第 5 讲：线性时不变系统的性质
本讲内容：线性时不变系统的性质，包括交换律、结合律和分配律；冲激的运算定义；系统的级联与并联；卷积的性质；离散时间累加器；一阶连续时间系统。
授课教师：艾伦·V. 奥本海姆教授
英文字幕稿
PROFESSOR: Last time, we talked about the
representation of linear time-invariant systems through
the convolution sum in the discrete-time case and the
convolution integral in the continuous-time case.
Now, although the derivation was relatively
straightforward, in fact, the result is really kind of
amazing because what it tells us is that for linear
time-invariant systems, if we know the response of the
system to a single impulse at t = 0, or in fact, at any
other time, then we can determine from that its
response to an arbitrary input through the use of
convolution.
Furthermore, what we'll see as the course develops is that,
in fact, the class of linear time-invariant systems is a
very rich class.
There are lots of systems that have that property.
And in addition, there are lots of very interesting
things that you can do with linear time-invariant systems.
In today's lecture, what I'd like to begin with is focusing
on convolution as an algebraic operation.
And we'll see that it has a number of algebraic properties
that in turn have important implications for linear
time-invariant systems.
Then we'll turn to a discussion of what the
characterization of linear time-invariant systems through
convolution implies, in terms of the relationship, of
various other system properties
to the impulse response.
Let me begin by reminding you of the basic result that we
developed last time, which is the convolution sum in
discrete time, as I indicate here, and the convolution
integral in continuous time.
And what the convolution sum, or the convolution integral,
tells us is how to relate the output to the input and to the
system impulse response.
And the expression looks basically the same in
continuous time and discrete time.
And I remind you also that we talked about a graphical
interpretation, where essentially, to graphically
interpret convolution required, or was developed, in
terms of taking the system impulse response, flipping it,
sliding it past the input, and positioned appropriately,
depending on the value of the independent variable for which
we're computing the convolution, and then
multiplying and summing in the discrete-time case, or
integrating in the continuous-time case.
Now convolution, as an algebraic operation, has a
number of important properties.
One of the properties of convolution is that it is what
is referred to as commutative.
Commutative means that we can think either of convolving x
with h, or h with x, and the order in which that's done
doesn't affect the output result.
So summarized here is what the commutative operation is in
discrete time, or in continuous time.
And it says, as I just indicated, that x[n]
convolved with h[n]
is equal to h[n]
convolved with x[n].
Or the same, of course, in continuous time.
And in fact, in the lecture last time, we worked an
example where we had, in discrete time, an impulse
response, which was an exponential, and an input,
which is a unit step.
And in the text, what you'll find is the same example
worked, except in that case, the input is the exponential.
And the system impulse response is the step.
And that corresponds to the example in the text, which is
example 3.1.
And what happens in that example is that, in fact, what
you'll see is that the same result occurs in example 3.1
as we generated in the lecture.
And there's a similar comparison in continuous time.
This example was worked in lecture.
And this example is worked in the text.
In other words, the text works the example where the system
impulse response is a unit step.
And the input is an exponential.
All right, now the commutative property, as I said, tells us
that the order in which we do convolution doesn't affect the
result of the convolution.
The same is true for continuous time
and discrete time.
And in fact, for the other algebraic properties that I'll
talk about, the results are exactly the same for
continuous time and discrete time.
So in fact, what we can do is drop the independent variable
as an argument so that we suppress any kind of
difference between continuous and discrete time.
And suppressing the independent variable, we then
state the commutative property as I've rewritten it here.
Just x convolved with h equals h convolved with x.
The same in continuous time and discrete time.
Now, the derivation of the commutative property is, more
or less, some algebra which you can follow
through in the book.
It involves some changes of variables and some
things of that sort.
What I'd like to focus on with that and other properties is
not the derivation, which you can see in the text, but
rather the interpretation.
So we have the commutative property, and now there are
two other important algebraic properties.
One being what is referred to as the associative property,
which tells us that if we have x convolved with the result of
convolving h1 with h2, that's exactly the same as x
convolved with h1, and that result convolved with h2.
And what this permits is for us to write, for example, x
convolved with h1 convolved with h2 without any ambiguity
because it doesn't matter from the associative property how
we group the terms together.
The third important property is what is referred to as the
distributive property, namely the fact that convolution
distributes over addition.
And what I mean by that is what I've indicated here on
the slide: that if I think of x convolved with the sum of h1
and h2, that's identical to first convolving x with h1,
also convolving x with h2, and then adding the two together.
And that result will be the same as this result.
So convolution is commutative, associative, and it
distributes over addition.
Three very important algebraic properties.
And by the way, there are other algebraic operations
that have that same property.
For example, multiplication of numbers is likewise
commutative, associative, and distributive.
Now let's look at what these three properties imply
specifically for linear time-invariant systems.
And as we'll see, the implications are both very
interesting and very important.
Let's begin with the commutative property.
And consider, in particular, a system with an impulse
response h.
And I represent that by simply writing the h inside the box.
An input x and an output, then, which is x * h.
Now, since this operation is commutative, I can write
instead of x * h, I can write h * x.
And that would correspond to a system with impulse response
x, and input h, and output then h * x.
So the commutative property tells us that for a linear
time-invariant system, the system output is independent
of which function we call the input and which function we
call the impulse response.
Kind of amazing actually.
We can interchange the role of input and impulse response.
And from an output point of view, the output or the system
doesn't care.
Now furthermore, if we combine the commutative property with
the associative property, we get another
very interesting result.
Namely that if we have two linear time-invariant systems
in cascade, the overall system is independent of the order in
which they're cascaded.
And in fact, in either case, the cascade can be collapsed
into a single system.
To see this, let's first consider the cascade of two
systems, one with impulse response h1, the other with
impulse response h2.
And the output of the first system is then x * h1.
And then that is the input to the second system.
And so the output of that is that result convolved with h2.
So this is the result of cascading the two.
And now we can use the associative property to
rewrite this as x * (h1 * h2), where we group
these two terms together.
And so using the associative property, we now can collapse
that into a single system with an input, which is x, and
impulse response, which is h1 * h2.
And the output is then x convolved with the result of
those two convolved.
Next, we can apply the commutative property.
And the commutative property says we could write this
impulse response that way, or we could write it this way.
And since convolution is commutative, the resulting
output will be exactly the same.
And so these resulting outputs will be exactly the same.
And now, once again we can use the associative property to
group these two terms together.
And x * h2 corresponds to putting x first through the
system h2 and then that output through the system h1.
And so finally applying the associative property again, as
I just outlined, we can expand that system back into two
systems in cascade with h2 first and h1 second,
OK, well that involves a certain amount of algebraic
manipulation.
And that is not the algebraic
manipulation that is important.
It's the result that it's important.
And what the result says, to reiterate, is if I have two
linear time-invariant systems in cascade, I can cascade them
in any order, and the result is the same.
Now you might think, well gee, maybe that actually applies to
systems in general, whether you put them
this way or that way.
But in fact, as we talked about last time, and I
illustrated with an example, in general, if the systems are
not linear and time-invariant, then the order in which
they're cascaded is important to the interpretation of the
overall system.
For example, if one system took the square root and the
other system doubled the input, taking the square root
and then doubling gives us a different answer than doubling
first and then taking the square root.
And of course, one can construct much more elaborate
examples than that.
So it's a property very particular to linear
time-invariant systems.
And also one that we will exploit many, many times as we
go through this material.
The final property related to an interconnection of systems
that I want to just indicate develops out of the
distributive property.
And what it applies to is an interpretation of the
interconnection of systems in parallel.
Recall that the parallel combination of systems
corresponds, as I indicate here, to a system in which we
simultaneously feed the input into h1 and h2, these
representing the impulse responses.
And then, the outputs are summed to
form the overall output.
And using the fact that convolution distributes over
addition, we can rewrite this as x * (h1 + h2).
And when we do that then, we can recognize this as the
output of a system with input x and impulse response, which
is the sum of these two impulse responses.
So for linear time-invariant systems in parallel, we can,
if we choose, replace that interconnection by a single
system whose impulse response is simply the sum of those
impulse responses.
OK, now we have this very powerful representation for
linear time-invariant systems in terms of convolution.
And we've seen so far in this lecture how convolution and
the representation through the impulse response leads to some
important implications for system interconnections.
What I'd like to turn to now are other system properties
and see how, for linear time-invariant systems in
particular, other system properties can be associated
with particular properties or characteristics of the system
impulse response.
And what we'll talk about are a variety of properties.
We'll talk about the issue of memory, we'll talk about the
issue of invertibility, and we'll talk about the issue of
causality and also stability.
Well, let's begin with the issue of memory.
And the question now is what are the implications for the
system impulse response for a linear time-invariant system?
Remember that we're always imposing that on the system.
What are the implications on the impulse response if the
system does or does not have memory?
Well, we can answer that by looking at
the convolution property.
And we have here, as a reminder, the convolution
integral, which tells us how x(tau) and h(t - tau) are
combined to give us y(t).
And what I've illustrated above is a
general kind of example.
Here is x(tau).
Here is h(t - tau).
And to compute the output at any time t, we would take
these two, multiply them together, and integrate from
-infinity to +infinity.
So the question then is what can we say about h(t), the
impulse response in order to guarantee, let's say, that the
output depends only on the input at time t.
Well, it's pretty much obvious from looking at the graphs.
If we only want the output to depend on x(tau) at tau = t,
then h(t - tau) better be non-zero only at tau = t.
And so the implication then is that for the system to be
memoryless, what we require is that h(t - tau) be non-zero
only at tau = t.
So we want the impulse response to be non-zero at
only one point.
We want it to contribute something after we multiply
and go through an integral.
And in effect, what that says is the only thing that it can
be and meet all those
conditions is a scaled impulse.
So if the system is to be memoryless, then that requires
that the impulse response be a scaled impulse.
Any other impulse response then, in essence, requires
that the system have memory, or implies that the system
have memory.
So for the continuous-time case then, memoryless would
correspond only to the impulse response being proportional to
an impulse.
And in the discrete-time case, a similar statement, in which
case, the output is just proportional to the input,
again either in the continuous-time or in the
discrete-time case.
All right.
Now we can turn our attention to the issue of system
invertibility.
And recall that what is meant by invertibility of a system,
or the inverse of a system.
The inverse of a system is a system, which when we cascade
it with the one that we're inquiring about, the overall
cascade is the identity system.
In other words, the output is equal to the input.
So let's consider a system with impulse
response h, input is x.
And let's say that the impulse response of the inverse system
is h_i, and the output is y.
Then, the output of this system is x * (h * h_i).
And we want this to come out equal to x.
And what that requires than is that this convolution just
simply be equal to an impulse, either in the discrete-time
case or in the continuous-time case.
And under those conditions then, h_i is equal to the
inverse of h.
Notationally, by the way, it's often convenient to write
instead of h_i as the impulse response of the inverse,
you'll find it convenient often and more typical to
write as the inverse, instead of h_i, h^(-1).
And we mean by that the inverse impulse response.
And one has to be careful not to misinterpret this as the
reciprocal of h(t) or h(n).
What's meant in this notation is the inverse system.
Now, if h_i is the inverse of h, is h the inverse of h_i?
Well, it seems like that ought to be plausible or perhaps
make sense.
The question, if you believe that the answer is yes, is
how, in fact, do you verify that?
And I'll leave it to you to think about it.
The answer is yes,
that if h_i is the inverse of h, then h is
the inverse of h_i.
And the key to showing that is to exploit the fact that when
we take these systems and cascade them, we can cascade
them in either order.
All right now let's turn to another system property, the
property of stability.
And again, we can tie that property directly to issues
related, in particular, to the system impulse response.
Now, stability is defined as we've chosen to define it and
as I've defined it previously, as bounded-input
bounded-output stability.
In other words, for every bounded input
is a bounded output.
What you can show--
and I won't go through the algebra here; it's gone
through in the book--
is that a necessary and sufficient condition for a
linear time-invariant system to be stable in the
bounded-input bounded-output sense is that the impulse
response be what is referred to as absolutely summable.
In other words, if you take the absolute values and sum
them over infinite limits, that's finite.
Or in the continuous-time case, that the impulse
response is absolutely integrable.
In other words, if you take the absolute values of h(t)
and integrate, that's finite.
And under those conditions, the system is stable.
If those conditions are violated, then for sure, as
you'll see in the text, the system is unstable.
So stability can also be tied to the
system impulse response.
Now, the next property that I want to talk about is the
property of causality.
And before I do, what I'd like to do is introduce a
peripheral result that we'll then use--
when we talked about causality--
namely what's referred to as the zero input response of a
linear system.
The basic result, which is a very interesting and useful
one, is that for a linear system--
and in fact, it's whether it's time-invariant or not, that
this applies--
if you put nothing into it, you get nothing out of it.
So if we have an input x(t) which is 0 for all t, and if
the output of that system is y(t), if the input is 0 for
all time, then the output likewise is 0 for all time.
That's true for continuous time, and it's also true for
discrete time.
And in fact, to show that result is pretty much
straightforward.
We could do it either by using convolution, which would, of
course, be associated with linearity and time invariance.
But in fact, we can show that property relatively easily by
simply using the fact that, for a linear system, what we
know is that if we have an input x(t) with an output
y(t), then if we scale the input, then the output scales
accordingly.
Well, we can simply choose, as the scale factor, a = 0.
And if we do that, it says put nothing in,
you get nothing out.
And what we'll see is that has some important implications in
terms of causality.
It's important, though, while we're on it, to stress that
not every system, obviously, has that property.
That if you put nothing in, you get nothing out.
A simple example is, let's say, a battery, let's say not
connected to anything.
The output is six volts no matter what the input is.
And it of course then doesn't have this zero response to a
zero input.
It's very particular to linear systems.
All right, well now let's see what this means for causality.
To remind you, causality says, in effect, that the system
can't anticipate the input.
That's what, basically, causality means.
When we talked about it previously, we defined it in a
variety of ways, one of which was the statement that if two
inputs are identical up until some time, then the outputs
must be identical up until the same time.
The reason, kind of intuitively, is that if the
system is causal--
so it can't anticipate the future--
it can't anticipate whether these two identical inputs are
sometime later going to change from each other or not.
So causality, in general, is simply this statement, either
continuous-time or discrete-time.
And now, so let's look at what that means
for a linear system.
For a linear system, what that corresponds to or could be
translated to is a statement that says that if x(t) is 0,
for t less than t_0, then y(t) must be 0 for t
less than t_0 also.
And so what that, in effect, says, is that the system--
for a linear system to be causal, it must have the
property sometimes referred to as initial rest, meaning it
doesn't respond until there's some input that happens.
That it's initially at rest until the
input becomes non-zero.
Now, why is this true?
Why is this a consequence of causality for linear systems?
Well, the reason is we know that if we put nothing in, we
get nothing out.
If we have an input that's 0 for t less than t_0, and the
system can't anticipate whether that input is going to
change from 0 or not, then the system must generate an output
that's 0 up until that time, following the principle that
if two inputs are identical up until some time, the outputs
must be identical up until the same time.
So this basic result for linear systems is essentially
a consequence of the statement that for a linear system, zero
in gives us zero out.
Now, that tells us how to interpret
causality for linear systems.
Now, let's proceed to linear time-invariant systems.
And in fact, we can carry the point one step further.
In particular, a necessary and sufficient condition for
causality in the case of linear time-invariant systems
is that the impulse response be 0, for t less than 0 in the
continuous-time case, or for n less than 0 in the
discrete-time case.
So for linear time-invariant systems, causality is
equivalent to the impulse response being 0 up until t or
n equal to 0.
Now, to show this essentially follows by first considering
why causality would imply that this is true.
And that follows because of the straightforward fact that
the impulse itself is 0 for t less than 0.
And what we saw before is that for any linear system,
causality requires that if the input is 0 up until some time,
the output must be 0 up until the same time.
And so that's showing the result in one direction.
To show the result in the other direction, namely to
show that if, in fact, the impulse response satisfies
that condition, then the system is causal.
While I won't work through it in detail, it essentially
boils down to recognizing that in the convolution sum, or in
the convolution integral, if, in fact, that condition is
satisfied on the impulse response, then the upper limit
on the sum, in the discrete-time
case, changes to n.
And in the continuous-time case, changes to t.
And that, in effect, says that values of the input only from
-infinity up to time n are used in computing y[n].
And a similar kind of result for the
continuous-time case y(t).
OK, so we've seen how the impulse response, or rather
how certain system properties in the linear time-invariant
case can, be converted into various requirements on the
impulse response of a linear time-invariant system, the
impulse response being a complete characterization.
Let's look at some particular examples just to kind of
cement the ideas further.
And let's begin with a system that you've seen previously,
which is an accumulator.
An accumulator, as you recall, has an output y[n], which is
the accumulated value of the input from -infinity up to n.
Now, you've seen in the impulse in a previous lecture,
or rather in the video course manual for a previous lecture,
that an accumulator is a linear time-invariant system.
And in fact, its impulse response is the accumulated
values of an impulse.
Namely, the impulse response is equal to a step.
So what we want to answer is, knowing what that impulse
response is, what some properties are of the
accumulator.
And let's first ask about memory.
Well, we recognize that the impulse response is not simply
an impulse.
In fact, it's a step.
And so this implies what?
Well, it implies that the system has memory.
Second, the impulse response is 0 for n less than 0.
That implies that the system is causal.
And finally, if we look at the sum of the absolute values of
the impulse response from -infinity to
+infinity, this is a step.
If we accumulate those values over infinite limits, then
that in fact comes out to be infinite.
And so what that implies, then, is that the accumulator
is not stable in the bounded-input
bounded-output sense.
Now I want to turn to some other systems.
But while we're on the accumulator, I just want to
draw your attention to the fact, which will kind of come
up in a variety of ways again later, that we can rewrite the
equation for an accumulator, the difference equation, by
recognizing that we could, in fact, write the output as the
accumulated values up to time n - 1 and then
add on the last value.
And in fact, if we do that, this corresponds to y[n-1].
And so we could rewrite this difference equation as y[n]
= y[n-1]
+ x[n].
So the output is the previously-computed output
plus the input.
Expressed that way, what that corresponds to is what is
called a recursive difference equation.
And different equations will be a topic of considerable
emphasis in the next lecture.
Now, does an accumulator have an inverse?
Well, the answer is, in fact, yes.
And let's look at what the inverse of the accumulator is.
The impulse response of the accumulator is a step.
To inquire about its inverse, we inquire about whether
there's a system, which when we cascade the accumulator
with that system, which I'm calling its inverse, we get an
impulse out.
Well, let's see.
The impulse response of the accumulator is a step.
We want to put the step into something
and get out an impulse.
And in fact, what you recall from the lecture in which we
introduced steps and impulses, the impulse is, in fact, the
first difference of the units step.
So we have a difference equation that describes for us
how the impulse is related to the step.
And so if this system does this, the output will be that,
an impulse.
And so if we think of x_2[n]
as the input and y_2[n]
as the output, then the difference equation for the
inverse system is what I've indicated here.
And if we want to look at the impulse response of that, we
can then inquire as to what the response is with an
impulse in.
And what develops in a straightforward way then is
delta[n], which is our impulse input, minus delta[n-1]
is equal to the impulse response
of the inverse system.
So I'll write that as h^(-1)[n]
(h-inverse of n).
Now, we have then that the accumulator has an inverse.
And this is the inverse.
And you can examine issues of memory, stability,
causality, et cetera.
What you'll find is that the system has memory, the inverse
accumulator.
It's stable, and it's causal.
And it's interesting to note, by the way, that although the
accumulator was an unstable system, the inverse of the
accumulator is a stable system.
In general, if the system is stable, its inverse does not
have to be stable or vice versa.
And the same thing with causality.
OK now, there are a number of other examples, which, of
course, we could discuss.
And let me just quickly point to one example, which is a
difference equation, as I've indicated here.
And as we'll talk about in more detail in our next
lecture, where we'll get involved in a fairly detailed
discussion of linear constant-coefficient
difference and differential equations, this falls into
that category.
And under the imposition of what's referred to as initial
rest, which corresponds to the response being 0 up until the
time that the input becomes non-zero, the impulse response
is a^n times u[n].
And something that you'll be asked to think about in the
video course manual is whether that system has memory,
whether it's causal, and whether it's stable.
And likewise, for a linear constant coefficient
differential equation, the specific one that I've
indicated here, under the assumption of initial rest,
the impulse response is e^(-2t) times u(t).
And in the video course manual again, you'll be asked to
examine whether the system has memory, whether it's causal,
and whether it's stable.
OK well, as I've indicated, in the next lecture we'll return
to a much more detailed discussion of linear
constant-coefficient differential
and difference equations.
Now, what I'd like to finally do in this lecture is use the
notion of convolution in a much different way to help us
with a problem that I alluded to earlier.
In particular, the issue of how to deal with some of the
mathematical difficulties associated with
impulses and steps.
Now, let me begin by illustrating kind of what the
problem is and an example of the kind of paradox that you
sort of run into when dealing with impulse functions and
step functions.
Let's consider, first of all, a system, which is the
identity system.
And so the output is, of course, equal to the input.
And again, we can talk about that either in continuous time
or in discrete time.
Well, we know that the function that you convolve
with a signal to retain the signal is an impulse.
And so that means that the impulse response of an
identity system is an impulse.
Makes logical sense.
Furthermore, if I take two identity systems and cascade
them, I put in an input, get the same thing out of the
first system.
That goes into the second system.
Get the same thing out of the second.
In other words, if I have two identity systems in cascade,
the cascade, likewise, is an identity system.
In other words, this overall system is
also an identity system.
And the implication there is that the impulse response of
this is an impulse.
The impulse response of this is an impulse.
And the convolution of those two is also an impulse.
So for continuous time, we require, then, that an impulse
convolved with itself is an impulse.
And the same thing for discrete time.
Now, in discrete time, we don't have any particular
problem with that.
If you think about convolving these together, it's a
straightforward mathematical operation since the impulse in
discrete time is very nicely defined.
However, in continuous time, we have to be somewhat careful
about the definition of the impulse because it was the
derivative of a step.
A step has a discontinuity.
You can't really differentiate at a discontinuity.
And the way that we dealt with that was to expand out the
discontinuity so that it had some finite time region in
which it happened.
When we did that, we ended up with a definition for the
impulse, which was the limiting form of this
function, which is a rectangle of width Delta, and height 1 /
Delta, and an area equal to 1.
Now, if we think of convolving this signal with itself, the
impulse being the limiting form of this, then the
convolution of this with itself is a triangle of width
2 Delta, height 1 / Delta, and area 1.
In other words, this triangular function is this
approximation delta_Delta(t) convolved with delta_Delta(t).
And since the limit of this would correspond to the
impulse response of the identity system convolved with
itself, the implication is that not only should the top
function, this one, correspond in its limiting form to an
impulse, but also this should correspond in its limiting
form to an impulse.
So one could wonder well, what is an impulse?
Is it this one in the limit?
Or is it this one in the limit?
Now, beyond that-- so kind of what this suggests is that in
the limiting form, you kind of run into a contradiction
unless you don't try to distinguish between this
rectangle and the triangle.
Things get even worse when you think about what happens when
you put an impulse into a differentiator.
And a differentiator is a very commonly occurring system.
In particular, suppose we had a system for which the output
was the derivative of the input.
So if we put in x(t), we got out dx(t) / dt.
If I put in an impulse, or if I talked about the impulse
response, what is that?
And of course, the problem is that if you think that the
impulse itself is very badly behaved, then what about its
derivative, which is not only infinitely big, but there's a
positive-going one, and a negative-going one, and the
difference between there has some area.
And you end up in a lot of difficulty.
Well, the way around this, formally, is through a set of
mathematics referred to as generalized functions.
We won't be quite that formal.
But I'd like to, at least, suggest what the essence of
that formality is.
And it really helps us in interpreting the impulses in
steps and functions of that type.
And what it is is an operational definition of
steps, impulses, and their derivatives in
the following sense.
Usually when we talk about a function, we talk about what
the value of the function is at any instant of time.
And of course, the trouble with an impulse is it's
infinitely big, in zero width, and has some area, et cetera.
What we can turn to is what is referred to as an operational
definition where the operational definition is
related not to what the impulse is, but to what the
impulse does under the operation of convolution.
So what is an impulse?
An impulse is something, which under
convolution, retains the function.
And that then can serve as a definition of the impulse.
Well, let's see where that gets us.
Suppose that we now want to talk about the derivative of
the impulse.
Well, what we ask about is what it is operationally.
And so if we have a system, which is a differentiator, and
we inquire about its impulse response, which let's say we
define notationally as u_1(t).
What's important about this function u_1(t) is not what it
is at each value of time but what it does under
convolution.
What does it do under convolution?
Well, the output of the differentiator is the
convolution of the input with the impulse response.
And so what u_1(t) does under convolution is to
differentiate.
And that is the operational definition.
And now, of course, we can think of extending that.
Not only would we want to think about differentiating an
impulse, but we would also want to think about
differentiating the derivative of an impulse.
We'll define that as a function u_2(t).
u_2(t)--
because we have this impulse response convolved with this
one is u_1(t) * u_1(t).
And what is u_2(t) operationally?
It is the operation such that when you convolve that with
x(t), what you get is the second derivative.
OK now, we can carry this further and, in fact, talk
about the result of convolving u_1(t) with itself more times.
In fact, if we think of the convulution of u_1(t) with
itself k times, then logically we would
define that as u_k(t).
Again, we would interpret that operationally.
And the operational definition is through convolution, where
this corresponds to u_k(t) being the impulse response of
k differentiators in cascade.
So what is the operational definition?
Well, it's simply that x(t) * u_k(t) is the k
derivative of x(t).
And this now gives us a set of what are referred to as
singularity functions.
Very badly behaved mathematically in a sense, but
as we've seen, reasonably well defined under an operational
definition.
With k = 0, incidentally, that's the same as what we
have referred to previously as the impulse.
So with k 0, that's just delta(t).
Now to be complete, we can also go the other way and talk
about the impulse response of a string of integrators
instead of a string of differentiators.
Of course, the impulse response of a single
integrator is a unit step.
Two integrators together is the integral of a
unit step, et cetera.
And that, likewise, corresponds to a set of what
are called singularity functions.
In particular, if I take a string of m integrators in
cascade, then the impulse response of that is denoted as
u sub minus m of t.
And for example, with a single integrator, u sub minus 1 of t
corresponds to our unit step as we talked about previously.
u sub minus 2 of t corresponds to a unit ramp, et cetera.
And there is, in fact, a reason for choosing negative
values of the argument when going in one direction near
integration as compared with positive values of the
argument when going in the other direction, namely
differentiation.
In particular, we know that with u sub minus m of t, the
operational definition is the mth running integral.
And likewise, u_k(t)--
so with a positive sub script--
has an operational definition, which is the derivative.
So it's the kth derivative of x(t).
And partly as a consequence of that, if we take u_k(t) and
convolve it with u_l(t), the result is the singularity
function with the subscript, which is the sum of k and l.
And that holds whether this is positive values of the
subscript or negative values of the subscript.
So just to summarize this last discussion, we've used an
operational definition to talk about derivatives of impulses
and integrals of impulses.
This led to a set of singularity functions-- what
I've called singularity functions--
of which the impulse and the step are two examples.
But using an operational definition through convolution
allows us to define, at least in an operational sense, these
functions that otherwise are very badly behaved.
OK now, in this lecture and previous lectures, for the
most part, our discussion has been about linear
time-invariant systems in fairly general terms.
And we've seen a variety of properties, representation
through convolution, and properties as they can be
associated with the impulse response.
In the next lecture, we'll turn our attention to a very
important subclass of those systems, namely systems that
are describable by linear constant-coefficient
difference equations in the discrete-time case, and linear
constant-coefficient differential equations in the
continuous-time case.
Those classes, while not forming all of the class of
linear time-invariant systems, are a very
important sub class.
And we'll focus in on those specifically next time.
Thank you.