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PROFESSOR STRANG: Well, hope
you had a good Thanksgiving.
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00:00:24 --> 00:00:30
So this is partly review
today, even, Wednesday
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even more review.
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Wednesday evening.
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Or Wednesday at 4 I'll be
here for any questions.
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00:00:38 --> 00:00:44
And then the exam is
Thursday at 7:30 in Walker.
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00:00:44 --> 00:00:49
Top floor of Walker this
time, not the same 54-100.
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00:00:49 --> 00:00:54
OK, and then, no
lectures after that.
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00:00:54 --> 00:00:56
Holiday, whatever.
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Yes.
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00:00:57 --> 00:01:00
Right, you get a chance
to do something.
20
00:01:00 --> 00:01:03
Catch up with all those other
courses that are being
21
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neglected in favor
of 18.085 Right.
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00:01:06 --> 00:01:11
OK, so here's a bit of
review right away.
23
00:01:11 --> 00:01:14
We really had four cases.
24
00:01:14 --> 00:01:19
We started with Fourier series,
that was periodic functions.
25
00:01:19 --> 00:01:23
And then discrete Fourier
series, also periodic in a way.
26
00:01:23 --> 00:01:26
Because w^n was one.
27
00:01:26 --> 00:01:32
So that we have n numbers
and then we could repeat
28
00:01:32 --> 00:01:33
them if we wanted.
29
00:01:33 --> 00:01:37
So those are the
two that repeat.
30
00:01:37 --> 00:01:41
This is the f(x), this is
all x, so that would be
31
00:01:41 --> 00:01:45
the Fourier integral that
we did just last week.
32
00:01:45 --> 00:01:47
Fourier integral transform.
33
00:01:47 --> 00:01:51
And this was the - well,
these are all, this is
34
00:01:51 --> 00:01:53
the discrete all the way.
35
00:01:53 --> 00:02:00
So that's - oh, you can see
these pair off, right?
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00:02:00 --> 00:02:08
The periodic function, the 2pi
periodic function has Fourier
37
00:02:08 --> 00:02:13
coefficients for all k, so
that's the pair that
38
00:02:13 --> 00:02:14
we started with.
39
00:02:14 --> 00:02:16
Section 4.1.
40
00:02:16 --> 00:02:19
This sort of pairs, I
don't know whether
41
00:02:19 --> 00:02:20
to say with itself.
42
00:02:20 --> 00:02:26
I mean, we start with n numbers
and we end with n numbers.
43
00:02:26 --> 00:02:29
We have n numbers
in physical space.
44
00:02:29 --> 00:02:33
And we have n numbers
in frequency space.
45
00:02:33 --> 00:02:37
Right, so we call those,
so those went to
46
00:02:37 --> 00:02:40
c_0 up to c_(N-1).
47
00:02:42 --> 00:02:47
And this, all x pair for
the function, paired
48
00:02:47 --> 00:02:48
off with itself.
49
00:02:48 --> 00:02:55
Or with this went to f -
well maybe I use small f.
50
00:02:55 --> 00:02:58
I guess I did in last week.
51
00:02:58 --> 00:03:02
So that, and I called
its Fourier transform
52
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f hat of k, all k.
53
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So that's the pairing kind of
inside n-dimensional space
54
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with the Fourier matrix.
55
00:03:12 --> 00:03:17
This is the pairing of the
formula for f, and its similar
56
00:03:17 --> 00:03:21
formula for f hat, and these
are the guys that connect
57
00:03:21 --> 00:03:22
with each other.
58
00:03:22 --> 00:03:25
OK, so that's what we know.
59
00:03:25 --> 00:03:31
What we haven't done is
anything into two dimensions.
60
00:03:31 --> 00:03:34
So I would like to
include that today.
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00:03:34 --> 00:03:37
I think my real message about
2-D, and I'm not going to
62
00:03:37 --> 00:03:43
include it on the exam, but
you might wonder, OK, can I
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have a function of x and y?
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00:03:45 --> 00:03:48
And will the whole setup work.
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00:03:48 --> 00:03:49
And the answer is yes.
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So really, my message is not to
be afraid in any way of 2-D.
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It's just the same formulas
with x,y or two indices, k,l.
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Yeah.
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You'll see that.
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OK, now for the new part.
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What's a convolution equation?
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That's my word for an equation
where instead of doing a
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convolution and finding the
right-hand side, instead we're
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00:04:19 --> 00:04:21
given the right-hand side.
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00:04:21 --> 00:04:26
And the unknown is
in the convolution.
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00:04:26 --> 00:04:30
So let me write examples
of convolution equation.
77
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Every one of these would
allow a convolution.
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So the convolution equation
would be something the integral
79
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of F(t)u, for the unknown,
at x-t is - oh no, sorry.
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F will be the right-hand side.
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00:04:50 --> 00:04:53
F of, well, can I - yeah,
better if I put it on
82
00:04:53 --> 00:04:55
the right-hand side.
83
00:04:55 --> 00:04:59
Wouldn't want to call it
the right-hand side.
84
00:04:59 --> 00:05:05
So this would be some, shall
I call it often K for kernel
85
00:05:05 --> 00:05:08
is sometimes the word.
86
00:05:08 --> 00:05:13
So what I'm saying is
equations come this way.
87
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This is really K
convolved with u.
88
00:05:19 --> 00:05:20
Equals F.
89
00:05:20 --> 00:05:24
You see, the only novelty
is the unknown is here.
90
00:05:24 --> 00:05:28
So that's why the word d
convolution is up there.
91
00:05:28 --> 00:05:29
Because that's what
we have to do.
92
00:05:29 --> 00:05:33
We have to undo the
convolution, this unknown
93
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function is convolved with a
known, K is known, some known
94
00:05:38 --> 00:05:42
kernel that tells us the point
spread of the telescope or
95
00:05:42 --> 00:05:44
whatever we're doing.
96
00:05:44 --> 00:05:47
And gives us the output
that we're looking at.
97
00:05:47 --> 00:05:50
And then we have to
find the input.
98
00:05:50 --> 00:05:53
OK, can I write down the
similar equations for
99
00:05:53 --> 00:05:55
the other three here?
100
00:05:55 --> 00:05:59
And then we'll just think
how would we find u, how
101
00:05:59 --> 00:06:00
would we solve them?
102
00:06:00 --> 00:06:07
So the equation here might
be that some kernel
103
00:06:07 --> 00:06:13
circle, convolved with
the unknown u is some y.
104
00:06:13 --> 00:06:16
These are now vectors.
105
00:06:16 --> 00:06:19
This is known.
106
00:06:19 --> 00:06:20
This is known.
107
00:06:20 --> 00:06:23
And those, the end-components
of u, are unknown.
108
00:06:23 --> 00:06:26
OK, so that would be
the same problem here.
109
00:06:26 --> 00:06:27
What would be here?
110
00:06:27 --> 00:06:28
Same thing.
111
00:06:28 --> 00:06:31
Now the integral will go from
- the only difference is the
112
00:06:31 --> 00:06:34
integral will go from minus
infinity to infinity,
113
00:06:34 --> 00:06:35
K(t)u(x-t)dt=f(x).
114
00:06:36 --> 00:06:48
u(x-t) And finally
regular convolution.
115
00:06:48 --> 00:06:50
What am I going to call it?
116
00:06:50 --> 00:06:56
K would be a sequence, maybe
I should call it a, known
117
00:06:56 --> 00:07:04
convolved with u, unknown,
is some c, known.
118
00:07:04 --> 00:07:04
Yeah.
119
00:07:04 --> 00:07:07
So those would be
four equations.
120
00:07:07 --> 00:07:10
You might say, wait a minute
where is Professor Strang come
121
00:07:10 --> 00:07:13
up with these problems at
the last week of the course.
122
00:07:13 --> 00:07:18
But, these are exactly the
type of problems that
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00:07:18 --> 00:07:20
we know and love.
124
00:07:20 --> 00:07:26
These come from constant
coefficient time invariant,
125
00:07:26 --> 00:07:29
shift invariant.
126
00:07:29 --> 00:07:31
Linear problems.
127
00:07:31 --> 00:07:33
LTI, linear time and variant.
128
00:07:33 --> 00:07:40
And my lecture Wednesday, just
before Thanksgiving, took a
129
00:07:40 --> 00:07:46
differential equation for u and
found, and put it in this form.
130
00:07:46 --> 00:07:48
I'll come back to that.
131
00:07:48 --> 00:07:52
So suddenly we're seeing, I
mean, we're actually seeing
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00:07:52 --> 00:07:56
some new things but also it
includes all the old ones.
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00:07:56 --> 00:08:00
These are all of the best
problems in the world.
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00:08:00 --> 00:08:02
These linear constant
coefficient problems.
135
00:08:02 --> 00:08:05
Time in variant, of
any of these types.
136
00:08:05 --> 00:08:10
This one was an integral from
minus pi to pi, where this
137
00:08:10 --> 00:08:12
one went all the way.
138
00:08:12 --> 00:08:16
So this is not brand new stuff.
139
00:08:16 --> 00:08:19
But it sort of looks new.
140
00:08:19 --> 00:08:26
And now the question is, so my
immediate question is, before
141
00:08:26 --> 00:08:31
doing any example, how would
you solve such an equation.
142
00:08:31 --> 00:08:36
And I saw on old exams, some
of this sort for example,
143
00:08:36 --> 00:08:39
let me focus on this one.
144
00:08:39 --> 00:08:43
Let me, instead of K there,
I'm not used to using K
145
00:08:43 --> 00:08:48
for a vector, I'm used to,
well maybe I'll use c.
146
00:08:48 --> 00:08:50
For the vector there.
147
00:08:50 --> 00:08:59
So this is N equations,
N unknowns.
148
00:08:59 --> 00:09:01
Oops, capital N is
our usual here.
149
00:09:01 --> 00:09:03
For the number.
150
00:09:03 --> 00:09:06
N u, N unknown u's.
151
00:09:06 --> 00:09:10
It's a matrix equation
with a circulant matrix.
152
00:09:10 --> 00:09:17
So all these equations are sort
of the special best kind.
153
00:09:17 --> 00:09:19
Because they're convolutions.
154
00:09:19 --> 00:09:22
And now tell me the main point.
155
00:09:22 --> 00:09:24
How do we solve
equations like this?
156
00:09:24 --> 00:09:30
How do we do a deconvolution,
so the unknown is convolved
157
00:09:30 --> 00:09:34
with c here, it's convolved
with K, it's convolved with
158
00:09:34 --> 00:09:40
a, how do we deconvolve
it to get u by itself?
159
00:09:40 --> 00:09:43
So what's the
central idea here?
160
00:09:43 --> 00:09:47
Central idea: go into
frequency space.
161
00:09:47 --> 00:09:50
Use the convolution rule.
162
00:09:50 --> 00:09:58
In frequency space, where these
transform, we're looking
163
00:09:58 --> 00:10:01
at multiplication.
164
00:10:01 --> 00:10:04
And multiplication,
we can undo.
165
00:10:04 --> 00:10:06
We can de-multiply.
166
00:10:06 --> 00:10:10
De-multiply is just a big
word for divide, right?
167
00:10:10 --> 00:10:12
So that's the point.
168
00:10:12 --> 00:10:14
Get into that space.
169
00:10:14 --> 00:10:15
That's what we've been
doing all the time.
170
00:10:15 --> 00:10:21
I better get one example, the
example from the problem
171
00:10:21 --> 00:10:23
Wednesday, just up here.
172
00:10:23 --> 00:10:24
Just so you see it.
173
00:10:24 --> 00:10:28
This won't look like a
convolution equation, but do
174
00:10:28 --> 00:10:33
you remember that it was -u''
plus a squared u
175
00:10:33 --> 00:10:34
equal some f(x)?
176
00:10:34 --> 00:10:37
177
00:10:37 --> 00:10:39
So that's a constant,
it's certainly constant
178
00:10:39 --> 00:10:41
coefficient linear.
179
00:10:41 --> 00:10:42
Time invariant.
180
00:10:42 --> 00:10:43
Right, OK.
181
00:10:43 --> 00:10:44
And how did we solve that?
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00:10:44 --> 00:10:46
We took Fourier transforms.
183
00:10:46 --> 00:10:52
So this was the second
derivative, the
184
00:10:52 --> 00:10:52
Fourier transform.
185
00:10:52 --> 00:10:55
What is the rule for
the Fourier transform
186
00:10:55 --> 00:10:56
of derivative?
187
00:10:56 --> 00:11:01
Every derivative brings down
an ik, in the transform.
188
00:11:01 --> 00:11:03
So we get ik twice.
189
00:11:03 --> 00:11:07
So it's k squared. i squared
cancels the minus one.
190
00:11:07 --> 00:11:11
So that's the
transform of -u''.
191
00:11:12 --> 00:11:16
This is the transform
of ordinary a squared
192
00:11:16 --> 00:11:17
u, just a squared.
193
00:11:17 --> 00:11:19
And this is f hat.
194
00:11:19 --> 00:11:23
So we've got into
frequency space.
195
00:11:23 --> 00:11:27
Where we are just seeing a
multiplication, K squared
196
00:11:27 --> 00:11:32
plus a squared, u hat.
197
00:11:32 --> 00:11:39
Of, this is u hat of k,
equals f hat of k, right?
198
00:11:39 --> 00:11:42
Oh well, sorry we were
- yeah, that's right.
199
00:11:42 --> 00:11:45
f hat of k, right.
200
00:11:45 --> 00:11:48
So we're in frequency
space, where we just
201
00:11:48 --> 00:11:49
see a multiplication.
202
00:11:49 --> 00:11:58
So again, this is now we just
demultiply, just divide.
203
00:11:58 --> 00:12:03
And then we have the answer,
but we have its transform.
204
00:12:03 --> 00:12:05
And then we have to
transform back.
205
00:12:05 --> 00:12:10
So we have to do the Fourier
transform to get to the,
206
00:12:10 --> 00:12:15
I'll say the inverse
Fourier transform.
207
00:12:15 --> 00:12:17
To get back to u(x).
208
00:12:19 --> 00:12:19
The answer.
209
00:12:19 --> 00:12:21
That's the model.
210
00:12:21 --> 00:12:22
That's the model.
211
00:12:22 --> 00:12:26
And that's maybe the one that
we've seen, now we're able
212
00:12:26 --> 00:12:32
to think about all
these four topics.
213
00:12:32 --> 00:12:34
Right.
214
00:12:34 --> 00:12:38
OK, so what was the key idea?
215
00:12:38 --> 00:12:42
Get into frequency space and
then it's just a, the equation
216
00:12:42 --> 00:12:45
is just a multiplication,
so the solution is
217
00:12:45 --> 00:12:46
just a division.
218
00:12:46 --> 00:12:52
So can I do that now with these
four examples, just see.
219
00:12:52 --> 00:12:54
So this is like bring
the pieces together.
220
00:12:54 --> 00:12:59
OK, and deconvolution is
a very key thing to do.
221
00:12:59 --> 00:13:01
OK, so I'll take all four
of those and bring them
222
00:13:01 --> 00:13:04
into frequency space.
223
00:13:04 --> 00:13:10
So this will be maybe, you'll
let me use k hat of, oh, k hat
224
00:13:10 --> 00:13:14
of k, that's not too good.
225
00:13:14 --> 00:13:27
Well, stuck with it.
226
00:13:27 --> 00:13:28
What am I doing here?
227
00:13:28 --> 00:13:31
In this 2pi periodic one?
228
00:13:31 --> 00:13:35
That's the one I started with,
but now I've got, I I'm
229
00:13:35 --> 00:13:37
using hats and so on.
230
00:13:37 --> 00:13:42
I didn't do that
in Section 4.1.
231
00:13:42 --> 00:13:46
What the heck am I going to,
what notation am I going to do?
232
00:13:46 --> 00:13:51
And I really didn't
do convolution that
233
00:13:51 --> 00:13:52
much, for functions.
234
00:13:52 --> 00:13:55
So let me jump to here.
235
00:13:55 --> 00:13:57
I'll come back.
236
00:13:57 --> 00:13:59
It follows exactly
the same pattern.
237
00:13:59 --> 00:14:01
So let me jump to this one.
238
00:14:01 --> 00:14:05
OK, so I have a
convolution equation now.
239
00:14:05 --> 00:14:10
This is one where you
could do this one.
240
00:14:10 --> 00:14:16
This could appear on the quiz
because I can do all of it.
241
00:14:16 --> 00:14:18
So what is this convolution?
242
00:14:18 --> 00:14:19
OK.
243
00:14:19 --> 00:14:21
I've got n equations,
n unknowns.
244
00:14:21 --> 00:14:24
Let me write them in
matrix form, just so you
245
00:14:24 --> 00:14:26
see it that way too.
246
00:14:26 --> 00:14:28
c_1, c_2, c_3>.
247
00:14:28 --> 00:14:29
I'll make n=4.
248
00:14:30 --> 00:14:38
And then these are, this
convolution has c_0, c_2, c_1.
249
00:14:39 --> 00:14:43
c_3, c_2, c_1, c_0.
250
00:14:44 --> 00:14:51
This'll be c_3, c_3, c_3, I'm
writing down all the right
251
00:14:51 --> 00:14:53
numbers in the right places.
252
00:14:53 --> 00:14:56
So that when I do that
multiplication with the
253
00:14:56 --> 00:15:08
unknown, ,
I get the right-hand, the
254
00:15:08 --> 00:15:10
known right-hand side.
255
00:15:10 --> 00:15:12
Maybe b would be
a little better.
256
00:15:12 --> 00:15:15
Because we're more used
to b as as a known.
257
00:15:15 --> 00:15:20
It's just an Ax=b problem,
or an Au=b problem.
258
00:15:20 --> 00:15:25
But it looks like a convolution
but now it's just a
259
00:15:25 --> 00:15:26
matrix multiplication.
260
00:15:26 --> 00:15:29
So this is just .
261
00:15:30 --> 00:15:32
OK.
262
00:15:32 --> 00:15:34
That's our equation.
263
00:15:34 --> 00:15:36
Special type of matrix.
264
00:15:36 --> 00:15:38
Circulant matrix.
265
00:15:38 --> 00:15:43
So this is just literally
the same as c circularly
266
00:15:43 --> 00:15:46
convolved with u equals b.
267
00:15:46 --> 00:15:50
I just wrote it out
in matrix language.
268
00:15:50 --> 00:15:59
So you could call MATLAB with
that matrix, and so one way to
269
00:15:59 --> 00:16:04
answer it would be get the
inverse of the matrix.
270
00:16:04 --> 00:16:13
But if it was large, a better
way would be switch over
271
00:16:13 --> 00:16:16
to frequency space.
272
00:16:16 --> 00:16:16
Think, now.
273
00:16:16 --> 00:16:22
What happens when I
switch these vectors
274
00:16:22 --> 00:16:23
to frequency space?
275
00:16:23 --> 00:16:25
It becomes a multiplication.
276
00:16:25 --> 00:16:29
So this becomes a
multiplication.
277
00:16:29 --> 00:16:38
Now so c, I want the Fourier,
from the c's, what am I going
278
00:16:38 --> 00:16:46
to - so these are all in the
space where it's a convolution.
279
00:16:46 --> 00:16:49
What am I going to call it
where it's in the space where
280
00:16:49 --> 00:16:51
it's a multiplication?
281
00:16:51 --> 00:16:54
I just need three new names.
282
00:16:54 --> 00:17:01
Maybe I'll use c hat, u hat,
and b hat just because there's
283
00:17:01 --> 00:17:05
no doubt in anybody's mind that
when you see that hat, you've
284
00:17:05 --> 00:17:07
gone into frequency space.
285
00:17:07 --> 00:17:11
Now, what's the equation
in frequency space?
286
00:17:11 --> 00:17:15
And then I'll do an example.
287
00:17:15 --> 00:17:21
It's a multiplication, but
I don't usually see a
288
00:17:21 --> 00:17:27
vector and nothing there.
289
00:17:27 --> 00:17:31
What's the multiplication
in frequency space?
290
00:17:31 --> 00:17:37
It's component by
component. c_0*u_0=b_0.
291
00:17:42 --> 00:17:50
c hat 1 u hat 1 equals b
hat 1. c hat 2, u hat
292
00:17:50 --> 00:17:53
2, equals b hat 2.
293
00:17:53 --> 00:18:00
And finally, c hat 3, u
hat 3, equals b hat 3.
294
00:18:00 --> 00:18:04
And there might be, I don't
swear that there isn't,
295
00:18:04 --> 00:18:06
a 1/4 somewhere.
296
00:18:06 --> 00:18:07
Right?
297
00:18:07 --> 00:18:12
But the point is, we're
in frequency space now.
298
00:18:12 --> 00:18:19
We just have a component by
component, each component of c
299
00:18:19 --> 00:18:22
hat times each component of u
hat gives us a component of b
300
00:18:22 --> 00:18:26
hat; now we're ready for a
deconvolution; just divide.
301
00:18:26 --> 00:18:31
So now u hat, obviously I don't
have to write all these,
302
00:18:31 --> 00:18:34
b hat 0 over c hat 0.
303
00:18:34 --> 00:18:35
Right?
304
00:18:35 --> 00:18:36
I just do a division.
305
00:18:36 --> 00:18:44
So on down to u hat 3 is b hat,
is the third component of b,
306
00:18:44 --> 00:18:47
divided by the third
component of c.
307
00:18:47 --> 00:18:50
OK, now don't forget here.
308
00:18:50 --> 00:18:54
That in going from here to
here, I had to figure out
309
00:18:54 --> 00:18:57
what the c hats were, right?
310
00:18:57 --> 00:19:03
I had to do the Fourier matrix,
or the inverse Fourier matrix
311
00:19:03 --> 00:19:08
to go from c to c hat, from b
to b hat, so everything
312
00:19:08 --> 00:19:10
got Fourier transforms.
313
00:19:10 --> 00:19:17
But the object was to
make the equation easy.
314
00:19:17 --> 00:19:21
And of course, now we've got
four trivial equations that
315
00:19:21 --> 00:19:24
we just solved that way.
316
00:19:24 --> 00:19:27
Alright, let me see if I
can just pull this down
317
00:19:27 --> 00:19:32
with some questions.
318
00:19:32 --> 00:19:34
Here's a good question.
319
00:19:34 --> 00:19:39
When is a circulant
matrix invertible.
320
00:19:39 --> 00:19:42
When will this method work?
321
00:19:42 --> 00:19:45
The circulant matrix could
fail to be invertible.
322
00:19:45 --> 00:19:50
How would I know that?
323
00:19:50 --> 00:19:54
If it's singular, and how would
I, if I proceed this way,
324
00:19:54 --> 00:19:57
here I've got an answer.
325
00:19:57 --> 00:19:59
But if it's singular I'm
not really expecting
326
00:19:59 --> 00:20:00
to get an answer.
327
00:20:00 --> 00:20:03
Let me left the board a little.
328
00:20:03 --> 00:20:09
So where would I get, oops.
329
00:20:09 --> 00:20:14
Have to stop this method.
330
00:20:14 --> 00:20:16
In solving those
four equations.
331
00:20:16 --> 00:20:20
Where would I learn
that it's is singular?
332
00:20:20 --> 00:20:22
What could go wrong in this?
333
00:20:22 --> 00:20:23
Yes.
334
00:20:23 --> 00:20:25
AUDIENCE: [INAUDIBLE]
335
00:20:25 --> 00:20:26
PROFESSOR STRANG: That's right.
336
00:20:26 --> 00:20:32
Always in math, the question
is are you dividing by zero.
337
00:20:32 --> 00:20:36
So the question of whether the
matrix is singular, is the same
338
00:20:36 --> 00:20:43
as the question of whether c_0
hat, c_01, c_02, and c_03, -
339
00:20:43 --> 00:20:48
sorry, c_0 hat, c_1 hat, c_2
hat, and c_3 hat,
340
00:20:48 --> 00:20:49
can't be zero.
341
00:20:49 --> 00:20:55
That's, in fact, even better
those four numbers, those four
342
00:20:55 --> 00:21:00
c hats, are actually the
eigenvalues of the matrix.
343
00:21:00 --> 00:21:04
We've switched, what the
Fourier transform did, was
344
00:21:04 --> 00:21:08
switch over to the eigenvalues
and eigenvectors.
345
00:21:08 --> 00:21:12
And there, that's the whole
message of those guys is, you
346
00:21:12 --> 00:21:14
follow each one separately.
347
00:21:14 --> 00:21:15
Just the way we're doing here.
348
00:21:15 --> 00:21:19
So this is the component
of the b in the for
349
00:21:19 --> 00:21:21
eigenvector directions.
350
00:21:21 --> 00:21:26
Those are the four eigenvalues,
and I have to divide by them.
351
00:21:26 --> 00:21:30
You see, the idea is, like,
we've diagonalized the matrix.
352
00:21:30 --> 00:21:35
We've had that matrix,
which is full.
353
00:21:35 --> 00:21:41
And we take by taking the
Fourier transforms, that's the
354
00:21:41 --> 00:21:47
same thing as as putting in the
eigenvectors, switching the
355
00:21:47 --> 00:21:51
matrix to this diagonal
matrix, right?
356
00:21:51 --> 00:21:58
Our problem has become like the
diagonalized form is c_0 hat
357
00:21:58 --> 00:22:01
down to c_3 hat, sitting
on the diagonal.
358
00:22:01 --> 00:22:03
All zeroes elsewhere.
359
00:22:03 --> 00:22:06
That's when we switched, when
we did Fourier transform we
360
00:22:06 --> 00:22:08
were switching to eigenvectors.
361
00:22:08 --> 00:22:10
OK, so that's the message.
362
00:22:10 --> 00:22:17
That the test for singularity
is the Fourier, the
363
00:22:17 --> 00:22:21
transform of c hits zero.
364
00:22:21 --> 00:22:22
Then we're in trouble.
365
00:22:22 --> 00:22:25
Let me do an example you know.
366
00:22:25 --> 00:22:27
Let me do an example you know.
367
00:22:27 --> 00:22:33
OK here's, so finally now we
get a numerical example.
368
00:22:33 --> 00:22:38
The example we really
know is this one, right?
369
00:22:38 --> 00:22:41
As I start writing that,
you may say in your
370
00:22:41 --> 00:22:43
mind, oh no not again.
371
00:22:43 --> 00:22:47
But give it to me, one more
week with these matrices.
372
00:22:47 --> 00:22:54
But it'll be the C matrix, so
it's going to be the circulant.
373
00:22:54 --> 00:22:57
Recognize this?
374
00:22:57 --> 00:23:02
And it's got those minus
ones in the corners, too.
375
00:23:02 --> 00:23:05
OK, let's go back to day one.
376
00:23:05 --> 00:23:09
Is that matrix invertible?
377
00:23:09 --> 00:23:12
Yes or no.
378
00:23:12 --> 00:23:14
Please, no.
379
00:23:14 --> 00:23:17
Everybody knows that
matrix is not invertible.
380
00:23:17 --> 00:23:21
And do you remember what's
in the null space?
381
00:23:21 --> 00:23:25
Yes, what's the vector in the
null space of that matrix?
382
00:23:25 --> 00:23:27
All ones.
383
00:23:27 --> 00:23:30
Now, when I take,
just think now.
384
00:23:30 --> 00:23:35
When I take Fourier transform,
that all ones is going
385
00:23:35 --> 00:23:38
to transform to what?
386
00:23:38 --> 00:23:40
It's going to transform
to the delta.
387
00:23:40 --> 00:23:45
It'll transform to the one
that is like, .
388
00:23:45 --> 00:23:47
Or maybe it's .
389
00:23:47 --> 00:23:56
But it's that, well, OK,
now I'm ready to take.
390
00:23:56 --> 00:24:01
So here's my c.
391
00:24:01 --> 00:24:02
So what's my method now?
392
00:24:02 --> 00:24:07
I'm going to do this method,
and I'm going to run into, this
393
00:24:07 --> 00:24:09
thing is going to be zero.
394
00:24:09 --> 00:24:13
Because that's the eigenvalue
that goes with the <1, 1, 1, 1,
395
00:24:13 --> 00:24:18
1> column, the constant, the
zero frequency in
396
00:24:18 --> 00:24:19
frequency space.
397
00:24:19 --> 00:24:20
You'll see it happen.
398
00:24:20 --> 00:24:24
So let's take the Fourier
transform of that.
399
00:24:24 --> 00:24:27
And then we would have to take
the Fourier transform of the
400
00:24:27 --> 00:24:29
right-hand side, b, whatever
that happened to be.
401
00:24:29 --> 00:24:31
But it's always the left side.
402
00:24:31 --> 00:24:33
The singular or not matrix.
403
00:24:33 --> 00:24:35
I believe we'll be
singular here.
404
00:24:35 --> 00:24:42
So, OK, just remind me how do I
take transforms of this guy?
405
00:24:42 --> 00:24:44
Gosh, we have to be
able to do that.
406
00:24:44 --> 00:24:48
That's Section 4.-
well, 4.3 isn't, yeah.
407
00:24:48 --> 00:24:53
The DFT of that vector.
408
00:24:53 --> 00:24:56
What do I get?
409
00:24:56 --> 00:24:58
Yes.
410
00:24:58 --> 00:25:02
How do I take the
DFT of a vector?
411
00:25:02 --> 00:25:05
I multiply by the
Fourier matrix, right?
412
00:25:05 --> 00:25:06
Yes.
413
00:25:06 --> 00:25:10
So I have to multiply that
thing by the Fourier matrix.
414
00:25:10 --> 00:25:17
So to get c hat, this was big C
for the matrix, little c for
415
00:25:17 --> 00:25:21
the vector that goes into
it, into column zero.
416
00:25:21 --> 00:25:24
And c hat for its transform.
417
00:25:24 --> 00:25:29
OK, so now here comes the
Fourier matrix that we know, 1,
418
00:25:29 --> 00:25:34
i, i^2, i^3, 1, i^2, i^4, i^6.
419
00:25:35 --> 00:25:38
1, i^3, i^6, and i^9.
420
00:25:39 --> 00:25:46
So I want to transform that c
to get, to find out c hat.
421
00:25:46 --> 00:25:52
OK, and what do I get up there?
422
00:25:52 --> 00:25:58
What's the first component, the
zeroth component I should say,
423
00:25:58 --> 00:26:03
when I take this guy, this
four, this vector with four
424
00:26:03 --> 00:26:07
components and I get back four
components, the frequency
425
00:26:07 --> 00:26:10
components, what's
the first one?
426
00:26:10 --> 00:26:13
Ones times this,
what am I getting?
427
00:26:13 --> 00:26:14
Zero.
428
00:26:14 --> 00:26:16
That's what I expected.
429
00:26:16 --> 00:26:20
That tells me the matrix is
not going to be invertible.
430
00:26:20 --> 00:26:27
Because in a different
language, I'm finding
431
00:26:27 --> 00:26:30
the eigenvalues and
that's one of them.
432
00:26:30 --> 00:26:34
And if an eigenvalue is zero,
that means the eigenvector is
433
00:26:34 --> 00:26:36
getting knocked out completely.
434
00:26:36 --> 00:26:40
And there's snow way a c
inverse could recover when
435
00:26:40 --> 00:26:42
that eigenvector is gone.
436
00:26:42 --> 00:26:43
OK, let's do the other ones.
437
00:26:43 --> 00:26:49
Two minus i, nothing.
438
00:26:49 --> 00:26:51
What's the other one here?
439
00:26:51 --> 00:26:56
Two, this is minus i, and
that's plus i, I think.
440
00:26:56 --> 00:26:59
So I think it's just two.
441
00:26:59 --> 00:27:02
Alright, this is 2 i squared,
can I write in some of these
442
00:27:02 --> 00:27:07
just so I have a little, i
squared is minus one, and i^4
443
00:27:07 --> 00:27:10
is one, and that's minus one.
444
00:27:10 --> 00:27:15
So that's two plus one,
plus one, I think is four.
445
00:27:15 --> 00:27:20
And then i^3 is the
same as minus i.
446
00:27:20 --> 00:27:22
And i^9 is the same as plus i.
447
00:27:22 --> 00:27:34
So I think I'm getting two plus
i, nothing, minus i, two.
448
00:27:34 --> 00:27:36
So what's my claim?
449
00:27:36 --> 00:27:39
My claim is that these are
the four eigenvalues that
450
00:27:39 --> 00:27:43
the Fourier - Fourier
diagonalizes these problems.
451
00:27:43 --> 00:27:44
That's what it comes to.
452
00:27:44 --> 00:27:49
Fourier diagonalizes all
constant coefficients, shift
453
00:27:49 --> 00:27:51
invariant, linear problems.
454
00:27:51 --> 00:27:55
And tells us here
are eigenvalues.
455
00:27:55 --> 00:27:57
So .
456
00:27:57 --> 00:28:00
Would you like to, how do
I check the eigenvalues
457
00:28:00 --> 00:28:00
of a matrix?
458
00:28:00 --> 00:28:02
Let's just remember.
459
00:28:02 --> 00:28:05
If I give you four numbers
and I say those are the
460
00:28:05 --> 00:28:09
eigenvalues, and you look
at that matrix, what quick
461
00:28:09 --> 00:28:12
check does everybody do?
462
00:28:12 --> 00:28:15
Compute the - the trace.
463
00:28:15 --> 00:28:19
Add up the diagonal of the
matrix, add up the proposed
464
00:28:19 --> 00:28:21
eigenvalues, they had
better be the same.
465
00:28:21 --> 00:28:22
And they are.
466
00:28:22 --> 00:28:23
I get eight both ways.
467
00:28:23 --> 00:28:26
That doesn't mean, of course,
that these four numbers are
468
00:28:26 --> 00:28:28
right, but I think they are.
469
00:28:28 --> 00:28:29
Yeah, yeah.
470
00:28:29 --> 00:28:31
So those added up to
eight, those numbers
471
00:28:31 --> 00:28:33
added up to eight.
472
00:28:33 --> 00:28:36
And yep.
473
00:28:36 --> 00:28:38
And these are real.
474
00:28:38 --> 00:28:40
They came out real, and
how did I know that would
475
00:28:40 --> 00:28:44
happen from the matrix?
476
00:28:44 --> 00:28:51
What matrices am I certain to
get real eigenvalues for?
477
00:28:51 --> 00:28:51
Symmetric.
478
00:28:51 --> 00:28:52
Right.
479
00:28:52 --> 00:28:55
Now, what about - I heard
the word positive.
480
00:28:55 --> 00:28:58
Of course, that's the other
question I have to ask.
481
00:28:58 --> 00:29:03
Is this matrix
positive definite?
482
00:29:03 --> 00:29:05
OK, everybody this is
the language we've
483
00:29:05 --> 00:29:06
learned in 18.085.
484
00:29:06 --> 00:29:09
Is that matrix positive
definite, yes or no?
485
00:29:09 --> 00:29:11
No.
486
00:29:11 --> 00:29:12
What is it?
487
00:29:12 --> 00:29:15
It's positive semi-definite.
488
00:29:15 --> 00:29:18
What does that tell me
about eigenvalues?
489
00:29:18 --> 00:29:22
Their one is zero, that's why
it's not positive definite.
490
00:29:22 --> 00:29:24
But the others are positive.
491
00:29:24 --> 00:29:28
So that sure enough, in other
words, what we've done here,
492
00:29:28 --> 00:29:32
for that matrix that came
on day one and now we're
493
00:29:32 --> 00:29:38
seeing it on day N-1 here.
494
00:29:38 --> 00:29:42
We're we're seeing sort of in a
new way, because at that time
495
00:29:42 --> 00:29:47
we didn't know these four were
the eigenvectors
496
00:29:47 --> 00:29:49
of that matrix.
497
00:29:49 --> 00:29:50
But they are.
498
00:29:50 --> 00:29:56
And we're coming to the
same conclusion we
499
00:29:56 --> 00:29:58
came to on day one.
500
00:29:58 --> 00:30:03
That the matrix is positive
semi-definite and that
501
00:30:03 --> 00:30:05
we know its eigenvalues.
502
00:30:05 --> 00:30:09
And we can, actually, let
me even take it one more
503
00:30:09 --> 00:30:13
step, just because this
example is so perfect.
504
00:30:13 --> 00:30:17
Some right-hand sides we
could solve for, right?
505
00:30:17 --> 00:30:22
If I have a matrix that's
singular, way, way back.
506
00:30:22 --> 00:30:25
Even, I think it was like a
worked example in Section 1.1,
507
00:30:25 --> 00:30:32
I could ask the question
when is Cx=b solvable.
508
00:30:32 --> 00:30:34
Because there are some
right-hand sides that'll work.
509
00:30:34 --> 00:30:37
Because if I just take an x and
multiply by C, I'll get a
510
00:30:37 --> 00:30:40
right-hand side that works.
511
00:30:40 --> 00:30:48
But for which vectors
right-hand sides, b,
512
00:30:48 --> 00:30:52
well my method work?
513
00:30:52 --> 00:30:54
The ones that have...?
514
00:30:54 --> 00:30:56
Yeah, the ones that have which?
515
00:30:56 --> 00:31:02
What do I need with these c's,
c_0, c_1, c_2, and c_3, for
516
00:31:02 --> 00:31:06
my solution to be possible.
517
00:31:06 --> 00:31:09
I need b_0 hat.
518
00:31:09 --> 00:31:11
Equals zero.
519
00:31:11 --> 00:31:13
I need b_0 hat equals zero.
520
00:31:13 --> 00:31:15
And then what does that say?
521
00:31:15 --> 00:31:20
That means that the b, the
vector b, has no constant
522
00:31:20 --> 00:31:22
term in the Fourier series.
523
00:31:22 --> 00:31:26
It means that the vector b
is orthogonal to the <1,
524
00:31:26 --> 00:31:29
1, 1, 1>, eigenvector.
525
00:31:29 --> 00:31:34
So this is like a subtle point
but just driving home the point
526
00:31:34 --> 00:31:39
that what Fourier does is
diagonalize everything.
527
00:31:39 --> 00:31:43
It diagonalizes all the
important problems of, all
528
00:31:43 --> 00:31:47
the simplest problems, of
differential equations.
529
00:31:47 --> 00:31:52
You know, I mean this is
like 18.03 looked at from
530
00:31:52 --> 00:31:53
Fourier's point of view.
531
00:31:53 --> 00:31:57
OK, what more could I
do with that equation?
532
00:31:57 --> 00:32:00
I think you really are seeing
all the good stuff here.
533
00:32:00 --> 00:32:02
You're seeing the matrix.
534
00:32:02 --> 00:32:05
We're recognizing
it as a circulant.
535
00:32:05 --> 00:32:08
We're realizing that we could
take its Fourier transform.
536
00:32:08 --> 00:32:11
We get the eigenvalues.
537
00:32:11 --> 00:32:15
We're diagonalizing the matrix,
the convolution becomes a
538
00:32:15 --> 00:32:19
multiplication, and the
solution becomes, inversion
539
00:32:19 --> 00:32:22
becomes division.
540
00:32:22 --> 00:32:26
I hope you see that.
541
00:32:26 --> 00:32:30
That's really a model
problem for this course.
542
00:32:30 --> 00:32:33
OK. yeah.
543
00:32:33 --> 00:32:33
Questions.
544
00:32:33 --> 00:32:41
Good.
545
00:32:41 --> 00:32:42
AUDIENCE: [INAUDIBLE]
546
00:32:42 --> 00:32:44
PROFESSOR STRANG: Would I give
you a six by six Fourier
547
00:32:44 --> 00:32:45
matrix on a test?
548
00:32:45 --> 00:32:46
Probably not.
549
00:32:46 --> 00:32:48
No.
550
00:32:48 --> 00:32:49
I just about could.
551
00:32:49 --> 00:32:55
I mean, it's, six by six, those
are pretty decent numbers.
552
00:32:55 --> 00:32:56
Right.
553
00:32:56 --> 00:32:59
Those six roots of unit
e, but not quite.
554
00:32:59 --> 00:33:00
Right, yeah.
555
00:33:00 --> 00:33:01
Yeah.
556
00:33:01 --> 00:33:01
Yeah.
557
00:33:01 --> 00:33:04
So four by four is,
five by five would not
558
00:33:04 --> 00:33:05
be nice, certainly.
559
00:33:05 --> 00:33:09
Who knows the cosine
of 72 degrees?
560
00:33:09 --> 00:33:10
Crazy.
561
00:33:10 --> 00:33:13
But, at 60 degrees we could do.
562
00:33:13 --> 00:33:17
So the Fourier matrix would
be full of square roots of
563
00:33:17 --> 00:33:20
three over two, and one over
two, an i's, and so on.
564
00:33:20 --> 00:33:24
But it wouldn't be, so
really four by four
565
00:33:24 --> 00:33:26
is sort of the model.
566
00:33:26 --> 00:33:28
Yeah, yeah.
567
00:33:28 --> 00:33:31
So four by four is that model.
568
00:33:31 --> 00:33:33
Other questions?
569
00:33:33 --> 00:33:35
Because this is really
a key example.
570
00:33:35 --> 00:33:37
Yeah.
571
00:33:37 --> 00:33:41
When I calculated the
eigenvalues, yeah.
572
00:33:41 --> 00:33:42
Ah.
573
00:33:42 --> 00:33:46
Because this matrix, I know
everything about that matrix
574
00:33:46 --> 00:33:47
when I know its first vector.
575
00:33:47 --> 00:33:49
AUDIENCE: [INAUDIBLE]
576
00:33:49 --> 00:33:51
PROFESSOR STRANG: Yeah,
it's because it's a
577
00:33:51 --> 00:33:52
circulant matrix.
578
00:33:52 --> 00:33:57
It's because that matrix is
expressing convolution with
579
00:33:57 --> 00:33:59
this vector. .
580
00:33:59 --> 00:34:03
581
00:34:03 --> 00:34:07
That circulant matrix
essentially is built from
582
00:34:07 --> 00:34:09
four numbers, right.
583
00:34:09 --> 00:34:09
Yeah.
584
00:34:09 --> 00:34:11
Yeah, and they go in
the zeroth column.
585
00:34:11 --> 00:34:12
Right, yeah.
586
00:34:12 --> 00:34:14
Yeah.
587
00:34:14 --> 00:34:20
Right, so there is an
example where we could
588
00:34:20 --> 00:34:21
like do everything.
589
00:34:21 --> 00:34:27
Now, and let me just remember
that with this example,
590
00:34:27 --> 00:34:29
we could do everything.
591
00:34:29 --> 00:34:37
So this is an example of, you
could say this type of problem.
592
00:34:37 --> 00:34:42
But with a very special kernel
there, so it turned out to be,
593
00:34:42 --> 00:34:45
it looks like an integral
equation here but if that
594
00:34:45 --> 00:34:50
kernel involves delta functions
and so on then it can be just
595
00:34:50 --> 00:34:51
a differential equation.
596
00:34:51 --> 00:34:53
And then that's
what we got there.
597
00:34:53 --> 00:34:59
So we took all the same steps
we did here, we did here.
598
00:34:59 --> 00:35:06
We took the Fourier transform,
and I emphasize there, just to
599
00:35:06 --> 00:35:09
remember Wednesday, this
was a delta function.
600
00:35:09 --> 00:35:13
When I took the Fourier
transform I got a one, so
601
00:35:13 --> 00:35:16
this was a one over this.
602
00:35:16 --> 00:35:21
And I did the inverse transform
and I got back to the
603
00:35:21 --> 00:35:24
function that I drew.
604
00:35:24 --> 00:35:28
Which was e^(-ax) over 2a.
605
00:35:29 --> 00:35:31
And even.
606
00:35:31 --> 00:35:36
So, yeah. this was
the answer u(x).
607
00:35:38 --> 00:35:41
So I was able to do that, I
mean this step was easy,
608
00:35:41 --> 00:35:43
that step is easy.
609
00:35:43 --> 00:35:45
That step is easy, the
division is easy.
610
00:35:45 --> 00:35:55
And then I just recognize this
as the transform of this one,
611
00:35:55 --> 00:35:57
this example that we had done.
612
00:35:57 --> 00:35:58
Once I divided by 2a.
613
00:35:59 --> 00:36:01
So you should be
able to do this.
614
00:36:01 --> 00:36:05
So those are two that you
should really be able to do.
615
00:36:05 --> 00:36:08
I'm not going to, obviously I'm
not going to, ask you a 2-D
616
00:36:08 --> 00:36:13
problem on the exam or
even on a homework.
617
00:36:13 --> 00:36:17
But now if you'll allow me,
I'd like to spend a few
618
00:36:17 --> 00:36:20
minutes to get into 2-D.
619
00:36:20 --> 00:36:25
Because really, you've got
the main thoughts here.
620
00:36:25 --> 00:36:28
That Fourier is the same
as finding eigenvectors
621
00:36:28 --> 00:36:30
and eigenvalues.
622
00:36:30 --> 00:36:35
That's the main thought
for these LTI problems.
623
00:36:35 --> 00:36:40
OK, now suppose I have, let's
just get the formalities
624
00:36:40 --> 00:36:41
straight here.
625
00:36:41 --> 00:36:45
Suppose I have a
function of x and y.
626
00:36:45 --> 00:36:48
2pi periodic in x, and in y.
627
00:36:48 --> 00:36:55
So if I bump x by 2pi, or if I
bump y by 2pi - oh, I'm using
628
00:36:55 --> 00:36:59
capital F for the
periodic guys.
629
00:36:59 --> 00:37:01
So let me stay with
capital F(x,y+2pi).
630
00:37:01 --> 00:37:04
631
00:37:04 --> 00:37:08
OK, so I have a function.
632
00:37:08 --> 00:37:10
This is given.
633
00:37:10 --> 00:37:13
This is, it's in 2-D now.
634
00:37:13 --> 00:37:16
And I want to write
its Fourier series.
635
00:37:16 --> 00:37:19
So I'm just asking the question
what does the Fourier series
636
00:37:19 --> 00:37:25
look like for a function
of two variables.
637
00:37:25 --> 00:37:28
The point is, it's going
to be a nice answer.
638
00:37:28 --> 00:37:33
And so everything, what
you know how to do in
639
00:37:33 --> 00:37:35
1-D you can do in 2-D.
640
00:37:35 --> 00:37:41
So let me write the complex
form, the e^(ik) stuff.
641
00:37:41 --> 00:37:44
So what would I write,
how would I write this?
642
00:37:44 --> 00:37:49
I would write that as a sum,
but it'll have, I'll make it a
643
00:37:49 --> 00:37:53
double sum, I'll write two
sigmas just to emphasize that
644
00:37:53 --> 00:37:58
we're summing from k equal
minus infinity, to infinity,
645
00:37:58 --> 00:38:02
and from l equal minus
infinity to infinity.
646
00:38:02 --> 00:38:03
We have coefficients c_kl.
647
00:38:03 --> 00:38:07
648
00:38:07 --> 00:38:11
They depend on two indices,
this is the pattern to know.
649
00:38:11 --> 00:38:15
Multiplying our e^(ikx),
and our e^(ily).
650
00:38:15 --> 00:38:19
651
00:38:19 --> 00:38:23
Right, good.
652
00:38:23 --> 00:38:25
So, alright.
653
00:38:25 --> 00:38:27
Let me ask you.
654
00:38:27 --> 00:38:29
How would I find c_23?
655
00:38:30 --> 00:38:35
Just to know that - we could
find all these coefficients,
656
00:38:35 --> 00:38:36
find formulas for them.
657
00:38:36 --> 00:38:39
We could do examples.
658
00:38:39 --> 00:38:40
How would I find c_23?
659
00:38:41 --> 00:38:45
So this is my F.
660
00:38:45 --> 00:38:47
I know F.
661
00:38:47 --> 00:38:48
I want to find c_23.
662
00:38:49 --> 00:38:52
What's the magic trick?
663
00:38:52 --> 00:38:55
Then and a 2pi periodic,
so all integrals.
664
00:38:55 --> 00:38:58
All the integrals, and I'm
giving you a hint, of course.
665
00:38:58 --> 00:39:01
I'm going to integrate.
666
00:39:01 --> 00:39:04
And the integrals will
all go from minus pi
667
00:39:04 --> 00:39:06
to pi in x, and in y.
668
00:39:06 --> 00:39:09
They'll integrate over
the period square.
669
00:39:09 --> 00:39:14
Here's the period square
from, there's the center. x
670
00:39:14 --> 00:39:21
direction, y direction, goes
out to pi and goes up to pi.
671
00:39:21 --> 00:39:24
So all integrals
will be over dxdy.
672
00:39:26 --> 00:39:28
But what do I integrate?
673
00:39:28 --> 00:39:29
To find c_23.
674
00:39:29 --> 00:39:33
675
00:39:33 --> 00:39:36
Well, these guys
are orthogonal.
676
00:39:36 --> 00:39:39
That's what's making everything
work, they're orthogonal
677
00:39:39 --> 00:39:40
and very special.
678
00:39:40 --> 00:39:43
So that by u's orthogonality,
what do I do?
679
00:39:43 --> 00:39:46
I multiply by?
680
00:39:46 --> 00:39:50
Just tell me what
to multiply by.
681
00:39:50 --> 00:39:57
By this and integrate.
682
00:39:57 --> 00:40:04
OK, what is it that I multiply
by if I'm shooting for c_23,
683
00:40:04 --> 00:40:13
for example? e^(i2x),
is it e^(i2x)?
684
00:40:15 --> 00:40:17
Minus, right.
685
00:40:17 --> 00:40:27
I multiply by e^(-i2x),
e^(-i3y), and integrate.
686
00:40:27 --> 00:40:28
Yeah.
687
00:40:28 --> 00:40:32
So when I multiply by that and
integrate, everything will
688
00:40:32 --> 00:40:35
go except the c_23 term.
689
00:40:35 --> 00:40:38
Which will be
multiplied by what?
690
00:40:38 --> 00:40:43
So I'll just have c_23 times
probably 2pi squared.
691
00:40:43 --> 00:40:46
I guess 2pi will come in
from both integrals, so
692
00:40:46 --> 00:40:48
the formula will be c_kl.
693
00:40:49 --> 00:40:52
c_kl will be, do you want
me to write this formula?
694
00:40:52 --> 00:40:56
I'll write it here and then
forget it right away. c_kl
695
00:40:56 --> 00:41:00
will be one over 2pi squared.
696
00:41:00 --> 00:41:04
The integral of my function.
697
00:41:04 --> 00:41:07
Times my e^(-ikx),
times my e^(-ily)dxdy.
698
00:41:07 --> 00:41:15
699
00:41:15 --> 00:41:17
So that just makes the point.
700
00:41:17 --> 00:41:22
That there's nothing new here,
it's just up a dimension.
701
00:41:22 --> 00:41:28
But the formulas all look the
same, and if f was a- well, if
702
00:41:28 --> 00:41:34
F is a delta function, if f
is now a 2-D delta function.
703
00:41:34 --> 00:41:38
We haven't done delta functions
in 2-D, why don't we?
704
00:41:38 --> 00:41:45
Suppose F is the delta
function in 2-D.
705
00:41:45 --> 00:41:49
Then what are the coefficients?
706
00:41:49 --> 00:41:51
What do you think you
this means, this delta
707
00:41:51 --> 00:41:53
function in 2-D?
708
00:41:53 --> 00:41:57
So if I put in the delta
here, and I integrate.
709
00:41:57 --> 00:42:01
And what do I get then?
710
00:42:01 --> 00:42:06
So if this guy is a delta, a
two-dimensional delta function,
711
00:42:06 --> 00:42:12
the rule is that when I
integrate over a region that
712
00:42:12 --> 00:42:17
includes the spike, so it's a
spike sitting up above a plane
713
00:42:17 --> 00:42:20
now, instead of sitting above a
line it's sitting
714
00:42:20 --> 00:42:21
above a plane.
715
00:42:21 --> 00:42:24
Then I get the value,
so this is the delta
716
00:42:24 --> 00:42:27
function at the origin.
717
00:42:27 --> 00:42:29
So I get the value of
this at the origin.
718
00:42:29 --> 00:42:32
So what answer do I get?
719
00:42:32 --> 00:42:34
I get one out of the
integral and then I just
720
00:42:34 --> 00:42:35
have this constant.
721
00:42:35 --> 00:42:38
So it's constant again.
722
00:42:38 --> 00:42:41
And it's just one.
723
00:42:41 --> 00:42:45
So the Fourier coefficients
of the delta function
724
00:42:45 --> 00:42:46
are constant.
725
00:42:46 --> 00:42:49
All frequencies
there are the same.
726
00:42:49 --> 00:42:53
What about a line of
delta functions?
727
00:42:53 --> 00:42:54
And what does that mean?
728
00:42:54 --> 00:43:00
What about, yeah let me
try to draw delta(x).
729
00:43:01 --> 00:43:07
Suppose I have a function of
x and y - it's just worth
730
00:43:07 --> 00:43:12
imagining a line of
delta functions.
731
00:43:12 --> 00:43:21
So I'm in the x,y, let me
look again at this thing.
732
00:43:21 --> 00:43:25
I have delta functions
all along this line.
733
00:43:25 --> 00:43:26
Now.
734
00:43:26 --> 00:43:29
Here is a crazy example,
just to say well there
735
00:43:29 --> 00:43:32
is something new in 2-D.
736
00:43:32 --> 00:43:36
So previously my delta function
was just at that point.
737
00:43:36 --> 00:43:38
And the all integrals just
picked out the value
738
00:43:38 --> 00:43:40
at that point.
739
00:43:40 --> 00:43:44
But now think of a delta
function's a sort
740
00:43:44 --> 00:43:46
of line of spikes.
741
00:43:46 --> 00:43:48
Going up here, and then
of course it's periodic.
742
00:43:48 --> 00:43:52
Everything's periodic so that
line continues and this line
743
00:43:52 --> 00:43:55
appears here, and this
line appears here.
744
00:43:55 --> 00:44:02
But I only have to focus
on one period square.
745
00:44:02 --> 00:44:04
What's my answer now?
746
00:44:04 --> 00:44:10
If this function suddenly
changes from a one point
747
00:44:10 --> 00:44:13
delta function to a line
of delta functions?
748
00:44:13 --> 00:44:17
Now tell me what the
coefficients are.
749
00:44:17 --> 00:44:21
What are the Fourier
coefficients in 2-D for a
750
00:44:21 --> 00:44:22
line of delta functions?
751
00:44:22 --> 00:44:28
A straight line of delta
functions going up the y axis?
752
00:44:28 --> 00:44:32
It'll be - let's see.
753
00:44:32 --> 00:44:34
What do I do?
754
00:44:34 --> 00:44:39
I'm go to integrate -
oh yeah, what is it?
755
00:44:39 --> 00:44:40
Good question.
756
00:44:40 --> 00:44:42
OK.
757
00:44:42 --> 00:44:47
So what is the - when do
I get zero and when do I
758
00:44:47 --> 00:44:49
not get zero out of this?
759
00:44:49 --> 00:44:54
Yeah, tell me first when do I
get zero out of this integral?
760
00:44:54 --> 00:44:56
And when do I not?
761
00:44:56 --> 00:45:00
What am I doing here?
762
00:45:00 --> 00:45:01
Help me.
763
00:45:01 --> 00:45:06
I said 2-D was easy and I've
got in over my head here.
764
00:45:06 --> 00:45:10
So look.
765
00:45:10 --> 00:45:12
I can do the x integral, right?
766
00:45:12 --> 00:45:15
We all know how to
do the x integral.
767
00:45:15 --> 00:45:18
Yes, is that right?
768
00:45:18 --> 00:45:22
If I integrate with respect
to x, what do I get?
769
00:45:22 --> 00:45:24
Let's see, I'll keep that
one over 2pi squared.
770
00:45:24 --> 00:45:28
Now I'm trying to
do this integral.
771
00:45:28 --> 00:45:34
Do I get a one?
772
00:45:34 --> 00:45:37
If I get a one from
the x integral.
773
00:45:37 --> 00:45:41
So then I'm down to one
integral, just the y integral
774
00:45:41 --> 00:45:47
is left. e^(-ily)dy, right?
775
00:45:47 --> 00:45:51
I did the x part, which
said, OK, take the value
776
00:45:51 --> 00:45:53
at x=0, which was one.
777
00:45:53 --> 00:45:57
So the x integral
was one, good.
778
00:45:57 --> 00:45:59
And now I've got
down to this part.
779
00:45:59 --> 00:46:03
Now what is that integral?
780
00:46:03 --> 00:46:10
It's two - wait a minute.
781
00:46:10 --> 00:46:16
Depends on l, doesn't it?
782
00:46:16 --> 00:46:18
When l - yeah.
783
00:46:18 --> 00:46:23
So it's going to depend on
whether l is zero or not.
784
00:46:23 --> 00:46:26
Is that right?
785
00:46:26 --> 00:46:28
Yeah, that's sort
of interesting.
786
00:46:28 --> 00:46:32
If l is zero, then I'm getting
- then this is a 2pi.
787
00:46:33 --> 00:46:40
So the answer, so I'm getting
c_k0, when l is zero I'm
788
00:46:40 --> 00:46:42
getting a 2pi out of that.
789
00:46:42 --> 00:46:45
If l is zero, I'm integrating
one, I get a 2pi
790
00:46:45 --> 00:46:46
cancels one of those.
791
00:46:46 --> 00:46:48
I get a one over 2pi.
792
00:46:48 --> 00:46:51
793
00:46:51 --> 00:46:59
And otherwise the other c_kl's,
when l is not zero, are what?
794
00:46:59 --> 00:47:00
Just zero, I think.
795
00:47:00 --> 00:47:03
The integral of this thing,
this is a periodic guy, if I
796
00:47:03 --> 00:47:07
integrate it from minus
pi to pi it's zero.
797
00:47:07 --> 00:47:12
What am I - I'm making a big
deal out of something that
798
00:47:12 --> 00:47:14
shouldn't be a big deal.
799
00:47:14 --> 00:47:20
The delta(x) function, this
is just, its Fourier series
800
00:47:20 --> 00:47:23
is just the one we know.
801
00:47:23 --> 00:47:25
Sum of e^(ikx)'s.
802
00:47:25 --> 00:47:31
803
00:47:31 --> 00:47:33
Do you see what's
happened here?
804
00:47:33 --> 00:47:36
It was supposed to
be a double sum.
805
00:47:36 --> 00:47:41
But the ones, when l
wasn't zero aren't there.
806
00:47:41 --> 00:47:42
The only ones - yeah.
807
00:47:42 --> 00:47:48
So I'm back to the, for a line
of spikes, a line of deltas,
808
00:47:48 --> 00:47:51
I'm back to - it so it only
depended on x, so the
809
00:47:51 --> 00:47:54
Fourier series is just
the one I already know.
810
00:47:54 --> 00:47:56
All ones.
811
00:47:56 --> 00:48:01
When there's no - when l is
zero, all ones or 1/2 2pi's,
812
00:48:01 --> 00:48:08
all constants when l is
zero, but there's no y.
813
00:48:08 --> 00:48:11
There's no oscillation
in the y direction.
814
00:48:11 --> 00:48:18
OK, I don't know why I got into
that example, because the
815
00:48:18 --> 00:48:22
conclusion was just it's the
Fourier series that we already
816
00:48:22 --> 00:48:26
know and it doesn't depend on
l, because the function
817
00:48:26 --> 00:48:28
didn't depend on y.
818
00:48:28 --> 00:48:33
OK, then we could imagine delta
functions in other positions,
819
00:48:33 --> 00:48:36
or a general function.
820
00:48:36 --> 00:48:39
OK, so that's 2-D.
821
00:48:39 --> 00:48:44
Would I want to
tackle a 2-D - ha.
822
00:48:44 --> 00:48:48
We've got two minutes. that's
one dimension a minute.
823
00:48:48 --> 00:48:50
Right, OK.
824
00:48:50 --> 00:48:57
What happens, what's a 2-D
discrete convolution?
825
00:48:57 --> 00:49:00
What's a 2-D discrete
convolution?
826
00:49:00 --> 00:49:02
Now, you might say OK,
why is Professor Strang
827
00:49:02 --> 00:49:04
inventing these problems?
828
00:49:04 --> 00:49:07
Because a 2-D discrete
convolution is the core
829
00:49:07 --> 00:49:10
idea of image processing.
830
00:49:10 --> 00:49:14
If I have an image, what
does image processing do?
831
00:49:14 --> 00:49:19
Image processing takes my
image, it separates it into
832
00:49:19 --> 00:49:23
pixels, right, that's all the
image is, bunch of pixels.
833
00:49:23 --> 00:49:30
Then many 2-D image processing
algorithms, we'll jpeg, all
834
00:49:30 --> 00:49:35
jpeg for example, would take
an eight by eight, eight by
835
00:49:35 --> 00:49:37
eight 2-D, in other words.
836
00:49:37 --> 00:49:41
Eight by eight, 2-D
is the main point.
837
00:49:41 --> 00:49:42
Set of pixels.
838
00:49:42 --> 00:49:43
And transform it.
839
00:49:43 --> 00:49:46
Do a 2-D transform.
840
00:49:46 --> 00:49:48
So what is a 2-D transform?
841
00:49:48 --> 00:49:56
What would be the 2-D transform
that would correspond to this?
842
00:49:56 --> 00:49:59
First of all how
big's the matrix?
843
00:49:59 --> 00:50:01
Just so we get an idea.
844
00:50:01 --> 00:50:04
I probably won't get to
the end of this example.
845
00:50:04 --> 00:50:10
But just, so in 1-D, my
matrix was four by four.
846
00:50:10 --> 00:50:15
Now I've got, so that was
for four points on a line.
847
00:50:15 --> 00:50:22
Now I've got a
square of points.
848
00:50:22 --> 00:50:25
So how big is my matrix?
849
00:50:25 --> 00:50:26
16, right?
850
00:50:26 --> 00:50:32
16 by 16, because it's
operating on 16 pixels.
851
00:50:32 --> 00:50:37
It's operating on 16 pixels,
where in 1-D it only
852
00:50:37 --> 00:50:38
had four to act on.
853
00:50:38 --> 00:50:42
So I'm going to end up with
a 16 by 16 matrix here.
854
00:50:42 --> 00:50:48
And I think - let me
see, what do I need?
855
00:50:48 --> 00:50:49
Oh, wait a minute.
856
00:50:49 --> 00:50:51
Uh-oh.
857
00:50:51 --> 00:50:54
Yeah, I think the
time's up here.
858
00:50:54 --> 00:50:59
Yeah, because my C, has my C
got to have 16 components?
859
00:50:59 --> 00:51:00
Yes.
860
00:51:00 --> 00:51:03
My u has to have 16, my
right-hand side has got
861
00:51:03 --> 00:51:05
these 16 components.
862
00:51:05 --> 00:51:05
Yeah.
863
00:51:05 --> 00:51:09
So I'm up to 16 but a
very special circulant
864
00:51:09 --> 00:51:11
of a circulant.
865
00:51:11 --> 00:51:13
It'll be a circulant of
a circulant somehow.
866
00:51:13 --> 00:51:16
OK, enough for 2-D.
867
00:51:16 --> 00:51:18
I'll see you Wednesday and
we're back to reality.
868
00:51:18 --> 00:51:19
OK.
869
00:51:19 --> 00:51:20