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PROFESSOR: --And go on with
lectures 8 to 10.
9
00:00:30,530 --> 00:00:33,090
First I want to briefly
review what we said
10
00:00:33,090 --> 00:00:36,320
about measurable functions.
11
00:00:36,320 --> 00:00:40,580
Again, I encourage you if you
hate this material and you
12
00:00:40,580 --> 00:00:43,700
think it's only for
mathematicians,
13
00:00:43,700 --> 00:00:45,950
please let me know.
14
00:00:45,950 --> 00:00:48,190
I don't know whether it's
appropriate to cover it in
15
00:00:48,190 --> 00:00:50,530
this class either.
16
00:00:50,530 --> 00:00:54,810
I think it probably is because
you need to know a little more
17
00:00:54,810 --> 00:01:00,340
mathematics when you deal with
these Fourier transforms and
18
00:01:00,340 --> 00:01:01,900
Fourier series.
19
00:01:04,490 --> 00:01:08,510
When you're dealing with
communication theory, then you
20
00:01:08,510 --> 00:01:10,330
need to know for signal
processing
21
00:01:10,330 --> 00:01:12,100
and things like that.
22
00:01:12,100 --> 00:01:15,750
When you get into learning a
little more, my sense at this
23
00:01:15,750 --> 00:01:21,070
point is it's much easier to
learn a good deal more than it
24
00:01:21,070 --> 00:01:23,660
is to learn just a little bit
more, because a little bit
25
00:01:23,660 --> 00:01:27,280
more you're always faced with
all of these questions of what
26
00:01:27,280 --> 00:01:29,610
does this really mean.
27
00:01:29,610 --> 00:01:33,250
It turns out that if you put
just a little bit of measure
28
00:01:33,250 --> 00:01:38,000
theory into your thinking, it
makes all these problems very
29
00:01:38,000 --> 00:01:39,710
much simpler.
30
00:01:39,710 --> 00:01:42,540
So you remember we talked about
what a measurable set
31
00:01:42,540 --> 00:01:45,400
was last time.
32
00:01:45,400 --> 00:01:50,380
What a measurable set is, is a
set which you can essentially
33
00:01:50,380 --> 00:01:55,150
break up into a countable
union of intervals, and
34
00:01:55,150 --> 00:02:03,030
something which is bounded by
a countble set of intervals
35
00:02:03,030 --> 00:02:05,340
which have arbitrarily
small measure.
36
00:02:05,340 --> 00:02:09,310
So you can take all of these
sets of zero measure --
37
00:02:09,310 --> 00:02:12,970
countable sets, cantor sets, all
of these things, and you
38
00:02:12,970 --> 00:02:16,140
can just throw all of those
out from consideration.
39
00:02:16,140 --> 00:02:19,960
That's what makes measure theory
useful and interesting
40
00:02:19,960 --> 00:02:22,530
and what simplifies things.
41
00:02:22,530 --> 00:02:26,750
So we then said that a function
is measurable if each
42
00:02:26,750 --> 00:02:31,930
of these sets here -- if the
set of times for which a is
43
00:02:31,930 --> 00:02:34,860
less than or equal to u of t is
less than or equal to b --
44
00:02:34,860 --> 00:02:37,500
these things are these
level sets here.
45
00:02:37,500 --> 00:02:42,920
It's a set of values, t, along
this axis in which the
46
00:02:42,920 --> 00:02:47,470
function is nailed between any
two of these points here.
47
00:02:52,230 --> 00:02:55,370
For a function to be measurable,
all of these sets
48
00:02:55,370 --> 00:02:57,910
in here have to be measurable.
49
00:02:57,910 --> 00:03:01,390
We haven't given you any
examples of sets which are not
50
00:03:01,390 --> 00:03:04,990
measurable, and therefore, we
haven't given you any examples
51
00:03:04,990 --> 00:03:07,720
of functions which are
not measurable.
52
00:03:07,720 --> 00:03:15,360
It's very, very difficult to
construct such functions, and
53
00:03:15,360 --> 00:03:18,480
since it's so difficult to
construct them, it's easier to
54
00:03:18,480 --> 00:03:23,510
just say that all of the
functions we can think of are
55
00:03:23,510 --> 00:03:24,540
measurable.
56
00:03:24,540 --> 00:03:27,550
That includes an awful lot more
functions than you ever
57
00:03:27,550 --> 00:03:30,640
deal with in engineering.
58
00:03:30,640 --> 00:03:37,410
If you ever make a serious
mistake in your engineering
59
00:03:37,410 --> 00:03:41,670
work because of thinking that a
function is measurable when
60
00:03:41,670 --> 00:03:45,750
it's not measurable, I
will give you $1,000.
61
00:03:45,750 --> 00:03:48,610
I make that promise that you
because I don't think it's
62
00:03:48,610 --> 00:03:51,290
ever happened in history, I
don't think it ever will
63
00:03:51,290 --> 00:03:58,440
happen in history, and that's
just how sure I am of it.
64
00:03:58,440 --> 00:04:01,580
Unless you try to deliberately
make such a mistake in order
65
00:04:01,580 --> 00:04:03,080
to collect $1,000.
66
00:04:03,080 --> 00:04:08,090
Of course, that's not fair.
67
00:04:08,090 --> 00:04:13,900
Then we said the way we define
this approximation to the
68
00:04:13,900 --> 00:04:17,650
integral, namely, we always
defined it from underneath,
69
00:04:17,650 --> 00:04:21,810
and therefore, if we started to
take these intervals here
70
00:04:21,810 --> 00:04:25,960
with epsilon and split them in
half, the thing that happens
71
00:04:25,960 --> 00:04:29,210
is you always get extra
components put in.
72
00:04:29,210 --> 00:04:33,560
So that as you make the scaling
finer and finer, what
73
00:04:33,560 --> 00:04:36,970
happens is the approximation
into the integral gets larger
74
00:04:36,970 --> 00:04:38,080
and larger.
75
00:04:38,080 --> 00:04:40,970
Because of that, if you're
dealing with a non-negative
76
00:04:40,970 --> 00:04:45,800
function, the only two things
that can happen, as you go to
77
00:04:45,800 --> 00:04:50,350
the limit of finer and finer
scaling, is one, you come to a
78
00:04:50,350 --> 00:04:53,830
finite limit, and two, you come
to an infinite limit.
79
00:04:53,830 --> 00:04:55,900
Nothing else can happen.
80
00:04:55,900 --> 00:05:00,440
You see that's remarkably simple
as mathematics go.
81
00:05:00,440 --> 00:05:04,340
This is dealing with all of
these functions you can ever
82
00:05:04,340 --> 00:05:06,980
think of, and those are the
only two things that can
83
00:05:06,980 --> 00:05:08,230
happen here.
84
00:05:12,660 --> 00:05:17,370
So then we went on to say that
a function as L1, if it's
85
00:05:17,370 --> 00:05:21,820
measurable, and if the integral
of its magnitude is
86
00:05:21,820 --> 00:05:23,830
less than infinity, and
so far we're still
87
00:05:23,830 --> 00:05:27,710
talking about real functions.
88
00:05:27,710 --> 00:05:38,520
If u of t is L1, then you can
take the integral of u of t
89
00:05:38,520 --> 00:05:41,210
and you can split it
up into two things.
90
00:05:41,210 --> 00:05:45,790
You can split it up into the set
of times over which u of t
91
00:05:45,790 --> 00:05:48,630
is positive, and the
set of times over
92
00:05:48,630 --> 00:05:51,270
which u of t is negative.
93
00:05:51,270 --> 00:05:54,880
The set of times over which u of
t is equal to zero doesn't
94
00:05:54,880 --> 00:05:56,900
contribute to the integral
at all so you can
95
00:05:56,900 --> 00:05:59,410
forget about that.
96
00:05:59,410 --> 00:06:03,040
This is well-defined as the
function is measurable.
97
00:06:03,040 --> 00:06:06,540
What's now happening is that
if the integral of this
98
00:06:06,540 --> 00:06:11,060
magnitude is less than infinity,
then both this has
99
00:06:11,060 --> 00:06:14,890
to be less than infinity, and
this has to be less than
100
00:06:14,890 --> 00:06:18,860
infinity, which says that so
long as you're dealing with L1
101
00:06:18,860 --> 00:06:24,540
functions, you never have to
worry about this nasty
102
00:06:24,540 --> 00:06:28,320
situation where the negative
part of the function is
103
00:06:28,320 --> 00:06:32,060
infinite, the positive part of
the function is infinite, and
104
00:06:32,060 --> 00:06:35,610
therefore, the difference, plus
infinity, minus infinity
105
00:06:35,610 --> 00:06:38,090
doesn't make any sense at all.
106
00:06:38,090 --> 00:06:41,360
And instead of having an
integral which is infinite or
107
00:06:41,360 --> 00:06:45,030
finite, you have a function
which might just be undefined
108
00:06:45,030 --> 00:06:46,040
completely.
109
00:06:46,040 --> 00:06:50,090
Well this says that if the
function is L1, you can't ever
110
00:06:50,090 --> 00:06:54,990
have any problem with integrals
of this sort thing
111
00:06:54,990 --> 00:06:55,710
being undefined.
112
00:06:55,710 --> 00:06:59,800
So, and in fact, u of t is
always integral and has a
113
00:06:59,800 --> 00:07:02,560
finite value in this case.
114
00:07:02,560 --> 00:07:06,340
Now, we say that a complex
function is measurable if both
115
00:07:06,340 --> 00:07:09,210
the real part is measurable
and the imaginary part is
116
00:07:09,210 --> 00:07:10,350
measurable.
117
00:07:10,350 --> 00:07:13,470
Therefore, you don't have to
worry about complex functions
118
00:07:13,470 --> 00:07:18,000
as really being any more than
twice as complicated as real
119
00:07:18,000 --> 00:07:20,530
functions, and conceptually
they aren't any more
120
00:07:20,530 --> 00:07:24,340
complicated at all because you
just treat them as two
121
00:07:24,340 --> 00:07:27,100
separate functions, and
everything you know about real
122
00:07:27,100 --> 00:07:32,170
functions you now know about
complex functions.
123
00:07:32,170 --> 00:07:34,570
Now, as far as Fourier theory
is concerned --
124
00:07:34,570 --> 00:07:39,490
Fourier series, Fourier
integrals, discrete time,
125
00:07:39,490 --> 00:07:45,200
Fourier transforms -- all of
these things, if u of t is an
126
00:07:45,200 --> 00:07:51,880
L1 function, then u of t times e
to the 2 pi ft has to be L1,
127
00:07:51,880 --> 00:07:55,350
and this has to be true for
all possible values of f.
128
00:07:55,350 --> 00:07:56,860
Why is that?
129
00:07:56,860 --> 00:08:00,330
Well, it's just that the
absolute value of u of t for
130
00:08:00,330 --> 00:08:04,650
any given t, is exactly the same
as the absolute value of
131
00:08:04,650 --> 00:08:08,510
u of t times e to
the 2 pi i ft.
132
00:08:08,510 --> 00:08:13,170
Namely, this is a quantity whose
magnitude is always 1,
133
00:08:13,170 --> 00:08:18,870
and therefore, this quantity,
this magnitude is equal to
134
00:08:18,870 --> 00:08:20,850
this magnitude.
135
00:08:20,850 --> 00:08:24,150
Therefore, when you integrate
this magnitude, you get the
136
00:08:24,150 --> 00:08:28,460
same thing as when you integrate
this magnitude.
137
00:08:28,460 --> 00:08:30,830
So, there's no problem here.
138
00:08:33,370 --> 00:08:37,640
So what that says, using this
idea, too, of positive and
139
00:08:37,640 --> 00:08:41,420
negative parts, it says that the
integral of u of t times e
140
00:08:41,420 --> 00:08:47,710
to the 2 pi i ft dt has to
exist for all real f.
141
00:08:47,710 --> 00:08:51,300
That covers Fourier series
and Fourier integral.
142
00:08:51,300 --> 00:08:54,980
We haven't talked about the
Fourier integral yet, but it
143
00:08:54,980 --> 00:09:00,510
says that just by defining u of
t to be L1, you avoid all
144
00:09:00,510 --> 00:09:04,570
of these problems of when these
Fourier integrals exist
145
00:09:04,570 --> 00:09:05,700
and when they don't exist.
146
00:09:05,700 --> 00:09:06,950
They always exist.
147
00:09:11,580 --> 00:09:14,100
So let's talk about the Fourier
transform then since
148
00:09:14,100 --> 00:09:18,140
we're backing into this
slowly anyway.
149
00:09:18,140 --> 00:09:21,540
What you've learned in probably
two or three classes
150
00:09:21,540 --> 00:09:25,490
by now is that the Fourier
transform is
151
00:09:25,490 --> 00:09:27,630
defined in this way.
152
00:09:27,630 --> 00:09:31,740
Namely, the transform, a
function of frequency, u hat
153
00:09:31,740 --> 00:09:37,120
of f, we use hats here instead
of capital letters because we
154
00:09:37,120 --> 00:09:41,160
like to use capital letters
for random variables.
155
00:09:41,160 --> 00:09:44,500
So that this transform here
is the integral from minus
156
00:09:44,500 --> 00:09:49,140
infinity to infinity of the
original function, we start
157
00:09:49,140 --> 00:09:53,950
with u of t times e to the
minus 2 pi ift dt.
158
00:09:53,950 --> 00:09:55,950
Now, what's the nice
part about this?
159
00:09:55,950 --> 00:10:02,600
The nice part about this is that
if u of t is L1, then in
160
00:10:02,600 --> 00:10:08,590
fact this Fourier transform
has to exist everywhere.
161
00:10:08,590 --> 00:10:12,710
Namely, what you've learned
before in all of these classes
162
00:10:12,710 --> 00:10:17,600
says essentially that if these
functions are well-behaved
163
00:10:17,600 --> 00:10:20,140
then these transforms exist.
164
00:10:20,140 --> 00:10:23,750
What you've learned about
well-behaved is completely
165
00:10:23,750 --> 00:10:27,060
circular, namely, a function
is well-behaved if the
166
00:10:27,060 --> 00:10:31,040
transform exists, and the
transform exists if the
167
00:10:31,040 --> 00:10:32,690
function is well-behaved.
168
00:10:32,690 --> 00:10:38,670
You have no idea of what kinds
of functions the Fourier
169
00:10:38,670 --> 00:10:40,740
transform exists and
what kinds of
170
00:10:40,740 --> 00:10:42,270
functions it doesn't exist.
171
00:10:42,270 --> 00:10:45,650
There's another thing buried
in here, in this transform
172
00:10:45,650 --> 00:10:46,980
relationship.
173
00:10:46,980 --> 00:10:50,660
Namely, the first thing is we're
trying to define the
174
00:10:50,660 --> 00:10:54,390
transform function of frequency
in terms of the
175
00:10:54,390 --> 00:10:56,020
function of time.
176
00:10:56,020 --> 00:10:57,860
Well and good.
177
00:10:57,860 --> 00:11:01,810
Then we try to define a function
of time in terms of a
178
00:11:01,810 --> 00:11:04,700
function of frequency.
179
00:11:04,700 --> 00:11:10,130
What's hidden here is if you
start with u of t, you then
180
00:11:10,130 --> 00:11:13,770
get a function of frequency,
and this sort of implicitly
181
00:11:13,770 --> 00:11:16,040
says that you get back
again the same
182
00:11:16,040 --> 00:11:18,680
thing you started with.
183
00:11:18,680 --> 00:11:21,730
That, in fact, is the most
ticklish part of all of this.
184
00:11:21,730 --> 00:11:27,270
So that if we start with an L1
function u of t, we wind up
185
00:11:27,270 --> 00:11:30,590
with a Fourier transform,
u hat of f.
186
00:11:30,590 --> 00:11:32,670
Then u hat of f goes in here.
187
00:11:32,670 --> 00:11:37,620
We hope we might get back
the sum transform here.
188
00:11:37,620 --> 00:11:40,420
The nasty thing, and one of the
reasons we're going to get
189
00:11:40,420 --> 00:11:47,080
away from L1 in just a minute,
is that if a function is L1,
190
00:11:47,080 --> 00:11:51,200
it's Fourier transform is
not necessarily L1.
191
00:11:51,200 --> 00:11:54,720
What that means is that you
have to learn all of this
192
00:11:54,720 --> 00:11:58,130
stuff about L1 functions, and
then as soon as you take the
193
00:11:58,130 --> 00:12:03,810
Fourier transform, bingo, it's
all gone up in a lot of smoke,
194
00:12:03,810 --> 00:12:07,730
and you have to start all over
again saying something about
195
00:12:07,730 --> 00:12:10,820
what properties the transform
might have.
196
00:12:10,820 --> 00:12:14,790
But anyway, it's nice to start
with because when u of t is
197
00:12:14,790 --> 00:12:18,745
L1, we know that this function
actually exists.
198
00:12:18,745 --> 00:12:21,120
It actually exists as
a complex number.
199
00:12:21,120 --> 00:12:23,750
It exists as a complex
number for every
200
00:12:23,750 --> 00:12:26,160
possible real f here.
201
00:12:26,160 --> 00:12:29,840
Namely, there aren't any if,
and's, but's or maybe's here,
202
00:12:29,840 --> 00:12:32,290
there's nothing like L2
convergence that we were
203
00:12:32,290 --> 00:12:35,260
talking about a little
bit before.
204
00:12:35,260 --> 00:12:37,160
This just exists, period.
205
00:12:40,340 --> 00:12:45,070
There's something more here,
that if this is L1, this not
206
00:12:45,070 --> 00:12:48,945
only exists but it's also a
continuous function, and those
207
00:12:48,945 --> 00:12:51,030
don't prove that.
208
00:12:51,030 --> 00:12:54,860
If you've taken a course in
analysis and you know what a
209
00:12:54,860 --> 00:12:58,850
complex function is and you're
quite patient, you can sit
210
00:12:58,850 --> 00:13:02,090
down and actually show
this yourselves, but
211
00:13:02,090 --> 00:13:03,340
it's a bit of a pain.
212
00:13:05,750 --> 00:13:07,890
So for a well-behaved function,
the first integral
213
00:13:07,890 --> 00:13:11,780
exists for all f, the second
exists for all t, and results
214
00:13:11,780 --> 00:13:13,670
in the original u of t.
215
00:13:13,670 --> 00:13:17,570
But then more specifically,
if u of t is L1, the first
216
00:13:17,570 --> 00:13:21,740
integral exists for all f and
it's a continuous function.
217
00:13:21,740 --> 00:13:24,950
It's also a bounded function, as
we'll see in a little bit.
218
00:13:24,950 --> 00:13:28,510
If u hat of f is L1, the second
integral exists for all
219
00:13:28,510 --> 00:13:31,410
t, and u of t is continuous.
220
00:13:31,410 --> 00:13:35,900
Therefore, if you assume at
the output at the onset of
221
00:13:35,900 --> 00:13:40,180
things that both this is
L1 and this is L1, then
222
00:13:40,180 --> 00:13:43,400
everything is also continuous
and you have a very nice
223
00:13:43,400 --> 00:13:46,840
theory, which doesn't apply to
too many of the things that
224
00:13:46,840 --> 00:13:48,730
you're interested in.
225
00:13:48,730 --> 00:13:50,720
And I'll explain why
in a little bit.
226
00:13:54,230 --> 00:13:58,510
Anyway, for these well-behaved
functions, we have all of
227
00:13:58,510 --> 00:14:02,640
these relationships that I'm
sure you've learned in
228
00:14:02,640 --> 00:14:04,820
whatever linear systems
course you've taken.
229
00:14:07,400 --> 00:14:09,980
Since you should know all of
these things, I just want to
230
00:14:09,980 --> 00:14:14,640
briefly talk about them I mean
this linearity idea is
231
00:14:14,640 --> 00:14:16,380
something you would
just use without
232
00:14:16,380 --> 00:14:19,210
thinking about it, I think.
233
00:14:19,210 --> 00:14:21,730
In other words, if you'd never
learned this and you were
234
00:14:21,730 --> 00:14:25,830
trying to work out a problem you
would just use it anyway,
235
00:14:25,830 --> 00:14:30,380
because anything respectable has
to have -- well, anything
236
00:14:30,380 --> 00:14:32,820
respectable, again,
means anything
237
00:14:32,820 --> 00:14:34,070
which has this property.
238
00:14:36,620 --> 00:14:39,590
This conjugate property you can
derive that easily from
239
00:14:39,590 --> 00:14:43,050
the Fourier transform
relationships also, if you
240
00:14:43,050 --> 00:14:45,010
have a well-behaved function.
241
00:14:45,010 --> 00:14:49,760
This quantity here, this
duality, is particularly
242
00:14:49,760 --> 00:14:57,890
interesting because it isn't
really duality, it's something
243
00:14:57,890 --> 00:15:00,820
called hermitian duality.
244
00:15:00,820 --> 00:15:04,650
You start out with this formula
here to go to there
245
00:15:04,650 --> 00:15:08,650
and you use almost the same
formula to get back again.
246
00:15:08,650 --> 00:15:13,710
The only difference is instead
of a minus 2 pi ift, you have
247
00:15:13,710 --> 00:15:15,960
a plus 2 pi ift.
248
00:15:15,960 --> 00:15:19,400
In other words, this is the
conjugate of this, which is
249
00:15:19,400 --> 00:15:22,480
why this is called hermitian
duality.
250
00:15:22,480 --> 00:15:27,100
But aside from that, everything
you learn about the
251
00:15:27,100 --> 00:15:31,480
Fourier transform, you also
automatically know about the
252
00:15:31,480 --> 00:15:35,740
inverse Fourier transform for
these well-behaved functions.
253
00:15:38,360 --> 00:15:40,980
Otherwise you don't know whether
the other one exists
254
00:15:40,980 --> 00:15:45,170
or not, and we'll certainly
get into that.
255
00:15:45,170 --> 00:15:49,290
So, the duality is expressed
this way, the Fourier
256
00:15:49,290 --> 00:15:54,520
transform of, bleah.
257
00:15:54,520 --> 00:15:58,900
If you take a function, u of
t, and regard that as a
258
00:15:58,900 --> 00:16:04,410
Fourier transform, then --
259
00:16:04,410 --> 00:16:07,800
I always have trouble
saying this.
260
00:16:07,800 --> 00:16:11,660
If you take a function, u of t,
and then you regard that as
261
00:16:11,660 --> 00:16:13,480
a function of frequency --
262
00:16:13,480 --> 00:16:16,840
OK, that's this -- and then you
regard it as a function of
263
00:16:16,840 --> 00:16:21,170
minus frequency, namely, you
substitute minus f for t in
264
00:16:21,170 --> 00:16:23,330
whatever time function
you start with.
265
00:16:23,330 --> 00:16:27,170
You start with a time function u
of t, you substitute minus f
266
00:16:27,170 --> 00:16:31,300
for t, which gives you a
function of frequency.
267
00:16:31,300 --> 00:16:35,780
The inverse Fourier transform
of that is what you get by
268
00:16:35,780 --> 00:16:40,660
taking the Fourier transform
of u of t, and then
269
00:16:40,660 --> 00:16:44,360
substituting t for f in it.
270
00:16:44,360 --> 00:16:47,990
It's much harder to
say it than to do.
271
00:16:47,990 --> 00:16:50,270
This time shift, you've
all seen that.
272
00:16:50,270 --> 00:16:53,880
If you shift a function in
time, the only thing that
273
00:16:53,880 --> 00:16:58,150
happens is you get this
rotating term in it.
274
00:16:58,150 --> 00:17:02,410
Same thing for a frequency
shift.
275
00:17:02,410 --> 00:17:05,200
You want to have an interesting
exercise, take
276
00:17:05,200 --> 00:17:09,320
time shift plus duality and
derive the scaling and
277
00:17:09,320 --> 00:17:13,380
frequency from it -- it has to
work, and of course, it does.
278
00:17:13,380 --> 00:17:14,640
Scaling --
279
00:17:14,640 --> 00:17:17,360
there's this relationship
here.
280
00:17:17,360 --> 00:17:20,440
This one is always
a funny one.
281
00:17:20,440 --> 00:17:25,480
It's a little strange because
when you scale here, it's not
282
00:17:25,480 --> 00:17:29,250
too surprising that when you
take a function and you squash
283
00:17:29,250 --> 00:17:32,120
it down that the Fourier
transform
284
00:17:32,120 --> 00:17:35,160
gets squashed upwards.
285
00:17:35,160 --> 00:17:38,840
Because in a sense what you're
doing when you squash a
286
00:17:38,840 --> 00:17:42,040
function down you're making
everything happen faster than
287
00:17:42,040 --> 00:17:45,080
it did before, which means that
all the frequencies in it
288
00:17:45,080 --> 00:17:47,420
are going to get higher
than they are before.
289
00:17:47,420 --> 00:17:50,780
But also when you squash it
down, the amount of energy in
290
00:17:50,780 --> 00:17:53,570
the function is going
to go down.
291
00:17:53,570 --> 00:17:56,600
One of the most important
properties that we're going to
292
00:17:56,600 --> 00:18:00,470
find and what you ought to know
already is that when you
293
00:18:00,470 --> 00:18:05,790
take the energy in a function,
you get the same answer as you
294
00:18:05,790 --> 00:18:09,190
get when you take the energy
in the Fourier transform.
295
00:18:09,190 --> 00:18:13,940
Namely, you integrate u of f
squared over frequency and you
296
00:18:13,940 --> 00:18:16,620
get the same as if you
integrate u of t
297
00:18:16,620 --> 00:18:18,450
squared over time.
298
00:18:18,450 --> 00:18:20,710
That's an important check
that you use on
299
00:18:20,710 --> 00:18:22,560
all sorts of things.
300
00:18:22,560 --> 00:18:25,490
The thing that happens now then
when you're scaling is
301
00:18:25,490 --> 00:18:28,550
when you scale a function, u
of t, you bring it down and
302
00:18:28,550 --> 00:18:30,950
you spread it out when
you bring it down.
303
00:18:30,950 --> 00:18:36,070
The frequency function goes up
so the energy in the time
304
00:18:36,070 --> 00:18:39,590
function goes down, the energy
in the frequency function goes
305
00:18:39,590 --> 00:18:44,400
up, and you need something in
order to keep the energy
306
00:18:44,400 --> 00:18:47,090
relationship working properly.
307
00:18:47,090 --> 00:18:48,480
This is what you need.
308
00:18:48,480 --> 00:18:50,260
Actually, if you derive
this, the t
309
00:18:50,260 --> 00:18:53,360
just falls out naturally.
310
00:18:53,360 --> 00:19:00,210
So we get the same thing if we
do scaling and frequency.
311
00:19:00,210 --> 00:19:03,010
I don't think I put that
down but it's the same
312
00:19:03,010 --> 00:19:04,090
relationship.
313
00:19:04,090 --> 00:19:06,110
There's differentiation.
314
00:19:06,110 --> 00:19:10,230
Differentiation we won't talk
about or use it a whole lot.
315
00:19:10,230 --> 00:19:14,120
All of these things turn out
to be remarkably robust.
316
00:19:14,120 --> 00:19:18,820
When you're dealing with L1
functions or L2 functions and
317
00:19:18,820 --> 00:19:21,300
you scale them or you shift
them or do any of those
318
00:19:21,300 --> 00:19:25,980
things, if they're L1, they're
still L1 after you're through,
319
00:19:25,980 --> 00:19:28,910
if they're L2, they're L2
after you're through.
320
00:19:28,910 --> 00:19:31,760
If you differentiate
a function, all
321
00:19:31,760 --> 00:19:33,000
those properties change.
322
00:19:33,000 --> 00:19:35,950
You can't be sure of
anything anymore.
323
00:19:35,950 --> 00:19:39,380
There's convolution, which I'm
sure you've derived many
324
00:19:39,380 --> 00:19:44,290
times, and which one of the
exercises derives again.
325
00:19:44,290 --> 00:19:52,370
There's correlation, and what
this sort of relationship says
326
00:19:52,370 --> 00:19:55,790
is taking products at the
frequency domain is the same
327
00:19:55,790 --> 00:20:00,220
as going through this
convolution relationship and
328
00:20:00,220 --> 00:20:01,470
the time domain.
329
00:20:01,470 --> 00:20:04,930
Of course, there's a dual
relation to that, which you
330
00:20:04,930 --> 00:20:07,680
don't use very often but
it still exists.
331
00:20:07,680 --> 00:20:16,280
Correlation is you actually get
correlation by using one
332
00:20:16,280 --> 00:20:20,420
of the conjugate properties on
the convolution and I'm sure
333
00:20:20,420 --> 00:20:22,060
you've all seen that.
334
00:20:25,640 --> 00:20:29,700
That's something you should
all have been using and
335
00:20:29,700 --> 00:20:31,910
familiar with for a long time.
336
00:20:31,910 --> 00:20:38,100
Two special cases of the Fourier
transform is that u of
337
00:20:38,100 --> 00:20:42,990
zero is what happens when you
take the Fourier transform and
338
00:20:42,990 --> 00:20:45,440
you evaluate it at
t equals zero.
339
00:20:45,440 --> 00:20:50,570
You get u of zero is just the
integral of u hat of f.
340
00:20:50,570 --> 00:20:55,090
u hat of zero is just the
integral of u of t.
341
00:20:55,090 --> 00:20:57,560
What do you use these for?
342
00:20:57,560 --> 00:21:01,350
Well the thing I use them for
is this half the time when
343
00:21:01,350 --> 00:21:06,630
you're working out a problem
it's obvious by inspection
344
00:21:06,630 --> 00:21:10,240
what this integral is or it's
obvious by inspection what
345
00:21:10,240 --> 00:21:16,210
this integral is, and by doing
that you can check whether you
346
00:21:16,210 --> 00:21:18,500
have all the constants
right in the
347
00:21:18,500 --> 00:21:20,880
transform that you've taken.
348
00:21:20,880 --> 00:21:24,270
I don't know anybody who can
take Fourier transforms
349
00:21:24,270 --> 00:21:27,480
without getting at least one
constant wrong at least half
350
00:21:27,480 --> 00:21:29,440
the time they do it.
351
00:21:29,440 --> 00:21:32,210
I probably get one constant
wrong about three-quarters of
352
00:21:32,210 --> 00:21:35,500
the time that I do it, and I'm
sure I'll do it here in class
353
00:21:35,500 --> 00:21:36,330
a number of times.
354
00:21:36,330 --> 00:21:43,720
I hope you find it, but it's one
of these things we all do.
355
00:21:43,720 --> 00:21:46,610
This is one of the best ways
of finding out what you've
356
00:21:46,610 --> 00:21:49,700
done and going back
and checking it.
357
00:21:49,700 --> 00:21:55,460
Parseval's Theorem, Parseval's
Theorem is really just this
358
00:21:55,460 --> 00:21:59,730
convolution equation which
we're applying
359
00:21:59,730 --> 00:22:02,520
at tau equals zero.
360
00:22:02,520 --> 00:22:05,490
You take the convolution, you
apply it at tau equals zero
361
00:22:05,490 --> 00:22:06,350
and what do you get?
362
00:22:06,350 --> 00:22:12,620
You get the integral of u of t
times some other conjugate --
363
00:22:12,620 --> 00:22:15,610
the conjugate of the other
function is equal to the
364
00:22:15,610 --> 00:22:21,560
integral of u hat of
f times the complex
365
00:22:21,560 --> 00:22:24,190
conjugate of v of f.
366
00:22:26,840 --> 00:22:31,050
Much more important than this
is what happens if v happens
367
00:22:31,050 --> 00:22:34,980
to be the same as u, and that
gives you the energy equation
368
00:22:34,980 --> 00:22:36,640
here, which is what I
was talking about.
369
00:22:36,640 --> 00:22:42,720
It says the integral in a
function you can find it two
370
00:22:42,720 --> 00:22:45,580
ways, either by looking
at it in time or by
371
00:22:45,580 --> 00:22:47,810
looking at it in frequency.
372
00:22:47,810 --> 00:22:50,450
I urge you to always think
about doing this whenever
373
00:22:50,450 --> 00:22:55,000
you're working problems, because
often the Fourier
374
00:22:55,000 --> 00:22:59,520
integral it's very easy to find
the integral and the and
375
00:22:59,520 --> 00:23:02,300
the time function is
very difficult.
376
00:23:02,300 --> 00:23:05,570
A good example of this
is sinc functions
377
00:23:05,570 --> 00:23:08,220
and rectangular functions.
378
00:23:08,220 --> 00:23:12,800
Anybody can take a rectangular
function, square it and
379
00:23:12,800 --> 00:23:14,470
integrate it.
380
00:23:14,470 --> 00:23:17,770
It takes a good deal of skill
if you don't use this
381
00:23:17,770 --> 00:23:21,890
relationship to take a sinc
function, to square it and to
382
00:23:21,890 --> 00:23:24,760
integrate it.
383
00:23:24,760 --> 00:23:28,630
You can do it if you're skillful
at integration, you
384
00:23:28,630 --> 00:23:31,170
might regard it as a challenge,
but after you get
385
00:23:31,170 --> 00:23:34,550
done you realize that you've
really wasted a lot of time
386
00:23:34,550 --> 00:23:37,170
because this is the right
way of doing it here.
387
00:23:43,980 --> 00:23:48,480
Now, as I mentioned before,
it's starting to look like
388
00:23:48,480 --> 00:23:52,750
Fourier series and Fourier
integrals are much nicer when
389
00:23:52,750 --> 00:23:58,450
you have L1 functions, and they
are, but L1 functions are
390
00:23:58,450 --> 00:24:01,880
not terribly useful
as far as most
391
00:24:01,880 --> 00:24:04,710
communication functions go.
392
00:24:04,710 --> 00:24:08,690
In other words, not enough
functions are L1 to provide
393
00:24:08,690 --> 00:24:11,860
suitable models for the
communication systems that we
394
00:24:11,860 --> 00:24:13,110
want to look at.
395
00:24:16,710 --> 00:24:19,820
In fact, most of the models that
we're going to look at,
396
00:24:19,820 --> 00:24:23,810
the functions that we're dealing
with are L2 functions.
397
00:24:23,810 --> 00:24:28,330
One of the reasons for this is
this sinc function, sine x
398
00:24:28,330 --> 00:24:31,120
over x function is not L1.
399
00:24:34,500 --> 00:24:38,660
A sinc function just goes down
as 1 over t, and since it goes
400
00:24:38,660 --> 00:24:42,350
down as 1 over t, you take the
absolute value of it and you
401
00:24:42,350 --> 00:24:45,810
integrate 1 over t and what do
you get when you integrate it
402
00:24:45,810 --> 00:24:48,450
from minus infinity
to infinity.
403
00:24:48,450 --> 00:24:53,340
A function that's 1 over t,
well, if you really want to go
404
00:24:53,340 --> 00:24:56,650
through the trouble of
integrating it, you can
405
00:24:56,650 --> 00:25:00,650
integrate it over limits and you
get limits where you have
406
00:25:00,650 --> 00:25:04,190
to evaluate the limits on
a logarithmic function.
407
00:25:04,190 --> 00:25:05,840
When you get all done
with that you
408
00:25:05,840 --> 00:25:06,950
get an infinite value.
409
00:25:06,950 --> 00:25:07,830
And you can see this.
410
00:25:07,830 --> 00:25:11,610
You could take 1 over t and you
just look at it, as you go
411
00:25:11,610 --> 00:25:14,050
further and further out it just
gets bigger and bigger
412
00:25:14,050 --> 00:25:15,570
without limit.
413
00:25:15,570 --> 00:25:18,220
So, sinc t is not L1.
414
00:25:18,220 --> 00:25:21,230
Sinc function is a function
we'd like to use.
415
00:25:21,230 --> 00:25:26,000
Any function with a
discontinuity can't be the
416
00:25:26,000 --> 00:25:30,390
Fourier transform of
any L1 function.
417
00:25:30,390 --> 00:25:32,620
In other words, we said that
if you take the Fourier
418
00:25:32,620 --> 00:25:36,920
transform of an L1 function, one
of the nice things about
419
00:25:36,920 --> 00:25:40,830
it is you get a continuous
function.
420
00:25:40,830 --> 00:25:43,520
One of the nasty things
about it is you get
421
00:25:43,520 --> 00:25:45,900
a continuous function.
422
00:25:45,900 --> 00:25:50,080
Since you get a continuous
function it says that any time
423
00:25:50,080 --> 00:25:52,970
you want to deal with transforms
which are not
424
00:25:52,970 --> 00:26:02,050
continuous, you can't be talking
about time functions
425
00:26:02,050 --> 00:26:03,420
which are L1.
426
00:26:03,420 --> 00:26:06,440
One of the frequency functions
we want to look at a great
427
00:26:06,440 --> 00:26:10,030
deal is a frequency function
corresponding to a
428
00:26:10,030 --> 00:26:12,020
band-limited function.
429
00:26:12,020 --> 00:26:15,370
When you take a band-limited
function you just chop it off
430
00:26:15,370 --> 00:26:19,000
at some frequency, and usually
when you chop it off, you chop
431
00:26:19,000 --> 00:26:22,000
it off and get a
discontinuity.
432
00:26:22,000 --> 00:26:26,300
When you chop it off and get a
discountinuity, bingo, the
433
00:26:26,300 --> 00:26:30,090
time function you're dealing
with cannot be L1.
434
00:26:30,090 --> 00:26:34,340
It has to dribble away
much too slowly as
435
00:26:34,340 --> 00:26:35,700
time goes to infinity.
436
00:26:35,700 --> 00:26:39,760
That's an extraordinarily
important thing to remember.
437
00:26:39,760 --> 00:26:44,210
Any time you get a function
which is discontinuous in the
438
00:26:44,210 --> 00:26:49,840
frequency domain, the function
cannot go to zero any faster
439
00:26:49,840 --> 00:26:53,660
in a time domain than 1 over
t and vice versa in the
440
00:26:53,660 --> 00:26:55,860
frequency domain.
441
00:26:55,860 --> 00:26:59,460
L1 functions sometimes
have infinite energy.
442
00:26:59,460 --> 00:27:03,080
In other words, sinc
t is not L1 --
443
00:27:03,080 --> 00:27:07,120
well, that's not a good example
because that's not L1,
444
00:27:07,120 --> 00:27:11,370
and it also has infinite energy,
but you can just as
445
00:27:11,370 --> 00:27:14,720
easily find functions which drop
off a little more slowly
446
00:27:14,720 --> 00:27:18,170
than sinc t, and which
have infinite
447
00:27:18,170 --> 00:27:21,300
energy because they--.
448
00:27:21,300 --> 00:27:22,980
Excuse me.
449
00:27:22,980 --> 00:27:27,720
Sometimes you have functions
which go off to infinity too
450
00:27:27,720 --> 00:27:32,760
fast as you approach time equals
zero, things which are
451
00:27:32,760 --> 00:27:34,960
a little bit like impulses
but not really.
452
00:27:34,960 --> 00:27:38,740
Impulses are awful and we'll
talk about them in a minute,
453
00:27:38,740 --> 00:27:42,490
because they don't have finite
energy as we said before.
454
00:27:42,490 --> 00:27:46,590
We have functions which just
slowly go off to infinity and
455
00:27:46,590 --> 00:27:50,540
they are L2 but they
aren't L1.
456
00:27:50,540 --> 00:27:52,800
Why do we care about those
weird functions?
457
00:27:52,800 --> 00:27:56,020
We care about them, as I said
before, because we would like
458
00:27:56,020 --> 00:27:59,680
to be able to make statements
which are simple which we can
459
00:27:59,680 --> 00:28:02,260
believe in.
460
00:28:02,260 --> 00:28:05,800
In other words, you don't want
to go through a course like
461
00:28:05,800 --> 00:28:10,550
this with a whole bunch of
things that you have to leave.
462
00:28:10,550 --> 00:28:12,930
It's very nice to have some
things that you really
463
00:28:12,930 --> 00:28:17,670
believe, and whether you believe
them or not, it's nice
464
00:28:17,670 --> 00:28:19,490
to have theorems about them.
465
00:28:19,490 --> 00:28:22,210
Even if you don't believe the
theorems, at least you have
466
00:28:22,210 --> 00:28:25,590
theorems so you can fool
other people about
467
00:28:25,590 --> 00:28:27,920
it, if nothing else.
468
00:28:27,920 --> 00:28:31,350
Well, it turns out that L2
functions are really the right
469
00:28:31,350 --> 00:28:33,530
class to look at here.
470
00:28:45,440 --> 00:28:47,480
Oh, I think I left out
one of the most
471
00:28:47,480 --> 00:28:48,540
important things here.
472
00:28:48,540 --> 00:28:51,360
Maybe I put it down here.
473
00:28:51,360 --> 00:28:53,130
No, probably not.
474
00:28:53,130 --> 00:28:57,480
One of the reasons we want to
deal with L2 functions is if
475
00:28:57,480 --> 00:29:04,460
you're dealing with compression,
for example, and
476
00:29:04,460 --> 00:29:09,220
you take a function, if the
function has infinite energy
477
00:29:09,220 --> 00:29:15,300
in it, then one of the things
that happens is that any time
478
00:29:15,300 --> 00:29:19,060
you expand it into any kind of
orthonormal expansion or
479
00:29:19,060 --> 00:29:22,260
orthongonal expansion, which
we'll talk about later, you
480
00:29:22,260 --> 00:29:26,000
have coefficients, which
have infinite energy.
481
00:29:26,000 --> 00:29:30,860
In other words, they have
infinite values, or the sum
482
00:29:30,860 --> 00:29:34,370
squared of the coefficients
is equal to infinity.
483
00:29:34,370 --> 00:29:37,850
When we try to compress those
we're going to find that no
484
00:29:37,850 --> 00:29:42,840
matter how we do it our
mean square error
485
00:29:42,840 --> 00:29:45,100
is going to be infinite.
486
00:29:45,100 --> 00:29:50,020
Yes, we will talk about that
later when we get to talking
487
00:29:50,020 --> 00:29:52,130
about expansions.
488
00:29:52,130 --> 00:29:56,220
So for all those reasons L1
isn't the right thing, L2 is
489
00:29:56,220 --> 00:29:58,800
the right thing.
490
00:29:58,800 --> 00:30:03,510
A function going from the reals
into the complexes, in
491
00:30:03,510 --> 00:30:07,350
other words, a complex valued
function is L2 if it's
492
00:30:07,350 --> 00:30:12,530
measurable, and if this
integral is less than
493
00:30:12,530 --> 00:30:15,950
infinity, in other words, if
that has finite energy.
494
00:30:15,950 --> 00:30:18,620
Primarily it means it has finite
energy because all the
495
00:30:18,620 --> 00:30:21,390
functions you can think
of are measurable.
496
00:30:21,390 --> 00:30:23,890
So it really says you're dealing
with functions that
497
00:30:23,890 --> 00:30:25,620
have finite energy here.
498
00:30:31,510 --> 00:30:34,040
So, let's go on to Fourier
transforms then.
499
00:30:37,670 --> 00:30:39,950
Interesting simple theorem.
500
00:30:39,950 --> 00:30:42,850
I think I stated this
last time, also.
501
00:30:42,850 --> 00:30:50,980
If a function is L2 and its time
limited, it's also L1.
502
00:30:50,980 --> 00:30:55,130
So we've already found that if
functions are L1, they have
503
00:30:55,130 --> 00:30:58,360
Fourier transforms that exist.
504
00:30:58,360 --> 00:31:03,070
The reason for this is if you
square u of t, take the
505
00:31:03,070 --> 00:31:07,320
magnitude squared of u of t for
any given t, it has to be
506
00:31:07,320 --> 00:31:11,550
less than or equal to the
sum of u of t plus 1.
507
00:31:11,550 --> 00:31:13,080
In fact, it has to be
less than that.
508
00:31:13,080 --> 00:31:14,330
Why?
509
00:31:19,530 --> 00:31:22,520
How would you prove this
if you had to prove it?
510
00:31:25,980 --> 00:31:29,770
Well, you say gee, this is
two separate terms here.
511
00:31:29,770 --> 00:31:32,730
Why don't I look at two
separate cases.
512
00:31:32,730 --> 00:31:38,320
The two separate cases are
first, suppose u of t itself
513
00:31:38,320 --> 00:31:40,930
as a magnitude less than 1.
514
00:31:40,930 --> 00:31:45,530
If u of t has and magnitude less
than 1 and you square it,
515
00:31:45,530 --> 00:31:47,630
you get something
even smaller.
516
00:31:47,630 --> 00:31:51,770
So, any time u of t has a
magnitude less than 1, u
517
00:31:51,770 --> 00:31:57,190
squared of t is less than
or equal to u of t.
518
00:31:57,190 --> 00:31:59,530
Blah blah blah blah blah
blah blah blah.
519
00:31:59,530 --> 00:32:00,930
If u of t --
520
00:32:10,020 --> 00:32:11,270
what did I do here?
521
00:32:19,320 --> 00:32:22,620
No wonder I couldn't explain
this to you.
522
00:32:27,330 --> 00:32:30,620
Let's try it that way
and see if it works.
523
00:32:30,620 --> 00:32:33,118
If you can prove something, turn
it around and see if you
524
00:32:33,118 --> 00:32:36,350
can prove it then.
525
00:32:36,350 --> 00:32:38,160
Now, two cases.
526
00:32:38,160 --> 00:32:41,940
First one, let's suppose
that u of t is less
527
00:32:41,940 --> 00:32:43,440
than or equal to 1.
528
00:32:43,440 --> 00:32:46,430
Well then, u of t is less
than or equal to 1.
529
00:32:46,430 --> 00:32:50,160
And this is positive so this is
less than or equal to that.
530
00:32:50,160 --> 00:32:53,420
Let's look at the other case.
u of t is greater than or
531
00:32:53,420 --> 00:32:55,560
equal to 1, magnitude.
532
00:32:55,560 --> 00:32:57,840
Well then, it's less
than or equal to u
533
00:32:57,840 --> 00:32:59,540
of t magnitude squared.
534
00:32:59,540 --> 00:33:02,530
So either way this is less
than or equal to that.
535
00:33:02,530 --> 00:33:07,390
What that says is the integral
over any finite limits of u of
536
00:33:07,390 --> 00:33:12,700
t dt is less than or equal
to the integral of this.
537
00:33:12,700 --> 00:33:15,870
Well, the integral of this
splits up into the integral
538
00:33:15,870 --> 00:33:22,940
magnitude u of t squared dt
plus the integral of 1.
539
00:33:22,940 --> 00:33:25,900
Now, the integral of 1 over
any finite limits is
540
00:33:25,900 --> 00:33:28,300
just b minus a.
541
00:33:28,300 --> 00:33:30,860
That's where the finite
limits come in.
542
00:33:30,860 --> 00:33:34,120
Finite limits say you don't have
to worry about this term
543
00:33:34,120 --> 00:33:36,050
because it's finite.
544
00:33:36,050 --> 00:33:42,650
So that says at any time you
have an L2 function over a
545
00:33:42,650 --> 00:33:46,710
finite range, that function
is also L1
546
00:33:46,710 --> 00:33:48,960
over that finite range.
547
00:33:48,960 --> 00:33:52,830
Which says that any time you
take a Fourier transform of an
548
00:33:52,830 --> 00:33:57,610
L2 function, which is only
non-zero over a finite range,
549
00:33:57,610 --> 00:34:00,910
bingo, it's L1 and you get all
these nice properties.
550
00:34:00,910 --> 00:34:03,910
It has to exist, it has to be
continuous, it has to be
551
00:34:03,910 --> 00:34:06,120
bounded, and all of
that neat stuff.
552
00:34:09,040 --> 00:34:14,060
So for any L2 function u of t,
what I'm going to try to do
553
00:34:14,060 --> 00:34:17,850
now, and I'm just copying what
a guy by the name of
554
00:34:17,850 --> 00:34:20,810
Plancherel did a
long time ago.
555
00:34:20,810 --> 00:34:24,530
The thing that Plancherel did
was he said how do I know when
556
00:34:24,530 --> 00:34:28,260
a Fourier transform
exists or not.
557
00:34:28,260 --> 00:34:31,040
I would like to make
it exist for L2
558
00:34:31,040 --> 00:34:33,900
functions, how do I do it?
559
00:34:33,900 --> 00:34:37,000
Well, he said OK, the thing I'm
going to do is to take the
560
00:34:37,000 --> 00:34:42,290
function u of t and I'm first
going to truncate it.
561
00:34:42,290 --> 00:34:45,290
In fact, if you think in terms
of Reimann integration and
562
00:34:45,290 --> 00:34:48,940
things like that, any time you
take an integral from minus
563
00:34:48,940 --> 00:34:53,140
infinity to plus infinity,
what do you mean by it?
564
00:34:53,140 --> 00:34:56,250
You mean the limit as you
integrate the function over
565
00:34:56,250 --> 00:35:00,510
finite limits and then you let
the limits go to infinity.
566
00:35:00,510 --> 00:35:04,230
So all we're doing is
the same trick here.
567
00:35:04,230 --> 00:35:09,180
So we're going to take u of t,
we're going to truncate it to
568
00:35:09,180 --> 00:35:13,900
some minus a to plus a over some
finite range, no matter
569
00:35:13,900 --> 00:35:15,720
how big a happens to be.
570
00:35:15,720 --> 00:35:20,570
We're going to call the function
b sub a of t, u of t
571
00:35:20,570 --> 00:35:22,580
truncated to these limits.
572
00:35:22,580 --> 00:35:27,690
In other words, u of t times a
rectangular function evaluated
573
00:35:27,690 --> 00:35:30,400
at t over 2a.
574
00:35:30,400 --> 00:35:34,650
Now, can all of you look at this
function and see that it
575
00:35:34,650 --> 00:35:38,560
just means truncate from
minus a to plus a?
576
00:35:38,560 --> 00:35:39,430
No.
577
00:35:39,430 --> 00:35:44,260
Well, you should learn
to do this.
578
00:35:44,260 --> 00:35:47,170
One of the ways to do it is
to say OK, the rectangular
579
00:35:47,170 --> 00:35:52,330
function is defined as having
the value 1 between minus 1/2
580
00:35:52,330 --> 00:35:55,330
and plus 1/2 and it's zero
everywhere else.
581
00:35:57,860 --> 00:36:00,570
I think I said that before
in class, didn't I?
582
00:36:00,570 --> 00:36:03,650
Certainly it's in the notes.
583
00:36:03,650 --> 00:36:16,490
Rectangle ft equals 1 for
t less than or equal
584
00:36:16,490 --> 00:36:19,960
to 1/2, zero else.
585
00:36:23,980 --> 00:36:27,360
So with this definition you
just evaluate what happens
586
00:36:27,360 --> 00:36:32,870
when t is equal to minus a, you
get rectangle of minus a
587
00:36:32,870 --> 00:36:36,150
over 2a, which is minus 1/2.
588
00:36:36,150 --> 00:36:39,910
When t is equal to a, you're up
to the other limit, so this
589
00:36:39,910 --> 00:36:44,650
function is 1 for t between
minus a and plus a and zero
590
00:36:44,650 --> 00:36:46,230
everywhere else.
591
00:36:46,230 --> 00:36:50,940
Please get used to using this
and become a little facile at
592
00:36:50,940 --> 00:36:55,800
sorting out what it means
because it's a very handy way
593
00:36:55,800 --> 00:36:59,300
to avoid writing awkward
things like this.
594
00:36:59,300 --> 00:37:05,870
So va of t by what we've said
is both L2 and its L1.
595
00:37:05,870 --> 00:37:10,770
We started out with a function
which is L2 and we truncated.
596
00:37:10,770 --> 00:37:16,360
Then according to this theorem
here, this function va of t is
597
00:37:16,360 --> 00:37:18,670
now time limited.
598
00:37:18,670 --> 00:37:24,300
It's also L2, and therefore, by
the theorem it's also L1.
599
00:37:24,300 --> 00:37:29,660
Therefore, it's continue and
you can take the Fourier
600
00:37:29,660 --> 00:37:35,700
transform of it -- v hat a of
f exists for all f and it's
601
00:37:35,700 --> 00:37:36,590
continuous.
602
00:37:36,590 --> 00:37:42,990
So this function is just the
Fourier transform that you get
603
00:37:42,990 --> 00:37:46,400
when you truncate the function,
which is what you
604
00:37:46,400 --> 00:37:48,780
would think of as a
way to find the
605
00:37:48,780 --> 00:37:51,840
Fourier transform anyway.
606
00:37:51,840 --> 00:37:54,810
I mean if this is not a
reasonable approximation to
607
00:37:54,810 --> 00:37:58,650
the Fourier transform a
function, you haven't modeled
608
00:37:58,650 --> 00:38:01,530
the function very well.
609
00:38:01,530 --> 00:38:05,470
Because when a gets
extraordinarily large, if
610
00:38:05,470 --> 00:38:08,320
there's anything of significance
that happens
611
00:38:08,320 --> 00:38:13,300
before year ten to the minus 6
or which happens after year
612
00:38:13,300 --> 00:38:17,120
ten to the plus 6, and you're
dealing with electronic
613
00:38:17,120 --> 00:38:21,530
speeds, your models don't
make any sense.
614
00:38:21,530 --> 00:38:26,050
So for anything of any interest,
these functions here
615
00:38:26,050 --> 00:38:29,550
are going to start approximating
u of t, and
616
00:38:29,550 --> 00:38:32,590
therefore we hope this will
start approximating the
617
00:38:32,590 --> 00:38:35,520
Fourier transform of u of t.
618
00:38:35,520 --> 00:38:37,500
Who can make that more
precise for me?
619
00:38:40,440 --> 00:38:44,500
What happens when u of
t has finite energy?
620
00:38:47,210 --> 00:38:51,900
If it has finite energy it means
that the integral of u
621
00:38:51,900 --> 00:38:54,410
of t magnitude squared over the
622
00:38:54,410 --> 00:38:58,460
infinite interval is finite.
623
00:38:58,460 --> 00:39:01,650
So you start integrating it
over bigger and bigger
624
00:39:01,650 --> 00:39:04,580
minus a to plus a.
625
00:39:04,580 --> 00:39:08,630
What happens is as the minus
a to plus a gets bigger and
626
00:39:08,630 --> 00:39:11,180
bigger and bigger, you're
including more and more of the
627
00:39:11,180 --> 00:39:16,760
function, so that the integral
of u of t squared over that
628
00:39:16,760 --> 00:39:19,460
bigger and bigger interval has
to be getting closer and
629
00:39:19,460 --> 00:39:23,780
closer to the overall
integral of u of t.
630
00:39:23,780 --> 00:39:29,800
Which says that the energy in u
of t minus the a of t has to
631
00:39:29,800 --> 00:39:33,760
get very, very small
as a gets large.
632
00:39:33,760 --> 00:39:36,560
That's one of the reasons why
we like to deal with finite
633
00:39:36,560 --> 00:39:38,200
energy functions.
634
00:39:38,200 --> 00:39:42,650
By definition they cannot have
a appreciable energy outside
635
00:39:42,650 --> 00:39:43,990
of very large limits.
636
00:39:43,990 --> 00:39:46,950
How large those large limits
have to be depends on the
637
00:39:46,950 --> 00:39:50,600
function, but if you make them
large enough you will always
638
00:39:50,600 --> 00:39:55,550
get negligible energy outside
of those limits.
639
00:39:55,550 --> 00:39:59,580
So then we can take the Fourier
transform of this
640
00:39:59,580 --> 00:40:03,750
function within those limits and
we get something which we
641
00:40:03,750 --> 00:40:07,500
hope is going to be a reasonable
approximation of
642
00:40:07,500 --> 00:40:10,300
the Fourier transform
of u of t.
643
00:40:16,400 --> 00:40:19,820
That's what Plancherel said.
644
00:40:19,820 --> 00:40:26,610
Plancherel said if we have an
L2 function, u of t, then
645
00:40:26,610 --> 00:40:30,900
there is an L2 function u hat
of f, which is really the
646
00:40:30,900 --> 00:40:33,270
Fourier transform of u of t.
647
00:40:33,270 --> 00:40:39,060
Some people call this the
Plancherel transform of u of t
648
00:40:39,060 --> 00:40:42,990
and say that indeed Plancherel
was the one that invented
649
00:40:42,990 --> 00:40:44,790
Fourier transforms
or Plancherel
650
00:40:44,790 --> 00:40:47,110
transforms for L2 functions.
651
00:40:47,110 --> 00:40:50,190
That's probably giving him a
little too much credit, and
652
00:40:50,190 --> 00:40:54,360
Fourier somewhat less
than due credit.
653
00:40:54,360 --> 00:40:57,590
But it was a neat theorem.
654
00:40:57,590 --> 00:41:03,140
What he said is that there is
a function u hat of f, which
655
00:41:03,140 --> 00:41:06,240
we'll call the Fourier
transform, which has the
656
00:41:06,240 --> 00:41:13,660
property that when you take the
integral of the difference
657
00:41:13,660 --> 00:41:20,830
between u hat of f and the
transform of b sub a of t,
658
00:41:20,830 --> 00:41:22,800
when you take the integral
of this dt --
659
00:41:22,800 --> 00:41:26,080
in other words, when you
evaluate the energy in the
660
00:41:26,080 --> 00:41:34,880
difference between u hat
of f and v sub a of f,
661
00:41:34,880 --> 00:41:37,570
that goes to zero.
662
00:41:37,570 --> 00:41:39,960
Well this isn't a big deal.
663
00:41:39,960 --> 00:41:43,090
In other words, this is
plausible, since this integral
664
00:41:43,090 --> 00:41:46,610
has to go to zero for an L2
function, that's what we just
665
00:41:46,610 --> 00:41:52,760
said, then therefore, using the
energy relation, this also
666
00:41:52,760 --> 00:41:55,570
has to go to zero.
667
00:41:55,570 --> 00:41:58,830
So is this another example where
Plancherel just came
668
00:41:58,830 --> 00:42:03,510
along at the right time and he
said something totally trivial
669
00:42:03,510 --> 00:42:05,960
and became famous
because of it?
670
00:42:05,960 --> 00:42:09,320
I mean as I've urged all of
you, work on problems that
671
00:42:09,320 --> 00:42:10,680
other people haven't
worked on.
672
00:42:10,680 --> 00:42:12,940
If you're lucky, you
will do something
673
00:42:12,940 --> 00:42:15,700
trivial and become famous.
674
00:42:15,700 --> 00:42:18,130
As another piece of philosophy,
you become far
675
00:42:18,130 --> 00:42:20,670
more famous for doing something
trivial than for
676
00:42:20,670 --> 00:42:23,990
doing something difficult,
because everybody remembers
677
00:42:23,990 --> 00:42:25,870
something trivial.
678
00:42:25,870 --> 00:42:28,390
And if you do something
difficult nobody even
679
00:42:28,390 --> 00:42:32,030
understands it.
680
00:42:32,030 --> 00:42:38,150
But no, it wasn't that because
there's something hidden here.
681
00:42:38,150 --> 00:42:42,500
He says a function
like this exists.
682
00:42:42,500 --> 00:42:47,740
In other words, the problem is
these functions get closer and
683
00:42:47,740 --> 00:42:50,690
closer to something.
684
00:42:50,690 --> 00:42:53,440
They get closer and closer to
each other as a gets bigger
685
00:42:53,440 --> 00:42:55,490
and bigger.
686
00:42:55,490 --> 00:42:57,030
You can show that because
you have a
687
00:42:57,030 --> 00:42:59,920
handle on these functions.
688
00:42:59,920 --> 00:43:03,000
Whether they get closer and
closer to a real bonafide
689
00:43:03,000 --> 00:43:05,900
function or not is
another question.
690
00:43:09,950 --> 00:43:15,320
Back when you studied
arithmetic, if you were in any
691
00:43:15,320 --> 00:43:18,730
kind of advanced class studying
arithmetic, you
692
00:43:18,730 --> 00:43:20,980
studied the rational
numbers and the
693
00:43:20,980 --> 00:43:23,440
real numbers you remember.
694
00:43:23,440 --> 00:43:28,500
And you remember the problem of
what happens if you take a
695
00:43:28,500 --> 00:43:34,760
sequence of rational numbers
which is approaching a limit.
696
00:43:34,760 --> 00:43:37,700
There's a big problem there
because when you take a
697
00:43:37,700 --> 00:43:40,900
sequence of rational numbers
that approaches a limit, the
698
00:43:40,900 --> 00:43:43,680
limit might not be rational.
699
00:43:43,680 --> 00:43:47,250
In other words, when you take
sequences of things you can
700
00:43:47,250 --> 00:43:50,790
get out of the domain of the
things you're working with.
701
00:43:50,790 --> 00:43:54,030
Now, we can't get out of the
domain of being L2, but we
702
00:43:54,030 --> 00:43:56,770
might get out of domain of
measurable functions, we might
703
00:43:56,770 --> 00:43:59,820
get out of the domain
of functions at all.
704
00:43:59,820 --> 00:44:03,480
We can have all sorts of
strange things happen.
705
00:44:03,480 --> 00:44:07,640
The nice thing here, which
was really a theorem by
706
00:44:07,640 --> 00:44:08,490
[? Resenage ?]
707
00:44:08,490 --> 00:44:10,470
a long time ago.
708
00:44:10,470 --> 00:44:13,740
It says that when you take
cosine sequences of L2
709
00:44:13,740 --> 00:44:17,950
functions, they converge
to an L2 function.
710
00:44:17,950 --> 00:44:21,200
So that's really what's
involved in here.
711
00:44:21,200 --> 00:44:24,420
So maybe this should be called
the [? Resenage ?]
712
00:44:24,420 --> 00:44:26,880
transform, I don't know.
713
00:44:26,880 --> 00:44:31,220
But anyway, whatever this says,
the theorem says, the
714
00:44:31,220 --> 00:44:34,080
first part of Plancherel's
theorem says that this
715
00:44:34,080 --> 00:44:39,050
function exists and you get a
handle on it by taking this
716
00:44:39,050 --> 00:44:43,260
transform, making a bigger and
bigger, and it says it will
717
00:44:43,260 --> 00:44:47,010
converge to something in
this energy sense.
718
00:44:47,010 --> 00:44:50,480
Bingo, when you're all done
this goes to zero.
719
00:44:50,480 --> 00:44:54,750
We're going to denote this
function as a limit and a mean
720
00:44:54,750 --> 00:44:57,230
of the Fourier transform.
721
00:44:57,230 --> 00:45:00,970
In other words, we do have a
Fourier transform in the same
722
00:45:00,970 --> 00:45:03,630
sense that we had a Fourier
series before.
723
00:45:03,630 --> 00:45:06,275
We didn't know weather the
Fourier series would converge
724
00:45:06,275 --> 00:45:09,600
at every point, but we knew that
it converged at enough
725
00:45:09,600 --> 00:45:13,190
points, namely, almost
everywhere, everywhere but on
726
00:45:13,190 --> 00:45:14,710
a set of measure zero.
727
00:45:14,710 --> 00:45:18,910
It converges so that, in fact,
you get this kind of
728
00:45:18,910 --> 00:45:21,400
relationship.
729
00:45:21,400 --> 00:45:23,890
Now, do you have to
worry about that?
730
00:45:23,890 --> 00:45:24,270
No.
731
00:45:24,270 --> 00:45:28,340
Again, this is one of these
very nice things that says
732
00:45:28,340 --> 00:45:32,700
there is a Fourier transform,
you don't have to worry about
733
00:45:32,700 --> 00:45:36,200
what goes on at these oddball
sets where the function has
734
00:45:36,200 --> 00:45:38,400
discountinuities and
things like that.
735
00:45:38,400 --> 00:45:40,010
You can forget all of that.
736
00:45:40,010 --> 00:45:43,560
You can be as careless as you've
ever been, and now you
737
00:45:43,560 --> 00:45:46,640
know that it all works out
mathematically, so long as you
738
00:45:46,640 --> 00:45:48,600
stick to L2 functions.
739
00:45:48,600 --> 00:45:53,010
So, sticking to L2 functions
says you can be a careless
740
00:45:53,010 --> 00:45:56,850
engineer, you can use lousy
mathematics and you'll always
741
00:45:56,850 --> 00:45:58,510
get the right answer.
742
00:45:58,510 --> 00:46:02,600
So, it's nice for engineers,
it's nice for me.
743
00:46:02,600 --> 00:46:05,730
I don't like to be careful
all the time.
744
00:46:05,730 --> 00:46:11,130
I like to be careful once and
then solve that and go on.
745
00:46:11,130 --> 00:46:13,850
Well, because of time frequency
duality, you can do
746
00:46:13,850 --> 00:46:17,860
exactly the same thing in
the frequency domain.
747
00:46:17,860 --> 00:46:22,170
So, you start out defining some
b, which is bigger than
748
00:46:22,170 --> 00:46:26,470
zero which is arbitrarily large,
you define a finite
749
00:46:26,470 --> 00:46:34,850
bandwidth approximation as w hat
sub b of f is u hat of f.
750
00:46:34,850 --> 00:46:36,820
We now know that u
hat of f exists
751
00:46:36,820 --> 00:46:38,900
and it's an L2 function.
752
00:46:38,900 --> 00:46:42,220
u hat of f times this
rectangular function,
753
00:46:42,220 --> 00:46:43,240
that's f over 2b.
754
00:46:43,240 --> 00:46:48,510
In other words, it's u hat of f
truncated to a big bandwidth
755
00:46:48,510 --> 00:46:49,810
minus b to plus b.
756
00:46:52,860 --> 00:46:56,510
Since w sub b of f
is L1, as well as
757
00:46:56,510 --> 00:47:00,140
L2, this always exists.
758
00:47:00,140 --> 00:47:02,800
So long as you deal with a
finite bandwidth, this
759
00:47:02,800 --> 00:47:04,520
quantity exists.
760
00:47:04,520 --> 00:47:06,950
It exists for all t and r.
761
00:47:06,950 --> 00:47:08,980
It's continuous.
762
00:47:08,980 --> 00:47:11,980
The second part of Plancherel's
theorem then says
763
00:47:11,980 --> 00:47:16,550
that the limit as b goes to
infinity of the integral of u
764
00:47:16,550 --> 00:47:21,130
of t minus this truncated
function, magnitude squared
765
00:47:21,130 --> 00:47:25,420
the energy in that,
goes to zero.
766
00:47:25,420 --> 00:47:29,060
This now is a little different
than what we did before.
767
00:47:29,060 --> 00:47:31,720
It's easier in the sense that
we don't have to worry about
768
00:47:31,720 --> 00:47:35,120
the existence of this function
because we started out with
769
00:47:35,120 --> 00:47:37,530
this to start with.
770
00:47:37,530 --> 00:47:40,710
It's a little harder because
we know that a function
771
00:47:40,710 --> 00:47:44,020
exists, but we don't know
that it's u of t.
772
00:47:44,020 --> 00:47:47,150
So, in fact, poor old Plancherel
had to do something
773
00:47:47,150 --> 00:47:52,040
other than just this very simple
argument that says all
774
00:47:52,040 --> 00:47:53,430
the energy works out right.
775
00:47:53,430 --> 00:47:56,720
He had to also show that you
really wind up with the right
776
00:47:56,720 --> 00:47:58,660
function when you get
all through with it.
777
00:47:58,660 --> 00:48:01,650
But again, this is the same
kind of energy convergence
778
00:48:01,650 --> 00:48:02,470
that we had before.
779
00:48:02,470 --> 00:48:02,840
Yeah?
780
00:48:02,840 --> 00:48:05,900
AUDIENCE: Could you discuss
non-uniqueness?
781
00:48:05,900 --> 00:48:07,940
Clearly, [INAUDIBLE]
782
00:48:07,940 --> 00:48:12,260
u hat f to satisfy Plancherel
1 and Plancherel 2.
783
00:48:12,260 --> 00:48:16,710
PROFESSOR: Yeah, in fact, any
two functions which are L2
784
00:48:16,710 --> 00:48:18,980
equivalent.
785
00:48:18,980 --> 00:48:23,270
But you see the nice thing is
when you take this finite
786
00:48:23,270 --> 00:48:26,370
bandwidth approximation
there's only one.
787
00:48:26,370 --> 00:48:28,370
It's only when you get
to the limit that all
788
00:48:28,370 --> 00:48:31,720
of this mess occurs.
789
00:48:31,720 --> 00:48:34,390
If you take these different
possible functions, u hat of
790
00:48:34,390 --> 00:48:40,880
f, which just differ in these
negligible sets of measure
791
00:48:40,880 --> 00:48:44,720
zero, those don't affect
this integral.
792
00:48:44,720 --> 00:48:48,190
Sets of measure zero don't
affect integrals at all.
793
00:48:48,190 --> 00:48:51,640
So the mathematicians deal
with L2 theory by talking
794
00:48:51,640 --> 00:48:55,280
about equivalence classes
of functions.
795
00:48:55,280 --> 00:48:57,920
I find it hard to think about
equivalence classes of
796
00:48:57,920 --> 00:49:01,560
functions and partitioning the
set of all functions into a
797
00:49:01,560 --> 00:49:04,120
bunch of equivalence classes.
798
00:49:04,120 --> 00:49:07,375
So I just sort of remember in
the back of my mind that there
799
00:49:07,375 --> 00:49:11,630
are all these functions which
differ in a strange way.
800
00:49:11,630 --> 00:49:13,990
We'll talk about that more when
we get to the sampling
801
00:49:13,990 --> 00:49:16,580
theorem later today,
because there it
802
00:49:16,580 --> 00:49:18,300
happens to be important.
803
00:49:18,300 --> 00:49:20,470
Here it's not really important,
here we don't have
804
00:49:20,470 --> 00:49:23,410
to worry about it.
805
00:49:23,410 --> 00:49:26,570
Anyway, we can always get back
to the u of t that we started
806
00:49:26,570 --> 00:49:29,670
with in this way.
807
00:49:29,670 --> 00:49:33,920
Now, this says that all L2
functions have Fourier
808
00:49:33,920 --> 00:49:37,470
transforms in this
very nice sense.
809
00:49:37,470 --> 00:49:40,720
In other words, at this point
you don't have to worry about
810
00:49:40,720 --> 00:49:43,750
continuity, you don't have to
worry about how fast things
811
00:49:43,750 --> 00:49:47,070
drop off, you don't have to
worry about anything.
812
00:49:47,070 --> 00:49:50,660
So long as you have finite
energy functions, this
813
00:49:50,660 --> 00:49:53,840
beautiful result always
holds true.
814
00:49:53,840 --> 00:49:58,390
There always is a Fourier
transform, it always has this
815
00:49:58,390 --> 00:50:02,350
nice property that it has the
same energy as the function
816
00:50:02,350 --> 00:50:04,240
you started with.
817
00:50:04,240 --> 00:50:09,820
The only nasty thing, as Dave
pointed out, is that, in fact,
818
00:50:09,820 --> 00:50:12,270
it might not be a unique
function, but it's close
819
00:50:12,270 --> 00:50:13,490
enough to unique.
820
00:50:13,490 --> 00:50:15,790
It's unique in an engineering
sense.
821
00:50:25,020 --> 00:50:29,570
The other thing is that L2 wave
forms don't include some
822
00:50:29,570 --> 00:50:32,880
of your favorite wave forms.
823
00:50:32,880 --> 00:50:37,510
They don't include constants,
they don't include sine waves,
824
00:50:37,510 --> 00:50:40,610
and they don't include Dirac
impulse functions.
825
00:50:40,610 --> 00:50:42,640
All of them have infinite
energy.
826
00:50:42,640 --> 00:50:45,790
I pointed out in class, spent
quite a bit of time explaining
827
00:50:45,790 --> 00:50:49,580
why an impulse function had
infinite energy by looking at
828
00:50:49,580 --> 00:50:54,040
it as a very narrow pulse of
width epsilon and a height 1
829
00:50:54,040 --> 00:50:59,330
over epsilon, and showing that
the energy in that is 1 over
830
00:50:59,330 --> 00:51:02,710
epsilon, and as epsilon goes
to zero and the pulse gets
831
00:51:02,710 --> 00:51:07,270
narrower and narrower, bingo,
the energy goes to infinity.
832
00:51:07,270 --> 00:51:11,060
Constants are the same way, they
extend on and on forever.
833
00:51:11,060 --> 00:51:14,870
Therefore, they have infinite
energy, except if the constant
834
00:51:14,870 --> 00:51:17,220
happens to be zero.
835
00:51:17,220 --> 00:51:20,780
Sine waves are the same way,
they dribble on forever.
836
00:51:23,400 --> 00:51:28,560
So the question is are these
good models of reality?
837
00:51:28,560 --> 00:51:31,450
The answer is they're good for
some things and they're very
838
00:51:31,450 --> 00:51:33,810
bad for other things.
839
00:51:33,810 --> 00:51:37,420
The point in this course is
that if you're looking for
840
00:51:37,420 --> 00:51:43,140
wave forms that are good models
for either the kinds of
841
00:51:43,140 --> 00:51:46,930
functions that we're going to
quantize, namely, source wave
842
00:51:46,930 --> 00:51:50,320
forms, or if you're looking for
the kinds of things that
843
00:51:50,320 --> 00:51:54,170
we're going to transmit
on channels, these
844
00:51:54,170 --> 00:51:56,720
are very lousy functions.
845
00:51:56,720 --> 00:51:59,860
They don't make any sense in
a communication context.
846
00:52:02,660 --> 00:52:06,370
But anyway, where did these
things come from?
847
00:52:06,370 --> 00:52:09,660
Constants and sine waves
result from refusing to
848
00:52:09,660 --> 00:52:11,720
explicitly model when very
849
00:52:11,720 --> 00:52:14,920
long-lasting functions terminate.
850
00:52:14,920 --> 00:52:17,260
In other words, if you're
looking at a carrier function
851
00:52:17,260 --> 00:52:26,640
in a communication's system,
sine of 2 pi, f carrier times
852
00:52:26,640 --> 00:52:31,620
t, it just keeps on wiggling
around forever.
853
00:52:31,620 --> 00:52:35,030
Since you want to talk about
that over the complete time of
854
00:52:35,030 --> 00:52:39,680
interest, you don't want to say
what the time of interest
855
00:52:39,680 --> 00:52:44,190
is, you don't want to admit to
your employer that this thing
856
00:52:44,190 --> 00:52:47,560
is going to stop working after
one month because you want to
857
00:52:47,560 --> 00:52:50,320
let him think that he's going
to make a profit off of this
858
00:52:50,320 --> 00:52:55,300
forever, and you don't want to
commit to putting it into use
859
00:52:55,300 --> 00:52:57,740
in one month when you know it's
going to get delayed for
860
00:52:57,740 --> 00:52:58,670
a whole year.
861
00:52:58,670 --> 00:52:59,940
So you want to think of this as
862
00:52:59,940 --> 00:53:01,400
something which is permanent.
863
00:53:01,400 --> 00:53:04,770
You don't want to answer the
question at what time does it
864
00:53:04,770 --> 00:53:08,450
start and at what time does it
end, because for many of the
865
00:53:08,450 --> 00:53:11,120
questions you ask, you
can just regard
866
00:53:11,120 --> 00:53:15,110
it as going on forever.
867
00:53:15,110 --> 00:53:17,440
You have the same thing
with impulses.
868
00:53:17,440 --> 00:53:25,420
Impulses are always models
of short pulses.
869
00:53:25,420 --> 00:53:28,300
If you put these short pulses
through a filter, the only
870
00:53:28,300 --> 00:53:30,680
thing which is of interest
in them is what
871
00:53:30,680 --> 00:53:32,670
their integral is.
872
00:53:32,670 --> 00:53:35,040
And since the only thing of
interest is their integral,
873
00:53:35,040 --> 00:53:39,120
you call it an impulse and you
don't worry about just how
874
00:53:39,120 --> 00:53:43,800
narrow it is, except that it
has infinite energy and,
875
00:53:43,800 --> 00:53:46,980
therefore, whenever you start
to deal with a situation in
876
00:53:46,980 --> 00:53:51,870
which energy is important, these
becomes lousy models.
877
00:53:51,870 --> 00:53:54,320
So we can't use these
when we're
878
00:53:54,320 --> 00:53:56,070
talking about L2 functions.
879
00:53:56,070 --> 00:53:59,540
That's the price we pay for
dealing with L2 functions.
880
00:53:59,540 --> 00:54:03,320
But it's a small price because
for almost all the things
881
00:54:03,320 --> 00:54:06,570
we'll be dealing with, it's the
energy of the functions
882
00:54:06,570 --> 00:54:09,810
that are really important.
883
00:54:09,810 --> 00:54:12,790
So as communication wave forms,
infinite energy wave
884
00:54:12,790 --> 00:54:16,090
forms make mean square error
quantization results
885
00:54:16,090 --> 00:54:17,860
meaningless.
886
00:54:17,860 --> 00:54:20,980
In other words, when you sample
these infinite energy
887
00:54:20,980 --> 00:54:24,985
wave forms you get results that
don't make any sense, and
888
00:54:24,985 --> 00:54:28,250
they make those channel
results meaningless.
889
00:54:28,250 --> 00:54:30,940
Therefore, from now on, whether
I remember to say it
890
00:54:30,940 --> 00:54:34,620
or not, everything we deal with,
unless we're looking at
891
00:54:34,620 --> 00:54:38,740
counter examples to something,
is going to be an L2 function.
892
00:54:45,630 --> 00:54:48,310
Let's go on.
893
00:54:48,310 --> 00:54:52,430
I'm starting to feel like I'm
back in our signals and
894
00:54:52,430 --> 00:54:56,570
systems course, because at this
point I'm defining my
895
00:54:56,570 --> 00:55:00,100
third different kind
of transform.
896
00:55:00,100 --> 00:55:02,990
Fortunately, this is the last
transform we will have to talk
897
00:55:02,990 --> 00:55:06,050
about, so we're all
done with this.
898
00:55:06,050 --> 00:55:09,860
The other nice thing is that
the discrete time Fourier
899
00:55:09,860 --> 00:55:15,020
transform happens to be just the
time frequency dual of the
900
00:55:15,020 --> 00:55:17,110
Fourier series.
901
00:55:17,110 --> 00:55:24,480
So that whether you've ever
studied the dtft or not, you
902
00:55:24,480 --> 00:55:27,460
already know everything there
is to know about it, because
903
00:55:27,460 --> 00:55:30,550
the only things there are to
know about it are the results
904
00:55:30,550 --> 00:55:33,050
about Fourier series.
905
00:55:33,050 --> 00:55:36,280
So the theorem is really the
same theorem that we had for
906
00:55:36,280 --> 00:55:38,570
Fourier series.
907
00:55:38,570 --> 00:55:43,360
Assume that you have a function
of frequency, u hat
908
00:55:43,360 --> 00:55:47,020
of f -- before we had a function
of time, now we have
909
00:55:47,020 --> 00:55:49,590
a function of frequency.
910
00:55:49,590 --> 00:55:52,680
Suppose it's defined over
the interval minus w
911
00:55:52,680 --> 00:55:56,260
to plus w into c.
912
00:55:56,260 --> 00:55:59,260
In other words, a way we often
say that a function is
913
00:55:59,260 --> 00:56:03,410
truncated is to say it goes
from some interval into c.
914
00:56:03,410 --> 00:56:08,510
This is a complex function which
is non-zero only for f
915
00:56:08,510 --> 00:56:12,290
in this finite bandwidth
range.
916
00:56:12,290 --> 00:56:16,240
We want to assume that this
is L2 and thus, we
917
00:56:16,240 --> 00:56:19,510
know it's also L1.
918
00:56:19,510 --> 00:56:23,380
Then we're going to take the
Fourier coefficients.
919
00:56:23,380 --> 00:56:26,710
Before we thought of the
Fourier coefficients as
920
00:56:26,710 --> 00:56:30,020
corresponding to what goes on at
different frequencies, now
921
00:56:30,020 --> 00:56:33,180
we're going to regard them as
time quantities, and we'll see
922
00:56:33,180 --> 00:56:35,360
exactly why later.
923
00:56:35,360 --> 00:56:37,240
So we'll define these
as the Fourier
924
00:56:37,240 --> 00:56:39,690
coefficients of this function.
925
00:56:39,690 --> 00:56:43,280
So they're 1 over 2w times
this integral here.
926
00:56:43,280 --> 00:56:46,650
You remember before when we
dealt with the Fourier series,
927
00:56:46,650 --> 00:56:50,220
we went from minus t over
2 to plus t over 2.
928
00:56:50,220 --> 00:56:53,250
Now we're going from
minus w to plus w.
929
00:56:53,250 --> 00:56:53,850
Why?
930
00:56:53,850 --> 00:56:57,150
It's just convention, there's
no real reason.
931
00:56:57,150 --> 00:57:02,790
So that what's happening here
is that this is a Fourier
932
00:57:02,790 --> 00:57:06,310
series formula for a coefficient
where we're
933
00:57:06,310 --> 00:57:11,710
substituting w for t over 2.
934
00:57:11,710 --> 00:57:15,130
We're putting a plus sign in
the exponent instead of a
935
00:57:15,130 --> 00:57:15,990
minus sign.
936
00:57:15,990 --> 00:57:21,500
In other words, we're doing this
hermitian duality bit.
937
00:57:21,500 --> 00:57:23,600
What's the other difference?
938
00:57:23,600 --> 00:57:26,090
We're interchanging time
and frequency.
939
00:57:26,090 --> 00:57:29,470
Aside from that it's exactly
the same theorem that we
940
00:57:29,470 --> 00:57:32,180
established -- well, that
we stated before.
941
00:57:32,180 --> 00:57:37,880
So we know that this quantity
here, since u hat of f is L1,
942
00:57:37,880 --> 00:57:41,690
this is finite and it exists
-- that's just a finite
943
00:57:41,690 --> 00:57:48,320
complex number and nothing more
-- for all integer k.
944
00:57:48,320 --> 00:57:54,580
Also, the convergence result
when we go back, this is the
945
00:57:54,580 --> 00:57:57,790
formula we try to use to go
back to the function we
946
00:57:57,790 --> 00:57:59,670
started with.
947
00:57:59,670 --> 00:58:02,020
It's just a finite
approximation
948
00:58:02,020 --> 00:58:04,300
to the Fourier series.
949
00:58:04,300 --> 00:58:08,350
We're saying that as k zero gets
larger and larger, this
950
00:58:08,350 --> 00:58:12,190
finite approximation approaches
the function in
951
00:58:12,190 --> 00:58:14,450
energy sense.
952
00:58:14,450 --> 00:58:19,420
So this is exactly what you
should mean by a Fourier
953
00:58:19,420 --> 00:58:20,430
series anyway.
954
00:58:20,430 --> 00:58:22,175
It's exactly what you
should mean by a
955
00:58:22,175 --> 00:58:24,110
Fourier transform anyway.
956
00:58:24,110 --> 00:58:28,610
As you go to the limit with more
and more terms you get
957
00:58:28,610 --> 00:58:34,010
something which is equal to what
you started with, except
958
00:58:34,010 --> 00:58:37,070
on the set of measure zero.
959
00:58:37,070 --> 00:58:40,240
In other words, it converges
everywhere where it matters.
960
00:58:40,240 --> 00:58:43,270
It converges to something
in the sense that the
961
00:58:43,270 --> 00:58:47,160
energy is the same.
962
00:58:47,160 --> 00:58:50,860
I said that in such a way that
makes it a little simpler than
963
00:58:50,860 --> 00:58:53,660
it really is.
964
00:58:53,660 --> 00:59:01,940
You can't always say that this
converges in any nice way.
965
00:59:01,940 --> 00:59:06,280
Next time I'm going to show
you a truly awful function
966
00:59:06,280 --> 00:59:10,910
which we'll use in the Fourier
series instead of dtft, which
967
00:59:10,910 --> 00:59:16,360
is time limited and which is
just incredibly messy and
968
00:59:16,360 --> 00:59:19,460
it'll show you why you have to
be a little bit careful about
969
00:59:19,460 --> 00:59:22,250
stating these results.
970
00:59:22,250 --> 00:59:26,250
But you don't have to worry
about it most of the time,
971
00:59:26,250 --> 00:59:29,560
because this theorem
is still true.
972
00:59:29,560 --> 00:59:31,960
It's just that you have to be a
little careful about how to
973
00:59:31,960 --> 00:59:35,230
interpret it because you don't
know whether this is going to
974
00:59:35,230 --> 00:59:37,340
reach a limit or not.
975
00:59:37,340 --> 00:59:39,720
All you know is that this will
be true, this energy
976
00:59:39,720 --> 00:59:42,960
difference goes to zero.
977
00:59:42,960 --> 00:59:48,060
Also, the energy in the function
is equal to the
978
00:59:48,060 --> 00:59:49,800
energy in the coefficients.
979
00:59:49,800 --> 00:59:52,610
This was the thing that we
found so useful with the
980
00:59:52,610 --> 00:59:54,810
Fourier series.
981
00:59:54,810 --> 01:00:00,420
It's why we can play this game
that we play with mean square
982
01:00:00,420 --> 01:00:06,550
quantization error of taking a
function and then turning it
983
01:00:06,550 --> 01:00:11,690
into a sequence of samples,
trying to quantize the samples
984
01:00:11,690 --> 01:00:16,050
for minimum mean square error
and associate the mean square
985
01:00:16,050 --> 01:00:18,590
error in the samples with
the mean square
986
01:00:18,590 --> 01:00:20,120
error on the function.
987
01:00:20,120 --> 01:00:24,280
You can't do that with anything
that I know of other
988
01:00:24,280 --> 01:00:25,830
than mean square error.
989
01:00:25,830 --> 01:00:28,790
If you want to deal with other
kinds of quantization errors,
990
01:00:28,790 --> 01:00:31,320
you have a real problem
going from
991
01:00:31,320 --> 01:00:34,170
coefficients to functions.
992
01:00:34,170 --> 01:00:43,310
And finally, for any set of
numbers u sub k, if the sum is
993
01:00:43,310 --> 01:00:46,470
less than infinity, in other
words, if you're dealing with
994
01:00:46,470 --> 01:00:52,770
a sequence of finite energy,
there always is such a
995
01:00:52,770 --> 01:00:54,540
frequency function.
996
01:00:54,540 --> 01:00:57,340
Many people when they use the
discrete time Fourier
997
01:00:57,340 --> 01:01:01,860
transform think of starting with
the sequence and taking a
998
01:01:01,860 --> 01:01:04,920
sequence and saying well it's
nice to think of this sequence
999
01:01:04,920 --> 01:01:09,270
in the frequency domain, and
then they say a function f
1000
01:01:09,270 --> 01:01:13,100
exists such that this is true,
or they just say that this is
1001
01:01:13,100 --> 01:01:16,300
equal to that without worrying
about the convergence at all,
1002
01:01:16,300 --> 01:01:18,150
which is more common.
1003
01:01:18,150 --> 01:01:20,620
But since we've already gone
through all of this for the
1004
01:01:20,620 --> 01:01:25,760
Fourier series, we might as well
say it right here also.
1005
01:01:25,760 --> 01:01:27,970
So there's really nothing
different here.
1006
01:01:27,970 --> 01:01:34,090
But now the question is why
do these u of k's --
1007
01:01:34,090 --> 01:01:38,060
I mean why do we think of those
as time coefficients?
1008
01:01:38,060 --> 01:01:40,420
I mean what's really going
on in this discrete
1009
01:01:40,420 --> 01:01:42,660
time Fourier transform.
1010
01:01:42,660 --> 01:01:47,050
At this point it just looks like
a lot of mathematics and
1011
01:01:47,050 --> 01:01:50,930
it's hard to interpret what
any of these things mean.
1012
01:01:50,930 --> 01:01:53,040
Well, the next thing I want
to do is to go into
1013
01:01:53,040 --> 01:01:55,760
the sampling theorem.
1014
01:01:55,760 --> 01:01:59,480
The sampling theorem, in fact,
is going to interpret for you
1015
01:01:59,480 --> 01:02:02,920
what this discrete time Fourier
transform is, because
1016
01:02:02,920 --> 01:02:05,760
the sampling theorem and the
discrete time Fourier
1017
01:02:05,760 --> 01:02:10,270
transform are just intimately
related, they're hand and
1018
01:02:10,270 --> 01:02:21,280
glove with each other,
and that's the next
1019
01:02:21,280 --> 01:02:22,800
thing we want to do.
1020
01:02:22,800 --> 01:02:27,230
But first we have to re-write
this a little bit.
1021
01:02:27,230 --> 01:02:32,180
We're going to say that this
frequency function is the
1022
01:02:32,180 --> 01:02:35,810
limit in the mean of this --
1023
01:02:35,810 --> 01:02:40,630
this rectangular function is
what we use just to make sure
1024
01:02:40,630 --> 01:02:43,190
we're only talking about
frequency between
1025
01:02:43,190 --> 01:02:47,250
minus w and plus w.
1026
01:02:47,250 --> 01:02:50,355
The limit in the mean, there's a
little notational trick that
1027
01:02:50,355 --> 01:02:55,510
we use so that we can think of
this as just a limit instead
1028
01:02:55,510 --> 01:02:59,310
of thinking of it as this
crazy thing that we just
1029
01:02:59,310 --> 01:03:02,120
derive, which is really
not so crazy.
1030
01:03:04,900 --> 01:03:08,540
That means we can talk about
this transform without always
1031
01:03:08,540 --> 01:03:14,060
rubbing our noses in all
of this mess here.
1032
01:03:14,060 --> 01:03:16,900
It just means that once in
awhile we go back and think
1033
01:03:16,900 --> 01:03:18,630
what does this really mean.
1034
01:03:18,630 --> 01:03:22,110
It means convergence in energy
rather than convergence
1035
01:03:22,110 --> 01:03:26,640
point-wise, because we might
not have convergence
1036
01:03:26,640 --> 01:03:27,890
point-wise.
1037
01:03:29,650 --> 01:03:34,660
So we're going to write this
also as the limit in the mean
1038
01:03:34,660 --> 01:03:37,580
of the sum over k of u of k.
1039
01:03:37,580 --> 01:03:40,480
We're going to glop all
of this together.
1040
01:03:40,480 --> 01:03:45,940
This is just some function
of k and a frequency.
1041
01:03:45,940 --> 01:03:50,920
So we're going to call that phi
sub k of f at some wave
1042
01:03:50,920 --> 01:03:56,150
form, and the wave
form is this.
1043
01:03:56,150 --> 01:04:01,940
What happens if you look at the
relationship between to
1044
01:04:01,940 --> 01:04:06,030
phi k of f and phi
k prime of f?
1045
01:04:06,030 --> 01:04:08,200
Namely, if you look at two
different functions.
1046
01:04:11,220 --> 01:04:14,490
These two functions are
orthongonal to each other for
1047
01:04:14,490 --> 01:04:21,570
the same reason that the
functions that the sinusoid
1048
01:04:21,570 --> 01:04:27,040
you used in the Fourier series
are orthongonal.
1049
01:04:27,040 --> 01:04:31,790
Namely, you take this function,
you multiply it by e
1050
01:04:31,790 --> 01:04:35,450
to the minus 2 pi ik prime
f over 2w times
1051
01:04:35,450 --> 01:04:37,070
this rectangular function.
1052
01:04:37,070 --> 01:04:40,880
You integrate from minus w
to w and what do you get?
1053
01:04:40,880 --> 01:04:43,490
You're just integrating
a sinusoid --
1054
01:04:43,490 --> 01:04:45,880
the whole thing is one
big sinusoid --
1055
01:04:45,880 --> 01:04:49,520
over one period of that sinusoid
or multiple periods
1056
01:04:49,520 --> 01:04:50,630
of the sinusoid.
1057
01:04:50,630 --> 01:04:54,650
Actually, k minus k prime
periods of the sinusoid.
1058
01:04:54,650 --> 01:04:57,360
And when you integrate a
sinusoid over a period, what
1059
01:04:57,360 --> 01:04:58,140
do you get?
1060
01:04:58,140 --> 01:05:00,200
You get zero.
1061
01:05:00,200 --> 01:05:03,330
So it says that these functions
are all orthongonal
1062
01:05:03,330 --> 01:05:04,460
to each other.
1063
01:05:04,460 --> 01:05:08,460
So, presto, we have another
orthongonal expansion just
1064
01:05:08,460 --> 01:05:11,850
like the Fourier series gave us
an orthongonal expansion.
1065
01:05:11,850 --> 01:05:14,020
And in fact, it's the same
orthongonal expansion.
1066
01:05:31,050 --> 01:05:36,400
Now the next thing to observe
is that we have done the
1067
01:05:36,400 --> 01:05:41,220
Fourier transform and we've also
done the discrete time
1068
01:05:41,220 --> 01:05:43,070
Fourier transform.
1069
01:05:43,070 --> 01:05:48,010
In both of them we're dealing
with some frequency function.
1070
01:05:48,010 --> 01:05:50,720
Now we're dealing with some
frequency function which is
1071
01:05:50,720 --> 01:05:54,560
limited to minus w to plus
w, but we have two
1072
01:05:54,560 --> 01:05:57,570
expansions for it.
1073
01:05:57,570 --> 01:06:00,740
We have the Fourier transform,
so we can go to a function u
1074
01:06:00,740 --> 01:06:03,710
of t, and we also have
this discrete
1075
01:06:03,710 --> 01:06:07,070
time Fourier transform.
1076
01:06:07,070 --> 01:06:13,070
So u of t is equal to this
Fourier transform here.
1077
01:06:13,070 --> 01:06:17,580
Again, I should write limit in
the mean here, but then I
1078
01:06:17,580 --> 01:06:19,390
think about and I say
do I have to write
1079
01:06:19,390 --> 01:06:21,220
limit in the mean?
1080
01:06:21,220 --> 01:06:22,090
No.
1081
01:06:22,090 --> 01:06:24,520
I don't need a limit in the
mean here because I'm
1082
01:06:24,520 --> 01:06:26,970
integrating this over
finite limits.
1083
01:06:26,970 --> 01:06:31,050
Since I'm taking a function over
finite limits, u hat of f
1084
01:06:31,050 --> 01:06:36,150
is over these limits, is an L1
function, therefore, this
1085
01:06:36,150 --> 01:06:37,010
integral exists.
1086
01:06:37,010 --> 01:06:41,510
This function is a continuous
function.
1087
01:06:41,510 --> 01:06:45,070
There aren't any sets of measure
zero involved here.
1088
01:06:45,070 --> 01:06:49,740
This is one specific function
which is always the same.
1089
01:06:49,740 --> 01:06:52,460
You know what it is exactly
at every point.
1090
01:06:52,460 --> 01:06:56,370
At every point t,
this converges.
1091
01:06:56,370 --> 01:06:58,890
So then what we're going to do,
what the sampling theorem
1092
01:06:58,890 --> 01:07:05,030
does is it relates this to
what you get with a dtft.
1093
01:07:05,030 --> 01:07:09,090
So the sampling theorem says
let u hat of f be an L2
1094
01:07:09,090 --> 01:07:20,050
function which goes from minus
ww to c, and let u of t be
1095
01:07:20,050 --> 01:07:23,730
this, namely, that, which we
now know exists and is
1096
01:07:23,730 --> 01:07:25,110
continuous.
1097
01:07:25,110 --> 01:07:28,750
Define capital T as 1 over 2w.
1098
01:07:28,750 --> 01:07:32,270
You don't have to do that if you
don't want to, but it's a
1099
01:07:32,270 --> 01:07:35,250
little easier to think in terms
of some increment of
1100
01:07:35,250 --> 01:07:37,880
time, T, here.
1101
01:07:37,880 --> 01:07:42,120
Then u of t is continuous,
L2 and bounded.
1102
01:07:42,120 --> 01:07:44,570
It's bounded by u of t less
than or equal to this.
1103
01:07:44,570 --> 01:07:46,030
Why is that?
1104
01:07:53,295 --> 01:07:57,670
It doesn't make any sense
as I stated it.
1105
01:07:57,670 --> 01:07:58,940
Now it makes sense.
1106
01:07:58,940 --> 01:08:01,720
OK, its magnitude is bounded.
1107
01:08:01,720 --> 01:08:04,880
Its magnitude is bounded because
if you take u of t
1108
01:08:04,880 --> 01:08:09,310
magnitude, it's equal to the
magnitude of this which is
1109
01:08:09,310 --> 01:08:12,740
less than or equal to the
integral of the magnitude of
1110
01:08:12,740 --> 01:08:18,130
this, which is equal to the
integral of the magnitude of
1111
01:08:18,130 --> 01:08:22,370
just u hat of f, which
is what we have here.
1112
01:08:22,370 --> 01:08:25,470
So all of that works nicely.
1113
01:08:25,470 --> 01:08:29,520
So, u of t is a nice,
well-defined function.
1114
01:08:29,520 --> 01:08:33,150
Then the other part of it is
that u of t is equal to the
1115
01:08:33,150 --> 01:08:38,800
sum if its values at these
sample points times sinc of t
1116
01:08:38,800 --> 01:08:40,810
minus kt over t.
1117
01:08:40,810 --> 01:08:43,490
Now you've probably seen this
sampling theorem before.
1118
01:08:43,490 --> 01:08:46,430
How many people haven't
seen this before?
1119
01:08:46,430 --> 01:08:50,130
I mean aside from the question
of trying to do it --
1120
01:08:50,130 --> 01:08:52,870
do it in a way that
makes sense.
1121
01:08:52,870 --> 01:08:53,890
OK, you've all seen it.
1122
01:08:53,890 --> 01:08:55,080
So good.
1123
01:08:55,080 --> 01:09:00,160
What it's saying is you can
represent a function in terms
1124
01:09:00,160 --> 01:09:03,310
of just knowing what
its samples are.
1125
01:09:03,310 --> 01:09:06,890
Or you can take the function,
you can sample it, and when
1126
01:09:06,890 --> 01:09:14,340
you sample it if you put these
little sinc hats around all
1127
01:09:14,340 --> 01:09:17,180
the samples, you get back
to the function again.
1128
01:09:17,180 --> 01:09:20,490
So you take all the samples,
you then put these sincs
1129
01:09:20,490 --> 01:09:22,630
around them, add them
all up, and
1130
01:09:22,630 --> 01:09:27,680
bingo, you got the function.
1131
01:09:27,680 --> 01:09:29,880
Let's see why that's true.
1132
01:09:35,820 --> 01:09:38,290
Here's the sinc function here.
1133
01:09:38,290 --> 01:09:43,290
The important thing about sinc
t, sine pi t over pi t, which
1134
01:09:43,290 --> 01:09:47,410
you can see by just looking at
the sine function, is it has
1135
01:09:47,410 --> 01:09:51,040
the value 1 when t
is equal to zero.
1136
01:09:51,040 --> 01:09:53,780
I mean to get that value 1, you
really have to go through
1137
01:09:53,780 --> 01:09:57,830
a limiting operation here to
think of sine pi t when t is
1138
01:09:57,830 --> 01:10:01,580
very small as being
approximately equal to pi t.
1139
01:10:01,580 --> 01:10:05,580
When you divide pi t by pi t you
get 1, so that's its value
1140
01:10:05,580 --> 01:10:08,490
there and value around there.
1141
01:10:08,490 --> 01:10:13,580
At every other sample point,
namely, at t equals 1, sine of
1142
01:10:13,580 --> 01:10:15,220
pi t is zero.
1143
01:10:15,220 --> 01:10:19,200
At t equals 2, sine of pi
t is equal to zero.
1144
01:10:19,200 --> 01:10:25,030
So the sinc function is 1 at
zero and is zero at every
1145
01:10:25,030 --> 01:10:27,660
other integer point.
1146
01:10:27,660 --> 01:10:32,240
Now to see why it's true and to
understand what the dtft is
1147
01:10:32,240 --> 01:10:39,270
all about, note that we have
said that a frequency
1148
01:10:39,270 --> 01:10:44,150
function, u hat of f, can be
expressed as the sum over k --
1149
01:10:44,150 --> 01:10:47,530
and I should use a limit in the
mean here but I'm not --
1150
01:10:47,530 --> 01:10:51,970
of uk times this transform
relationship here.
1151
01:10:54,560 --> 01:10:59,450
Well, these are these functions
that we talked about
1152
01:10:59,450 --> 01:11:00,700
awhile ago.
1153
01:11:09,010 --> 01:11:09,980
They're these functions.
1154
01:11:09,980 --> 01:11:15,840
They're the sinusoids, periodic
sinusoids in k
1155
01:11:15,840 --> 01:11:18,510
truncated in frequency.
1156
01:11:24,430 --> 01:11:31,880
So we know that u hat of f is
equal to that dtft expansion.
1157
01:11:31,880 --> 01:11:35,360
If I take the inverse Fourier
transform of that, I can take
1158
01:11:35,360 --> 01:11:38,490
the inverse Fourier transform
of all these functions and
1159
01:11:38,490 --> 01:11:43,590
I'll get u of t is equal to the
sum over k, of uk pk of t.
1160
01:11:43,590 --> 01:11:45,710
I'm being careless about
the mathematics here.
1161
01:11:45,710 --> 01:11:48,510
I've been careful about
it all along.
1162
01:11:48,510 --> 01:11:52,450
The notes does it carefully,
particularly in the appendix.
1163
01:11:52,450 --> 01:11:54,750
I'm not going to worry
about that here.
1164
01:11:54,750 --> 01:11:58,520
If I take the function pk of
f, which is this truncated
1165
01:11:58,520 --> 01:12:03,340
sinusoid, and I take the inverse
transform of that --
1166
01:12:03,340 --> 01:12:07,350
take the transform of
this, I get this.
1167
01:12:07,350 --> 01:12:10,870
Can you see that just
by inspection?
1168
01:12:10,870 --> 01:12:13,090
If you were really hot on these
things, if you just
1169
01:12:13,090 --> 01:12:16,830
finished taking 6.003, you could
probably see that by
1170
01:12:16,830 --> 01:12:18,630
inspection.
1171
01:12:18,630 --> 01:12:23,490
If you remember all of those
relationships that we went
1172
01:12:23,490 --> 01:12:26,650
through before, you can
see it by inspection.
1173
01:12:26,650 --> 01:12:31,260
The Fourier transform of erect
function is a sinc function.
1174
01:12:31,260 --> 01:12:35,520
This exponential here when you
go into the time domain
1175
01:12:35,520 --> 01:12:39,010
corresponds to a time shift,
so that gives rise to this
1176
01:12:39,010 --> 01:12:41,350
time shift here.
1177
01:12:41,350 --> 01:12:45,150
The 1 over t is just one of
these constants you have to
1178
01:12:45,150 --> 01:12:48,250
keep straight, and which I would
do just by integrating
1179
01:12:48,250 --> 01:12:51,110
the things to see what I get.
1180
01:12:51,110 --> 01:13:00,970
Finally, u of kt, if I look at
this, is just 1 over t times u
1181
01:13:00,970 --> 01:13:05,220
of k, because these functions
here are these sinc functions,
1182
01:13:05,220 --> 01:13:08,350
which are zero everywhere
but on their own point.
1183
01:13:08,350 --> 01:13:17,240
So if I look at u of kt, it's a
sum over k of ck of kt, and
1184
01:13:17,240 --> 01:13:26,160
ck of kt is only 1 when little
t is equal to k times capital
1185
01:13:26,160 --> 01:13:30,690
T, and therefore, I get
that point there.
1186
01:13:30,690 --> 01:13:35,740
Therefore, u of kt is just
1 over t times u of k.
1187
01:13:35,740 --> 01:13:39,950
That finishes the sampling
theorem except for really
1188
01:13:39,950 --> 01:13:47,110
tracing through all of these
things about convergence, but
1189
01:13:47,110 --> 01:13:50,350
it also tells you what the
discrete time Fourier
1190
01:13:50,350 --> 01:13:52,380
transform is.
1191
01:13:52,380 --> 01:13:56,420
Because the discrete time
Fourier transform is just
1192
01:13:56,420 --> 01:13:59,560
scaled samples of u of t.
1193
01:13:59,560 --> 01:14:02,380
In other words, you start out
with this frequency function,
1194
01:14:02,380 --> 01:14:05,110
you take the inverse Fourier
transform of it, you get a
1195
01:14:05,110 --> 01:14:06,280
time function.
1196
01:14:06,280 --> 01:14:10,050
You take the samples of that,
you scale them, and those are
1197
01:14:10,050 --> 01:14:12,800
the coefficients in the
discrete time Fourier
1198
01:14:12,800 --> 01:14:15,410
transform, which is what you
use discrete time Fourier
1199
01:14:15,410 --> 01:14:17,100
transforms for.
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01:14:17,100 --> 01:14:20,520
You think of sampling of
function, then you represent
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01:14:20,520 --> 01:14:23,450
the function in terms of those
samples and then you want to
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01:14:23,450 --> 01:14:26,820
go into the frequency domain as
a way of dealing with the
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01:14:26,820 --> 01:14:30,110
properties of those samples, and
all you're doing is just
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01:14:30,110 --> 01:14:34,690
going into the Fourier transform
of u of t, and all
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01:14:34,690 --> 01:14:37,550
of that works out.
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01:14:42,780 --> 01:14:46,370
There's one bizarre thing here,
and I'm going to talk
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01:14:46,370 --> 01:14:50,170
about that more next time.
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01:14:50,170 --> 01:14:54,340
That is that when you look at
this time frequency limited
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01:14:54,340 --> 01:15:03,080
function, u hat of f, u hat of
f can be very badly behaved.
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01:15:03,080 --> 01:15:06,340
You can have a frequency limited
function which does
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01:15:06,340 --> 01:15:08,330
all sorts of crazy things.
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01:15:08,330 --> 01:15:11,770
Since it's frequency limited
as inverse transform, it's
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01:15:11,770 --> 01:15:12,820
beautifully behaved.
1214
01:15:12,820 --> 01:15:15,460
It's just the sum of sinc
functions -- it's bounded,
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01:15:15,460 --> 01:15:18,580
it's continuous and
everything else.
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01:15:18,580 --> 01:15:22,180
When we go back into the
frequency domain it is just as
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01:15:22,180 --> 01:15:23,980
ugly as can be.
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01:15:23,980 --> 01:15:29,050
So what we have in the sampling
theorem, it comes out
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01:15:29,050 --> 01:15:31,570
particularly clearly.
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01:15:31,570 --> 01:15:36,240
Is that L1 functions in one
domain are nice, continuous,
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01:15:36,240 --> 01:15:39,910
beautiful functions which are
not L1 in the other domain.
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01:15:39,910 --> 01:15:44,260
So you sort of go from
L1 to continuous.
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01:15:44,260 --> 01:15:47,360
Now when we're dealing with
L2 functions, they're not
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01:15:47,360 --> 01:15:50,630
continuous, they're not anything
else, but you always
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01:15:50,630 --> 01:15:52,380
go from L2 to L2.
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01:15:52,380 --> 01:15:56,240
In other words, you can't get
out of the L2 domain, and
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01:15:56,240 --> 01:16:01,140
therefore, when you're dealing
with L2 functions, all you
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01:16:01,140 --> 01:16:04,700
have to worry about is L2
functions, because you always
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01:16:04,700 --> 01:16:06,890
stay there no matter what.
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01:16:06,890 --> 01:16:10,120
When we start talking about
stochastic processes, we'll
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01:16:10,120 --> 01:16:12,510
find out you always
stay there also.
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01:16:12,510 --> 01:16:15,450
In other words, you only have
to know about one thing.
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01:16:15,450 --> 01:16:18,380
We've seen here that to
interpret what these Fourier
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01:16:18,380 --> 01:16:23,230
transforms mean, it's nice to
have a little idea about L1
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01:16:23,230 --> 01:16:28,400
functions also because when we
think of going to the limit,
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01:16:28,400 --> 01:16:33,450
when we go to the limit we get
something which is badly
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01:16:33,450 --> 01:16:37,390
behaved, and for these finite
time and finite frequency
1238
01:16:37,390 --> 01:16:41,000
approximations, we have things
which are beautifully behaved.
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01:16:41,000 --> 01:16:43,280
I'm going to stop now so we
can pass out the quizzes.