:03,940 --> 00:30:07,360
if two inputs are identical up
until some time, the outputs
433
00:30:07,360 --> 00:30:10,660
must be identical up until
the same time.
434
00:30:10,660 --> 00:30:18,750
So this basic result for linear
systems is essentially
435
00:30:18,750 --> 00:30:24,590
a consequence of the statement
that for a linear system, zero
436
00:30:24,590 --> 00:30:29,010
in gives us zero out.
437
00:30:33,490 --> 00:30:40,390
Now, that tells us
how to interpret
438
00:30:40,390 --> 00:30:43,190
causality for linear systems.
439
00:30:43,190 --> 00:30:46,360
Now, let's proceed to linear
time-invariant systems.
440
00:30:46,360 --> 00:30:50,900
And in fact, we can carry the
point one step further.
441
00:30:50,900 --> 00:30:55,380
In particular, a necessary and
sufficient condition for
442
00:30:55,380 --> 00:30:59,840
causality in the case of linear
time-invariant systems
443
00:30:59,840 --> 00:31:08,570
is that the impulse response be
0, for t less than 0 in the
444
00:31:08,570 --> 00:31:13,030
continuous-time case, or for
n less than 0 in the
445
00:31:13,030 --> 00:31:14,630
discrete-time case.
446
00:31:14,630 --> 00:31:18,660
So for linear time-invariant
systems, causality is
447
00:31:18,660 --> 00:31:24,310
equivalent to the impulse
response being 0 up until t or
448
00:31:24,310 --> 00:31:27,090
n equal to 0.
449
00:31:27,090 --> 00:31:35,330
Now, to show this essentially
follows by first considering
450
00:31:35,330 --> 00:31:42,710
why causality would imply
that this is true.
451
00:31:42,710 --> 00:31:47,790
And that follows because of the
straightforward fact that
452
00:31:47,790 --> 00:31:50,910
the impulse itself is
0 for t less than 0.
453
00:31:53,610 --> 00:31:58,480
And what we saw before is that
for any linear system,
454
00:31:58,480 --> 00:32:02,280
causality requires that if the
input is 0 up until some time,
455
00:32:02,280 --> 00:32:06,170
the output must be 0 up
until the same time.
456
00:32:06,170 --> 00:32:11,740
And so that's showing the
result in one direction.
457
00:32:11,740 --> 00:32:15,775
To show the result in the other
direction, namely to
458
00:32:15,775 --> 00:32:20,820
show that if, in fact, the
impulse response satisfies
459
00:32:20,820 --> 00:32:23,860
that condition, then the
system is causal.
460
00:32:23,860 --> 00:32:28,170
While I won't work through it
in detail, it essentially
461
00:32:28,170 --> 00:32:36,830
boils down to recognizing that
in the convolution sum, or in
462
00:32:36,830 --> 00:32:41,880
the convolution integral, if,
in fact, that condition is
463
00:32:41,880 --> 00:32:46,950
satisfied on the impulse
response, then the upper limit
464
00:32:46,950 --> 00:32:48,880
on the sum, in the
discrete-time
465
00:32:48,880 --> 00:32:50,900
case, changes to n.
466
00:32:50,900 --> 00:32:54,370
And in the continuous-time
case, changes to t.
467
00:32:54,370 --> 00:33:00,940
And that, in effect, says that
values of the input only from
468
00:33:00,940 --> 00:33:05,840
-infinity up to time n are
used in computing y[n].
469
00:33:05,840 --> 00:33:09,770
And a similar kind
of result for the
470
00:33:09,770 --> 00:33:11,140
continuous-time case y(t).
471
00:33:15,380 --> 00:33:20,330
OK, so we've seen how the
impulse response, or rather
472
00:33:20,330 --> 00:33:23,170
how certain system properties
in the linear time-invariant
473
00:33:23,170 --> 00:33:28,920
case can, be converted into
various requirements on the
474
00:33:28,920 --> 00:33:31,670
impulse response of a linear
time-invariant system, the
475
00:33:31,670 --> 00:33:35,440
impulse response being a
complete characterization.
476
00:33:35,440 --> 00:33:39,390
Let's look at some particular
examples just to kind of
477
00:33:39,390 --> 00:33:42,140
cement the ideas further.
478
00:33:42,140 --> 00:33:46,190
And let's begin with a system
that you've seen previously,
479
00:33:46,190 --> 00:33:48,165
which is an accumulator.
480
00:33:48,165 --> 00:33:55,860
An accumulator, as you recall,
has an output y[n], which is
481
00:33:55,860 --> 00:34:03,080
the accumulated value of the
input from -infinity up to n.
482
00:34:03,080 --> 00:34:07,660
Now, you've seen in the impulse
in a previous lecture,
483
00:34:07,660 --> 00:34:11,239
or rather in the video course
manual for a previous lecture,
484
00:34:11,239 --> 00:34:15,429
that an accumulator is a linear
time-invariant system.
485
00:34:15,429 --> 00:34:19,760
And in fact, its impulse
response is the accumulated
486
00:34:19,760 --> 00:34:21,370
values of an impulse.
487
00:34:21,370 --> 00:34:24,855
Namely, the impulse response
is equal to a step.
488
00:34:27,650 --> 00:34:34,010
So what we want to answer is,
knowing what that impulse
489
00:34:34,010 --> 00:34:36,929
response is, what some
properties are of the
490
00:34:36,929 --> 00:34:38,280
accumulator.
491
00:34:38,280 --> 00:34:41,420
And let's first ask
about memory.
492
00:34:41,420 --> 00:34:46,679
Well, we recognize that the
impulse response is not simply
493
00:34:46,679 --> 00:34:47,300
an impulse.
494
00:34:47,300 --> 00:34:48,830
In fact, it's a step.
495
00:34:48,830 --> 00:34:51,560
And so this implies what?
496
00:34:51,560 --> 00:34:55,530
Well, it implies that the
system has memory.
497
00:35:00,050 --> 00:35:06,780
Second, the impulse response
is 0 for n less than 0.
498
00:35:06,780 --> 00:35:11,180
That implies that the
system is causal.
499
00:35:15,180 --> 00:35:22,530
And finally, if we look at the
sum of the absolute values of
500
00:35:22,530 --> 00:35:24,930
the impulse response
from -infinity to
501
00:35:24,930 --> 00:35:27,380
+infinity, this is a step.
502
00:35:27,380 --> 00:35:30,790
If we accumulate those values
over infinite limits, then
503
00:35:30,790 --> 00:35:35,210
that in fact comes out
to be infinite.
504
00:35:35,210 --> 00:35:40,140
And so what that implies, then,
is that the accumulator
505
00:35:40,140 --> 00:35:43,490
is not stable in the
bounded-input
506
00:35:43,490 --> 00:35:44,780
bounded-output sense.
507
00:35:48,240 --> 00:35:50,410
Now I want to turn to
some other systems.
508
00:35:50,410 --> 00:35:52,630
But while we're on the
accumulator, I just want to
509
00:35:52,630 --> 00:35:55,900
draw your attention to the fact,
which will kind of come
510
00:35:55,900 --> 00:36:01,350
up in a variety of ways again
later, that we can rewrite the
511
00:36:01,350 --> 00:36:04,840
equation for an accumulator,
the difference equation, by
512
00:36:04,840 --> 00:36:10,460
recognizing that we could, in
fact, write the output as the
513
00:36:10,460 --> 00:36:14,540
accumulated values up to
time n - 1 and then
514
00:36:14,540 --> 00:36:17,120
add on the last value.
515
00:36:17,120 --> 00:36:23,510
And in fact, if we do that, this
corresponds to y[n-1].
516
00:36:23,510 --> 00:36:28,330
And so we could rewrite this
difference equation as y[n]
517
00:36:28,330 --> 00:36:30,090
= y[n-1]
518
00:36:30,090 --> 00:36:31,340
+ x[n].
519
00:36:31,340 --> 00:36:34,630
So the output is the
previously-computed output
520
00:36:34,630 --> 00:36:36,800
plus the input.
521
00:36:36,800 --> 00:36:42,170
Expressed that way, what that
corresponds to is what is
522
00:36:42,170 --> 00:36:44,400
called a recursive difference
equation.
523
00:36:44,400 --> 00:36:47,980
And different equations will
be a topic of considerable
524
00:36:47,980 --> 00:36:51,290
emphasis in the next lecture.
525
00:36:51,290 --> 00:36:53,520
Now, does an accumulator
have an inverse?
526
00:36:53,520 --> 00:36:57,100
Well, the answer is,
in fact, yes.
527
00:36:57,100 --> 00:37:02,190
And let's look at what the
inverse of the accumulator is.
528
00:37:02,190 --> 00:37:06,290
The impulse response of the
accumulator is a step.
529
00:37:06,290 --> 00:37:11,100
To inquire about its inverse,
we inquire about whether
530
00:37:11,100 --> 00:37:15,780
there's a system, which when
we cascade the accumulator
531
00:37:15,780 --> 00:37:20,510
with that system, which I'm
calling its inverse, we get an
532
00:37:20,510 --> 00:37:22,410
impulse out.
533
00:37:22,410 --> 00:37:23,800
Well, let's see.
534
00:37:23,800 --> 00:37:28,060
The impulse response of the
accumulator is a step.
535
00:37:28,060 --> 00:37:29,820
We want to put the step
into something
536
00:37:29,820 --> 00:37:32,470
and get out an impulse.
537
00:37:32,470 --> 00:37:37,500
And in fact, what you recall
from the lecture in which we
538
00:37:37,500 --> 00:37:42,120
introduced steps and impulses,
the impulse is, in fact, the
539
00:37:42,120 --> 00:37:44,630
first difference of
the units step.
540
00:37:44,630 --> 00:37:50,210
So we have a difference equation
that describes for us
541
00:37:50,210 --> 00:37:54,660
how the impulse is related
to the step.
542
00:37:54,660 --> 00:37:59,770
And so if this system does this,
the output will be that,
543
00:37:59,770 --> 00:38:01,420
an impulse.
544
00:38:01,420 --> 00:38:04,465
And so if we think of x_2[n]
545
00:38:07,580 --> 00:38:09,470
as the input and y_2[n]
546
00:38:09,470 --> 00:38:14,410
as the output, then the
difference equation for the
547
00:38:14,410 --> 00:38:18,330
inverse system is what
I've indicated here.
548
00:38:18,330 --> 00:38:22,200
And if we want to look at the
impulse response of that, we
549
00:38:22,200 --> 00:38:25,120
can then inquire as to what
the response is with an
550
00:38:25,120 --> 00:38:26,830
impulse in.
551
00:38:26,830 --> 00:38:31,290
And what develops in a
straightforward way then is
552
00:38:31,290 --> 00:38:37,580
delta[n], which is our impulse
input, minus delta[n-1]
553
00:38:37,580 --> 00:38:40,530
is equal to the impulse
response
554
00:38:40,530 --> 00:38:42,730
of the inverse system.
555
00:38:42,730 --> 00:38:45,260
So I'll write that
as h^(-1)[n]
556
00:38:45,260 --> 00:38:47,410
(h-inverse of n).
557
00:38:47,410 --> 00:38:52,420
Now, we have then that the
accumulator has an inverse.
558
00:38:52,420 --> 00:38:54,420
And this is the inverse.
559
00:38:54,420 --> 00:38:57,910
And you can examine issues
of memory, stability,
560
00:38:57,910 --> 00:38:59,430
causality, et cetera.
561
00:38:59,430 --> 00:39:05,170
What you'll find is that the
system has memory, the inverse
562
00:39:05,170 --> 00:39:05,810
accumulator.
563
00:39:05,810 --> 00:39:09,360
It's stable, and it's causal.
564
00:39:09,360 --> 00:39:12,360
And it's interesting to note, by
the way, that although the
565
00:39:12,360 --> 00:39:17,620
accumulator was an unstable
system, the inverse of the
566
00:39:17,620 --> 00:39:19,930
accumulator is a
stable system.
567
00:39:19,930 --> 00:39:24,680
In general, if the system is
stable, its inverse does not
568
00:39:24,680 --> 00:39:26,940
have to be stable
or vice versa.
569
00:39:26,940 --> 00:39:28,365
And the same thing
with causality.
570
00:39:32,210 --> 00:39:38,360
OK now, there are a number of
other examples, which, of
571
00:39:38,360 --> 00:39:40,200
course, we could discuss.
572
00:39:40,200 --> 00:39:46,830
And let me just quickly point
to one example, which is a
573
00:39:46,830 --> 00:39:50,640
difference equation, as
I've indicated here.
574
00:39:50,640 --> 00:39:55,980
And as we'll talk about in
more detail in our next
575
00:39:55,980 --> 00:39:59,480
lecture, where we'll get
involved in a fairly detailed
576
00:39:59,480 --> 00:40:01,360
discussion of linear
constant-coefficient
577
00:40:01,360 --> 00:40:04,690
difference and differential
equations, this falls into
578
00:40:04,690 --> 00:40:06,290
that category.
579
00:40:06,290 --> 00:40:10,960
And under the imposition of
what's referred to as initial
580
00:40:10,960 --> 00:40:17,970
rest, which corresponds to the
response being 0 up until the
581
00:40:17,970 --> 00:40:22,450
time that the input becomes
non-zero, the impulse response
582
00:40:22,450 --> 00:40:25,410
is a^n times u[n].
583
00:40:25,410 --> 00:40:29,490
And something that you'll be
asked to think about in the
584
00:40:29,490 --> 00:40:33,490
video course manual is whether
that system has memory,
585
00:40:33,490 --> 00:40:36,550
whether it's causal, and
whether it's stable.
586
00:40:36,550 --> 00:40:41,700
And likewise, for a linear
constant coefficient
587
00:40:41,700 --> 00:40:46,070
differential equation, the
specific one that I've
588
00:40:46,070 --> 00:40:50,020
indicated here, under the
assumption of initial rest,
589
00:40:50,020 --> 00:40:54,710
the impulse response is
e^(-2t) times u(t).
590
00:40:54,710 --> 00:40:58,810
And in the video course manual
again, you'll be asked to
591
00:40:58,810 --> 00:41:02,930
examine whether the system has
memory, whether it's causal,
592
00:41:02,930 --> 00:41:06,410
and whether it's stable.
593
00:41:06,410 --> 00:41:10,420
OK well, as I've indicated, in
the next lecture we'll return
594
00:41:10,420 --> 00:41:13,270
to a much more detailed
discussion of linear
595
00:41:13,270 --> 00:41:15,180
constant-coefficient
differential
596
00:41:15,180 --> 00:41:17,190
and difference equations.
597
00:41:17,190 --> 00:41:22,480
Now, what I'd like to finally do
in this lecture is use the
598
00:41:22,480 --> 00:41:26,410
notion of convolution in a much
different way to help us
599
00:41:26,410 --> 00:41:30,030
with a problem that I
alluded to earlier.
600
00:41:30,030 --> 00:41:33,800
In particular, the issue of how
to deal with some of the
601
00:41:33,800 --> 00:41:37,200
mathematical difficulties
associated with
602
00:41:37,200 --> 00:41:39,620
impulses and steps.
603
00:41:39,620 --> 00:41:43,540
Now, let me begin by
illustrating kind of what the
604
00:41:43,540 --> 00:41:49,845
problem is and an example of the
kind of paradox that you
605
00:41:49,845 --> 00:41:53,480
sort of run into when dealing
with impulse functions and
606
00:41:53,480 --> 00:41:54,730
step functions.
607
00:41:57,000 --> 00:42:00,980
Let's consider, first of all,
a system, which is the
608
00:42:00,980 --> 00:42:02,520
identity system.
609
00:42:02,520 --> 00:42:07,090
And so the output is, of course,
equal to the input.
610
00:42:07,090 --> 00:42:12,180
And again, we can talk about
that either in continuous time
611
00:42:12,180 --> 00:42:14,800
or in discrete time.
612
00:42:14,800 --> 00:42:18,990
Well, we know that the function
that you convolve
613
00:42:18,990 --> 00:42:23,610
with a signal to retain the
signal is an impulse.
614
00:42:23,610 --> 00:42:26,650
And so that means that the
impulse response of an
615
00:42:26,650 --> 00:42:28,740
identity system is an impulse.
616
00:42:28,740 --> 00:42:32,220
Makes logical sense.
617
00:42:32,220 --> 00:42:36,300
Furthermore, if I take two
identity systems and cascade
618
00:42:36,300 --> 00:42:40,240
them, I put in an input, get
the same thing out of the
619
00:42:40,240 --> 00:42:40,900
first system.
620
00:42:40,900 --> 00:42:42,190
That goes into the
second system.
621
00:42:42,190 --> 00:42:44,125
Get the same thing out
of the second.
622
00:42:44,125 --> 00:42:51,540
In other words, if I have two
identity systems in cascade,
623
00:42:51,540 --> 00:42:55,840
the cascade, likewise, is
an identity system.
624
00:42:55,840 --> 00:42:59,020
In other words, this
overall system is
625
00:42:59,020 --> 00:43:03,280
also an identity system.
626
00:43:03,280 --> 00:43:07,610
And the implication there is
that the impulse response of
627
00:43:07,610 --> 00:43:09,110
this is an impulse.
628
00:43:09,110 --> 00:43:11,750
The impulse response of
this is an impulse.
629
00:43:11,750 --> 00:43:16,560
And the convolution of those
two is also an impulse.
630
00:43:16,560 --> 00:43:21,860
So for continuous time, we
require, then, that an impulse
631
00:43:21,860 --> 00:43:24,550
convolved with itself
is an impulse.
632
00:43:24,550 --> 00:43:28,850
And the same thing for
discrete time.
633
00:43:28,850 --> 00:43:33,280
Now, in discrete time, we don't
have any particular
634
00:43:33,280 --> 00:43:34,220
problem with that.
635
00:43:34,220 --> 00:43:37,130
If you think about convolving
these together, it's a
636
00:43:37,130 --> 00:43:42,850
straightforward mathematical
operation since the impulse in
637
00:43:42,850 --> 00:43:47,300
discrete time is very
nicely defined.
638
00:43:47,300 --> 00:43:51,070
However, in continuous time, we
have to be somewhat careful
639
00:43:51,070 --> 00:43:54,710
about the definition of the
impulse because it was the
640
00:43:54,710 --> 00:43:55,870
derivative of a step.
641
00:43:55,870 --> 00:43:57,930
A step has a discontinuity.
642
00:43:57,930 --> 00:44:01,190
You can't really differentiate
at a discontinuity.
643
00:44:01,190 --> 00:44:06,020
And the way that we dealt with
that was to expand out the
644
00:44:06,020 --> 00:44:09,570
discontinuity so that it had
some finite time region in
645
00:44:09,570 --> 00:44:11,090
which it happened.
646
00:44:11,090 --> 00:44:14,080
When we did that, we ended up
with a definition for the
647
00:44:14,080 --> 00:44:18,200
impulse, which was the
limiting form of this
648
00:44:18,200 --> 00:44:23,510
function, which is a rectangle
of width Delta, and height 1 /
649
00:44:23,510 --> 00:44:27,120
Delta, and an area equal to 1.
650
00:44:27,120 --> 00:44:33,680
Now, if we think of convolving
this signal with itself, the
651
00:44:33,680 --> 00:44:37,480
impulse being the limiting
form of this, then the
652
00:44:37,480 --> 00:44:42,690
convolution of this with itself
is a triangle of width
653
00:44:42,690 --> 00:44:47,290
2 Delta, height 1 /
Delta, and area 1.
654
00:44:47,290 --> 00:44:51,560
In other words, this triangular
function is this
655
00:44:51,560 --> 00:44:59,030
approximation delta_Delta(t)
convolved with delta_Delta(t).
656
00:44:59,030 --> 00:45:05,000
And since the limit of this
would correspond to the
657
00:45:05,000 --> 00:45:11,480
impulse response of the identity
system convolved with
658
00:45:11,480 --> 00:45:17,470
itself, the implication is that
not only should the top
659
00:45:17,470 --> 00:45:23,600
function, this one, correspond
in its limiting form to an
660
00:45:23,600 --> 00:45:28,130
impulse, but also this should
correspond in its limiting
661
00:45:28,130 --> 00:45:29,990
form to an impulse.
662
00:45:29,990 --> 00:45:32,750
So one could wonder well,
what is an impulse?
663
00:45:32,750 --> 00:45:34,320
Is it this one in the limit?
664
00:45:34,320 --> 00:45:35,650
Or is it this one
in the limit?
665
00:45:42,880 --> 00:45:45,420
Now, beyond that-- so kind of
what this suggests is that in
666
00:45:45,420 --> 00:45:48,360
the limiting form, you kind of
run into a contradiction
667
00:45:48,360 --> 00:45:50,920
unless you don't try to
distinguish between this
668
00:45:50,920 --> 00:45:53,170
rectangle and the triangle.
669
00:45:53,170 --> 00:45:57,050
Things get even worse when you
think about what happens when
670
00:45:57,050 --> 00:45:59,160
you put an impulse into
a differentiator.
671
00:45:59,160 --> 00:46:03,630
And a differentiator is a very
commonly occurring system.
672
00:46:03,630 --> 00:46:08,350
In particular, suppose we had a
system for which the output
673
00:46:08,350 --> 00:46:11,010
was the derivative
of the input.
674
00:46:11,010 --> 00:46:17,500
So if we put in x(t), we
got out dx(t) / dt.
675
00:46:17,500 --> 00:46:20,360
If I put in an impulse, or if
I talked about the impulse
676
00:46:20,360 --> 00:46:23,130
response, what is that?
677
00:46:23,130 --> 00:46:27,610
And of course, the problem is
that if you think that the
678
00:46:27,610 --> 00:46:31,880
impulse itself is very badly
behaved, then what about its
679
00:46:31,880 --> 00:46:36,110
derivative, which is not only
infinitely big, but there's a
680
00:46:36,110 --> 00:46:38,350
positive-going one, and a
negative-going one, and the
681
00:46:38,350 --> 00:46:40,440
difference between there
has some area.
682
00:46:40,440 --> 00:46:42,910
And you end up in a
lot of difficulty.
683
00:46:45,930 --> 00:46:51,250
Well, the way around this,
formally, is through a set of
684
00:46:51,250 --> 00:46:56,140
mathematics referred to as
generalized functions.
685
00:46:56,140 --> 00:46:58,110
We won't be quite that formal.
686
00:46:58,110 --> 00:47:01,340
But I'd like to, at least,
suggest what the essence of
687
00:47:01,340 --> 00:47:02,530
that formality is.
688
00:47:02,530 --> 00:47:07,360
And it really helps us in
interpreting the impulses in
689
00:47:07,360 --> 00:47:09,620
steps and functions
of that type.
690
00:47:09,620 --> 00:47:13,660
And what it is is an operational
definition of
691
00:47:13,660 --> 00:47:16,560
steps, impulses, and their
derivatives in
692
00:47:16,560 --> 00:47:18,360
the following sense.
693
00:47:18,360 --> 00:47:21,600
Usually when we talk about a
function, we talk about what
694
00:47:21,600 --> 00:47:24,910
the value of the function is
at any instant of time.
695
00:47:24,910 --> 00:47:26,810
And of course, the trouble
with an impulse is it's
696
00:47:26,810 --> 00:47:30,930
infinitely big, in zero width,
and has some area, et cetera.
697
00:47:30,930 --> 00:47:35,310
What we can turn to is what is
referred to as an operational
698
00:47:35,310 --> 00:47:41,550
definition where the operational
definition is
699
00:47:41,550 --> 00:47:45,790
related not to what the impulse
is, but to what the
700
00:47:45,790 --> 00:47:50,760
impulse does under the operation
of convolution.
701
00:47:50,760 --> 00:47:52,540
So what is an impulse?
702
00:47:52,540 --> 00:47:56,110
An impulse is something,
which under
703
00:47:56,110 --> 00:47:59,460
convolution, retains the function.
704
00:47:59,460 --> 00:48:04,200
And that then can serve as a
definition of the impulse.
705
00:48:04,200 --> 00:48:06,410
Well, let's see where
that gets us.
706
00:48:06,410 --> 00:48:11,060
Suppose that we now want to talk
about the derivative of
707
00:48:11,060 --> 00:48:13,370
the impulse.
708
00:48:13,370 --> 00:48:18,970
Well, what we ask about is
what it is operationally.
709
00:48:18,970 --> 00:48:25,510
And so if we have a system,
which is a differentiator, and
710
00:48:25,510 --> 00:48:28,600
we inquire about its impulse
response, which let's say we
711
00:48:28,600 --> 00:48:32,630
define notationally as u_1(t).
712
00:48:32,630 --> 00:48:37,650
What's important about this
function u_1(t) is not what it
713
00:48:37,650 --> 00:48:42,620
is at each value of time
but what it does under
714
00:48:42,620 --> 00:48:43,840
convolution.
715
00:48:43,840 --> 00:48:45,690
What does it do under
convolution?
716
00:48:45,690 --> 00:48:49,610
Well, the output of the
differentiator is the
717
00:48:49,610 --> 00:48:53,190
convolution of the input with
the impulse response.
718
00:48:53,190 --> 00:48:58,100
And so what u_1(t) does under
convolution is to
719
00:48:58,100 --> 00:48:59,780
differentiate.
720
00:48:59,780 --> 00:49:03,770
And that is the operational
definition.
721
00:49:03,770 --> 00:49:07,450
And now, of course, we can
think of extending that.
722
00:49:07,450 --> 00:49:11,540
Not only would we want to think
about differentiating an
723
00:49:11,540 --> 00:49:15,960
impulse, but we would also
want to think about
724
00:49:15,960 --> 00:49:18,740
differentiating the derivative
of an impulse.
725
00:49:18,740 --> 00:49:23,000
We'll define that as
a function u_2(t).
726
00:49:23,000 --> 00:49:24,690
u_2(t)--
727
00:49:24,690 --> 00:49:28,050
because we have this impulse
response convolved with this
728
00:49:28,050 --> 00:49:31,700
one is u_1(t) * u_1(t).
729
00:49:31,700 --> 00:49:36,800
And what is u_2(t)
operationally?
730
00:49:36,800 --> 00:49:42,800
It is the operation such that
when you convolve that with
731
00:49:42,800 --> 00:49:48,540
x(t), what you get is the
second derivative.
732
00:49:48,540 --> 00:49:52,110
OK now, we can carry this
further and, in fact, talk
733
00:49:52,110 --> 00:49:59,410
about the result of convolving
u_1(t) with itself more times.
734
00:49:59,410 --> 00:50:03,870
In fact, if we think of the
convulution of u_1(t) with
735
00:50:03,870 --> 00:50:07,460
itself k times, then
logically we would
736
00:50:07,460 --> 00:50:11,330
define that as u_k(t).
737
00:50:11,330 --> 00:50:14,800
Again, we would interpret
that operationally.
738
00:50:14,800 --> 00:50:20,440
And the operational definition
is through convolution, where
739
00:50:20,440 --> 00:50:26,600
this corresponds to u_k(t) being
the impulse response of
740
00:50:26,600 --> 00:50:29,630
k differentiators in cascade.
741
00:50:29,630 --> 00:50:32,090
So what is the operational
definition?
742
00:50:32,090 --> 00:50:39,800
Well, it's simply that x(t)
* u_k(t) is the k
743
00:50:39,800 --> 00:50:42,440
derivative of x(t).
744
00:50:45,080 --> 00:50:49,480
And this now gives us a set
of what are referred to as
745
00:50:49,480 --> 00:50:50,720
singularity functions.
746
00:50:50,720 --> 00:50:55,360
Very badly behaved
mathematically in a sense, but
747
00:50:55,360 --> 00:50:58,700
as we've seen, reasonably well
defined under an operational
748
00:50:58,700 --> 00:51:00,780
definition.
749
00:51:00,780 --> 00:51:06,010
With k = 0, incidentally, that's
the same as what we
750
00:51:06,010 --> 00:51:08,560
have referred to previously
as the impulse.
751
00:51:08,560 --> 00:51:13,260
So with k 0, that's
just delta(t).
752
00:51:13,260 --> 00:51:17,150
Now to be complete, we can also
go the other way and talk
753
00:51:17,150 --> 00:51:20,940
about the impulse response of
a string of integrators
754
00:51:20,940 --> 00:51:23,460
instead of a string of
differentiators.
755
00:51:23,460 --> 00:51:25,280
Of course, the impulse
response of a single
756
00:51:25,280 --> 00:51:27,580
integrator is a unit step.
757
00:51:27,580 --> 00:51:30,120
Two integrators together
is the integral of a
758
00:51:30,120 --> 00:51:32,680
unit step, et cetera.
759
00:51:32,680 --> 00:51:38,110
And that, likewise, corresponds
to a set of what
760
00:51:38,110 --> 00:51:40,290
are called singularity
functions.
761
00:51:40,290 --> 00:51:46,410
In particular, if I take a
string of m integrators in
762
00:51:46,410 --> 00:51:52,610
cascade, then the impulse
response of that is denoted as
763
00:51:52,610 --> 00:51:56,000
u sub minus m of t.
764
00:51:56,000 --> 00:52:00,380
And for example, with a single
integrator, u sub minus 1 of t
765
00:52:00,380 --> 00:52:06,950
corresponds to our unit step as
we talked about previously.
766
00:52:06,950 --> 00:52:14,430
u sub minus 2 of t corresponds
to a unit ramp, et cetera.
767
00:52:14,430 --> 00:52:18,060
And there is, in fact, a reason
for choosing negative
768
00:52:18,060 --> 00:52:22,850
values of the argument when
going in one direction near
769
00:52:22,850 --> 00:52:25,370
integration as compared with
positive values of the
770
00:52:25,370 --> 00:52:28,130
argument when going in the
other direction, namely
771
00:52:28,130 --> 00:52:30,120
differentiation.
772
00:52:30,120 --> 00:52:38,110
In particular, we know that with
u sub minus m of t, the
773
00:52:38,110 --> 00:52:44,520
operational definition is the
mth running integral.
774
00:52:44,520 --> 00:52:48,370
And likewise, u_k(t)--
775
00:52:48,370 --> 00:52:51,040
so with a positive
sub script--
776
00:52:51,040 --> 00:52:56,240
has an operational definition,
which is the derivative.
777
00:52:56,240 --> 00:53:01,290
So it's the kth derivative
of x(t).
778
00:53:01,290 --> 00:53:08,070
And partly as a consequence of
that, if we take u_k(t) and
779
00:53:08,070 --> 00:53:13,400
convolve it with u_l(t), the
result is the singularity
780
00:53:13,400 --> 00:53:17,170
function with the subscript,
which is the sum of k and l.
781
00:53:17,170 --> 00:53:21,470
And that holds whether this
is positive values of the
782
00:53:21,470 --> 00:53:24,610
subscript or negative values
of the subscript.
783
00:53:24,610 --> 00:53:28,550
So just to summarize this last
discussion, we've used an
784
00:53:28,550 --> 00:53:35,520
operational definition to talk
about derivatives of impulses
785
00:53:35,520 --> 00:53:37,940
and integrals of impulses.
786
00:53:37,940 --> 00:53:40,705
This led to a set of singularity
functions-- what
787
00:53:40,705 --> 00:53:42,270
I've called singularity
functions--
788
00:53:42,270 --> 00:53:45,400
of which the impulse and the
step are two examples.
789
00:53:45,400 --> 00:53:49,700
But using an operational
definition through convolution
790
00:53:49,700 --> 00:53:55,500
allows us to define, at least in
an operational sense, these
791
00:53:55,500 --> 00:53:57,880
functions that otherwise
are very badly behaved.
792
00:54:01,110 --> 00:54:06,330
OK now, in this lecture and
previous lectures, for the
793
00:54:06,330 --> 00:54:10,890
most part, our discussion
has been about linear
794
00:54:10,890 --> 00:54:14,910
time-invariant systems in
fairly general terms.
795
00:54:14,910 --> 00:54:18,480
And we've seen a variety of
properties, representation
796
00:54:18,480 --> 00:54:22,390
through convolution, and
properties as they can be
797
00:54:22,390 --> 00:54:25,750
associated with the
impulse response.
798
00:54:25,750 --> 00:54:28,970
In the next lecture, we'll turn
our attention to a very
799
00:54:28,970 --> 00:54:34,150
important subclass of those
systems, namely systems that
800
00:54:34,150 --> 00:54:37,410
are describable by linear
constant-coefficient
801
00:54:37,410 --> 00:54:40,640
difference equations in the
discrete-time case, and linear
802
00:54:40,640 --> 00:54:43,540
constant-coefficient
differential equations in the
803
00:54:43,540 --> 00:54:45,410
continuous-time case.
804
00:54:45,410 --> 00:54:50,800
Those classes, while not forming
all of the class of
805
00:54:50,800 --> 00:54:53,080
linear time-invariant
systems, are a very
806
00:54:53,080 --> 00:54:54,930
important sub class.
807
00:54:54,930 --> 00:54:57,470
And we'll focus in on those
specifically next time.
808
00:54:57,470 --> 00:54:58,720
Thank you.