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[MUSIC PLAYING]
PROFESSOR: Last time, we introduced the Laplace
transform as a generalization of the Fourier transform, and,
just as a reminder, the Laplace transform expression
as we developed it is this integral, very much similar to
the Fourier transform integral, except with a more
general complex variable.
And, in fact, we developed and talked about the relationship
between the Laplace transform and the Fourier transform.
In particular, the Laplace transform with the Laplace
transform variable s, purely imaginary, in fact, reduces to
the Fourier transform.
Or, more generally, with the Laplace transform variable as
a complex number, the Laplace transform is the Fourier
transform of the corresponding time function with an
exponential weighting.
And, also, as you should recall, the exponential
waiting introduced the notion that the Laplace transform may
converge for some values of sigma and perhaps not for
other values of sigma.
So associated with the Laplace transform was what we refer to
as the region of convergence.
Now just as with the Fourier transform, there are a number
of properties of the Laplace transform that are extremely
useful in describing and analyzing signals and systems.
For example, one of the properties that we, in fact,
took advantage of in our discussion last time was the
linearly the linearity property, which says, in
essence, that the Laplace transform of the linear
combination of two time functions is the same linear
combination of the associated Laplace transforms.
Also, there is a very important and useful property,
which tells us how the
derivative of a time function--
rather, the Laplace transform of the derivative--
is related to the Laplace transform.
In particular, the Laplace transform of the derivative is
the Laplace transform x of t multiplied by s.
And, as you can see by just setting s equal to j omega, in
fact, this reduces to the corresponding Fourier
transform property.
And a third property that we'll make frequent use of is
referred to as the convolution property.
Again, a generalization of the convolution property for
Fourier transforms.
Here the convolution property says that the Laplace
transform of the convolution of two time functions is the
product of the associated Laplace transforms.
Now it's important at some point to think carefully about
the region of convergence as we discuss these properties.
And let me just draw your attention to the fact that in
discussing properties fully and in detail, one has to pay
attention not just to how the algebraic expression changes,
but also what the consequences are for the region of
convergence, and that's discussed in somewhat more
detail in the text and I won't do that here.
Now the convolution property leads to, of course, a very
important and useful mechanism for dealing with linear time
invariant systems, very much as the Fourier transform did.
In particular, the convolution property tells us that if we
have a linear time invariant system, the output in the time
domain is the convolution of the input
and the impulse response.
In the Laplace transform domain, the Laplace transform
of the output is the Laplace transform of the impulse
response times the Laplace transform of the input.
And again, this is a generalization of the
corresponding property for Fourier transforms.
In the case of the Fourier transform, the Fourier
transform [? of ?] the impulse response we refer to as the
frequency response.
In the more general case with Laplace transforms, it's
typical to refer to the Laplace transform of the
impulse response as the system function.
Now in talking about the system function, some issues
of the region of convergence--
and for that matter, location of poles
of the system function--
are closely tied in and related to issues of whether
the system is stable and causal.
And in fact, there's some useful statements that can be
made that play an important role throughout the further
discussion.
For example, we know from previous discussions that
there's a condition for stability of a system, which
is absolute integrability of the impulse response.
And that, in fact, is the same condition for convergence of
the Fourier transform of the impulse response.
What that says, really, is that if a system is stable,
then the region of convergence of the system function must
include the j omega axis.
Which, of course, is where the Laplace transform reduces to
the Fourier transform.
So that relates the region of convergence and stability.
Also, you recall from last time that we talked about the
region of convergence associated with right sided
time functions.
In particular for a right sided time function, the
region of convergence must be to the right of
the rightmost pole.
Well, if, in fact, we have a system that's causal, then
that causality imposes the condition that the impulse
response be right sided.
And so, in fact, for causality, we would have a
region of convergence associated with the system
function, which is to the right of the rightmost pole.
Now interestingly and very important is the consequence,
if you put those two statements together, in
particular, you're led to the conclusion that for stable
causal systems, all the poles must be in the left half of
the s-plane.
What's the reason?
The reason, of course, is that if the system is stable and
causal, the region of convergence must be to the
right of the rightmost pole.
It must include the j omega axis.
Obviously then, all the poles must be in the left half of
the s-plane.
And again, that's an issue that is discussed somewhat
more carefully and in more detail in the text.
Now, the properties that we're talking about here are not the
only properties, there are many others.
But these properties, in particular, provide the
mechanism--
as they did with Fourier transforms--
for turning linear constant coefficient differential
equations into algebraic equations and, corresponding,
lead to a mechanism for dealing with and solving
linear constant coefficient differential equations.
And I'd like to illustrate that by looking at both first
order and second order differential equations.
Let's begin, first of all, with a first order
differential equation.
So what we're talking about is a first order system.
What I mean by that is a system that's characterized by
a first order differential equation.
And if we apply to this equation the differentiation
property, then the derivative--
the Laplace transform of the derivative is s times the
Laplace transform of the time function.
The linearity property allows us to combine these together.
And so, consequently, applying the Laplace transform to this
equation leads us to this algebraic equation, and
following that through, leads us to the statement that the
Laplace transform of the output is one over s plus a
times the Laplace transform of the input.
We know from the convolution property that this Laplace
transform is the system function times x of s.
And so, one over s plus a is the system function or
equivalently, the Laplace transform
of the impulse response.
So, we can determine the impulse response by taking the
inverse Laplace transform of h of s given by
one over s plus a.
Well, we can do that using the inspection method, which is
one way that we have of doing inverse Laplace transforms.
The question is then, what time function has a Laplace
transform which is one over s plus a?
The problem that we run into is that there are
two answers to that.
one over s plus a is the Laplace transform of an
exponential for positive time, but one over s plus a is also
the Laplace transform of an exponential for negative time.
Which one of these do we end up picking?
Well, recall that the difference between these was
in their region of convergence.
And in fact, in this case, this corresponded to a region
of convergence, which was the real part of s
greater than minus a.
In this case, this was the corresponding Laplace
transform, provided that the real part of s is
less than minus a.
So we have to decide which region of convergence that we
pick and it's not the differential equation that
will tell us that, it's something else that has to
give us that information.
What could it be?
Well, what it might be is the additional information that
the system is either stable or causal.
So for example, if the system was causal, we would know that
the region of convergence is to the right of the pole and
that would correspond, then, to this
being the impulse response.
Whereas, with a negative--
I'm sorry with a positive--
if we knew that the system, let's say, was non-causal,
then we would associate with this region of convergence and
we would know then that this is the impulse response.
So a very important point is that what we see is that the
linear constant coefficient differential equation gives us
the algebraic expression for the system function, but does
not tell us about the region of convergence.
We get the reach of convergence from some
auxiliary information.
What is that information?
Well, it might, for example, be knowledge that the system
is perhaps stable, which tells us that the region of
convergence includes the j omega axis, or perhaps causal,
which tells us that the region of convergence is to the right
of the rightmost pole.
So it's the auxiliary information that specifies for
us the region of convergence.
Very important point.
The differential equation by itself does not completely
specify the system, it only essentially tells us what the
algebraic expression is for the system function.
Alright that's a first order example.
Let's now look at a second order system and the
differential equation that I picked in this case.
I've parameterized in a certain way, which we'll see
will be useful.
In particular, it's a second order differential equation
and I chosen, just for simplicity, to not include any
derivatives on the right hand side, although we could have.
In fact, if we did, that would insert zeros into the system
function, as well as the poles inserted by
the left hand side.
We can determine the system function in exactly the same
way, namely, apply the Laplace transform to this equation.
That would convert this differential equation to an
algebraic equation.
And now when we solve this algebraic equation for y of s,
in terms of x of s, it will come out in the form of y of
s, equal to h of s, times x of s.
And h of s, in that case, we would get simply by dividing
out by this polynomial [? in ?] s, and so the system
function then is the expression that I have here.
So this is the form for a second order system where
there are two poles.
Since this is a second order polynomial, there are no zeros
associated with the fact that I had no derivatives of the
input on the right hand side of the equation.
Well, let's look at this example--
namely the second order system--
in a little more detail.
And what we'll want to look at is the location of the poles
and some issues such as, for example,
the frequency response.
So here again I have the algebraic expression for the
system function.
And as I indicated, this is a second order polynomial, which
means that we can factor it into two roots.
So c1 and c2 represent the poles of the system function.
And in particular, in relation to the two parameters zeta and
omega sub n--
if we look at what these roots are, then what we get are the
two expressions that I have below.
And notice, incidentally, that if zeta is less than one, then
what's under the square root is negative.
And so this, in fact, corresponds to
an imaginary part--
an imaginary term for zeta less than one.
And so the two roots, then, have a real part which is
given by minus zeta omega sub n, and an imaginary part-- if
I were to rewrite this and then express it in terms of j
or the square root of minus one.
Looking below, we'll have a real part which is minus zeta
omega sub n--
an imaginary part which is omega sub n
times this square root.
So that's for zeta less than one and for zeta greater than
one, the two roots, of course, will be real.
Alright, so let's examine this for the case where zeta is
less than one.
And what that corresponds to, then, are two poles in the
complex plane.
And they have a real part and an imaginary part.
And you can explore this in somewhat more detail on your
own, but, essentially what happens is that as you keep
the parameter omega sub n fixed and vary zeta, these
poles trace out a circle.
And, for example, where zeta equal to zero, the poles are
on the j omega axis at omega sub n.
As zeta increases and gets closer to one, the poles
converge toward the real axis and then, in particular, for
zeta greater than one, what we end up with are two poles on
the real axis.
Well, actually, the case that we want to look at a little
more carefully is when the poles are complex.
And what this becomes is a second order system, which as
we'll see as the discussion goes on, has an impulse
response which oscillates with time and correspondingly a
frequency response that has a resonance.
Well let's examine the frequency response a little
more carefully.
And what I'm assuming in the discussion is that, first of
all, the poles are in the left half plane corresponding to
zeta omega sub n being positive--
and so this is-- minus that is negative.
And furthermore, I'm assuming that the poles are complex.
And in that case, the algebraic expression for the
system function is omega sub n squared in the numerator and
two poles in the denominator, which are complex conjugates.
Now, what we want to look at is the frequency response of
the system.
And
that corresponds to looking at the Fourier transform of the
impulse response, which is the Laplace transform on the j
omega axis.
So we want to examine what h of s is as we move along the j
omega axis.
And notice, that to do that, in this algebraic expression,
we want to set s equal to j omega and then evaluate--
for example, if we want to look at the magnitude of the
frequency response--
evaluate the magnitude of the complex number.
Well, there's a very convenient way of doing that
geometrically by recognizing that in the complex plane,
this complex number minus that complex number
represents a vector.
And essentially, to look at the magnitude of this complex
number corresponds to taking omega sub n squared and
dividing it by the product of the lengths of these vectors.
So let's look, for example, at the vector s minus c1, where s
is on the j omega axis.
And doing that, here is the vector c1, and here is the
vector s-- which is j omega if we're looking, let's say, at
this value of frequency--
and this vector, then, is the vector which is
j omega minus c1.
So in fact, it's the length of this vector that we want to
observe as we change omega--
namely as we move along the j omega axis.
We want to take this vector and this vector, take the
lengths of those vectors, multiply them together, divide
that into omega sub n squared, and that will give us the
frequency response.
Now that's a little hard to see how the frequency response
will work out just looking at one point.
Although notice that as we move along the j omega axis,
as we get closer to this pole, this vector, in fact, gets
shorter, and so we might expect , that
the frequency response--
as we're moving along the j omega axis in the vicinity of
that pole--
would start to peak.
Well, I think that all of this is much better seen
dynamically on the computer display, so let's go to the
computer display and what we'll look at is a second
order system--
the frequency response of it-- as we move
along the j omega axis.
So here we see the pole pair in the complex plane and to
generate the frequency response, we want to look at
the behavior of the pole vectors as we move vertically
along the j omega axis.
So we'll show the pole vectors and let's begin at omega
equals zero.
So here we have the pole vectors from the poles to the
point omega equal to zero.
And, as we move vertically along the j omega axis, we'll
see how those pole vectors change in length.
The magnitude of the frequency response is the reciprocal of
the product of the lengths of those vectors.
Shown below is the frequency response where we've begun
just at omega equal to zero.
And as we move vertically along the j omega axis and the
pole vector lengths change, that will, then, influence
what the frequency response looks like.
We've started here to move a little bit away from omega
equal to zero and notice that in the upper half plane the
pole vector has gotten shorter.
The pole vector for the pole in the lower half plane has
gotten longer.
And now, as omega increases further, that
process will continue.
And in particular, the pole vector associated with the
pole in the upper half plane will be its
shortest in the vicinity--
at a frequency in the vicinity of that pole--
and so, for that frequency, then, the frequency response
will peak and we see that here.
From this point as the frequency increases,
corresponding to moving further vertically along the j
omega axis, both pole vectors will increase in length.
And that means, then, that the magnitude of the frequency
response will decrease.
For this specific example, the magnitude of the frequency
response will asymptotically go to zero.
So what we see here is that the frequency response has a
resonance and as we see geometrically from the way the
vectors behaved, that resonance in frequency is very
clearly associated with the position of the poles.
And so, in fact, to illustrate that further and dramatize it
as long as we're focused on it, let's now look at the
frequency response for the second order example as we
change the pole positions.
And first, what we'll do is let the polls move vertically
parallel to the j omega axis and see how the frequency
response changes, and then we'll have the polls move
horizontally parallel to the real axis and see how the
frequency response changes.
To display the behavior of the frequency response as the
poles move, we've changed the vertical scale on the
frequency response somewhat.
And now what we want to do is move the poles, first,
parallel to the j omega axis, and then
parallel to the real axis.
Here we see the effect of moving the poles parallel to
the j omega axis.
And what we observe is that, in fact, the frequency
location of the resonance shifts, basically tracking the
location of the pole.
If we now move the pole back down closer to the real axis,
then this resonance will shift back toward its original
location and so let's now see that.
And here we are back at the frequency that we started at.
Now we'll move the poles even closer to the real axis.
The frequency location of the resonance will continue to
shift toward lower frequencies.
And also in the process, incidentally, the height over
the resonant peak will increase because, of course,
the lengths of the pole vectors are getting shorter.
And so, we see now the resonance shifting down toward
lower and lower frequency.
And, finally, what we'll now do is move the poles back to
their original position and the resonant peak will, of
course, shift back up.
And correspondingly the height or amplitude of the resonance
will decrease.
And now we're back at the frequency response that we had
generated previously.
Next we'd like to look at the behavior as the polls move
parallel to the real axis.
First closer to the j omega axis and then further away.
As they move closer to the j omega axis, the resonance
sharpens because of the fact that the pole vector gets
shorter and responds--
or changes in length more quickly as we move past it
moving along the j omega axis.
So here we see the effect of moving the poles closer to the
j omega axis.
The resonance has gotten narrower in frequency and
higher in amplitude, associated with the fact that
the pole vector gets shorter.
Next as we move back to the original location, the
resonance will broaden once again and the
amplitude will decrease.
And then, if we continue to move the poles even further
away from the real axis, the resonance will broaden even
further and the amplitude of the peak
will become even smaller.
And finally, let's now look just move the poles back to
their original position and we'll see the resonance narrow
again and become higher.
And so what we see then is that for a second order
system, the behavior of the resonance basically is
associated with the pole locations, the frequency of
the resonance associated with the vertical position of the
poles, and the sharpness of the resonance associated with
the real part of the poles-- in other words, their position
closer or further away from the j omega axis.
OK, so for complex poles, then, for the second order
system, what we see is that we get a resonant kind of
behavior, and, in particular, then that resonate behavior
tends to peak, or get peakier, as the value
of zeta gets smaller.
And here, just to remind you of what you saw, here is the
frequency response with one particular choice of values--
well, this is normalized so that omega sub n is one-- one
particular choice for zeta, namely 0.4.
Here is what we have with zeta smaller, and, finally, here is
an example where zeta has gotten even smaller than that.
And what that corresponds to is the poles moving closer to
the j omega axis, the corresponding frequency
response getting peakier.
Now in the time domain what happens is that we have, of
course, these complex roots, which I indicated previously,
where this represents the imaginary part because zeta is
less than one.
And in the time domain, we will have a form for the
behavior, which is a e to the c one t, plus a conjugate, e
to the c one conjugate t.
And so, in fact, as the poles get closer
to the j omega axis--
corresponding to zeta getting smaller--
as the polls get closer to the j omega axis, in the frequency
domain the resonances get sharper.
In the time domain, the real part of the poles has gotten
smaller, and that means, in fact, that in the time domain,
the behavior will be more oscillatory and less damped.
And so just looking at that again.
Here is, in the time domain, what happens.
First of all, with the parameter zeta equal to 0.4,
and it oscillates and exponentially dies out.
Here is the second order system where zeta is now 0.2
instead of 0.4.
And, finally, the second order system where zeta is 0.1.
And what we see as zeta gets smaller and smaller is that
the oscillations are basically the same, but the exponential
damping becomes less and less.
Alright, now, this is a somewhat more detailed look at
second order systems.
And second order systems-- and for that
matter, first order systems--
are systems that are important in their own right, but they
also are important as basic building blocks for more
general, in particular, for higher order systems.
And the way in which that's done typically is by combining
first and second order systems together in such a way that
they implement higher order systems.
And two very common connections are connections
which are cascade connections, and connections which are
parallel connections.
In a cascade connection, we would think of combining the
individual systems together as I indicate here in series.
And, of course, from the convolution property, the
overall system function is the product of the individual
system functions.
So, for example, if these were all second order systems, and
I combine n of them together in cascade, the overall system
would be a system that would have to n poles-- in other
words, it would be a two n order system.
That's one very common kind of connection.
Another very common kind of connection for first and
second order systems is a parallel connection, where, in
that case, we connect the systems together
as I indicate here.
The overall system function is just simply the sum of these,
and that follows from the linearity property.
And so the overall system function would be as I
indicate algebraically here.
And notice that if each of these are second order
systems, and I had capital N of them in parallel, when you
think of putting the overall system function over one
common denominator, that common denominator, in
general, is going to be of order two N. So either the
parallel connection or the cascade connection could be
used to implement higher order systems.
One very common context in which second order systems are
combined together, either in parallel or in cascade, to
form a more interesting system is, in
fact, in speech synthesis.
And what I'd like to do is demonstrate a speech
synthesizer, which I have here, which in fact is a
parallel combination of four second order systems, very
much of the type that we've just talked about.
I'll return to the synthesizer in a minute.
Let me first just indicate what the basic idea is.
In speech synthesis, what we're trying to represent or
implement is something that corresponds
to the vocal tract.
The vocal tract is characterized by a set of
resonances.
And we can think of representing each of those
resonances by a second order system.
And then the higher order system corresponding to the
vocal tract is built by, in this case, a parallel
combination of those second order systems.
So for the synthesizer, what we have connected together in
parallel is four second order systems.
And a control on each one of them that controls the center
frequency or the resonant frequency of each of the
second order systems.
The excitation is an excitation that would
represent the air flow through the vocal cords.
The vocal cords vibrate and there are puffs of air through
the vocal cords as they open and close.
And so the excitation for the synthesizer corresponds to a
pulse train representing the air flow
through the vocal cords.
The fundamental frequency of this representing the
fundamental frequency of the synthesized voice.
So that's the basic structure of the synthesizer
And what we have in this analog synthesizer are
separate controls on the individual center frequencies.
There is a control representing the center
frequency of the third resonator and the fourth
resonator, and those are represented
by these two knobs.
And then the first and second resonators are controlled by
moving this joystick.
The first resonator by moving the joystick along this axis
and the second resonator by moving the
joystick along this axis.
And then, in addition to controls on the four
resonators, we can control the fundamental frequency of the
excitation, and we do that with this knob.
So let's, first of all, just listen to one of the
resonators, and the resonator that I'll play
is the fourth resonator.
And what you'll hear first is the output as I vary the
center frequency of that resonator.
[BUZZING]
So I'm lowering the center frequency.
And then, bringing the center frequency back up.
And then, as I indicated, I can also control the
fundamental frequency of the excitation by
turning this knob.
[BUZZING]
Lowering the fundamental frequency.
And then, increasing the fundamental frequency.
Alright, now, if the four resonators in parallel are an
implementation of the vocal cavity, then, presumably, what
we can synthesize when we put them all in are vowel sounds
and let's do that.
I'll now switch in the other resonators.
When we do that, then, depending on what choice we
have for the individual resonant frequencies, we
should be able to synthesize vowel sounds.
So here, for example, is the vowel e.
[BUZZING "E"].
Here is
[BUZZING "AH"]
--ah.
A.
[BUZZING FLAT A]
And, of course, we can--
[BUZZING OO]
--generate
[BUZZING "I"]
--lots of other vowel sounds.
[BUZZING "AH"]
--and change the fundamental frequency at the same time.
[CHANGES FREQUENCY UP AND DOWN]
Now, if we want to synthesize speech it's not enough to just
synthesize steady state vowels-- that gets boring
after a while.
Of course what happens with the vocal cavity is that it
moves as a function of time and that's what generates the
speech that we want to generate.
And so, presumably then, if we change these resonant
frequencies as a function of time appropriately, then we
should be able to synthesize speech.
And so by moving these resonances around, we can
generate synthesized speech.
And let's try it with some phrase.
And I'll do that by simply adjusting the center
frequencies appropriately.
[BUZZING "HOW ARE YOU"]
Well, hopefully you understood that.
As you could imagine, I spent at least a few minutes before
the lecture trying to practice that so that it would come out
to be more or less intelligible.
Now the system as I've just demonstrated it is, of course,
a continuous time system or an analog speech synthesizer.
There are many versions of digital or discrete time
synthesizers.
One of the first, in fact, being a device that many of
you are very likely familiar with, which is the Texas
Instruments Speak and Spell, which I show here.
And what's very interesting and rather dramatic about this
device is the fact that it implements the speech
synthesis in very much the same way as I've demonstrated
with the analog synthesizer.
In this case, it's five second order filters in a
configuration that's slightly different than a parallel
configuration but conceptually very closely related.
And let's take a look inside the box.
And what we see there, with a slide that was kindly supplied
by Texas Instruments, is the fact that there really are
only four chips in there--
a controller chip, some storage.
And the important point is the chip that's labeled as the
speech synthesis chip, in fact, is what embodies or
implements the five second order filters and, in
addition, incorporates some other things-- some memory and
also the [? DDA ?]
converters.
So, in fact, the implementation of the
synthesizer is pretty much done on a single chip.
Well that's a discrete time system.
We've been talking for the last several lectures about
continuous time systems and the Laplace transform.
Hopefully what you've seen in this lecture and the previous
lecture is the powerful tool that the Laplace transform
affords us in analyzing and understanding system behavior.
In the next lecture what I'd like to do is parallel the
discussion for discrete time, turn our attention to the z
transform, and, as you can imagine simply by virtue of
the fact that I have shown you a digital and analog version
of very much the same kind of system, the discussions
parallel themselves very strongly and the z transform
will play very much the same role in discrete time that the
Laplace transform does in continuous time.
Thank you.