第 3 讲：信号与系统（二）
本讲内容：单位阶跃信号与单位冲激（脉冲）信号；系统的框图表示与互连；系统的性质，包括记忆性、可逆性与逆系统、因果性、稳定性、时不变性和线性。
授课教师：艾伦·V. 奥本海姆教授
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PROFESSOR: in the last lecture, we discussed
sinusoidal and real and complex exponential signals
both for continuous time and discrete time.
And those signals will form very important building blocks
when we return to a discussion of Fourier
analysis in a later lecture.
In today's lecture, I'd like to introduce some additional
basic signals, specifically the unit step and
unit impulse signal.
Let's begin with discrete time and the discrete-time unit
step and unit impulse.
The discrete-time unit step is a sequence as I've indicated
here, specifically a sequence which is 0 for negative values
of its argument, and equal to 1 for positive values of its
argument and 0.
So mathematically, the unit step sequence is 1 for n
greater than or equal to 0 and 0 for n less than 0.
The unit impulse sequence, likewise, is defined in a
straightforward way.
The unit impulse sequence is a sequence which is 0 for all
values of its argument except for n = 0.
So the unit step and unit impulse sequence are defined
in a straightforward way mathematically.
And in fact, they are also related to each other in a
straightforward way mathematically.
Specifically, the unit impulse can be related to the unit
step through the relationship that I've indicated here--
delta of n, the unit impulse, equal to a unit step minus the
unit step delayed.
So mathematically, the relationship is what is
referred to as a first difference.
And to see the validity of this expression, we can simply
look at the unit step and its delayed version.
So here, we show the unit step, u[n].
Here, we show the unit step delayed by 1.
So it's 0 for n less than or equal to 0.
And clearly, if we subtract the delayed step from the
original unit step, everything subtracts out except at n = 0,
at which point the difference is equal to 1.
And so the difference between u[n]
and u[n-1]
is simply the unit impulse, sometimes incidentally also
referred to as the unit sample.
Now, in a similar way, we can express the unit step in terms
of the unit impulse.
And there are several ways of doing this.
One way is through a relationship referred to as a
running sum.
What I mean by that is the following expression.
If we think of forming the sum from minus infinity up to some
value n of a unit impulse or unit sample, then this running
sum, in fact, is equal to the unit step.
And we can see that in a fairly straightforward way, by
simply observing that in this expression, when n is less
than 0, there's nothing accumulated in the sum.
And we can see that graphically
as I've shown here.
So for n less than 0, so we accumulate no
terms in the sum.
Whereas for n greater than 0, we accumulate 1 non-zero value
in the sum, namely the value of the unit sample at n = 0.
So we have, then, one expression for the
relationship between the unit step and the unit sample.
We can also develop another relationship by observing in
essence that if we look at the unit step sequence, as I've
returned to here, we can, in effect, think of the unit step
sequence as a succession of unit impulses,
one following another.
So if we consider forming a sum of delayed impulses, as I
indicate mathematically here, and as I indicate graphically
down below, we have an impulse here at n = 0 and an impulse
here at n = 1, an impulse here at n = 2, et cetera.
And when we continue to add these up, then what they add
up to is the unit step sequence.
And so mathematically, then, that would correspond to an
impulse at n = 0 plus an impulse at n = 1 plus an
impulse at n = 2, et cetera.
Now, in continuous time, we have a very more or less
similar situation.
We will find it equally useful to talk about a unit step
continuous-time signal and a unit impulse
continuous-time signal.
Let's begin with the continuous-time unit step.
The continuous-time unit step function is graphically
indicated as I've shown here.
It's a time function which is 0 for t less than 0.
And it's 1 for t greater than 0.
And so mathematically, what it corresponds to is u[t]
t defined as a time function which is 0 for t less than 0,
1 for t greater than 0.
Now, an obvious question is, what happens at t = 0?
And the difficulty here-- which is not a difficulty that
arises in the discrete-time case--
is that at t = 0, the units step function is in fact
discontinuous, which generates a variety of
mathematical problems.
And one can define the unit step at t = 0 in a variety of
ways, but the essential point is that the unit step function
is discontinuous.
So in effect, what we need to do is think of the unit step
function as the limit of a continuous function.
And so we can define a function, which I specify here
as u_delta(t) (u sub delta of t).
And u_delta(t) is a time function which is 0 for t less
than 0, linearly increases to time delta which would
correspond to this break point, and
then 1 following that.
And so we can think then of the discontinuous unit step as
the limiting form of u_delta(t) as delta goes to 0.
Now, we also want to define a unit impulse function.
And it had a fairly straightforward definition in
discrete time.
In continuous time, things get slightly more difficult.
And to motivate the definition, let me return to
the discrete-time definition--
or rather, the discrete-time relationship between the unit
step and the unit impulse function.
In discrete time, we saw that the unit impulse function is
the first difference of the unit step function.
Well, similarly in continuous time, we can talk about an
impulse function, which is the first derivative of a unit
step function.
So the unit impulse, as we want to define it, is the
derivative of the unit step.
Of course, we just finished discussing the fact that the
unit step function is, in fact, discontinuous at t = 0.
But we can think of its derivative as related to this
approximation to the unit step.
And specifically, we will think of the continuous-time
impulse as the derivative of u_delta(t) as delta goes to 0.
So to define the unit impulse, we think of the derivative of
this approximation to the unit step and then observe what
happens as delta goes to 0.
We have, then, the definition of the unit impulse function
more or less formally defined as the first derivative of the
unit step, or thought of as the limiting form of the
derivative of the approximation to the unit step
in the limit, as delta, the duration of the
discontinuity, goes to 0.
Well, let's look at that.
If we think about the derivative of u_delta(t), the
derivative, of course, is 0 for t less than 0.
It's equal to a constant during this linear slope, and
then, it's 0 for t greater than delta.
So the derivative of u_delta(t) will then be as
I've indicated here.
And it's simply a rectangle with a height, which is 1 /
delta, and a width, which is equal to delta.
And observe that no matter what the value of delta is,
the area is always equal to 1.
Now, as we let delta go to 0, what happens is that the width
of the rectangle gets smaller, the height of the rectangle
gets bigger, the area still remains 1.
As delta goes to 0, of course, the width goes to 0, and the
height goes to infinity.
And graphically, we choose to depict that as an arrow, where
the arrow indicates the fact that we have an impulse
occurring at t = 0, and the height of the impulse is used
to represent what the area of the impulse is.
And in this case, since we took the derivative of a unit
step, the height is equal to 1.
So the impulse has 0 width, infinite height, area 1.
It's mathematically not terribly comfortable because
what we've done is taken the derivative of the unit step,
which has a discontinuity at the origin, and there's some
mathematical difficulties in doing that.
We'll in fact return to another interpretation of the
impulse later to emphasize some of the discomfort with
the impulse.
I remember something that Sam Mason used to say that his
students said about the unit impulse.
His definition was the unit impulse is something that's so
small every place I can't see it except at one point where
it's so big I can't see it.
In other words, I can't see it at.
Well, accept some informality with the unit impulse function
in the continuous time case, and generally, we'll see that
we won't get into particular difficulty.
OK, now the impulse is the derivative of the step.
We saw in the discrete-time case that the step could be
recovered from the impulse through a running sum.
In continuous time, if the impulse is the derivative of a
step, we would more or less reasonably expect that the
step would be like an integral of the impulse.
And indeed, that's true.
In fact, mathematically the relationship is that the unit
step function in continuous time is the running integral
of the unit impulse function.
And so it's the integral from minus infinity up to time t,
where t is the argument at which we're examining u(t).
And so, just as with the discrete-time case, if we are
looking at t less than 0, there's no area accumulated in
the integral.
If we're looking at t greater than 0, then we accumulate the
area under the impulse.
OK, now we'll shortly be returning to a further
discussion on use of impulses and step functions.
And in particular, what we'll see is that they provide a
very convenient and powerful mechanism for describing a
particular class of systems referred to as linear
time-invariant systems.
To lead up to that discussion, which will be the principal
focus of the next lecture, let's for the remainder of
this lecture talk about systems in general and then
some properties of systems.
And as the lecture proceeds, the specific properties that I
want to get to are properties of linearity and time
invariance.
So let's first talk about systems in general.
And a system in general, in its most general definition,
is simply a transformation from an input signal to an
output signal.
So in a continuous-time case, we would have a
continuous-time system, the input, x[t],
and the output, y[t].
And the box, in essence, is used to denote a
transformation from x[t]
to y[t].
And sometimes, we'll also use a shorthand notation along the
lines of indicating that the input, x[t], is transformed to
the output, y[t].
Now, we have exactly the same kind of definition in the
discrete-time case.
In the discrete-time case, of course, the inputs are
sequences, and the outputs are sequences.
And our shorthand notation is similar.
Often when we talk about systems, we'll want to talk
about interconnections of systems.
And we'll see again in later lectures that interconnections
become very important and powerful.
And in the way of introducing terminology, let me introduce
the terminology for a few basic and important
interconnections of systems.
The first is what's referred to as a cascade of systems, or
sometimes as a series interconnection of systems.
And putting two systems in cascade, as I've indicated
here, means taking the output of one system--
let's say system 1--
and using that as the input to the second system, which I've
denoted as system 2.
So in this cascade with system 1 first and system 2 second,
we have the output of system 1 going into the
input of system 2.
Now, we could, of course, take those two systems and cascade
them in the reverse order.
Namely, take the output of system 2 and put it into the
input of system 1.
And I've indicated that here.
And we have system 2 first in the cascade
followed by system 1.
It's important to keep in mind that in general, for general
systems, the order in which you cascade the systems is
very important.
And in fact, the overall system transformation will be
different depending on which system came first and which
system came second.
For example, if system 1 was, let's say, a system for which
the output doubles the input, and system 2, the output is
the square root of the input, clearly doubling first and
then taking the square root is different than taking the
square root first and then doubling.
Now, kind of amazingly, what we'll see again when we get to
this issue of linearity and time invariance--
which are a class of systems that we'll focus in on--
somewhat amazingly, it turns out that for that specific
class of systems, the overall system transformation is
independent of the order in which
the systems are cascaded.
And we'll see that, of course, in more detail later on.
All right, well, that's a cascade or a series
interconnection.
Let's look at another interconnection, which is a
parallel interconnection of systems.
And here, what's meant by the interconnection is that the
input is fed simultaneously into system 1
and into system 2.
And then, the outputs of these two systems are added to give
the overall system output.
Now, in this particular case, in contrast to a cascade, no
matter what the systems are, it turns out that a parallel
combination, the order in which they're
put in parallel is--
the overall transformation is independent of the order.
And of course, that follows from the fact that we're
simply adding up outputs.
And the outputs can be added in any order because of the
fact that addition doesn't care in which order you're
adding things up.
OK, so that's the parallel interconnection.
And let's look at one more interconnection, which is
what's referred to as a feedback interconnection.
And this, again, is an interconnection that will
become a very important topic much later in the course.
But let me just indicate at this point what I mean by it.
What a feedback interconnection means is that
we have one system, system 1, with an input and an output.
The output of system 1 is the output of the overall system.
And that output is fed into system 2.
The output of system 2 is then added to the input to the
overall system.
So in essence, what happens in a feedback interconnection is
we have one system.
The output of that system is fed back through system 2,
added to the overall input.
And then, that sum is what forms the input of system 1.
And we'll see that there are lots of uses for feedback and,
of course, also lots of ways that feedback gets in the way.
In fact, probably some of you are already familiar with some
of the issues in feedback--
for example, in audio systems or whatever.
But this will be a topic that we'll devote a considerable
amount of time to later in the course.
And then, of course, there are lots of other interconnections
of systems.
And as the course progresses, we'll see lots of ways in
which systems get interconnected both in series
and in parallel and feedback interconnections, et cetera,
to achieve a wide variety of things.
Now, what I've said so far relates to systems in general.
And you can't say much about systems when you try to treat
them in their most general form.
So it's useful and important to focus in on properties that
a system may or may not have.
So what I'd like to do now is turn our attention to system
properties.
And we'll define a number of them.
Some of them, we'll want to impose on a system.
Some of them, we may not want to impose on a system.
But as things progress, we'll tend to find it useful to ask
whether a system does or doesn't
have a certain property.
Well, let's begin with a property which I refer to here
as memoryless.
And what I mean by a system being memoryless is that the
output at any given time, t_0, depends only on the input at
the same time.
And so what I'm suggesting is that a memoryless system is
one for which the output at a specific time depends only on
the input at that time.
And that statement is true, or that definition applies, both
for continuous time, as I've indicated here, and also for
discrete time, as I've indicated here.
And so we have a similar definition in discrete time,
that the system is memoryless if the output at any given
time depends only on the input at that time.
Well, I have a number of examples here.
Let's look at the first example in which the output is
the square of the input.
And that is a possible system either in continuous time or
discrete time.
And that system, as you would expect, is commonly referred
to a squarer.
And since the square of a signal at any time depends
only on the value of the signal at that time, clearly a
squarer is a memoryless system.
So in fact, we can indicate here that this system is
memoryless.
Another important system is what is referred to as an
integrator.
The output is equal to the integral of the input.
Here, as I've indicated it, it's not just integrating
x(t), but it's integrating the square of x(t).
And whether we square before we integrate or not, the
essential point is that since we're integrating the input,
the value of the output at any time is an accumulation of
past history of the input.
Well, an accumulation of past history in
essence implies memory.
To get the output at a given time requires the input over
an interval, specifically for longer than that time.
So this system, in fact, is not memoryless.
And now, I have a third system defined here.
The third system is a system, a discrete-time system, in
which the output is equal to the input but not quite.
The output at some time-- let's say for
example at n = 0--
is equal to the input at one time sample, or instant, or
value of the index before that.
And so this, in fact, is a system for which the output is
simply the input delayed or shifted by one sample.
So this system is referred to as a unit delay.
And now, the question is, is that system memoryless?
Well, the output depends only on the input--
the output at any instant depends only on the input at
one instant.
But since it depends on an instant prior to the time at
which we're looking, or different than the time at
which we're looking, it violates the definition of
memoryless that we introduced.
And so, in fact, the unit delay is a system that has
memory, and so let's indicate that here.
So this is not a memoryless system because of the fact
that there is 1 unit of delay.
And in essence, delay requires memory.
OK, so that's the issue of memory and memoryless systems.
Let's now turn to another property which a system may or
may not have, the property referred to as invertibility.
Now, essentially what invertibility means is that
given the output of the system, you can figure out
uniquely what the input was.
That's one definition for invertibility.
Said another way, invertibility means that given
the output, there's only one input that
could have caused it.
That's another common definition for invertibility.
Another way of looking at it is, in fact, to look at it in
the context of a cascade of systems.
So let's consider a system.
And here is a system which I refer to as system A. And this
could be continuous-time or discrete-time.
It has an input, x_1(t) or x_1[n], depending on whether
it's continuous-time or discrete-time that we're
talking about, and an associated output.
And here, we have system B with its
associated input and output.
And now, let's put these two systems in cascade.
So we'll take the output of system 1 and feed it into the
input of system 2--
or system B. So the output of system A goes into the input
of system B. And if system A is invertible and system B is
its inverse, then the consequence is that the output
of system B is equal to the input of system A.
Now, I know there are a lot of inputs and outputs and
inverses in there.
But essentially, what we mean by what I've just said is that
if we have system A and it's invertible, and if we cascade
it with its inverse, system B, then the overall cascade of
these two systems is simply what's referred to as the
identity system.
And the identity system is simply a system which if you
put a signal into it, you get the same signal out of it.
In other words, the overall system is no
transformation at all.
And clearly, of course, for a system to be able to be
cascaded with another system to generate the identity
system requires that the first system, system A, be
invertible.
So let's look at some examples.
If we had system A as I've indicated here, where now the
output is the running integral of the input--
and remember that we saw the running integral when we
talked about the relationship between steps and impulses--
if this happened to be an impulse, then the
output would be a step.
This system is referred to, of course, as an integrator since
the output is the running integral of the input.
And the integrator, in fact, is an invertible system.
And its inverse is a system for which the output is the
derivative of the input.
So the inverse of system A, if system A is an integrator, is
a system for which the output is equal to the derivative of
the input which, not surprisingly, is referred to
as a differentiator.
So an integrator is invertible.
Its inverse is a differentiator.
What you might want to think about is the question of
whether a differentiator is invertible.
Now, to answer that, what you would ask yourself is, if you
always knew what the derivative of the signal is,
would you necessarily know what the signal was?
In other words, if you have a differentiator and you have
the output of the differentiator, could you
always figure out what the input was?
If you could, the system would be invertible.
If you couldn't, the system would not be invertible.
So you might just want to think about that.
I guess I won't tell you right now.
But I'm sure that you'll think about that more, particularly
with the guidance of the video manual.
OK, well let's look at one last system.
I've indicated here a system for which the output is
related to the input through this curve.
And what I mean by this curve, which wasn't done quite as
well as I might have, is that the output is equal to the
square of the input.
So this system is our squarer as we talked about before.
We saw previously, or discussed the fact previously,
that the squarer is a memoryless system.
And now the question is, is a squarer an invertible system?
Well, the question then is, if you're given the square of a
signal, can you figure out what the signal is?
And as I'm sure you've already suspected, the answer to that
is no, because obviously if the signal was negative or
positive, you wouldn't be able to figure that out after
you've squared.
So in fact, the squarer is not invertible.
All right, so we've introduced several properties.
And by the way, as we've gone through it, also introduced
some systems that will turn out to be useful
and important systems.
And now, let's continue with some other properties.
A property that we'll find useful to make reference to,
from time to time, and will, in fact, play a fairly
important role in a variety of discussions during the course,
is a property which is referred to as causality.
Now, in essence what causality means is the following.
A system is set to be causal if, as one way of saying it,
it only responds when you kick it.
Is another way of saying it, its response at any time only
depends on values of the input prior to that time.
So a causal system, both continuous time and discrete
time, is defined sometimes as a system which has the
property that the output at any time depends only on the
input prior or equal to that time.
Essentially what we're saying is that the system can't
anticipate future inputs.
And so that, in fact, is another possible definition
for causality, that the system is causal if it can't
anticipate future inputs.
Finally, an alternative way of saying it
mathematically is as follows.
If I have two signals, x_1(t) and x_2(t), with their
associated outputs, y_1(t) and y_2(t), then a system is said
to be causal if and only if it has the property that if those
two inputs are identical up until some time, then the
outputs are identical up until the same time.
So if we have two signals that are exactly the same up until
some time and perhaps do something different later on,
causality requires that the outputs not anticipate the
fact that those inputs are at some future time going to do
something different.
And that, in fact, is the most useful mathematical definition
of causality.
And of course, I've written that here for continuous time.
And the same definition of causality also applies for
discrete time.
OK, well let's look at an example.
Let's take an example which is a system which is the
following discrete-time system.
The output at any given time is the sum of x[n], x[n]
delayed, and x[n]
anticipated.
And this is, in fact, a system that's very useful and
referred to as a moving average.
And so if we think of a moving average, if we have here a
sequence, x[n]
and x[n]
with other values going off in both directions, for any
value, n_0, at which we're computing the output--
and this is y[n], the output--
we take x[n 0], x[n 0-1], and x[n 0+1].
And so to form that moving average, we would take these
three values and combine them together, adding them and then
dividing by 3 to get that.
Well, is the system causal?
One way to answer that is to determine whether the output
at any given time depends on future values of the input.
And clearly, if you look at this, what you see is that the
output at time n_0 depends both on past values and on
future values.
As opposed to another moving average, which I've indicated
here, where I simply shifted the values
that I combined together.
And in this case, because of the way in which I picked the
values, the output depends on the value at n_0, the value at
n_0 - 1, and the value at n_0 - 2.
So y[n 0]
would depend here on x[n 0], x[n 0-1], and x[n 0-2].
And so in this case, the system is not causal.
And in this case, the system is causal.
All right, now let's turn to another system property, the
property referred to as stability.
Now, there are lots of definitions of stability, and
some of them get very mathematical and formal.
But we've chosen, and what we'll use as our definition of
stability, is what's called bounded-input
bounded-output stability.
And essentially, the definition is that a system is
stable if and only if for every bounded input, the
output is bounded.
So the notion is if you have a system and the input never
gets above some finite value, then stability requires that
the output also stay within some bounded values.
And I'm sure that stability and instability are things
that you're kind of informally familiar with.
Let me just emphasize the point with something that I
borrowed actually from my son with some
reluctance on his part.
If we, for example, take a system like this, which is in
essence a pendulum, this system as I'm holding it here
is stable because if I put in a bounded input, which is a
displacement, the output, which is the movement of it,
remains bounded.
Now on the other hand, if I put the system like this,
which is, in fact, what's referred to as an inverted
pendulum, although we could conceivably get this to
balance, just a slight displacement because of the
fact that the pendulum is inverted, a slight
displacement and the output becomes unbounded.
Now, an interesting thing with the inverted pendulum, by the
way, which I'm sure all of you, if you were anything like
me, were intrigued with as a kid, was the notion that you
could take an inverted pendulum and in effect turn it
back into a stable system by using what I'm doing right
now, which is feedback.
What I've done in that case is I've stabilized the system by
using feedback, visual feedback, from
my eye to my hand.
And in fact, one of the very important things that we'll
see about feedback when we talk about feedback systems
much later in the course is that one of their very
important applications is in stabilizing unstable systems.
By the way, one of their problems is that if not used
correctly, it can destabilize stable systems.
OK, well let's continue on with
our property of stability.
I have here, again, the example of an integrator.
And as I indicate here, if we have an integrator and we put
a step function into it or a step signal into it, the
output is what's referred to as a ramp signal.
It linearly increases.
Now, the question is, is a ramp unbounded?
The input is bounded.
The step input is bounded.
The ramp is unbounded because if you try to establish any
bound on it, you can always go out far enough in time so that
the output will exceed that bound.
So in fact, the integrator is not stable.
OK, now finally, I'd like to turn to two properties that
we'll make considerable use of as the course goes on, the
properties of time invariance and linearity.
Time invariance, in essence, says that the system doesn't
really care what you call the origin.
In other words, it says if you take the input and you shift
it in time, all that you've done is taken the output and
shifted it in time by the same amount.
Somewhat more formally as I've indicated here, if in
continuous time we have an input, x(t), which generate an
output, y(t), then time invariance requires that if
the input is shifted by any amount of time, the output is
shifted by the same amount of time.
And exactly the same applies in discrete time.
For example, if we have a system which is a system I've
shown here, which by the way is the system that we talked
about previously to go from a step sequence--
I'm sorry, from an impulse sequence to a step sequence.
We called it a running sum.
And actually, what it's also often called is an
accumulator.
What it does is accumulate past values of the input.
Well, is the accumulator time-invariant?
The best way to establish that is to work through the
equations and verify that it either does or doesn't satisfy
the formal definition of time invariance.
Informally, if you think about it, it makes intuitive sense
that the accumulator is time-invariant because if
you're accumulating values and if you delay the values that
you're putting into the accumulator, then the
associated values that come out will be delayed by the
same amount.
The accumulator doesn't care really if you shift the input.
It'll just simply shift the associated output.
But more generally, if you're trying to test time
invariance, it's important to return to the definition.
And that's what you're required to do in the examples
in the video manual.
OK, well, I indicate another example.
We had the example of an accumulator.
Here's another example which, in fact, as we'll see later is
a system which is a modulator.
The output is the input, modulated.
And although you might think at first that this system is
time invariant, in fact it is not, because the input shifted
generates an output which is the input shifted times the
same modulation function.
Whereas if we were to take the output of the system, we have
x(t) is the input, then what that would correspond to is
sin(t-t_0) * x(t-t_0).
And since these two are not equal, this
system is not time invariant.
And this is an example that often causes a slight amount
of difficulty because it seems like when you look
at it ought to be.
And so I strongly encourage you, in the context of working
problems in the manual, that you think very carefully about
this and at least believe that what I told you
is the right answer.
OK, now the final property that I want to introduce today
is the property of linearity.
And linearity is defined in a manner similar for continuous
time and discrete time.
And what it says is that if we have some inputs with
associated outputs, let's say x_1(t) and x_2(t), then a
system is linear if it has the property that the output to a
linear combination of those inputs is the same linear
combination of the associated outputs.
And so that's what I've indicated here, that if we now
put into the system a linear combination of those inputs,
then for linearity, we require that the output is the same
linear combination.
And exactly the same applies in discrete time.
And you can show from this definition that if a system is
linear with two inputs, then it's linear in terms of an
arbitrary number of inputs.
I have a number of examples.
And these are examples that, again, I ask you to think
about as you look at the video manual.
Just to suggest the answer--
well, not to suggest but to tell you the answer, the
integrator as we have here is linear.
This system in which the output is double the input
plus a constant, you would kind of think it's linear
because it's a straight line.
But one has to be careful.
And in fact, as it turns out, this is not linear.
There is a qualifier attached to it because it has a
property referred to as incrementally linear, which is
discussed somewhat more in the text.
And finally, we have a system which is the squarer that I've
indicated again here.
And squaring it is definitely not a linear operation.
OK, so what we've done, then, is to introduce a number of
properties of systems.
And we've also, by the way, as I've stressed previously, as
we've gone along introduced also a number of important and
useful systems, like the accumulator, the integrator,
the differentiator, et cetera.
What we'll see is that the properties of linearity and
time invariance in particular become central and important
properties.
And in the next lecture, what we'll show is that with
systems that are linear and time-invariant, the use of the
impulse function, both in continuous time and discrete
time, provides an extraordinarily important and
useful mechanism for characterizing those systems.
Thank you.