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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.
