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PROFESSOR: I want to spend a
little bit of time today

9
00:00:25,110 --> 00:00:29,860
finishing up what we were
saying last time, about

10
00:00:29,860 --> 00:00:32,910
orthonormal expansions.

11
00:00:32,910 --> 00:00:39,600
Just, essentially, to get a
better understanding of what

12
00:00:39,600 --> 00:00:41,740
it means to go from finite

13
00:00:41,740 --> 00:00:45,030
dimension to infinite dimension.

14
00:00:45,030 --> 00:00:48,280
For the most part, you can just
ignore the question of

15
00:00:48,280 --> 00:00:52,780
being at an infinite dimensional
space, and

16
00:00:52,780 --> 00:00:55,610
everything is pretty much the
same as it is in finite

17
00:00:55,610 --> 00:00:56,780
dimensions.

18
00:00:56,780 --> 00:00:59,500
But there are a few things
you have to be a

19
00:00:59,500 --> 00:01:01,700
little careful about.

20
00:01:01,700 --> 00:01:07,020
So, let's suppose that we have
a set of orthonormal vectors.

21
00:01:07,020 --> 00:01:10,310
Like the sinc vectors for
example, that we were using

22
00:01:10,310 --> 00:01:13,490
for the sinc functions that we
were using, to talk about the

23
00:01:13,490 --> 00:01:16,040
Fourier series.

24
00:01:16,040 --> 00:01:26,830
And the thing that we said last
time at the end of the

25
00:01:26,830 --> 00:01:32,710
hour, we quoted this theorem
which was almost obvious, I

26
00:01:32,710 --> 00:01:36,410
think, in terms of the other
things that we did.

27
00:01:36,410 --> 00:01:40,220
Which said that if you start
out with some vector v, you

28
00:01:40,220 --> 00:01:47,010
can project it successively onto
the set of orthonormal

29
00:01:47,010 --> 00:01:51,350
functions in such a way that you
get closer and closer to

30
00:01:51,350 --> 00:01:53,560
the vector that you're
looking for.

31
00:01:53,560 --> 00:01:58,530
And the theorem that we had was
that the projection on the

32
00:01:58,530 --> 00:02:04,950
whole space, which we call u,
in fact is the same as the

33
00:02:04,950 --> 00:02:10,510
limit of this sum of orthonormal
functions as we

34
00:02:10,510 --> 00:02:15,780
take the limit adding more and
more functions into this sum.

35
00:02:15,780 --> 00:02:18,790
So this was sort of the same
kind of thing that we were

36
00:02:18,790 --> 00:02:21,900
doing when we were going --

37
00:02:27,570 --> 00:02:30,230
it's the same sort of thing that
we did when we were doing

38
00:02:30,230 --> 00:02:34,070
the projection theorem for
finite dimensions.

39
00:02:34,070 --> 00:02:38,110
You remember the thing that we
did there was to first project

40
00:02:38,110 --> 00:02:43,440
this function onto a
single waveform.

41
00:02:43,440 --> 00:02:47,550
Then we found the part of the
waveform that was orthogonal

42
00:02:47,550 --> 00:02:49,130
to the first waveform.

43
00:02:49,130 --> 00:02:52,150
That gave us our second
waveform.

44
00:02:52,150 --> 00:02:57,820
And the thing that we're doing
here, which is sort of

45
00:02:57,820 --> 00:03:02,000
different, is we're starting
out with the orthogonal

46
00:03:02,000 --> 00:03:05,830
sequence to start with.

47
00:03:05,830 --> 00:03:09,720
Anyway, the thing that we showed
was that the vector we

48
00:03:09,720 --> 00:03:15,600
start with, minus this
approximation, which is in the

49
00:03:15,600 --> 00:03:19,320
space generated by all of these
p sub n's, that this

50
00:03:19,320 --> 00:03:22,600
difference has to be orthogonal
to each p sub n.

51
00:03:22,600 --> 00:03:26,150
Because that was a way
we constructed it.

52
00:03:26,150 --> 00:03:31,780
So this difference is -- well,
the limit goes to zero, but

53
00:03:31,780 --> 00:03:34,770
also the difference is
orthogonal to each phi sub x.

54
00:03:34,770 --> 00:03:38,880
This is the thing which is new,
which we didn't get from

55
00:03:38,880 --> 00:03:41,130
looking at the Fourier series.

56
00:03:41,130 --> 00:03:44,030
We could have gotten it from
looking at the Fourier series,

57
00:03:44,030 --> 00:03:48,640
but this is the thing which is
general for every orthonormal

58
00:03:48,640 --> 00:03:50,770
expansion in the world.

59
00:03:50,770 --> 00:03:55,440
Not just a Fourier series,
except all of them.

60
00:03:55,440 --> 00:03:59,600
And then we say that an inner
product space has accountably

61
00:03:59,600 --> 00:04:02,230
infinite dimension.

62
00:04:02,230 --> 00:04:06,070
If accountably infinite set of
orthonormal vectors exist,

63
00:04:06,070 --> 00:04:08,240
such that only the
zero vectors are

64
00:04:08,240 --> 00:04:10,520
orthogonal to each other.

65
00:04:10,520 --> 00:04:14,530
This comes back to one of the
issues that we faced when we

66
00:04:14,530 --> 00:04:16,380
were going through the
Fourier series.

67
00:04:16,380 --> 00:04:23,850
Namely, we showed all this nice
stuff, which was really

68
00:04:23,850 --> 00:04:28,660
about approximating a waveform
by the Fourier series.

69
00:04:28,660 --> 00:04:31,630
The thing that we never showed,
and the thing which is

70
00:04:31,630 --> 00:04:35,700
messy and difficult, is that
if you take a time-limited

71
00:04:35,700 --> 00:04:39,590
function and expand it in a
Fourier series, how do you

72
00:04:39,590 --> 00:04:42,450
know that when you get all done,
you're actually going to

73
00:04:42,450 --> 00:04:45,220
get the function back again?

74
00:04:45,220 --> 00:04:48,610
This is the part of it which
we never talked about.

75
00:04:48,610 --> 00:04:51,540
We talked about how you generate
all of the Fourier

76
00:04:51,540 --> 00:04:55,180
coefficients, and all
of that was fine.

77
00:04:55,180 --> 00:04:58,480
And we showed that when you
were all done, this

78
00:04:58,480 --> 00:05:03,210
representation function that you
had was in fact orthogonal

79
00:05:03,210 --> 00:05:04,770
the thing that was left over.

80
00:05:04,770 --> 00:05:06,790
But you never showed that
the thing that was left

81
00:05:06,790 --> 00:05:09,680
over went to 0.

82
00:05:09,680 --> 00:05:12,820
And the thing this is saying
is you don't have to worry

83
00:05:12,820 --> 00:05:14,220
about that too much.

84
00:05:14,220 --> 00:05:17,470
Because the thing which is left
over has to be orthogonal

85
00:05:17,470 --> 00:05:19,560
to everything you
started with.

86
00:05:19,560 --> 00:05:24,030
And this sum here has
to go to some limit.

87
00:05:24,030 --> 00:05:27,230
What is this limit in the case
of the Fourier series?

88
00:05:27,230 --> 00:05:31,030
Suppose we start out with an
arbitrary L2 function.

89
00:05:31,030 --> 00:05:35,260
We now try to expand that
arbitrary L2 function in a

90
00:05:35,260 --> 00:05:39,940
Fourier series from minus t
over 2 to plus t over 2.

91
00:05:39,940 --> 00:05:42,430
And we can do that.

92
00:05:42,430 --> 00:05:48,380
When we're all done, this series
here is in fact going

93
00:05:48,380 --> 00:05:52,080
to be the part of the function
which lies between minus t

94
00:05:52,080 --> 00:05:54,890
over 2 and plus t over 2.

95
00:05:54,890 --> 00:06:01,420
And this difference here -- well
v minus u, the difference

96
00:06:01,420 --> 00:06:05,500
between the original vector and
this representation vector

97
00:06:05,500 --> 00:06:08,220
is going to be this part of
the function which lies

98
00:06:08,220 --> 00:06:11,550
outside of minus t over
2 to plus t over 2.

99
00:06:15,700 --> 00:06:18,580
So the next thing that we did
then was to show that the

100
00:06:18,580 --> 00:06:23,080
truncated sinusoids and the sinc
weighted sinusoids both

101
00:06:23,080 --> 00:06:25,250
in fact span L2.

102
00:06:25,250 --> 00:06:32,350
In other words, any function in
L2, after you represent it

103
00:06:32,350 --> 00:06:36,450
in terms of that series, what
you have left over has to be

104
00:06:36,450 --> 00:06:37,770
orthogonal.

105
00:06:37,770 --> 00:06:42,250
In other words, if you take this
entire set of truncated

106
00:06:42,250 --> 00:06:48,060
sinusoids, every non-zero
function is

107
00:06:48,060 --> 00:06:50,820
orthogonal to all of that.

108
00:06:50,820 --> 00:06:56,130
Every non-zero function in
this L2 equivalent sense.

109
00:06:56,130 --> 00:07:01,580
So in fact, the thing you get
out of this is this part of

110
00:07:01,580 --> 00:07:05,460
looking at functions which
we always ignored.

111
00:07:05,460 --> 00:07:07,890
There's another issue that we
ignored, when we were looking

112
00:07:07,890 --> 00:07:12,100
at functions, which
comes out of this.

113
00:07:12,100 --> 00:07:17,020
Since v, by assumption
is an L2 vector.

114
00:07:17,020 --> 00:07:20,450
In other words, it's an L2
function aside from this L2

115
00:07:20,450 --> 00:07:25,940
equivalent stuff, and phi sub
m is an L2 vector also, this

116
00:07:25,940 --> 00:07:28,980
inner product has to be
finite, by the Schwarz

117
00:07:28,980 --> 00:07:31,280
inequality.

118
00:07:31,280 --> 00:07:33,440
How did we get around this when
we were dealing with the

119
00:07:33,440 --> 00:07:34,195
Fourier series?

120
00:07:34,195 --> 00:07:36,080
Does anybody remember?

121
00:07:36,080 --> 00:07:37,400
Of course you don't remember.

122
00:07:37,400 --> 00:07:40,120
I hardly remembered that.

123
00:07:40,120 --> 00:07:45,500
The way we got around it
was by saying that this

124
00:07:45,500 --> 00:07:50,230
orthonormal function which was
in a truncated sinusoid.

125
00:07:50,230 --> 00:07:54,590
It was truncated to minus t over
2 and plus t over 2, and

126
00:07:54,590 --> 00:07:59,810
the waveform was just
e to the i 2 pi f t.

127
00:07:59,810 --> 00:08:02,405
The magnitude of that function
was always less

128
00:08:02,405 --> 00:08:03,655
than or equal to 1.

129
00:08:06,470 --> 00:08:09,330
And because the magnitude was
always less than or equal to

130
00:08:09,330 --> 00:08:14,130
1, you could just take the
integral of v of t times phi

131
00:08:14,130 --> 00:08:17,750
sub n of t, and you could show
directly that that integral

132
00:08:17,750 --> 00:08:19,400
always existed.

133
00:08:19,400 --> 00:08:22,660
Because of the special property
of the sinusoids.

134
00:08:22,660 --> 00:08:25,160
Here what we're saying is,
you don't have to worry

135
00:08:25,160 --> 00:08:27,160
about that any more.

136
00:08:27,160 --> 00:08:32,110
You can use arbitrary L2
functions as the components of

137
00:08:32,110 --> 00:08:36,350
an L2 orthonormal expansion.

138
00:08:36,350 --> 00:08:37,960
And it still works.

139
00:08:37,960 --> 00:08:41,480
So every one of these things
has to be finite also.

140
00:08:41,480 --> 00:08:44,040
So, in fact we're buying
something out of this.

141
00:08:44,040 --> 00:08:48,690
At the end of the notes on
Lectures Eight to Ten, you

142
00:08:48,690 --> 00:08:52,140
will notice something that I'm
certainly not going to hold

143
00:08:52,140 --> 00:08:55,140
you responsible for.

144
00:08:55,140 --> 00:08:59,350
Something called the prolate
spiroidal expansion.

145
00:08:59,350 --> 00:09:02,940
And it's just given there
primarily as one more example

146
00:09:02,940 --> 00:09:05,600
of an orthonormal expansion.

147
00:09:05,600 --> 00:09:08,950
It's a very interesting
orthonormal expansion because

148
00:09:08,950 --> 00:09:15,760
it has the property that if you
start out asking how much

149
00:09:15,760 --> 00:09:20,170
energy can I concentrate within
a fixed time interval,

150
00:09:20,170 --> 00:09:22,540
and a fixed bandwidth
interval.

151
00:09:22,540 --> 00:09:25,610
Which is one of the things which
make this whole subject

152
00:09:25,610 --> 00:09:29,880
a little bit fishy
in a whole and

153
00:09:29,880 --> 00:09:31,530
certainly very, very messy.

154
00:09:31,530 --> 00:09:34,790
If I start out with a time
limited function and go

155
00:09:34,790 --> 00:09:38,970
through the Fourier series,
these truncated sinusoids

156
00:09:38,970 --> 00:09:44,050
spill their energy out of
the band enormously.

157
00:09:44,050 --> 00:09:47,840
In fact, one of the problems at
the end of Lectures Eight

158
00:09:47,840 --> 00:09:52,500
to Ten carries you through the
process of just how much

159
00:09:52,500 --> 00:09:55,990
energy you can have outside
of band by using

160
00:09:55,990 --> 00:09:57,780
these truncated sinusoids.

161
00:09:57,780 --> 00:10:02,260
Because truncated sinusoids,
when you look at them in a

162
00:10:02,260 --> 00:10:05,160
Fourier series, have a lot
of energy outside of

163
00:10:05,160 --> 00:10:07,910
where it should be.

164
00:10:07,910 --> 00:10:14,300
So that this particular set of
prolate spiroidal functions

165
00:10:14,300 --> 00:10:18,010
gives you the answer to the
following question.

166
00:10:18,010 --> 00:10:22,540
I would like to find that
function, which is limited to

167
00:10:22,540 --> 00:10:27,410
minus t over 2 plus t over 2,
which has the largest amount

168
00:10:27,410 --> 00:10:31,220
of energy, largest fraction of
its energy, within the band

169
00:10:31,220 --> 00:10:33,870
minus w to plus w.

170
00:10:33,870 --> 00:10:35,530
What is that function?

171
00:10:35,530 --> 00:10:38,850
Well, that function happens to
be the zero order prolate

172
00:10:38,850 --> 00:10:41,460
spheroidal function.

173
00:10:41,460 --> 00:10:42,970
It's the nice property
that it has.

174
00:10:42,970 --> 00:10:46,330
And it's a nice function, which
almost looks like a

175
00:10:46,330 --> 00:10:47,590
rectangular function.

176
00:10:47,590 --> 00:10:50,370
But it just trails off to 0.

177
00:10:50,370 --> 00:10:54,480
And I say, OK, if I want to
construct an orthonormal

178
00:10:54,480 --> 00:10:59,920
expansion, and I want to find
another function which is

179
00:10:59,920 --> 00:11:03,870
orthogonal to that function
and has the next biggest

180
00:11:03,870 --> 00:11:08,880
amount fraction of its energy,
inside of this -- strictly

181
00:11:08,880 --> 00:11:12,700
inside of this time limit, and
as much of the energy within

182
00:11:12,700 --> 00:11:14,640
the frequency limit
as possible,

183
00:11:14,640 --> 00:11:16,820
what's that next function?

184
00:11:16,820 --> 00:11:18,590
Well, you solve that problem.

185
00:11:18,590 --> 00:11:20,830
If you're very good at
interval equations.

186
00:11:20,830 --> 00:11:23,010
And I certainly couldn't
solve it.

187
00:11:23,010 --> 00:11:25,480
But, anyway, it has
been solved.

188
00:11:25,480 --> 00:11:28,610
I mean, physicists for a long
time have worried about that

189
00:11:28,610 --> 00:11:30,200
particular kind of question.

190
00:11:30,200 --> 00:11:32,610
Because it comes up in
physics all the time.

191
00:11:32,610 --> 00:11:35,600
If you time-limit something
and you take the Fourier

192
00:11:35,600 --> 00:11:39,390
integral, how can you also come
as close to frequency

193
00:11:39,390 --> 00:11:41,140
limiting as possible?

194
00:11:41,140 --> 00:11:45,430
So this particular prolate
spheroidal set of orthogonal

195
00:11:45,430 --> 00:11:50,190
functions, in fact, exactly
solves the problem of how do

196
00:11:50,190 --> 00:11:53,390
you generate a set of
orthonormal functions which in

197
00:11:53,390 --> 00:11:56,680
fact have as much energy
as possible both

198
00:11:56,680 --> 00:11:59,900
frequency-limited and
time-limited?

199
00:11:59,900 --> 00:12:04,930
And what you find when you do
that is, when you take 2 w t

200
00:12:04,930 --> 00:12:11,310
of them, at that point the
energy in these orthonormal

201
00:12:11,310 --> 00:12:14,670
functions, the part of it that's
inside the band, really

202
00:12:14,670 --> 00:12:17,540
starts to cut off very,
very sharply,

203
00:12:17,540 --> 00:12:19,940
when w and t are large.

204
00:12:19,940 --> 00:12:23,710
So that in fact you get 2 w t of
these functions, which have

205
00:12:23,710 --> 00:12:26,890
almost all of their energy
in this band.

206
00:12:26,890 --> 00:12:30,690
And everything else has almost
no energy in the band.

207
00:12:30,690 --> 00:12:34,740
So if you ever get interested
in the question of how many

208
00:12:34,740 --> 00:12:39,020
waveforms really are there,
which are concentrated in time

209
00:12:39,020 --> 00:12:43,410
and frequency, that's the
answer to the problem.

210
00:12:43,410 --> 00:12:46,380
And don't bother to read it now,
but just remember that if

211
00:12:46,380 --> 00:12:48,860
you ever get interested in that
problem, which I'm sure

212
00:12:48,860 --> 00:12:51,610
you will at some point or
other, that's where the

213
00:12:51,610 --> 00:12:53,470
solution lies.

214
00:12:53,470 --> 00:12:55,430
And I hope I gave a reference
to it there.

215
00:12:55,430 --> 00:12:56,680
I think I did.

216
00:13:00,260 --> 00:13:03,930
Anyway, we have these functions
which span L2.

217
00:13:03,930 --> 00:13:05,940
And there's at least one
other orthonormal

218
00:13:05,940 --> 00:13:08,560
expansion that we have.

219
00:13:08,560 --> 00:13:13,260
Which is both time- and
frequency-limited.

220
00:13:13,260 --> 00:13:17,920
And either at the end of today
or at the beginning of Monday,

221
00:13:17,920 --> 00:13:21,890
we're going to find another
particularly important

222
00:13:21,890 --> 00:13:25,180
sequence of orthonormal
functions.

223
00:13:25,180 --> 00:13:28,470
Which we actually use when
we're transmitting data.

224
00:13:31,050 --> 00:13:33,560
So to give an example of what
we were just talking about,

225
00:13:33,560 --> 00:13:37,430
the Fourier series functions
span the space of functions

226
00:13:37,430 --> 00:13:40,480
over minus t over
2 to t over 2.

227
00:13:40,480 --> 00:13:44,560
And when normalized, these
functions become 1 over the

228
00:13:44,560 --> 00:13:48,340
square root of t times what
we started with before.

229
00:13:48,340 --> 00:13:51,410
Namely, a sinusoid truncated.

230
00:13:51,410 --> 00:13:54,530
Before we were dealing with
the orthogonal functions

231
00:13:54,530 --> 00:13:57,210
without the 1 over square
root of t in it.

232
00:13:57,210 --> 00:14:00,210
If you want to make them
orthonormal, you get this

233
00:14:00,210 --> 00:14:03,430
square root of 1 over t, because
when you take the

234
00:14:03,430 --> 00:14:07,120
energy in this function,
you get t.

235
00:14:07,120 --> 00:14:11,030
If you don't believe me, set k
equal to 0 and look at that.

236
00:14:11,030 --> 00:14:13,720
And even I can integrate that.

237
00:14:13,720 --> 00:14:16,830
And when you integrate 1 from
minus t over 2 to t

238
00:14:16,830 --> 00:14:19,360
over 2, you get 1.

239
00:14:19,360 --> 00:14:22,670
So, then you have to multiply
by the square

240
00:14:22,670 --> 00:14:24,010
root of 1 over t.

241
00:14:24,010 --> 00:14:29,930
When you view the Fourier series
functions in this way,

242
00:14:29,930 --> 00:14:32,700
it's nice because they're
orthonormal.

243
00:14:32,700 --> 00:14:35,790
It's not nice because the
square root of 1 over t

244
00:14:35,790 --> 00:14:37,250
appears everyplace.

245
00:14:37,250 --> 00:14:40,930
But the nice thing about it is
that then when you expand in

246
00:14:40,930 --> 00:14:44,800
the Fourier series, you don't
find this t anywhere.

247
00:14:44,800 --> 00:14:50,900
Namely, v is equal to the limit
of these approximations

248
00:14:50,900 --> 00:14:54,970
where the n'th approximation is
just the sum from minus m

249
00:14:54,970 --> 00:15:00,440
to m of alpha sub k. phi sub k
and alpha sub k is just this

250
00:15:00,440 --> 00:15:02,640
inner product.

251
00:15:02,640 --> 00:15:05,420
So, again, you got something
nicer by looking at these

252
00:15:05,420 --> 00:15:09,840
things in terms of vectors You
get nice statements about how

253
00:15:09,840 --> 00:15:14,740
these things converge, and you
also get a very compact way of

254
00:15:14,740 --> 00:15:18,200
writing out what the
expressions are.

255
00:15:18,200 --> 00:15:22,330
You also get something a little
bit fishy, which a lot

256
00:15:22,330 --> 00:15:25,790
of people in the communication
field, especially ones who do

257
00:15:25,790 --> 00:15:30,230
theoretical work, run
into problems with.

258
00:15:30,230 --> 00:15:34,570
And the problem is the
following: when you start

259
00:15:34,570 --> 00:15:37,890
dealing all the time with
vectors, and you forget about

260
00:15:37,890 --> 00:15:42,620
the underlying functions and the
underlying integrals, you

261
00:15:42,620 --> 00:15:45,760
start to think that the
subject is simpler

262
00:15:45,760 --> 00:15:47,330
than it really is.

263
00:15:47,330 --> 00:15:50,180
Because you forget about all
the limiting issues.

264
00:15:50,180 --> 00:15:53,260
And when you forget about all
the limiting issues, it's fine

265
00:15:53,260 --> 00:15:55,680
almost all the time.

266
00:15:55,680 --> 00:15:58,070
But every once in a while,
you get trapped.

267
00:15:58,070 --> 00:16:03,130
And when you get trapped, you
then have to go back behind

268
00:16:03,130 --> 00:16:05,890
all of the vector stuff and you
have to start looking at

269
00:16:05,890 --> 00:16:08,800
these integrals again and it
gets rather frustrating.

270
00:16:08,800 --> 00:16:10,750
So you ought to keep both
of these things in mind.

271
00:16:15,490 --> 00:16:22,440
Let's go on, let's get back from
mathematics into worrying

272
00:16:22,440 --> 00:16:26,960
about the question of how
do you send data over

273
00:16:26,960 --> 00:16:29,950
communication channels.

274
00:16:29,950 --> 00:16:34,020
And this is just a picture that
we saw starting on Day

275
00:16:34,020 --> 00:16:40,830
One of this course, which says
the usual way of doing this

276
00:16:40,830 --> 00:16:48,210
is, you start out with --

277
00:16:48,210 --> 00:16:49,860
you start out with the source.

278
00:16:49,860 --> 00:16:54,050
You break the source down into
binary data, and then you take

279
00:16:54,050 --> 00:16:57,340
the binary data and you transmit
it over a channel.

280
00:16:57,340 --> 00:16:59,640
And this is the picture of
what you get when you're

281
00:16:59,640 --> 00:17:03,550
trying to transmit binary
data over a channel.

282
00:17:03,550 --> 00:17:08,060
And here, we've broken down the
encoder, as we called it,

283
00:17:08,060 --> 00:17:09,410
into two pieces.

284
00:17:09,410 --> 00:17:13,080
One of which we call the
discrete encoder and one of

285
00:17:13,080 --> 00:17:16,260
which we call modulation.

286
00:17:16,260 --> 00:17:18,860
Now, this is a little
bit fishy also.

287
00:17:18,860 --> 00:17:22,810
Because there are a lot of
people who now talk about

288
00:17:22,810 --> 00:17:27,240
coded modulation, where it
turns out to be nice to

289
00:17:27,240 --> 00:17:32,620
combine this discrete encoder
with the modulation function.

290
00:17:32,620 --> 00:17:38,160
And when you actually build
modern-day full encoder,

291
00:17:38,160 --> 00:17:41,640
namely the whole thing from
binary digits to what goes out

292
00:17:41,640 --> 00:17:44,990
on the channel, you very
often combine these

293
00:17:44,990 --> 00:17:47,240
two functions together.

294
00:17:47,240 --> 00:17:50,320
What is it that allows
people to do that?

295
00:17:50,320 --> 00:17:53,320
The fact that they first studied
how to do it when they

296
00:17:53,320 --> 00:17:57,530
separate the problems into
two separate problems.

297
00:17:57,530 --> 00:18:01,180
And when you separate the two
problems, what we wind up with

298
00:18:01,180 --> 00:18:03,450
is binary digits coming in.

299
00:18:03,450 --> 00:18:08,940
You massage those binary digits
strictly digitally.

300
00:18:08,940 --> 00:18:12,970
And what comes out is a
sequence of symbols.

301
00:18:12,970 --> 00:18:16,430
And, usually, the sequence of
symbols comes out at a slower

302
00:18:16,430 --> 00:18:18,690
rate, and the binary
digits come in.

303
00:18:18,690 --> 00:18:22,590
For example, if you take two
binary digits and you convert

304
00:18:22,590 --> 00:18:27,820
them into one symbol, from a
symbol alphabet with force of

305
00:18:27,820 --> 00:18:33,670
size four, then for every
two in you get one out.

306
00:18:33,670 --> 00:18:38,990
If you take three binary digits
coming in, then you

307
00:18:38,990 --> 00:18:42,090
need a symbol alphabet
of size eight here.

308
00:18:42,090 --> 00:18:50,220
If you move down in rate from
four binary digits to one

309
00:18:50,220 --> 00:18:55,020
symbol, then of course you need
an alphabet of size 16.

310
00:18:55,020 --> 00:18:58,340
So the size of the alphabet
here is going up

311
00:18:58,340 --> 00:19:03,330
exponentially, as the great
advantage that you get is

312
00:19:03,330 --> 00:19:05,190
going down.

313
00:19:05,190 --> 00:19:08,880
We will talk about that
more as we move on.

314
00:19:08,880 --> 00:19:12,120
You should sort of keep in mind
that there's a strange

315
00:19:12,120 --> 00:19:18,090
relationship between size of
alphabet and the rate at which

316
00:19:18,090 --> 00:19:19,840
you can transmit.

317
00:19:19,840 --> 00:19:23,910
As you try to transmit at higher
and higher rates here,

318
00:19:23,910 --> 00:19:27,660
the size of your alphabet goes
up exponentially, with the

319
00:19:27,660 --> 00:19:31,000
rate gain that you get.

320
00:19:31,000 --> 00:19:34,900
In fact, when you look at
Shannon's famous formula of

321
00:19:34,900 --> 00:19:38,690
how fast you can communicate
on a channel, you find a

322
00:19:38,690 --> 00:19:43,270
logarithm of a signal
to noise ratio.

323
00:19:43,270 --> 00:19:46,110
Of 1 plus a signal
to noise ratio.

324
00:19:46,110 --> 00:19:49,400
When you look at that signal to
noise ratio and you look at

325
00:19:49,400 --> 00:19:53,070
this alphabet size in here, you
can almost see just from

326
00:19:53,070 --> 00:19:56,360
that why in fact that
logarithm in

327
00:19:56,360 --> 00:19:58,230
these capacity formulas.

328
00:19:58,230 --> 00:19:59,860
And we'll come back
to talk about that

329
00:19:59,860 --> 00:20:02,110
again more later also.

330
00:20:02,110 --> 00:20:06,040
But, anyway, what we're going
to do at this point is to

331
00:20:06,040 --> 00:20:10,660
separate this into the problem
of discrete encoding and

332
00:20:10,660 --> 00:20:11,690
modulation.

333
00:20:11,690 --> 00:20:14,870
And in modulation, you start
out with some arbitrary

334
00:20:14,870 --> 00:20:16,360
alphabet of symbols.

335
00:20:16,360 --> 00:20:21,450
You turn the symbols in that
alphabet into waveforms.

336
00:20:21,450 --> 00:20:23,340
You transmit the waveforms.

337
00:20:23,340 --> 00:20:26,730
You then get the symbol back,
and then you put it into a

338
00:20:26,730 --> 00:20:30,960
digital encoder, which gets
the binary digits back.

339
00:20:30,960 --> 00:20:34,270
Now, what order are we going
to study these in?

340
00:20:34,270 --> 00:20:38,140
Well, we're going to study
the modulation first.

341
00:20:38,140 --> 00:20:41,710
And we're not going to study
this at all, essentially,

342
00:20:41,710 --> 00:20:45,260
except for a few very, very
simple-minded examples.

343
00:20:45,260 --> 00:20:50,220
6.451, which is the course that
follows after this, which

344
00:20:50,220 --> 00:20:54,480
might or might not be taught
in the Spring term, and

345
00:20:54,480 --> 00:20:57,150
incidentally those of you who
want it to be taught should

346
00:20:57,150 --> 00:21:01,100
send me an email saying you
really need to take this next

347
00:21:01,100 --> 00:21:04,110
term for some reason or other.

348
00:21:04,110 --> 00:21:08,160
Because otherwise it will be
given in Spring of the

349
00:21:08,160 --> 00:21:11,640
following year instead of
Spring of this year.

350
00:21:11,640 --> 00:21:15,070
There's another side to that
issue, which is a wireless

351
00:21:15,070 --> 00:21:19,580
course is going to be given in
the Spring of this year.

352
00:21:19,580 --> 00:21:23,140
And if you want to take a
wireless course and postpone

353
00:21:23,140 --> 00:21:26,770
taking the coding course until
the following year, then in

354
00:21:26,770 --> 00:21:31,140
fact it might be better to
do the postponing job.

355
00:21:31,140 --> 00:21:34,290
I would recommend that those of
you who are interested in

356
00:21:34,290 --> 00:21:35,890
wireless take the course now.

357
00:21:35,890 --> 00:21:39,220
Because if you're looking for
research problems in a

358
00:21:39,220 --> 00:21:44,040
communication area, wireless is
probably the hottest area

359
00:21:44,040 --> 00:21:45,900
around, and the area
where the most

360
00:21:45,900 --> 00:21:47,780
interesting problems occur.

361
00:21:47,780 --> 00:21:49,710
So, you have to make
that tradeoff.

362
00:21:49,710 --> 00:21:51,470
If you want to take both
of the courses, great.

363
00:21:51,470 --> 00:21:53,690
Send me an email and
say you want to

364
00:21:53,690 --> 00:21:54,850
take the coding course.

365
00:21:54,850 --> 00:21:57,830
But anyway, there'll be very
little coding in this course,

366
00:21:57,830 --> 00:22:00,540
very little discrete coding.

367
00:22:00,540 --> 00:22:03,400
And mostly we're going to look
at simple ways of taking

368
00:22:03,400 --> 00:22:07,870
simple symbol sequences, turning
them into waveforms

369
00:22:07,870 --> 00:22:12,416
and then transmitting
them on channels.

370
00:22:12,416 --> 00:22:16,211
AUDIENCE: So, basically, when
you do the source coding, you

371
00:22:16,211 --> 00:22:19,910
have to [UNINTELLIGIBLE]

372
00:22:19,910 --> 00:22:24,240
binary --

373
00:22:27,390 --> 00:22:30,990
PROFESSOR: Binary to
symbol to waveform.

374
00:22:30,990 --> 00:22:32,240
Yes.

375
00:22:33,938 --> 00:22:34,970
AUDIENCE: [UNINTELLIGIBLE]

376
00:22:34,970 --> 00:22:37,280
PROFESSOR: OK, why don't I go
from waveforms directly to

377
00:22:37,280 --> 00:22:40,750
waveforms instead of going from
waveforms to symbols to

378
00:22:40,750 --> 00:22:44,950
binary digits, and then I just
have to go back up again.

379
00:22:44,950 --> 00:22:47,080
A bunch of reasons.

380
00:22:47,080 --> 00:22:49,400
One of the reasons is that some
of the stuff that you

381
00:22:49,400 --> 00:22:53,940
transmit is already digital
to start with.

382
00:22:53,940 --> 00:22:56,530
When you look at what goes
over a network today, for

383
00:22:56,530 --> 00:22:59,500
example, it's carrying
digital data, it's

384
00:22:59,500 --> 00:23:01,130
carrying analog data.

385
00:23:01,130 --> 00:23:02,790
It's carrying images.

386
00:23:02,790 --> 00:23:05,800
It's carrying everything
you can imagine.

387
00:23:05,800 --> 00:23:11,670
If you want to design a
modulation system which goes

388
00:23:11,670 --> 00:23:16,800
directly from analog data to
channel data, and you have a

389
00:23:16,800 --> 00:23:21,280
hundred different kinds of
analog source data, and you

390
00:23:21,280 --> 00:23:25,500
have a hundred different kinds
of channels, then you're going

391
00:23:25,500 --> 00:23:32,070
to have to build one encoder
for every combination of

392
00:23:32,070 --> 00:23:33,740
source and channel.

393
00:23:33,740 --> 00:23:37,220
In other words, you've got to
built ten thousand devices.

394
00:23:37,220 --> 00:23:42,470
If your interest is in keeping
engineers employed, which I

395
00:23:42,470 --> 00:23:46,620
think is a very good interest,
that's a very good

396
00:23:46,620 --> 00:23:48,300
philosophy to take.

397
00:23:48,300 --> 00:23:51,910
But, unfortunately, most people
who build communication

398
00:23:51,910 --> 00:23:56,270
equipment say, we would really
rather have just a hundred --

399
00:23:56,270 --> 00:23:58,640
well, two hundred different
devices.

400
00:23:58,640 --> 00:24:02,430
One hundred devices which turn
all these different sources

401
00:24:02,430 --> 00:24:06,190
into binary digits, and another
hundred devices which

402
00:24:06,190 --> 00:24:09,420
turn the binary digits into
something that can be

403
00:24:09,420 --> 00:24:12,620
transmitted over any channel
you want to.

404
00:24:12,620 --> 00:24:14,530
I mean, this is just a
general example of

405
00:24:14,530 --> 00:24:16,430
what people call layering.

406
00:24:16,430 --> 00:24:20,810
You want to turn systems into
systems with a bunch of layers

407
00:24:20,810 --> 00:24:24,840
in them, where each layer is
standardized in some way.

408
00:24:24,840 --> 00:24:29,310
And only has to take care of
a few particular functions.

409
00:24:29,310 --> 00:24:31,530
And, in fact, that's what
we're doing here also.

410
00:24:31,530 --> 00:24:35,990
Because we're dealing with
one layer, which is

411
00:24:35,990 --> 00:24:38,530
doing discrete encoding.

412
00:24:38,530 --> 00:24:42,190
Which in fact is sort of
generating, for the most part,

413
00:24:42,190 --> 00:24:45,680
binary digits out of the
discrete encoder.

414
00:24:45,680 --> 00:24:49,220
Where, as you go through all of
this stuff and you come out

415
00:24:49,220 --> 00:24:52,540
with binary digits, some of
which are wrong, you can do

416
00:24:52,540 --> 00:24:56,150
the decoding and get the correct
binary digits out.

417
00:24:56,150 --> 00:24:59,880
If you look at an awful lot of
coding theory, you'll find out

418
00:24:59,880 --> 00:25:02,890
it doesn't pay any attention at
all to what's going on in

419
00:25:02,890 --> 00:25:05,160
the channel.

420
00:25:05,160 --> 00:25:07,880
Lots of people who've studied
coding all of their lives, and

421
00:25:07,880 --> 00:25:11,030
decoding all of their lives,
live in this mathematical

422
00:25:11,030 --> 00:25:13,190
theory of abstract algebra.

423
00:25:13,190 --> 00:25:16,170
And they have no idea of
what channels are.

424
00:25:16,170 --> 00:25:20,010
And they survive because of
this layering principle.

425
00:25:20,010 --> 00:25:22,260
So if you want to employ
engineers, it's nice to have

426
00:25:22,260 --> 00:25:24,310
layering also.

427
00:25:24,310 --> 00:25:27,650
Because engineers don't have
to know as much then.

428
00:25:27,650 --> 00:25:29,550
Of course, you people
should know it all.

429
00:25:29,550 --> 00:25:32,540
So because, then you
can do anything.

430
00:25:32,540 --> 00:25:36,510
And you can be part of those
very rare people who can put

431
00:25:36,510 --> 00:25:38,640
it all together.

432
00:25:38,640 --> 00:25:42,350
I was just at the 70th birthday
party of Irwin Jacobs

433
00:25:42,350 --> 00:25:44,550
this past week.

434
00:25:44,550 --> 00:25:50,360
And Irwin Jacobs is the CEO of
Qualcomm, which is the company

435
00:25:50,360 --> 00:25:56,590
that started to build CDMA
wireless systems.

436
00:25:56,590 --> 00:26:02,690
For a long time, the CDMA
systems were thought of as

437
00:26:02,690 --> 00:26:09,500
probably better than TDMA and
FDMA, but much more expensive.

438
00:26:09,500 --> 00:26:13,380
And eventually, we're in the
situation where all the new

439
00:26:13,380 --> 00:26:16,790
systems being designed
are all using CMDA.

440
00:26:16,790 --> 00:26:19,350
As a result of this,
Irwin Jacobs is a

441
00:26:19,350 --> 00:26:21,320
very wealthy person.

442
00:26:21,320 --> 00:26:26,540
We went to a symphony Friday
night, out in San Diego, which

443
00:26:26,540 --> 00:26:28,730
was given specifically
for him.

444
00:26:28,730 --> 00:26:31,690
Partly because he'd just
given $110 million to

445
00:26:31,690 --> 00:26:32,940
the San Diego Symphony.

446
00:26:35,400 --> 00:26:38,370
So you can figure from that
that he's fairly wealthy.

447
00:26:38,370 --> 00:26:40,750
Well the point of all of that
is, he started out as a

448
00:26:40,750 --> 00:26:43,220
faculty member here.

449
00:26:43,220 --> 00:26:48,390
He was one of the authors of
Rosencraft and Jacobs, which

450
00:26:48,390 --> 00:26:53,700
is sort of the Bible of digital
communications systems

451
00:26:53,700 --> 00:26:54,800
from the `60s.

452
00:26:54,800 --> 00:26:58,050
And people still use it, it's
still an excellent book.

453
00:26:58,050 --> 00:27:00,970
And in doing that, he really
learned the communication

454
00:27:00,970 --> 00:27:05,960
trade from soup to --

455
00:27:05,960 --> 00:27:09,800
I guess, soup to nuts is
the way we put it now.

456
00:27:09,800 --> 00:27:11,630
And he could do the
whole thing.

457
00:27:11,630 --> 00:27:13,730
And because he could do the
whole thing, because he

458
00:27:13,730 --> 00:27:17,460
understood coding, because he
understood modulation, because

459
00:27:17,460 --> 00:27:21,000
he understood channels, this is
why he could design systems

460
00:27:21,000 --> 00:27:23,400
that really work.

461
00:27:23,400 --> 00:27:27,320
I was talking to the Chief
Technology Officer out there,

462
00:27:27,320 --> 00:27:29,690
and the Chief Technology Officer
said his job was

463
00:27:29,690 --> 00:27:34,330
really very easy because Irwin
did all of that himself.

464
00:27:34,330 --> 00:27:36,730
And he still does it.

465
00:27:36,730 --> 00:27:39,580
So it really makes sense to know
something about all of

466
00:27:39,580 --> 00:27:41,040
these pieces.

467
00:27:41,040 --> 00:27:42,970
If you want to become
a billionaire.

468
00:27:42,970 --> 00:27:47,510
And if you want to become a
billionaire without cheating.

469
00:27:47,510 --> 00:27:50,780
Now, many people become
billionaires by cheating, and

470
00:27:50,780 --> 00:27:53,160
you've heard of many
of them but.

471
00:27:53,160 --> 00:27:54,170
Well, anyway.

472
00:27:54,170 --> 00:27:55,860
Enough of that.

473
00:27:55,860 --> 00:27:59,780
I mean, none of us really want
to be billionaires anyway, --

474
00:27:59,780 --> 00:28:01,750
I mean, there's nothing you
can do with more than a

475
00:28:01,750 --> 00:28:04,240
billion dollars, right?

476
00:28:04,240 --> 00:28:05,490
It's all wasted.

477
00:28:09,600 --> 00:28:11,960
Let's move on.

478
00:28:11,960 --> 00:28:14,740
We've gotten rid of digital
coding, saying we're not going

479
00:28:14,740 --> 00:28:15,990
to deal with that here.

480
00:28:19,460 --> 00:28:22,110
Let me take the PAM out of here
because I don't want to

481
00:28:22,110 --> 00:28:23,400
even say what it is yet.

482
00:28:26,950 --> 00:28:30,840
If we take this box I called
modulation before, which was

483
00:28:30,840 --> 00:28:37,280
one of the two main pieces of
a channel encoder, we can

484
00:28:37,280 --> 00:28:40,080
break it down into two pieces.

485
00:28:40,080 --> 00:28:43,120
Namely, mainly we're going to
be layering things again.

486
00:28:43,120 --> 00:28:46,090
We start out with this symbol
sequence, which is what comes

487
00:28:46,090 --> 00:28:50,500
out of the encoder, which is
usually just a short sequence

488
00:28:50,500 --> 00:28:52,590
of binary digits.

489
00:28:52,590 --> 00:28:57,850
So the symbols can be thought
of as two binary digits in a

490
00:28:57,850 --> 00:29:01,260
sequence, or four binary digits,
or six tuples of

491
00:29:01,260 --> 00:29:03,690
binary digits, or
what have you.

492
00:29:03,690 --> 00:29:06,420
And all of those are common.

493
00:29:06,420 --> 00:29:09,920
There's then a signal
consolation that turns symbols

494
00:29:09,920 --> 00:29:11,390
into signals.

495
00:29:11,390 --> 00:29:17,770
Now, notation in the field is
totally non-uniform, because

496
00:29:17,770 --> 00:29:24,290
most people use symbols and
sequences and waveforms and

497
00:29:24,290 --> 00:29:29,810
binary n-tuples, all totally
synonymously.

498
00:29:29,810 --> 00:29:33,930
And the distinction we'll try
to make here is that symbols

499
00:29:33,930 --> 00:29:39,790
are things like binary digits,
that don't have any numerical

500
00:29:39,790 --> 00:29:41,190
meaning to them.

501
00:29:41,190 --> 00:29:45,480
I mean, a 1 and a 0, the only
thing important there is that

502
00:29:45,480 --> 00:29:48,150
1 and 0 are different
from each other.

503
00:29:48,150 --> 00:29:54,030
All computer scientists call
1 and 0's Alices and Bobs.

504
00:29:54,030 --> 00:29:56,130
Lots of other people
call them plus-1's.

505
00:29:56,130 --> 00:29:57,310
and minus 1's.

506
00:29:57,310 --> 00:29:59,640
Doesn't make any difference
what you'd call them, it's

507
00:29:59,640 --> 00:30:03,830
just an alphabet with
two values in it.

508
00:30:03,830 --> 00:30:06,130
When we talk about signals,
we're talking

509
00:30:06,130 --> 00:30:08,650
about numerical values.

510
00:30:08,650 --> 00:30:12,780
So in fact, you could have a
signal constellation with two

511
00:30:12,780 --> 00:30:16,320
elements in it, where what
you're doing is mapping 1 and

512
00:30:16,320 --> 00:30:19,490
0 into plus 1 and minus 1.

513
00:30:19,490 --> 00:30:22,850
Now, that's a pretty silly
kind of situation, but it

514
00:30:22,850 --> 00:30:27,320
still has the value of saying,
you're turning things where

515
00:30:27,320 --> 00:30:32,000
there's just an alphabeticized
2 into something where you're

516
00:30:32,000 --> 00:30:34,820
saying, these are numerical
values and you're interested

517
00:30:34,820 --> 00:30:37,280
in the difference between
the numerical values.

518
00:30:37,280 --> 00:30:41,630
And the difference is measured
in the ordinary way of

519
00:30:41,630 --> 00:30:45,310
measuring difference for
numerical values.

520
00:30:45,310 --> 00:30:48,650
Or, these can be vectors also.

521
00:30:48,650 --> 00:30:51,300
But vectors in an inner product
space where, again,

522
00:30:51,300 --> 00:30:52,650
you have length.

523
00:30:52,650 --> 00:30:53,530
And you have distance.

524
00:30:53,530 --> 00:30:55,520
So you have numerical
values here.

525
00:30:55,520 --> 00:30:57,890
You don't have numerical
values here.

526
00:30:57,890 --> 00:31:00,830
So we're going to talk about
how you do this.

527
00:31:00,830 --> 00:31:04,710
And then, for the most part,
we're going to define

528
00:31:04,710 --> 00:31:08,200
modulation as what happens when
you go from the signal

529
00:31:08,200 --> 00:31:11,790
sequence to the waveform.

530
00:31:11,790 --> 00:31:16,100
You might realize that our
notation is lousy here.

531
00:31:16,100 --> 00:31:19,070
Because I'm calling modulation
this whole thing.

532
00:31:19,070 --> 00:31:23,050
And I'm also calling modulation
just this thing.

533
00:31:23,050 --> 00:31:26,250
And I'm doing that because there
doesn't seem to be any

534
00:31:26,250 --> 00:31:29,810
other reasonable word for either
one of these things.

535
00:31:29,810 --> 00:31:32,120
So, and I'm going to
later split this

536
00:31:32,120 --> 00:31:34,430
into two pieces also.

537
00:31:34,430 --> 00:31:35,730
But that will come later.

538
00:31:35,730 --> 00:31:37,870
So, anyway, we're going
from symbols

539
00:31:37,870 --> 00:31:41,080
to signals to waveforms.

540
00:31:41,080 --> 00:31:44,090
Which might look remarkably
similar to

541
00:31:44,090 --> 00:31:45,590
what we did with sources.

542
00:31:45,590 --> 00:31:47,420
But we just did it backwards
with sources.

543
00:31:47,420 --> 00:31:52,580
We went from waveforms to
signals to symbols there.

544
00:31:52,580 --> 00:31:56,030
And here we're doing
the same thing.

545
00:31:56,030 --> 00:32:01,120
The modulator often converts a
signal sequence to a baseband

546
00:32:01,120 --> 00:32:05,000
waveform, and then converts
the baseband waveform to a

547
00:32:05,000 --> 00:32:07,110
passband waveform.

548
00:32:07,110 --> 00:32:12,220
And, just historically, people
used to think of modulation as

549
00:32:12,220 --> 00:32:16,060
the process of taking something
at baseband and

550
00:32:16,060 --> 00:32:19,820
converting it up to
some passband.

551
00:32:19,820 --> 00:32:23,670
Now it's recognized that the
interesting problem is not,

552
00:32:23,670 --> 00:32:26,120
how do you go from baseband
to passband.

553
00:32:26,120 --> 00:32:29,610
Which is just multiplying by
cosine wave, not much more to

554
00:32:29,610 --> 00:32:32,270
it than that.

555
00:32:32,270 --> 00:32:36,320
Well, it's a little more
to it, but not much.

556
00:32:36,320 --> 00:32:42,320
And the interesting problem is,
how do you convert signals

557
00:32:42,320 --> 00:32:44,180
into waveforms.

558
00:32:44,180 --> 00:32:46,930
Which is why we went, one of
the reasons we went through

559
00:32:46,930 --> 00:32:52,040
all of this stuff about L2
waveforms and orthonormal

560
00:32:52,040 --> 00:32:55,810
expansions and all
of this stuff.

561
00:32:55,810 --> 00:32:59,360
So, for the time being, we're
going to look at modulation

562
00:32:59,360 --> 00:33:03,400
and demodulation, without
worrying about what bandwidth

563
00:33:03,400 --> 00:33:04,720
any of this occurs at.

564
00:33:04,720 --> 00:33:06,750
So we'll just look at
it at baseband.

565
00:33:13,140 --> 00:33:16,520
So the simplest example of all
of this, so simple that it

566
00:33:16,520 --> 00:33:21,460
almost looks like it's silly, is
to map a sequence of binary

567
00:33:21,460 --> 00:33:26,260
digits into a sequence of
signals from the constellation

568
00:33:26,260 --> 00:33:28,480
1 and minus 1.

569
00:33:28,480 --> 00:33:35,000
So all you're doing there is
mapping 0 into 1 and mapping 1

570
00:33:35,000 --> 00:33:35,960
into minus 1.

571
00:33:35,960 --> 00:33:39,290
In other words, the 0 and 1
binary digits are mapped into

572
00:33:39,290 --> 00:33:40,480
1 and minus 1.

573
00:33:40,480 --> 00:33:43,580
Why do you do it this way, which
is a little confusing,

574
00:33:43,580 --> 00:33:47,600
mapping 0 into 1 instead
of mapping 1 into 1?

575
00:33:47,600 --> 00:33:51,750
Well, primarily, so you can
look at multiplication of

576
00:33:51,750 --> 00:33:56,830
signals in the same way as you
look at modular to addition of

577
00:33:56,830 --> 00:33:58,400
binary digits.

578
00:33:58,400 --> 00:34:00,030
It just turns out to
be a little more

579
00:34:00,030 --> 00:34:01,990
convenient that way.

580
00:34:01,990 --> 00:34:03,870
And it doesn't make a whole
lot of difference.

581
00:34:03,870 --> 00:34:08,800
The point is, we're going from
0 1's to signals which are 1

582
00:34:08,800 --> 00:34:10,650
and minus 1.

583
00:34:10,650 --> 00:34:15,630
Then this sequence of signals
is mapped into a waveform

584
00:34:15,630 --> 00:34:19,820
which is the sum of u of k times
the sinc function t over

585
00:34:19,820 --> 00:34:22,300
capital T minus k.

586
00:34:22,300 --> 00:34:25,040
In other words, you're thinking
of each one of these

587
00:34:25,040 --> 00:34:31,780
signals, now, think of it as
being a sum of impulses,

588
00:34:31,780 --> 00:34:33,220
delayed impulses.

589
00:34:33,220 --> 00:34:37,600
And then think of taking a
little sinc, sinc t over t and

590
00:34:37,600 --> 00:34:39,710
putting it around each one
of those impulses.

591
00:34:39,710 --> 00:34:42,060
In other words, think of
passing each of those

592
00:34:42,060 --> 00:34:46,900
impulses, which is weighted by
one of these values, u sub k,

593
00:34:46,900 --> 00:34:55,150
into a linear filter whose
response is sine of x over x.

594
00:34:55,150 --> 00:34:55,550
Anybody

595
00:34:55,550 --> 00:34:57,520
see anything a little bit
fishy about that?

596
00:35:01,510 --> 00:35:03,520
Have you ever built a filter
whose response

597
00:35:03,520 --> 00:35:05,978
was sine x over x?

598
00:35:05,978 --> 00:35:08,400
AUDIENCE: It's hard to.

599
00:35:08,400 --> 00:35:10,890
PROFESSOR: It's hard to, why?

600
00:35:10,890 --> 00:35:13,850
AUDIENCE: [UNINTELLIGIBLE]

601
00:35:13,850 --> 00:35:16,170
PROFESSOR: Because it's
not causal, yes.

602
00:35:16,170 --> 00:35:18,830
We're going to talk about
that more later today.

603
00:35:18,830 --> 00:35:24,340
Communication engineers hardly
ever talk about causality.

604
00:35:24,340 --> 00:35:26,045
They hardly ever talk about
whether something

605
00:35:26,045 --> 00:35:28,940
is realized or not.

606
00:35:28,940 --> 00:35:33,600
And the reason I want to say
this a number of times is that

607
00:35:33,600 --> 00:35:36,170
one of the parts of
any receiver is a

608
00:35:36,170 --> 00:35:38,370
timing recovery circuit.

609
00:35:38,370 --> 00:35:42,140
And what the timing recovery
circuit does is, it tries to

610
00:35:42,140 --> 00:35:45,070
figure out what the transmitter
timing is, in

611
00:35:45,070 --> 00:35:47,640
terms of what it's receiving.

612
00:35:47,640 --> 00:35:50,160
And when you're figuring out
what the transmitter timing is

613
00:35:50,160 --> 00:35:53,910
in terms of what you're
receiving, the most convenient

614
00:35:53,910 --> 00:35:57,740
way of doing that, if you're
sending a pulse or something,

615
00:35:57,740 --> 00:36:02,140
you would like the receiver
timing to peak up at the same

616
00:36:02,140 --> 00:36:05,310
time that the transmitter timing
is peaking up, in terms

617
00:36:05,310 --> 00:36:05,960
of the pulse.

618
00:36:05,960 --> 00:36:08,650
In other words, you want to
get rid of the propagation

619
00:36:08,650 --> 00:36:11,240
delay and just ignore that.

620
00:36:11,240 --> 00:36:14,430
The receiver timing is going
to be delayed from the

621
00:36:14,430 --> 00:36:19,970
transmitter timing exactly
by the propagation delay.

622
00:36:19,970 --> 00:36:23,970
Now, if we always do that,
causality becomes totally

623
00:36:23,970 --> 00:36:25,220
unimportant.

624
00:36:27,720 --> 00:36:31,070
Now, one of the problems with a
sinc function is, it starts

625
00:36:31,070 --> 00:36:33,830
at time minus infinity.

626
00:36:33,830 --> 00:36:37,770
And even if you delay the
receiver timing by an infinite

627
00:36:37,770 --> 00:36:41,440
amount of time you can never use
the communications system

628
00:36:41,440 --> 00:36:45,910
until it's time to tear it down
and bring in a new one.

629
00:36:45,910 --> 00:36:47,880
Which, unfortunately, is what
happened to the third

630
00:36:47,880 --> 00:36:49,590
generation wireless systems.

631
00:36:49,590 --> 00:36:52,870
By the time, people figured
out how to build them,

632
00:36:52,870 --> 00:36:56,940
everybody was saying, ah,
old-fashioned stuff, we're

633
00:36:56,940 --> 00:36:59,580
going to go directly onto
to fourth generation.

634
00:36:59,580 --> 00:37:02,430
And they might go directly
from fourth generation to

635
00:37:02,430 --> 00:37:03,300
fifth generation.

636
00:37:03,300 --> 00:37:04,650
Who knows.

637
00:37:04,650 --> 00:37:09,900
Anyway, you can't implement
these even with the receiver

638
00:37:09,900 --> 00:37:13,360
timing different than the
transmitter timing.

639
00:37:13,360 --> 00:37:17,430
But you can always approximate
things with enough delay

640
00:37:17,430 --> 00:37:19,610
between transmitter
and receiver.

641
00:37:19,610 --> 00:37:22,840
So that you can approximate
any filter you want to,

642
00:37:22,840 --> 00:37:26,580
without worrying about
causality at all.

643
00:37:26,580 --> 00:37:29,520
And it's hard enough designing
good filters without worrying

644
00:37:29,520 --> 00:37:33,530
about casuality that you don't
want to bring that into your

645
00:37:33,530 --> 00:37:36,420
picture at the beginning.

646
00:37:36,420 --> 00:37:41,030
So communication engineers
usually say that those timing

647
00:37:41,030 --> 00:37:43,850
issues are not important.

648
00:37:43,850 --> 00:37:48,060
And a little bit of delay is
not going to hurt anything.

649
00:37:48,060 --> 00:37:53,250
All of Shannon's theory was as
successful as it has been, and

650
00:37:53,250 --> 00:37:56,530
transformed the whole
communication industry,

651
00:37:56,530 --> 00:38:00,020
because he never talked
about delay at all.

652
00:38:00,020 --> 00:38:02,190
He just ignored the question
of delay from

653
00:38:02,190 --> 00:38:04,840
beginning to end.

654
00:38:04,840 --> 00:38:09,000
And when you look at his
capacity formulas, they are

655
00:38:09,000 --> 00:38:12,300
assuming that you have as much
delay as you need between

656
00:38:12,300 --> 00:38:14,050
transmitter and receiver.

657
00:38:14,050 --> 00:38:17,210
And, assuming that that isn't
important, and in terms of the

658
00:38:17,210 --> 00:38:22,330
propagation delays, and the
filtering delays in most

659
00:38:22,330 --> 00:38:25,090
modern communication systems,
those delays

660
00:38:25,090 --> 00:38:26,500
are not really important.

661
00:38:26,500 --> 00:38:29,300
The propagation delays
you can't avoid.

662
00:38:29,300 --> 00:38:31,130
So you might as well
ignore them.

663
00:38:31,130 --> 00:38:34,650
Which is why you build timing
recovery circuits.

664
00:38:34,650 --> 00:38:38,840
And the filtering delays are
usually unimportant.

665
00:38:38,840 --> 00:38:40,880
Relative to all of the
other delays that

666
00:38:40,880 --> 00:38:42,700
come into these system.

667
00:38:42,700 --> 00:38:44,840
So for the most part,
we're just going to

668
00:38:44,840 --> 00:38:47,240
ignore those questions.

669
00:38:47,240 --> 00:38:52,130
If you have no noise, no delay,
and no attenuation, the

670
00:38:52,130 --> 00:38:55,180
received waveform is going
to be the same as

671
00:38:55,180 --> 00:38:57,390
the transmitted waveform.

672
00:38:57,390 --> 00:39:00,300
And then what you would like
to do is to sample that and

673
00:39:00,300 --> 00:39:02,370
convert it back to binary.

674
00:39:02,370 --> 00:39:04,430
So, that's what the receiving
side of this

675
00:39:04,430 --> 00:39:07,140
trivial system is doing.

676
00:39:07,140 --> 00:39:09,060
Now, why can you ignore
attenuation?

677
00:39:11,700 --> 00:39:14,260
Anybody have any idea why
we might want to ignore

678
00:39:14,260 --> 00:39:15,510
attenuation?

679
00:39:20,380 --> 00:39:23,020
Well, it's like all of
these other things.

680
00:39:23,020 --> 00:39:27,340
You don't ignore it, but the
question you ask is, can I

681
00:39:27,340 --> 00:39:30,500
separate the issue of
attentuation from the other

682
00:39:30,500 --> 00:39:32,570
issues that I want
to look at now.

683
00:39:32,570 --> 00:39:36,220
In other words, can I layer
this problem in such a way

684
00:39:36,220 --> 00:39:39,830
that we can deal with our
problems one by one.

685
00:39:39,830 --> 00:39:42,570
And the problem of attentuation
is something that

686
00:39:42,570 --> 00:39:47,590
communication engineers have had
to deal with from day one.

687
00:39:47,590 --> 00:39:51,690
In dealing with it from day one,
they have dealt with all

688
00:39:51,690 --> 00:39:56,060
of the different ways of losing
power that you have.

689
00:39:56,060 --> 00:40:01,370
Which includes attenuation in
the actual medium between

690
00:40:01,370 --> 00:40:03,180
transmitter and receiver.

691
00:40:03,180 --> 00:40:06,840
It deals with the attenuation
you get in the receiver by

692
00:40:06,840 --> 00:40:10,400
building filters and by all the
other things you do there.

693
00:40:10,400 --> 00:40:13,760
When you get all done with that,
there's only one thing

694
00:40:13,760 --> 00:40:15,010
that's important.

695
00:40:18,430 --> 00:40:23,620
Because you can deal digitally
with very, very small signals.

696
00:40:23,620 --> 00:40:25,960
And the only thing that's
important is what

697
00:40:25,960 --> 00:40:27,050
happens to the noise.

698
00:40:27,050 --> 00:40:29,600
You get noise in the
communication medium.

699
00:40:29,600 --> 00:40:32,520
That comes into the receiver.

700
00:40:32,520 --> 00:40:36,410
And now, any time you amplify
what you receive, they're

701
00:40:36,410 --> 00:40:39,730
going to be amplifying the noise
as well as amplifying

702
00:40:39,730 --> 00:40:41,490
the signal.

703
00:40:41,490 --> 00:40:45,130
An easy way to deal with that is
to assume that there isn't

704
00:40:45,130 --> 00:40:48,410
any attentuation in what you're
calling the signal,

705
00:40:48,410 --> 00:40:50,830
because you're going to amplify
that to a reasonable

706
00:40:50,830 --> 00:40:54,040
value to operate on it anyway.

707
00:40:54,040 --> 00:40:57,350
So in fact you're amplifying
the noise.

708
00:40:57,350 --> 00:41:00,380
So the only thing you're
interested in is, what's the

709
00:41:00,380 --> 00:41:04,580
ratio between the signal power
and the noise power.

710
00:41:07,490 --> 00:41:10,980
So, those issues, we can deal
with completely separately

711
00:41:10,980 --> 00:41:12,920
from these issues of what
kind of waveforms

712
00:41:12,920 --> 00:41:14,460
do we want to choose.

713
00:41:14,460 --> 00:41:19,390
What should we choose in place
of this sinc function, which

714
00:41:19,390 --> 00:41:24,500
is impractical because it causes
too much delay and it's

715
00:41:24,500 --> 00:41:27,770
also hard to build
the filters.

716
00:41:27,770 --> 00:41:30,530
So we'd like to deal with that
question separately from the

717
00:41:30,530 --> 00:41:32,010
question of attenuation.

718
00:41:32,010 --> 00:41:34,350
There's a short section in the
notes, which I'm not going to

719
00:41:34,350 --> 00:41:38,150
go into, because you can read
it just as easily as I can

720
00:41:38,150 --> 00:41:41,950
talk about it, about dB.

721
00:41:41,950 --> 00:41:47,550
And why people use dB to talk
about all of these questions

722
00:41:47,550 --> 00:41:49,750
of energy losses.

723
00:41:49,750 --> 00:41:53,120
And why, in fact, there's a
whole set of engineers whose

724
00:41:53,120 --> 00:41:56,030
function is to deal
with link budgets.

725
00:41:56,030 --> 00:41:58,810
In other words, when you build
a communications system,

726
00:41:58,810 --> 00:42:02,710
you're losing power everywhere
along the way.

727
00:42:02,710 --> 00:42:04,570
You're losing it
in the median.

728
00:42:04,570 --> 00:42:07,710
You're losing it by the way
you build the antennas.

729
00:42:07,710 --> 00:42:11,320
You're losing some here, you're
losing some there.

730
00:42:11,320 --> 00:42:14,920
And what these people do is,
they look at all of these

731
00:42:14,920 --> 00:42:16,550
attenuations together.

732
00:42:16,550 --> 00:42:18,530
And they multiply together.

733
00:42:18,530 --> 00:42:22,270
So, in fact, it's easier to take
the logarithm of all of

734
00:42:22,270 --> 00:42:23,520
these terms.

735
00:42:23,520 --> 00:42:26,590
And when you take the logarithm,
that's where you

736
00:42:26,590 --> 00:42:28,570
get decibels from.

737
00:42:28,570 --> 00:42:31,830
Because these people, instead of
taking natural logs, which

738
00:42:31,830 --> 00:42:35,820
would seem like a much more
reasonable thing, always take

739
00:42:35,820 --> 00:42:40,610
the logarithm to the base 10
and then divide by 10.

740
00:42:40,610 --> 00:42:45,420
And what the section in the
notes says is that that's a

741
00:42:45,420 --> 00:42:50,520
practice which has grown up
because it's very easy to do

742
00:42:50,520 --> 00:42:52,920
mental arithmetic with that.

743
00:42:52,920 --> 00:42:59,380
The logarithm to the base 10 of
the number 2 which is one

744
00:42:59,380 --> 00:43:02,400
of the biggest numbers
that ever appears.

745
00:43:02,400 --> 00:43:06,050
2's are more important
than thousands, OK?

746
00:43:06,050 --> 00:43:16,930
And the logarithm to the
base 2 of 10 is 0.3.

747
00:43:16,930 --> 00:43:20,450
And when you divide it
by 10 you get 3 dB.

748
00:43:20,450 --> 00:43:23,720
So a factor of 2
is called 3 dB.

749
00:43:23,720 --> 00:43:28,160
This means that a factor of 4,
which is 2 squared, is 6 dB.

750
00:43:28,160 --> 00:43:32,200
Factor of 8 is 9 dB,
and so forth.

751
00:43:32,200 --> 00:43:36,010
And this gives this table of
all these numbers which all

752
00:43:36,010 --> 00:43:38,990
communication engineers
memorize.

753
00:43:38,990 --> 00:43:42,180
The other reason for it is that
if you're talking to a

754
00:43:42,180 --> 00:43:45,570
communication engineer, he will
recognize you as a member

755
00:43:45,570 --> 00:43:48,790
of his fraternity if
you let the word dB

756
00:43:48,790 --> 00:43:51,290
slip several times.

757
00:43:51,290 --> 00:43:52,520
Don't even have to know
what it means.

758
00:43:52,520 --> 00:43:54,980
You just talk about dB.

759
00:43:54,980 --> 00:43:59,800
And you say, my income is 3
dB lower than your income.

760
00:43:59,800 --> 00:44:01,050
And that makes him very happy.

761
00:44:03,910 --> 00:44:04,580
So.

762
00:44:04,580 --> 00:44:07,360
Anyway.

763
00:44:07,360 --> 00:44:10,230
That's all we need for
this simple example.

764
00:44:10,230 --> 00:44:16,050
We now want to go into more
complicated things.

765
00:44:16,050 --> 00:44:19,570
OK pulse amplitude
and modulation.

766
00:44:19,570 --> 00:44:26,110
This is one of the major ways of
turning bits into signals,

767
00:44:26,110 --> 00:44:30,170
and then turning the signals
into waveforms again.

768
00:44:30,170 --> 00:44:33,430
Again, I'm doing this first
not because it's the most

769
00:44:33,430 --> 00:44:36,460
important scheme to talk about,
but because we want to

770
00:44:36,460 --> 00:44:39,630
understand these things
one by one.

771
00:44:39,630 --> 00:44:43,330
And when we understand PAM,
we'll then talk about QAM.

772
00:44:43,330 --> 00:44:44,800
And you'll understand that.

773
00:44:44,800 --> 00:44:48,050
And then we can go on
to look at other

774
00:44:48,050 --> 00:44:50,040
variations of these things.

775
00:44:50,040 --> 00:44:54,030
So the signals in PAM
are one dimensional.

776
00:44:54,030 --> 00:44:57,470
The constellation, the only
thing it can be, since it's

777
00:44:57,470 --> 00:45:00,700
one dimensional, one
real dimension, is

778
00:45:00,700 --> 00:45:03,570
a set of real numbers.

779
00:45:03,570 --> 00:45:09,600
So you're going to modulate
these real numbers.

780
00:45:09,600 --> 00:45:11,230
And that, we're going
to talk about later.

781
00:45:11,230 --> 00:45:13,120
How do you find this
function here?

782
00:45:13,120 --> 00:45:15,580
We're just going to take these
real numbers, which are coming

783
00:45:15,580 --> 00:45:18,030
into the transmitter
one by one out

784
00:45:18,030 --> 00:45:19,790
of the digital encoder.

785
00:45:19,790 --> 00:45:21,030
You're going to take
these numbers.

786
00:45:21,030 --> 00:45:24,130
You're going to view them as
being multiplied by delayed

787
00:45:24,130 --> 00:45:27,560
impulses, and then pass
through a filter.

788
00:45:27,560 --> 00:45:31,290
And the filter's response is
just, impulse response is

789
00:45:31,290 --> 00:45:32,400
just, p of t.

790
00:45:32,400 --> 00:45:35,060
In other words, what we're doing
is saying, we don't like

791
00:45:35,060 --> 00:45:37,430
the sinc function.

792
00:45:37,430 --> 00:45:40,640
Therefore, simplest thing to
do is replace the sinc

793
00:45:40,640 --> 00:45:43,130
function by something
we do like.

794
00:45:43,130 --> 00:45:46,380
So we're going to replace the
sinc function by some filter

795
00:45:46,380 --> 00:45:49,440
characteristic, which we like.

796
00:45:49,440 --> 00:45:54,230
And that's the way to modify
this previous example into

797
00:45:54,230 --> 00:45:56,160
something that makes
better sense.

798
00:45:56,160 --> 00:45:59,240
So we're doing two things here
when we're talking about PAM.

799
00:45:59,240 --> 00:46:03,020
One, we're talking about
generalizing this binary

800
00:46:03,020 --> 00:46:04,540
signal set.

801
00:46:04,540 --> 00:46:08,510
And, two, we're talking about
generalizing the sinc function

802
00:46:08,510 --> 00:46:12,920
into some arbitrary
impulse response.

803
00:46:12,920 --> 00:46:18,640
So a standard PAM signal set
uses equispaced signals

804
00:46:18,640 --> 00:46:22,260
symmetric around 0.

805
00:46:22,260 --> 00:46:24,250
And if you look at the picture,
it makes it clear

806
00:46:24,250 --> 00:46:26,070
what this means.

807
00:46:26,070 --> 00:46:28,840
It's the same thing that we were
using all along when we

808
00:46:28,840 --> 00:46:31,450
were talking about
quantization.

809
00:46:31,450 --> 00:46:34,750
If you're looking at one
dimensional quantization,

810
00:46:34,750 --> 00:46:37,510
that's a very natural
way to choose the

811
00:46:37,510 --> 00:46:39,560
representation points.

812
00:46:39,560 --> 00:46:42,450
Here, we're doing everything
in the opposite way.

813
00:46:42,450 --> 00:46:46,370
So we're starting out
with these points.

814
00:46:46,370 --> 00:46:50,130
Well, we're starting out with
eight symbols and turning them

815
00:46:50,130 --> 00:46:51,980
into eight signals.

816
00:46:51,980 --> 00:46:54,620
And when you take eight symbols
and turn them into

817
00:46:54,620 --> 00:46:59,660
eight signals, perfectly natural
thing to do is to make

818
00:46:59,660 --> 00:47:03,480
each of these signals equispaced
from each other,

819
00:47:03,480 --> 00:47:05,130
and center them on the origin.

820
00:47:09,100 --> 00:47:12,450
This is not something we
have to see later.

821
00:47:12,450 --> 00:47:17,340
I think you can see that if
these symbols all are used

822
00:47:17,340 --> 00:47:22,750
with equal probability, and
you're trying to reduce the

823
00:47:22,750 --> 00:47:28,680
amount of energy that the
signals use, which will then

824
00:47:28,680 --> 00:47:31,970
go through into reducing the
amount of energy in the

825
00:47:31,970 --> 00:47:37,470
waveforms that we're
transmitting, it's nice to

826
00:47:37,470 --> 00:47:39,760
have them centered around 0.

827
00:47:39,760 --> 00:47:42,460
Because if they have a mean,
that mean is just going to

828
00:47:42,460 --> 00:47:47,850
contribute directly to the
expected mean square value of

829
00:47:47,850 --> 00:47:49,790
the signal that you're using.

830
00:47:49,790 --> 00:47:51,870
So why not center it around 0.

831
00:47:54,730 --> 00:47:56,800
Later on we'll see many
reasons for not

832
00:47:56,800 --> 00:47:58,260
centering it around 0.

833
00:47:58,260 --> 00:48:01,350
But for now we're not going to
worry about any of those, and

834
00:48:01,350 --> 00:48:03,780
we're going to center
it around 0.

835
00:48:03,780 --> 00:48:05,700
And the other thing is,
why do you want them

836
00:48:05,700 --> 00:48:07,270
to be equally spaced?

837
00:48:07,270 --> 00:48:09,320
Well, I'll talk about that
in just a second.

838
00:48:13,730 --> 00:48:19,520
Anyway, the signal energy in
these equally spaced signals,

839
00:48:19,520 --> 00:48:23,280
you can calculate it to be d
squared times m squared minus

840
00:48:23,280 --> 00:48:26,100
1 divided by 12.

841
00:48:26,100 --> 00:48:28,630
I think we sort of did this when
we were worrying about

842
00:48:28,630 --> 00:48:30,600
quantization.

843
00:48:30,600 --> 00:48:38,870
There's a problem at the end
of lectures 11 and 12 which

844
00:48:38,870 --> 00:48:42,800
guides you by the hand in
how to calculate this.

845
00:48:42,800 --> 00:48:45,710
Or you can sit down and just
calculate it by hand, it's not

846
00:48:45,710 --> 00:48:46,620
hard to do.

847
00:48:46,620 --> 00:48:55,620
But, anyway, that's the mean
square value of these signals.

848
00:48:55,620 --> 00:48:58,760
If you assume that they're
equiprobable.

849
00:48:58,760 --> 00:49:06,090
Now, if you take m to be 2 to
the b, now what's b? b is the

850
00:49:06,090 --> 00:49:13,360
number of binary digits which
come into this signal former

851
00:49:13,360 --> 00:49:18,450
when you produce
one signal out.

852
00:49:18,450 --> 00:49:23,660
Namely, if you bring in b binary
digits, you need an

853
00:49:23,660 --> 00:49:27,730
alphabet of size 2 to the b.

854
00:49:27,730 --> 00:49:30,745
If you have an alphabet of size
2 to b, then you're going

855
00:49:30,745 --> 00:49:33,850
to need m equals 2 to the
b different signals.

856
00:49:33,850 --> 00:49:40,340
So, usually when you have a
standard PAM system, that

857
00:49:40,340 --> 00:49:46,360
number there, 8 PAM,
means 8 signals.

858
00:49:46,360 --> 00:49:51,570
8 is usually going to be
replaced by 4, or 16, or 32,

859
00:49:51,570 --> 00:49:54,360
or 64, or what have you.

860
00:49:54,360 --> 00:49:56,140
And, usually, not
anything else.

861
00:49:56,140 --> 00:49:58,810
Because you're usually going to
deal with something which

862
00:49:58,810 --> 00:50:00,180
is a power of 2.

863
00:50:00,180 --> 00:50:04,710
Because the logarithm of this to
the base 2 is the number of

864
00:50:04,710 --> 00:50:08,730
bits which are coming into the
signal former for each signal

865
00:50:08,730 --> 00:50:09,980
that comes out.

866
00:50:16,830 --> 00:50:22,520
This goes up very rapidly
as m squared goes up.

867
00:50:22,520 --> 00:50:26,790
In other words, you try to
transmit data faster by

868
00:50:26,790 --> 00:50:32,280
bringing more and more bits in
per signal that you transmit,

869
00:50:32,280 --> 00:50:35,870
it's a losing proposition
very, very quickly.

870
00:50:35,870 --> 00:50:38,350
It's this business of a
logarithm which comes into

871
00:50:38,350 --> 00:50:41,300
everything here.

872
00:50:41,300 --> 00:50:43,010
We're going to talk
about noise later.

873
00:50:43,010 --> 00:50:45,540
We're not going to talk
about it now.

874
00:50:45,540 --> 00:50:50,230
But, we have to recognize the
existence of noise enough to

875
00:50:50,230 --> 00:50:54,730
realize that when you look at
this diagram here, when you

876
00:50:54,730 --> 00:50:58,960
look at generating a waveform
around this, or a waveform

877
00:50:58,960 --> 00:51:03,790
around this, however you receive
these things, noise is

878
00:51:03,790 --> 00:51:07,940
going to corrupt what you
receive here by a little bit.

879
00:51:07,940 --> 00:51:10,440
Usually it's Gaussian,
which means it tails

880
00:51:10,440 --> 00:51:12,920
off very, very quickly.

881
00:51:12,920 --> 00:51:14,750
With larger amplitudes.

882
00:51:14,750 --> 00:51:20,350
And what that means is, when
you send a three, the most

883
00:51:20,350 --> 00:51:24,490
likely thing to happen is
that you're going to

884
00:51:24,490 --> 00:51:26,770
detect a three again.

885
00:51:26,770 --> 00:51:28,950
The next most likely thing
is you'll detect

886
00:51:28,950 --> 00:51:31,440
either a four or a two.

887
00:51:31,440 --> 00:51:33,670
In other words, what's important
here is this

888
00:51:33,670 --> 00:51:34,550
distance here.

889
00:51:34,550 --> 00:51:36,260
And hardly anything else.

890
00:51:36,260 --> 00:51:40,690
If you send these signals with
equal probability, and the

891
00:51:40,690 --> 00:51:44,000
noise is additive, the noise
does the same thing no matter

892
00:51:44,000 --> 00:51:46,710
where you are along here.

893
00:51:46,710 --> 00:51:51,100
Which says that this standard
PAM set is almost the only

894
00:51:51,100 --> 00:51:53,170
thing you want to look at.

895
00:51:53,170 --> 00:51:56,180
So long as you assume that the
noise is going to be additive.

896
00:51:56,180 --> 00:51:58,950
And the noise is going to affect
everything along this

897
00:51:58,950 --> 00:52:00,490
line equally.

898
00:52:00,490 --> 00:52:03,950
It says that you just want to
make the spacing between

899
00:52:03,950 --> 00:52:09,110
points big enough that it will
pretty much avoid the noise.

900
00:52:09,110 --> 00:52:12,830
So, the point of all of
that is that d is

901
00:52:12,830 --> 00:52:14,910
fixed, ahead of time.

902
00:52:14,910 --> 00:52:16,420
You can't play with that.

903
00:52:16,420 --> 00:52:17,710
You can play with m.

904
00:52:17,710 --> 00:52:19,840
When you play with m,
you're playing a

905
00:52:19,840 --> 00:52:22,510
losing game with energy.

906
00:52:22,510 --> 00:52:25,590
So that's why standard PAM
is the thing which

907
00:52:25,590 --> 00:52:28,020
is used with PAM.

908
00:52:28,020 --> 00:52:29,530
Not much you can do about it.

909
00:52:32,330 --> 00:52:38,380
And say here that the noise is
independent of the signal.

910
00:52:38,380 --> 00:52:40,020
We will talk about this later.

911
00:52:40,020 --> 00:52:42,150
The noise being additive.

912
00:52:42,150 --> 00:52:45,770
The noise being independent of
the signal are both saying

913
00:52:45,770 --> 00:52:47,300
almost the same thing.

914
00:52:47,300 --> 00:52:50,410
Which will be obvious to
you in a little while.

915
00:52:50,410 --> 00:52:52,540
But not until we start
talking about noise.

916
00:52:52,540 --> 00:52:55,250
We don't want to start talking
about noise now.

917
00:52:55,250 --> 00:52:58,260
So, for now, we deal with the
noise just by saying that

918
00:52:58,260 --> 00:53:01,600
every signal must be separated
by some minimum amount.

919
00:53:09,280 --> 00:53:11,740
Again, what we said about
delay and attenuation.

920
00:53:11,740 --> 00:53:14,940
Let me say it again, because
it's important enough that you

921
00:53:14,940 --> 00:53:16,270
have to understand it.

922
00:53:19,300 --> 00:53:22,960
After you go through two or
three problem sets, you will

923
00:53:22,960 --> 00:53:24,320
not understand it again.

924
00:53:24,320 --> 00:53:28,360
Because you'll get so used to
dealing the receive signal as

925
00:53:28,360 --> 00:53:32,020
the same as the transmitted
signal that you'll forget the

926
00:53:32,020 --> 00:53:34,670
weirdness in that.

927
00:53:34,670 --> 00:53:37,150
I mean, people become accustomed
to extraordinarily

928
00:53:37,150 --> 00:53:40,180
weird things very,
very easily.

929
00:53:40,180 --> 00:53:42,300
And especially when you're
taking a course and trying to

930
00:53:42,300 --> 00:53:45,090
get the problems done, you just
take things which are

931
00:53:45,090 --> 00:53:48,270
totally ridiculous and accept
them if it lets you get

932
00:53:48,270 --> 00:53:49,320
through the problem set.

933
00:53:49,320 --> 00:53:53,420
So I want to tell you right up
front that there is some

934
00:53:53,420 --> 00:53:55,180
weirdness associated here.

935
00:53:55,180 --> 00:53:57,450
It is something you have
to think about.

936
00:53:57,450 --> 00:54:00,820
After you think about it once,
you then accept this is a

937
00:54:00,820 --> 00:54:02,140
layering decision.

938
00:54:02,140 --> 00:54:05,330
You ignore delay, since the
timing recovery locks the

939
00:54:05,330 --> 00:54:08,210
receiver clock to the
transmitter clock plus the

940
00:54:08,210 --> 00:54:09,730
propagation delay.

941
00:54:09,730 --> 00:54:12,510
And in fact, it can lock the
receive clock to any place

942
00:54:12,510 --> 00:54:15,300
that wants to lock it to.

943
00:54:15,300 --> 00:54:18,070
So we're going to lock it in
such a way that the receive

944
00:54:18,070 --> 00:54:21,950
signal looks like the
transmitted signal.

945
00:54:21,950 --> 00:54:26,090
And the attenuation is really
part of the link budget.

946
00:54:26,090 --> 00:54:30,470
We can separate that from all
the things we're going to do.

947
00:54:30,470 --> 00:54:33,500
I mean, if we don't separate
that, you have to go into

948
00:54:33,500 --> 00:54:35,380
antenna design.

949
00:54:35,380 --> 00:54:36,550
And all this other stuff.

950
00:54:36,550 --> 00:54:39,060
And who wants to do that?

951
00:54:39,060 --> 00:54:41,660
I mean, we have enough
to do in this course.

952
00:54:41,660 --> 00:54:44,990
It's pretty full anyway.

953
00:54:44,990 --> 00:54:48,970
So we're just going to scale the
signal and noise together.

954
00:54:48,970 --> 00:54:51,030
And that's a separate issue.

955
00:54:54,370 --> 00:54:57,170
So now we want to look at the
thing which is called PAM

956
00:54:57,170 --> 00:54:58,010
modulation.

957
00:54:58,010 --> 00:55:00,840
In other words, in this one
slide we separated the

958
00:55:00,840 --> 00:55:07,040
question of choosing the signal
constellation, which

959
00:55:07,040 --> 00:55:10,000
we've now solved by saying, we
want to use signals that are

960
00:55:10,000 --> 00:55:12,160
equally spaced.

961
00:55:12,160 --> 00:55:13,450
So that's an easy one.

962
00:55:13,450 --> 00:55:17,160
From the question of, how do
you choose the filter.

963
00:55:17,160 --> 00:55:21,640
So the PAM modulation is going
to go by taking a sequence of

964
00:55:21,640 --> 00:55:26,800
signals, mapping it into a
waveform, which is this

965
00:55:26,800 --> 00:55:27,970
expansion here.

966
00:55:27,970 --> 00:55:31,420
We're not assuming that these
functions are orthogonal to

967
00:55:31,420 --> 00:55:33,500
each other.

968
00:55:33,500 --> 00:55:38,030
Although later, we will find
out that they should be.

969
00:55:38,030 --> 00:55:43,810
But for now, p of t is just
some arbitrary waveform.

970
00:55:43,810 --> 00:55:49,520
And we will try to figure out
how to choose this waveform in

971
00:55:49,520 --> 00:55:54,350
such a way as to replace the
sinc waveform with something

972
00:55:54,350 --> 00:55:58,770
which is better in terms of
delay and almost as good in

973
00:55:58,770 --> 00:56:00,620
all other ways.

974
00:56:00,620 --> 00:56:05,080
We're not going to worry about
the fact that p of t has to be

975
00:56:05,080 --> 00:56:05,930
realizable.

976
00:56:05,930 --> 00:56:08,630
Because with our arbitrary
time reference at the

977
00:56:08,630 --> 00:56:10,930
receiver, it doesn't have
to be realizable.

978
00:56:10,930 --> 00:56:12,630
It doesn't have to be causal.

979
00:56:15,150 --> 00:56:19,220
So, p of t could be
sinc of t over t.

980
00:56:19,220 --> 00:56:23,160
Which would give us a nice
baseband limited function.

981
00:56:23,160 --> 00:56:24,680
But it could be anything
else at all.

982
00:56:28,080 --> 00:56:31,260
As we said before, sinc
t over t a dies out

983
00:56:31,260 --> 00:56:33,960
impractically slowly.

984
00:56:33,960 --> 00:56:38,830
And it requires infinite delay
at the transmitter.

985
00:56:38,830 --> 00:56:42,740
You can't even send these
signals with a

986
00:56:42,740 --> 00:56:46,040
sinc t over t signal.

987
00:56:46,040 --> 00:56:50,040
Because to send them, you have
to start out at the beginning

988
00:56:50,040 --> 00:56:52,360
of this little bit
of wiggling.

989
00:56:52,360 --> 00:56:54,950
Now, you say, OK I'm going
to truncate that

990
00:56:54,950 --> 00:56:56,430
when it's very small.

991
00:56:56,430 --> 00:56:58,620
And I don't worry about that.

992
00:56:58,620 --> 00:57:01,740
The point of what we're starting
to look at now,

993
00:57:01,740 --> 00:57:04,210
though, is we're saying, OK,
you truncate it, you do all

994
00:57:04,210 --> 00:57:05,660
these practical things.

995
00:57:05,660 --> 00:57:09,170
But it turns out that this
problem of choosing this

996
00:57:09,170 --> 00:57:12,240
filter response has
a very elegant

997
00:57:12,240 --> 00:57:14,240
and a very nice solution.

998
00:57:14,240 --> 00:57:18,220
And when we put noise in, it
fits in perfectly with the

999
00:57:18,220 --> 00:57:24,270
idea of also choosing this
filter in a particular way.

1000
00:57:24,270 --> 00:57:26,620
Now, we've talked about many
problems here which were

1001
00:57:26,620 --> 00:57:30,630
solved almost immediately after
Shannon came out with

1002
00:57:30,630 --> 00:57:34,880
his way of looking at
communication in 1948.

1003
00:57:34,880 --> 00:57:38,280
Guess when this problem
was solved?

1004
00:57:38,280 --> 00:57:41,960
It was solved 20 years earlier
than that by a guy by the name

1005
00:57:41,960 --> 00:57:44,340
of Nyquist, who was at
Bell Labs back when

1006
00:57:44,340 --> 00:57:46,610
Bell Labs meant something.

1007
00:57:49,150 --> 00:57:51,000
I mean, in `28 it was
a great place.

1008
00:57:51,000 --> 00:57:56,150
It was a great place until
seven or eight years ago.

1009
00:57:56,150 --> 00:57:59,630
Nyquist is important
in feedback theory.

1010
00:57:59,630 --> 00:58:02,880
He's done some of the most
important things there.

1011
00:58:02,880 --> 00:58:06,360
His Nyquist criterion in dealing
with how do you choose

1012
00:58:06,360 --> 00:58:10,450
these filters to avoid
intersymbol interference is

1013
00:58:10,450 --> 00:58:11,640
fairly simple.

1014
00:58:11,640 --> 00:58:14,700
But it's a very, very
nice result.

1015
00:58:14,700 --> 00:58:17,000
So we're going to
talk about it.

1016
00:58:17,000 --> 00:58:20,070
And then we will use that to
say, ok, we don't have to

1017
00:58:20,070 --> 00:58:23,230
worry about intersymbol
interference any more, so all

1018
00:58:23,230 --> 00:58:24,620
we have to worry
about is noise.

1019
00:58:24,620 --> 00:58:27,430
So we're getting rid of these
problems one by one.

1020
00:58:31,280 --> 00:58:34,820
Our main problem is to choose
this filter, so that we get

1021
00:58:34,820 --> 00:58:38,070
some kind of reasonable
compromise between time delay

1022
00:58:38,070 --> 00:58:39,830
and bandwidth.

1023
00:58:39,830 --> 00:58:44,990
That's -- that's basically the
problem that we're facing.

1024
00:58:44,990 --> 00:58:47,610
And Nyquist's solution
to this was to say,

1025
00:58:47,610 --> 00:58:49,330
forget about that also.

1026
00:58:49,330 --> 00:58:53,270
Let's look at what set of
filters work, and what set of

1027
00:58:53,270 --> 00:58:55,320
filters don't work.

1028
00:58:55,320 --> 00:58:57,510
And then you can take your
choice among this set of

1029
00:58:57,510 --> 00:58:58,760
folders that work.

1030
00:59:05,900 --> 00:59:14,630
So our problem is, how do you
take the waveform u of t,

1031
00:59:14,630 --> 00:59:21,330
which you receive, and find
these samples out of it?

1032
00:59:21,330 --> 00:59:24,600
Now, we already know that if
you use sinc functions, all

1033
00:59:24,600 --> 00:59:27,920
you have to do is sample this
and you get these u sub k's

1034
00:59:27,920 --> 00:59:29,040
back again.

1035
00:59:29,040 --> 00:59:32,810
But if this thing is some
absolutely wild waveform,

1036
00:59:32,810 --> 00:59:35,090
maybe that's not what you get.

1037
00:59:35,090 --> 00:59:42,250
So we say, OK, how do we
retrieve these signals from

1038
00:59:42,250 --> 00:59:45,650
the waveform that was
transmitted and therefore from

1039
00:59:45,650 --> 00:59:47,610
the waveform that
was received.

1040
00:59:47,610 --> 00:59:50,520
We're separating this from
the noise question.

1041
00:59:50,520 --> 00:59:53,630
Even if there's no noise at all,
we still have a question

1042
00:59:53,630 --> 00:59:56,490
and how do you find
those input values

1043
00:59:56,490 --> 00:59:59,600
directly from the function.

1044
00:59:59,600 --> 01:00:01,940
What we're going to do is to
assume that the receiver

1045
01:00:01,940 --> 01:00:06,200
filters u of t, with a linear
time invariant filter, with

1046
01:00:06,200 --> 01:00:08,970
impulse response q of t.

1047
01:00:08,970 --> 01:00:11,530
Now, since we've said that
p of t doesn't have to be

1048
01:00:11,530 --> 01:00:15,800
causal, we might as well say
that q of t doesn't have to be

1049
01:00:15,800 --> 01:00:16,370
causal either.

1050
01:00:16,370 --> 01:00:21,150
Because we've already thrown
these details of

1051
01:00:21,150 --> 01:00:24,080
delay to the winds.

1052
01:00:24,080 --> 01:00:27,310
So the filtered waveform, then
is going to be r of t.

1053
01:00:27,310 --> 01:00:32,910
Will be the integral of what
was transmitted, convolved

1054
01:00:32,910 --> 01:00:34,380
with q of t.

1055
01:00:34,380 --> 01:00:37,330
So this is what you receive.

1056
01:00:37,330 --> 01:00:40,300
And then what we're going to do
is, we're going to sample

1057
01:00:40,300 --> 01:00:42,810
this waveform.

1058
01:00:42,810 --> 01:00:47,370
So Nyquist said, let's restrict
ourselves to looking

1059
01:00:47,370 --> 01:00:51,570
at receivers which first filter
by some filter we're

1060
01:00:51,570 --> 01:00:55,200
going to decide on,
and then sample.

1061
01:00:55,200 --> 01:00:56,600
Why do you want to do that?

1062
01:00:56,600 --> 01:00:58,950
Well, an interesting question.

1063
01:00:58,950 --> 01:01:01,830
And Nyquist said, that's
what we're going to do.

1064
01:01:01,830 --> 01:01:06,060
And since Nyquist was famous,
that's what we're going to do.

1065
01:01:06,060 --> 01:01:11,500
One of the problems in the
homework this week is to show

1066
01:01:11,500 --> 01:01:15,160
that if you relax this a little
bit and you say, well,

1067
01:01:15,160 --> 01:01:18,710
I don't want to do what Nyquist
said, what I want to

1068
01:01:18,710 --> 01:01:22,690
do is to look at an arbitrary
linear receiver which takes

1069
01:01:22,690 --> 01:01:25,760
this received waveform, goes
through any old linear

1070
01:01:25,760 --> 01:01:29,960
operations that I want
to go through, and

1071
01:01:29,960 --> 01:01:31,600
solves for what --

1072
01:01:34,170 --> 01:01:37,820
and from that, tries to pick
out these coefficients.

1073
01:01:37,820 --> 01:01:41,560
And the question is, can you
do anything more than what

1074
01:01:41,560 --> 01:01:42,880
Nyquist did.

1075
01:01:42,880 --> 01:01:45,250
And the answer is, no.

1076
01:01:45,250 --> 01:01:49,460
If you look at it in another
way, you will find that in

1077
01:01:49,460 --> 01:01:53,980
fact, if you know what the
signal constellation is, you

1078
01:01:53,980 --> 01:01:55,790
can look at non-linear
receivers.

1079
01:01:55,790 --> 01:01:59,700
Which in the absence of noise
will let you pick out these

1080
01:01:59,700 --> 01:02:03,270
coefficients in a much
broader context than

1081
01:02:03,270 --> 01:02:05,170
what Nyquist said.

1082
01:02:05,170 --> 01:02:07,090
And why don't we do that?

1083
01:02:07,090 --> 01:02:10,660
Well, because when noise comes
in, that doesn't work at all.

1084
01:02:10,660 --> 01:02:13,930
That's a lousy solution.

1085
01:02:13,930 --> 01:02:17,320
So what we're going to do is
say, OK, Mr. Nyquist, we'll

1086
01:02:17,320 --> 01:02:19,400
play your silly game.

1087
01:02:19,400 --> 01:02:22,920
We will have this filter
at the transmitter.

1088
01:02:22,920 --> 01:02:25,680
We'll have this filter
at the receiver.

1089
01:02:25,680 --> 01:02:28,690
We'll have the sampler and we'll
look at what conditions

1090
01:02:28,690 --> 01:02:32,360
we need in order to make
the whole thing work.

1091
01:02:32,360 --> 01:02:34,980
And then we will fit it in with
noise and everything.

1092
01:02:34,980 --> 01:02:37,070
It will fit in, in
a very nice way.

1093
01:02:37,070 --> 01:02:40,500
So we have a nice layered
solution when we do that.

1094
01:02:40,500 --> 01:02:42,910
And we will find --

1095
01:02:42,910 --> 01:02:44,930
I mean, Nyquist had
some of Shannon's

1096
01:02:44,930 --> 01:02:47,030
genes in him, I think.

1097
01:02:47,030 --> 01:02:50,640
Because what we find when we're
all finished with this

1098
01:02:50,640 --> 01:02:54,600
is that by avoiding -- by being
able to receive these

1099
01:02:54,600 --> 01:02:58,710
coefficients perfectly, which
we'll refer to as avoiding

1100
01:02:58,710 --> 01:03:03,530
intersymbol interference, it
doesn't hurt us at all as far

1101
01:03:03,530 --> 01:03:05,190
as taking care of the noise.

1102
01:03:05,190 --> 01:03:07,410
In other words, you can have
your cake and you can eat it

1103
01:03:07,410 --> 01:03:10,080
too as far as intersymbol
interference

1104
01:03:10,080 --> 01:03:11,330
and noise is concerned.

1105
01:03:18,790 --> 01:03:23,260
We wind up with a received
waveform, which is the

1106
01:03:23,260 --> 01:03:27,290
integral of the transmitted
waveform times some filter.

1107
01:03:27,290 --> 01:03:29,350
And we don't know how to
choose this filter.

1108
01:03:29,350 --> 01:03:32,830
But let's just -- this
is a filter.

1109
01:03:32,830 --> 01:03:37,340
This can be represented now in
terms of u of t is equal to

1110
01:03:37,340 --> 01:03:41,630
this transmitted waveform in
terms of this other filter

1111
01:03:41,630 --> 01:03:42,760
that we don't understand.

1112
01:03:42,760 --> 01:03:46,270
So we now have two filters
that we don't understand.

1113
01:03:46,270 --> 01:03:49,000
And we have this
integral here.

1114
01:03:49,000 --> 01:03:50,890
Well, we can take this sum.

1115
01:03:50,890 --> 01:03:54,380
We can bring this sum outside
of the integral and have the

1116
01:03:54,380 --> 01:03:59,010
sum of u sub k times, just some
composite filter g of t

1117
01:03:59,010 --> 01:04:01,840
where g of t is the
convolution of p

1118
01:04:01,840 --> 01:04:03,510
of t and q of t.

1119
01:04:03,510 --> 01:04:06,860
Now, when you look at the notes,
the notes are fairly

1120
01:04:06,860 --> 01:04:10,850
careful in dealing with all
these questions of L2 and

1121
01:04:10,850 --> 01:04:12,860
convergence and all of
this stuff that we've

1122
01:04:12,860 --> 01:04:14,860
been talking about.

1123
01:04:14,860 --> 01:04:18,780
Again, when you're trying to
understand something for the

1124
01:04:18,780 --> 01:04:22,970
first time, ignore all those
mathematical issues.

1125
01:04:22,970 --> 01:04:25,120
Try to find out what's
going on.

1126
01:04:25,120 --> 01:04:28,170
When you get an intuitive sense
of what's going on, go

1127
01:04:28,170 --> 01:04:30,150
back and look at the
mathematics then.

1128
01:04:30,150 --> 01:04:34,230
But don't fuss about the
mathematics at this point.

1129
01:04:34,230 --> 01:04:38,840
OK, so r of t, then, is just
going to be the sum over k of

1130
01:04:38,840 --> 01:04:45,440
these sample values that are
coming in, times this

1131
01:04:45,440 --> 01:04:50,040
composite filter, which is the
convolution of the transmit

1132
01:04:50,040 --> 01:04:53,550
filter and the recieve filter.

1133
01:04:53,550 --> 01:04:55,770
Now, this shouldn't
be surprising.

1134
01:04:55,770 --> 01:05:00,320
If you think of u of t as
being formed by taking a

1135
01:05:00,320 --> 01:05:04,200
sequence of impulses, and then
passing that sequence of

1136
01:05:04,200 --> 01:05:08,530
impulses through a filter p of
t, and then passing the output

1137
01:05:08,530 --> 01:05:12,670
through a filter q of t, all
you're doing is passing this

1138
01:05:12,670 --> 01:05:16,005
sequence of impulses through
the convolution of p

1139
01:05:16,005 --> 01:05:19,130
of t and q of t.

1140
01:05:19,130 --> 01:05:22,780
In other words, in terms of
this received waveform, it

1141
01:05:22,780 --> 01:05:25,750
couldn't care less what
filtering you do at the

1142
01:05:25,750 --> 01:05:29,490
transmitter and what filtering
you do at the receiver.

1143
01:05:29,490 --> 01:05:34,290
It's all one big filter as far
as the receiver is concerned.

1144
01:05:34,290 --> 01:05:37,300
When we study noise, what
happens with the transmitter

1145
01:05:37,300 --> 01:05:39,200
and what happens with
the receiver will

1146
01:05:39,200 --> 01:05:40,870
become important again.

1147
01:05:40,870 --> 01:05:45,420
But, so far, none of this
makes any difference.

1148
01:05:45,420 --> 01:05:50,070
And this is all we need
to worry about.

1149
01:05:50,070 --> 01:05:55,040
Then, Nyquist said, a waveform
g of t is ideal Nyquist.

1150
01:05:55,040 --> 01:05:56,430
He didn't call it ideal
Nyquist, he

1151
01:05:56,430 --> 01:05:58,080
was very modest person.

1152
01:05:58,080 --> 01:06:01,850
But since then, people call it
ideal Nyquist because he was

1153
01:06:01,850 --> 01:06:05,230
the guy that sorted
it all out.

1154
01:06:05,230 --> 01:06:08,700
It's ideal Nyquist with period
t, and I usually leave out the

1155
01:06:08,700 --> 01:06:12,880
period t because that's usually
understood, if g of k

1156
01:06:12,880 --> 01:06:16,780
t is equal to delta of k.

1157
01:06:16,780 --> 01:06:25,380
In other words, if g of 0 is
equal to 1 and g of k times t,

1158
01:06:25,380 --> 01:06:29,470
for every non-zero integer
k is equal to 0.

1159
01:06:29,470 --> 01:06:32,880
In other words, if g has the
same property that the sine

1160
01:06:32,880 --> 01:06:36,000
function has, sine
function is 1.

1161
01:06:36,000 --> 01:06:37,020
It's 0.

1162
01:06:37,020 --> 01:06:41,770
And it's 0 at every integer
point beyond that.

1163
01:06:41,770 --> 01:06:46,520
And Nyquist said, well, gee, all
we need to do is make this

1164
01:06:46,520 --> 01:06:48,750
filter has a property.

1165
01:06:48,750 --> 01:06:52,010
And when you look at this, it's
fairly obvious that that

1166
01:06:52,010 --> 01:06:55,200
works, right?

1167
01:06:55,200 --> 01:06:57,260
I mean, we want a
sample RFT --

1168
01:06:57,260 --> 01:07:03,150
say, at j times capital T. Or,
if j times capital T, it's

1169
01:07:03,150 --> 01:07:15,760
going to be the sum here of u
sub k times g of j t r of t.

1170
01:07:15,760 --> 01:07:29,150
r of j t is equal to sum
over k of u k times g

1171
01:07:29,150 --> 01:07:33,860
of j t minus k t.

1172
01:07:40,200 --> 01:07:45,680
If the waveform is ideal
Nyquist, then this quantity is

1173
01:07:45,680 --> 01:07:50,800
0 for all integer k except
when k is equal to j.

1174
01:07:50,800 --> 01:07:57,090
So, this is just equal
to u sub j.

1175
01:07:57,090 --> 01:08:02,570
And conversely, if this waveform
is not ideal Nyquist,

1176
01:08:02,570 --> 01:08:08,430
you have the problem that you
can pick two values. u sub k

1177
01:08:08,430 --> 01:08:10,890
and u sub j, in such
a way that they

1178
01:08:10,890 --> 01:08:13,140
interfere with each other.

1179
01:08:13,140 --> 01:08:16,880
In other words, that they add up
at some sample point to the

1180
01:08:16,880 --> 01:08:17,610
wrong value.

1181
01:08:17,610 --> 01:08:22,150
One of them is going to come
in and clobber the other.

1182
01:08:22,150 --> 01:08:26,590
So, this is a necessary and
sufficient condition for

1183
01:08:26,590 --> 01:08:29,540
avoiding intersymbol
interference.

1184
01:08:29,540 --> 01:08:32,130
So long as you're not smart
enough to look at what those

1185
01:08:32,130 --> 01:08:33,950
actual values are.

1186
01:08:33,950 --> 01:08:36,010
In other words, so long as
you're only going to use a

1187
01:08:36,010 --> 01:08:40,620
linear receiver, which is
what that amounts to.

1188
01:08:40,620 --> 01:08:44,560
While ignoring noise, r of t is
determined by g of t, p of

1189
01:08:44,560 --> 01:08:47,050
t and q of t otherwise
irrelevant.

1190
01:08:47,050 --> 01:08:49,320
That's what we said.

1191
01:08:49,320 --> 01:08:50,280
We said this.

1192
01:08:50,280 --> 01:08:54,600
If t of t is ideal Nyquist and
r of k t equals u of k

1193
01:08:54,600 --> 01:08:55,540
for all k and z.

1194
01:08:55,540 --> 01:08:59,910
If f of t is not ideal Nyquist,
and r f k t unequal

1195
01:08:59,910 --> 01:09:05,320
to u k for some k and some
choice of the sequence.

1196
01:09:05,320 --> 01:09:07,740
Now, so far there's
no big deal here.

1197
01:09:10,350 --> 01:09:12,070
This is pretty easy
to figure out.

1198
01:09:12,070 --> 01:09:16,040
You don't need rocket science
to say, once I pose the

1199
01:09:16,040 --> 01:09:18,990
problem this why, which
is where part of

1200
01:09:18,990 --> 01:09:20,870
Nyquist's genius came from.

1201
01:09:20,870 --> 01:09:22,920
The hard thing is always to
post the right problem.

1202
01:09:22,920 --> 01:09:25,160
It's not to solve it.

1203
01:09:25,160 --> 01:09:30,140
Those of you who want to do Ph.D
theses, believe me, 80%

1204
01:09:30,140 --> 01:09:33,080
of the problem is finding
the problem.

1205
01:09:33,080 --> 01:09:35,670
20% of the problem
is doing it.

1206
01:09:35,670 --> 01:09:40,890
If you do a really outstanding
thesis, 99% is finding the

1207
01:09:40,890 --> 01:09:45,430
problem and 1% of
it is doing it.

1208
01:09:45,430 --> 01:09:46,620
And I believe that.

1209
01:09:46,620 --> 01:09:47,870
I'm not exaggerating.

1210
01:09:52,890 --> 01:09:56,710
An ideal Nyquist g of t implies
that no intersymbol

1211
01:09:56,710 --> 01:09:59,500
interference occurs at
the above receiver.

1212
01:09:59,500 --> 01:10:03,400
In other words, you have a
receiver that actually works.

1213
01:10:03,400 --> 01:10:05,850
We're going to see that choosing
g of t to be ideal

1214
01:10:05,850 --> 01:10:09,440
Nyquist fits in nicely when
looking at the real problem,

1215
01:10:09,440 --> 01:10:12,160
which is coping with both
noise and intersymbol

1216
01:10:12,160 --> 01:10:13,880
interference.

1217
01:10:13,880 --> 01:10:18,180
And we've also seen that if g of
t is sync of t over capital

1218
01:10:18,180 --> 01:10:19,770
T, that works.

1219
01:10:19,770 --> 01:10:22,390
It has no intersymbol
interference because that's,

1220
01:10:22,390 --> 01:10:28,380
one, at t equals 0, and it's 0
at every other sample point.

1221
01:10:28,380 --> 01:10:32,150
We don't like that, because
it has too much delay.

1222
01:10:32,150 --> 01:10:34,490
We want to make g of
t strictly baseband

1223
01:10:34,490 --> 01:10:37,370
limited to 1 over 2t.

1224
01:10:37,370 --> 01:10:39,660
And this turns out to be
the only solution.

1225
01:10:39,660 --> 01:10:42,390
And we'll see that in
a little while.

1226
01:10:42,390 --> 01:10:45,860
In other words, if you want to
do something which has better

1227
01:10:45,860 --> 01:10:50,190
delay characteristics than the
sinc function, your only way

1228
01:10:50,190 --> 01:10:52,860
of doing it is spilling out
into slightly higher

1229
01:10:52,860 --> 01:10:54,170
frequencies.

1230
01:10:54,170 --> 01:10:58,360
So, what Nyquist really wanted
to find out, although you

1231
01:10:58,360 --> 01:11:01,480
won't find that out by reading
his paper because it's all

1232
01:11:01,480 --> 01:11:05,300
this nice mathematical proof,
is how much do you have to

1233
01:11:05,300 --> 01:11:11,130
expand the bandwidth in order to
get nice delay things which

1234
01:11:11,130 --> 01:11:12,380
do the same thing.

1235
01:11:17,830 --> 01:11:20,150
Well, we think about
it a little bit.

1236
01:11:20,150 --> 01:11:23,140
And we have an advantage that
Nyquist didn't have.

1237
01:11:23,140 --> 01:11:26,460
Because we understand
about aliasing.

1238
01:11:26,460 --> 01:11:28,340
Nyquist didn't understand
about aliasing.

1239
01:11:28,340 --> 01:11:30,855
Aliasing hadn't been
done at that time.

1240
01:11:30,855 --> 01:11:33,270
It was probably done
as a result of

1241
01:11:33,270 --> 01:11:36,710
what Nyquist had done.

1242
01:11:36,710 --> 01:11:40,700
And if we look at the
reconstruction from the

1243
01:11:40,700 --> 01:11:46,940
samples, g of k t of g, we get
this function s of t, which is

1244
01:11:46,940 --> 01:11:51,180
the sum of all the samples
times sinc of t

1245
01:11:51,180 --> 01:11:53,100
over t, minus k.

1246
01:11:53,100 --> 01:11:55,980
That's for an arbitrary
filter.

1247
01:11:55,980 --> 01:12:00,500
This baseband reconstruction
by definition, when we were

1248
01:12:00,500 --> 01:12:04,360
talking about alising, is
just this function.

1249
01:12:04,360 --> 01:12:07,880
We've said that g of t is ideal
Nyquist, if and only if

1250
01:12:07,880 --> 01:12:12,180
s of t is equal to sinc
of t over capital t.

1251
01:12:12,180 --> 01:12:14,190
Why is that?

1252
01:12:14,190 --> 01:12:17,140
You look at this
function here.

1253
01:12:17,140 --> 01:12:19,540
You look at s of 0.

1254
01:12:19,540 --> 01:12:21,970
And what do you get?

1255
01:12:21,970 --> 01:12:25,940
You get the sum of g of
k t times sinc of t

1256
01:12:25,940 --> 01:12:28,500
over t minus k.

1257
01:12:28,500 --> 01:12:32,190
If we have an ideal Nyquist
filter, then all of these

1258
01:12:32,190 --> 01:12:36,250
terms are 0 except when
k is equal to 0.

1259
01:12:43,120 --> 01:12:48,830
s of t, in general, if you have
a ideal Nyquist filter,

1260
01:12:48,830 --> 01:12:50,760
you only have one
term in here.

1261
01:12:50,760 --> 01:12:54,720
So s of t is just equal
to sinc of t over t.

1262
01:12:54,720 --> 01:12:58,390
Because all those other
terms go away.

1263
01:12:58,390 --> 01:13:01,950
If we take the Fourier transform
of s of t equals

1264
01:13:01,950 --> 01:13:06,820
sinc t over capital T, the
Fourier transform is s s tilde

1265
01:13:06,820 --> 01:13:10,340
of f is equal to t times
the rectangle

1266
01:13:10,340 --> 01:13:13,070
function of f times t.

1267
01:13:13,070 --> 01:13:19,070
A sinc function of a Fourier
transform is a rectangle.

1268
01:13:19,070 --> 01:13:23,350
And the aliasing theorem then
says that this Fourier

1269
01:13:23,350 --> 01:13:28,480
transform of this low pass
representation has to be equal

1270
01:13:28,480 --> 01:13:32,330
to this sum of different
frequency terms.

1271
01:13:32,330 --> 01:13:34,410
That's what aliasing says.

1272
01:13:34,410 --> 01:13:38,750
It says that this baseband
representation is aliased into

1273
01:13:38,750 --> 01:13:41,030
by all of these other
frequency bands.

1274
01:13:41,030 --> 01:13:44,340
And each of them come in and
add to what you get in

1275
01:13:44,340 --> 01:13:45,480
frequency here.

1276
01:13:45,480 --> 01:13:48,740
Remember that diagram that we
drew where we took this

1277
01:13:48,740 --> 01:13:55,080
arbitrary frequency function and
we picked up what was in

1278
01:13:55,080 --> 01:13:58,240
each band, then we stuck it into
the center band and then

1279
01:13:58,240 --> 01:14:01,010
we added all these things up?

1280
01:14:01,010 --> 01:14:03,630
That's what the aliasing
thing says.

1281
01:14:03,630 --> 01:14:08,380
So it says that g of t is ideal
Nyquist if and only if

1282
01:14:08,380 --> 01:14:12,210
this sum is equal to that.

1283
01:14:12,210 --> 01:14:15,220
And that's the Nyquist
criteria.

1284
01:14:15,220 --> 01:14:16,470
That's what Nyquist did.

1285
01:14:21,340 --> 01:14:25,190
But he did it long before
anybody had heard of aliasing.

1286
01:14:38,260 --> 01:14:43,390
There's a slightly easier way to
look at aliasing criterion,

1287
01:14:43,390 --> 01:14:46,810
which now becomes the
Nyquist criterion.

1288
01:14:46,810 --> 01:14:53,100
When what you're interested in
is a waveform g of t, which is

1289
01:14:53,100 --> 01:14:56,580
almost band-limited
but not quite.

1290
01:14:56,580 --> 01:14:59,870
So what we're going to assume
is that it's limited to, at

1291
01:14:59,870 --> 01:15:05,940
most, twice this -- this w here
is 1 over 2t, which is

1292
01:15:05,940 --> 01:15:08,900
called the Nyquist bandwidth.

1293
01:15:08,900 --> 01:15:12,860
Everything is called
Nyquist here, so.

1294
01:15:12,860 --> 01:15:17,850
This value here is the minimum
bandwidth you could be using.

1295
01:15:17,850 --> 01:15:19,810
It's 1 over capital 2t.

1296
01:15:19,810 --> 01:15:22,620
It's what you would get if you
were using the sinc function.

1297
01:15:22,620 --> 01:15:25,830
If you were using the sinc
function, what you would get

1298
01:15:25,830 --> 01:15:28,420
is something which is
a rectangle here.

1299
01:15:28,420 --> 01:15:30,730
Cut off, right at this point.

1300
01:15:30,730 --> 01:15:34,260
And cut off right
at this point.

1301
01:15:34,260 --> 01:15:41,440
Nyquist is saying, suppose it's
limited to at most 2w.

1302
01:15:41,440 --> 01:15:44,650
In other words, suppose you
have a slopover into other

1303
01:15:44,650 --> 01:15:50,280
frequencies but, at most into
the next frequency band and no

1304
01:15:50,280 --> 01:15:51,870
more than that.

1305
01:15:51,870 --> 01:15:57,330
Then, if you look at this thing,
which is spilling out,

1306
01:15:57,330 --> 01:16:00,740
and we take the same picture we
were looking at before, we

1307
01:16:00,740 --> 01:16:02,230
take this quantity.

1308
01:16:02,230 --> 01:16:04,550
Bring it back down here.

1309
01:16:04,550 --> 01:16:07,630
We take this quantity,
bring it up here.

1310
01:16:07,630 --> 01:16:09,630
And what do we get?

1311
01:16:09,630 --> 01:16:12,860
This, added into here.

1312
01:16:12,860 --> 01:16:17,140
This thing adds up
right there.

1313
01:16:22,210 --> 01:16:27,110
In other words -- well, let's
take this one here.

1314
01:16:30,740 --> 01:16:35,160
This thing here gets translated
over to here.

1315
01:16:38,850 --> 01:16:42,150
And added to this.

1316
01:16:42,150 --> 01:16:53,840
This is assuming that g hat of
f is real, and we're ignoring

1317
01:16:53,840 --> 01:16:55,260
the complex part of it.

1318
01:16:55,260 --> 01:16:57,830
So this gets added to this.

1319
01:16:57,830 --> 01:17:03,090
This is just enough to turn
this into something, which

1320
01:17:03,090 --> 01:17:07,140
goes across here
and down here.

1321
01:17:09,850 --> 01:17:10,520
Down here.

1322
01:17:10,520 --> 01:17:12,730
My finger is not perfect.

1323
01:17:12,730 --> 01:17:16,990
But, anyway, when you add this
to this, you get this ideal

1324
01:17:16,990 --> 01:17:18,790
rectangular shape.

1325
01:17:18,790 --> 01:17:22,130
What this is saying, in terms
of just this upper side band

1326
01:17:22,130 --> 01:17:26,320
here, this is going to be the
same as this, from symmetry.

1327
01:17:26,320 --> 01:17:29,940
So it's saying, if you take
what's on the positive side of

1328
01:17:29,940 --> 01:17:35,790
w, and you rotate it around this
way, you rotate it around

1329
01:17:35,790 --> 01:17:40,340
up here, if it's just enough to
fill that in, then you've

1330
01:17:40,340 --> 01:17:43,410
satisfied the Nyquist
criteria.

1331
01:17:43,410 --> 01:17:46,250
In other words, you want band
edge symmetry here.

1332
01:17:46,250 --> 01:17:50,340
You want this to be symmetrical
to this.

1333
01:17:50,340 --> 01:17:54,030
In that rotated 180
degree sense.

1334
01:17:54,030 --> 01:17:56,820
So it says that anything which
has this band edged symmetry

1335
01:17:56,820 --> 01:18:01,430
condition satisfies the property
of no intersymbol

1336
01:18:01,430 --> 01:18:03,820
interference.

1337
01:18:03,820 --> 01:18:06,220
So this makes the problem
easy for us.

1338
01:18:06,220 --> 01:18:09,020
It says, we would like to have
a rectangular function.

1339
01:18:09,020 --> 01:18:11,960
That has too much delay, because
in particular the

1340
01:18:11,960 --> 01:18:15,240
inverse Fourier transform of
that, because we have a

1341
01:18:15,240 --> 01:18:20,080
discontinuity, can drop off,
at most, as 1 over t.

1342
01:18:20,080 --> 01:18:22,890
Which is pretty slow.

1343
01:18:22,890 --> 01:18:26,710
We've gotten rid of
the discontinuity.

1344
01:18:26,710 --> 01:18:29,980
We've also gotten rid of the
slope discontinuity, the way I

1345
01:18:29,980 --> 01:18:31,910
drew the figure here.

1346
01:18:31,910 --> 01:18:34,510
And, therefore, we can wind up
with a function which decays

1347
01:18:34,510 --> 01:18:36,970
as 1 over t cubed.

1348
01:18:36,970 --> 01:18:38,940
Which is a whole lot better
than 1 over t.

1349
01:18:42,160 --> 01:18:47,930
And, excuse me for going
a little over, but.

1350
01:18:51,710 --> 01:18:54,210
The things people build in
practice usually have

1351
01:18:54,210 --> 01:18:57,500
something called a raised
cosine filter.

1352
01:18:57,500 --> 01:19:00,920
Which is this messy expression
here, for

1353
01:19:00,920 --> 01:19:02,690
the frequency response.

1354
01:19:02,690 --> 01:19:04,780
But it really isn't that bad.

1355
01:19:04,780 --> 01:19:08,120
In fact, it's just
what this is.

1356
01:19:12,070 --> 01:19:18,090
This frequency response is t
over most of the frequency

1357
01:19:18,090 --> 01:19:21,560
interval, up to 1 over 2t.

1358
01:19:21,560 --> 01:19:27,150
It's t times a raised cosine
over this part of the

1359
01:19:27,150 --> 01:19:29,550
frequency band here.

1360
01:19:29,550 --> 01:19:33,420
t times cosine squared, and the
cosine squared just raises

1361
01:19:33,420 --> 01:19:37,350
things up to be average
out at 1/2, and it's

1362
01:19:37,350 --> 01:19:39,810
0 everywhere else.

1363
01:19:39,810 --> 01:19:45,370
So what you're doing here in
that formula is simply making

1364
01:19:45,370 --> 01:19:47,340
things t up to here.

1365
01:19:50,080 --> 01:19:55,560
Making it a raised
cosine here.

1366
01:19:55,560 --> 01:19:58,200
And making it 0 everywhere
else.

1367
01:20:00,890 --> 01:20:06,410
And, depending on what do choose
as alpha, that makes

1368
01:20:06,410 --> 01:20:08,830
this sharper or less sharp.

1369
01:20:08,830 --> 01:20:11,520
And people usually choose it
to be about, alpha to be

1370
01:20:11,520 --> 01:20:14,080
about 15% or so.

1371
01:20:14,080 --> 01:20:18,000
Which means these filters chop
off very, very rapidly.

1372
01:20:18,000 --> 01:20:19,960
Is that hard to build?

1373
01:20:19,960 --> 01:20:22,420
Doesn't make any difference.

1374
01:20:22,420 --> 01:20:25,370
I mean, these days, anything
which you can figure out how

1375
01:20:25,370 --> 01:20:28,550
to compute, you can put it on
a little chip and it costs

1376
01:20:28,550 --> 01:20:30,720
nothing if you make
enough of them.

1377
01:20:30,720 --> 01:20:34,540
So you want to raise the cosine
filter which cuts off

1378
01:20:34,540 --> 01:20:35,830
at 15%, fine.

1379
01:20:35,830 --> 01:20:37,950
Somebody spends a year
designing it.

1380
01:20:37,950 --> 01:20:42,400
And then you cookie-cut
it forever after.

1381
01:20:42,400 --> 01:20:45,240
So it doesn't cost anything.

1382
01:20:45,240 --> 01:20:51,340
And, well, g of t also has an
inverse transform which we

1383
01:20:51,340 --> 01:20:52,590
won't worry about.