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[MUSIC PLAYING]
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PROFESSOR: Over the last
series of lectures, in
10
00:00:58,350 --> 00:01:00,940
discussing filtering,
modulation, and sampling,
11
00:01:00,940 --> 00:01:03,720
we've seen how powerful
and useful the
12
00:01:03,720 --> 00:01:06,490
Fourier transform is.
13
00:01:06,490 --> 00:01:09,210
Beginning with this lecture,
and over the next several
14
00:01:09,210 --> 00:01:14,400
lectures, I'd like to develop
and exploit a generalization
15
00:01:14,400 --> 00:01:18,600
of the Fourier transform, which
will not only lead to
16
00:01:18,600 --> 00:01:22,240
some important new insights
about signals and systems, but
17
00:01:22,240 --> 00:01:27,110
also will remove some of the
restrictions that we've had
18
00:01:27,110 --> 00:01:30,300
with the Fourier transform.
19
00:01:30,300 --> 00:01:34,510
The generalization that we'll
be talking about in the
20
00:01:34,510 --> 00:01:38,330
continuous time case is referred
to as the Laplace
21
00:01:38,330 --> 00:01:42,060
transform, and in the discrete
time case, is referred to as
22
00:01:42,060 --> 00:01:43,680
the z transform.
23
00:01:43,680 --> 00:01:46,810
What I'd like to do in today's
lecture is begin on the
24
00:01:46,810 --> 00:01:49,110
continuous time case,
namely a discussion
25
00:01:49,110 --> 00:01:51,010
of the Laplace transform.
26
00:01:51,010 --> 00:01:55,130
Continue that into the next
lecture, and following that
27
00:01:55,130 --> 00:01:58,690
develop the z transform
for discrete time.
28
00:01:58,690 --> 00:02:02,150
And also, as we go
along, exploit
29
00:02:02,150 --> 00:02:05,080
the two notions together.
30
00:02:05,080 --> 00:02:09,630
Now, to introduce the notion of
the Laplace transform, let
31
00:02:09,630 --> 00:02:12,250
me remind you again
of what led us
32
00:02:12,250 --> 00:02:15,380
into the Fourier transform.
33
00:02:15,380 --> 00:02:19,910
We developed the Fourier
transform by considering the
34
00:02:19,910 --> 00:02:26,210
idea of representing signals
as linear combinations of
35
00:02:26,210 --> 00:02:27,660
basic signals.
36
00:02:27,660 --> 00:02:33,060
And in the Fourier transform,
in the continuous time case,
37
00:02:33,060 --> 00:02:36,190
the basic signals that we picked
in the representation
38
00:02:36,190 --> 00:02:38,800
were complex exponentials.
39
00:02:38,800 --> 00:02:43,650
And in what we had referred to
as the synthesis equation, the
40
00:02:43,650 --> 00:02:48,020
synthesis equation corresponded
to, in effect, a
41
00:02:48,020 --> 00:02:52,230
decomposition as a linear
combination, a decomposition
42
00:02:52,230 --> 00:02:55,350
of x of t as a linear
combination of complex
43
00:02:55,350 --> 00:02:56,980
exponentials.
44
00:02:56,980 --> 00:03:00,440
And of course, associated with
this was the corresponding
45
00:03:00,440 --> 00:03:05,010
analysis equation that, in
effect, gave us the amplitudes
46
00:03:05,010 --> 00:03:08,850
associated with the complex
exponentials.
47
00:03:08,850 --> 00:03:11,550
Now, why did we pick complex
exponentials?
48
00:03:11,550 --> 00:03:16,760
Well, recall that the reason was
that complex exponentials
49
00:03:16,760 --> 00:03:20,250
are eigenfunctions of linear
time-invariant systems, and
50
00:03:20,250 --> 00:03:21,900
that was very convenient.
51
00:03:21,900 --> 00:03:27,900
Specifically, if we have a
linear time-invariant system
52
00:03:27,900 --> 00:03:34,080
with an impulse response h of
t, what we had shown is that
53
00:03:34,080 --> 00:03:37,750
that class of systems has the
property that if we put in a
54
00:03:37,750 --> 00:03:43,070
complex exponential, we get out
a complex exponential at
55
00:03:43,070 --> 00:03:47,100
the same frequency and with
a change in amplitude.
56
00:03:47,100 --> 00:03:51,790
And this change in amplitude,
in fact, corresponded as we
57
00:03:51,790 --> 00:03:55,180
showed as the discussion went
along, to the Fourier
58
00:03:55,180 --> 00:04:00,380
transform of the system
impulse response.
59
00:04:00,380 --> 00:04:06,630
So the notion of decomposing
signals into complex
60
00:04:06,630 --> 00:04:09,710
exponentials was very intimately
connected, and the
61
00:04:09,710 --> 00:04:13,470
Fourier transform was very
intimately connected, with the
62
00:04:13,470 --> 00:04:16,810
eigenfunction property of
complex exponentials for
63
00:04:16,810 --> 00:04:20,070
linear time-invariant systems.
64
00:04:20,070 --> 00:04:24,370
Well, complex exponentials of
that type are not the only
65
00:04:24,370 --> 00:04:28,470
eigenfunctions for linear
time-invariant systems.
66
00:04:28,470 --> 00:04:33,210
In fact, what you've seen
previously is that if we took
67
00:04:33,210 --> 00:04:39,290
a more general exponential, e
to the st, where s is a more
68
00:04:39,290 --> 00:04:40,760
general complex number.
69
00:04:40,760 --> 00:04:45,670
Not just j omega, but in fact
sigma plus j omega.
70
00:04:45,670 --> 00:04:50,110
For any value of s, the complex
exponential is an
71
00:04:50,110 --> 00:04:51,370
eigenfunction.
72
00:04:51,370 --> 00:04:55,350
And we can justify that simply
by substitution into the
73
00:04:55,350 --> 00:04:57,040
convolution integral.
74
00:04:57,040 --> 00:05:01,060
In other words, the response to
this complex exponential is
75
00:05:01,060 --> 00:05:05,570
the convolution of the impulse
response with the excitation.
76
00:05:05,570 --> 00:05:12,070
And notice that we can break
this term into a product, e to
77
00:05:12,070 --> 00:05:16,030
the st e to the minus s tau.
78
00:05:16,030 --> 00:05:21,300
And the e to the st term can
come outside the integration.
79
00:05:21,300 --> 00:05:25,160
And consequently, just carrying
through that algebra,
80
00:05:25,160 --> 00:05:32,760
would reduce this integral to
an integral with an e to the
81
00:05:32,760 --> 00:05:35,040
st factor outside.
82
00:05:35,040 --> 00:05:38,130
So just simply carrying through
the algebra, what we
83
00:05:38,130 --> 00:05:42,360
would conclude is that a complex
exponential with any
84
00:05:42,360 --> 00:05:47,110
complex number s would generate,
as an output, a
85
00:05:47,110 --> 00:05:51,960
complex exponential of the
same form multiplied by
86
00:05:51,960 --> 00:05:53,460
whatever this integral is.
87
00:05:53,460 --> 00:05:56,520
And this integral, of course,
will depend on what
88
00:05:56,520 --> 00:05:58,110
the value of s is.
89
00:05:58,110 --> 00:06:00,030
But that's all that
it will depend on.
90
00:06:00,030 --> 00:06:06,340
Or said another way, what this
all can be denoted as is some
91
00:06:06,340 --> 00:06:11,560
function h of s that depends
on the value of s.
92
00:06:11,560 --> 00:06:16,370
So finally then, e to the st as
an excitation to a linear
93
00:06:16,370 --> 00:06:20,430
time-invariant system generates
a response, which is
94
00:06:20,430 --> 00:06:26,840
a complex constant depending
on s, multiplying the same
95
00:06:26,840 --> 00:06:30,340
function that excited
the system.
96
00:06:30,340 --> 00:06:35,840
So what we have then is the
eigenfunction property, more
97
00:06:35,840 --> 00:06:41,300
generally, in terms of a more
general complex exponential
98
00:06:41,300 --> 00:06:46,660
where the complex factor is
given by this integral.
99
00:06:46,660 --> 00:06:51,830
Well, in fact, what that
integral corresponds to is
100
00:06:51,830 --> 00:06:57,120
what we will define as the
Laplace transform of the
101
00:06:57,120 --> 00:06:58,860
impulse response.
102
00:06:58,860 --> 00:07:04,710
And in fact, we can apply this
transformation to a more
103
00:07:04,710 --> 00:07:08,240
general time function that may
or may not be the impulse
104
00:07:08,240 --> 00:07:12,370
response of a linear
time-invariant system.
105
00:07:12,370 --> 00:07:17,130
And so, in general, it is this
transformation on a time
106
00:07:17,130 --> 00:07:22,020
function which is the Laplace
transform of that time
107
00:07:22,020 --> 00:07:25,050
function, and it's
a function of s.
108
00:07:25,050 --> 00:07:31,120
So the definition of the Laplace
transform is that the
109
00:07:31,120 --> 00:07:37,120
Laplace transform of a time
function x of t is the result
110
00:07:37,120 --> 00:07:39,990
of this transformation
on x of t.
111
00:07:39,990 --> 00:07:45,280
It's denoted as x of s, and as a
shorthand notation as we had
112
00:07:45,280 --> 00:07:49,040
with the Fourier transform,
then we have in the time
113
00:07:49,040 --> 00:07:52,640
domain, the time function x
of t, and in the Laplace
114
00:07:52,640 --> 00:07:57,170
transform domain, the
function x of s.
115
00:07:57,170 --> 00:08:01,560
And these then represent
a transform pair.
116
00:08:01,560 --> 00:08:07,190
Now, let me remind you that
the development of that
117
00:08:07,190 --> 00:08:11,170
mapping is exactly the process
the we went through initially
118
00:08:11,170 --> 00:08:14,670
in developing a mapping
that ended up giving
119
00:08:14,670 --> 00:08:17,260
us the Fourier transform.
120
00:08:17,260 --> 00:08:20,710
Essentially, what we've done is
just broadened our horizon
121
00:08:20,710 --> 00:08:24,140
somewhat, or our notation
somewhat.
122
00:08:24,140 --> 00:08:27,750
And rather than pushing just a
complex exponential through
123
00:08:27,750 --> 00:08:31,830
the system, we've pushed a more
general time function e
124
00:08:31,830 --> 00:08:35,760
to the st, where s is a complex
number with both a
125
00:08:35,760 --> 00:08:39,080
real part and an
imaginary part.
126
00:08:39,080 --> 00:08:41,440
Well, the discussion that we've
gone through so far, of
127
00:08:41,440 --> 00:08:45,440
course, is very closely related
to what we went
128
00:08:45,440 --> 00:08:48,020
through for the Fourier
transform.
129
00:08:48,020 --> 00:08:49,860
The mapping that we've
ended up with is
130
00:08:49,860 --> 00:08:51,850
called the Laplace transform.
131
00:08:51,850 --> 00:08:54,490
And as you can well imagine
and perhaps, may have
132
00:08:54,490 --> 00:08:59,200
recognized already, there's a
very close connection between
133
00:08:59,200 --> 00:09:03,500
the Laplace transform and
the Fourier transform.
134
00:09:03,500 --> 00:09:06,440
Well, to see one of the
connections, what we can
135
00:09:06,440 --> 00:09:11,240
observe is that if we look
at the Fourier transform
136
00:09:11,240 --> 00:09:16,020
expression and if we look
at the Laplace transform
137
00:09:16,020 --> 00:09:21,720
expression, where s is now a
general complex number sigma
138
00:09:21,720 --> 00:09:25,720
plus j omega, these two
expressions, in fact, are
139
00:09:25,720 --> 00:09:31,730
identical if, in fact,
sigma is equal to 0.
140
00:09:31,730 --> 00:09:36,530
If sigma is equal to 0 so that
s is just j omega, then all
141
00:09:36,530 --> 00:09:40,550
that this transformation is,
is the same as that.
142
00:09:40,550 --> 00:09:44,490
Substitute in s equals j omega
and this is what we get.
143
00:09:47,530 --> 00:09:51,190
What this then tells us is that
if we have the Laplace
144
00:09:51,190 --> 00:09:57,390
transform, and if we look at
the Laplace transform at s
145
00:09:57,390 --> 00:10:05,350
equals j omega, then that, in
fact, corresponds to the
146
00:10:05,350 --> 00:10:09,420
Fourier transform of x of t.
147
00:10:09,420 --> 00:10:13,910
Now, there is a slight
notational issue that this
148
00:10:13,910 --> 00:10:15,650
raises, and it's very
149
00:10:15,650 --> 00:10:17,530
straightforward to clean it up.
150
00:10:17,530 --> 00:10:19,520
But it's something that it's--
151
00:10:19,520 --> 00:10:23,980
you have to just kind of focus
on for a second to understand
152
00:10:23,980 --> 00:10:25,470
what the issue is.
153
00:10:25,470 --> 00:10:30,510
Notice that on the left-hand
side of this equation, x of s
154
00:10:30,510 --> 00:10:33,550
representing the Laplace
transform.
155
00:10:33,550 --> 00:10:36,690
When we look at that with sigma
equal to 0 or s equal to
156
00:10:36,690 --> 00:10:41,100
j omega, our natural inclination
is to write that
157
00:10:41,100 --> 00:10:44,140
as x of j omega, of course.
158
00:10:44,140 --> 00:10:46,470
On the other hand, the
right-hand side of the
159
00:10:46,470 --> 00:10:49,970
equation, namely the Fourier
transform of x of t, we've
160
00:10:49,970 --> 00:10:55,500
typically written
as x of omega.
161
00:10:55,500 --> 00:10:58,210
Focusing on the fact that
it's a function of
162
00:10:58,210 --> 00:11:00,580
this variable omega.
163
00:11:00,580 --> 00:11:03,250
Well, there's a slight
awkwardness here because here
164
00:11:03,250 --> 00:11:05,740
we're talking about an argument
j omega, here we're
165
00:11:05,740 --> 00:11:07,920
talking about an
argument omega.
166
00:11:07,920 --> 00:11:11,370
And a very straightforward way
of dealing with that is to
167
00:11:11,370 --> 00:11:16,760
simply change our notation for
the Fourier transform,
168
00:11:16,760 --> 00:11:19,850
recognizing that the Fourier
transform, of course, is a
169
00:11:19,850 --> 00:11:22,740
function of omega, but
it's also, in fact, a
170
00:11:22,740 --> 00:11:24,470
function of j omega.
171
00:11:24,470 --> 00:11:28,060
And if we write it that
way, then the two
172
00:11:28,060 --> 00:11:29,440
notations come together.
173
00:11:29,440 --> 00:11:33,520
In other words, the Laplace
transform at s equals j omega
174
00:11:33,520 --> 00:11:36,260
just simply reduces both
mathematically and
175
00:11:36,260 --> 00:11:39,100
notationally to the
Fourier transform.
176
00:11:39,100 --> 00:11:43,400
So the notation that we'll now
be adopting for the Fourier
177
00:11:43,400 --> 00:11:48,110
transform is the notation
whereby we express the Fourier
178
00:11:48,110 --> 00:11:53,190
transform no longer simply as
x of omega, but choosing as
179
00:11:53,190 --> 00:11:55,760
the argument j omega.
180
00:11:55,760 --> 00:11:59,620
Simple notational change.
181
00:11:59,620 --> 00:12:02,360
Now, here we see one
relationship between the
182
00:12:02,360 --> 00:12:06,560
Fourier transform and the
Laplace transform.
183
00:12:06,560 --> 00:12:10,180
Namely that the Laplace
transform for s equals j omega
184
00:12:10,180 --> 00:12:12,770
reduces to the Fourier
transform.
185
00:12:12,770 --> 00:12:17,340
We also have another important
relationship.
186
00:12:17,340 --> 00:12:22,880
In particular, the fact that the
Laplace transform can be
187
00:12:22,880 --> 00:12:28,160
interpreted as the Fourier
transform of a modified
188
00:12:28,160 --> 00:12:29,710
version of x of t.
189
00:12:29,710 --> 00:12:32,820
Let me show you what I mean.
190
00:12:32,820 --> 00:12:35,000
Here, of course, we have
the relationship
191
00:12:35,000 --> 00:12:36,190
that we just developed.
192
00:12:36,190 --> 00:12:38,225
Namely that s equals j omega.
193
00:12:38,225 --> 00:12:42,540
The Laplace transform reduces
to the Fourier transform.
194
00:12:42,540 --> 00:12:45,810
But now let's look at the more
general Laplace transform
195
00:12:45,810 --> 00:12:47,440
expression.
196
00:12:47,440 --> 00:12:51,800
And if we substitute in s equals
sigma plus j omega,
197
00:12:51,800 --> 00:12:57,500
which is the general form for
this complex variable s, and
198
00:12:57,500 --> 00:13:04,820
we carry through some of the
algebra, breaking this into
199
00:13:04,820 --> 00:13:06,840
the product of two exponentials,
z to the minus
200
00:13:06,840 --> 00:13:10,360
sigma t times z to the
minus j omega t.
201
00:13:10,360 --> 00:13:15,530
We now have this expression
where, of course, in both of
202
00:13:15,530 --> 00:13:20,140
these there is a dt.
203
00:13:20,140 --> 00:13:26,380
And now when we look at this,
what we observe is that this,
204
00:13:26,380 --> 00:13:30,930
in fact, is the Fourier
transform of something.
205
00:13:30,930 --> 00:13:32,160
What's the something?
206
00:13:32,160 --> 00:13:35,830
It's not x of t anymore, it's
the Fourier transform of x of
207
00:13:35,830 --> 00:13:40,730
t multiplied by e to
the minus sigma t.
208
00:13:40,730 --> 00:13:44,790
So if we think of these two
terms together, this integral
209
00:13:44,790 --> 00:13:46,930
is just the Fourier transform.
210
00:13:46,930 --> 00:13:51,070
It's the Fourier transform of
x of t multiplied by an
211
00:13:51,070 --> 00:13:52,170
exponential.
212
00:13:52,170 --> 00:13:57,600
If sigma is greater than 0,
it's an exponential that
213
00:13:57,600 --> 00:13:59,720
decays with time.
214
00:13:59,720 --> 00:14:02,540
If sigma is less than 0,
it's an exponential
215
00:14:02,540 --> 00:14:05,380
that grows with time.
216
00:14:05,380 --> 00:14:09,720
So we have then this additional
relationship, which
217
00:14:09,720 --> 00:14:13,010
tells us that the Laplace
transform is the Fourier
218
00:14:13,010 --> 00:14:19,630
transform of an exponentially
weighted time function.
219
00:14:19,630 --> 00:14:24,290
Now, this exponential weighting
has some important
220
00:14:24,290 --> 00:14:26,000
significance.
221
00:14:26,000 --> 00:14:29,800
In particular, recall that
there were issues of
222
00:14:29,800 --> 00:14:31,840
convergence with the
Fourier transform.
223
00:14:31,840 --> 00:14:35,590
In particular, the
Fourier transform
224
00:14:35,590 --> 00:14:36,940
may or may not converge.
225
00:14:36,940 --> 00:14:40,700
And for convergence, in fact,
what's required is that the
226
00:14:40,700 --> 00:14:43,250
time function that we're
transforming be absolutely
227
00:14:43,250 --> 00:14:45,140
integrable.
228
00:14:45,140 --> 00:14:49,920
Now, we can have a time function
that isn't absolutely
229
00:14:49,920 --> 00:14:52,930
integrable because, let's
say, it grows
230
00:14:52,930 --> 00:14:56,760
exponentially as time increases.
231
00:14:56,760 --> 00:15:00,940
But when we multiply it by this
exponential factor that's
232
00:15:00,940 --> 00:15:05,050
embodied in the Laplace
transform, in fact that brings
233
00:15:05,050 --> 00:15:07,970
the function back down
for positive time.
234
00:15:07,970 --> 00:15:13,430
And we'll impose absolute
integrability on the product
235
00:15:13,430 --> 00:15:17,190
of x of t times e to
the minus sigma t.
236
00:15:17,190 --> 00:15:22,300
And so the conclusion, an
important point is that the
237
00:15:22,300 --> 00:15:25,610
Laplace transform, the Fourier
transform of this product may
238
00:15:25,610 --> 00:15:27,490
converge, even though
the Fourier
239
00:15:27,490 --> 00:15:29,530
transform of x of t doesn't.
240
00:15:29,530 --> 00:15:33,460
In other words, the Laplace
transform may converge even
241
00:15:33,460 --> 00:15:35,970
when the Fourier transform
doesn't converge.
242
00:15:35,970 --> 00:15:38,030
And we'll see that and we'll
see examples of it as the
243
00:15:38,030 --> 00:15:40,830
discussion goes along.
244
00:15:40,830 --> 00:15:43,500
Now let me also draw your
attention to the fact,
245
00:15:43,500 --> 00:15:48,720
although we won't be working
through this in detail.
246
00:15:48,720 --> 00:15:53,520
To the fact that this equation,
in effect, provides
247
00:15:53,520 --> 00:16:00,990
the basis for us to figure out
how to express x of t in terms
248
00:16:00,990 --> 00:16:03,100
of the Laplace transform.
249
00:16:03,100 --> 00:16:06,350
In effect, we can apply the
inverse Fourier transform to
250
00:16:06,350 --> 00:16:10,870
this, thereby to this, account
for the exponential factor by
251
00:16:10,870 --> 00:16:12,980
bringing it over to
the other side.
252
00:16:12,980 --> 00:16:17,260
And if you go through this, and
in fact, you'll have an
253
00:16:17,260 --> 00:16:19,860
opportunity to go through this
both in the video course
254
00:16:19,860 --> 00:16:26,180
manual and also it's carried
through in the text, what you
255
00:16:26,180 --> 00:16:30,600
end up with is a synthesis
equation, an expression for x
256
00:16:30,600 --> 00:16:34,070
of t in terms of x of
s which corresponds
257
00:16:34,070 --> 00:16:35,730
to a synthesis equation.
258
00:16:35,730 --> 00:16:40,070
And which now builds x of t out
of a linear combination of
259
00:16:40,070 --> 00:16:44,570
not necessarily functions of the
form e to the j omega t,
260
00:16:44,570 --> 00:16:50,930
but in terms of functions or
basic signals which are more
261
00:16:50,930 --> 00:16:56,160
general exponentials
e to the st.
262
00:16:56,160 --> 00:17:01,720
OK, well, let's just look at
some examples of the Laplace
263
00:17:01,720 --> 00:17:03,320
transform of some
time functions.
264
00:17:03,320 --> 00:17:07,980
And these examples that I'll
go through are all examples
265
00:17:07,980 --> 00:17:10,410
that are worked out
in the text.
266
00:17:10,410 --> 00:17:12,790
And so I don't want to
focus on the algebra.
267
00:17:12,790 --> 00:17:15,990
What I'd like to focus on are
some of the issues and the
268
00:17:15,990 --> 00:17:18,450
interpretation.
269
00:17:18,450 --> 00:17:24,010
Let's first of all, look at the
example in the text, which
270
00:17:24,010 --> 00:17:26,329
is Example 9.1.
271
00:17:26,329 --> 00:17:31,720
If we take the Fourier transform
of this exponential,
272
00:17:31,720 --> 00:17:35,745
then, as you well know, the
result we have is 1 over j
273
00:17:35,745 --> 00:17:37,500
omega plus a.
274
00:17:37,500 --> 00:17:41,010
And that can't converge
for any a.
275
00:17:41,010 --> 00:17:44,680
In particular, it's only
for a greater than 0.
276
00:17:44,680 --> 00:17:48,090
What that really means is that
for convergence of the Fourier
277
00:17:48,090 --> 00:17:52,840
transform, this has to be
a decaying exponential.
278
00:17:52,840 --> 00:17:56,620
It can't be an increasing
exponential.
279
00:17:56,620 --> 00:18:01,620
If instead we apply the Laplace
transform to this,
280
00:18:01,620 --> 00:18:06,030
applying the Laplace transform
is the same as taking the
281
00:18:06,030 --> 00:18:11,430
Fourier transform of x of t
times an exponential, and the
282
00:18:11,430 --> 00:18:14,160
exponent that we would
multiply by is e to
283
00:18:14,160 --> 00:18:15,710
the minus sigma t.
284
00:18:15,710 --> 00:18:21,100
So in effect, taking the Laplace
transform of this is
285
00:18:21,100 --> 00:18:26,032
like taking the Fourier
transform of e to the minus at
286
00:18:26,032 --> 00:18:29,860
e to the minus sigma t.
287
00:18:29,860 --> 00:18:34,490
And if we carry that through,
just working through the
288
00:18:34,490 --> 00:18:38,750
integral, we end up with a
Laplace transform, which is 1
289
00:18:38,750 --> 00:18:42,200
over s plus a.
290
00:18:42,200 --> 00:18:47,710
But just as in the Fourier
transform, the Fourier
291
00:18:47,710 --> 00:18:50,980
transform won't converge
for any a.
292
00:18:50,980 --> 00:18:56,110
Now what happens is that the
Laplace transform will only
293
00:18:56,110 --> 00:19:00,530
converge when the Fourier
transform of this converges.
294
00:19:00,530 --> 00:19:04,430
Said another way, it's when
the combination of a plus
295
00:19:04,430 --> 00:19:07,490
sigma is greater than 0.
296
00:19:07,490 --> 00:19:11,860
So we would require that, if I
write it over here, a plus
297
00:19:11,860 --> 00:19:14,000
sigma is greater than 0.
298
00:19:14,000 --> 00:19:19,510
Or that sigma is greater
than minus a.
299
00:19:19,510 --> 00:19:24,460
So in fact, in the Laplace
transform of this, we have an
300
00:19:24,460 --> 00:19:26,880
expression 1 over s plus a.
301
00:19:26,880 --> 00:19:32,260
But we also require, in
interpreting that, that the
302
00:19:32,260 --> 00:19:37,430
real part of s be greater
than minus a.
303
00:19:37,430 --> 00:19:42,120
So that, essentially, the
Fourier transform of x of t
304
00:19:42,120 --> 00:19:45,300
times e to the minus
sigma t converges.
305
00:19:45,300 --> 00:19:49,960
So it's important to recognize
that the algebraic expression
306
00:19:49,960 --> 00:19:55,040
that we get is only valid for
certain values of the
307
00:19:55,040 --> 00:19:56,580
real part of s.
308
00:19:56,580 --> 00:20:01,180
And so, for this example, we
can summarize it as this
309
00:20:01,180 --> 00:20:05,110
exponential has a Laplace
transform, which is 1 over s
310
00:20:05,110 --> 00:20:09,560
plus a, where s is restricted to
the range the real part of
311
00:20:09,560 --> 00:20:11,430
s greater than minus a.
312
00:20:14,020 --> 00:20:18,900
Now, we haven't had this issue
before of restrictions on what
313
00:20:18,900 --> 00:20:20,210
the value of s is.
314
00:20:20,210 --> 00:20:22,600
With the Fourier transform,
either it converged or it
315
00:20:22,600 --> 00:20:23,880
didn't converge.
316
00:20:23,880 --> 00:20:27,760
With the Laplace transform,
there are certain values of s.
317
00:20:27,760 --> 00:20:30,620
We now have more flexibility,
and so there's certain values
318
00:20:30,620 --> 00:20:33,690
of the real part of s for which
it converges and certain
319
00:20:33,690 --> 00:20:36,360
values for which it doesn't.
320
00:20:36,360 --> 00:20:39,110
The values of s for
which the Laplace
321
00:20:39,110 --> 00:20:42,880
transform converges is--
322
00:20:42,880 --> 00:20:47,570
the values are referred to as
the region of convergence of
323
00:20:47,570 --> 00:20:50,580
the Laplace transform.
324
00:20:50,580 --> 00:20:56,130
And it's important to recognize
that in specifying
325
00:20:56,130 --> 00:20:59,370
the Laplace transform, what's
required is not only the
326
00:20:59,370 --> 00:21:05,370
algebraic expression, but also
the domain or set of values of
327
00:21:05,370 --> 00:21:12,210
s for which that algebraic
expression is valid.
328
00:21:12,210 --> 00:21:16,670
Just to underscore that point,
let me draw your attention to
329
00:21:16,670 --> 00:21:21,890
another example in the text,
which is Example 9.2.
330
00:21:21,890 --> 00:21:27,445
In Example 9.2, we have an
exponential for negative time,
331
00:21:27,445 --> 00:21:30,040
0 for positive time.
332
00:21:30,040 --> 00:21:33,380
And if you carry through the
algebra there, you end up with
333
00:21:33,380 --> 00:21:35,950
a Laplace transform expression,
which is again 1
334
00:21:35,950 --> 00:21:38,660
over s plus a.
335
00:21:38,660 --> 00:21:42,280
Exactly the same algebraic
expression as we had for the
336
00:21:42,280 --> 00:21:44,490
previous example.
337
00:21:44,490 --> 00:21:49,660
The important distinction is
that now the real part of s is
338
00:21:49,660 --> 00:21:52,170
restricted to be less
than minus a.
339
00:21:52,170 --> 00:21:56,750
And so, in fact, if you compare
this example with the
340
00:21:56,750 --> 00:22:00,180
one above it, and let's just
look back at the answer that
341
00:22:00,180 --> 00:22:03,350
we had there.
342
00:22:03,350 --> 00:22:07,240
If you compare those two
examples, here the algebraic
343
00:22:07,240 --> 00:22:11,820
expression is 1 over s plus
a with a certain region of
344
00:22:11,820 --> 00:22:13,130
convergence.
345
00:22:13,130 --> 00:22:18,260
Here the algebraic expression
is 1 over s plus a.
346
00:22:18,260 --> 00:22:21,260
And the only difference between
those two is the
347
00:22:21,260 --> 00:22:23,180
domain or region
of convergence.
348
00:22:23,180 --> 00:22:27,840
So there is another
complication, or twist, now.
349
00:22:27,840 --> 00:22:32,040
Not only do we need to generate
the algebraic
350
00:22:32,040 --> 00:22:37,200
expression, but we also have to
be careful to specify the
351
00:22:37,200 --> 00:22:41,500
region of convergence over
which that algebraic
352
00:22:41,500 --> 00:22:44,280
expression is valid.
353
00:22:44,280 --> 00:22:49,450
Now, later on in this lecture,
and actually also as the
354
00:22:49,450 --> 00:22:52,830
discussion of the Laplace
transform goes on, we'll begin
355
00:22:52,830 --> 00:22:56,350
to see and understand more
about how the region of
356
00:22:56,350 --> 00:22:59,360
convergence relates to various
357
00:22:59,360 --> 00:23:03,420
properties of the time function.
358
00:23:03,420 --> 00:23:07,070
Well, let's finally look at one
additional example from
359
00:23:07,070 --> 00:23:10,550
the text, And this
is Example 9.3.
360
00:23:10,550 --> 00:23:15,420
And what it consists of is the
time function, which is the
361
00:23:15,420 --> 00:23:18,620
sum of two exponentials.
362
00:23:18,620 --> 00:23:22,190
And although we haven't
formally talked about
363
00:23:22,190 --> 00:23:24,920
properties of the Laplace
transform yet, one of the
364
00:23:24,920 --> 00:23:26,210
properties that we'll
see-- and it's
365
00:23:26,210 --> 00:23:28,050
relatively easy to develop--
366
00:23:28,050 --> 00:23:32,260
is the fact that the Laplace
transform of a sum is the sum
367
00:23:32,260 --> 00:23:33,430
of the Laplace transform.
368
00:23:33,430 --> 00:23:38,330
So, in fact, we can get the
Laplace transform of the sum
369
00:23:38,330 --> 00:23:44,910
of these two terms as the sum
of the Laplace transforms.
370
00:23:44,910 --> 00:23:49,390
So for this one, we know from
the example that we looked at
371
00:23:49,390 --> 00:23:53,810
previously, Example 9.1, that
this is of the form 1 over s
372
00:23:53,810 --> 00:23:57,940
plus 1 with a region of
convergence, which is the real
373
00:23:57,940 --> 00:24:01,180
part of s greater
than minus 1.
374
00:24:01,180 --> 00:24:04,030
For this one, we have a Laplace
transform which is 1
375
00:24:04,030 --> 00:24:08,810
over s plus 2 with a region of
convergence which is the real
376
00:24:08,810 --> 00:24:12,270
part of s greater
than minus 2.
377
00:24:12,270 --> 00:24:16,010
So for the two of them together,
we have to take the
378
00:24:16,010 --> 00:24:18,140
overlap of those two regions.
379
00:24:18,140 --> 00:24:21,540
In other words, we have to
take the region that
380
00:24:21,540 --> 00:24:24,810
encompasses both the real part
of s greater than minus 1 and
381
00:24:24,810 --> 00:24:27,230
the real part of s greater
than minus 2.
382
00:24:27,230 --> 00:24:31,870
And if we put those together,
then we have a combined region
383
00:24:31,870 --> 00:24:34,140
of convergence, which is
the real part of s
384
00:24:34,140 --> 00:24:36,610
greater than minus 1.
385
00:24:36,610 --> 00:24:39,140
So this is the expression.
386
00:24:39,140 --> 00:24:44,270
And for this particular example,
what we have is a
387
00:24:44,270 --> 00:24:46,450
ratio of polynomials.
388
00:24:46,450 --> 00:24:49,830
The ratio of polynomials,
there's a numerator polynomial
389
00:24:49,830 --> 00:24:52,080
and a denominator polynomial.
390
00:24:52,080 --> 00:24:59,020
And it's convenient to summarize
these by plotting
391
00:24:59,020 --> 00:25:02,180
the roots of the numerator
polynomial and the roots of
392
00:25:02,180 --> 00:25:06,110
the denominator polynomial
in the complex plane.
393
00:25:06,110 --> 00:25:10,650
And the complex plane which
they're plotted is referred to
394
00:25:10,650 --> 00:25:13,350
the s-plane.
395
00:25:13,350 --> 00:25:18,590
So we can, for example, take the
denominator polynomial and
396
00:25:18,590 --> 00:25:24,230
summarize it by specifying the
fact, or by representing the
397
00:25:24,230 --> 00:25:28,710
fact that it has roots at s
equals minus 1 and at s
398
00:25:28,710 --> 00:25:30,190
equals minus 2.
399
00:25:30,190 --> 00:25:33,950
And I've done that in this
picture by putting an x where
400
00:25:33,950 --> 00:25:37,850
the roots of the denominator
polynomial are.
401
00:25:37,850 --> 00:25:42,720
The numerator polynomial has a
root at s equals minus 3/2,
402
00:25:42,720 --> 00:25:45,070
and I've represented
that by a circle.
403
00:25:45,070 --> 00:25:48,510
So these are the roots of the
denominator polynomial and
404
00:25:48,510 --> 00:25:50,070
this is the root of
the numerator
405
00:25:50,070 --> 00:25:52,220
polynomial for this example.
406
00:25:52,220 --> 00:25:56,540
And also, for this example, we
can represent the region of
407
00:25:56,540 --> 00:26:00,300
convergence, which is
the real part of s
408
00:26:00,300 --> 00:26:01,600
greater than minus 1.
409
00:26:01,600 --> 00:26:06,720
And so that's, in fact,
the region over here.
410
00:26:06,720 --> 00:26:09,850
There is also, if I draw these,
just the roots of the
411
00:26:09,850 --> 00:26:13,180
numerator and denominator of
polynomials, I would need an
412
00:26:13,180 --> 00:26:16,820
additional piece of information
to specify the
413
00:26:16,820 --> 00:26:18,460
algebraic expression
completely.
414
00:26:18,460 --> 00:26:21,250
Namely, a multiplying constant
out in front
415
00:26:21,250 --> 00:26:24,190
of the whole thing.
416
00:26:24,190 --> 00:26:31,220
Well, this particular example,
has the Laplace transform as a
417
00:26:31,220 --> 00:26:32,600
rational function.
418
00:26:32,600 --> 00:26:35,950
Namely, it's one polynomial in
the numerator and another
419
00:26:35,950 --> 00:26:38,040
polynomial in the denominator.
420
00:26:38,040 --> 00:26:41,610
And in fact, as we'll see,
Laplace transforms, which are
421
00:26:41,610 --> 00:26:45,170
ratios of polynomials, form
a very important class.
422
00:26:45,170 --> 00:26:48,430
They, in fact, represent systems
that are describable
423
00:26:48,430 --> 00:26:51,810
by linear constant coefficient
differential equations.
424
00:26:51,810 --> 00:26:53,510
You shouldn't necessarily--
425
00:26:53,510 --> 00:26:55,420
in fact, for sure you
shouldn't see
426
00:26:55,420 --> 00:26:56,850
why that's true now.
427
00:26:56,850 --> 00:26:59,050
We'll see that later.
428
00:26:59,050 --> 00:27:06,060
But that means that Laplace
transforms that are rational
429
00:27:06,060 --> 00:27:10,280
functions, namely, the ratio
of a numerator polynomial
430
00:27:10,280 --> 00:27:14,180
divided by the denominator
polynomial, become very
431
00:27:14,180 --> 00:27:18,640
important in the discussion
that follows.
432
00:27:18,640 --> 00:27:22,760
And in fact, we have some
terminology for this.
433
00:27:22,760 --> 00:27:27,030
The roots of the numerator
polynomial are referred to as
434
00:27:27,030 --> 00:27:30,750
the zeroes of the Laplace
transform.
435
00:27:30,750 --> 00:27:34,165
Because, of course, those are
the values of s at which x of
436
00:27:34,165 --> 00:27:36,790
s becomes 0.
437
00:27:36,790 --> 00:27:40,130
And the roots of the denominator
polynomial are
438
00:27:40,130 --> 00:27:44,900
referred to as the poles of
the Laplace transform.
439
00:27:44,900 --> 00:27:49,840
And those are the values of
s at which the Laplace
440
00:27:49,840 --> 00:27:52,090
transform blows up.
441
00:27:52,090 --> 00:27:53,600
Namely, becomes infinite.
442
00:27:53,600 --> 00:27:57,430
If you think of setting s equal
to a value where this
443
00:27:57,430 --> 00:28:00,780
denominator polynomial goes
to 0, of course, x
444
00:28:00,780 --> 00:28:02,880
of s becomes infinite.
445
00:28:02,880 --> 00:28:06,860
And what we would expect and,
of course, we'll see
446
00:28:06,860 --> 00:28:09,360
that this is true.
447
00:28:09,360 --> 00:28:12,670
What we would expect is that
wherever that happens, there
448
00:28:12,670 --> 00:28:14,750
must be some problem
with convergence
449
00:28:14,750 --> 00:28:15,930
of the Laplace transform.
450
00:28:15,930 --> 00:28:18,160
And indeed, the Laplace
transform doesn't
451
00:28:18,160 --> 00:28:20,960
converge at the poles.
452
00:28:20,960 --> 00:28:24,390
Namely, at the roots of the
denominator polynomial.
453
00:28:24,390 --> 00:28:28,120
So, in fact, let's focus in
on that a little further.
454
00:28:28,120 --> 00:28:33,980
Let's examine and talk about the
region of convergence of
455
00:28:33,980 --> 00:28:38,200
the Laplace transform, and how
it's associated both with
456
00:28:38,200 --> 00:28:41,970
properties of the time function,
and also with the
457
00:28:41,970 --> 00:28:46,720
location of the poles of
the Laplace transform.
458
00:28:46,720 --> 00:28:50,320
And as we'll see, there are
some very specific and
459
00:28:50,320 --> 00:28:54,500
important relationships and
conclusions that we can draw
460
00:28:54,500 --> 00:28:58,790
about how the region of
convergence is constrained and
461
00:28:58,790 --> 00:29:05,790
associated with the locations
of the poles in the s-plane.
462
00:29:05,790 --> 00:29:11,310
Well, to begin with, we can, of
course, make the statement
463
00:29:11,310 --> 00:29:14,350
as I've just made that
the region of
464
00:29:14,350 --> 00:29:17,260
convergence contains no poles.
465
00:29:17,260 --> 00:29:22,600
In particular, if I think
of this general rational
466
00:29:22,600 --> 00:29:29,310
function, the poles of x of s
are the values of s at which
467
00:29:29,310 --> 00:29:30,870
the denominator is 0.
468
00:29:30,870 --> 00:29:34,560
Or equivalently, x
of s blows up.
469
00:29:34,560 --> 00:29:38,160
And of course then, that implies
that the expression
470
00:29:38,160 --> 00:29:41,460
has no longer converged.
471
00:29:41,460 --> 00:29:43,370
Well, that's one statement
that we can make.
472
00:29:43,370 --> 00:29:46,160
Now, there are some others.
473
00:29:46,160 --> 00:29:51,170
And one, for example, is the
statement that if I have a
474
00:29:51,170 --> 00:29:55,820
point in the s-plane that
corresponds to convergence,
475
00:29:55,820 --> 00:30:00,680
then in fact any line in the
s-plane with that same real
476
00:30:00,680 --> 00:30:05,100
part will also be a set of
values for which the Laplace
477
00:30:05,100 --> 00:30:07,110
transform converges.
478
00:30:07,110 --> 00:30:09,450
And what's the reason
for that?
479
00:30:09,450 --> 00:30:15,910
The reason for that is that s
is sigma plus j omega and
480
00:30:15,910 --> 00:30:20,840
convergence of the Laplace
transform is associated with
481
00:30:20,840 --> 00:30:25,580
convergence of the Fourier
transform of e to the minus
482
00:30:25,580 --> 00:30:28,390
sigma t times x of t.
483
00:30:28,390 --> 00:30:31,950
And so the convergence only
depends on sigma.
484
00:30:31,950 --> 00:30:37,010
If it only depends on sigma,
then if it converges for one
485
00:30:37,010 --> 00:30:39,200
value of sigma--
486
00:30:39,200 --> 00:30:43,980
I'm sorry, for a value of sigma
for some value of omega,
487
00:30:43,980 --> 00:30:47,840
then it will converge for
that same sigma for
488
00:30:47,840 --> 00:30:50,130
any value of omega.
489
00:30:50,130 --> 00:30:55,650
The conclusion then is that the
region of convergence, if
490
00:30:55,650 --> 00:30:58,110
I have a point, then
I also have a line.
491
00:30:58,110 --> 00:31:01,080
And so what that suggests is
that as we look at the region
492
00:31:01,080 --> 00:31:05,930
of convergence, it in fact
corresponds to strips in the
493
00:31:05,930 --> 00:31:09,360
complex plane.
494
00:31:09,360 --> 00:31:12,930
Now, finally we can tie
together the region of
495
00:31:12,930 --> 00:31:17,980
convergence to the convergence
of the Fourier transform.
496
00:31:17,980 --> 00:31:23,220
In particular, since we know
that the Laplace transform
497
00:31:23,220 --> 00:31:28,480
reduces to the Fourier transform
when the complex
498
00:31:28,480 --> 00:31:34,770
variable s is equal to j omega,
the implication is that
499
00:31:34,770 --> 00:31:40,000
if we have the Laplace transform
and if the Laplace
500
00:31:40,000 --> 00:31:43,300
transform reduces to the
Fourier transform when
501
00:31:43,300 --> 00:31:44,440
sigma equals 0.
502
00:31:44,440 --> 00:31:48,140
In other words, when s is equal
to j omega, then the
503
00:31:48,140 --> 00:31:52,600
Fourier transform of x of t
converging is equivalent to
504
00:31:52,600 --> 00:31:57,530
the statement that the Laplace
transform converges for sigma
505
00:31:57,530 --> 00:31:58,930
equal to 0.
506
00:31:58,930 --> 00:32:00,310
In other words, that
the region of
507
00:32:00,310 --> 00:32:04,340
convergence includes what?
508
00:32:04,340 --> 00:32:06,690
The j omega axis
in the s-plane.
509
00:32:10,880 --> 00:32:14,230
So we have then some statements
that kind of tie
510
00:32:14,230 --> 00:32:17,680
together the location of the
poles and the region of
511
00:32:17,680 --> 00:32:18,270
convergence.
512
00:32:18,270 --> 00:32:21,340
Let me make one other statement,
which is a much
513
00:32:21,340 --> 00:32:22,800
harder statement to justify.
514
00:32:22,800 --> 00:32:25,240
And I won't try to, I'll
just simply state it.
515
00:32:25,240 --> 00:32:27,810
And that is that the region of
convergence of the Laplace
516
00:32:27,810 --> 00:32:31,320
transform is a connected
region.
517
00:32:31,320 --> 00:32:37,330
In other words, if the entire
region consists of a single
518
00:32:37,330 --> 00:32:42,480
strip in the s-plane, it can't
consist of a strip over here,
519
00:32:42,480 --> 00:32:44,650
for example, and a
strip over there.
520
00:32:44,650 --> 00:32:50,420
Well, let me emphasize some of
those points a little further.
521
00:32:50,420 --> 00:32:59,060
Let's suppose that I have a
Laplace transform, and the
522
00:32:59,060 --> 00:33:03,280
Laplace transform that I'm
talking about is a rational
523
00:33:03,280 --> 00:33:09,610
function, which is 1 over
s plus 1 times s plus 2.
524
00:33:09,610 --> 00:33:14,460
Then the pole-zero pattern, as
it's referred to, in the
525
00:33:14,460 --> 00:33:18,160
s-plane, the location of the
roots of the numerator and
526
00:33:18,160 --> 00:33:19,490
denominator polynomials.
527
00:33:19,490 --> 00:33:22,230
Of course, there is no
numerator polynomial.
528
00:33:22,230 --> 00:33:25,850
The denominator polynomial
roots, which I've represented
529
00:33:25,850 --> 00:33:28,440
by these x's, are shown here.
530
00:33:28,440 --> 00:33:33,430
And so this is the pole-zero
pattern.
531
00:33:33,430 --> 00:33:37,690
And from what I've said, the
region of convergence can't
532
00:33:37,690 --> 00:33:42,590
include any poles and it
must correspond to
533
00:33:42,590 --> 00:33:44,860
strips in the s-plane.
534
00:33:44,860 --> 00:33:49,160
And furthermore, it must be
just one connected region
535
00:33:49,160 --> 00:33:51,640
rather than multiple regions.
536
00:33:51,640 --> 00:33:57,090
And so with this algebraic
expression then, the possible
537
00:33:57,090 --> 00:34:00,250
choices for the region of
convergence consistent with
538
00:34:00,250 --> 00:34:02,550
those properties are
the following.
539
00:34:02,550 --> 00:34:05,830
One of them would be a region
of convergence to the
540
00:34:05,830 --> 00:34:09,489
right of this pole.
541
00:34:09,489 --> 00:34:16,190
A second would be a region of
convergence which lies between
542
00:34:16,190 --> 00:34:21,570
the two poles as I show here.
543
00:34:21,570 --> 00:34:27,350
And a third is a region of
convergence which is to the
544
00:34:27,350 --> 00:34:31,199
left of this pole.
545
00:34:31,199 --> 00:34:36,010
And because of the fact that I
said without proof that the
546
00:34:36,010 --> 00:34:39,870
region of convergence must be
a single strip, it can't be
547
00:34:39,870 --> 00:34:41,159
multiple strips.
548
00:34:41,159 --> 00:34:44,460
In fact, we could not consider,
as a possible region
549
00:34:44,460 --> 00:34:48,060
of convergence, what
I show here.
550
00:34:48,060 --> 00:34:52,810
So, in fact, this is not a valid
region of convergence.
551
00:34:52,810 --> 00:34:56,750
There are only three
possibilities associated with
552
00:34:56,750 --> 00:34:58,900
this pole-zero pattern.
553
00:34:58,900 --> 00:35:02,270
Namely, to the right of this
pole, between the two poles,
554
00:35:02,270 --> 00:35:06,500
and to the left of this pole.
555
00:35:06,500 --> 00:35:11,250
Now, to carry the discussion
further, we can, in fact,
556
00:35:11,250 --> 00:35:16,490
associate the region of
convergence of the Laplace
557
00:35:16,490 --> 00:35:21,180
transform with some very
specific characteristics of
558
00:35:21,180 --> 00:35:22,540
the time function.
559
00:35:22,540 --> 00:35:28,790
And what this will do is to
help us understand how for
560
00:35:28,790 --> 00:35:32,320
various choices of the region
of convergence, the
561
00:35:32,320 --> 00:35:35,740
interpretation that we
can impose on the
562
00:35:35,740 --> 00:35:37,310
related time function.
563
00:35:37,310 --> 00:35:40,360
Let me show you what I mean.
564
00:35:40,360 --> 00:35:46,860
Suppose that we start with a
time function as I indicate
565
00:35:46,860 --> 00:35:53,820
here, which is a finite duration
time function.
566
00:35:53,820 --> 00:35:58,230
In other words, it's 0 except
in some time interval.
567
00:35:58,230 --> 00:36:03,200
Now, recall that the Fourier
transform converges if the
568
00:36:03,200 --> 00:36:06,580
time function has the property
that it's absolutely
569
00:36:06,580 --> 00:36:07,450
integrable.
570
00:36:07,450 --> 00:36:10,870
And as long as everything's
stays finite in terms of
571
00:36:10,870 --> 00:36:14,130
amplitudes in a finite duration
signal, there's no
572
00:36:14,130 --> 00:36:17,830
difficulty that we're going
to run into here.
573
00:36:17,830 --> 00:36:21,040
Now, here the Fourier transform
will converge.
574
00:36:21,040 --> 00:36:25,920
And now the question is, what
can we say about the region of
575
00:36:25,920 --> 00:36:29,770
convergence of the Laplace
transform?
576
00:36:29,770 --> 00:36:34,700
Well, the Laplace transform is
the Fourier transform of the
577
00:36:34,700 --> 00:36:37,810
time function multiplied
by an exponential.
578
00:36:37,810 --> 00:36:41,820
And so we can ask about whether
we can destroy the
579
00:36:41,820 --> 00:36:45,560
absolute integrability of this
by multiplying by an
580
00:36:45,560 --> 00:36:49,510
exponential that grows
to fast or decays
581
00:36:49,510 --> 00:36:50,490
too fast, or whatever.
582
00:36:50,490 --> 00:36:53,130
And let's take a look at that.
583
00:36:53,130 --> 00:36:57,310
Suppose that this time function
is absolutely
584
00:36:57,310 --> 00:36:59,230
integrable.
585
00:36:59,230 --> 00:37:05,550
And let's multiply it by
a decaying exponential.
586
00:37:05,550 --> 00:37:09,540
So this is now x of t times z
to the minus sigma t if I
587
00:37:09,540 --> 00:37:11,960
think of multiplying
these two together.
588
00:37:11,960 --> 00:37:16,260
And what you can see is that
for positive time, sort of
589
00:37:16,260 --> 00:37:20,140
thinking informally, I'm helping
the integrability of
590
00:37:20,140 --> 00:37:23,680
the product because I'm pushing
this part down.
591
00:37:23,680 --> 00:37:25,960
For negative time,
unfortunately, I'm making
592
00:37:25,960 --> 00:37:28,030
things grow.
593
00:37:28,030 --> 00:37:31,710
But I don't let them grow
indefinitely because there's
594
00:37:31,710 --> 00:37:36,180
some time before which
this is equal to 0.
595
00:37:36,180 --> 00:37:44,660
Likewise, if I had a growing
exponential, then for a
596
00:37:44,660 --> 00:37:48,960
growing exponential for negative
time, or for this
597
00:37:48,960 --> 00:37:52,720
part, I'm making
things smaller.
598
00:37:52,720 --> 00:37:55,580
For positive time, eventually
this exponential is growing
599
00:37:55,580 --> 00:37:56,930
without bound.
600
00:37:56,930 --> 00:38:01,370
But the time function
stops at some point.
601
00:38:01,370 --> 00:38:08,580
So the idea then kind of is
that for a finite duration
602
00:38:08,580 --> 00:38:12,760
time function, no matter what
kind of exponential I multiply
603
00:38:12,760 --> 00:38:16,720
by, whether it's going this way
or going this way, because
604
00:38:16,720 --> 00:38:19,870
of the fact that essentially the
limits on the integral are
605
00:38:19,870 --> 00:38:24,940
finite, I'm guaranteed that I'll
always maintain absolute
606
00:38:24,940 --> 00:38:26,370
integrability.
607
00:38:26,370 --> 00:38:30,890
And so, in fact then, for a
finite duration time function,
608
00:38:30,890 --> 00:38:33,800
the region of convergence
is the entire s-plane.
609
00:38:37,100 --> 00:38:41,890
Now, we can also make statements
about other kinds
610
00:38:41,890 --> 00:38:43,470
of time functions.
611
00:38:43,470 --> 00:38:52,280
And let's look at a time
function which I define as a
612
00:38:52,280 --> 00:38:55,470
right-sided time function.
613
00:38:55,470 --> 00:39:01,720
And a right-sided time function
is one which is 0 up
614
00:39:01,720 --> 00:39:06,200
until some time, and then it
goes on after that, presumably
615
00:39:06,200 --> 00:39:08,050
off to infinity.
616
00:39:08,050 --> 00:39:12,850
Now, let me remind you that the
whole issue here with the
617
00:39:12,850 --> 00:39:18,540
region of convergence has to do
with exponentials that we
618
00:39:18,540 --> 00:39:23,850
can multiply a time function by
and have the product end up
619
00:39:23,850 --> 00:39:26,430
being absolutely integrable.
620
00:39:26,430 --> 00:39:32,020
Well, suppose that when I
multiply this time function by
621
00:39:32,020 --> 00:39:35,400
an exponential which,
let's say decays.
622
00:39:35,400 --> 00:39:39,860
But an exponential e to the
minus sigma 0 t, what you can
623
00:39:39,860 --> 00:39:43,730
see sort of intuitively is
that if this product is
624
00:39:43,730 --> 00:39:49,030
absolutely integrable, if I were
to increase sigma 0, then
625
00:39:49,030 --> 00:39:52,090
I'm making things even better
for positive time because I'm
626
00:39:52,090 --> 00:39:53,640
pushing them down.
627
00:39:53,640 --> 00:39:57,730
And whereas they might be worse
for negative time, that
628
00:39:57,730 --> 00:40:01,170
doesn't matter because
before some time the
629
00:40:01,170 --> 00:40:02,540
product is equal to 0.
630
00:40:02,540 --> 00:40:08,200
So if this product is absolutely
integrable, then if
631
00:40:08,200 --> 00:40:12,840
I chose an exponential e to the
minus sigma 1t where sigma
632
00:40:12,840 --> 00:40:17,370
1 is greater than sigma 0, then
that product will also be
633
00:40:17,370 --> 00:40:19,470
absolutely integrable.
634
00:40:19,470 --> 00:40:23,270
And we can draw an important
conclusion about that, about
635
00:40:23,270 --> 00:40:25,420
the region of convergence
from that.
636
00:40:25,420 --> 00:40:30,700
In particular, we can make the
statement that if the time
637
00:40:30,700 --> 00:40:38,310
function is right-sided and if
convergence occurs for some
638
00:40:38,310 --> 00:40:44,250
value sigma 0, then in fact,
we will have convergence of
639
00:40:44,250 --> 00:40:49,700
the Laplace transform for all
values of the real part of s
640
00:40:49,700 --> 00:40:51,770
greater than sigma 0.
641
00:40:51,770 --> 00:40:56,610
The reason, of course, being
that if sigma 0 increases,
642
00:40:56,610 --> 00:41:03,210
then the exponential decays even
faster for positive time.
643
00:41:03,210 --> 00:41:06,070
Now what that says then thinking
another way, in terms
644
00:41:06,070 --> 00:41:09,700
of the region of convergence
as we might draw it in the
645
00:41:09,700 --> 00:41:13,080
s-plane, is that if we have a
point that's in the region of
646
00:41:13,080 --> 00:41:17,450
convergence corresponding to
some value sigma 0, then all
647
00:41:17,450 --> 00:41:21,580
values of s to the right of that
in the s-plane will also
648
00:41:21,580 --> 00:41:24,330
be in the region
of convergence.
649
00:41:24,330 --> 00:41:26,800
We can also combine that with
the statement that for
650
00:41:26,800 --> 00:41:31,490
rational functions we know that
there can't be any poles
651
00:41:31,490 --> 00:41:33,190
in the region of convergence.
652
00:41:33,190 --> 00:41:36,190
If you put those two statements
together, then we
653
00:41:36,190 --> 00:41:41,830
end up with a statement that if
x of t is right-sided and
654
00:41:41,830 --> 00:41:46,060
if its Laplace transform is
rational, then the region of
655
00:41:46,060 --> 00:41:51,280
convergence is to the right
of the rightmost pole.
656
00:41:51,280 --> 00:41:55,130
So we have here a very important
insight, which tells
657
00:41:55,130 --> 00:41:59,980
us that we can infer some
property about the time
658
00:41:59,980 --> 00:42:02,640
function from the region
of convergence.
659
00:42:02,640 --> 00:42:05,880
Or conversely, if we know
something about the time
660
00:42:05,880 --> 00:42:09,440
function, namely being
right-sided, then we can infer
661
00:42:09,440 --> 00:42:12,570
something about the region
of convergence.
662
00:42:12,570 --> 00:42:17,090
Well, in addition to right-sided
signals, we can
663
00:42:17,090 --> 00:42:19,510
also have left-sided signals.
664
00:42:19,510 --> 00:42:22,370
And a left-sided signal is
essentially a right-sided
665
00:42:22,370 --> 00:42:23,910
signal turned around.
666
00:42:23,910 --> 00:42:27,480
In other words, a left-sided
signal is one that is
667
00:42:27,480 --> 00:42:30,460
0 after some time.
668
00:42:30,460 --> 00:42:32,850
Well, we can carry out
exactly the same
669
00:42:32,850 --> 00:42:34,580
kind of argument there.
670
00:42:34,580 --> 00:42:37,800
Namely, if the signal goes off
to infinity in the negative
671
00:42:37,800 --> 00:42:42,690
time direction and stops some
place for positive time, if I
672
00:42:42,690 --> 00:42:46,110
have an exponential that I can
multiply it by and have that
673
00:42:46,110 --> 00:42:48,620
product be absolutely
integrable.
674
00:42:48,620 --> 00:42:51,590
And if I choose an exponential
that decays even faster for
675
00:42:51,590 --> 00:42:54,560
negative time so that I'm
pushing the stuff way out
676
00:42:54,560 --> 00:42:58,830
there down even further,
then I enhance the
677
00:42:58,830 --> 00:43:00,850
integrability even more.
678
00:43:00,850 --> 00:43:03,830
And you might have to think
through that a little bit, but
679
00:43:03,830 --> 00:43:07,170
it's exactly the flip
side of the argument
680
00:43:07,170 --> 00:43:08,890
for right-sided signals.
681
00:43:08,890 --> 00:43:14,470
And the conclusion then is that
if we have a left-sided
682
00:43:14,470 --> 00:43:20,010
signal and we have a point, a
value of the real part of s
683
00:43:20,010 --> 00:43:23,400
which is in the region of
convergence, then in fact, all
684
00:43:23,400 --> 00:43:29,140
values to the left of that point
in the s-plane will also
685
00:43:29,140 --> 00:43:32,500
be in the region
of convergence.
686
00:43:32,500 --> 00:43:35,330
Now, similar to the statement
that we made for right-sided
687
00:43:35,330 --> 00:43:39,230
signals, if x of t is left-sided
and, in fact, we're
688
00:43:39,230 --> 00:43:41,720
talking about a rational Laplace
transform, which we
689
00:43:41,720 --> 00:43:43,830
most typically will.
690
00:43:43,830 --> 00:43:47,730
Then, in fact, we can make the
statement that the region of
691
00:43:47,730 --> 00:43:53,290
convergence is to the left of
the leftmost pole because we
692
00:43:53,290 --> 00:43:56,730
know if we find a point that's
in the region of convergence,
693
00:43:56,730 --> 00:43:58,930
everything to the left of that
has to be in the region of
694
00:43:58,930 --> 00:43:59,870
convergence.
695
00:43:59,870 --> 00:44:02,480
We can't have any poles in the
region of convergence.
696
00:44:02,480 --> 00:44:04,960
You put those two statements
together and it says it's to
697
00:44:04,960 --> 00:44:07,710
the left of the leftmost pole.
698
00:44:07,710 --> 00:44:12,340
Now the final situation is the
situation where we have a
699
00:44:12,340 --> 00:44:14,680
signal which is neither
right-sided nor left-sided.
700
00:44:14,680 --> 00:44:18,010
It goes off to infinity for
positive time and it goes off
701
00:44:18,010 --> 00:44:20,100
to infinity for negative time.
702
00:44:20,100 --> 00:44:23,460
And there the thing to kind of
recognize is that if you
703
00:44:23,460 --> 00:44:28,370
multiply by an exponential, and
it's decaying very fast
704
00:44:28,370 --> 00:44:30,750
for positive time, it's going
to be growing very fast for
705
00:44:30,750 --> 00:44:32,060
negative time.
706
00:44:32,060 --> 00:44:35,060
Conversely, if it's decaying
very fast for negative time,
707
00:44:35,060 --> 00:44:37,530
it's growing very fast
for positive time.
708
00:44:37,530 --> 00:44:39,840
And there's this notion
of trying to balance
709
00:44:39,840 --> 00:44:41,710
the value of sigma.
710
00:44:41,710 --> 00:44:44,620
And in effect, what that says
is that the region of
711
00:44:44,620 --> 00:44:47,810
convergence can't extent
too far to the left or
712
00:44:47,810 --> 00:44:49,450
too far to the right.
713
00:44:49,450 --> 00:44:55,790
Said another way for a two-sided
signal, if we have a
714
00:44:55,790 --> 00:44:59,560
point which is in the region
of convergence, then that
715
00:44:59,560 --> 00:45:06,530
point defines a strip in the
s-plane that takes that point
716
00:45:06,530 --> 00:45:09,960
and extends it to the left until
you bump into a pole,
717
00:45:09,960 --> 00:45:12,990
and extends it to the right
until you bump it into a pole.
718
00:45:15,845 --> 00:45:20,460
So you begin to then see that
we can tie together some
719
00:45:20,460 --> 00:45:24,040
properties of the region
of convergence and the
720
00:45:24,040 --> 00:45:25,940
right-sidedness, or
left-sidedness, or
721
00:45:25,940 --> 00:45:29,030
two-sidedness of the
time function.
722
00:45:29,030 --> 00:45:32,750
And you'll have a chance to
examine that in more detail in
723
00:45:32,750 --> 00:45:35,010
the video course manual.
724
00:45:35,010 --> 00:45:41,440
Let's conclude this lecture by
talking about how we might get
725
00:45:41,440 --> 00:45:46,040
the time function given the
appliance transform.
726
00:45:46,040 --> 00:45:49,830
Well, if we have a Laplace
transform, we can, in
727
00:45:49,830 --> 00:45:53,010
principle, get the time
function back again by
728
00:45:53,010 --> 00:45:56,230
recognizing this relationship
between the Laplace transform
729
00:45:56,230 --> 00:45:59,360
and the Fourier transform, and
using the formal Fourier
730
00:45:59,360 --> 00:46:00,650
transform expression.
731
00:46:00,650 --> 00:46:04,510
Or equivalently, the formal
inverse Laplace transform
732
00:46:04,510 --> 00:46:07,190
expression, which
is in the text.
733
00:46:07,190 --> 00:46:11,040
But more typically what we would
do is what we've done
734
00:46:11,040 --> 00:46:15,910
also with the Fourier transform,
which is to use
735
00:46:15,910 --> 00:46:21,580
simple Laplace transform pairs
together with the notion of
736
00:46:21,580 --> 00:46:23,490
the partial fraction
expansion.
737
00:46:23,490 --> 00:46:28,020
And let's just go through
that with an example.
738
00:46:28,020 --> 00:46:34,350
Let's suppose that I have
a Laplace transform as I
739
00:46:34,350 --> 00:46:38,380
indicated here in its pole-zero
plot and a region of
740
00:46:38,380 --> 00:46:42,060
convergence which is to the
right of this pole.
741
00:46:42,060 --> 00:46:45,140
And what we can identify from
the region of convergence, in
742
00:46:45,140 --> 00:46:47,980
fact, is that we're
talking about a
743
00:46:47,980 --> 00:46:50,660
right-sided time function.
744
00:46:50,660 --> 00:46:54,300
So the region of convergence is
the real part of s greater
745
00:46:54,300 --> 00:46:55,900
than minus 1.
746
00:46:55,900 --> 00:47:01,050
And now looking down at the
algebraic expression, we have
747
00:47:01,050 --> 00:47:04,610
the algebraic expression for
this, as I indicated here,
748
00:47:04,610 --> 00:47:09,560
equivalently expanded in a
partial fraction expansion, as
749
00:47:09,560 --> 00:47:10,250
I show below.
750
00:47:10,250 --> 00:47:13,590
So if you just simply combine
these together, that's the
751
00:47:13,590 --> 00:47:15,030
same as this.
752
00:47:15,030 --> 00:47:19,360
And the region of convergence is
the real part of s greater
753
00:47:19,360 --> 00:47:21,410
than minus 1.
754
00:47:21,410 --> 00:47:24,830
Now, the region of
convergence of--
755
00:47:24,830 --> 00:47:28,360
this is the sum of two terms,
so the time function is the
756
00:47:28,360 --> 00:47:30,320
sum of two time functions.
757
00:47:30,320 --> 00:47:37,760
And the region of convergence of
the combination must be the
758
00:47:37,760 --> 00:47:39,850
intersection of the region
of convergence
759
00:47:39,850 --> 00:47:41,960
associated with each one.
760
00:47:41,960 --> 00:47:46,990
Recognizing that this is to the
right of the poles, that
761
00:47:46,990 --> 00:47:51,290
tells us immediately that each
of these two then would
762
00:47:51,290 --> 00:47:55,110
correspond to the Laplace
transform of a right-sided
763
00:47:55,110 --> 00:47:57,180
time function.
764
00:47:57,180 --> 00:48:00,910
Well, let's look at
it term by term.
765
00:48:00,910 --> 00:48:06,620
The first term is the factor 1
over s plus 1 with a region of
766
00:48:06,620 --> 00:48:10,620
convergence to the right
of this pole.
767
00:48:10,620 --> 00:48:15,070
And this algebraically
corresponds
768
00:48:15,070 --> 00:48:16,510
to what I've indicated.
769
00:48:16,510 --> 00:48:21,610
And this, in fact, is similar
to, or a special case of the
770
00:48:21,610 --> 00:48:24,940
example that we pointed to at
the beginning of the lecture.
771
00:48:24,940 --> 00:48:27,010
Namely, Example 9.1.
772
00:48:27,010 --> 00:48:29,370
And so we can just simply
use that result.
773
00:48:29,370 --> 00:48:32,120
If you think back to that
example or refer to your
774
00:48:32,120 --> 00:48:37,570
notes, we know that time
function of the form e to the
775
00:48:37,570 --> 00:48:41,360
minus a t gives us the Laplace
transform, which is 1 over s
776
00:48:41,360 --> 00:48:47,530
plus a with the real part of
s greater than minus a.
777
00:48:47,530 --> 00:48:50,620
And so this is the Laplace
transform of the first.
778
00:48:50,620 --> 00:48:53,080
Or, I'm sorry, this is the
inverse Laplace transform of
779
00:48:53,080 --> 00:48:55,310
the first term.
780
00:48:55,310 --> 00:49:03,610
If we now consider the pole at
s equals minus 2, and here is
781
00:49:03,610 --> 00:49:06,790
the region of convergence that
we originally began with.
782
00:49:06,790 --> 00:49:12,370
In fact, we can having removed
the pole at minus 1, extend
783
00:49:12,370 --> 00:49:16,370
this region of convergence
to this pole.
784
00:49:16,370 --> 00:49:21,820
And we now have an algebraic
expression, which is minus 1
785
00:49:21,820 --> 00:49:26,370
over s plus 2, the real part
of s greater than minus 1.
786
00:49:26,370 --> 00:49:29,190
Although, in fact, we can
extend the region of
787
00:49:29,190 --> 00:49:31,870
convergence up to the pole.
788
00:49:31,870 --> 00:49:37,250
And the inverse transform of
this is now, again, referring
789
00:49:37,250 --> 00:49:40,840
to the same example, minus
e to the minus 2t
790
00:49:40,840 --> 00:49:43,210
times the unit step.
791
00:49:43,210 --> 00:49:49,530
And if we simply put the two
terms together then, adding
792
00:49:49,530 --> 00:49:54,100
the one that we have here to
what we had before, we have a
793
00:49:54,100 --> 00:50:00,860
total inverse Laplace transform,
which is that.
794
00:50:00,860 --> 00:50:05,890
So essentially, what's happened
is that each of the
795
00:50:05,890 --> 00:50:09,540
poles has contributed an
exponential factor.
796
00:50:09,540 --> 00:50:13,030
And because of the region of
convergence being to the right
797
00:50:13,030 --> 00:50:18,000
of all those poles, that is
consistent with the notion
798
00:50:18,000 --> 00:50:20,080
that both of those terms
correspond to
799
00:50:20,080 --> 00:50:22,720
right-sided time functions.
800
00:50:22,720 --> 00:50:29,260
Well, let's just focus for a
second or two on the same
801
00:50:29,260 --> 00:50:31,260
pole-zero pattern.
802
00:50:31,260 --> 00:50:36,620
But instead of a region of
convergence which is to the
803
00:50:36,620 --> 00:50:40,030
right of the poles as we had
before, we'll now take a
804
00:50:40,030 --> 00:50:43,310
region of convergence which
is between the two poles.
805
00:50:43,310 --> 00:50:47,700
And I'll let you work through
this more leisurely in the
806
00:50:47,700 --> 00:50:49,510
video course manual.
807
00:50:49,510 --> 00:50:53,200
But when we carry out the
partial fraction expansion, as
808
00:50:53,200 --> 00:51:00,060
I've done below, we would now
associate with this pole a
809
00:51:00,060 --> 00:51:02,490
region of convergence
to the right.
810
00:51:02,490 --> 00:51:07,200
With this pole, a region of
convergence to the left.
811
00:51:07,200 --> 00:51:13,610
And so what we would have is the
sum of a right-sided time
812
00:51:13,610 --> 00:51:15,520
function due to this pole.
813
00:51:15,520 --> 00:51:20,160
And in fact it's of the form e
to the minus t for t positive.
814
00:51:20,160 --> 00:51:26,330
And a left-sided time function
due to this pole.
815
00:51:26,330 --> 00:51:28,590
And in fact, that's of
the form e to the
816
00:51:28,590 --> 00:51:30,970
minus 2t for t negative.
817
00:51:30,970 --> 00:51:35,280
And so, in fact, the answer
that we will get when we
818
00:51:35,280 --> 00:51:39,120
decompose this, use the partial
fraction expansion,
819
00:51:39,120 --> 00:51:42,870
being very careful about
associating the region of
820
00:51:42,870 --> 00:51:46,230
convergence of this pole to the
right and of this pole to
821
00:51:46,230 --> 00:51:50,530
the left, we'll have then, when
we're all done, a time
822
00:51:50,530 --> 00:51:55,570
function which will be of the
form e to the minus t times
823
00:51:55,570 --> 00:51:58,660
the unit step for t positive.
824
00:51:58,660 --> 00:52:03,380
And then we'll have a term
of the form e to the--
825
00:52:03,380 --> 00:52:06,750
I'm sorry, this would be e to
the minus 2t since this is at
826
00:52:06,750 --> 00:52:09,150
minus 2 and this
is at minus 1.
827
00:52:09,150 --> 00:52:11,960
This would be a plus sign and
this would be minus e to the
828
00:52:11,960 --> 00:52:15,300
minus t for t negative.
829
00:52:15,300 --> 00:52:20,120
And you'll look at that a little
more carefully when you
830
00:52:20,120 --> 00:52:21,970
sit down with the video
course manual.
831
00:52:24,950 --> 00:52:28,920
OK, well, what we've gone
through, rather quickly, is an
832
00:52:28,920 --> 00:52:32,510
introduction to the
Laplace transform.
833
00:52:32,510 --> 00:52:36,970
And a couple of points to
underscore again, is the fact
834
00:52:36,970 --> 00:52:41,190
that the Laplace transform is
very closely associated with
835
00:52:41,190 --> 00:52:42,690
the Fourier transform.
836
00:52:42,690 --> 00:52:47,280
And in fact, the Laplace
transform for s equals j omega
837
00:52:47,280 --> 00:52:49,570
reduces to the Fourier
transform.
838
00:52:49,570 --> 00:52:52,790
But more generally, the Laplace
transform is the
839
00:52:52,790 --> 00:52:59,760
Fourier transform of x of t with
an exponential weighting.
840
00:52:59,760 --> 00:53:02,870
And there are some exponentials
for which that
841
00:53:02,870 --> 00:53:05,010
product converges.
842
00:53:05,010 --> 00:53:08,430
There are other exponentials for
which that product has a
843
00:53:08,430 --> 00:53:10,780
Fourier transform that
doesn't converge.
844
00:53:10,780 --> 00:53:14,370
That then imposes on the
discussion of the Laplace
845
00:53:14,370 --> 00:53:18,860
transform what we refer to as
the region of convergence.
846
00:53:18,860 --> 00:53:22,670
And it's very important to
understand that in specifying
847
00:53:22,670 --> 00:53:28,670
a Laplace transform, it's
important to identify not only
848
00:53:28,670 --> 00:53:34,870
the algebraic expression, but
also the values of s for which
849
00:53:34,870 --> 00:53:35,620
it's valid.
850
00:53:35,620 --> 00:53:37,740
Namely, the region
of convergence
851
00:53:37,740 --> 00:53:39,490
of the Laplace transform.
852
00:53:39,490 --> 00:53:44,350
Finally what we did was to tie
together some properties of a
853
00:53:44,350 --> 00:53:48,200
time function with things that
we can say about the region of
854
00:53:48,200 --> 00:53:52,060
convergence of its Laplace
transform.
855
00:53:52,060 --> 00:53:54,910
Now, just as with the Fourier
transform, the Laplace
856
00:53:54,910 --> 00:53:58,670
transform has some very
important properties.
857
00:53:58,670 --> 00:54:04,140
And out of these properties,
both are some mechanisms for
858
00:54:04,140 --> 00:54:09,760
using the Laplace transform
for such systems as those
859
00:54:09,760 --> 00:54:12,280
described by linear constant
coefficient
860
00:54:12,280 --> 00:54:14,080
differential equations.
861
00:54:14,080 --> 00:54:17,900
But more importantly, the
properties will help us.
862
00:54:17,900 --> 00:54:23,550
As we understand them further,
will help us in using and
863
00:54:23,550 --> 00:54:27,890
exploiting the Laplace transform
to study and
864
00:54:27,890 --> 00:54:31,700
understand linear time-invariant
systems.
865
00:54:31,700 --> 00:54:34,970
And that's what we'll
go on to next time.
866
00:54:34,970 --> 00:54:38,530
In particular, talking about
properties, and then
867
00:54:38,530 --> 00:54:42,610
associating with linear
time-invariant systems much of
868
00:54:42,610 --> 00:54:46,610
the discussion that we've
had today relating
869
00:54:46,610 --> 00:54:48,040
to the Laplace transform.
870
00:54:48,040 --> 00:54:49,290
Thank you.