1
00:00:00,000 --> 00:00:00,030
2
00:00:00,030 --> 00:00:02,285
The following content is
provided under a Creative
3
00:00:02,285 --> 00:00:03,610
Commons license.
4
00:00:03,610 --> 00:00:05,460
Your support will help
MIT OpenCourseWare
5
00:00:05,460 --> 00:00:09,940
continue to offer high-quality
educational resources for free.
6
00:00:09,940 --> 00:00:12,530
To make a donation, or to
view additional materials
7
00:00:12,530 --> 00:00:15,840
from hundreds of MIT courses,
visit MIT OpenCourseWare
8
00:00:15,840 --> 00:00:20,150
at ocw.mit.edu.
9
00:00:20,150 --> 00:00:21,840
PROFESSOR STRANG: OK, hi.
10
00:00:21,840 --> 00:00:25,810
So I've got homework
nine for you.
11
00:00:25,810 --> 00:00:28,020
Ready to return at the end.
12
00:00:28,020 --> 00:00:31,440
Also, the department asked
me to do evaluations,
13
00:00:31,440 --> 00:00:33,490
but that's the end
of the lecture.
14
00:00:33,490 --> 00:00:39,250
Then, so everybody knows
there's a quiz tomorrow night.
15
00:00:39,250 --> 00:00:45,030
And shall I just remember the
four questions on the quiz?
16
00:00:45,030 --> 00:00:49,580
I mean, not the details but the
general idea of the questions.
17
00:00:49,580 --> 00:00:51,110
Details OK too.
18
00:00:51,110 --> 00:00:53,070
Yes.
19
00:00:53,070 --> 00:00:58,270
Yeah, so there'll be one
question on a Fourier series.
20
00:00:58,270 --> 00:01:03,870
And you should know
the energy equality
21
00:01:03,870 --> 00:01:09,460
for all of these possibilities,
connecting the function squared
22
00:01:09,460 --> 00:01:11,740
with the coefficient squared.
23
00:01:11,740 --> 00:01:17,630
A second one on the
discrete Fourier transform.
24
00:01:17,630 --> 00:01:19,200
Cyclic stuff.
25
00:01:19,200 --> 00:01:25,060
The third one on the
Fourier integral.
26
00:01:25,060 --> 00:01:35,490
And have a look at the
applications to solving an ODE.
27
00:01:35,490 --> 00:01:38,920
I did one in class.
28
00:01:38,920 --> 00:01:42,310
The one in class was
the one in the book
29
00:01:42,310 --> 00:01:47,800
-u''+a^2*u=f(x),
so this will be.
30
00:01:47,800 --> 00:01:50,610
So have a look at
that application.
31
00:01:50,610 --> 00:01:53,770
This is, of course, on
minus infinity to infinity.
32
00:01:53,770 --> 00:02:00,310
and then a fourth
question on convolution.
33
00:02:00,310 --> 00:02:01,000
OK.
34
00:02:01,000 --> 00:02:03,610
And this afternoon,
of course, I'll
35
00:02:03,610 --> 00:02:07,950
be here to answer any
questions from the homework,
36
00:02:07,950 --> 00:02:13,270
from any source,
for these topics.
37
00:02:13,270 --> 00:02:15,060
Are there any questions
just now, though?
38
00:02:15,060 --> 00:02:19,950
I'm OK to take questions.
39
00:02:19,950 --> 00:02:24,660
I thought I'd discussed
today a topic that
40
00:02:24,660 --> 00:02:27,870
involves both Fourier series
and Fourier integrals.
41
00:02:27,870 --> 00:02:32,650
It's a kind of cool
connection and it's
42
00:02:32,650 --> 00:02:36,300
linked to the name
of Claude Shannon who
43
00:02:36,300 --> 00:02:41,440
created information theory, who
was a Bell Labs guy and then
44
00:02:41,440 --> 00:02:43,790
an MIT professor.
45
00:02:43,790 --> 00:02:49,560
So I should put his name in.
46
00:02:49,560 --> 00:02:50,060
Shannon.
47
00:02:50,060 --> 00:02:54,400
OK, so this is, yeah.
48
00:02:54,400 --> 00:02:57,160
You'll see.
49
00:02:57,160 --> 00:02:59,680
So it's not on the
quiz but it gives me
50
00:02:59,680 --> 00:03:03,790
a chance to say something
important, and at the same time
51
00:03:03,790 --> 00:03:07,140
review Fourier series
and Fourier integrals.
52
00:03:07,140 --> 00:03:11,130
So let me start
with the problem.
53
00:03:11,130 --> 00:03:17,250
The problem comes for
an A to D converter.
54
00:03:17,250 --> 00:03:19,620
So what does that mean?
55
00:03:19,620 --> 00:03:22,550
That means, this A is analog.
56
00:03:22,550 --> 00:03:28,080
That means we have a
function, A for analog.
57
00:03:28,080 --> 00:03:30,590
And D for digital.
58
00:03:30,590 --> 00:03:37,270
So we have a function,
like-- So f(x), say,
59
00:03:37,270 --> 00:03:39,650
all the way minus
infinity to infinity,
60
00:03:39,650 --> 00:03:43,300
so we'll be doing,
that's where the Fourier
61
00:03:43,300 --> 00:03:45,190
integral's going to come up.
62
00:03:45,190 --> 00:03:47,380
So that's analog.
63
00:03:47,380 --> 00:03:51,210
All x, it's some curve.
64
00:03:51,210 --> 00:03:55,940
And people build,
and you can buy,
65
00:03:55,940 --> 00:04:01,530
and they're sold in large
quantities, something that
66
00:04:01,530 --> 00:04:03,900
just samples that function.
67
00:04:03,900 --> 00:04:05,750
Say, at the integers.
68
00:04:05,750 --> 00:04:11,300
So now I'll sample
that function and let
69
00:04:11,300 --> 00:04:14,880
me take the period of the
sample to be one, so that I'm
70
00:04:14,880 --> 00:04:20,010
going to take the values f(n).
71
00:04:20,010 --> 00:04:22,350
So now I've got
something digital
72
00:04:22,350 --> 00:04:26,510
that I can work with,
that I can compute with.
73
00:04:26,510 --> 00:04:36,370
And, so the sampling theorem--
Well, I mean, the question is--
74
00:04:36,370 --> 00:04:40,450
Yeah, the sampling theorem
is about this question,
75
00:04:40,450 --> 00:04:45,410
and it seems a crazy question,
when do these numbers --
76
00:04:45,410 --> 00:04:47,420
That's just a
sequence of numbers.
77
00:04:47,420 --> 00:04:52,690
This x was all the way from
minus infinity to infinity.
78
00:04:52,690 --> 00:04:58,520
And similarly, n is numbers
all the way from minus infinity
79
00:04:58,520 --> 00:05:01,520
to infinity, they're
just samples.
80
00:05:01,520 --> 00:05:06,800
When does that tell
me the function?
81
00:05:06,800 --> 00:05:10,260
When can I learn
from those samples,
82
00:05:10,260 --> 00:05:13,950
when do I have total
information about the function?
83
00:05:13,950 --> 00:05:16,170
Now, you'll say impossible.
84
00:05:16,170 --> 00:05:16,990
Right?
85
00:05:16,990 --> 00:05:24,410
So suppose I draw a
function f(x), OK?
86
00:05:24,410 --> 00:05:31,560
And I'm going to sample
it at these points.
87
00:05:31,560 --> 00:05:37,150
All the way, so these are the
numbers, these are my f(n).
88
00:05:37,150 --> 00:05:42,780
Sampling at equal
intervals, because if we
89
00:05:42,780 --> 00:05:45,510
want to use Fourier
ideas, equal spacing
90
00:05:45,510 --> 00:05:47,220
is the right thing to have.
91
00:05:47,220 --> 00:05:50,340
So when could I recover
the function in between?
92
00:05:50,340 --> 00:05:52,580
Well, you'd say, never.
93
00:05:52,580 --> 00:05:55,640
Because how do I know
what that function
94
00:05:55,640 --> 00:05:57,530
could be doing in between.
95
00:05:57,530 --> 00:06:04,640
So let me take the case when
all the samples are zero.
96
00:06:04,640 --> 00:06:06,540
And let's think about that case.
97
00:06:06,540 --> 00:06:12,700
Suppose, what could the
function be if all the samples,
98
00:06:12,700 --> 00:06:18,500
if these are all
zeroes, forever.
99
00:06:18,500 --> 00:06:21,900
OK, well there's one
leading candidate
100
00:06:21,900 --> 00:06:26,830
for the function,
the zero function.
101
00:06:26,830 --> 00:06:31,370
Now, you'll see the whole
point of the sampling theorem
102
00:06:31,370 --> 00:06:36,140
if you think about other,
what other functions?
103
00:06:36,140 --> 00:06:37,460
Familiar functions, yeah.
104
00:06:37,460 --> 00:06:42,030
I mean, we could, obviously,
any, all sorts of things.
105
00:06:42,030 --> 00:06:45,670
But since we're
doing Fourier, we
106
00:06:45,670 --> 00:06:51,110
like to pick on the sines,
cosines, the special functions,
107
00:06:51,110 --> 00:06:53,440
and think about
those in particular.
108
00:06:53,440 --> 00:06:57,100
So, somebody said sines.
109
00:06:57,100 --> 00:07:02,850
Now, what function, so a
sine function certainly,
110
00:07:02,850 --> 00:07:07,300
the sine function hits
zero infinitely often.
111
00:07:07,300 --> 00:07:14,160
What frequency, so sine of what
would give me the same answer?
112
00:07:14,160 --> 00:07:16,440
The same samples.
113
00:07:16,440 --> 00:07:19,930
If I put this sine function
that you're going to tell me,
114
00:07:19,930 --> 00:07:24,450
so you're going to tell
me it's sine of something
115
00:07:24,450 --> 00:07:27,540
will have these
zero values at all
116
00:07:27,540 --> 00:07:32,390
the integers, at zero, one, two,
minus one, minus two, and so
117
00:07:32,390 --> 00:07:32,940
on.
118
00:07:32,940 --> 00:07:36,780
So what would do the job?
119
00:07:36,780 --> 00:07:37,730
Sine of?
120
00:07:37,730 --> 00:07:45,390
Of what will hit zero.
121
00:07:45,390 --> 00:07:48,790
So I'm looking for
a sine function,
122
00:07:48,790 --> 00:07:51,620
I guess I'm looking first
for the function that
123
00:07:51,620 --> 00:07:54,130
just does that.
124
00:07:54,130 --> 00:07:56,880
And what is it? sin(pi*x).
125
00:07:56,880 --> 00:08:01,160
And now tell me some more.
126
00:08:01,160 --> 00:08:03,150
Tell me another function.
127
00:08:03,150 --> 00:08:07,360
Which will also, it
won't be that graph.
128
00:08:07,360 --> 00:08:09,620
sin(2pi*x).
129
00:08:09,620 --> 00:08:14,800
And all the rest, OK?
130
00:08:14,800 --> 00:08:18,500
Let me just use a word
that's kind of a handy word.
131
00:08:18,500 --> 00:08:22,300
Of course, let's put
zero on the list here.
132
00:08:22,300 --> 00:08:24,210
OK.
133
00:08:24,210 --> 00:08:29,000
So this is where k, the
frequency, usually appears.
134
00:08:29,000 --> 00:08:33,790
This is where k--
135
00:08:33,790 --> 00:08:37,390
The word I want to
introduce is alias.
136
00:08:37,390 --> 00:08:44,670
This frequency, pi, is an alias
for this at frequency zero.
137
00:08:44,670 --> 00:08:49,110
Here's the, it's a
different function
138
00:08:49,110 --> 00:08:51,370
but yet the samples
are the same.
139
00:08:51,370 --> 00:08:55,200
So if you're only
looking at the samples
140
00:08:55,200 --> 00:08:59,360
you're getting the
same answer but somehow
141
00:08:59,360 --> 00:09:01,070
the function has
a different name.
142
00:09:01,070 --> 00:09:05,360
So that frequency and this
frequency, and all those others
143
00:09:05,360 --> 00:09:06,880
would be alias.
144
00:09:06,880 --> 00:09:11,010
Can I just write that word
down, because you see it often.
145
00:09:11,010 --> 00:09:13,760
Alias.
146
00:09:13,760 --> 00:09:16,850
That means two frequencies,
like pi and 2pi,
147
00:09:16,850 --> 00:09:20,580
and zero or whatever, that
give you the same samples.
148
00:09:20,580 --> 00:09:27,980
OK, so now comes
Shannon's question.
149
00:09:27,980 --> 00:09:32,810
So we have to make some
assumption on the function.
150
00:09:32,810 --> 00:09:38,730
To knock out those
possibilities.
151
00:09:38,730 --> 00:09:44,160
We want to know a limited
class of functions.
152
00:09:44,160 --> 00:09:47,370
Which don't include these guys.
153
00:09:47,370 --> 00:09:50,290
So that within this
limited class of functions,
154
00:09:50,290 --> 00:09:54,900
this is the only candidate
and we have this possibility
155
00:09:54,900 --> 00:09:57,110
of doing the impossible.
156
00:09:57,110 --> 00:10:02,340
Of determining that
if I know zeroes here,
157
00:10:02,340 --> 00:10:05,670
the function has to
be zero everywhere.
158
00:10:05,670 --> 00:10:08,620
OK, now the question is
what class of functions?
159
00:10:08,620 --> 00:10:13,140
We want to eliminate these
guys, and sort of, your instinct
160
00:10:13,140 --> 00:10:19,260
is, you want to eliminate
functions that, you know,
161
00:10:19,260 --> 00:10:21,770
if it's not zero then it's
got to get up and back
162
00:10:21,770 --> 00:10:24,400
down in every thing.
163
00:10:24,400 --> 00:10:26,980
It could do different things
in different intervals.
164
00:10:26,980 --> 00:10:33,160
But somehow it's got to have
some of these frequencies.
165
00:10:33,160 --> 00:10:36,370
Pi or higher.
166
00:10:36,370 --> 00:10:37,810
Would have to be in there.
167
00:10:37,810 --> 00:10:40,450
So this is the instinct.
168
00:10:40,450 --> 00:10:48,870
That if I limit the frequency
band, so I'm going to say f(x)
169
00:10:48,870 --> 00:10:53,560
is band-limited, can
I introduce that word?
170
00:10:53,560 --> 00:10:55,740
I'll maybe take a
moment just ask you
171
00:10:55,740 --> 00:10:59,480
if you've seen that word before.
172
00:10:59,480 --> 00:11:01,850
How many have seen this
word, band-limited?
173
00:11:01,850 --> 00:11:04,410
Quite a few but not half.
174
00:11:04,410 --> 00:11:05,070
OK.
175
00:11:05,070 --> 00:11:10,600
Band-limited means the band
is a band of frequencies.
176
00:11:10,600 --> 00:11:14,290
So the function's
band-limited when
177
00:11:14,290 --> 00:11:19,270
its transform, this tells me
how much of each frequency
178
00:11:19,270 --> 00:11:20,190
there is.
179
00:11:20,190 --> 00:11:23,420
If this is zero, in some band.
180
00:11:23,420 --> 00:11:32,480
In some band, let's say, all
frequencies below something.
181
00:11:32,480 --> 00:11:36,110
And let's not even
put equal in there.
182
00:11:36,110 --> 00:11:37,580
OK.
183
00:11:37,580 --> 00:11:40,190
But that's not critical.
184
00:11:40,190 --> 00:11:44,120
Band-limited, I have to tell
you the size of the band.
185
00:11:44,120 --> 00:11:48,590
And the size of the band,
the limiting frequency
186
00:11:48,590 --> 00:11:51,130
is this famous
Nyquist frequency,
187
00:11:51,130 --> 00:11:53,270
so Nyquist is a guy's name.
188
00:11:53,270 --> 00:11:57,420
And the Nyquist frequency
in our problem here is pi.
189
00:11:57,420 --> 00:12:00,090
This is the Nyquist frequency.
190
00:12:00,090 --> 00:12:08,780
If we let that frequency,
that's the borderline frequency.
191
00:12:08,780 --> 00:12:12,510
And there would be a similar
Nyquist sampling rate.
192
00:12:12,510 --> 00:12:16,680
So Nyquist is the
guy who studied
193
00:12:16,680 --> 00:12:20,220
the sort of borderline case.
194
00:12:20,220 --> 00:12:26,940
So the point is that if our,
say, band-limited by pi,
195
00:12:26,940 --> 00:12:29,710
I have to tell you, so
band-limited means there's
196
00:12:29,710 --> 00:12:31,550
some limit on the band.
197
00:12:31,550 --> 00:12:37,030
And our interest is when that
limit is the Nyquist frequency.
198
00:12:37,030 --> 00:12:43,310
The one we don't want to allow,
so we-- This is the point.
199
00:12:43,310 --> 00:12:45,320
So this will be the idea.
200
00:12:45,320 --> 00:12:52,570
That if we take this
class of function,
201
00:12:52,570 --> 00:12:56,600
that band-limited-- Those are
called band-limited functions,
202
00:12:56,600 --> 00:12:58,770
and they're band-limited
specifically
203
00:12:58,770 --> 00:13:01,940
by the Nyquist limit.
204
00:13:01,940 --> 00:13:05,050
If we take those,
then the idea is
205
00:13:05,050 --> 00:13:13,990
that then we can reconstruct
from the samples.
206
00:13:13,990 --> 00:13:17,980
Because the only function
that has zero samples in that
207
00:13:17,980 --> 00:13:20,560
class is the zero function.
208
00:13:20,560 --> 00:13:22,490
You see that class
has knocked out,
209
00:13:22,490 --> 00:13:26,130
is not allowing these guys.
210
00:13:26,130 --> 00:13:29,270
Of course, haven't
proved anything yet.
211
00:13:29,270 --> 00:13:31,920
And I haven't shown
how to reconstruct.
212
00:13:31,920 --> 00:13:35,790
Well, of course, we quickly
reconstructed the zero function
213
00:13:35,790 --> 00:13:41,200
out of those zeroes, but now
let me take another obviously
214
00:13:41,200 --> 00:13:43,570
important possible sample.
215
00:13:43,570 --> 00:13:48,560
Suppose I get zero samples
except at that point, where
216
00:13:48,560 --> 00:13:51,910
it's one.
217
00:13:51,910 --> 00:14:00,270
OK, now the question is
what function, f(x)--
218
00:14:00,270 --> 00:14:04,490
Can I fill in, in between zero,
zero, zero, zero, one, zero,
219
00:14:04,490 --> 00:14:11,010
zero, zero, can I fill in
exactly one function that comes
220
00:14:11,010 --> 00:14:14,040
from this class?
221
00:14:14,040 --> 00:14:19,030
So that I have now the answer
for this highly important
222
00:14:19,030 --> 00:14:19,930
sample?
223
00:14:19,930 --> 00:14:23,310
The sample that's all zeroes
except for the delta sample,
224
00:14:23,310 --> 00:14:24,420
you could say.
225
00:14:24,420 --> 00:14:27,130
OK, so I'm looking
for the function now
226
00:14:27,130 --> 00:14:30,760
which is one at that point.
227
00:14:30,760 --> 00:14:32,780
And zero at the others.
228
00:14:32,780 --> 00:14:36,150
So here's a key function.
229
00:14:36,150 --> 00:14:39,370
And I'll show you what it is.
230
00:14:39,370 --> 00:14:43,980
So the function, this function
that I'm going to mention,
231
00:14:43,980 --> 00:14:49,860
will get down here, it'll
oscillate, it'll go forever.
232
00:14:49,860 --> 00:14:52,450
It's not like a spline.
233
00:14:52,450 --> 00:14:58,030
Splines made it to zero
and stayed there, right?
234
00:14:58,030 --> 00:15:00,140
The cubic spline, for example.
235
00:15:00,140 --> 00:15:08,480
OK, but I guess, yeah,
that somehow that function,
236
00:15:08,480 --> 00:15:11,160
we're not in that league.
237
00:15:11,160 --> 00:15:13,650
We're in this
band-limited league.
238
00:15:13,650 --> 00:15:20,080
In a way you could
say that, I mean,
239
00:15:20,080 --> 00:15:24,140
what's the key connection
between dropoff of the Fourier
240
00:15:24,140 --> 00:15:25,280
transform?
241
00:15:25,280 --> 00:15:28,790
So if the Fourier
transform drops off fast,
242
00:15:28,790 --> 00:15:31,640
what does that tell
me about the function?
243
00:15:31,640 --> 00:15:32,850
It's smooth, thanks.
244
00:15:32,850 --> 00:15:34,260
That's exactly the right word.
245
00:15:34,260 --> 00:15:36,280
If the Fourier transform
drops off fast,
246
00:15:36,280 --> 00:15:38,080
the function is smooth.
247
00:15:38,080 --> 00:15:42,060
OK, this is really
an extreme case.
248
00:15:42,060 --> 00:15:46,470
That it's dropped off totally.
249
00:15:46,470 --> 00:15:50,770
You know, it's not just decay
rate, it's just zonk, out.
250
00:15:50,770 --> 00:15:53,870
Beyond this band of frequencies.
251
00:15:53,870 --> 00:15:57,700
And so that gives,
you could say, sort
252
00:15:57,700 --> 00:15:59,620
of a hyper-smooth function.
253
00:15:59,620 --> 00:16:01,890
I mean, so smooth that
you know everything
254
00:16:01,890 --> 00:16:03,130
by knowing the sample.
255
00:16:03,130 --> 00:16:07,710
OK, now I'm ready to write
down the key function,
256
00:16:07,710 --> 00:16:11,420
a famous function that
has those samples.
257
00:16:11,420 --> 00:16:17,960
And that function
is sin(pi*x)/(pi*x).
258
00:16:17,960 --> 00:16:21,820
259
00:16:21,820 --> 00:16:24,450
I don't know if you ever
thought about this function,
260
00:16:24,450 --> 00:16:25,640
and it has a name.
261
00:16:25,640 --> 00:16:27,700
Do you know its name?
262
00:16:27,700 --> 00:16:28,220
Sinc.
263
00:16:28,220 --> 00:16:29,980
It's the sinc function.
264
00:16:29,980 --> 00:16:32,060
Which is a little--
you know, the name's
265
00:16:32,060 --> 00:16:34,390
a little unfortunate.
266
00:16:34,390 --> 00:16:37,680
Mainly because you know,
you're using those same letters
267
00:16:37,680 --> 00:16:41,550
S I N but it's a
c that turns it,
268
00:16:41,550 --> 00:16:46,120
that gives it-- So this is
called the sinc function.
269
00:16:46,120 --> 00:16:47,660
sinc(x).
270
00:16:47,660 --> 00:16:50,530
But the main thing
is its formula.
271
00:16:50,530 --> 00:16:56,410
OK, well everybody sees that
at x=1, the sin(pi) is zero,
272
00:16:56,410 --> 00:17:00,600
at x=2, the sin(pi) is zero, all
these ones we've seen already.
273
00:17:00,600 --> 00:17:03,230
And now what happens
at equals zero?
274
00:17:03,230 --> 00:17:07,460
Do you recognize that this
function, as x goes to zero,
275
00:17:07,460 --> 00:17:10,230
is a perfectly good function?
276
00:17:10,230 --> 00:17:14,750
I mean, it becomes 0/0 at x=0.
277
00:17:14,750 --> 00:17:17,860
But the limit, we
know to be one.
278
00:17:17,860 --> 00:17:18,360
Right?
279
00:17:18,360 --> 00:17:25,550
This sin(theta)/theta is one as
theta approaches zero, right?
280
00:17:25,550 --> 00:17:28,630
So that does have
that correct sample.
281
00:17:28,630 --> 00:17:32,130
And now what's the,
I claim that that
282
00:17:32,130 --> 00:17:33,690
is a band-limited function.
283
00:17:33,690 --> 00:17:40,650
And you'll see that that
function pushes the limit.
284
00:17:40,650 --> 00:17:46,620
It's right-- Nyquist
barely lets it in.
285
00:17:46,620 --> 00:17:48,190
Now here's a calculation.
286
00:17:48,190 --> 00:17:50,470
So this is our practice.
287
00:17:50,470 --> 00:17:59,360
What is f hat of k
for that function?
288
00:17:59,360 --> 00:18:01,450
Now, let me think
how to do this one.
289
00:18:01,450 --> 00:18:07,720
So just to understand
this better,
290
00:18:07,720 --> 00:18:10,390
I want to see that that
is a band-limited function
291
00:18:10,390 --> 00:18:14,130
and what is its
Fourier transform.
292
00:18:14,130 --> 00:18:18,980
OK, now once again here we have
a function where if I want to
293
00:18:18,980 --> 00:18:25,220
do-- How best to do this?
294
00:18:25,220 --> 00:18:33,010
You could say well, just do it.
295
00:18:33,010 --> 00:18:35,830
As I would say on the quiz,
just go ahead and do it.
296
00:18:35,830 --> 00:18:41,470
But you'll see I'm going
to have a problem, I think.
297
00:18:41,470 --> 00:18:44,040
But this gives us a chance
to remember the formula.
298
00:18:44,040 --> 00:18:47,450
So what's the formula for the
Fourier integral transform
299
00:18:47,450 --> 00:18:48,680
of this particular?
300
00:18:48,680 --> 00:18:51,640
So my function is
the sinc function,
301
00:18:51,640 --> 00:18:55,400
sin(pi*x) over sin(pi*x).
302
00:18:55,400 --> 00:19:03,000
So how do I get its, I
do a what here? e^(-ikx),
303
00:19:03,000 --> 00:19:05,650
and am I doing dx or dk?
304
00:19:05,650 --> 00:19:07,290
dx, right?
305
00:19:07,290 --> 00:19:10,790
And I'm going from minus
infinity to infinity.
306
00:19:10,790 --> 00:19:14,640
And am I, do I have a 2pi?
307
00:19:14,640 --> 00:19:16,160
Yes or no?
308
00:19:16,160 --> 00:19:17,850
Who knows, anyway?
309
00:19:17,850 --> 00:19:21,150
Right.
310
00:19:21,150 --> 00:19:22,960
In the book I didn't?
311
00:19:22,960 --> 00:19:25,750
OK.
312
00:19:25,750 --> 00:19:33,100
Now, well, I don't
know the answer.
313
00:19:33,100 --> 00:19:37,640
But so let's, it's much better
to start with the answer,
314
00:19:37,640 --> 00:19:42,350
right, and check
that-- So let me
315
00:19:42,350 --> 00:19:44,470
say what I think the answer is.
316
00:19:44,470 --> 00:19:51,340
I think the answer is,
it's a function that's
317
00:19:51,340 --> 00:19:59,060
exactly as I say, it pushes
the limit from, this is k.
318
00:19:59,060 --> 00:20:03,770
It's the square wave, it's
zero, the height is one.
319
00:20:03,770 --> 00:20:08,010
It's the function that's
zero all the way here,
320
00:20:08,010 --> 00:20:09,660
all the way there.
321
00:20:09,660 --> 00:20:13,740
I think that that's the Fourier
transform of that function.
322
00:20:13,740 --> 00:20:20,370
And just before we check it,
see how is Nyquist got really
323
00:20:20,370 --> 00:20:22,550
pushed up to the wall, right?
324
00:20:22,550 --> 00:20:26,590
Because the frequency
is non-zero right,
325
00:20:26,590 --> 00:20:30,170
all the way through pi.
326
00:20:30,170 --> 00:20:36,020
But pi is just one point there.
327
00:20:36,020 --> 00:20:41,460
And anyway, I think we won't get
into philosophical discussion
328
00:20:41,460 --> 00:20:45,460
about whether, you know,
is that limited to pi?
329
00:20:45,460 --> 00:20:49,440
I don't know whether to put,
you saw me chicken out here.
330
00:20:49,440 --> 00:20:52,620
I didn't know whether to
put less or equal or not,
331
00:20:52,620 --> 00:20:55,640
and I still don't.
332
00:20:55,640 --> 00:21:00,770
But this is making the point
that that's the key frequency.
333
00:21:00,770 --> 00:21:03,260
So this particular
function, what I'm saying
334
00:21:03,260 --> 00:21:07,080
is this particular function
has all the frequencies
335
00:21:07,080 --> 00:21:12,480
in equal amounts over a band,
and nothing outside that band.
336
00:21:12,480 --> 00:21:14,260
And that's the Nyquist band.
337
00:21:14,260 --> 00:21:18,250
OK, now why is this the
correct answer here?
338
00:21:18,250 --> 00:21:25,080
I guess the smart way would
be, this is a good function,
339
00:21:25,080 --> 00:21:26,140
easy function.
340
00:21:26,140 --> 00:21:32,270
So let's take the transform
in the other direction.
341
00:21:32,270 --> 00:21:34,710
Start from here and get to here.
342
00:21:34,710 --> 00:21:35,210
Right?
343
00:21:35,210 --> 00:21:36,500
That would be convincing.
344
00:21:36,500 --> 00:21:41,470
Because we do know that
that pair of formulas
345
00:21:41,470 --> 00:21:46,220
for f connected to f
hat connected to f,
346
00:21:46,220 --> 00:21:47,620
they go together.
347
00:21:47,620 --> 00:21:54,750
So if I can show that I go
from here, that that Fourier
348
00:21:54,750 --> 00:21:56,710
integral takes me
from here to there,
349
00:21:56,710 --> 00:21:59,080
then this guy will take me back.
350
00:21:59,080 --> 00:22:00,550
So let me just do that.
351
00:22:00,550 --> 00:22:05,120
Because that's a very very
important one that you
352
00:22:05,120 --> 00:22:07,470
should be prepared for.
353
00:22:07,470 --> 00:22:07,970
Right.
354
00:22:07,970 --> 00:22:10,580
So now what do I want to do?
355
00:22:10,580 --> 00:22:23,780
Here's my function of k, and
I'm hoping that I recall-- Now,
356
00:22:23,780 --> 00:22:27,240
what do I do when I
want to do the transform
357
00:22:27,240 --> 00:22:29,010
in the opposite direction?
358
00:22:29,010 --> 00:22:37,120
It'll be an e^(+ikx),
right? d what? dk, now.
359
00:22:37,120 --> 00:22:39,700
From k equal minus
infinity to to infinity.
360
00:22:39,700 --> 00:22:43,800
And now I think I do put
in the 2pi, is that right?
361
00:22:43,800 --> 00:22:50,740
And the question is, does that
bring back the sinc function?
362
00:22:50,740 --> 00:22:55,640
If it does, then this was
OK in the other direction.
363
00:22:55,640 --> 00:22:57,790
If the transform's correct
in one direction then
364
00:22:57,790 --> 00:22:59,430
the inverse transform
will be correct.
365
00:22:59,430 --> 00:23:03,370
So I just plan to
do this integral. f
366
00:23:03,370 --> 00:23:07,380
hat of k, of course, is an
easy integral now. f hat of k
367
00:23:07,380 --> 00:23:13,320
is one over, between
minus pi and pi,
368
00:23:13,320 --> 00:23:17,040
so I only have to do over
that range where f hat of k
369
00:23:17,040 --> 00:23:20,480
is just a one.
370
00:23:20,480 --> 00:23:23,660
And now that's an integral
we can certainly do.
371
00:23:23,660 --> 00:23:30,490
So I have 1/(2pi), integrating
e^(ikx) will give me e^(ikx)
372
00:23:30,490 --> 00:23:33,100
over ix.
373
00:23:33,100 --> 00:23:35,680
Now, remember I'm
integrating dk.
374
00:23:35,680 --> 00:23:36,510
Oh, look.
375
00:23:36,510 --> 00:23:40,690
See, we're showing this
x now in the denominator.
376
00:23:40,690 --> 00:23:44,340
That we're hoping for.
377
00:23:44,340 --> 00:23:50,360
And now I have to do that
between k is minus pi and pi.
378
00:23:50,360 --> 00:23:56,550
So this is like,
so I get 1/(2pi),
379
00:23:56,550 --> 00:24:06,390
e^(i*pi*k) minus
e^(-i*pi*x), right?
380
00:24:06,390 --> 00:24:07,790
Over the ix.
381
00:24:07,790 --> 00:24:10,350
382
00:24:10,350 --> 00:24:13,560
OK so far?
383
00:24:13,560 --> 00:24:17,710
I was doing a k integral
and I get an x answer.
384
00:24:17,710 --> 00:24:21,350
And I want to be sure that
this x answer is the x answer I
385
00:24:21,350 --> 00:24:23,890
want, it's the sinc function.
386
00:24:23,890 --> 00:24:24,890
OK, it is.
387
00:24:24,890 --> 00:24:25,900
Right?
388
00:24:25,900 --> 00:24:31,290
I recognize the sine,
e^(i*theta)-e^(-i*theta),
389
00:24:31,290 --> 00:24:36,530
divided by two, I guess.
390
00:24:36,530 --> 00:24:38,260
Is the sine, right?
391
00:24:38,260 --> 00:24:40,700
So I have 1/(2pi).
392
00:24:40,700 --> 00:24:46,740
And here is ix-- Well, no,
the i is part of that sine.
393
00:24:46,740 --> 00:24:54,930
So I'm just using the fact that
e^(i*theta)-e^(-i*theta) is,
394
00:24:54,930 --> 00:24:59,650
this is cosine plus i
sine, subtract cosine.
395
00:24:59,650 --> 00:25:05,630
But subtract minus i sine, so
that will be 2i*sin(theta),
396
00:25:05,630 --> 00:25:07,080
right?
397
00:25:07,080 --> 00:25:11,530
We all know, and now
theta is pi*x here.
398
00:25:11,530 --> 00:25:17,780
So I have two-- let me keep
the i there, and 2i*sin(pi*x).
399
00:25:17,780 --> 00:25:20,780
You see it works.
400
00:25:20,780 --> 00:25:25,170
Just using this, replacing
this by the sine,
401
00:25:25,170 --> 00:25:27,440
the two cancels the two.
402
00:25:27,440 --> 00:25:32,060
The i cancels the i, and I
have sin(pi*x) over pi*x,
403
00:25:32,060 --> 00:25:34,010
that's the sinc function.
404
00:25:34,010 --> 00:25:42,380
OK, so that's the function
we've now checked.
405
00:25:42,380 --> 00:25:44,860
We've checked two things
about this function.
406
00:25:44,860 --> 00:25:48,190
It has the right samples,
zero, zero, zero, zero,
407
00:25:48,190 --> 00:25:51,790
one at x=0 and
then back to zero.
408
00:25:51,790 --> 00:25:55,650
It is band-limited,
so it's the guy,
409
00:25:55,650 --> 00:26:00,570
if this was my sample, if this
was my f(n), zero, zero, zero,
410
00:26:00,570 --> 00:26:05,820
one, zero, zero, zero,
then I've got it.
411
00:26:05,820 --> 00:26:10,950
It's the right function.
412
00:26:10,950 --> 00:26:15,840
OK, we can create
Shannon's sampling formula.
413
00:26:15,840 --> 00:26:20,830
Shannon's sampling formula
gives me the f(x) for any f(n).
414
00:26:20,830 --> 00:26:23,380
415
00:26:23,380 --> 00:26:25,370
Maybe you can spot that, now.
416
00:26:25,370 --> 00:26:28,200
So this is going to use
the shift invariance.
417
00:26:28,200 --> 00:26:34,120
Oh yeah, let's-- Tell me what
the, let's take one step here.
418
00:26:34,120 --> 00:26:39,300
Suppose my f(n)'s were zero,
zero, zero, zero, and a one
419
00:26:39,300 --> 00:26:40,780
there.
420
00:26:40,780 --> 00:26:46,720
So suppose-- This is, for the
exam too, this idea of shifting
421
00:26:46,720 --> 00:26:49,890
is simple and basic.
422
00:26:49,890 --> 00:26:52,270
And it's a great thing
to be able to do.
423
00:26:52,270 --> 00:26:56,050
So this wouldn't be
the right answer.
424
00:26:56,050 --> 00:26:59,230
That function produced
the one at zero.
425
00:26:59,230 --> 00:27:04,500
Tell me what function,
copying this idea,
426
00:27:04,500 --> 00:27:11,570
will produce all zeroes except
for a one at this point.
427
00:27:11,570 --> 00:27:17,400
So I'll put this
question over here.
428
00:27:17,400 --> 00:27:24,220
Suppose I'm all zeroes at that
point but at this point I'm up
429
00:27:24,220 --> 00:27:26,400
and then I go back to zeroes.
430
00:27:26,400 --> 00:27:33,440
What function is giving me that?
431
00:27:33,440 --> 00:27:36,820
You see it does decay because
of the x in the denominator,
432
00:27:36,820 --> 00:27:40,580
it kind of goes to
zero but not very fast.
433
00:27:40,580 --> 00:27:45,370
OK, what's that function?
434
00:27:45,370 --> 00:27:48,810
I replace x by x-1, right.
435
00:27:48,810 --> 00:27:53,940
So the function here
is sin(pi(x-1)),
436
00:27:53,940 --> 00:27:56,360
divided by pi(x-1).
437
00:27:56,360 --> 00:27:59,230
I just change the x to x-1.
438
00:27:59,230 --> 00:28:05,940
Now again, at x=0, this
is sin(pi), sin(-pi),
439
00:28:05,940 --> 00:28:10,500
it's safely zero, but at x=1
that's now the point where
440
00:28:10,500 --> 00:28:13,660
I'm getting 0/0.
441
00:28:13,660 --> 00:28:18,080
And the numbers are right to
give me the exact answer, one.
442
00:28:18,080 --> 00:28:24,110
OK, so now we see what to do
if the sampling turned out
443
00:28:24,110 --> 00:28:25,790
to give us this answer.
444
00:28:25,790 --> 00:28:29,440
And now can you tell
me the whole formula?
445
00:28:29,440 --> 00:28:31,660
Can you tell me
the whole formula,
446
00:28:31,660 --> 00:28:35,210
so now I'm ready for
Shannon's sampling theorem.
447
00:28:35,210 --> 00:28:41,200
Is that f(x), if
f(x) is band-limited,
448
00:28:41,200 --> 00:28:45,040
then I can tell you
what it is at all x.
449
00:28:45,040 --> 00:28:48,730
So I'm going back to the
beginning of this lecture.
450
00:28:48,730 --> 00:28:52,880
It's a miracle that
this is possible.
451
00:28:52,880 --> 00:28:57,690
That we can write down a
formula for f(x) at all x,
452
00:28:57,690 --> 00:29:01,530
only using f(n)'s.
453
00:29:01,530 --> 00:29:05,550
OK, now it's going to be a sum.
454
00:29:05,550 --> 00:29:09,250
From n equal minus
infinity to infinity,
455
00:29:09,250 --> 00:29:12,970
because I'm going to use all
the f(n)'s to produce the f(x).
456
00:29:12,970 --> 00:29:17,910
And now what do I put in there?
457
00:29:17,910 --> 00:29:23,500
So I want my formula
to be correct.
458
00:29:23,500 --> 00:29:28,810
I want my formula to be
correct in this case,
459
00:29:28,810 --> 00:29:34,130
so that if all the f's were
zero except for the middle one,
460
00:29:34,130 --> 00:29:37,200
then I want to put
sin(pi*x)/(pi*x) in there.
461
00:29:37,200 --> 00:29:39,120
The sinc function.
462
00:29:39,120 --> 00:29:41,500
And I also want to
get this one right.
463
00:29:41,500 --> 00:29:45,650
If all the f's are zero, so
there'll only be one term.
464
00:29:45,650 --> 00:29:48,920
If this is the term, I
want that to show up.
465
00:29:48,920 --> 00:29:53,710
OK, what do I-- Yeah,
you can tell me.
466
00:29:53,710 --> 00:29:57,740
Suppose this is hitting at n.
467
00:29:57,740 --> 00:30:00,640
We'll just fix this
and then you'll see it.
468
00:30:00,640 --> 00:30:06,460
Suppose that all the others, n,
n-1, n+1, all those, it's zero,
469
00:30:06,460 --> 00:30:09,630
but at x=n, it's one.
470
00:30:09,630 --> 00:30:11,730
Now what should I have chosen?
471
00:30:11,730 --> 00:30:14,420
What's the correct--
I'm just going
472
00:30:14,420 --> 00:30:19,190
to make it easy for
all of us to, yes.
473
00:30:19,190 --> 00:30:25,460
What's the good sinc function
which peaks at a point n?
474
00:30:25,460 --> 00:30:27,790
Again, I'm just
shifting it over.
475
00:30:27,790 --> 00:30:30,810
So what do I do?
476
00:30:30,810 --> 00:30:34,920
Put in, what do I write here? n.
477
00:30:34,920 --> 00:30:36,500
I shift the whole thing by n.
478
00:30:36,500 --> 00:30:39,660
So that's the right answer
when this hits at n.
479
00:30:39,660 --> 00:30:42,150
So now maybe you
see that this is
480
00:30:42,150 --> 00:30:51,970
going to be the right
answer for all of them.
481
00:30:51,970 --> 00:30:56,130
You see that, we're using
linearity and shift invariance.
482
00:30:56,130 --> 00:31:00,240
The shift invariance is
telling us this answer
483
00:31:00,240 --> 00:31:03,620
for wherever the one hits.
484
00:31:03,620 --> 00:31:05,350
That's what we need.
485
00:31:05,350 --> 00:31:08,740
And then by linearity,
I put together
486
00:31:08,740 --> 00:31:12,200
whatever the f is
at that point, that
487
00:31:12,200 --> 00:31:14,320
would just amplify the sinc.
488
00:31:14,320 --> 00:31:17,800
And then I have to put them
in for all the other values.
489
00:31:17,800 --> 00:31:20,410
That's the Shannon formula.
490
00:31:20,410 --> 00:31:25,940
That's the Shannon formula, and
this function is band-limited,
491
00:31:25,940 --> 00:31:27,160
let's see.
492
00:31:27,160 --> 00:31:29,630
What's the-- Oh, yeah.
493
00:31:29,630 --> 00:31:34,120
What's the, do you
see that this one,
494
00:31:34,120 --> 00:31:36,180
that this guy is band-limited?
495
00:31:36,180 --> 00:31:37,970
We checked, right?
496
00:31:37,970 --> 00:31:42,640
We checked that this
one, sin(pi*x)/(pi*x),
497
00:31:42,640 --> 00:31:47,190
that was band-limited because
we actually found the band.
498
00:31:47,190 --> 00:31:50,510
Now, that just gives us
another chance to think.
499
00:31:50,510 --> 00:31:51,560
Allowed.
500
00:31:51,560 --> 00:31:55,280
What's the Fourier
transform of this guy?
501
00:31:55,280 --> 00:31:58,560
My claim is that it's
also in this band.
502
00:31:58,560 --> 00:32:04,200
Non-zero only in the band, and
zero outside the Nyquist band.
503
00:32:04,200 --> 00:32:08,440
What is the transform of that?
504
00:32:08,440 --> 00:32:11,360
What happens if you
shift a function, what
505
00:32:11,360 --> 00:32:14,280
happens to its transform?
506
00:32:14,280 --> 00:32:23,670
So that's one of the key rules
that makes Fourier so special.
507
00:32:23,670 --> 00:32:28,300
If I took this sine, let
me write this guy again.
508
00:32:28,300 --> 00:32:31,710
This was the un-shifted one.
509
00:32:31,710 --> 00:32:35,180
That connected to
the, what am I going
510
00:32:35,180 --> 00:32:40,440
to call that, the box function.
511
00:32:40,440 --> 00:32:42,950
The box function,
the square wave.
512
00:32:42,950 --> 00:32:45,380
Well, box is good.
513
00:32:45,380 --> 00:32:49,040
Now, what if I
shift the function?
514
00:32:49,040 --> 00:32:50,670
If I shift a
function, what happens
515
00:32:50,670 --> 00:32:53,120
to its Fourier transform?
516
00:32:53,120 --> 00:32:55,300
Anybody remember?
517
00:32:55,300 --> 00:33:02,090
You multiply it by, so
if I shift the function,
518
00:33:02,090 --> 00:33:04,410
I just multiply
this box function,
519
00:33:04,410 --> 00:33:07,790
this is a box function in
the k, times something,
520
00:33:07,790 --> 00:33:19,140
e to the i shift, and
the shift was one, right?
521
00:33:19,140 --> 00:33:24,210
Is it just e to the ik, d
being the shift distance.
522
00:33:24,210 --> 00:33:26,340
Oh, the shift distance was n.
523
00:33:26,340 --> 00:33:27,970
Right.
524
00:33:27,970 --> 00:33:30,430
And possibly minus, who knows.
525
00:33:30,430 --> 00:33:33,500
OK, but what's the point here?
526
00:33:33,500 --> 00:33:38,100
The point is that it's
still zero outside the box.
527
00:33:38,100 --> 00:33:43,700
Inside the box, instead of being
one, it's this complex guy.
528
00:33:43,700 --> 00:33:45,660
But no change.
529
00:33:45,660 --> 00:33:47,810
It's still zero outside the box.
530
00:33:47,810 --> 00:33:49,370
It's still band-limited.
531
00:33:49,370 --> 00:33:54,010
So this is the
transform of this guy.
532
00:33:54,010 --> 00:33:56,820
And then the transform
of this combination
533
00:33:56,820 --> 00:34:00,960
would be still in
the box, multiplied
534
00:34:00,960 --> 00:34:05,130
by some messy expression.
535
00:34:05,130 --> 00:34:07,000
So what was I doing there?
536
00:34:07,000 --> 00:34:10,000
I was just checking that,
sure enough, this guy
537
00:34:10,000 --> 00:34:11,420
is band-limited.
538
00:34:11,420 --> 00:34:20,770
And it's band-limited, it gives
us the right f(n)'s, of course.
539
00:34:20,770 --> 00:34:29,180
Everybody sees that at x=n,
let's just have a look now.
540
00:34:29,180 --> 00:34:30,900
We've got this great formula.
541
00:34:30,900 --> 00:34:33,330
Plug in x=n.
542
00:34:33,330 --> 00:34:37,660
What happens when you plug
in at one of the samples,
543
00:34:37,660 --> 00:34:43,710
you look to see what this A to
D converter produced at time n,
544
00:34:43,710 --> 00:34:46,210
and let's just see.
545
00:34:46,210 --> 00:34:49,200
So at x=n, the
left side is f(n).
546
00:34:49,200 --> 00:34:52,130
Why is the right
side f of that n?
547
00:34:52,130 --> 00:34:53,440
That particular n?
548
00:34:53,440 --> 00:34:59,070
Maybe I should give a
specific letter to that n.
549
00:34:59,070 --> 00:35:04,840
So at that particular sample,
this left side is f(N),
550
00:35:04,840 --> 00:35:06,940
and I hope that the
right side gives me
551
00:35:06,940 --> 00:35:10,670
f at that capital N.
That particular one.
552
00:35:10,670 --> 00:35:12,400
Why does it?
553
00:35:12,400 --> 00:35:14,510
You're all seeing that.
554
00:35:14,510 --> 00:35:19,170
At x equal capital N,
these guys are all zero,
555
00:35:19,170 --> 00:35:21,920
except for one of them.
556
00:35:21,920 --> 00:35:27,700
Except for the one when little
n and capital N are the same.
557
00:35:27,700 --> 00:35:29,880
Then that becomes the one.
558
00:35:29,880 --> 00:35:34,440
And I'm getting f at capital
N. So it will give me that,
559
00:35:34,440 --> 00:35:39,030
for the n=1, the one term, yeah.
560
00:35:39,030 --> 00:35:41,900
I don't know if it was
necessary to say that.
561
00:35:41,900 --> 00:35:48,390
You've got the idea of
the sampling formula.
562
00:35:48,390 --> 00:35:51,770
I could say more
about the sampling,
563
00:35:51,770 --> 00:36:03,570
just to realize that the
technology, communication
564
00:36:03,570 --> 00:36:07,340
theory is always
trying to, like,
565
00:36:07,340 --> 00:36:10,480
to have a greater bandwidth.
566
00:36:10,480 --> 00:36:12,510
You always want a
greater bandwidth.
567
00:36:12,510 --> 00:36:17,220
But if the bandwidth, which
is this, is increased,
568
00:36:17,220 --> 00:36:19,240
well by the way,
what does happen?
569
00:36:19,240 --> 00:36:30,190
Suppose it's band-limited
by pi, oh, by pi/T.
570
00:36:30,190 --> 00:36:35,610
Let's just, I normalize
things to choose samples
571
00:36:35,610 --> 00:36:37,400
every integer.
572
00:36:37,400 --> 00:36:39,380
Zero, one, two, three.
573
00:36:39,380 --> 00:36:43,290
And that turned out that the
Nyquist frequency was pi.
574
00:36:43,290 --> 00:36:50,260
Now, what sampling
rate would correspond
575
00:36:50,260 --> 00:36:54,980
to this band, which
could be-- Well, let
576
00:36:54,980 --> 00:36:56,860
me just say what it is.
577
00:36:56,860 --> 00:36:59,730
That would be the
Nyquist frequency
578
00:36:59,730 --> 00:37:10,710
for sampling every T. Instead
of a sampling interval of one,
579
00:37:10,710 --> 00:37:17,920
if I sample every T, 2T, 3T,
-T, my sampling rate is T,
580
00:37:17,920 --> 00:37:21,870
so if T is small, I'm
sampling much more.
581
00:37:21,870 --> 00:37:24,910
Suppose T is 1/4.
582
00:37:24,910 --> 00:37:29,430
If T is 1/4, then I'm
doing four samplings.
583
00:37:29,430 --> 00:37:33,040
I'm taking four samples,
I'm paying more for this A
584
00:37:33,040 --> 00:37:35,720
to D converter, because it's
taking four samples where
585
00:37:35,720 --> 00:37:39,780
previously it took one.
586
00:37:39,780 --> 00:37:41,420
How do I get paid back?
587
00:37:41,420 --> 00:37:43,240
What's the reward?
588
00:37:43,240 --> 00:37:51,850
The reward is if T is 1/4,
then the Nyquist limit is 4pi.
589
00:37:51,850 --> 00:37:55,390
I can get a broader
band of signals
590
00:37:55,390 --> 00:37:58,390
by sampling them more often.
591
00:37:58,390 --> 00:38:00,490
Let me just say that
again, because that's
592
00:38:00,490 --> 00:38:03,070
the fundamental idea behind it.
593
00:38:03,070 --> 00:38:08,350
If I sample more often, say, so
fast sampling would be small--
594
00:38:08,350 --> 00:38:20,850
Fast samples would be small
t and then a higher Nyquist.
595
00:38:20,850 --> 00:38:23,140
A higher band limit.
596
00:38:23,140 --> 00:38:24,390
More functions allowed.
597
00:38:24,390 --> 00:38:29,270
If I sample more often I'm able
to catch on to more functions.
598
00:38:29,270 --> 00:38:33,190
If I sample-- And
that's what, I mean,
599
00:38:33,190 --> 00:38:37,460
now, communications
want wide bands.
600
00:38:37,460 --> 00:38:41,280
And this is where
they get limited.
601
00:38:41,280 --> 00:38:46,060
I mean, this is, you could
say, the fundamental,
602
00:38:46,060 --> 00:38:48,830
I don't know whether to
say physical limit, sort
603
00:38:48,830 --> 00:38:53,610
of maybe Fourier limit
on sampling theory.
604
00:38:53,610 --> 00:38:56,810
Is exactly this
Nyquist frequency.
605
00:38:56,810 --> 00:39:00,160
OK, questions or
discussion about that.
606
00:39:00,160 --> 00:39:04,110
OK.
607
00:39:04,110 --> 00:39:07,350
So that's an
example that allowed
608
00:39:07,350 --> 00:39:13,110
us to do a lot of things.
609
00:39:13,110 --> 00:39:16,380
I did want to ask for your
help doing these evaluations.
610
00:39:16,380 --> 00:39:19,840
Let me say what I'm going
to do this afternoon.
611
00:39:19,840 --> 00:39:24,260
I'm going to answer all
the questions I can,
612
00:39:24,260 --> 00:39:27,180
and I planned, when
there is a pause,
613
00:39:27,180 --> 00:39:34,990
and nobody else asks, I plan
to compute the Fourier integral
614
00:39:34,990 --> 00:39:38,580
and Fourier series,
say, Fourier series,
615
00:39:38,580 --> 00:39:48,510
for a function that has, it's
going to be like the one today
616
00:39:48,510 --> 00:39:53,820
except this is going to
have a height of 1/h,
617
00:39:53,820 --> 00:39:56,490
and a width of h.
618
00:39:56,490 --> 00:40:03,630
So that's, in case you're not
able to be here this afternoon,
619
00:40:03,630 --> 00:40:06,520
I thought I'd just say in
advance what calculations
620
00:40:06,520 --> 00:40:07,740
I thought I would do.
621
00:40:07,740 --> 00:40:11,570
So there's a particular
function f(x),
622
00:40:11,570 --> 00:40:13,880
it happens to be
an even function.
623
00:40:13,880 --> 00:40:16,520
We'll compute its
Fourier coefficients, in
624
00:40:16,520 --> 00:40:20,870
and we'll let h go to zero.
625
00:40:20,870 --> 00:40:21,930
To see what happens.
626
00:40:21,930 --> 00:40:23,540
It's just a good
example that you
627
00:40:23,540 --> 00:40:27,180
may have seen on older exams.
628
00:40:27,180 --> 00:40:32,740
OK, well can I just
say a personal word
629
00:40:32,740 --> 00:40:36,210
before I pass out--
So evaluations,
630
00:40:36,210 --> 00:40:39,640
if you're willing to help, and
just leave them on the table,
631
00:40:39,640 --> 00:40:41,930
would be much appreciated.
632
00:40:41,930 --> 00:40:45,630
I'll stretch out the homeworks.
633
00:40:45,630 --> 00:40:48,570
I just want to say I've
enjoyed teaching you guys.
634
00:40:48,570 --> 00:40:49,960
Very much.
635
00:40:49,960 --> 00:40:52,730
Thank you all, and-- Thanks.
636
00:40:52,730 --> 00:40:56,990