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PROFESSOR STRANG: OK, hi.
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00:00:21 --> 00:00:25
So I've got homework
nine for you.
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Ready to return at the end.
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Also, the department asked me
to do evaluations, but that's
13
00:00:32 --> 00:00:33
the end of the lecture.
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00:00:33 --> 00:00:39
Then, so everybody knows
there's a quiz tomorrow night.
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00:00:39 --> 00:00:45
And shall I just remember the
four questions on the quiz?
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00:00:45 --> 00:00:49
I mean, not the details but the
general idea of the questions.
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00:00:49 --> 00:00:51
Details OK too.
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00:00:51 --> 00:00:53
Yes.
19
00:00:53 --> 00:00:58
Yeah, so there'll be one
question on a Fourier series.
20
00:00:58 --> 00:01:05
And you should know the energy
equality for all of these
21
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possibilities, connecting the
function squared with the
22
00:01:10 --> 00:01:11
coefficient squared.
23
00:01:11 --> 00:01:17
A second one on the discrete
Fourier transform.
24
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Cyclic stuff.
25
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The third one on the
Fourier integral.
26
00:01:25 --> 00:01:35
And have a look at the
applications to solving an ODE.
27
00:01:35 --> 00:01:38
I did one in class.
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00:01:38 --> 00:01:45
The one in class was the one
in the book -u''+a^2*u=f(x),
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00:01:45 --> 00:01:47
so this will be.
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00:01:47 --> 00:01:50
So have a look at
that application.
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00:01:50 --> 00:01:54
This is, of course, on minus
infinity to infinity, and
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00:01:54 --> 00:02:00
then a fourth question
on convolution.
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OK.
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00:02:01 --> 00:02:06
And this afternoon, of course,
I'll be here to answer any
35
00:02:06 --> 00:02:10
questions from the
homework, from any source
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00:02:10 --> 00:02:13
for these topics.
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00:02:13 --> 00:02:15
Are there any questions
just now, though?
38
00:02:15 --> 00:02:19
I'm OK to take questions.
39
00:02:19 --> 00:02:25
I thought I'd discussed today
a topic that involves both
40
00:02:25 --> 00:02:27
Fourier series and
Fourier integrals.
41
00:02:27 --> 00:02:34
It's a kind of cool connection
and it's linked the name of
42
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Claude Shannon who created
information theory, who was a
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Bell Labs guy and then
an MIT professor.
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00:02:43 --> 00:02:49
So I should put his name in.
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00:02:49 --> 00:02:50
Shannon.
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OK, so this is, yeah.
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You'll see.
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00:02:57 --> 00:03:01
So it's not on the quiz but it
gives me a chance to say
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something important, and at the
same time review Fourier
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00:03:05 --> 00:03:07
series and Fourier integrals.
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00:03:07 --> 00:03:11
So let me start
with the problem.
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The problem comes for
a A to D converter.
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So what does that mean?
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That means this a is analog.
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That means we have a
function, A for analog.
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And D for digital.
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So we have a function.
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00:03:32 --> 00:03:34
Like, so f(x).
59
00:03:34 --> 00:03:36
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00:03:36 --> 00:03:42
Say, all the way minus infinity
to infinity, so we'll be doing,
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that's where the Fourier
integral's going to come up.
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So that's analog.
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All x, it's some curve.
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And people build, and you can
buy, and they're sold in large
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quantities, something that
just samples that function.
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Say, at the integers.
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So now I'll sample that
function and let me take the
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period of the sample to
be one, so that I'm going
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to take the values f(n).
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so now I've got something
digital like I work with I can
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come here with and so the
saplings here I'm well he I
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mean and they're going to
question it yeah selling
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sampling say army is about this
question and it seems like
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crazy question when do these
numbers that's just a
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sequence of numbers.
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This x was all the way from
minus infinity to infinity.
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And similarly, n is numbers all
the way from minus infinity to
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infinity, they're just samples.
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When does that tell
me the function?
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When can I learn from those
samples, when do I have total
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information about the function?
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Now, you'll say impossible.
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Right?
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00:05:16 --> 00:05:24
So suppose I draw
function f(x), OK?
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And I'm going to sample
it at these points.
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All the way, so these are the
numbers, these are my f(n).
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00:05:37 --> 00:05:43
Sampling it, equal intervals
because if we want to use
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Fourier ideas, equal spacing
is the right thing to have.
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00:05:47 --> 00:05:50
So when could I recover
the function in between?
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Well, you'd say, never.
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Because how do I know
what that function could
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be doing in between.
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00:05:57 --> 00:06:04
So let me take the case when
all the samples are zero.
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And let's think
about that case.
96
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Suppose, what could the
function be if all the
97
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samples, if these are
all zeroes, forever.
98
00:06:18 --> 00:06:23
OK, well there's one leading
candidate for the function,
99
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the zero function.
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00:06:26 --> 00:06:31
Now, you'll see the whole
point of the sampling theorem
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if you think about other.
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00:06:33 --> 00:06:36
What other functions?
103
00:06:36 --> 00:06:37
Familiar functions, yeah.
104
00:06:37 --> 00:06:42
I mean, we could, obviously,
any, all sorts of things.
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00:06:42 --> 00:06:48
But since we're doing Fourier,
we like to pick on the sines,
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00:06:48 --> 00:06:52
cosines, the special function,
and think about those
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00:06:52 --> 00:06:53
in particular.
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00:06:53 --> 00:06:57
So, somebody said sines.
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Now, what function, so a sine
function certainly, the sine
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00:07:03 --> 00:07:07
function hits zero at
infinitely often.
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00:07:07 --> 00:07:11
What frequency, so what
sine of what would give
112
00:07:11 --> 00:07:14
me the same answer?
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The same samples. i put this
sine function that you're going
114
00:07:19 --> 00:07:24
to tell me, so you're going to
tell me it's sine of something.
115
00:07:24 --> 00:07:29
Will have these zero values at
all the integers, at zero, one,
116
00:07:29 --> 00:07:32
two, minus one, minus
two, and so on.
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00:07:32 --> 00:07:36
So what would do the job?
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Sine of?
119
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Of what will hit zero.
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00:07:45 --> 00:07:49
So I'm looking for a sine
function, I guess I'm looking
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00:07:49 --> 00:07:54
first for the function
that just does that.
122
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And what is it? sin(pi*x).
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00:07:56 --> 00:08:01
And now tell me some more.
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00:08:01 --> 00:08:03
Tell me another function.
125
00:08:03 --> 00:08:07
Which will also, it won't
be that graph. sin(2pi*x).
126
00:08:09 --> 00:08:14
And all the rest, OK?
127
00:08:14 --> 00:08:18
Let me just use a word that's
kind of a handy word.
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00:08:18 --> 00:08:22
Of course, let's put
zero on the list here.
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OK.
130
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So this is where k, the
frequency, usually appears.
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This is where k.
132
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The word I want to
introduce is alias.
133
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This frequency, pi, is an alias
for this at frequency zero.
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Here's the, it's a different
function but yet the
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samples are the same.
136
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So if you're only looking at
the samples you're getting the
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00:08:55 --> 00:09:01
same answer but somehow the
function has a different name.
138
00:09:01 --> 00:09:04
So that frequency and this
frequency, and all those
139
00:09:04 --> 00:09:06
others would be alias.
140
00:09:06 --> 00:09:11
Can I just write that word
down, because you it often.
141
00:09:11 --> 00:09:13
Alias.
142
00:09:13 --> 00:09:17
That means two frequencies,
like pi and 2 pi, and zero
143
00:09:17 --> 00:09:20
or whatever, that give
you the same samples.
144
00:09:20 --> 00:09:27
OK, so now comes
Shannon's question.
145
00:09:27 --> 00:09:32
So we have to make some
assumption on the function.
146
00:09:32 --> 00:09:38
To knock out those
possibilities.
147
00:09:38 --> 00:09:44
We want to know a limited
class of functions.
148
00:09:44 --> 00:09:47
Which don't include these guys.
149
00:09:47 --> 00:09:50
So that within this limited
class of functions, this is the
150
00:09:50 --> 00:09:55
only candidate and we have this
possibility of doing
151
00:09:55 --> 00:09:57
the impossible.
152
00:09:57 --> 00:10:02
Of determining that if I know
zeroes here, the function
153
00:10:02 --> 00:10:05
has to be zero everywhere.
154
00:10:05 --> 00:10:08
OK, now the question is
what class of functions?
155
00:10:08 --> 00:10:12
We want to eliminate these
guys, and sort of, your
156
00:10:12 --> 00:10:19
instinct is, you want to
eliminate functions that, you
157
00:10:19 --> 00:10:21
know if it's not zero then
got to get up and back
158
00:10:21 --> 00:10:24
down in everything.
159
00:10:24 --> 00:10:26
It could do different things
in different intervals.
160
00:10:26 --> 00:10:33
But somehow it's got to have
some of these frequencies.
161
00:10:33 --> 00:10:36
Pi or higher.
162
00:10:36 --> 00:10:37
Would have to be in there.
163
00:10:37 --> 00:10:40
So this is the instinct.
164
00:10:40 --> 00:10:48
That if I limit the frequency
band, so I'm going to say f(x)
165
00:10:48 --> 00:10:53
is band-limited, can I
introduce that word?
166
00:10:53 --> 00:10:56
I'll maybe take a moment
just ask you if you've
167
00:10:56 --> 00:10:59
seen that word before.
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00:10:59 --> 00:11:01
How many have seen this
word, band-limited?
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00:11:01 --> 00:11:04
Quite a few but not half.
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00:11:04 --> 00:11:05
OK.
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00:11:05 --> 00:11:10
Band-limited means the band
is a band of frequencies.
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00:11:10 --> 00:11:17
So the function's band-limited
when its transform, this tells
173
00:11:17 --> 00:11:20
me how much of each
frequency there is.
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00:11:20 --> 00:11:23
If this is zero, in some band.
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00:11:23 --> 00:11:32
In some band, let's say, all
frequencies below something.
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00:11:32 --> 00:11:36
And let's not even
put equal in there.
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00:11:36 --> 00:11:37
OK.
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00:11:37 --> 00:11:40
But that's not critical.
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00:11:40 --> 00:11:44
Band-limited, I have to tell
you the size of the band.
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00:11:44 --> 00:11:48
And the size of the band, the
limiting frequency is this
181
00:11:48 --> 00:11:53
famous Nyquist frequency, so
Nyquist is a guy's name.
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And the Nyquist frequency
in our problem here is pi.
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00:11:57 --> 00:12:00
This is the Nyquist frequency.
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00:12:00 --> 00:12:04
If we let that frequency,
that's the borderline
185
00:12:04 --> 00:12:08
frequency.
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00:12:08 --> 00:12:12
And there would be a similar
Nyquist sampling rate.
187
00:12:12 --> 00:12:18
So Nyquist is the guy who
studied the sort of borderline
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00:12:18 --> 00:12:25
case and so the point is that
if our, say, band-limited by
189
00:12:25 --> 00:12:29
pi, I have to tell you, so
band-limited means there's
190
00:12:29 --> 00:12:31
some limit on the band.
191
00:12:31 --> 00:12:37
And our interest is when that
limit is the Nyquist frequency.
192
00:12:37 --> 00:12:43
The one we don't want to allow,
so we, this is the point.
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00:12:43 --> 00:12:45
So this will be the idea.
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00:12:45 --> 00:12:54
That if we take this class of
function, that band limit at
195
00:12:54 --> 00:12:56
that, those are called
band-limited functions, and
196
00:12:56 --> 00:12:59
they're band-limited
specifically by the
197
00:12:59 --> 00:13:01
Nyquist limit.
198
00:13:01 --> 00:13:11
If we take those, then the
idea is that then we can
199
00:13:11 --> 00:13:13
reconstruct from the samples.
200
00:13:13 --> 00:13:17
Because the only function that
has zero samples in that
201
00:13:17 --> 00:13:20
class is the zero function.
202
00:13:20 --> 00:13:22
You see that class has
knocked out, is not
203
00:13:22 --> 00:13:26
allowing these guys.
204
00:13:26 --> 00:13:29
Of course, haven't
proved anything yet.
205
00:13:29 --> 00:13:31
And I haven't shown
how to reconstruct.
206
00:13:31 --> 00:13:35
Well, of course, we quickly
reconstructed the zero function
207
00:13:35 --> 00:13:41
out of those zeroes, but now
let me take another obviously
208
00:13:41 --> 00:13:43
important possible sample.
209
00:13:43 --> 00:13:48
Suppose I get zero samples
except at that point,
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00:13:48 --> 00:13:51
where it's one.
211
00:13:51 --> 00:14:01
OK, now the question is what
function, f(x), can I fill in,
212
00:14:01 --> 00:14:05
in between zero, zero, zero,
zero, one, zero, zero, zero,
213
00:14:05 --> 00:14:10
can I fill in exactly
one function.
214
00:14:10 --> 00:14:14
That comes from this class.
215
00:14:14 --> 00:14:18
So that I have now the
answer for this highly
216
00:14:18 --> 00:14:19
important sample?
217
00:14:19 --> 00:14:22
The sample that's all zeroes
except for the delta
218
00:14:22 --> 00:14:24
sample, you could say.
219
00:14:24 --> 00:14:28
OK, so I'm looking for the
function now which is
220
00:14:28 --> 00:14:30
one at that point.
221
00:14:30 --> 00:14:32
And zero at the others.
222
00:14:32 --> 00:14:36
So here's a key function.
223
00:14:36 --> 00:14:39
And I'll show you what it is.
224
00:14:39 --> 00:14:43
So the function, this function
that I'm going to mention,
225
00:14:43 --> 00:14:49
will get down here, it'll
oscillate, it'll go forever.
226
00:14:49 --> 00:14:52
It's not like a spline.
227
00:14:52 --> 00:14:58
Splines made it to zero
and stayed there, right?
228
00:14:58 --> 00:15:00
The cubic spline, for example.
229
00:15:00 --> 00:15:08
OK, but I guess, yeah, that
somehow that function,
230
00:15:08 --> 00:15:11
we're not in that league.
231
00:15:11 --> 00:15:13
We're in this
band-limited league.
232
00:15:13 --> 00:15:21
In a way you could say that, I
mean, what's the key connection
233
00:15:21 --> 00:15:25
between dropoff of the
Fourier transform?
234
00:15:25 --> 00:15:29
So if the Fourier transform
drops off fast, what does that
235
00:15:29 --> 00:15:31
tell me about the function?
236
00:15:31 --> 00:15:32
It's smooth, thanks.
237
00:15:32 --> 00:15:34
That's exactly the right word.
238
00:15:34 --> 00:15:36
If the Fourier transform
drops off fast, the
239
00:15:36 --> 00:15:38
function is smooth.
240
00:15:38 --> 00:15:42
OK, this is really
an extreme case.
241
00:15:42 --> 00:15:46
That it's dropped off totally.
242
00:15:46 --> 00:15:50
You know, it's not just decay
rate, it's just zonk, out.
243
00:15:50 --> 00:15:53
Beyond this band
of frequencies.
244
00:15:53 --> 00:15:57
And so that gives, you
could say, sort of a
245
00:15:57 --> 00:15:59
hyper-smooth function.
246
00:15:59 --> 00:16:02
I mean, so smooth that
you know everything by
247
00:16:02 --> 00:16:03
knowing the sample.
248
00:16:03 --> 00:16:08
OK, now I'm ready to write down
the key function, a famous
249
00:16:08 --> 00:16:11
function that has
those samples.
250
00:16:11 --> 00:16:14
And that function is
sin(pi*x)/(pi*x).
251
00:16:14 --> 00:16:21
252
00:16:21 --> 00:16:24
I don't know if you ever
thought about this function,
253
00:16:24 --> 00:16:25
and it has a name.
254
00:16:25 --> 00:16:28
Do you know its name? sinc.
255
00:16:28 --> 00:16:29
It's the sinc function.
256
00:16:29 --> 00:16:34
Which is a little - you know, a
name's a little unfortunate.
257
00:16:34 --> 00:16:38
Mainly because you know, you're
using those same letters S I N,
258
00:16:38 --> 00:16:44
but it's a c that turns it,
that gives it, so this
259
00:16:44 --> 00:16:46
is called the sinc
function. sinc(x).
260
00:16:47 --> 00:16:50
But the main thing
is its formula.
261
00:16:50 --> 00:16:56
OK, well everybody sees that at
x=1, the sin(pi) is zero, at
262
00:16:56 --> 00:17:00
x=2, the sin(pi) is zero, all
these ones we've seen already.
263
00:17:00 --> 00:17:03
And now what happens
at equals zero?
264
00:17:03 --> 00:17:07
Do you recognize that this
function, as x goes to zero,
265
00:17:07 --> 00:17:10
is a perfectly good function?
266
00:17:10 --> 00:17:14
I mean, it becomes 0/0 at x=0.
267
00:17:14 --> 00:17:18
But the limit, we
know to be one.
268
00:17:18 --> 00:17:18
Right?
269
00:17:18 --> 00:17:22
This sin(theta)/theta is one.
270
00:17:22 --> 00:17:25
As theta approaches
zero, right?
271
00:17:25 --> 00:17:28
So that does have
that correct sample.
272
00:17:28 --> 00:17:32
And now what's the, I
claim that that is a
273
00:17:32 --> 00:17:33
band-limited function.
274
00:17:33 --> 00:17:40
And you'll see that that
function pushes the limit.
275
00:17:40 --> 00:17:46
It's right, Nyquist
barely lets it in.
276
00:17:46 --> 00:17:48
Now here's a calculation.
277
00:17:48 --> 00:17:50
So this is our practice.
278
00:17:50 --> 00:17:59
What is f hat of k
for that function?
279
00:17:59 --> 00:18:01
Now, let me think
how to do this one.
280
00:18:01 --> 00:18:08
So just to understand this
better, I want to see that that
281
00:18:08 --> 00:18:14
is a band-limited function and
what is its Fourier transform.
282
00:18:14 --> 00:18:18
OK, now once again here we have
a function where if I want
283
00:18:18 --> 00:18:25
to do - how best to do this.
284
00:18:25 --> 00:18:33
You could say well, just do it.
285
00:18:33 --> 00:18:35
As I would say on the quiz,
just go ahead and do it.
286
00:18:35 --> 00:18:41
But you'll see I'm going to
have a problem, I think.
287
00:18:41 --> 00:18:44
But this gives us a chance
to remember the formula.
288
00:18:44 --> 00:18:47
So what's the formula for the
Fourier integral transform
289
00:18:47 --> 00:18:48
of this particular?
290
00:18:48 --> 00:18:52
So my function is the
sinc function, sin(pi*x)
291
00:18:52 --> 00:18:53
over sin(pi*x).
292
00:18:55 --> 00:19:03
So how do I get its, I do a
what here? e^(-ikx), and
293
00:19:03 --> 00:19:04
am I doing dx or dk?
294
00:19:05 --> 00:19:07
dx, right?
295
00:19:07 --> 00:19:10
And I'm going from minus
infinity to infinity.
296
00:19:10 --> 00:19:12
And am I, do I have a 2pi?
297
00:19:14 --> 00:19:16
Yes or no?
298
00:19:16 --> 00:19:17
Who knows, anyway?
299
00:19:17 --> 00:19:21
Right.
300
00:19:21 --> 00:19:22
In the book I didn't?
301
00:19:22 --> 00:19:25
OK.
302
00:19:25 --> 00:19:33
Now, well, I don't
know the answer.
303
00:19:33 --> 00:19:37
But so let's - it's much better
to start with the answer,
304
00:19:37 --> 00:19:43
right, and check that - so let
me say what I think
305
00:19:43 --> 00:19:44
the answer is.
306
00:19:44 --> 00:19:54
I think the answer is, it's
a function that's exactly
307
00:19:54 --> 00:19:59
as I say, it pushes the
limit from, this is k.
308
00:19:59 --> 00:20:03
It's the square wave, it's
zero, the height is one.
309
00:20:03 --> 00:20:08
It's the function that's
zero all the way here,
310
00:20:08 --> 00:20:09
all the way there.
311
00:20:09 --> 00:20:13
I think that that's the Fourier
transform of that function.
312
00:20:13 --> 00:20:20
And just before we check it,
see how is Nyquist got really
313
00:20:20 --> 00:20:22
pushed up to the wall, right?
314
00:20:22 --> 00:20:26
Because the frequency
is non-zero right all
315
00:20:26 --> 00:20:30
the way through pi.
316
00:20:30 --> 00:20:36
But pi is just one point there.
317
00:20:36 --> 00:20:40
And anyway, I think we won't
get into philosophical
318
00:20:40 --> 00:20:45
discussion about whether is
that is that limited to pi, I
319
00:20:45 --> 00:20:49
don't know whether to put, you
saw me chicken out here.
320
00:20:49 --> 00:20:52
I didn't know whether to
put less or equal or
321
00:20:52 --> 00:20:55
not, and I still don't.
322
00:20:55 --> 00:21:00
But this is making the point
that that's the key frequency.
323
00:21:00 --> 00:21:03
So this particular function,
what I'm saying is this
324
00:21:03 --> 00:21:08
particular function has all the
frequencies in equal amounts
325
00:21:08 --> 00:21:12
over a band, and nothing
outside that band.
326
00:21:12 --> 00:21:14
And that's the Nyquist band.
327
00:21:14 --> 00:21:18
OK, now why is this the
correct answer here?
328
00:21:18 --> 00:21:24
I guess the smart way
would be, this is a good
329
00:21:24 --> 00:21:26
function, easy function.
330
00:21:26 --> 00:21:32
So let's take the transform
in the other direction.
331
00:21:32 --> 00:21:34
Start from here
and get to here.
332
00:21:34 --> 00:21:35
Right?
333
00:21:35 --> 00:21:36
That would be convincing.
334
00:21:36 --> 00:21:42
Because we do know that that
pair of formulas for f
335
00:21:42 --> 00:21:47
connected to f hat, connected
to f, they go together.
336
00:21:47 --> 00:21:54
So if I can show that I go from
here, that that Fourier
337
00:21:54 --> 00:21:58
integral takes me from here to
there, then this guy
338
00:21:58 --> 00:21:59
will take me back.
339
00:21:59 --> 00:22:00
So let me just do that.
340
00:22:00 --> 00:22:05
Because that's a very very
important one that you
341
00:22:05 --> 00:22:07
should be prepared for.
342
00:22:07 --> 00:22:07
Right.
343
00:22:07 --> 00:22:10
So now what do I want to do?
344
00:22:10 --> 00:22:23
Here's my function of k, and
I I'm hoping that I recall.
345
00:22:23 --> 00:22:27
Now, what do I do when I
want to do the transform
346
00:22:27 --> 00:22:29
in the opposite direction?
347
00:22:29 --> 00:22:37
It'll be an e^(+ikx),
right? d what? dk, now.
348
00:22:37 --> 00:22:39
From k equal minus
infinity to to infinity.
349
00:22:39 --> 00:22:43
And now I think I do put in
the 2pi, is that right?
350
00:22:43 --> 00:22:50
And the question is, does that
bring back the sinc function?
351
00:22:50 --> 00:22:55
If it does, then this was
OK in the other direction.
352
00:22:55 --> 00:22:58
If the transform's correct in
one direction then the inverse
353
00:22:58 --> 00:22:59
transform will be correct.
354
00:22:59 --> 00:23:04
So I just plan to do this
integral. f hat of k, of
355
00:23:04 --> 00:23:10
course, is an easy integral
now. f hat of k is one over,
356
00:23:10 --> 00:23:15
between minus pi and pi, so I
only have to do over that range
357
00:23:15 --> 00:23:20
where f hat of k is just a one.
358
00:23:20 --> 00:23:23
And now that's an integral
we can certainly do.
359
00:23:23 --> 00:23:29
So I have 1/2pi, integrating
e^(ikx) will give
360
00:23:29 --> 00:23:29
me e^(ikx)/ix.
361
00:23:29 --> 00:23:33
362
00:23:33 --> 00:23:34
Now, remember I'm
integrating dk.
363
00:23:35 --> 00:23:36
Oh, look.
364
00:23:36 --> 00:23:40
See, we're showing this x
now in the denominator.
365
00:23:40 --> 00:23:44
That we're hoping for.
366
00:23:44 --> 00:23:50
And now I have to do that
between k is minus pi and pi.
367
00:23:50 --> 00:24:01
So this is like, so I
get 1/2pi, e^(i*pi*k)
368
00:24:01 --> 00:24:06
-e^(-i*pi*x), right?
369
00:24:06 --> 00:24:07
Over the ix.
370
00:24:07 --> 00:24:10
371
00:24:10 --> 00:24:13
OK so far?
372
00:24:13 --> 00:24:17
I was doing a k integral
and I get an x answer.
373
00:24:17 --> 00:24:21
And I want to be sure that this
x answer is the x answer I
374
00:24:21 --> 00:24:23
want, it's the sinc function.
375
00:24:23 --> 00:24:24
OK, it is.
376
00:24:24 --> 00:24:25
Right?
377
00:24:25 --> 00:24:31
I recognize the sine,
e^(i*theta)-e^(-i*theta),
378
00:24:31 --> 00:24:36
divided by two, I guess.
379
00:24:36 --> 00:24:38
Is the sine, right?
380
00:24:38 --> 00:24:39
So I have 1/2pi.
381
00:24:40 --> 00:24:46
And here is ix - well, no,
the i is part of that sine.
382
00:24:46 --> 00:24:53
So I'm just using the fact
that e^(i*theta)-e^(-i*theta)
383
00:24:53 --> 00:24:59
is, this is cosine plus
i sine, subtract cosine.
384
00:24:59 --> 00:25:03
But subtract minus i sine,
so that will be two i
385
00:25:03 --> 00:25:07
sin(theta), right?.
386
00:25:07 --> 00:25:11
We all know, and now
theta is pi*x here.
387
00:25:11 --> 00:25:16
So I have two - let me keep the
i there, and 2i*sin(pi*x).
388
00:25:17 --> 00:25:20
You see it works.
389
00:25:20 --> 00:25:25
Just using this, replacing
this by the sine, the
390
00:25:25 --> 00:25:27
two cancels the two.
391
00:25:27 --> 00:25:32
The i cancels the i, and I
have sin(pi*x) over pi*x,
392
00:25:32 --> 00:25:34
that's the sinc function.
393
00:25:34 --> 00:25:42
OK, so that's the function
we've now checked.
394
00:25:42 --> 00:25:44
We've checked two things
about this function.
395
00:25:44 --> 00:25:49
It has the right samples, zero,
zero, zero, zero, one at
396
00:25:49 --> 00:25:51
x=0 and then back to zero.
397
00:25:51 --> 00:25:58
It is band-limited, so it's the
guy, if this was my sample, if
398
00:25:58 --> 00:26:01
this was my f(n), zero, zero,
zero, one, zero, zero,
399
00:26:01 --> 00:26:05
zero, then I've got it.
400
00:26:05 --> 00:26:10
It's the right function.
401
00:26:10 --> 00:26:15
OK, we can create Shannon's
sampling formula.
402
00:26:15 --> 00:26:20
Shannon's sampling formula
gives me the f(x) for any f(n).
403
00:26:20 --> 00:26:23
404
00:26:23 --> 00:26:25
Maybe you can spot that, now.
405
00:26:25 --> 00:26:28
So this is going to use
the shift in variance.
406
00:26:28 --> 00:26:34
Oh yeah, let's - tell me what
the, let's take one step here.
407
00:26:34 --> 00:26:37
Suppose my f(n), were
zero, zero, zero,
408
00:26:37 --> 00:26:40
zero, and a one there.
409
00:26:40 --> 00:26:45
So suppose this is, for the
exam too, this idea of
410
00:26:45 --> 00:26:49
shifting is simple and basic.
411
00:26:49 --> 00:26:52
And it's a great thing
to be able to do.
412
00:26:52 --> 00:26:56
So this wouldn't be the right
answer that that function
413
00:26:56 --> 00:26:59
turn it was the one at zero.
414
00:26:59 --> 00:27:02
Tell me what function.
415
00:27:02 --> 00:27:06
Copying this idea will produce
all zeroes except for
416
00:27:06 --> 00:27:11
a one at this point.
417
00:27:11 --> 00:27:17
So I'll put this
question over here.
418
00:27:17 --> 00:27:24
Suppose I'm all zeroes at that
point but at this point I'm up
419
00:27:24 --> 00:27:26
and then I go back to zeroes.
420
00:27:26 --> 00:27:33
What function is
giving me that?
421
00:27:33 --> 00:27:36
You see it does decay because
of the x in the denominator,
422
00:27:36 --> 00:27:40
it kind of goes to zero
but not very fast.
423
00:27:40 --> 00:27:45
OK, what's that function?
424
00:27:45 --> 00:27:48
I replace x by x-1, right.
425
00:27:48 --> 00:27:53
So the function here
is sin(pi(x-1)),
426
00:27:53 --> 00:27:55
divided by pi(x-1).
427
00:27:56 --> 00:27:57
I just change the x to x-1.
428
00:27:59 --> 00:28:06
Now again, at x=0, this is
sin(pi), sin(-pi) is safely
429
00:28:06 --> 00:28:11
zero, but at x=1 that's now the
point where I'm getting 0/0.
430
00:28:13 --> 00:28:18
And the numbers are right to
give me the exact answer, one.
431
00:28:18 --> 00:28:24
OK, so now we see what to do
if the sampling turned out
432
00:28:24 --> 00:28:25
to give us this answer.
433
00:28:25 --> 00:28:29
And now can you tell
me the whole formula?
434
00:28:29 --> 00:28:32
Can you tell me the whole
formula, so now I'm ready for
435
00:28:32 --> 00:28:35
Shannon's sampling theorem.
436
00:28:35 --> 00:28:43
Is that f(x), if f(x) is
band-limited, then I can tell
437
00:28:43 --> 00:28:45
you what it is at all x.
438
00:28:45 --> 00:28:48
So i'm going back to the
beginning of this lecture.
439
00:28:48 --> 00:28:52
It's a miracle that
this is possible.
440
00:28:52 --> 00:28:56
That we can write down a
formula for f(x) at all
441
00:28:56 --> 00:28:59
x, only using f(n)'s.
442
00:29:01 --> 00:29:05
OK, now it's going to be a sum.
443
00:29:05 --> 00:29:09
From n equal minus infinity to
infinity, because I'm going to
444
00:29:09 --> 00:29:11
use all the f(n)'s to
produce the f(x).
445
00:29:12 --> 00:29:17
And now what do I put in there?
446
00:29:17 --> 00:29:23
So I want my formula
to be correct.
447
00:29:23 --> 00:29:30
I want my formula to be correct
in this case, so that if all
448
00:29:30 --> 00:29:34
the f's were zero except for
the middle one, then I want to
449
00:29:34 --> 00:29:37
put sin(pi*x)/(pi*x) in there.
450
00:29:37 --> 00:29:39
The sinc function.
451
00:29:39 --> 00:29:41
And I also want to
get this one right.
452
00:29:41 --> 00:29:45
If all the f's are zero, so
there'll only be one term.
453
00:29:45 --> 00:29:48
If this is the term, I
want that to show up.
454
00:29:48 --> 00:29:53
Ok what do I - yeah,
you can tell me.
455
00:29:53 --> 00:29:57
Suppose this is hitting at n.
456
00:29:57 --> 00:30:00
We'll just fix this and
then you'll see it.
457
00:30:00 --> 00:30:05
Suppose that all the others,
n, n-1, n+1, all those at
458
00:30:05 --> 00:30:09
zero, but at x=n, it's one.
459
00:30:09 --> 00:30:11
Now what should I have chosen?
460
00:30:11 --> 00:30:16
What's the correct - I'm
just going to make it easy
461
00:30:16 --> 00:30:19
for all of us to, yes.
462
00:30:19 --> 00:30:25
What's the good sinc function
which peaks at a point n?
463
00:30:25 --> 00:30:27
Again, I'm just
shifting it over.
464
00:30:27 --> 00:30:30
So what do I do?
465
00:30:30 --> 00:30:34
Put in, what do I
write here? n.
466
00:30:34 --> 00:30:36
I shift the whole thing by n.
467
00:30:36 --> 00:30:39
So that's the right answer
when this hits at n.
468
00:30:39 --> 00:30:43
So now maybe you see that this
is going to be the right
469
00:30:43 --> 00:30:51
answer for all of them.
470
00:30:51 --> 00:30:56
You see that, we're using
linearity shift invariance.
471
00:30:56 --> 00:31:00
The shift invariance is
telling us this answer for
472
00:31:00 --> 00:31:03
wherever the one hits.
473
00:31:03 --> 00:31:05
That's what we need.
474
00:31:05 --> 00:31:11
And then by linearity, I put
together whatever the f is at
475
00:31:11 --> 00:31:14
that point, that would
just amplify the sinc.
476
00:31:14 --> 00:31:17
And then I have to put them
in for all the other values.
477
00:31:17 --> 00:31:20
That's the Shannon formula.
478
00:31:20 --> 00:31:23
That's the Shannon formula,
and this function is
479
00:31:23 --> 00:31:27
band-limited, let's see.
480
00:31:27 --> 00:31:29
What's the - oh, yeah.
481
00:31:29 --> 00:31:34
What's the, do you see
that this one, that this
482
00:31:34 --> 00:31:36
guy is band-limited?
483
00:31:36 --> 00:31:37
We checked, right?
484
00:31:37 --> 00:31:43
We checked that this one,
sin(pi*x)/(pi*x), that was
485
00:31:43 --> 00:31:47
band-limited because we
actually found the band.
486
00:31:47 --> 00:31:50
Now, that just gives us
another chance to think.
487
00:31:50 --> 00:31:51
Allowed.
488
00:31:51 --> 00:31:55
What's the Fourier
transform of this guy?
489
00:31:55 --> 00:31:58
My claim is that it's
also in this band.
490
00:31:58 --> 00:32:04
Non-zero only in the band, and
zero outside the Nyquist band.
491
00:32:04 --> 00:32:08
What is the transform of that?
492
00:32:08 --> 00:32:11
What happens if you shift
a function, what happens
493
00:32:11 --> 00:32:14
to its transform?
494
00:32:14 --> 00:32:23
So that's one of the key rules
that makes Fourier so special.
495
00:32:23 --> 00:32:28
If I took this sine, let
me write this guy again.
496
00:32:28 --> 00:32:31
This was the un-shifted one.
497
00:32:31 --> 00:32:35
That connected to the,
what am I going to call
498
00:32:35 --> 00:32:40
that, the box function.
499
00:32:40 --> 00:32:42
The box function,
the square wave.
500
00:32:42 --> 00:32:45
Well, box is good.
501
00:32:45 --> 00:32:49
Now, what if I shift
the function?
502
00:32:49 --> 00:32:50
If I shift a function,
what happens to its
503
00:32:50 --> 00:32:53
Fourier transform?
504
00:32:53 --> 00:32:55
Anybody remember?
505
00:32:55 --> 00:33:02
You multiply it by, so if I
shift the function, I just
506
00:33:02 --> 00:33:05
multiply this box function,
this is a box function in the
507
00:33:05 --> 00:33:14
k, times something e to the i
shift, and the shift
508
00:33:14 --> 00:33:19
was one, right?
509
00:33:19 --> 00:33:24
Is it just e^(ik), d being
the shift distance.
510
00:33:24 --> 00:33:26
Oh, the shift distance was n.
511
00:33:26 --> 00:33:27
Right.
512
00:33:27 --> 00:33:30
And possibly minus, who knows.
513
00:33:30 --> 00:33:33
OK, but what's the point here?
514
00:33:33 --> 00:33:38
The point is that it's still
zero outside the box.
515
00:33:38 --> 00:33:41
Inside the box, instead
of being one, it's
516
00:33:41 --> 00:33:43
this complex guy.
517
00:33:43 --> 00:33:45
But no change.
518
00:33:45 --> 00:33:47
It's still zero
outside the box.
519
00:33:47 --> 00:33:49
It's still band-limited.
520
00:33:49 --> 00:33:54
So this is the
transform of this guy.
521
00:33:54 --> 00:33:59
And then the transform of this
combination would be still in
522
00:33:59 --> 00:34:05
the box, multiplied by
some messy expression.
523
00:34:05 --> 00:34:07
So what was I doing there?
524
00:34:07 --> 00:34:10
I was just checking that,
sure enough, this guy is
525
00:34:10 --> 00:34:15
band-limited, and it's
band-limited it, gives us the
526
00:34:15 --> 00:34:20
right f(n)'s, of course.
527
00:34:20 --> 00:34:29
Everybody sees that at x=n,
let's just have a look now.
528
00:34:29 --> 00:34:30
We've got this great formula.
529
00:34:30 --> 00:34:31
Plug in x=n.
530
00:34:33 --> 00:34:37
What happens when you plug in
at one of the samples, you
531
00:34:37 --> 00:34:41
look to see what this a to d
converter produced at time
532
00:34:41 --> 00:34:46
n, and let's just see.
533
00:34:46 --> 00:34:48
So at x=n, the left
side is f(n).
534
00:34:49 --> 00:34:51
Why is the right
side f of that N?
535
00:34:52 --> 00:34:53
That particular N?
536
00:34:53 --> 00:34:59
Maybe I should give a
specific letter to that N.
537
00:34:59 --> 00:35:05
So at that particular sample,
this left side is f(N), and I
538
00:35:05 --> 00:35:09
hope that the right side gives
me f at that capital N.
539
00:35:09 --> 00:35:10
That particular one.
540
00:35:10 --> 00:35:12
Why does it?
541
00:35:12 --> 00:35:17
You're all seeing that at x
equal capital N, these guys
542
00:35:17 --> 00:35:21
are all zero, except
for one of them.
543
00:35:21 --> 00:35:27
Except for the one when little
n and capital N are the same.
544
00:35:27 --> 00:35:29
Then that becomes the one.
545
00:35:29 --> 00:35:32
And I'm getting f at capital N.
546
00:35:32 --> 00:35:39
So it will give me that, for
the n=1, the one term, yeah.
547
00:35:39 --> 00:35:41
I don't know if it was
necessary to say that.
548
00:35:41 --> 00:35:48
You've got the idea of
the sampling formula.
549
00:35:48 --> 00:35:58
I could say more about the
sampling, just to realize that
550
00:35:58 --> 00:36:06
the technology, communication
theory is always trying
551
00:36:06 --> 00:36:10
to, like, to have a
greater bandwidth.
552
00:36:10 --> 00:36:12
You always want a
greater bandwidth.
553
00:36:12 --> 00:36:17
But if the bandwidth, which is
this, is increased, well by
554
00:36:17 --> 00:36:19
the way, what does happen?
555
00:36:19 --> 00:36:25
Suppose it's band-limited
by pi, oh, by pi/T.
556
00:36:25 --> 00:36:30
557
00:36:30 --> 00:36:34
Let's just, I normalize
things to choose
558
00:36:34 --> 00:36:37
samples every integer.
559
00:36:37 --> 00:36:39
Zero, one, two, three.
560
00:36:39 --> 00:36:43
And that turned out that the
Nyquist frequency was pi.
561
00:36:43 --> 00:36:53
Now, what sampling rate would
correspond to this band, which
562
00:36:53 --> 00:36:56
could be - well, let me
just say what it is.
563
00:36:56 --> 00:37:05
That would be the Nyquist
frequency for sampling every T.
564
00:37:05 --> 00:37:13
Instead of a sampling interval
of one, if I sample every T,
565
00:37:13 --> 00:37:20
2T, 3T, -T, my sampling rate
is T, so if T is small,
566
00:37:20 --> 00:37:21
I'm sampling much more.
567
00:37:21 --> 00:37:24
Suppose T is 1/4.
568
00:37:24 --> 00:37:29
If T is 1/4, then I'm
doing four samplings.
569
00:37:29 --> 00:37:32
I'm taking four samples,
I'm paying more.
570
00:37:32 --> 00:37:35
For this A to D you converter,
because it's taking four
571
00:37:35 --> 00:37:39
samples where previously
it took one.
572
00:37:39 --> 00:37:41
How do I get paid back?
573
00:37:41 --> 00:37:43
What's the reward?
574
00:37:43 --> 00:37:51
The reward is if T is 1/4, then
the Nyquist limit is 4pi.
575
00:37:51 --> 00:37:56
I can get a broader band
of singles by sampling
576
00:37:56 --> 00:37:58
them more often.
577
00:37:58 --> 00:38:01
Let me just say that
again, because that's the
578
00:38:01 --> 00:38:03
fundamental idea behind it.
579
00:38:03 --> 00:38:08
If I sample more often, say, so
fast sampling would be small,
580
00:38:08 --> 00:38:20
fast samples, would be small
t and then a higher Nyquist.
581
00:38:20 --> 00:38:23
A higher band limit.
582
00:38:23 --> 00:38:24
More functions allowed.
583
00:38:24 --> 00:38:29
If I sample more often I'm able
to catch on to more functions.
584
00:38:29 --> 00:38:35
If I sample, and that's what,
I mean, now, communications
585
00:38:35 --> 00:38:37
want wide bands.
586
00:38:37 --> 00:38:41
And this is where
they get limited.
587
00:38:41 --> 00:38:46
I mean, this is, could say,
the fundamental, I don't
588
00:38:46 --> 00:38:50
know whether to say physical
limit, sort of maybe Fourier
589
00:38:50 --> 00:38:53
limit on sampling theory.
590
00:38:53 --> 00:38:56
Is exactly this
Nyquist frequency.
591
00:38:56 --> 00:39:00
OK, questions or
discussion about that.
592
00:39:00 --> 00:39:04
OK.
593
00:39:04 --> 00:39:07
So that's an example
that allowed us to
594
00:39:07 --> 00:39:13
do a lot of things.
595
00:39:13 --> 00:39:16
I did want to ask for your
help doing these evaluations.
596
00:39:16 --> 00:39:19
Let me say what I'm going
to do this afternoon.
597
00:39:19 --> 00:39:25
I'm going to answer all the
questions I can, and I planned,
598
00:39:25 --> 00:39:30
when there is a pause, and
nobody else asks, I plan to
599
00:39:30 --> 00:39:37
compute the Fourier integral
and Fourier series, say,
600
00:39:37 --> 00:39:47
Fourier series, for a function
that has, it's going to be like
601
00:39:47 --> 00:39:52
the one today except this is
going to have a height of
602
00:39:52 --> 00:39:56
1/h, and a width of h.
603
00:39:56 --> 00:40:03
So that's, in case you're not
able to be here this afternoon,
604
00:40:03 --> 00:40:06
I thought I'd just say in
advance what calculations
605
00:40:06 --> 00:40:07
I thought I would do.
606
00:40:07 --> 00:40:12
So there's a particular
function f(x), it happens
607
00:40:12 --> 00:40:13
to be an even function.
608
00:40:13 --> 00:40:16
We'll compute its Fourier
coefficients, in and
609
00:40:16 --> 00:40:20
we'll let h go to zero.
610
00:40:20 --> 00:40:21
To see what happens.
611
00:40:21 --> 00:40:23
It's just a good example
that you may have
612
00:40:23 --> 00:40:27
seen on older exams.
613
00:40:27 --> 00:40:32
OK, well can I just
say a personal word
614
00:40:32 --> 00:40:34
before I pass out.
615
00:40:34 --> 00:40:38
So evaluations, if you're
willing to help, and just
616
00:40:38 --> 00:40:41
leave them on the table
would be much appreciated.
617
00:40:41 --> 00:40:45
I'll stretch out the homeworks.
618
00:40:45 --> 00:40:48
I just want to say I've
enjoyed teaching you guys.
619
00:40:48 --> 00:40:49
Very much.
620
00:40:49 --> 00:40:52
Thank you all, and, thanks.
621
00:40:52 --> 00:40:56