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[MUSIC PLAYING]
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PROFESSOR: Last time, we
introduced the Laplace
00:00:58.660 --> 00:01:03.310
transform as a generalization of
the Fourier transform, and,
00:01:03.310 --> 00:01:07.530
just as a reminder, the Laplace
transform expression
00:01:07.530 --> 00:01:13.720
as we developed it is this
integral, very much similar to
00:01:13.720 --> 00:01:16.870
the Fourier transform integral,
except with a more
00:01:16.870 --> 00:01:19.210
general complex variable.
00:01:19.210 --> 00:01:22.940
And, in fact, we developed and
talked about the relationship
00:01:22.940 --> 00:01:26.260
between the Laplace transform
and the Fourier transform.
00:01:26.260 --> 00:01:30.880
In particular, the Laplace
transform with the Laplace
00:01:30.880 --> 00:01:36.590
transform variable s, purely
imaginary, in fact, reduces to
00:01:36.590 --> 00:01:38.970
the Fourier transform.
00:01:38.970 --> 00:01:43.320
Or, more generally, with the
Laplace transform variable as
00:01:43.320 --> 00:01:48.550
a complex number, the Laplace
transform is the Fourier
00:01:48.550 --> 00:01:53.430
transform of the corresponding
time function with an
00:01:53.430 --> 00:01:55.280
exponential weighting.
00:01:55.280 --> 00:01:59.730
And, also, as you should
recall, the exponential
00:01:59.730 --> 00:02:04.870
waiting introduced the notion
that the Laplace transform may
00:02:04.870 --> 00:02:08.820
converge for some values of
sigma and perhaps not for
00:02:08.820 --> 00:02:10.360
other values of sigma.
00:02:10.360 --> 00:02:15.890
So associated with the Laplace
transform was what we refer to
00:02:15.890 --> 00:02:17.765
as the region of convergence.
00:02:20.400 --> 00:02:25.120
Now just as with the Fourier
transform, there are a number
00:02:25.120 --> 00:02:28.470
of properties of the Laplace
transform that are extremely
00:02:28.470 --> 00:02:35.250
useful in describing and
analyzing signals and systems.
00:02:35.250 --> 00:02:39.200
For example, one of the
properties that we, in fact,
00:02:39.200 --> 00:02:42.740
took advantage of in our
discussion last time was the
00:02:42.740 --> 00:02:46.250
linearly the linearity property,
which says, in
00:02:46.250 --> 00:02:49.980
essence, that the Laplace
transform of the linear
00:02:49.980 --> 00:02:54.830
combination of two time
functions is the same linear
00:02:54.830 --> 00:03:00.260
combination of the associated
Laplace transforms.
00:03:00.260 --> 00:03:03.640
Also, there is a very important
and useful property,
00:03:03.640 --> 00:03:05.980
which tells us how the
00:03:05.980 --> 00:03:09.650
derivative of a time function--
00:03:09.650 --> 00:03:12.090
rather, the Laplace transform
of the derivative--
00:03:12.090 --> 00:03:14.710
is related to the Laplace
transform.
00:03:14.710 --> 00:03:19.350
In particular, the Laplace
transform of the derivative is
00:03:19.350 --> 00:03:23.320
the Laplace transform x
of t multiplied by s.
00:03:23.320 --> 00:03:27.200
And, as you can see by just
setting s equal to j omega, in
00:03:27.200 --> 00:03:30.230
fact, this reduces to the
corresponding Fourier
00:03:30.230 --> 00:03:32.740
transform property.
00:03:32.740 --> 00:03:37.830
And a third property that we'll
make frequent use of is
00:03:37.830 --> 00:03:40.390
referred to as the convolution
property.
00:03:40.390 --> 00:03:43.910
Again, a generalization of the
convolution property for
00:03:43.910 --> 00:03:45.700
Fourier transforms.
00:03:45.700 --> 00:03:49.270
Here the convolution property
says that the Laplace
00:03:49.270 --> 00:03:53.390
transform of the convolution of
two time functions is the
00:03:53.390 --> 00:03:58.620
product of the associated
Laplace transforms.
00:03:58.620 --> 00:04:04.610
Now it's important at some point
to think carefully about
00:04:04.610 --> 00:04:09.400
the region of convergence as we
discuss these properties.
00:04:09.400 --> 00:04:13.430
And let me just draw your
attention to the fact that in
00:04:13.430 --> 00:04:17.970
discussing properties fully and
in detail, one has to pay
00:04:17.970 --> 00:04:25.090
attention not just to how the
algebraic expression changes,
00:04:25.090 --> 00:04:27.990
but also what the consequences
are for the region of
00:04:27.990 --> 00:04:30.990
convergence, and that's
discussed in somewhat more
00:04:30.990 --> 00:04:35.130
detail in the text and
I won't do that here.
00:04:35.130 --> 00:04:40.750
Now the convolution property
leads to, of course, a very
00:04:40.750 --> 00:04:46.300
important and useful mechanism
for dealing with linear time
00:04:46.300 --> 00:04:50.330
invariant systems, very much as
the Fourier transform did.
00:04:50.330 --> 00:04:54.990
In particular, the convolution
property tells us that if we
00:04:54.990 --> 00:04:59.280
have a linear time invariant
system, the output in the time
00:04:59.280 --> 00:05:02.520
domain is the convolution
of the input
00:05:02.520 --> 00:05:04.390
and the impulse response.
00:05:04.390 --> 00:05:08.450
In the Laplace transform domain,
the Laplace transform
00:05:08.450 --> 00:05:12.530
of the output is the Laplace
transform of the impulse
00:05:12.530 --> 00:05:16.760
response times the Laplace
transform of the input.
00:05:16.760 --> 00:05:20.700
And again, this is a
generalization of the
00:05:20.700 --> 00:05:24.220
corresponding property for
Fourier transforms.
00:05:24.220 --> 00:05:28.020
In the case of the Fourier
transform, the Fourier
00:05:28.020 --> 00:05:30.720
transform [? of ?] the impulse
response we refer to as the
00:05:30.720 --> 00:05:32.430
frequency response.
00:05:32.430 --> 00:05:36.290
In the more general case with
Laplace transforms, it's
00:05:36.290 --> 00:05:38.950
typical to refer to the Laplace
transform of the
00:05:38.950 --> 00:05:43.530
impulse response as the
system function.
00:05:43.530 --> 00:05:52.500
Now in talking about the system
function, some issues
00:05:52.500 --> 00:05:54.120
of the region of convergence--
00:05:54.120 --> 00:05:55.920
and for that matter,
location of poles
00:05:55.920 --> 00:05:57.940
of the system function--
00:05:57.940 --> 00:06:03.010
are closely tied in and related
to issues of whether
00:06:03.010 --> 00:06:05.940
the system is stable
and causal.
00:06:05.940 --> 00:06:08.570
And in fact, there's some useful
statements that can be
00:06:08.570 --> 00:06:12.610
made that play an important role
throughout the further
00:06:12.610 --> 00:06:13.640
discussion.
00:06:13.640 --> 00:06:19.030
For example, we know from
previous discussions that
00:06:19.030 --> 00:06:21.380
there's a condition for
stability of a system, which
00:06:21.380 --> 00:06:25.390
is absolute integrability
of the impulse response.
00:06:25.390 --> 00:06:29.650
And that, in fact, is the same
condition for convergence of
00:06:29.650 --> 00:06:33.180
the Fourier transform of
the impulse response.
00:06:33.180 --> 00:06:39.350
What that says, really, is that
if a system is stable,
00:06:39.350 --> 00:06:43.530
then the region of convergence
of the system function must
00:06:43.530 --> 00:06:45.710
include the j omega axis.
00:06:45.710 --> 00:06:49.390
Which, of course, is where the
Laplace transform reduces to
00:06:49.390 --> 00:06:52.380
the Fourier transform.
00:06:52.380 --> 00:06:56.540
So that relates the region of
convergence and stability.
00:06:56.540 --> 00:07:00.790
Also, you recall from last time
that we talked about the
00:07:00.790 --> 00:07:03.740
region of convergence associated
with right sided
00:07:03.740 --> 00:07:05.050
time functions.
00:07:05.050 --> 00:07:07.300
In particular for a right
sided time function, the
00:07:07.300 --> 00:07:10.030
region of convergence must
be to the right of
00:07:10.030 --> 00:07:12.330
the rightmost pole.
00:07:12.330 --> 00:07:17.970
Well, if, in fact, we have a
system that's causal, then
00:07:17.970 --> 00:07:22.040
that causality imposes the
condition that the impulse
00:07:22.040 --> 00:07:23.820
response be right sided.
00:07:23.820 --> 00:07:27.510
And so, in fact, for causality,
we would have a
00:07:27.510 --> 00:07:29.960
region of convergence associated
with the system
00:07:29.960 --> 00:07:33.780
function, which is to the right
of the rightmost pole.
00:07:33.780 --> 00:07:37.710
Now interestingly and very
important is the consequence,
00:07:37.710 --> 00:07:40.380
if you put those two statements
together, in
00:07:40.380 --> 00:07:45.620
particular, you're led to the
conclusion that for stable
00:07:45.620 --> 00:07:51.770
causal systems, all the poles
must be in the left half of
00:07:51.770 --> 00:07:53.700
the s-plane.
00:07:53.700 --> 00:07:54.870
What's the reason?
00:07:54.870 --> 00:07:59.010
The reason, of course, is that
if the system is stable and
00:07:59.010 --> 00:08:01.720
causal, the region of
convergence must be to the
00:08:01.720 --> 00:08:03.520
right of the rightmost pole.
00:08:03.520 --> 00:08:06.460
It must include the
j omega axis.
00:08:06.460 --> 00:08:09.170
Obviously then, all the poles
must be in the left half of
00:08:09.170 --> 00:08:10.660
the s-plane.
00:08:10.660 --> 00:08:14.400
And again, that's an issue that
is discussed somewhat
00:08:14.400 --> 00:08:18.650
more carefully and in more
detail in the text.
00:08:18.650 --> 00:08:23.840
Now, the properties that we're
talking about here are not the
00:08:23.840 --> 00:08:26.620
only properties, there
are many others.
00:08:26.620 --> 00:08:30.200
But these properties, in
particular, provide the
00:08:30.200 --> 00:08:30.880
mechanism--
00:08:30.880 --> 00:08:33.409
as they did with Fourier
transforms--
00:08:33.409 --> 00:08:37.280
for turning linear constant
coefficient differential
00:08:37.280 --> 00:08:41.840
equations into algebraic
equations and, corresponding,
00:08:41.840 --> 00:08:46.250
lead to a mechanism for dealing
with and solving
00:08:46.250 --> 00:08:48.890
linear constant coefficient
differential equations.
00:08:48.890 --> 00:08:53.280
And I'd like to illustrate that
by looking at both first
00:08:53.280 --> 00:08:56.010
order and second order
differential equations.
00:08:56.010 --> 00:08:59.400
Let's begin, first of all,
with a first order
00:08:59.400 --> 00:09:01.940
differential equation.
00:09:01.940 --> 00:09:06.500
So what we're talking about
is a first order system.
00:09:06.500 --> 00:09:11.930
What I mean by that is a system
that's characterized by
00:09:11.930 --> 00:09:14.870
a first order differential
equation.
00:09:14.870 --> 00:09:19.460
And if we apply to this equation
the differentiation
00:09:19.460 --> 00:09:22.580
property, then the
derivative--
00:09:22.580 --> 00:09:25.910
the Laplace transform of the
derivative is s times the
00:09:25.910 --> 00:09:28.540
Laplace transform of
the time function.
00:09:28.540 --> 00:09:32.580
The linearity property allows us
to combine these together.
00:09:32.580 --> 00:09:36.520
And so, consequently, applying
the Laplace transform to this
00:09:36.520 --> 00:09:40.780
equation leads us to this
algebraic equation, and
00:09:40.780 --> 00:09:43.990
following that through, leads
us to the statement that the
00:09:43.990 --> 00:09:47.430
Laplace transform of the output
is one over s plus a
00:09:47.430 --> 00:09:50.620
times the Laplace transform
of the input.
00:09:50.620 --> 00:09:54.530
We know from the convolution
property that this Laplace
00:09:54.530 --> 00:09:58.020
transform is the system
function times x of s.
00:09:58.020 --> 00:10:02.530
And so, one over s plus a is
the system function or
00:10:02.530 --> 00:10:04.670
equivalently, the Laplace
transform
00:10:04.670 --> 00:10:07.420
of the impulse response.
00:10:07.420 --> 00:10:12.400
So, we can determine the impulse
response by taking the
00:10:12.400 --> 00:10:17.560
inverse Laplace transform
of h of s given by
00:10:17.560 --> 00:10:20.890
one over s plus a.
00:10:20.890 --> 00:10:24.600
Well, we can do that using the
inspection method, which is
00:10:24.600 --> 00:10:28.620
one way that we have of doing
inverse Laplace transforms.
00:10:28.620 --> 00:10:32.380
The question is then, what time
function has a Laplace
00:10:32.380 --> 00:10:35.640
transform which is one
over s plus a?
00:10:35.640 --> 00:10:38.280
The problem that we run
into is that there are
00:10:38.280 --> 00:10:40.910
two answers to that.
00:10:40.910 --> 00:10:46.020
one over s plus a is the
Laplace transform of an
00:10:46.020 --> 00:10:52.250
exponential for positive time,
but one over s plus a is also
00:10:52.250 --> 00:10:57.550
the Laplace transform of an
exponential for negative time.
00:10:57.550 --> 00:11:01.340
Which one of these do
we end up picking?
00:11:01.340 --> 00:11:05.200
Well, recall that the difference
between these was
00:11:05.200 --> 00:11:06.810
in their region of
convergence.
00:11:06.810 --> 00:11:11.080
And in fact, in this case, this
corresponded to a region
00:11:11.080 --> 00:11:14.280
of convergence, which was
the real part of s
00:11:14.280 --> 00:11:16.740
greater than minus a.
00:11:16.740 --> 00:11:19.850
In this case, this was the
corresponding Laplace
00:11:19.850 --> 00:11:24.100
transform, provided that
the real part of s is
00:11:24.100 --> 00:11:26.970
less than minus a.
00:11:26.970 --> 00:11:30.400
So we have to decide which
region of convergence that we
00:11:30.400 --> 00:11:33.980
pick and it's not the
differential equation that
00:11:33.980 --> 00:11:37.650
will tell us that, it's
something else that has to
00:11:37.650 --> 00:11:39.640
give us that information.
00:11:39.640 --> 00:11:40.520
What could it be?
00:11:40.520 --> 00:11:45.310
Well, what it might be is the
additional information that
00:11:45.310 --> 00:11:49.210
the system is either
stable or causal.
00:11:49.210 --> 00:11:52.560
So for example, if the system
was causal, we would know that
00:11:52.560 --> 00:11:55.950
the region of convergence is to
the right of the pole and
00:11:55.950 --> 00:11:59.320
that would correspond,
then, to this
00:11:59.320 --> 00:12:02.630
being the impulse response.
00:12:02.630 --> 00:12:06.270
Whereas, with a negative--
00:12:06.270 --> 00:12:08.320
I'm sorry with a positive--
00:12:08.320 --> 00:12:12.820
if we knew that the system,
let's say, was non-causal,
00:12:12.820 --> 00:12:17.280
then we would associate with
this region of convergence and
00:12:17.280 --> 00:12:20.600
we would know then that this
is the impulse response.
00:12:20.600 --> 00:12:26.060
So a very important point is
that what we see is that the
00:12:26.060 --> 00:12:30.850
linear constant coefficient
differential equation gives us
00:12:30.850 --> 00:12:36.770
the algebraic expression for the
system function, but does
00:12:36.770 --> 00:12:39.190
not tell us about the region
of convergence.
00:12:39.190 --> 00:12:41.690
We get the reach of convergence
from some
00:12:41.690 --> 00:12:43.540
auxiliary information.
00:12:43.540 --> 00:12:44.860
What is that information?
00:12:44.860 --> 00:12:49.090
Well, it might, for example, be
knowledge that the system
00:12:49.090 --> 00:12:52.930
is perhaps stable, which tells
us that the region of
00:12:52.930 --> 00:12:57.290
convergence includes the j omega
axis, or perhaps causal,
00:12:57.290 --> 00:13:00.920
which tells us that the region
of convergence is to the right
00:13:00.920 --> 00:13:02.340
of the rightmost pole.
00:13:02.340 --> 00:13:06.190
So it's the auxiliary
information that specifies for
00:13:06.190 --> 00:13:08.010
us the region of convergence.
00:13:08.010 --> 00:13:09.340
Very important point.
00:13:09.340 --> 00:13:11.980
The differential equation by
itself does not completely
00:13:11.980 --> 00:13:16.250
specify the system, it only
essentially tells us what the
00:13:16.250 --> 00:13:20.310
algebraic expression is for
the system function.
00:13:20.310 --> 00:13:21.950
Alright that's a first
order example.
00:13:21.950 --> 00:13:26.900
Let's now look at a second
order system and the
00:13:26.900 --> 00:13:30.080
differential equation that
I picked in this case.
00:13:30.080 --> 00:13:32.860
I've parameterized in a certain
way, which we'll see
00:13:32.860 --> 00:13:34.150
will be useful.
00:13:34.150 --> 00:13:37.160
In particular, it's a second
order differential equation
00:13:37.160 --> 00:13:40.800
and I chosen, just for
simplicity, to not include any
00:13:40.800 --> 00:13:44.030
derivatives on the right hand
side, although we could have.
00:13:44.030 --> 00:13:47.860
In fact, if we did, that would
insert zeros into the system
00:13:47.860 --> 00:13:50.600
function, as well as the
poles inserted by
00:13:50.600 --> 00:13:52.950
the left hand side.
00:13:52.950 --> 00:13:56.750
We can determine the system
function in exactly the same
00:13:56.750 --> 00:14:00.840
way, namely, apply the Laplace
transform to this equation.
00:14:00.840 --> 00:14:04.200
That would convert this
differential equation to an
00:14:04.200 --> 00:14:06.290
algebraic equation.
00:14:06.290 --> 00:14:11.590
And now when we solve this
algebraic equation for y of s,
00:14:11.590 --> 00:14:15.590
in terms of x of s, it will come
out in the form of y of
00:14:15.590 --> 00:14:19.890
s, equal to h of s,
times x of s.
00:14:19.890 --> 00:14:23.430
And h of s, in that case, we
would get simply by dividing
00:14:23.430 --> 00:14:27.240
out by this polynomial [? in ?]
s, and so the system
00:14:27.240 --> 00:14:32.120
function then is the expression
that I have here.
00:14:32.120 --> 00:14:37.030
So this is the form for a second
order system where
00:14:37.030 --> 00:14:38.240
there are two poles.
00:14:38.240 --> 00:14:42.980
Since this is a second order
polynomial, there are no zeros
00:14:42.980 --> 00:14:47.090
associated with the fact that
I had no derivatives of the
00:14:47.090 --> 00:14:51.120
input on the right hand
side of the equation.
00:14:51.120 --> 00:14:54.110
Well, let's look at
this example--
00:14:54.110 --> 00:14:55.620
namely the second
order system--
00:14:55.620 --> 00:14:57.190
in a little more detail.
00:14:57.190 --> 00:15:01.100
And what we'll want to look at
is the location of the poles
00:15:01.100 --> 00:15:02.880
and some issues such
as, for example,
00:15:02.880 --> 00:15:05.720
the frequency response.
00:15:05.720 --> 00:15:11.170
So here again I have the
algebraic expression for the
00:15:11.170 --> 00:15:13.020
system function.
00:15:13.020 --> 00:15:16.820
And as I indicated, this is a
second order polynomial, which
00:15:16.820 --> 00:15:21.830
means that we can factor
it into two roots.
00:15:21.830 --> 00:15:27.320
So c1 and c2 represent the poles
of the system function.
00:15:27.320 --> 00:15:31.970
And in particular, in relation
to the two parameters zeta and
00:15:31.970 --> 00:15:33.500
omega sub n--
00:15:33.500 --> 00:15:39.160
if we look at what these roots
are, then what we get are the
00:15:39.160 --> 00:15:42.670
two expressions that
I have below.
00:15:42.670 --> 00:15:50.270
And notice, incidentally, that
if zeta is less than one, then
00:15:50.270 --> 00:15:52.410
what's under the square
root is negative.
00:15:52.410 --> 00:15:54.830
And so this, in fact,
corresponds to
00:15:54.830 --> 00:15:56.600
an imaginary part--
00:15:56.600 --> 00:15:59.410
an imaginary term for
zeta less than one.
00:15:59.410 --> 00:16:04.680
And so the two roots, then,
have a real part which is
00:16:04.680 --> 00:16:09.390
given by minus zeta omega sub n,
and an imaginary part-- if
00:16:09.390 --> 00:16:14.020
I were to rewrite this and then
express it in terms of j
00:16:14.020 --> 00:16:16.270
or the square root
of minus one.
00:16:16.270 --> 00:16:21.200
Looking below, we'll have a real
part which is minus zeta
00:16:21.200 --> 00:16:22.560
omega sub n--
00:16:22.560 --> 00:16:25.140
an imaginary part which
is omega sub n
00:16:25.140 --> 00:16:27.290
times this square root.
00:16:27.290 --> 00:16:31.700
So that's for zeta less than one
and for zeta greater than
00:16:31.700 --> 00:16:35.400
one, the two roots, of
course, will be real.
00:16:35.400 --> 00:16:40.260
Alright, so let's examine this
for the case where zeta is
00:16:40.260 --> 00:16:41.720
less than one.
00:16:41.720 --> 00:16:46.690
And what that corresponds to,
then, are two poles in the
00:16:46.690 --> 00:16:49.310
complex plane.
00:16:49.310 --> 00:16:54.670
And they have a real part
and an imaginary part.
00:16:54.670 --> 00:16:58.570
And you can explore this in
somewhat more detail on your
00:16:58.570 --> 00:17:02.220
own, but, essentially what
happens is that as you keep
00:17:02.220 --> 00:17:07.980
the parameter omega sub n fixed
and vary zeta, these
00:17:07.980 --> 00:17:11.619
poles trace out a circle.
00:17:11.619 --> 00:17:16.910
And, for example, where zeta
equal to zero, the poles are
00:17:16.910 --> 00:17:21.810
on the j omega axis
at omega sub n.
00:17:21.810 --> 00:17:31.370
As zeta increases and gets
closer to one, the poles
00:17:31.370 --> 00:17:37.430
converge toward the real axis
and then, in particular, for
00:17:37.430 --> 00:17:42.060
zeta greater than one, what we
end up with are two poles on
00:17:42.060 --> 00:17:44.430
the real axis.
00:17:44.430 --> 00:17:48.310
Well, actually, the case that
we want to look at a little
00:17:48.310 --> 00:17:51.460
more carefully is when the
poles are complex.
00:17:51.460 --> 00:17:55.410
And what this becomes is a
second order system, which as
00:17:55.410 --> 00:17:58.580
we'll see as the discussion
goes on, has an impulse
00:17:58.580 --> 00:18:02.450
response which oscillates with
time and correspondingly a
00:18:02.450 --> 00:18:06.380
frequency response that
has a resonance.
00:18:06.380 --> 00:18:09.640
Well let's examine the frequency
response a little
00:18:09.640 --> 00:18:11.250
more carefully.
00:18:11.250 --> 00:18:14.640
And what I'm assuming in the
discussion is that, first of
00:18:14.640 --> 00:18:20.110
all, the poles are in the left
half plane corresponding to
00:18:20.110 --> 00:18:22.640
zeta omega sub n being
positive--
00:18:22.640 --> 00:18:25.490
and so this is-- minus
that is negative.
00:18:25.490 --> 00:18:29.710
And furthermore, I'm assuming
that the poles are complex.
00:18:29.710 --> 00:18:33.770
And in that case, the algebraic
expression for the
00:18:33.770 --> 00:18:38.110
system function is omega sub n
squared in the numerator and
00:18:38.110 --> 00:18:42.700
two poles in the denominator,
which are complex conjugates.
00:18:42.700 --> 00:18:48.360
Now, what we want to look at is
the frequency response of
00:18:48.360 --> 00:18:49.720
the system.
00:18:49.720 --> 00:18:50.620
And
00:18:50.620 --> 00:18:55.370
that corresponds to looking at
the Fourier transform of the
00:18:55.370 --> 00:18:59.450
impulse response, which is the
Laplace transform on the j
00:18:59.450 --> 00:19:00.930
omega axis.
00:19:00.930 --> 00:19:05.190
So we want to examine what h of
s is as we move along the j
00:19:05.190 --> 00:19:06.680
omega axis.
00:19:06.680 --> 00:19:11.810
And notice, that to do that, in
this algebraic expression,
00:19:11.810 --> 00:19:15.920
we want to set s equal to j
omega and then evaluate--
00:19:15.920 --> 00:19:17.830
for example, if we want to look
at the magnitude of the
00:19:17.830 --> 00:19:19.470
frequency response--
00:19:19.470 --> 00:19:24.740
evaluate the magnitude of
the complex number.
00:19:24.740 --> 00:19:27.100
Well, there's a very convenient
way of doing that
00:19:27.100 --> 00:19:33.420
geometrically by recognizing
that in the complex plane,
00:19:33.420 --> 00:19:36.000
this complex number minus
that complex number
00:19:36.000 --> 00:19:37.870
represents a vector.
00:19:37.870 --> 00:19:41.360
And essentially, to look at the
magnitude of this complex
00:19:41.360 --> 00:19:46.060
number corresponds to taking
omega sub n squared and
00:19:46.060 --> 00:19:51.720
dividing it by the product of
the lengths of these vectors.
00:19:51.720 --> 00:19:57.820
So let's look, for example, at
the vector s minus c1, where s
00:19:57.820 --> 00:20:00.980
is on the j omega axis.
00:20:00.980 --> 00:20:08.920
And doing that, here is the
vector c1, and here is the
00:20:08.920 --> 00:20:12.530
vector s-- which is j omega if
we're looking, let's say, at
00:20:12.530 --> 00:20:14.650
this value of frequency--
00:20:14.650 --> 00:20:18.120
and this vector, then, is
the vector which is
00:20:18.120 --> 00:20:21.100
j omega minus c1.
00:20:21.100 --> 00:20:24.630
So in fact, it's the length of
this vector that we want to
00:20:24.630 --> 00:20:28.240
observe as we change omega--
00:20:28.240 --> 00:20:31.540
namely as we move along
the j omega axis.
00:20:31.540 --> 00:20:35.470
We want to take this vector
and this vector, take the
00:20:35.470 --> 00:20:39.140
lengths of those vectors,
multiply them together, divide
00:20:39.140 --> 00:20:42.400
that into omega sub n squared,
and that will give us the
00:20:42.400 --> 00:20:43.590
frequency response.
00:20:43.590 --> 00:20:46.920
Now that's a little hard to see
how the frequency response
00:20:46.920 --> 00:20:48.750
will work out just looking
at one point.
00:20:48.750 --> 00:20:53.610
Although notice that as we move
along the j omega axis,
00:20:53.610 --> 00:20:57.230
as we get closer to this pole,
this vector, in fact, gets
00:20:57.230 --> 00:21:01.160
shorter, and so we might
expect , that
00:21:01.160 --> 00:21:02.490
the frequency response--
00:21:02.490 --> 00:21:05.120
as we're moving along the j
omega axis in the vicinity of
00:21:05.120 --> 00:21:06.160
that pole--
00:21:06.160 --> 00:21:08.010
would start to peak.
00:21:08.010 --> 00:21:11.320
Well, I think that all of
this is much better seen
00:21:11.320 --> 00:21:14.850
dynamically on the computer
display, so let's go to the
00:21:14.850 --> 00:21:17.630
computer display and what we'll
look at is a second
00:21:17.630 --> 00:21:19.180
order system--
00:21:19.180 --> 00:21:21.720
the frequency response
of it-- as we move
00:21:21.720 --> 00:21:25.340
along the j omega axis.
00:21:25.340 --> 00:21:31.220
So here we see the pole pair
in the complex plane and to
00:21:31.220 --> 00:21:33.890
generate the frequency response,
we want to look at
00:21:33.890 --> 00:21:37.660
the behavior of the pole vectors
as we move vertically
00:21:37.660 --> 00:21:39.660
along the j omega axis.
00:21:39.660 --> 00:21:45.620
So we'll show the pole vectors
and let's begin at omega
00:21:45.620 --> 00:21:46.800
equals zero.
00:21:46.800 --> 00:21:49.510
So here we have the pole vectors
from the poles to the
00:21:49.510 --> 00:21:52.130
point omega equal to zero.
00:21:52.130 --> 00:21:56.480
And, as we move vertically along
the j omega axis, we'll
00:21:56.480 --> 00:22:01.190
see how those pole vectors
change in length.
00:22:01.190 --> 00:22:04.310
The magnitude of the frequency
response is the reciprocal of
00:22:04.310 --> 00:22:07.440
the product of the lengths
of those vectors.
00:22:07.440 --> 00:22:12.090
Shown below is the frequency
response where we've begun
00:22:12.090 --> 00:22:14.800
just at omega equal to zero.
00:22:14.800 --> 00:22:22.800
And as we move vertically along
the j omega axis and the
00:22:22.800 --> 00:22:27.000
pole vector lengths change,
that will, then, influence
00:22:27.000 --> 00:22:28.980
what the frequency response
looks like.
00:22:28.980 --> 00:22:34.400
We've started here to move a
little bit away from omega
00:22:34.400 --> 00:22:39.640
equal to zero and notice that
in the upper half plane the
00:22:39.640 --> 00:22:41.840
pole vector has gotten
shorter.
00:22:41.840 --> 00:22:44.450
The pole vector for the pole
in the lower half plane has
00:22:44.450 --> 00:22:45.830
gotten longer.
00:22:45.830 --> 00:22:50.070
And now, as omega increases
further, that
00:22:50.070 --> 00:22:51.910
process will continue.
00:22:51.910 --> 00:22:55.690
And in particular, the pole
vector associated with the
00:22:55.690 --> 00:22:59.530
pole in the upper half
plane will be its
00:22:59.530 --> 00:23:02.200
shortest in the vicinity--
00:23:02.200 --> 00:23:04.810
at a frequency in the vicinity
of that pole--
00:23:04.810 --> 00:23:08.830
and so, for that frequency,
then, the frequency response
00:23:08.830 --> 00:23:13.450
will peak and we
see that here.
00:23:13.450 --> 00:23:17.640
From this point as the
frequency increases,
00:23:17.640 --> 00:23:20.655
corresponding to moving further
vertically along the j
00:23:20.655 --> 00:23:25.280
omega axis, both pole vectors
will increase in length.
00:23:25.280 --> 00:23:29.530
And that means, then, that the
magnitude of the frequency
00:23:29.530 --> 00:23:31.530
response will decrease.
00:23:31.530 --> 00:23:35.250
For this specific example, the
magnitude of the frequency
00:23:35.250 --> 00:23:39.900
response will asymptotically
go to zero.
00:23:39.900 --> 00:23:44.490
So what we see here is that the
frequency response has a
00:23:44.490 --> 00:23:49.510
resonance and as we see
geometrically from the way the
00:23:49.510 --> 00:23:53.940
vectors behaved, that resonance
in frequency is very
00:23:53.940 --> 00:23:58.630
clearly associated with the
position of the poles.
00:23:58.630 --> 00:24:03.240
And so, in fact, to illustrate
that further and dramatize it
00:24:03.240 --> 00:24:07.070
as long as we're focused on
it, let's now look at the
00:24:07.070 --> 00:24:12.310
frequency response for the
second order example as we
00:24:12.310 --> 00:24:14.440
change the pole positions.
00:24:14.440 --> 00:24:19.260
And first, what we'll do is let
the polls move vertically
00:24:19.260 --> 00:24:22.610
parallel to the j omega axis
and see how the frequency
00:24:22.610 --> 00:24:26.120
response changes, and then
we'll have the polls move
00:24:26.120 --> 00:24:29.260
horizontally parallel to the
real axis and see how the
00:24:29.260 --> 00:24:31.950
frequency response changes.
00:24:31.950 --> 00:24:35.560
To display the behavior of the
frequency response as the
00:24:35.560 --> 00:24:40.020
poles move, we've changed the
vertical scale on the
00:24:40.020 --> 00:24:42.890
frequency response somewhat.
00:24:42.890 --> 00:24:47.950
And now what we want to do
is move the poles, first,
00:24:47.950 --> 00:24:51.030
parallel to the j omega
axis, and then
00:24:51.030 --> 00:24:53.380
parallel to the real axis.
00:24:53.380 --> 00:24:57.210
Here we see the effect of moving
the poles parallel to
00:24:57.210 --> 00:24:59.030
the j omega axis.
00:24:59.030 --> 00:25:02.880
And what we observe is that,
in fact, the frequency
00:25:02.880 --> 00:25:07.390
location of the resonance
shifts, basically tracking the
00:25:07.390 --> 00:25:10.520
location of the pole.
00:25:10.520 --> 00:25:16.770
If we now move the pole back
down closer to the real axis,
00:25:16.770 --> 00:25:21.430
then this resonance will shift
back toward its original
00:25:21.430 --> 00:25:25.100
location and so let's
now see that.
00:25:39.060 --> 00:25:43.480
And here we are back at the
frequency that we started at.
00:25:43.480 --> 00:25:47.790
Now we'll move the poles even
closer to the real axis.
00:25:47.790 --> 00:25:52.760
The frequency location of the
resonance will continue to
00:25:52.760 --> 00:25:55.340
shift toward lower
frequencies.
00:25:55.340 --> 00:25:59.500
And also in the process,
incidentally, the height over
00:25:59.500 --> 00:26:03.350
the resonant peak will increase
because, of course,
00:26:03.350 --> 00:26:08.780
the lengths of the pole vectors
are getting shorter.
00:26:08.780 --> 00:26:12.350
And so, we see now the resonance
shifting down toward
00:26:12.350 --> 00:26:14.600
lower and lower frequency.
00:26:14.600 --> 00:26:20.960
And, finally, what we'll now do
is move the poles back to
00:26:20.960 --> 00:26:25.400
their original position and
the resonant peak will, of
00:26:25.400 --> 00:26:27.970
course, shift back up.
00:26:27.970 --> 00:26:33.000
And correspondingly the height
or amplitude of the resonance
00:26:33.000 --> 00:26:34.250
will decrease.
00:26:39.480 --> 00:26:43.000
And now we're back at the
frequency response that we had
00:26:43.000 --> 00:26:45.180
generated previously.
00:26:45.180 --> 00:26:49.160
Next we'd like to look at the
behavior as the polls move
00:26:49.160 --> 00:26:50.740
parallel to the real axis.
00:26:50.740 --> 00:26:54.530
First closer to the j omega axis
and then further away.
00:26:54.530 --> 00:26:58.710
As they move closer to the j
omega axis, the resonance
00:26:58.710 --> 00:27:03.810
sharpens because of the fact
that the pole vector gets
00:27:03.810 --> 00:27:06.060
shorter and responds--
00:27:06.060 --> 00:27:10.070
or changes in length more
quickly as we move past it
00:27:10.070 --> 00:27:13.470
moving along the j omega axis.
00:27:13.470 --> 00:27:18.360
So here we see the effect of
moving the poles closer to the
00:27:18.360 --> 00:27:20.020
j omega axis.
00:27:20.020 --> 00:27:24.000
The resonance has gotten
narrower in frequency and
00:27:24.000 --> 00:27:28.380
higher in amplitude, associated
with the fact that
00:27:28.380 --> 00:27:29.715
the pole vector gets shorter.
00:27:33.330 --> 00:27:38.320
Next as we move back to the
original location, the
00:27:38.320 --> 00:27:40.760
resonance will broaden
once again and the
00:27:40.760 --> 00:27:43.035
amplitude will decrease.
00:27:54.160 --> 00:27:57.810
And then, if we continue to move
the poles even further
00:27:57.810 --> 00:28:02.770
away from the real axis, the
resonance will broaden even
00:28:02.770 --> 00:28:05.580
further and the amplitude
of the peak
00:28:05.580 --> 00:28:07.305
will become even smaller.
00:28:13.830 --> 00:28:18.330
And finally, let's now look just
move the poles back to
00:28:18.330 --> 00:28:23.310
their original position and
we'll see the resonance narrow
00:28:23.310 --> 00:28:24.610
again and become higher.
00:28:34.540 --> 00:28:38.800
And so what we see then is
that for a second order
00:28:38.800 --> 00:28:42.530
system, the behavior of the
resonance basically is
00:28:42.530 --> 00:28:46.310
associated with the pole
locations, the frequency of
00:28:46.310 --> 00:28:48.760
the resonance associated with
the vertical position of the
00:28:48.760 --> 00:28:54.900
poles, and the sharpness of the
resonance associated with
00:28:54.900 --> 00:28:59.020
the real part of the poles-- in
other words, their position
00:28:59.020 --> 00:29:02.365
closer or further away from
the j omega axis.
00:29:05.240 --> 00:29:10.460
OK, so for complex poles, then,
for the second order
00:29:10.460 --> 00:29:15.690
system, what we see is that
we get a resonant kind of
00:29:15.690 --> 00:29:21.020
behavior, and, in particular,
then that resonate behavior
00:29:21.020 --> 00:29:26.010
tends to peak, or get peakier,
as the value
00:29:26.010 --> 00:29:28.510
of zeta gets smaller.
00:29:28.510 --> 00:29:32.960
And here, just to remind you of
what you saw, here is the
00:29:32.960 --> 00:29:38.080
frequency response with one
particular choice of values--
00:29:38.080 --> 00:29:40.320
well, this is normalized so that
omega sub n is one-- one
00:29:40.320 --> 00:29:43.910
particular choice for
zeta, namely 0.4.
00:29:43.910 --> 00:29:52.390
Here is what we have with zeta
smaller, and, finally, here is
00:29:52.390 --> 00:29:56.200
an example where zeta has gotten
even smaller than that.
00:29:56.200 --> 00:29:59.960
And what that corresponds to is
the poles moving closer to
00:29:59.960 --> 00:30:02.980
the j omega axis, the
corresponding frequency
00:30:02.980 --> 00:30:06.290
response getting peakier.
00:30:06.290 --> 00:30:12.620
Now in the time domain what
happens is that we have, of
00:30:12.620 --> 00:30:17.660
course, these complex roots,
which I indicated previously,
00:30:17.660 --> 00:30:21.180
where this represents the
imaginary part because zeta is
00:30:21.180 --> 00:30:23.230
less than one.
00:30:23.230 --> 00:30:25.900
And in the time domain, we
will have a form for the
00:30:25.900 --> 00:30:33.960
behavior, which is a e to the
c one t, plus a conjugate, e
00:30:33.960 --> 00:30:38.670
to the c one conjugate t.
00:30:38.670 --> 00:30:43.270
And so, in fact, as the
poles get closer
00:30:43.270 --> 00:30:44.740
to the j omega axis--
00:30:44.740 --> 00:30:47.450
corresponding to zeta
getting smaller--
00:30:47.450 --> 00:30:53.110
as the polls get closer to the j
omega axis, in the frequency
00:30:53.110 --> 00:30:55.840
domain the resonances
get sharper.
00:30:55.840 --> 00:30:59.650
In the time domain, the real
part of the poles has gotten
00:30:59.650 --> 00:31:04.320
smaller, and that means, in
fact, that in the time domain,
00:31:04.320 --> 00:31:08.330
the behavior will be more
oscillatory and less damped.
00:31:08.330 --> 00:31:12.850
And so just looking
at that again.
00:31:12.850 --> 00:31:17.280
Here is, in the time domain,
what happens.
00:31:17.280 --> 00:31:24.170
First of all, with the parameter
zeta equal to 0.4,
00:31:24.170 --> 00:31:29.130
and it oscillates and
exponentially dies out.
00:31:29.130 --> 00:31:35.890
Here is the second order system
where zeta is now 0.2
00:31:35.890 --> 00:31:37.730
instead of 0.4.
00:31:37.730 --> 00:31:44.250
And, finally, the second order
system where zeta is 0.1.
00:31:44.250 --> 00:31:49.930
And what we see as zeta gets
smaller and smaller is that
00:31:49.930 --> 00:31:53.810
the oscillations are basically
the same, but the exponential
00:31:53.810 --> 00:31:58.610
damping becomes less and less.
00:31:58.610 --> 00:32:02.150
Alright, now, this is a somewhat
more detailed look at
00:32:02.150 --> 00:32:03.950
second order systems.
00:32:03.950 --> 00:32:07.010
And second order systems--
and for that
00:32:07.010 --> 00:32:08.390
matter, first order systems--
00:32:08.390 --> 00:32:13.100
are systems that are important
in their own right, but they
00:32:13.100 --> 00:32:18.600
also are important as basic
building blocks for more
00:32:18.600 --> 00:32:21.750
general, in particular, for
higher order systems.
00:32:21.750 --> 00:32:25.850
And the way in which that's done
typically is by combining
00:32:25.850 --> 00:32:29.400
first and second order systems
together in such a way that
00:32:29.400 --> 00:32:31.750
they implement higher
order systems.
00:32:31.750 --> 00:32:35.330
And two very common connections
are connections
00:32:35.330 --> 00:32:39.470
which are cascade connections,
and connections which are
00:32:39.470 --> 00:32:42.170
parallel connections.
00:32:42.170 --> 00:32:47.980
In a cascade connection, we
would think of combining the
00:32:47.980 --> 00:32:52.060
individual systems together as
I indicate here in series.
00:32:52.060 --> 00:32:55.150
And, of course, from the
convolution property, the
00:32:55.150 --> 00:32:59.430
overall system function is the
product of the individual
00:32:59.430 --> 00:33:01.270
system functions.
00:33:01.270 --> 00:33:06.840
So, for example, if these were
all second order systems, and
00:33:06.840 --> 00:33:11.500
I combine n of them together in
cascade, the overall system
00:33:11.500 --> 00:33:15.010
would be a system that would
have to n poles-- in other
00:33:15.010 --> 00:33:18.190
words, it would be a
two n order system.
00:33:18.190 --> 00:33:21.140
That's one very common
kind of connection.
00:33:21.140 --> 00:33:24.010
Another very common kind of
connection for first and
00:33:24.010 --> 00:33:28.180
second order systems is a
parallel connection, where, in
00:33:28.180 --> 00:33:31.360
that case, we connect
the systems together
00:33:31.360 --> 00:33:33.990
as I indicate here.
00:33:33.990 --> 00:33:38.900
The overall system function is
just simply the sum of these,
00:33:38.900 --> 00:33:41.710
and that follows from the
linearity property.
00:33:41.710 --> 00:33:44.720
And so the overall system
function would be as I
00:33:44.720 --> 00:33:47.160
indicate algebraically here.
00:33:47.160 --> 00:33:51.130
And notice that if each of
these are second order
00:33:51.130 --> 00:33:56.020
systems, and I had capital N of
them in parallel, when you
00:33:56.020 --> 00:33:59.080
think of putting the overall
system function over one
00:33:59.080 --> 00:34:02.070
common denominator, that
common denominator, in
00:34:02.070 --> 00:34:08.050
general, is going to be of order
two N. So either the
00:34:08.050 --> 00:34:11.639
parallel connection or the
cascade connection could be
00:34:11.639 --> 00:34:16.370
used to implement higher
order systems.
00:34:16.370 --> 00:34:20.870
One very common context in which
second order systems are
00:34:20.870 --> 00:34:25.739
combined together, either in
parallel or in cascade, to
00:34:25.739 --> 00:34:30.219
form a more interesting
system is, in
00:34:30.219 --> 00:34:31.929
fact, in speech synthesis.
00:34:31.929 --> 00:34:35.639
And what I'd like to do is
demonstrate a speech
00:34:35.639 --> 00:34:40.780
synthesizer, which I have
here, which in fact is a
00:34:40.780 --> 00:34:46.460
parallel combination of four
second order systems, very
00:34:46.460 --> 00:34:49.960
much of the type that we've
just talked about.
00:34:49.960 --> 00:34:53.170
I'll return to the synthesizer
in a minute.
00:34:53.170 --> 00:34:56.949
Let me first just indicate
what the basic idea is.
00:34:56.949 --> 00:35:00.930
In speech synthesis, what we're
trying to represent or
00:35:00.930 --> 00:35:04.140
implement is something
that corresponds
00:35:04.140 --> 00:35:06.450
to the vocal tract.
00:35:06.450 --> 00:35:09.570
The vocal tract is characterized
by a set of
00:35:09.570 --> 00:35:10.760
resonances.
00:35:10.760 --> 00:35:12.700
And we can think of representing
each of those
00:35:12.700 --> 00:35:15.160
resonances by a second
order system.
00:35:15.160 --> 00:35:17.700
And then the higher order system
corresponding to the
00:35:17.700 --> 00:35:21.350
vocal tract is built by, in
this case, a parallel
00:35:21.350 --> 00:35:25.210
combination of those second
order systems.
00:35:25.210 --> 00:35:30.470
So for the synthesizer, what we
have connected together in
00:35:30.470 --> 00:35:35.710
parallel is four second
order systems.
00:35:35.710 --> 00:35:41.130
And a control on each one of
them that controls the center
00:35:41.130 --> 00:35:45.850
frequency or the resonant
frequency of each of the
00:35:45.850 --> 00:35:48.930
second order systems.
00:35:48.930 --> 00:35:52.880
The excitation is an excitation
that would
00:35:52.880 --> 00:35:56.270
represent the air flow through
the vocal cords.
00:35:56.270 --> 00:35:59.950
The vocal cords vibrate and
there are puffs of air through
00:35:59.950 --> 00:36:02.630
the vocal cords as they
open and close.
00:36:02.630 --> 00:36:08.430
And so the excitation for the
synthesizer corresponds to a
00:36:08.430 --> 00:36:12.640
pulse train representing
the air flow
00:36:12.640 --> 00:36:14.270
through the vocal cords.
00:36:14.270 --> 00:36:17.850
The fundamental frequency
of this representing the
00:36:17.850 --> 00:36:21.740
fundamental frequency of
the synthesized voice.
00:36:21.740 --> 00:36:25.890
So that's the basic structure
of the synthesizer
00:36:25.890 --> 00:36:30.380
And what we have in this
analog synthesizer are
00:36:30.380 --> 00:36:35.600
separate controls on the
individual center frequencies.
00:36:35.600 --> 00:36:38.250
There is a control representing
the center
00:36:38.250 --> 00:36:40.930
frequency of the third resonator
and the fourth
00:36:40.930 --> 00:36:42.660
resonator, and those
are represented
00:36:42.660 --> 00:36:44.570
by these two knobs.
00:36:44.570 --> 00:36:48.860
And then the first and second
resonators are controlled by
00:36:48.860 --> 00:36:50.650
moving this joystick.
00:36:50.650 --> 00:36:56.280
The first resonator by moving
the joystick along this axis
00:36:56.280 --> 00:36:58.220
and the second resonator
by moving the
00:36:58.220 --> 00:37:01.030
joystick along this axis.
00:37:01.030 --> 00:37:05.570
And then, in addition to
controls on the four
00:37:05.570 --> 00:37:09.630
resonators, we can control the
fundamental frequency of the
00:37:09.630 --> 00:37:13.360
excitation, and we do
that with this knob.
00:37:13.360 --> 00:37:17.070
So let's, first of all, just
listen to one of the
00:37:17.070 --> 00:37:20.490
resonators, and the resonator
that I'll play
00:37:20.490 --> 00:37:22.310
is the fourth resonator.
00:37:22.310 --> 00:37:26.840
And what you'll hear first is
the output as I vary the
00:37:26.840 --> 00:37:28.730
center frequency of
that resonator.
00:37:28.730 --> 00:37:30.320
[BUZZING]
00:37:30.320 --> 00:37:32.190
So I'm lowering the
center frequency.
00:37:35.310 --> 00:37:37.700
And then, bringing the center
frequency back up.
00:37:40.790 --> 00:37:44.200
And then, as I indicated,
I can also control the
00:37:44.200 --> 00:37:47.400
fundamental frequency
of the excitation by
00:37:47.400 --> 00:37:48.541
turning this knob.
00:37:48.541 --> 00:37:51.310
[BUZZING]
00:37:51.310 --> 00:37:52.720
Lowering the fundamental
frequency.
00:37:55.510 --> 00:37:58.486
And then, increasing the
fundamental frequency.
00:38:01.960 --> 00:38:07.010
Alright, now, if the four
resonators in parallel are an
00:38:07.010 --> 00:38:11.640
implementation of the vocal
cavity, then, presumably, what
00:38:11.640 --> 00:38:15.600
we can synthesize when we put
them all in are vowel sounds
00:38:15.600 --> 00:38:17.360
and let's do that.
00:38:17.360 --> 00:38:22.500
I'll now switch in the
other resonators.
00:38:22.500 --> 00:38:27.530
When we do that, then, depending
on what choice we
00:38:27.530 --> 00:38:30.920
have for the individual resonant
frequencies, we
00:38:30.920 --> 00:38:32.990
should be able to synthesize
vowel sounds.
00:38:32.990 --> 00:38:35.210
So here, for example,
is the vowel e.
00:38:35.210 --> 00:38:37.640
[BUZZING "E"].
00:38:37.640 --> 00:38:38.280
Here is
00:38:38.280 --> 00:38:39.025
[BUZZING "AH"]
00:38:39.025 --> 00:38:40.275
--ah.
00:38:41.875 --> 00:38:42.662
A.
00:38:42.662 --> 00:38:44.390
[BUZZING FLAT A]
00:38:44.390 --> 00:38:45.360
And, of course, we can--
00:38:45.360 --> 00:38:46.662
[BUZZING OO]
00:38:46.662 --> 00:38:47.526
--generate
00:38:47.526 --> 00:38:47.902
[BUZZING "I"]
00:38:47.902 --> 00:38:50.160
--lots of other vowel sounds.
00:38:50.160 --> 00:38:51.170
[BUZZING "AH"]
00:38:51.170 --> 00:38:53.910
--and change the fundamental
frequency at the same time.
00:38:53.910 --> 00:38:55.160
[CHANGES FREQUENCY
UP AND DOWN]
00:39:01.600 --> 00:39:05.380
Now, if we want to synthesize
speech it's not enough to just
00:39:05.380 --> 00:39:08.100
synthesize steady state vowels--
that gets boring
00:39:08.100 --> 00:39:08.920
after a while.
00:39:08.920 --> 00:39:13.360
Of course what happens with the
vocal cavity is that it
00:39:13.360 --> 00:39:19.330
moves as a function of time and
that's what generates the
00:39:19.330 --> 00:39:21.580
speech that we want
to generate.
00:39:21.580 --> 00:39:26.520
And so, presumably then, if
we change these resonant
00:39:26.520 --> 00:39:30.330
frequencies as a function of
time appropriately, then we
00:39:30.330 --> 00:39:32.500
should be able to synthesize
speech.
00:39:32.500 --> 00:39:36.650
And so by moving these
resonances around, we can
00:39:36.650 --> 00:39:38.800
generate synthesized speech.
00:39:38.800 --> 00:39:43.600
And let's try it with
some phrase.
00:39:43.600 --> 00:39:46.380
And I'll do that by simply
adjusting the center
00:39:46.380 --> 00:39:47.630
frequencies appropriately.
00:39:50.310 --> 00:39:57.030
[BUZZING "HOW ARE YOU"]
00:39:57.030 --> 00:39:59.740
Well, hopefully you
understood that.
00:39:59.740 --> 00:40:03.910
As you could imagine, I spent at
least a few minutes before
00:40:03.910 --> 00:40:06.820
the lecture trying to practice
that so that it would come out
00:40:06.820 --> 00:40:10.090
to be more or less
intelligible.
00:40:10.090 --> 00:40:14.370
Now the system as I've just
demonstrated it is, of course,
00:40:14.370 --> 00:40:19.490
a continuous time system or an
analog speech synthesizer.
00:40:19.490 --> 00:40:23.540
There are many versions of
digital or discrete time
00:40:23.540 --> 00:40:24.980
synthesizers.
00:40:24.980 --> 00:40:30.000
One of the first, in fact, being
a device that many of
00:40:30.000 --> 00:40:33.310
you are very likely familiar
with, which is the Texas
00:40:33.310 --> 00:40:36.680
Instruments Speak and Spell,
which I show here.
00:40:36.680 --> 00:40:41.290
And what's very interesting and
rather dramatic about this
00:40:41.290 --> 00:40:44.740
device is the fact that it
implements the speech
00:40:44.740 --> 00:40:49.500
synthesis in very much the same
way as I've demonstrated
00:40:49.500 --> 00:40:51.620
with the analog synthesizer.
00:40:51.620 --> 00:40:55.670
In this case, it's five second
order filters in a
00:40:55.670 --> 00:40:58.170
configuration that's slightly
different than a parallel
00:40:58.170 --> 00:41:02.130
configuration but conceptually
very closely related.
00:41:02.130 --> 00:41:05.510
And let's take a look
inside the box.
00:41:05.510 --> 00:41:09.520
And what we see there, with a
slide that was kindly supplied
00:41:09.520 --> 00:41:13.720
by Texas Instruments, is the
fact that there really are
00:41:13.720 --> 00:41:15.470
only four chips in there--
00:41:15.470 --> 00:41:17.600
a controller chip,
some storage.
00:41:17.600 --> 00:41:21.370
And the important point is the
chip that's labeled as the
00:41:21.370 --> 00:41:25.860
speech synthesis chip, in fact,
is what embodies or
00:41:25.860 --> 00:41:30.540
implements the five second
order filters and, in
00:41:30.540 --> 00:41:34.470
addition, incorporates some
other things-- some memory and
00:41:34.470 --> 00:41:36.310
also the [? DDA ?]
00:41:36.310 --> 00:41:37.270
converters.
00:41:37.270 --> 00:41:40.260
So, in fact, the implementation
of the
00:41:40.260 --> 00:41:45.040
synthesizer is pretty much
done on a single chip.
00:41:45.040 --> 00:41:48.940
Well that's a discrete
time system.
00:41:48.940 --> 00:41:53.600
We've been talking for the last
several lectures about
00:41:53.600 --> 00:41:57.230
continuous time systems and
the Laplace transform.
00:41:57.230 --> 00:41:59.950
Hopefully what you've seen in
this lecture and the previous
00:41:59.950 --> 00:42:07.110
lecture is the powerful tool
that the Laplace transform
00:42:07.110 --> 00:42:11.830
affords us in analyzing and
understanding system behavior.
00:42:14.440 --> 00:42:18.560
In the next lecture what I'd
like to do is parallel the
00:42:18.560 --> 00:42:21.530
discussion for discrete time,
turn our attention to the z
00:42:21.530 --> 00:42:26.330
transform, and, as you can
imagine simply by virtue of
00:42:26.330 --> 00:42:31.670
the fact that I have shown you
a digital and analog version
00:42:31.670 --> 00:42:35.040
of very much the same kind of
system, the discussions
00:42:35.040 --> 00:42:38.960
parallel themselves very
strongly and the z transform
00:42:38.960 --> 00:42:42.560
will play very much the same
role in discrete time that the
00:42:42.560 --> 00:42:45.090
Laplace transform does
in continuous time.
00:42:45.090 --> 00:42:46.340
Thank you.