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[MUSIC PLAYING]
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PROFESSOR: Last time, we
introduced the Laplace
11
00:00:58,660 --> 00:01:03,310
transform as a generalization of
the Fourier transform, and,
12
00:01:03,310 --> 00:01:07,530
just as a reminder, the Laplace
transform expression
13
00:01:07,530 --> 00:01:13,720
as we developed it is this
integral, very much similar to
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00:01:13,720 --> 00:01:16,870
the Fourier transform integral,
except with a more
15
00:01:16,870 --> 00:01:19,210
general complex variable.
16
00:01:19,210 --> 00:01:22,940
And, in fact, we developed and
talked about the relationship
17
00:01:22,940 --> 00:01:26,260
between the Laplace transform
and the Fourier transform.
18
00:01:26,260 --> 00:01:30,880
In particular, the Laplace
transform with the Laplace
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00:01:30,880 --> 00:01:36,590
transform variable s, purely
imaginary, in fact, reduces to
20
00:01:36,590 --> 00:01:38,970
the Fourier transform.
21
00:01:38,970 --> 00:01:43,320
Or, more generally, with the
Laplace transform variable as
22
00:01:43,320 --> 00:01:48,550
a complex number, the Laplace
transform is the Fourier
23
00:01:48,550 --> 00:01:53,430
transform of the corresponding
time function with an
24
00:01:53,430 --> 00:01:55,280
exponential weighting.
25
00:01:55,280 --> 00:01:59,730
And, also, as you should
recall, the exponential
26
00:01:59,730 --> 00:02:04,870
waiting introduced the notion
that the Laplace transform may
27
00:02:04,870 --> 00:02:08,820
converge for some values of
sigma and perhaps not for
28
00:02:08,820 --> 00:02:10,360
other values of sigma.
29
00:02:10,360 --> 00:02:15,890
So associated with the Laplace
transform was what we refer to
30
00:02:15,890 --> 00:02:17,765
as the region of convergence.
31
00:02:17,765 --> 00:02:20,400
32
00:02:20,400 --> 00:02:25,120
Now just as with the Fourier
transform, there are a number
33
00:02:25,120 --> 00:02:28,470
of properties of the Laplace
transform that are extremely
34
00:02:28,470 --> 00:02:35,250
useful in describing and
analyzing signals and systems.
35
00:02:35,250 --> 00:02:39,200
For example, one of the
properties that we, in fact,
36
00:02:39,200 --> 00:02:42,740
took advantage of in our
discussion last time was the
37
00:02:42,740 --> 00:02:46,250
linearly the linearity property,
which says, in
38
00:02:46,250 --> 00:02:49,980
essence, that the Laplace
transform of the linear
39
00:02:49,980 --> 00:02:54,830
combination of two time
functions is the same linear
40
00:02:54,830 --> 00:03:00,260
combination of the associated
Laplace transforms.
41
00:03:00,260 --> 00:03:03,640
Also, there is a very important
and useful property,
42
00:03:03,640 --> 00:03:05,980
which tells us how the
43
00:03:05,980 --> 00:03:09,650
derivative of a time function--
44
00:03:09,650 --> 00:03:12,090
rather, the Laplace transform
of the derivative--
45
00:03:12,090 --> 00:03:14,710
is related to the Laplace
transform.
46
00:03:14,710 --> 00:03:19,350
In particular, the Laplace
transform of the derivative is
47
00:03:19,350 --> 00:03:23,320
the Laplace transform x
of t multiplied by s.
48
00:03:23,320 --> 00:03:27,200
And, as you can see by just
setting s equal to j omega, in
49
00:03:27,200 --> 00:03:30,230
fact, this reduces to the
corresponding Fourier
50
00:03:30,230 --> 00:03:32,740
transform property.
51
00:03:32,740 --> 00:03:37,830
And a third property that we'll
make frequent use of is
52
00:03:37,830 --> 00:03:40,390
referred to as the convolution
property.
53
00:03:40,390 --> 00:03:43,910
Again, a generalization of the
convolution property for
54
00:03:43,910 --> 00:03:45,700
Fourier transforms.
55
00:03:45,700 --> 00:03:49,270
Here the convolution property
says that the Laplace
56
00:03:49,270 --> 00:03:53,390
transform of the convolution of
two time functions is the
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00:03:53,390 --> 00:03:58,620
product of the associated
Laplace transforms.
58
00:03:58,620 --> 00:04:04,610
Now it's important at some point
to think carefully about
59
00:04:04,610 --> 00:04:09,400
the region of convergence as we
discuss these properties.
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00:04:09,400 --> 00:04:13,430
And let me just draw your
attention to the fact that in
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00:04:13,430 --> 00:04:17,970
discussing properties fully and
in detail, one has to pay
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00:04:17,970 --> 00:04:25,090
attention not just to how the
algebraic expression changes,
63
00:04:25,090 --> 00:04:27,990
but also what the consequences
are for the region of
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00:04:27,990 --> 00:04:30,990
convergence, and that's
discussed in somewhat more
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00:04:30,990 --> 00:04:35,130
detail in the text and
I won't do that here.
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00:04:35,130 --> 00:04:40,750
Now the convolution property
leads to, of course, a very
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00:04:40,750 --> 00:04:46,300
important and useful mechanism
for dealing with linear time
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00:04:46,300 --> 00:04:50,330
invariant systems, very much as
the Fourier transform did.
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00:04:50,330 --> 00:04:54,990
In particular, the convolution
property tells us that if we
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00:04:54,990 --> 00:04:59,280
have a linear time invariant
system, the output in the time
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00:04:59,280 --> 00:05:02,520
domain is the convolution
of the input
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00:05:02,520 --> 00:05:04,390
and the impulse response.
73
00:05:04,390 --> 00:05:08,450
In the Laplace transform domain,
the Laplace transform
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00:05:08,450 --> 00:05:12,530
of the output is the Laplace
transform of the impulse
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00:05:12,530 --> 00:05:16,760
response times the Laplace
transform of the input.
76
00:05:16,760 --> 00:05:20,700
And again, this is a
generalization of the
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00:05:20,700 --> 00:05:24,220
corresponding property for
Fourier transforms.
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00:05:24,220 --> 00:05:28,020
In the case of the Fourier
transform, the Fourier
79
00:05:28,020 --> 00:05:30,720
transform [? of ?] the impulse
response we refer to as the
80
00:05:30,720 --> 00:05:32,430
frequency response.
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00:05:32,430 --> 00:05:36,290
In the more general case with
Laplace transforms, it's
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00:05:36,290 --> 00:05:38,950
typical to refer to the Laplace
transform of the
83
00:05:38,950 --> 00:05:43,530
impulse response as the
system function.
84
00:05:43,530 --> 00:05:52,500
Now in talking about the system
function, some issues
85
00:05:52,500 --> 00:05:54,120
of the region of convergence--
86
00:05:54,120 --> 00:05:55,920
and for that matter,
location of poles
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00:05:55,920 --> 00:05:57,940
of the system function--
88
00:05:57,940 --> 00:06:03,010
are closely tied in and related
to issues of whether
89
00:06:03,010 --> 00:06:05,940
the system is stable
and causal.
90
00:06:05,940 --> 00:06:08,570
And in fact, there's some useful
statements that can be
91
00:06:08,570 --> 00:06:12,610
made that play an important role
throughout the further
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00:06:12,610 --> 00:06:13,640
discussion.
93
00:06:13,640 --> 00:06:19,030
For example, we know from
previous discussions that
94
00:06:19,030 --> 00:06:21,380
there's a condition for
stability of a system, which
95
00:06:21,380 --> 00:06:25,390
is absolute integrability
of the impulse response.
96
00:06:25,390 --> 00:06:29,650
And that, in fact, is the same
condition for convergence of
97
00:06:29,650 --> 00:06:33,180
the Fourier transform of
the impulse response.
98
00:06:33,180 --> 00:06:39,350
What that says, really, is that
if a system is stable,
99
00:06:39,350 --> 00:06:43,530
then the region of convergence
of the system function must
100
00:06:43,530 --> 00:06:45,710
include the j omega axis.
101
00:06:45,710 --> 00:06:49,390
Which, of course, is where the
Laplace transform reduces to
102
00:06:49,390 --> 00:06:52,380
the Fourier transform.
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00:06:52,380 --> 00:06:56,540
So that relates the region of
convergence and stability.
104
00:06:56,540 --> 00:07:00,790
Also, you recall from last time
that we talked about the
105
00:07:00,790 --> 00:07:03,740
region of convergence associated
with right sided
106
00:07:03,740 --> 00:07:05,050
time functions.
107
00:07:05,050 --> 00:07:07,300
In particular for a right
sided time function, the
108
00:07:07,300 --> 00:07:10,030
region of convergence must
be to the right of
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00:07:10,030 --> 00:07:12,330
the rightmost pole.
110
00:07:12,330 --> 00:07:17,970
Well, if, in fact, we have a
system that's causal, then
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00:07:17,970 --> 00:07:22,040
that causality imposes the
condition that the impulse
112
00:07:22,040 --> 00:07:23,820
response be right sided.
113
00:07:23,820 --> 00:07:27,510
And so, in fact, for causality,
we would have a
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00:07:27,510 --> 00:07:29,960
region of convergence associated
with the system
115
00:07:29,960 --> 00:07:33,780
function, which is to the right
of the rightmost pole.
116
00:07:33,780 --> 00:07:37,710
Now interestingly and very
important is the consequence,
117
00:07:37,710 --> 00:07:40,380
if you put those two statements
together, in
118
00:07:40,380 --> 00:07:45,620
particular, you're led to the
conclusion that for stable
119
00:07:45,620 --> 00:07:51,770
causal systems, all the poles
must be in the left half of
120
00:07:51,770 --> 00:07:53,700
the s-plane.
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00:07:53,700 --> 00:07:54,870
What's the reason?
122
00:07:54,870 --> 00:07:59,010
The reason, of course, is that
if the system is stable and
123
00:07:59,010 --> 00:08:01,720
causal, the region of
convergence must be to the
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00:08:01,720 --> 00:08:03,520
right of the rightmost pole.
125
00:08:03,520 --> 00:08:06,460
It must include the
j omega axis.
126
00:08:06,460 --> 00:08:09,170
Obviously then, all the poles
must be in the left half of
127
00:08:09,170 --> 00:08:10,660
the s-plane.
128
00:08:10,660 --> 00:08:14,400
And again, that's an issue that
is discussed somewhat
129
00:08:14,400 --> 00:08:18,650
more carefully and in more
detail in the text.
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00:08:18,650 --> 00:08:23,840
Now, the properties that we're
talking about here are not the
131
00:08:23,840 --> 00:08:26,620
only properties, there
are many others.
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00:08:26,620 --> 00:08:30,200
But these properties, in
particular, provide the
133
00:08:30,200 --> 00:08:30,880
mechanism--
134
00:08:30,880 --> 00:08:33,409
as they did with Fourier
transforms--
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00:08:33,409 --> 00:08:37,280
for turning linear constant
coefficient differential
136
00:08:37,280 --> 00:08:41,840
equations into algebraic
equations and, corresponding,
137
00:08:41,840 --> 00:08:46,250
lead to a mechanism for dealing
with and solving
138
00:08:46,250 --> 00:08:48,890
linear constant coefficient
differential equations.
139
00:08:48,890 --> 00:08:53,280
And I'd like to illustrate that
by looking at both first
140
00:08:53,280 --> 00:08:56,010
order and second order
differential equations.
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00:08:56,010 --> 00:08:59,400
Let's begin, first of all,
with a first order
142
00:08:59,400 --> 00:09:01,940
differential equation.
143
00:09:01,940 --> 00:09:06,500
So what we're talking about
is a first order system.
144
00:09:06,500 --> 00:09:11,930
What I mean by that is a system
that's characterized by
145
00:09:11,930 --> 00:09:14,870
a first order differential
equation.
146
00:09:14,870 --> 00:09:19,460
And if we apply to this equation
the differentiation
147
00:09:19,460 --> 00:09:22,580
property, then the
derivative--
148
00:09:22,580 --> 00:09:25,910
the Laplace transform of the
derivative is s times the
149
00:09:25,910 --> 00:09:28,540
Laplace transform of
the time function.
150
00:09:28,540 --> 00:09:32,580
The linearity property allows us
to combine these together.
151
00:09:32,580 --> 00:09:36,520
And so, consequently, applying
the Laplace transform to this
152
00:09:36,520 --> 00:09:40,780
equation leads us to this
algebraic equation, and
153
00:09:40,780 --> 00:09:43,990
following that through, leads
us to the statement that the
154
00:09:43,990 --> 00:09:47,430
Laplace transform of the output
is one over s plus a
155
00:09:47,430 --> 00:09:50,620
times the Laplace transform
of the input.
156
00:09:50,620 --> 00:09:54,530
We know from the convolution
property that this Laplace
157
00:09:54,530 --> 00:09:58,020
transform is the system
function times x of s.
158
00:09:58,020 --> 00:10:02,530
And so, one over s plus a is
the system function or
159
00:10:02,530 --> 00:10:04,670
equivalently, the Laplace
transform
160
00:10:04,670 --> 00:10:07,420
of the impulse response.
161
00:10:07,420 --> 00:10:12,400
So, we can determine the impulse
response by taking the
162
00:10:12,400 --> 00:10:17,560
inverse Laplace transform
of h of s given by
163
00:10:17,560 --> 00:10:20,890
one over s plus a.
164
00:10:20,890 --> 00:10:24,600
Well, we can do that using the
inspection method, which is
165
00:10:24,600 --> 00:10:28,620
one way that we have of doing
inverse Laplace transforms.
166
00:10:28,620 --> 00:10:32,380
The question is then, what time
function has a Laplace
167
00:10:32,380 --> 00:10:35,640
transform which is one
over s plus a?
168
00:10:35,640 --> 00:10:38,280
The problem that we run
into is that there are
169
00:10:38,280 --> 00:10:40,910
two answers to that.
170
00:10:40,910 --> 00:10:46,020
one over s plus a is the
Laplace transform of an
171
00:10:46,020 --> 00:10:52,250
exponential for positive time,
but one over s plus a is also
172
00:10:52,250 --> 00:10:57,550
the Laplace transform of an
exponential for negative time.
173
00:10:57,550 --> 00:11:01,340
Which one of these do
we end up picking?
174
00:11:01,340 --> 00:11:05,200
Well, recall that the difference
between these was
175
00:11:05,200 --> 00:11:06,810
in their region of
convergence.
176
00:11:06,810 --> 00:11:11,080
And in fact, in this case, this
corresponded to a region
177
00:11:11,080 --> 00:11:14,280
of convergence, which was
the real part of s
178
00:11:14,280 --> 00:11:16,740
greater than minus a.
179
00:11:16,740 --> 00:11:19,850
In this case, this was the
corresponding Laplace
180
00:11:19,850 --> 00:11:24,100
transform, provided that
the real part of s is
181
00:11:24,100 --> 00:11:26,970
less than minus a.
182
00:11:26,970 --> 00:11:30,400
So we have to decide which
region of convergence that we
183
00:11:30,400 --> 00:11:33,980
pick and it's not the
differential equation that
184
00:11:33,980 --> 00:11:37,650
will tell us that, it's
something else that has to
185
00:11:37,650 --> 00:11:39,640
give us that information.
186
00:11:39,640 --> 00:11:40,520
What could it be?
187
00:11:40,520 --> 00:11:45,310
Well, what it might be is the
additional information that
188
00:11:45,310 --> 00:11:49,210
the system is either
stable or causal.
189
00:11:49,210 --> 00:11:52,560
So for example, if the system
was causal, we would know that
190
00:11:52,560 --> 00:11:55,950
the region of convergence is to
the right of the pole and
191
00:11:55,950 --> 00:11:59,320
that would correspond,
then, to this
192
00:11:59,320 --> 00:12:02,630
being the impulse response.
193
00:12:02,630 --> 00:12:06,270
Whereas, with a negative--
194
00:12:06,270 --> 00:12:08,320
I'm sorry with a positive--
195
00:12:08,320 --> 00:12:12,820
if we knew that the system,
let's say, was non-causal,
196
00:12:12,820 --> 00:12:17,280
then we would associate with
this region of convergence and
197
00:12:17,280 --> 00:12:20,600
we would know then that this
is the impulse response.
198
00:12:20,600 --> 00:12:26,060
So a very important point is
that what we see is that the
199
00:12:26,060 --> 00:12:30,850
linear constant coefficient
differential equation gives us
200
00:12:30,850 --> 00:12:36,770
the algebraic expression for the
system function, but does
201
00:12:36,770 --> 00:12:39,190
not tell us about the region
of convergence.
202
00:12:39,190 --> 00:12:41,690
We get the reach of convergence
from some
203
00:12:41,690 --> 00:12:43,540
auxiliary information.
204
00:12:43,540 --> 00:12:44,860
What is that information?
205
00:12:44,860 --> 00:12:49,090
Well, it might, for example, be
knowledge that the system
206
00:12:49,090 --> 00:12:52,930
is perhaps stable, which tells
us that the region of
207
00:12:52,930 --> 00:12:57,290
convergence includes the j omega
axis, or perhaps causal,
208
00:12:57,290 --> 00:13:00,920
which tells us that the region
of convergence is to the right
209
00:13:00,920 --> 00:13:02,340
of the rightmost pole.
210
00:13:02,340 --> 00:13:06,190
So it's the auxiliary
information that specifies for
211
00:13:06,190 --> 00:13:08,010
us the region of convergence.
212
00:13:08,010 --> 00:13:09,340
Very important point.
213
00:13:09,340 --> 00:13:11,980
The differential equation by
itself does not completely
214
00:13:11,980 --> 00:13:16,250
specify the system, it only
essentially tells us what the
215
00:13:16,250 --> 00:13:20,310
algebraic expression is for
the system function.
216
00:13:20,310 --> 00:13:21,950
Alright that's a first
order example.
217
00:13:21,950 --> 00:13:26,900
Let's now look at a second
order system and the
218
00:13:26,900 --> 00:13:30,080
differential equation that
I picked in this case.
219
00:13:30,080 --> 00:13:32,860
I've parameterized in a certain
way, which we'll see
220
00:13:32,860 --> 00:13:34,150
will be useful.
221
00:13:34,150 --> 00:13:37,160
In particular, it's a second
order differential equation
222
00:13:37,160 --> 00:13:40,800
and I chosen, just for
simplicity, to not include any
223
00:13:40,800 --> 00:13:44,030
derivatives on the right hand
side, although we could have.
224
00:13:44,030 --> 00:13:47,860
In fact, if we did, that would
insert zeros into the system
225
00:13:47,860 --> 00:13:50,600
function, as well as the
poles inserted by
226
00:13:50,600 --> 00:13:52,950
the left hand side.
227
00:13:52,950 --> 00:13:56,750
We can determine the system
function in exactly the same
228
00:13:56,750 --> 00:14:00,840
way, namely, apply the Laplace
transform to this equation.
229
00:14:00,840 --> 00:14:04,200
That would convert this
differential equation to an
230
00:14:04,200 --> 00:14:06,290
algebraic equation.
231
00:14:06,290 --> 00:14:11,590
And now when we solve this
algebraic equation for y of s,
232
00:14:11,590 --> 00:14:15,590
in terms of x of s, it will come
out in the form of y of
233
00:14:15,590 --> 00:14:19,890
s, equal to h of s,
times x of s.
234
00:14:19,890 --> 00:14:23,430
And h of s, in that case, we
would get simply by dividing
235
00:14:23,430 --> 00:14:27,240
out by this polynomial [? in ?]
s, and so the system
236
00:14:27,240 --> 00:14:32,120
function then is the expression
that I have here.
237
00:14:32,120 --> 00:14:37,030
So this is the form for a second
order system where
238
00:14:37,030 --> 00:14:38,240
there are two poles.
239
00:14:38,240 --> 00:14:42,980
Since this is a second order
polynomial, there are no zeros
240
00:14:42,980 --> 00:14:47,090
associated with the fact that
I had no derivatives of the
241
00:14:47,090 --> 00:14:51,120
input on the right hand
side of the equation.
242
00:14:51,120 --> 00:14:54,110
Well, let's look at
this example--
243
00:14:54,110 --> 00:14:55,620
namely the second
order system--
244
00:14:55,620 --> 00:14:57,190
in a little more detail.
245
00:14:57,190 --> 00:15:01,100
And what we'll want to look at
is the location of the poles
246
00:15:01,100 --> 00:15:02,880
and some issues such
as, for example,
247
00:15:02,880 --> 00:15:05,720
the frequency response.
248
00:15:05,720 --> 00:15:11,170
So here again I have the
algebraic expression for the
249
00:15:11,170 --> 00:15:13,020
system function.
250
00:15:13,020 --> 00:15:16,820
And as I indicated, this is a
second order polynomial, which
251
00:15:16,820 --> 00:15:21,830
means that we can factor
it into two roots.
252
00:15:21,830 --> 00:15:27,320
So c1 and c2 represent the poles
of the system function.
253
00:15:27,320 --> 00:15:31,970
And in particular, in relation
to the two parameters zeta and
254
00:15:31,970 --> 00:15:33,500
omega sub n--
255
00:15:33,500 --> 00:15:39,160
if we look at what these roots
are, then what we get are the
256
00:15:39,160 --> 00:15:42,670
two expressions that
I have below.
257
00:15:42,670 --> 00:15:50,270
And notice, incidentally, that
if zeta is less than one, then
258
00:15:50,270 --> 00:15:52,410
what's under the square
root is negative.
259
00:15:52,410 --> 00:15:54,830
And so this, in fact,
corresponds to
260
00:15:54,830 --> 00:15:56,600
an imaginary part--
261
00:15:56,600 --> 00:15:59,410
an imaginary term for
zeta less than one.
262
00:15:59,410 --> 00:16:04,680
And so the two roots, then,
have a real part which is
263
00:16:04,680 --> 00:16:09,390
given by minus zeta omega sub n,
and an imaginary part-- if
264
00:16:09,390 --> 00:16:14,020
I were to rewrite this and then
express it in terms of j
265
00:16:14,020 --> 00:16:16,270
or the square root
of minus one.
266
00:16:16,270 --> 00:16:21,200
Looking below, we'll have a real
part which is minus zeta
267
00:16:21,200 --> 00:16:22,560
omega sub n--
268
00:16:22,560 --> 00:16:25,140
an imaginary part which
is omega sub n
269
00:16:25,140 --> 00:16:27,290
times this square root.
270
00:16:27,290 --> 00:16:31,700
So that's for zeta less than one
and for zeta greater than
271
00:16:31,700 --> 00:16:35,400
one, the two roots, of
course, will be real.
272
00:16:35,400 --> 00:16:40,260
Alright, so let's examine this
for the case where zeta is
273
00:16:40,260 --> 00:16:41,720
less than one.
274
00:16:41,720 --> 00:16:46,690
And what that corresponds to,
then, are two poles in the
275
00:16:46,690 --> 00:16:49,310
complex plane.
276
00:16:49,310 --> 00:16:54,670
And they have a real part
and an imaginary part.
277
00:16:54,670 --> 00:16:58,570
And you can explore this in
somewhat more detail on your
278
00:16:58,570 --> 00:17:02,220
own, but, essentially what
happens is that as you keep
279
00:17:02,220 --> 00:17:07,980
the parameter omega sub n fixed
and vary zeta, these
280
00:17:07,980 --> 00:17:11,619
poles trace out a circle.
281
00:17:11,619 --> 00:17:16,910
And, for example, where zeta
equal to zero, the poles are
282
00:17:16,910 --> 00:17:21,810
on the j omega axis
at omega sub n.
283
00:17:21,810 --> 00:17:31,370
As zeta increases and gets
closer to one, the poles
284
00:17:31,370 --> 00:17:37,430
converge toward the real axis
and then, in particular, for
285
00:17:37,430 --> 00:17:42,060
zeta greater than one, what we
end up with are two poles on
286
00:17:42,060 --> 00:17:44,430
the real axis.
287
00:17:44,430 --> 00:17:48,310
Well, actually, the case that
we want to look at a little
288
00:17:48,310 --> 00:17:51,460
more carefully is when the
poles are complex.
289
00:17:51,460 --> 00:17:55,410
And what this becomes is a
second order system, which as
290
00:17:55,410 --> 00:17:58,580
we'll see as the discussion
goes on, has an impulse
291
00:17:58,580 --> 00:18:02,450
response which oscillates with
time and correspondingly a
292
00:18:02,450 --> 00:18:06,380
frequency response that
has a resonance.
293
00:18:06,380 --> 00:18:09,640
Well let's examine the frequency
response a little
294
00:18:09,640 --> 00:18:11,250
more carefully.
295
00:18:11,250 --> 00:18:14,640
And what I'm assuming in the
discussion is that, first of
296
00:18:14,640 --> 00:18:20,110
all, the poles are in the left
half plane corresponding to
297
00:18:20,110 --> 00:18:22,640
zeta omega sub n being
positive--
298
00:18:22,640 --> 00:18:25,490
and so this is-- minus
that is negative.
299
00:18:25,490 --> 00:18:29,710
And furthermore, I'm assuming
that the poles are complex.
300
00:18:29,710 --> 00:18:33,770
And in that case, the algebraic
expression for the
301
00:18:33,770 --> 00:18:38,110
system function is omega sub n
squared in the numerator and
302
00:18:38,110 --> 00:18:42,700
two poles in the denominator,
which are complex conjugates.
303
00:18:42,700 --> 00:18:48,360
Now, what we want to look at is
the frequency response of
304
00:18:48,360 --> 00:18:49,720
the system.
305
00:18:49,720 --> 00:18:50,620
And
306
00:18:50,620 --> 00:18:55,370
that corresponds to looking at
the Fourier transform of the
307
00:18:55,370 --> 00:18:59,450
impulse response, which is the
Laplace transform on the j
308
00:18:59,450 --> 00:19:00,930
omega axis.
309
00:19:00,930 --> 00:19:05,190
So we want to examine what h of
s is as we move along the j
310
00:19:05,190 --> 00:19:06,680
omega axis.
311
00:19:06,680 --> 00:19:11,810
And notice, that to do that, in
this algebraic expression,
312
00:19:11,810 --> 00:19:15,920
we want to set s equal to j
omega and then evaluate--
313
00:19:15,920 --> 00:19:17,830
for example, if we want to look
at the magnitude of the
314
00:19:17,830 --> 00:19:19,470
frequency response--
315
00:19:19,470 --> 00:19:24,740
evaluate the magnitude of
the complex number.
316
00:19:24,740 --> 00:19:27,100
Well, there's a very convenient
way of doing that
317
00:19:27,100 --> 00:19:33,420
geometrically by recognizing
that in the complex plane,
318
00:19:33,420 --> 00:19:36,000
this complex number minus
that complex number
319
00:19:36,000 --> 00:19:37,870
represents a vector.
320
00:19:37,870 --> 00:19:41,360
And essentially, to look at the
magnitude of this complex
321
00:19:41,360 --> 00:19:46,060
number corresponds to taking
omega sub n squared and
322
00:19:46,060 --> 00:19:51,720
dividing it by the product of
the lengths of these vectors.
323
00:19:51,720 --> 00:19:57,820
So let's look, for example, at
the vector s minus c1, where s
324
00:19:57,820 --> 00:20:00,980
is on the j omega axis.
325
00:20:00,980 --> 00:20:08,920
And doing that, here is the
vector c1, and here is the
326
00:20:08,920 --> 00:20:12,530
vector s-- which is j omega if
we're looking, let's say, at
327
00:20:12,530 --> 00:20:14,650
this value of frequency--
328
00:20:14,650 --> 00:20:18,120
and this vector, then, is
the vector which is
329
00:20:18,120 --> 00:20:21,100
j omega minus c1.
330
00:20:21,100 --> 00:20:24,630
So in fact, it's the length of
this vector that we want to
331
00:20:24,630 --> 00:20:28,240
observe as we change omega--
332
00:20:28,240 --> 00:20:31,540
namely as we move along
the j omega axis.
333
00:20:31,540 --> 00:20:35,470
We want to take this vector
and this vector, take the
334
00:20:35,470 --> 00:20:39,140
lengths of those vectors,
multiply them together, divide
335
00:20:39,140 --> 00:20:42,400
that into omega sub n squared,
and that will give us the
336
00:20:42,400 --> 00:20:43,590
frequency response.
337
00:20:43,590 --> 00:20:46,920
Now that's a little hard to see
how the frequency response
338
00:20:46,920 --> 00:20:48,750
will work out just looking
at one point.
339
00:20:48,750 --> 00:20:53,610
Although notice that as we move
along the j omega axis,
340
00:20:53,610 --> 00:20:57,230
as we get closer to this pole,
this vector, in fact, gets
341
00:20:57,230 --> 00:21:01,160
shorter, and so we might
expect , that
342
00:21:01,160 --> 00:21:02,490
the frequency response--
343
00:21:02,490 --> 00:21:05,120
as we're moving along the j
omega axis in the vicinity of
344
00:21:05,120 --> 00:21:06,160
that pole--
345
00:21:06,160 --> 00:21:08,010
would start to peak.
346
00:21:08,010 --> 00:21:11,320
Well, I think that all of
this is much better seen
347
00:21:11,320 --> 00:21:14,850
dynamically on the computer
display, so let's go to the
348
00:21:14,850 --> 00:21:17,630
computer display and what we'll
look at is a second
349
00:21:17,630 --> 00:21:19,180
order system--
350
00:21:19,180 --> 00:21:21,720
the frequency response
of it-- as we move
351
00:21:21,720 --> 00:21:25,340
along the j omega axis.
352
00:21:25,340 --> 00:21:31,220
So here we see the pole pair
in the complex plane and to
353
00:21:31,220 --> 00:21:33,890
generate the frequency response,
we want to look at
354
00:21:33,890 --> 00:21:37,660
the behavior of the pole vectors
as we move vertically
355
00:21:37,660 --> 00:21:39,660
along the j omega axis.
356
00:21:39,660 --> 00:21:45,620
So we'll show the pole vectors
and let's begin at omega
357
00:21:45,620 --> 00:21:46,800
equals zero.
358
00:21:46,800 --> 00:21:49,510
So here we have the pole vectors
from the poles to the
359
00:21:49,510 --> 00:21:52,130
point omega equal to zero.
360
00:21:52,130 --> 00:21:56,480
And, as we move vertically along
the j omega axis, we'll
361
00:21:56,480 --> 00:22:01,190
see how those pole vectors
change in length.
362
00:22:01,190 --> 00:22:04,310
The magnitude of the frequency
response is the reciprocal of
363
00:22:04,310 --> 00:22:07,440
the product of the lengths
of those vectors.
364
00:22:07,440 --> 00:22:12,090
Shown below is the frequency
response where we've begun
365
00:22:12,090 --> 00:22:14,800
just at omega equal to zero.
366
00:22:14,800 --> 00:22:22,800
And as we move vertically along
the j omega axis and the
367
00:22:22,800 --> 00:22:27,000
pole vector lengths change,
that will, then, influence
368
00:22:27,000 --> 00:22:28,980
what the frequency response
looks like.
369
00:22:28,980 --> 00:22:34,400
We've started here to move a
little bit away from omega
370
00:22:34,400 --> 00:22:39,640
equal to zero and notice that
in the upper half plane the
371
00:22:39,640 --> 00:22:41,840
pole vector has gotten
shorter.
372
00:22:41,840 --> 00:22:44,450
The pole vector for the pole
in the lower half plane has
373
00:22:44,450 --> 00:22:45,830
gotten longer.
374
00:22:45,830 --> 00:22:50,070
And now, as omega increases
further, that
375
00:22:50,070 --> 00:22:51,910
process will continue.
376
00:22:51,910 --> 00:22:55,690
And in particular, the pole
vector associated with the
377
00:22:55,690 --> 00:22:59,530
pole in the upper half
plane will be its
378
00:22:59,530 --> 00:23:02,200
shortest in the vicinity--
379
00:23:02,200 --> 00:23:04,810
at a frequency in the vicinity
of that pole--
380
00:23:04,810 --> 00:23:08,830
and so, for that frequency,
then, the frequency response
381
00:23:08,830 --> 00:23:13,450
will peak and we
see that here.
382
00:23:13,450 --> 00:23:17,640
From this point as the
frequency increases,
383
00:23:17,640 --> 00:23:20,655
corresponding to moving further
vertically along the j
384
00:23:20,655 --> 00:23:25,280
omega axis, both pole vectors
will increase in length.
385
00:23:25,280 --> 00:23:29,530
And that means, then, that the
magnitude of the frequency
386
00:23:29,530 --> 00:23:31,530
response will decrease.
387
00:23:31,530 --> 00:23:35,250
For this specific example, the
magnitude of the frequency
388
00:23:35,250 --> 00:23:39,900
response will asymptotically
go to zero.
389
00:23:39,900 --> 00:23:44,490
So what we see here is that the
frequency response has a
390
00:23:44,490 --> 00:23:49,510
resonance and as we see
geometrically from the way the
391
00:23:49,510 --> 00:23:53,940
vectors behaved, that resonance
in frequency is very
392
00:23:53,940 --> 00:23:58,630
clearly associated with the
position of the poles.
393
00:23:58,630 --> 00:24:03,240
And so, in fact, to illustrate
that further and dramatize it
394
00:24:03,240 --> 00:24:07,070
as long as we're focused on
it, let's now look at the
395
00:24:07,070 --> 00:24:12,310
frequency response for the
second order example as we
396
00:24:12,310 --> 00:24:14,440
change the pole positions.
397
00:24:14,440 --> 00:24:19,260
And first, what we'll do is let
the polls move vertically
398
00:24:19,260 --> 00:24:22,610
parallel to the j omega axis
and see how the frequency
399
00:24:22,610 --> 00:24:26,120
response changes, and then
we'll have the polls move
400
00:24:26,120 --> 00:24:29,260
horizontally parallel to the
real axis and see how the
401
00:24:29,260 --> 00:24:31,950
frequency response changes.
402
00:24:31,950 --> 00:24:35,560
To display the behavior of the
frequency response as the
403
00:24:35,560 --> 00:24:40,020
poles move, we've changed the
vertical scale on the
404
00:24:40,020 --> 00:24:42,890
frequency response somewhat.
405
00:24:42,890 --> 00:24:47,950
And now what we want to do
is move the poles, first,
406
00:24:47,950 --> 00:24:51,030
parallel to the j omega
axis, and then
407
00:24:51,030 --> 00:24:53,380
parallel to the real axis.
408
00:24:53,380 --> 00:24:57,210
Here we see the effect of moving
the poles parallel to
409
00:24:57,210 --> 00:24:59,030
the j omega axis.
410
00:24:59,030 --> 00:25:02,880
And what we observe is that,
in fact, the frequency
411
00:25:02,880 --> 00:25:07,390
location of the resonance
shifts, basically tracking the
412
00:25:07,390 --> 00:25:10,520
location of the pole.
413
00:25:10,520 --> 00:25:16,770
If we now move the pole back
down closer to the real axis,
414
00:25:16,770 --> 00:25:21,430
then this resonance will shift
back toward its original
415
00:25:21,430 --> 00:25:25,100
location and so let's
now see that.
416
00:25:25,100 --> 00:25:39,060
417
00:25:39,060 --> 00:25:43,480
And here we are back at the
frequency that we started at.
418
00:25:43,480 --> 00:25:47,790
Now we'll move the poles even
closer to the real axis.
419
00:25:47,790 --> 00:25:52,760
The frequency location of the
resonance will continue to
420
00:25:52,760 --> 00:25:55,340
shift toward lower
frequencies.
421
00:25:55,340 --> 00:25:59,500
And also in the process,
incidentally, the height over
422
00:25:59,500 --> 00:26:03,350
the resonant peak will increase
because, of course,
423
00:26:03,350 --> 00:26:08,780
the lengths of the pole vectors
are getting shorter.
424
00:26:08,780 --> 00:26:12,350
And so, we see now the resonance
shifting down toward
425
00:26:12,350 --> 00:26:14,600
lower and lower frequency.
426
00:26:14,600 --> 00:26:20,960
And, finally, what we'll now do
is move the poles back to
427
00:26:20,960 --> 00:26:25,400
their original position and
the resonant peak will, of
428
00:26:25,400 --> 00:26:27,970
course, shift back up.
429
00:26:27,970 --> 00:26:33,000
And correspondingly the height
or amplitude of the resonance
430
00:26:33,000 --> 00:26:34,250
will decrease.
431
00:26:34,250 --> 00:26:39,480
432
00:26:39,480 --> 00:26:43,000
And now we're back at the
frequency response that we had
433
00:26:43,000 --> 00:26:45,180
generated previously.
434
00:26:45,180 --> 00:26:49,160
Next we'd like to look at the
behavior as the polls move
435
00:26:49,160 --> 00:26:50,740
parallel to the real axis.
436
00:26:50,740 --> 00:26:54,530
First closer to the j omega axis
and then further away.
437
00:26:54,530 --> 00:26:58,710
As they move closer to the j
omega axis, the resonance
438
00:26:58,710 --> 00:27:03,810
sharpens because of the fact
that the pole vector gets
439
00:27:03,810 --> 00:27:06,060
shorter and responds--
440
00:27:06,060 --> 00:27:10,070
or changes in length more
quickly as we move past it
441
00:27:10,070 --> 00:27:13,470
moving along the j omega axis.
442
00:27:13,470 --> 00:27:18,360
So here we see the effect of
moving the poles closer to the
443
00:27:18,360 --> 00:27:20,020
j omega axis.
444
00:27:20,020 --> 00:27:24,000
The resonance has gotten
narrower in frequency and
445
00:27:24,000 --> 00:27:28,380
higher in amplitude, associated
with the fact that
446
00:27:28,380 --> 00:27:29,715
the pole vector gets shorter.
447
00:27:29,715 --> 00:27:33,330
448
00:27:33,330 --> 00:27:38,320
Next as we move back to the
original location, the
449
00:27:38,320 --> 00:27:40,760
resonance will broaden
once again and the
450
00:27:40,760 --> 00:27:43,035
amplitude will decrease.
451
00:27:43,035 --> 00:27:54,160
452
00:27:54,160 --> 00:27:57,810
And then, if we continue to move
the poles even further
453
00:27:57,810 --> 00:28:02,770
away from the real axis, the
resonance will broaden even
454
00:28:02,770 --> 00:28:05,580
further and the amplitude
of the peak
455
00:28:05,580 --> 00:28:07,305
will become even smaller.
456
00:28:07,305 --> 00:28:13,830
457
00:28:13,830 --> 00:28:18,330
And finally, let's now look just
move the poles back to
458
00:28:18,330 --> 00:28:23,310
their original position and
we'll see the resonance narrow
459
00:28:23,310 --> 00:28:24,610
again and become higher.
460
00:28:24,610 --> 00:28:34,540
461
00:28:34,540 --> 00:28:38,800
And so what we see then is
that for a second order
462
00:28:38,800 --> 00:28:42,530
system, the behavior of the
resonance basically is
463
00:28:42,530 --> 00:28:46,310
associated with the pole
locations, the frequency of
464
00:28:46,310 --> 00:28:48,760
the resonance associated with
the vertical position of the
465
00:28:48,760 --> 00:28:54,900
poles, and the sharpness of the
resonance associated with
466
00:28:54,900 --> 00:28:59,020
the real part of the poles-- in
other words, their position
467
00:28:59,020 --> 00:29:02,365
closer or further away from
the j omega axis.
468
00:29:02,365 --> 00:29:05,240
469
00:29:05,240 --> 00:29:10,460
OK, so for complex poles, then,
for the second order
470
00:29:10,460 --> 00:29:15,690
system, what we see is that
we get a resonant kind of
471
00:29:15,690 --> 00:29:21,020
behavior, and, in particular,
then that resonate behavior
472
00:29:21,020 --> 00:29:26,010
tends to peak, or get peakier,
as the value
473
00:29:26,010 --> 00:29:28,510
of zeta gets smaller.
474
00:29:28,510 --> 00:29:32,960
And here, just to remind you of
what you saw, here is the
475
00:29:32,960 --> 00:29:38,080
frequency response with one
particular choice of values--
476
00:29:38,080 --> 00:29:40,320
well, this is normalized so that
omega sub n is one-- one
477
00:29:40,320 --> 00:29:43,910
particular choice for
zeta, namely 0.4.
478
00:29:43,910 --> 00:29:52,390
Here is what we have with zeta
smaller, and, finally, here is
479
00:29:52,390 --> 00:29:56,200
an example where zeta has gotten
even smaller than that.
480
00:29:56,200 --> 00:29:59,960
And what that corresponds to is
the poles moving closer to
481
00:29:59,960 --> 00:30:02,980
the j omega axis, the
corresponding frequency
482
00:30:02,980 --> 00:30:06,290
response getting peakier.
483
00:30:06,290 --> 00:30:12,620
Now in the time domain what
happens is that we have, of
484
00:30:12,620 --> 00:30:17,660
course, these complex roots,
which I indicated previously,
485
00:30:17,660 --> 00:30:21,180
where this represents the
imaginary part because zeta is
486
00:30:21,180 --> 00:30:23,230
less than one.
487
00:30:23,230 --> 00:30:25,900
And in the time domain, we
will have a form for the
488
00:30:25,900 --> 00:30:33,960
behavior, which is a e to the
c one t, plus a conjugate, e
489
00:30:33,960 --> 00:30:38,670
to the c one conjugate t.
490
00:30:38,670 --> 00:30:43,270
And so, in fact, as the
poles get closer
491
00:30:43,270 --> 00:30:44,740
to the j omega axis--
492
00:30:44,740 --> 00:30:47,450
corresponding to zeta
getting smaller--
493
00:30:47,450 --> 00:30:53,110
as the polls get closer to the j
omega axis, in the frequency
494
00:30:53,110 --> 00:30:55,840
domain the resonances
get sharper.
495
00:30:55,840 --> 00:30:59,650
In the time domain, the real
part of the poles has gotten
496
00:30:59,650 --> 00:31:04,320
smaller, and that means, in
fact, that in the time domain,
497
00:31:04,320 --> 00:31:08,330
the behavior will be more
oscillatory and less damped.
498
00:31:08,330 --> 00:31:12,850
And so just looking
at that again.
499
00:31:12,850 --> 00:31:17,280
Here is, in the time domain,
what happens.
500
00:31:17,280 --> 00:31:24,170
First of all, with the parameter
zeta equal to 0.4,
501
00:31:24,170 --> 00:31:29,130
and it oscillates and
exponentially dies out.
502
00:31:29,130 --> 00:31:35,890
Here is the second order system
where zeta is now 0.2
503
00:31:35,890 --> 00:31:37,730
instead of 0.4.
504
00:31:37,730 --> 00:31:44,250
And, finally, the second order
system where zeta is 0.1.
505
00:31:44,250 --> 00:31:49,930
And what we see as zeta gets
smaller and smaller is that
506
00:31:49,930 --> 00:31:53,810
the oscillations are basically
the same, but the exponential
507
00:31:53,810 --> 00:31:58,610
damping becomes less and less.
508
00:31:58,610 --> 00:32:02,150
Alright, now, this is a somewhat
more detailed look at
509
00:32:02,150 --> 00:32:03,950
second order systems.
510
00:32:03,950 --> 00:32:07,010
And second order systems--
and for that
511
00:32:07,010 --> 00:32:08,390
matter, first order systems--
512
00:32:08,390 --> 00:32:13,100
are systems that are important
in their own right, but they
513
00:32:13,100 --> 00:32:18,600
also are important as basic
building blocks for more
514
00:32:18,600 --> 00:32:21,750
general, in particular, for
higher order systems.
515
00:32:21,750 --> 00:32:25,850
And the way in which that's done
typically is by combining
516
00:32:25,850 --> 00:32:29,400
first and second order systems
together in such a way that
517
00:32:29,400 --> 00:32:31,750
they implement higher
order systems.
518
00:32:31,750 --> 00:32:35,330
And two very common connections
are connections
519
00:32:35,330 --> 00:32:39,470
which are cascade connections,
and connections which are
520
00:32:39,470 --> 00:32:42,170
parallel connections.
521
00:32:42,170 --> 00:32:47,980
In a cascade connection, we
would think of combining the
522
00:32:47,980 --> 00:32:52,060
individual systems together as
I indicate here in series.
523
00:32:52,060 --> 00:32:55,150
And, of course, from the
convolution property, the
524
00:32:55,150 --> 00:32:59,430
overall system function is the
product of the individual
525
00:32:59,430 --> 00:33:01,270
system functions.
526
00:33:01,270 --> 00:33:06,840
So, for example, if these were
all second order systems, and
527
00:33:06,840 --> 00:33:11,500
I combine n of them together in
cascade, the overall system
528
00:33:11,500 --> 00:33:15,010
would be a system that would
have to n poles-- in other
529
00:33:15,010 --> 00:33:18,190
words, it would be a
two n order system.
530
00:33:18,190 --> 00:33:21,140
That's one very common
kind of connection.
531
00:33:21,140 --> 00:33:24,010
Another very common kind of
connection for first and
532
00:33:24,010 --> 00:33:28,180
second order systems is a
parallel connection, where, in
533
00:33:28,180 --> 00:33:31,360
that case, we connect
the systems together
534
00:33:31,360 --> 00:33:33,990
as I indicate here.
535
00:33:33,990 --> 00:33:38,900
The overall system function is
just simply the sum of these,
536
00:33:38,900 --> 00:33:41,710
and that follows from the
linearity property.
537
00:33:41,710 --> 00:33:44,720
And so the overall system
function would be as I
538
00:33:44,720 --> 00:33:47,160
indicate algebraically here.
539
00:33:47,160 --> 00:33:51,130
And notice that if each of
these are second order
540
00:33:51,130 --> 00:33:56,020
systems, and I had capital N of
them in parallel, when you
541
00:33:56,020 --> 00:33:59,080
think of putting the overall
system function over one
542
00:33:59,080 --> 00:34:02,070
common denominator, that
common denominator, in
543
00:34:02,070 --> 00:34:08,050
general, is going to be of order
two N. So either the
544
00:34:08,050 --> 00:34:11,639
parallel connection or the
cascade connection could be
545
00:34:11,639 --> 00:34:16,370
used to implement higher
order systems.
546
00:34:16,370 --> 00:34:20,870
One very common context in which
second order systems are
547
00:34:20,870 --> 00:34:25,739
combined together, either in
parallel or in cascade, to
548
00:34:25,739 --> 00:34:30,219
form a more interesting
system is, in
549
00:34:30,219 --> 00:34:31,929
fact, in speech synthesis.
550
00:34:31,929 --> 00:34:35,639
And what I'd like to do is
demonstrate a speech
551
00:34:35,639 --> 00:34:40,780
synthesizer, which I have
here, which in fact is a
552
00:34:40,780 --> 00:34:46,460
parallel combination of four
second order systems, very
553
00:34:46,460 --> 00:34:49,960
much of the type that we've
just talked about.
554
00:34:49,960 --> 00:34:53,170
I'll return to the synthesizer
in a minute.
555
00:34:53,170 --> 00:34:56,949
Let me first just indicate
what the basic idea is.
556
00:34:56,949 --> 00:35:00,930
In speech synthesis, what we're
trying to represent or
557
00:35:00,930 --> 00:35:04,140
implement is something
that corresponds
558
00:35:04,140 --> 00:35:06,450
to the vocal tract.
559
00:35:06,450 --> 00:35:09,570
The vocal tract is characterized
by a set of
560
00:35:09,570 --> 00:35:10,760
resonances.
561
00:35:10,760 --> 00:35:12,700
And we can think of representing
each of those
562
00:35:12,700 --> 00:35:15,160
resonances by a second
order system.
563
00:35:15,160 --> 00:35:17,700
And then the higher order system
corresponding to the
564
00:35:17,700 --> 00:35:21,350
vocal tract is built by, in
this case, a parallel
565
00:35:21,350 --> 00:35:25,210
combination of those second
order systems.
566
00:35:25,210 --> 00:35:30,470
So for the synthesizer, what we
have connected together in
567
00:35:30,470 --> 00:35:35,710
parallel is four second
order systems.
568
00:35:35,710 --> 00:35:41,130
And a control on each one of
them that controls the center
569
00:35:41,130 --> 00:35:45,850
frequency or the resonant
frequency of each of the
570
00:35:45,850 --> 00:35:48,930
second order systems.
571
00:35:48,930 --> 00:35:52,880
The excitation is an excitation
that would
572
00:35:52,880 --> 00:35:56,270
represent the air flow through
the vocal cords.
573
00:35:56,270 --> 00:35:59,950
The vocal cords vibrate and
there are puffs of air through
574
00:35:59,950 --> 00:36:02,630
the vocal cords as they
open and close.
575
00:36:02,630 --> 00:36:08,430
And so the excitation for the
synthesizer corresponds to a
576
00:36:08,430 --> 00:36:12,640
pulse train representing
the air flow
577
00:36:12,640 --> 00:36:14,270
through the vocal cords.
578
00:36:14,270 --> 00:36:17,850
The fundamental frequency
of this representing the
579
00:36:17,850 --> 00:36:21,740
fundamental frequency of
the synthesized voice.
580
00:36:21,740 --> 00:36:25,890
So that's the basic structure
of the synthesizer
581
00:36:25,890 --> 00:36:30,380
And what we have in this
analog synthesizer are
582
00:36:30,380 --> 00:36:35,600
separate controls on the
individual center frequencies.
583
00:36:35,600 --> 00:36:38,250
There is a control representing
the center
584
00:36:38,250 --> 00:36:40,930
frequency of the third resonator
and the fourth
585
00:36:40,930 --> 00:36:42,660
resonator, and those
are represented
586
00:36:42,660 --> 00:36:44,570
by these two knobs.
587
00:36:44,570 --> 00:36:48,860
And then the first and second
resonators are controlled by
588
00:36:48,860 --> 00:36:50,650
moving this joystick.
589
00:36:50,650 --> 00:36:56,280
The first resonator by moving
the joystick along this axis
590
00:36:56,280 --> 00:36:58,220
and the second resonator
by moving the
591
00:36:58,220 --> 00:37:01,030
joystick along this axis.
592
00:37:01,030 --> 00:37:05,570
And then, in addition to
controls on the four
593
00:37:05,570 --> 00:37:09,630
resonators, we can control the
fundamental frequency of the
594
00:37:09,630 --> 00:37:13,360
excitation, and we do
that with this knob.
595
00:37:13,360 --> 00:37:17,070
So let's, first of all, just
listen to one of the
596
00:37:17,070 --> 00:37:20,490
resonators, and the resonator
that I'll play
597
00:37:20,490 --> 00:37:22,310
is the fourth resonator.
598
00:37:22,310 --> 00:37:26,840
And what you'll hear first is
the output as I vary the
599
00:37:26,840 --> 00:37:28,730
center frequency of
that resonator.
600
00:37:28,730 --> 00:37:30,320
[BUZZING]
601
00:37:30,320 --> 00:37:32,190
So I'm lowering the
center frequency.
602
00:37:32,190 --> 00:37:35,310
603
00:37:35,310 --> 00:37:37,700
And then, bringing the center
frequency back up.
604
00:37:37,700 --> 00:37:40,790
605
00:37:40,790 --> 00:37:44,200
And then, as I indicated,
I can also control the
606
00:37:44,200 --> 00:37:47,400
fundamental frequency
of the excitation by
607
00:37:47,400 --> 00:37:48,541
turning this knob.
608
00:37:48,541 --> 00:37:51,310
[BUZZING]
609
00:37:51,310 --> 00:37:52,720
Lowering the fundamental
frequency.
610
00:37:52,720 --> 00:37:55,510
611
00:37:55,510 --> 00:37:58,486
And then, increasing the
fundamental frequency.
612
00:37:58,486 --> 00:38:01,960
613
00:38:01,960 --> 00:38:07,010
Alright, now, if the four
resonators in parallel are an
614
00:38:07,010 --> 00:38:11,640
implementation of the vocal
cavity, then, presumably, what
615
00:38:11,640 --> 00:38:15,600
we can synthesize when we put
them all in are vowel sounds
616
00:38:15,600 --> 00:38:17,360
and let's do that.
617
00:38:17,360 --> 00:38:22,500
I'll now switch in the
other resonators.
618
00:38:22,500 --> 00:38:27,530
When we do that, then, depending
on what choice we
619
00:38:27,530 --> 00:38:30,920
have for the individual resonant
frequencies, we
620
00:38:30,920 --> 00:38:32,990
should be able to synthesize
vowel sounds.
621
00:38:32,990 --> 00:38:35,210
So here, for example,
is the vowel e.
622
00:38:35,210 --> 00:38:37,640
[BUZZING "E"].
623
00:38:37,640 --> 00:38:38,280
Here is
624
00:38:38,280 --> 00:38:39,025
[BUZZING "AH"]
625
00:38:39,025 --> 00:38:40,275
--ah.
626
00:38:40,275 --> 00:38:41,875
627
00:38:41,875 --> 00:38:42,662
A.
628
00:38:42,662 --> 00:38:44,390
[BUZZING FLAT A]
629
00:38:44,390 --> 00:38:45,360
And, of course, we can--
630
00:38:45,360 --> 00:38:46,662
[BUZZING OO]
631
00:38:46,662 --> 00:38:47,526
--generate
632
00:38:47,526 --> 00:38:47,902
[BUZZING "I"]
633
00:38:47,902 --> 00:38:50,160
--lots of other vowel sounds.
634
00:38:50,160 --> 00:38:51,170
[BUZZING "AH"]
635
00:38:51,170 --> 00:38:53,910
--and change the fundamental
frequency at the same time.
636
00:38:53,910 --> 00:38:55,160
[CHANGES FREQUENCY
UP AND DOWN]
637
00:38:55,160 --> 00:39:01,600
638
00:39:01,600 --> 00:39:05,380
Now, if we want to synthesize
speech it's not enough to just
639
00:39:05,380 --> 00:39:08,100
synthesize steady state vowels--
that gets boring
640
00:39:08,100 --> 00:39:08,920
after a while.
641
00:39:08,920 --> 00:39:13,360
Of course what happens with the
vocal cavity is that it
642
00:39:13,360 --> 00:39:19,330
moves as a function of time and
that's what generates the
643
00:39:19,330 --> 00:39:21,580
speech that we want
to generate.
644
00:39:21,580 --> 00:39:26,520
And so, presumably then, if
we change these resonant
645
00:39:26,520 --> 00:39:30,330
frequencies as a function of
time appropriately, then we
646
00:39:30,330 --> 00:39:32,500
should be able to synthesize
speech.
647
00:39:32,500 --> 00:39:36,650
And so by moving these
resonances around, we can
648
00:39:36,650 --> 00:39:38,800
generate synthesized speech.
649
00:39:38,800 --> 00:39:43,600
And let's try it with
some phrase.
650
00:39:43,600 --> 00:39:46,380
And I'll do that by simply
adjusting the center
651
00:39:46,380 --> 00:39:47,630
frequencies appropriately.
652
00:39:47,630 --> 00:39:50,310
653
00:39:50,310 --> 00:39:57,030
[BUZZING "HOW ARE YOU"]
654
00:39:57,030 --> 00:39:59,740
Well, hopefully you
understood that.
655
00:39:59,740 --> 00:40:03,910
As you could imagine, I spent at
least a few minutes before
656
00:40:03,910 --> 00:40:06,820
the lecture trying to practice
that so that it would come out
657
00:40:06,820 --> 00:40:10,090
to be more or less
intelligible.
658
00:40:10,090 --> 00:40:14,370
Now the system as I've just
demonstrated it is, of course,
659
00:40:14,370 --> 00:40:19,490
a continuous time system or an
analog speech synthesizer.
660
00:40:19,490 --> 00:40:23,540
There are many versions of
digital or discrete time
661
00:40:23,540 --> 00:40:24,980
synthesizers.
662
00:40:24,980 --> 00:40:30,000
One of the first, in fact, being
a device that many of
663
00:40:30,000 --> 00:40:33,310
you are very likely familiar
with, which is the Texas
664
00:40:33,310 --> 00:40:36,680
Instruments Speak and Spell,
which I show here.
665
00:40:36,680 --> 00:40:41,290
And what's very interesting and
rather dramatic about this
666
00:40:41,290 --> 00:40:44,740
device is the fact that it
implements the speech
667
00:40:44,740 --> 00:40:49,500
synthesis in very much the same
way as I've demonstrated
668
00:40:49,500 --> 00:40:51,620
with the analog synthesizer.
669
00:40:51,620 --> 00:40:55,670
In this case, it's five second
order filters in a
670
00:40:55,670 --> 00:40:58,170
configuration that's slightly
different than a parallel
671
00:40:58,170 --> 00:41:02,130
configuration but conceptually
very closely related.
672
00:41:02,130 --> 00:41:05,510
And let's take a look
inside the box.
673
00:41:05,510 --> 00:41:09,520
And what we see there, with a
slide that was kindly supplied
674
00:41:09,520 --> 00:41:13,720
by Texas Instruments, is the
fact that there really are
675
00:41:13,720 --> 00:41:15,470
only four chips in there--
676
00:41:15,470 --> 00:41:17,600
a controller chip,
some storage.
677
00:41:17,600 --> 00:41:21,370
And the important point is the
chip that's labeled as the
678
00:41:21,370 --> 00:41:25,860
speech synthesis chip, in fact,
is what embodies or
679
00:41:25,860 --> 00:41:30,540
implements the five second
order filters and, in
680
00:41:30,540 --> 00:41:34,470
addition, incorporates some
other things-- some memory and
681
00:41:34,470 --> 00:41:36,310
also the [? DDA ?]
682
00:41:36,310 --> 00:41:37,270
converters.
683
00:41:37,270 --> 00:41:40,260
So, in fact, the implementation
of the
684
00:41:40,260 --> 00:41:45,040
synthesizer is pretty much
done on a single chip.
685
00:41:45,040 --> 00:41:48,940
Well that's a discrete
time system.
686
00:41:48,940 --> 00:41:53,600
We've been talking for the last
several lectures about
687
00:41:53,600 --> 00:41:57,230
continuous time systems and
the Laplace transform.
688
00:41:57,230 --> 00:41:59,950
Hopefully what you've seen in
this lecture and the previous
689
00:41:59,950 --> 00:42:07,110
lecture is the powerful tool
that the Laplace transform
690
00:42:07,110 --> 00:42:11,830
affords us in analyzing and
understanding system behavior.
691
00:42:11,830 --> 00:42:14,440
692
00:42:14,440 --> 00:42:18,560
In the next lecture what I'd
like to do is parallel the
693
00:42:18,560 --> 00:42:21,530
discussion for discrete time,
turn our attention to the z
694
00:42:21,530 --> 00:42:26,330
transform, and, as you can
imagine simply by virtue of
695
00:42:26,330 --> 00:42:31,670
the fact that I have shown you
a digital and analog version
696
00:42:31,670 --> 00:42:35,040
of very much the same kind of
system, the discussions
697
00:42:35,040 --> 00:42:38,960
parallel themselves very
strongly and the z transform
698
00:42:38,960 --> 00:42:42,560
will play very much the same
role in discrete time that the
699
00:42:42,560 --> 00:42:45,090
Laplace transform does
in continuous time.
700
00:42:45,090 --> 00:42:46,340
Thank you.
701
00:42:46,340 --> 00:42:52,651