1
00:00:00,040 --> 00:00:02,470
The following content is
provided under a Creative

2
00:00:02,470 --> 00:00:03,880
Commons license.

3
00:00:03,880 --> 00:00:06,920
Your support will help MIT
OpenCourseWare continue to

4
00:00:06,920 --> 00:00:10,570
offer high quality educational
resources for free.

5
00:00:10,570 --> 00:00:13,470
To make a donation, or view
additional materials from

6
00:00:13,470 --> 00:00:19,290
hundreds of MIT courses, visit
MIT OpenCourseWare at

7
00:00:19,290 --> 00:00:55,940
ocw.mit.edu

8
00:00:55,940 --> 00:00:58,420
PROFESSOR: In the last
lecture, we began the

9
00:00:58,420 --> 00:01:00,360
discussion of modulation.

10
00:01:00,360 --> 00:01:03,930
And in particular, what
we focused on, was the

11
00:01:03,930 --> 00:01:05,710
continuous time case.

12
00:01:05,710 --> 00:01:08,420
And in talking about continuous
time modulation,

13
00:01:08,420 --> 00:01:09,950
we've covered a number
of topics.

14
00:01:09,950 --> 00:01:15,330
We talked about the properties
and analysis of modulation

15
00:01:15,330 --> 00:01:20,160
when we had a complex
exponential carrier signal.

16
00:01:20,160 --> 00:01:23,960
We talked about the properties
and analysis in the case of a

17
00:01:23,960 --> 00:01:26,130
sinusoidal carrier.

18
00:01:26,130 --> 00:01:31,180
And in that context and related
to the application

19
00:01:31,180 --> 00:01:33,900
associated with communications,
we talked

20
00:01:33,900 --> 00:01:39,660
about synchronous modulation,
asynchronous modulation and

21
00:01:39,660 --> 00:01:45,360
also the notion of single
side band modulation.

22
00:01:45,360 --> 00:01:49,960
In the lecture today, there are
two issues that I'd like

23
00:01:49,960 --> 00:01:51,920
to address, broad topics.

24
00:01:51,920 --> 00:01:57,270
One is a parallel discussion,
particularly, as associated

25
00:01:57,270 --> 00:02:00,940
with complex exponential and
sinusoidal modulation for

26
00:02:00,940 --> 00:02:03,260
discrete time signals.

27
00:02:03,260 --> 00:02:07,940
And the second is the
introduction and analysis of

28
00:02:07,940 --> 00:02:11,710
another kind of carriers,
specifically a pulse kind of

29
00:02:11,710 --> 00:02:15,680
carrier in continuous time
leading to the notions of

30
00:02:15,680 --> 00:02:19,150
pulse amplitude modulation
and, eventually, a very

31
00:02:19,150 --> 00:02:22,990
powerful theorem and result
called the sampling theorem.

32
00:02:22,990 --> 00:02:27,100
Well, let me begin the lecture,
though, focusing on

33
00:02:27,100 --> 00:02:31,380
the discrete time modulation
to essentially draw your

34
00:02:31,380 --> 00:02:36,090
attention to the fact that the
analysis in discrete time very

35
00:02:36,090 --> 00:02:40,200
much parallels the analysis
in continuous time.

36
00:02:40,200 --> 00:02:44,330
Well, let's consider, in the
discrete time case, just as we

37
00:02:44,330 --> 00:02:49,810
had in continuous time, a signal
modulating a carrier

38
00:02:49,810 --> 00:02:56,170
signal and the resulting
modulated signal is y of n.

39
00:02:56,170 --> 00:03:01,510
And it was in continuous time
the modulation property

40
00:03:01,510 --> 00:03:06,460
associated with the Fourier
Transform that provided the

41
00:03:06,460 --> 00:03:08,340
basis for the analysis.

42
00:03:08,340 --> 00:03:10,610
And exactly the same thing
is true in the

43
00:03:10,610 --> 00:03:12,120
discrete time case.

44
00:03:12,120 --> 00:03:16,200
In particular, what we have
in discrete time, is the

45
00:03:16,200 --> 00:03:19,050
modulation property as it
relates to the Fourier

46
00:03:19,050 --> 00:03:23,820
Transform, which tells us that
the Fourier Transform of the

47
00:03:23,820 --> 00:03:30,080
modulated signal is the
convolution of the Fourier

48
00:03:30,080 --> 00:03:34,840
Transform of the carrier and the
Fourier Transform of the

49
00:03:34,840 --> 00:03:36,460
modulated signal.

50
00:03:36,460 --> 00:03:41,070
And the only real difference at
issue here is that, in the

51
00:03:41,070 --> 00:03:45,590
discrete time case, what we're
talking about is a periodic

52
00:03:45,590 --> 00:03:47,970
convolution because
the specter,

53
00:03:47,970 --> 00:03:49,490
of course is periodic.

54
00:03:49,490 --> 00:03:53,080
Whereas, in the continuous time
case, it was an aperiodic

55
00:03:53,080 --> 00:03:55,010
convolution.

56
00:03:55,010 --> 00:03:58,860
So let's parallel the
discussion, and in particular,

57
00:03:58,860 --> 00:04:05,300
what we'll focus on is, first,
a complex exponential carrier

58
00:04:05,300 --> 00:04:08,820
and second a sinusoidal
carrier.

59
00:04:08,820 --> 00:04:11,470
And we'll see how this parallels
our discussion in

60
00:04:11,470 --> 00:04:16,029
continuous time, and we'll make
fairly brief reference as

61
00:04:16,029 --> 00:04:20,350
we introduce the pulse carrier
for continuous time.

62
00:04:20,350 --> 00:04:22,900
We'll make very brief reference
to the pulse carrier

63
00:04:22,900 --> 00:04:26,790
for discrete time indicating
that, again, the analysis and

64
00:04:26,790 --> 00:04:30,280
discrete time and continuous
time is very parallel.

65
00:04:30,280 --> 00:04:33,990
So let's, first, consider
complex exponential and

66
00:04:33,990 --> 00:04:37,650
sinusoidal carriers for the
discrete time case,

67
00:04:37,650 --> 00:04:41,960
emphasizing the very strong
parallel and similarity

68
00:04:41,960 --> 00:04:46,170
between discrete time
and continuous time.

69
00:04:46,170 --> 00:04:50,740
Well we have, once, again
the modulation property.

70
00:04:50,740 --> 00:04:55,000
And the modulation property
tells us that the spectrum of

71
00:04:55,000 --> 00:05:00,280
the modulated signal is the
periodic convolution of the

72
00:05:00,280 --> 00:05:02,120
two spectra.

73
00:05:02,120 --> 00:05:06,200
And let's consider, for
example, an input, or

74
00:05:06,200 --> 00:05:09,480
modulating spectrum, as
I've indicated here.

75
00:05:09,480 --> 00:05:14,530
And since we want to consider,
first of all, a complex

76
00:05:14,530 --> 00:05:19,670
exponential carrier, we'll
consider the case of c of n

77
00:05:19,670 --> 00:05:23,940
equal to e to the
j omega sub cn.

78
00:05:23,940 --> 00:05:29,040
And let me stress, by the way,
as I did in the continuous

79
00:05:29,040 --> 00:05:33,510
time case, that I'll tend to
suppress the phase angle

80
00:05:33,510 --> 00:05:35,260
which, of course, can
be associated

81
00:05:35,260 --> 00:05:37,610
with the carrier also.

82
00:05:37,610 --> 00:05:41,260
All right, so we have,
then, the spectrum of

83
00:05:41,260 --> 00:05:43,320
the modulated signal.

84
00:05:43,320 --> 00:05:46,130
The spectra, the carrier
signal, if this is the

85
00:05:46,130 --> 00:05:50,160
carrier, then it's spectrum is
an impulse train, and that

86
00:05:50,160 --> 00:05:54,690
impulse train, I've
indicated here.

87
00:05:54,690 --> 00:05:59,110
And let me stress, also, that in
the discrete time case, of

88
00:05:59,110 --> 00:06:03,820
course, these spectra and all
of the spectra involved, are

89
00:06:03,820 --> 00:06:06,500
periodic with a period
of 2 pi.

90
00:06:06,500 --> 00:06:10,000
So this then is the spectrum
of the carrier signal.

91
00:06:10,000 --> 00:06:12,440
This is the spectrum of
the input signal.

92
00:06:12,440 --> 00:06:15,730
The periodic convolution of
these two is the spectrum the

93
00:06:15,730 --> 00:06:17,360
modulated signal.

94
00:06:17,360 --> 00:06:22,940
And the result is, then, this
spectrum shifted to a center

95
00:06:22,940 --> 00:06:28,170
frequency, which is the carrier
frequency omega sub c.

96
00:06:28,170 --> 00:06:34,120
So the result of modulation with
a complex exponential is

97
00:06:34,120 --> 00:06:39,940
a straightforward shift of the
spectrum so that what occurred

98
00:06:39,940 --> 00:06:43,820
around zero frequency now occurs
around the frequency

99
00:06:43,820 --> 00:06:47,040
omega sub c.

100
00:06:47,040 --> 00:06:51,670
Now, in the continuous time
case, we demodulated, when we

101
00:06:51,670 --> 00:06:55,600
had a complex exponential
carrier, we demodulated by,

102
00:06:55,600 --> 00:06:58,930
essentially, just shifting
the spectrum back.

103
00:06:58,930 --> 00:07:01,950
And in fact, in the discrete
time case, were able to do

104
00:07:01,950 --> 00:07:04,180
exactly the same thing.

105
00:07:04,180 --> 00:07:11,660
So if we were to replace c of n
which is either j omega sub

106
00:07:11,660 --> 00:07:17,490
cn by c of n equals e to the
minus j omega sub cn, the

107
00:07:17,490 --> 00:07:23,920
resulting spectra would be an
impulse train, as I indicate

108
00:07:23,920 --> 00:07:32,340
here, and the result of multiply
y of n by that new

109
00:07:32,340 --> 00:07:36,240
carrier, in the frequency domain
as a convolution of

110
00:07:36,240 --> 00:07:40,300
these two, and it's relatively
straightforward to verify that

111
00:07:40,300 --> 00:07:44,160
if you convolve these with a
periodic convolution, then

112
00:07:44,160 --> 00:07:47,300
that will get us back to the
original spectrum that we

113
00:07:47,300 --> 00:07:49,460
started with.

114
00:07:49,460 --> 00:07:52,270
So what's happened in the
discrete time case, with the

115
00:07:52,270 --> 00:07:55,970
complex exponential, is
exactly the same as in

116
00:07:55,970 --> 00:07:57,420
continuous time.

117
00:07:57,420 --> 00:08:00,040
Namely, we modulate that
corresponds to

118
00:08:00,040 --> 00:08:01,470
shifting the spectrum.

119
00:08:01,470 --> 00:08:05,960
We demodulate by multiplying
by the complex conjugate of

120
00:08:05,960 --> 00:08:10,710
the original modulated carrier
and that shifts the spectrum

121
00:08:10,710 --> 00:08:14,290
back to where it
was originally.

122
00:08:14,290 --> 00:08:14,760
OK.

123
00:08:14,760 --> 00:08:18,540
Now let's consider the case
of a sinusoidal carrier in

124
00:08:18,540 --> 00:08:19,910
discrete time.

125
00:08:19,910 --> 00:08:24,390
And again, things very much
parallel what we saw in

126
00:08:24,390 --> 00:08:26,190
continuous time.

127
00:08:26,190 --> 00:08:31,460
And again, as we look at the
spectra, I will choose a phase

128
00:08:31,460 --> 00:08:35,770
angle of zero, mainly for
notational and analytical

129
00:08:35,770 --> 00:08:37,460
convenience.

130
00:08:37,460 --> 00:08:42,190
So in this case, now, rather
than a carrier signal, which

131
00:08:42,190 --> 00:08:45,740
is a single complex exponential,
it's now a

132
00:08:45,740 --> 00:08:49,470
sinusoidal carrier and the
sinusoidal carrier is the sum

133
00:08:49,470 --> 00:08:52,320
of two complex exponential.

134
00:08:52,320 --> 00:08:56,960
And so if we consider a
modulated spectrum, that is

135
00:08:56,960 --> 00:09:01,510
the spectrum of x of n,
something of the type that I

136
00:09:01,510 --> 00:09:09,430
indicate here, and the spectrum
of the carrier, now,

137
00:09:09,430 --> 00:09:12,460
since the carrier is sinusoidal
rather than a

138
00:09:12,460 --> 00:09:16,750
complex exponential consists of
two impulses, one at plus

139
00:09:16,750 --> 00:09:22,070
omega sub c and one at minus
omega sub c, convolving this

140
00:09:22,070 --> 00:09:26,440
spectrum with this spectrum
gives us a replication of x of

141
00:09:26,440 --> 00:09:32,450
omega around plus and
minus omega sub c.

142
00:09:32,450 --> 00:09:36,800
And incidentally, with an
amplitude change of a half.

143
00:09:36,800 --> 00:09:41,540
So again, things have
worked as they did

144
00:09:41,540 --> 00:09:43,640
in continuous time.

145
00:09:43,640 --> 00:09:47,690
In continuous time or in
discrete time, modulating with

146
00:09:47,690 --> 00:09:52,590
a sinusoidal carrier would
correspond to a replication of

147
00:09:52,590 --> 00:09:56,800
the spectrum around, plus the
carrier frequency and a

148
00:09:56,800 --> 00:09:59,450
replication of the spectrum
around minus the carrier

149
00:09:59,450 --> 00:10:05,350
frequency, in both cases, as
long as the carrier frequency

150
00:10:05,350 --> 00:10:09,060
is large enough compared with
the bandwidth of the signal so

151
00:10:09,060 --> 00:10:13,920
that those two replication
don't overlap, then it's

152
00:10:13,920 --> 00:10:17,850
reasonable to suppose that we
should be able to recover the

153
00:10:17,850 --> 00:10:19,330
original signal.

154
00:10:19,330 --> 00:10:25,470
Well, in fact, to demodulate in
the discrete time case, we

155
00:10:25,470 --> 00:10:30,170
would again follow very much
the strategy that we did in

156
00:10:30,170 --> 00:10:32,080
continuous time.

157
00:10:32,080 --> 00:10:37,180
In particular, let's consider
demodulating by taking the

158
00:10:37,180 --> 00:10:40,040
modulated signal and, again,
putting that through a

159
00:10:40,040 --> 00:10:43,220
modulator, again, with
the carrier which is

160
00:10:43,220 --> 00:10:45,690
cosine omega sub cn.

161
00:10:45,690 --> 00:10:50,700
If we do that, we have a
demodulator or what will turn

162
00:10:50,700 --> 00:10:53,010
out to be part of a demodulator,
as I indicate

163
00:10:53,010 --> 00:11:00,320
here, the spectrum of the input
signal is, as I had just

164
00:11:00,320 --> 00:11:04,040
developed, a replication of the
original spectrum around

165
00:11:04,040 --> 00:11:09,760
plus and minus omega sub c with
an amplitude of a half.

166
00:11:09,760 --> 00:11:14,410
When this is, again, convolved
with the spectrum of the

167
00:11:14,410 --> 00:11:18,860
carrier, then we get a
replication of the original

168
00:11:18,860 --> 00:11:25,650
spectrum, first around zero
frequency, as I indicate here,

169
00:11:25,650 --> 00:11:30,680
and then around twice the
carrier frequency and minus

170
00:11:30,680 --> 00:11:33,010
twice the carrier frequency.

171
00:11:33,010 --> 00:11:37,030
And as long as the carrier
frequency is large enough

172
00:11:37,030 --> 00:11:42,610
compared with the width of the
original signal, then, as you

173
00:11:42,610 --> 00:11:48,360
can see, by extracting this part
of the spectrum with a

174
00:11:48,360 --> 00:11:51,780
low pass filter, we can, in
principle, recover the

175
00:11:51,780 --> 00:11:54,810
spectrum associated with
the original signal.

176
00:11:54,810 --> 00:11:58,060
And again, just as in continuous
time, because this

177
00:11:58,060 --> 00:12:02,860
amplitude is a half, we would
want to choose, for scaling

178
00:12:02,860 --> 00:12:08,850
purposes, a low pass filter
amplitude which is 2 to

179
00:12:08,850 --> 00:12:12,630
compensate for this
factor of a half.

180
00:12:12,630 --> 00:12:18,400
So once again things work out
basically the same way as they

181
00:12:18,400 --> 00:12:20,350
had in continuous time.

182
00:12:20,350 --> 00:12:26,250
We have sinusoidal modulation
which consists of using a

183
00:12:26,250 --> 00:12:28,010
sinusoidal carrier.

184
00:12:28,010 --> 00:12:33,100
And we have the demodulator
which consists of taking a

185
00:12:33,100 --> 00:12:38,260
modulated signal, multiplying
by the carrier, and then

186
00:12:38,260 --> 00:12:44,650
processing that with a low pass
filter to extract the

187
00:12:44,650 --> 00:12:49,070
portion of the spectrum, which
is around zero frequency, as I

188
00:12:49,070 --> 00:12:55,220
indicate in the spectrum below
and the result, then, that

189
00:12:55,220 --> 00:12:59,730
this low pass filter having
a gain of 2 is that we've

190
00:12:59,730 --> 00:13:06,020
recovered the original spectrum,
x of omega, which is

191
00:13:06,020 --> 00:13:09,390
the spectrum that
we started with.

192
00:13:09,390 --> 00:13:12,060
Now several other things
to stress.

193
00:13:12,060 --> 00:13:17,310
This is a fairly quick tour
through sinusoidal modulation

194
00:13:17,310 --> 00:13:19,000
for discrete time.

195
00:13:19,000 --> 00:13:22,720
There are very similar issues
that arise in the discrete

196
00:13:22,720 --> 00:13:27,040
time case in terms of having
phase synchronization and

197
00:13:27,040 --> 00:13:29,820
frequency synchronization
between the modulator and

198
00:13:29,820 --> 00:13:31,110
demodulator.

199
00:13:31,110 --> 00:13:34,120
And we had discussed that in a
fair amount of detail for the

200
00:13:34,120 --> 00:13:36,900
continuous time case.

201
00:13:36,900 --> 00:13:40,680
In some sense, in practical
terms, that becomes much more

202
00:13:40,680 --> 00:13:45,860
of an issue in continuous time
than it does in discrete time,

203
00:13:45,860 --> 00:13:50,550
in part, because synchronization
between a

204
00:13:50,550 --> 00:13:55,050
modulator and demodulator is
often much harder in a

205
00:13:55,050 --> 00:13:59,570
continuous time system, which
is essentially an analog

206
00:13:59,570 --> 00:14:03,490
system as compared with
a digital system.

207
00:14:03,490 --> 00:14:06,520
Another very important reason
and it's important to stress

208
00:14:06,520 --> 00:14:11,550
this at the outset is that,
whereas the theory involving

209
00:14:11,550 --> 00:14:15,750
the use of complex exponential
and sinusoidal modulation

210
00:14:15,750 --> 00:14:19,480
parallels very strongly in the
continuous time and the

211
00:14:19,480 --> 00:14:21,240
discrete time case.

212
00:14:21,240 --> 00:14:25,350
In practical terms, it has
much more significance in

213
00:14:25,350 --> 00:14:28,810
continuous time than it
does in discrete time.

214
00:14:28,810 --> 00:14:33,140
That is, the notion called
sinusidal modulation, in the

215
00:14:33,140 --> 00:14:36,410
context of communication
systems, is extremely

216
00:14:36,410 --> 00:14:40,740
important for continuous time
systems, and less so in

217
00:14:40,740 --> 00:14:42,720
discrete time systems.

218
00:14:42,720 --> 00:14:47,690
Now as a preview of a point to
be raised later on, I should

219
00:14:47,690 --> 00:14:52,040
modify that slightly with the
statement that one very

220
00:14:52,040 --> 00:14:56,700
important place in which
sinusoidal modulation in a

221
00:14:56,700 --> 00:15:01,420
discrete time context arises,
is in a class of systems

222
00:15:01,420 --> 00:15:06,410
called transmultiplexers or
transmodulation systems.

223
00:15:06,410 --> 00:15:11,190
And this surface is basically
because so many communication

224
00:15:11,190 --> 00:15:16,080
systems are now becoming digital
and, specifically,

225
00:15:16,080 --> 00:15:18,400
discrete time, although the
actual transmission is

226
00:15:18,400 --> 00:15:22,510
continuous time, the signal
processing manipulation and

227
00:15:22,510 --> 00:15:25,080
switching is discrete time.

228
00:15:25,080 --> 00:15:28,510
And so, in fact, it turns out
to be very important and

229
00:15:28,510 --> 00:15:35,200
useful to take a discrete time
representation of the analog

230
00:15:35,200 --> 00:15:39,010
signals or continuous time
signals and, in a discrete

231
00:15:39,010 --> 00:15:42,380
time, or digital representation,
to convert

232
00:15:42,380 --> 00:15:46,240
them from one modulation scheme
or one multiplexing

233
00:15:46,240 --> 00:15:48,070
scheme to another.

234
00:15:48,070 --> 00:15:53,280
And although I said a lot there
that really requires

235
00:15:53,280 --> 00:16:00,900
much more detail to develop in
any sense at all, you should

236
00:16:00,900 --> 00:16:06,530
get the notion that discrete
time modulation systems become

237
00:16:06,530 --> 00:16:09,560
very important, in
part, because of

238
00:16:09,560 --> 00:16:13,110
implementational issues.

239
00:16:13,110 --> 00:16:17,720
OK, now, there is another
application that we have

240
00:16:17,720 --> 00:16:20,820
discussed for both continuous
time and actually, previously,

241
00:16:20,820 --> 00:16:25,510
for discrete time, amplitude
modulation with sinusoidal

242
00:16:25,510 --> 00:16:27,630
complex exponential carriers.

243
00:16:27,630 --> 00:16:30,800
And let me just remind you of
that, because, in fact, it

244
00:16:30,800 --> 00:16:34,930
becomes a very important
one in the case of

245
00:16:34,930 --> 00:16:36,980
discrete time systems.

246
00:16:36,980 --> 00:16:42,360
And that is the notion of using
modulation together with

247
00:16:42,360 --> 00:16:48,170
fixed filtering to implement a
filter, which either has a

248
00:16:48,170 --> 00:16:52,940
variable cut off or converts,
let's say, a low pass filter

249
00:16:52,940 --> 00:16:54,580
to a high pass filter.

250
00:16:54,580 --> 00:16:57,240
We had originally talked about
this when we introduce the

251
00:16:57,240 --> 00:17:02,260
modulation property in the
context of converting a low

252
00:17:02,260 --> 00:17:05,579
pass filter to a high
pass filter.

253
00:17:05,579 --> 00:17:12,180
And the notion was that, if we
modulate the signal with a

254
00:17:12,180 --> 00:17:15,520
carrier which is minus 1 to the
n, and that's just simply

255
00:17:15,520 --> 00:17:19,540
a complex exponential or
sinusoidal carrier with a

256
00:17:19,540 --> 00:17:25,240
carrier frequency of pi, then
that, in effect, interchanges

257
00:17:25,240 --> 00:17:28,000
the low frequencies and
the high frequencies.

258
00:17:28,000 --> 00:17:34,450
And if, after modulation, that
is processed with a low pass

259
00:17:34,450 --> 00:17:42,150
filter, and then the result is
demodulated, using exactly the

260
00:17:42,150 --> 00:17:47,840
same carrier, namely a carrier
which is minus 1 to the n,

261
00:17:47,840 --> 00:17:52,900
then the effect of that is
equivalent to high pass

262
00:17:52,900 --> 00:17:56,170
filtering on the original
signal.

263
00:17:56,170 --> 00:18:00,210
And a generalization of that
would involve, instead of this

264
00:18:00,210 --> 00:18:04,530
specific choice of minus 1 to
the n, would involve a choice,

265
00:18:04,530 --> 00:18:08,830
in general, of e to the j omega
sub cn, that is a more

266
00:18:08,830 --> 00:18:14,000
general carrier frequency, and
a demodulator which is e to

267
00:18:14,000 --> 00:18:16,590
the minus j omega sub cn.

268
00:18:16,590 --> 00:18:20,360
And as I've represented it here,
and as we had talked

269
00:18:20,360 --> 00:18:23,780
about it when we talked about
the modulation property for

270
00:18:23,780 --> 00:18:27,020
discrete time signals, we had
specifically chosen the

271
00:18:27,020 --> 00:18:30,060
conversion of a low pass
to a high pass filter.

272
00:18:30,060 --> 00:18:34,530
Well, let me continue the
review of that just by

273
00:18:34,530 --> 00:18:37,090
reminding you of the details
of what happens with the

274
00:18:37,090 --> 00:18:43,390
spectra, and, specifically, the
notion, if we take this

275
00:18:43,390 --> 00:18:48,200
particular case of omega sub
c is equal to pi, or

276
00:18:48,200 --> 00:18:51,950
equivalently, a carrier signal
which is minus 1 to the n,

277
00:18:51,950 --> 00:19:01,090
then if we have the original
spectra and the spectrum of

278
00:19:01,090 --> 00:19:06,100
the carrier signal, the spectrum
of the carrier signal

279
00:19:06,100 --> 00:19:10,690
convolved with this spectrum
will then, in effect, shift

280
00:19:10,690 --> 00:19:13,900
this by pi.

281
00:19:13,900 --> 00:19:19,410
And so, after modulating, the
result that we have is a shift

282
00:19:19,410 --> 00:19:23,260
of that spectrum so that what
happened in low frequencies

283
00:19:23,260 --> 00:19:26,740
now happens at high frequencies,
namely around pi,

284
00:19:26,740 --> 00:19:32,280
and what happened at high
frequencies now happens at low

285
00:19:32,280 --> 00:19:33,900
frequencies.

286
00:19:33,900 --> 00:19:38,480
Well, if that's processed now,
with a low pass filter, and

287
00:19:38,480 --> 00:19:42,690
this dashed line indicates the
low pass filter, then the

288
00:19:42,690 --> 00:19:47,090
result that we get is shown
here, where we've extracted

289
00:19:47,090 --> 00:19:50,630
the low frequency portion
of the modulated signal.

290
00:19:50,630 --> 00:19:56,990
And now when we modulate or
demodulate back, then this

291
00:19:56,990 --> 00:20:00,640
spectrum is shifted back
to where it belongs.

292
00:20:00,640 --> 00:20:06,580
Namely, it's shifted back to
be centered around minus pi

293
00:20:06,580 --> 00:20:08,780
and around plus pi.

294
00:20:08,780 --> 00:20:13,660
So if we just compare this
spectrum with the original

295
00:20:13,660 --> 00:20:18,240
spectrum at the top, what we
can see is that, in effect,

296
00:20:18,240 --> 00:20:23,690
what we've done is to extract
a portion of the spectrum

297
00:20:23,690 --> 00:20:28,380
equivalent to processing with
a high pass filter.

298
00:20:28,380 --> 00:20:32,350
And, again, this is very similar
to what we did in

299
00:20:32,350 --> 00:20:38,090
continuous time and all of the
analytical processes and

300
00:20:38,090 --> 00:20:41,180
convolution involved are
very much the same.

301
00:20:41,180 --> 00:20:43,610
Really, the biggest difference
between continuous time

302
00:20:43,610 --> 00:20:47,520
discrete time has to do, not
so much with the details of

303
00:20:47,520 --> 00:20:53,670
the analysis, but perhaps has
more to do with issues of

304
00:20:53,670 --> 00:20:54,920
practical applications.

305
00:20:59,130 --> 00:21:05,020
OK, well, so what we've done,
so far, for continuous time

306
00:21:05,020 --> 00:21:10,250
and discrete time, is to talk
about modulation, amplitude

307
00:21:10,250 --> 00:21:13,730
modulation with complex
exponential

308
00:21:13,730 --> 00:21:15,540
and sinusoidal carriers.

309
00:21:15,540 --> 00:21:19,010
We saw that the analysis is
very similar, although

310
00:21:19,010 --> 00:21:21,190
applications are slightly
different.

311
00:21:21,190 --> 00:21:26,480
And now what I'd like to turn
to is a different choice of

312
00:21:26,480 --> 00:21:27,650
carrier signal.

313
00:21:27,650 --> 00:21:31,400
And the carrier signal, in this
particular case, is a

314
00:21:31,400 --> 00:21:36,450
pulse train rather than
a sinusoidal signal.

315
00:21:36,450 --> 00:21:42,930
Now the idea is the following.

316
00:21:42,930 --> 00:21:47,700
In general, of course, the
modulator consists of all of

317
00:21:47,700 --> 00:21:52,490
multiplying x of t by whatever
the carrier signal is.

318
00:21:52,490 --> 00:21:55,960
And previously, we've talked
about a carrier signal which

319
00:21:55,960 --> 00:21:58,435
is sinusoidal signal.

320
00:21:58,435 --> 00:22:00,740
The carrier signal that we
want to consider now is a

321
00:22:00,740 --> 00:22:05,860
carrier signal which, in
fact, is a pulse train.

322
00:22:05,860 --> 00:22:10,530
And so, in fact, what we want
to do is multiply the input

323
00:22:10,530 --> 00:22:17,160
signal by a pulse train and, in
effect, then, the modulated

324
00:22:17,160 --> 00:22:22,070
signal consists of the original
signal, simply with

325
00:22:22,070 --> 00:22:25,250
time slices pulled out of it,
as I've indicated in the

326
00:22:25,250 --> 00:22:27,030
bottom curve.

327
00:22:27,030 --> 00:22:33,930
So what we have now is a
modulated signal that is a

328
00:22:33,930 --> 00:22:40,880
chopped or sliced version of the
original waveform and that

329
00:22:40,880 --> 00:22:46,610
is what's referred to as pulse
amplitude modulation.

330
00:22:46,610 --> 00:22:49,430
Now it seems like it's
kind of a crazy idea.

331
00:22:49,430 --> 00:22:55,000
The idea is to chop out slices
of the wave form and hope that

332
00:22:55,000 --> 00:22:57,680
you could put things back
together again.

333
00:22:57,680 --> 00:23:01,480
And the amazing thing about it,
as we'll see, is that, in

334
00:23:01,480 --> 00:23:06,140
fact, under fairly broad
general and applicable

335
00:23:06,140 --> 00:23:10,800
conditions, you really can put
the waveform back together

336
00:23:10,800 --> 00:23:14,720
again if you just have
these time slices.

337
00:23:14,720 --> 00:23:20,350
Not only that, but that basic
notion, as we'll see, is

338
00:23:20,350 --> 00:23:22,820
independent, in fact,
of what the width of

339
00:23:22,820 --> 00:23:24,370
those time slices are.

340
00:23:24,370 --> 00:23:25,900
In fact the width
can go to zero.

341
00:23:25,900 --> 00:23:29,530
And, in fact, we're going to
make it go to zero, and really

342
00:23:29,530 --> 00:23:32,920
only dependent on what
the frequency of

343
00:23:32,920 --> 00:23:35,700
the pulse train is.

344
00:23:35,700 --> 00:23:38,950
So let's explore that
in some detail.

345
00:23:38,950 --> 00:23:45,010
And what we want to look at is
the analysis, but let me,

346
00:23:45,010 --> 00:23:48,680
first, just comment, very
briefly, that all of the

347
00:23:48,680 --> 00:23:51,520
analysis we go through, as has
been true in the case of

348
00:23:51,520 --> 00:23:55,140
sinusoidal modulation, all of
the analysis then we go

349
00:23:55,140 --> 00:24:00,660
through holds just as well
with, essentially minor

350
00:24:00,660 --> 00:24:05,510
analytical modifications, to
discrete time pulse amplitude

351
00:24:05,510 --> 00:24:09,190
modulation as it does to
continuous time pulse

352
00:24:09,190 --> 00:24:10,920
amplitude modulation.

353
00:24:10,920 --> 00:24:14,390
And so we'll really only go
through this in terms of

354
00:24:14,390 --> 00:24:16,900
tracking the wave forms
and spectra for the

355
00:24:16,900 --> 00:24:18,320
continuous time case.

356
00:24:18,320 --> 00:24:22,370
But bear in mind that the
results are basically similar

357
00:24:22,370 --> 00:24:25,940
for discrete time.

358
00:24:25,940 --> 00:24:30,350
OK, well, let's see how so we
get the basic result that we

359
00:24:30,350 --> 00:24:33,110
want to get.

360
00:24:33,110 --> 00:24:38,860
What we have is modulated signal
which is a pulse train,

361
00:24:38,860 --> 00:24:42,620
basically a square wave, and
as we've seen in previous

362
00:24:42,620 --> 00:24:47,870
lectures, the spectra or Fourier
transform associated

363
00:24:47,870 --> 00:24:50,420
with that is an impulse train.

364
00:24:50,420 --> 00:24:54,840
And the envelope of that impulse
train is on the form

365
00:24:54,840 --> 00:24:57,430
of a sine x over x function.

366
00:24:57,430 --> 00:25:00,220
The Fourier transform
is impulses.

367
00:25:00,220 --> 00:25:05,970
And the spacing of the impulses
is associated with

368
00:25:05,970 --> 00:25:09,470
the fundamental frequency of
the pulse train and that's

369
00:25:09,470 --> 00:25:10,790
omega sub p.

370
00:25:10,790 --> 00:25:13,630
So omega sub p is pi divided
by the period

371
00:25:13,630 --> 00:25:15,570
of the pulse train.

372
00:25:15,570 --> 00:25:23,350
And the amplitude and shape of
this envelope is dictated by

373
00:25:23,350 --> 00:25:26,570
the parameter delta, which
has to do with how

374
00:25:26,570 --> 00:25:29,340
wide the pulses are.

375
00:25:29,340 --> 00:25:32,100
OK, so we have a
time function.

376
00:25:32,100 --> 00:25:34,750
It's multiplied by
this pulse train.

377
00:25:34,750 --> 00:25:37,780
Now we're talking
continuous time.

378
00:25:37,780 --> 00:25:41,800
So, in the frequency domain, we
have the Fourier transform

379
00:25:41,800 --> 00:25:45,350
of the time function convolved
with this Fourier transform

380
00:25:45,350 --> 00:25:46,720
for the pulse train.

381
00:25:46,720 --> 00:25:48,550
And let's see what
that looks like.

382
00:25:48,550 --> 00:25:52,830
If we were to consider, let's
say, a Fourier transform,

383
00:25:52,830 --> 00:25:57,910
which I have chosen as more or
less a general one, then in

384
00:25:57,910 --> 00:26:03,370
fact, when we convert all of
this with this impulse train,

385
00:26:03,370 --> 00:26:11,240
what we end up with is a
replication of this spectrum

386
00:26:11,240 --> 00:26:16,700
at the places in the frequency
domain where the individual

387
00:26:16,700 --> 00:26:19,260
impulses occurred.

388
00:26:19,260 --> 00:26:23,200
So we can see that this spectrum
is replicated at each

389
00:26:23,200 --> 00:26:25,530
of these locations.

390
00:26:25,530 --> 00:26:33,220
And as long as the frequency
of the pulse train is large

391
00:26:33,220 --> 00:26:38,410
enough, compared with the
maximum frequency in the

392
00:26:38,410 --> 00:26:42,750
original signal, x of t, so
that there's no overlap

393
00:26:42,750 --> 00:26:47,570
between these triangles, then
what you can see, in fact,

394
00:26:47,570 --> 00:26:52,130
somewhat amazingly is that,
simply by low pass filtering

395
00:26:52,130 --> 00:26:57,100
the result, we can get back,
except for amplitude factor,

396
00:26:57,100 --> 00:27:00,490
we can get back to the
original signal.

397
00:27:00,490 --> 00:27:02,870
Now it's amazing.

398
00:27:02,870 --> 00:27:08,470
It really is amazing that all
that this depends on is the

399
00:27:08,470 --> 00:27:12,530
original signal being band
limited and the frequency of

400
00:27:12,530 --> 00:27:15,870
the pulse train being high
enough so that when you

401
00:27:15,870 --> 00:27:19,700
replicate the spectrum the
frequency domain, there's no

402
00:27:19,700 --> 00:27:22,480
overlap between these individual
replications.

403
00:27:22,480 --> 00:27:26,360
And we'll have address that a
little more in a few minutes.

404
00:27:26,360 --> 00:27:30,550
But let me, first of all, point
out that this has a

405
00:27:30,550 --> 00:27:33,300
whole variety of very important
implications.

406
00:27:33,300 --> 00:27:37,350
One is, in the context of
communications, it leads to

407
00:27:37,350 --> 00:27:41,290
another very important
multiplexing scheme for

408
00:27:41,290 --> 00:27:42,510
communications.

409
00:27:42,510 --> 00:27:45,530
We had talked last time about
frequency division

410
00:27:45,530 --> 00:27:49,010
multiplexing, where individual
signals were put into

411
00:27:49,010 --> 00:27:53,250
individual frequencies slots by
choosing different carrier

412
00:27:53,250 --> 00:27:56,790
frequencies for a sinusoidal
modulating signal.

413
00:27:56,790 --> 00:28:05,400
What this suggests is that
what we can put different

414
00:28:05,400 --> 00:28:12,050
signals into, non-overlapping
time slots and, in fact, be

415
00:28:12,050 --> 00:28:14,960
able to recover the original
signals back again.

416
00:28:14,960 --> 00:28:19,460
So in particular, suppose that
I had a signal which I

417
00:28:19,460 --> 00:28:25,070
modulated with a pulse train
and I chose another signal,

418
00:28:25,070 --> 00:28:28,660
modulated with another pulse
train, where the time slot was

419
00:28:28,660 --> 00:28:32,470
different, and I continued
this process.

420
00:28:32,470 --> 00:28:37,330
And after I'd done this with
some number of channels,

421
00:28:37,330 --> 00:28:43,910
simply added all those together
as I indicate here.

422
00:28:43,910 --> 00:28:47,220
Then as long as I knew what time
slots to associate with

423
00:28:47,220 --> 00:28:53,370
what signal, I could get the
original modulated signals

424
00:28:53,370 --> 00:28:54,920
back again.

425
00:28:54,920 --> 00:28:59,590
And then as long as the
frequency of the impulse train

426
00:28:59,590 --> 00:29:05,400
was such that I was able to do
this reconstruction by simply

427
00:29:05,400 --> 00:29:10,100
low pass filtering, then I would
be able to demodulate.

428
00:29:10,100 --> 00:29:12,770
So it's a very different very
important modulation scheme

429
00:29:12,770 --> 00:29:16,890
called time division
multiplexing in contrast to

430
00:29:16,890 --> 00:29:19,060
frequency division multiplexing
as we had talked

431
00:29:19,060 --> 00:29:20,740
about last time.

432
00:29:20,740 --> 00:29:23,750
I had made reference earlier
to the concept of

433
00:29:23,750 --> 00:29:24,570
trans-multiplexing.

434
00:29:24,570 --> 00:29:29,760
And in fact, what happens in
many communication systems is

435
00:29:29,760 --> 00:29:33,530
that the signals are
represented, in fact, in

436
00:29:33,530 --> 00:29:34,760
discrete time.

437
00:29:34,760 --> 00:29:37,380
The analog and continuous time
signals are represented in

438
00:29:37,380 --> 00:29:39,750
discrete time.

439
00:29:39,750 --> 00:29:42,330
And very often the conversion
from frequency division

440
00:29:42,330 --> 00:29:45,970
multiplexing to time division
multiplexing and back is done,

441
00:29:45,970 --> 00:29:50,160
in fact, in the discrete
time domain.

442
00:29:50,160 --> 00:30:01,800
OK, so what we have then, is
the notion that we can

443
00:30:01,800 --> 00:30:08,670
multiply a time function
by a pulse train,

444
00:30:08,670 --> 00:30:10,940
as I indicate here.

445
00:30:10,940 --> 00:30:17,020
And from the output I can, if
the frequency of this pulse

446
00:30:17,020 --> 00:30:19,770
train is high enough in relation
to this bandwidth,

447
00:30:19,770 --> 00:30:25,640
from the output, which consists
of time slices, from

448
00:30:25,640 --> 00:30:31,450
those time slices I can recover
the original signal.

449
00:30:31,450 --> 00:30:37,770
Stressing again the reason it
relates to the spectra, and

450
00:30:37,770 --> 00:30:41,760
the reason is that the original
spectra is simply

451
00:30:41,760 --> 00:30:48,170
replicated at multiples of the
fundamental frequency of the

452
00:30:48,170 --> 00:30:49,420
pulse train.

453
00:30:51,960 --> 00:30:55,320
Now there's a very important
thing to observe here, which

454
00:30:55,320 --> 00:31:01,620
is that the ability to do the
reconstruction is associated

455
00:31:01,620 --> 00:31:03,690
with the notion of whether
we can extract

456
00:31:03,690 --> 00:31:05,360
that central triangle.

457
00:31:05,360 --> 00:31:08,030
I happened to choose a
triangular shape but obviously

458
00:31:08,030 --> 00:31:10,260
I could be talking about any
shape, as long as it's band

459
00:31:10,260 --> 00:31:13,230
limited, the ability
to extract that.

460
00:31:13,230 --> 00:31:19,270
And notice that, in this
modulated output spectrum, the

461
00:31:19,270 --> 00:31:23,800
ability to recover this is
totally independent called

462
00:31:23,800 --> 00:31:26,880
what the value of delta is.

463
00:31:26,880 --> 00:31:35,520
In other words, if we look back
at the modulator, then,

464
00:31:35,520 --> 00:31:40,360
in fact, we can make delta,
the width of these pulses,

465
00:31:40,360 --> 00:31:42,470
arbitrarily small.

466
00:31:42,470 --> 00:31:46,710
And, in theory, that doesn't
affect our ability to do the

467
00:31:46,710 --> 00:31:48,150
reconstruction.

468
00:31:48,150 --> 00:31:50,590
Now in practical
terms it might.

469
00:31:50,590 --> 00:31:55,360
Looking back once more at the
spectrum of the output, notice

470
00:31:55,360 --> 00:31:59,060
that this amplitude is
proportional to delta.

471
00:31:59,060 --> 00:32:03,470
And what that suggests is that,
as we make delta smaller

472
00:32:03,470 --> 00:32:06,320
and smaller, which we might,
in fact, want to do, if you

473
00:32:06,320 --> 00:32:10,370
want to time division multiplex
lots of channels, in

474
00:32:10,370 --> 00:32:12,210
principle, in theory, you could
make it an infinite

475
00:32:12,210 --> 00:32:13,680
number of channels just
by making that

476
00:32:13,680 --> 00:32:15,780
infinitesimally small.

477
00:32:15,780 --> 00:32:19,030
The smaller it is, in
some sense, the less

478
00:32:19,030 --> 00:32:20,140
energy there is.

479
00:32:20,140 --> 00:32:23,340
And again, in practical terms,
this one of those things if

480
00:32:23,340 --> 00:32:27,310
you push down here pops up
there, namely, you eventually

481
00:32:27,310 --> 00:32:30,480
run into issues such
as noise problems.

482
00:32:30,480 --> 00:32:36,510
So, more typically what's done
is to, in fact, eliminate this

483
00:32:36,510 --> 00:32:38,540
scale factor of delta.

484
00:32:38,540 --> 00:32:42,570
And the way that that's
done is very simply.

485
00:32:42,570 --> 00:32:51,600
It's done by choosing the width
of the pulses, and the

486
00:32:51,600 --> 00:32:56,240
height of the pulses, in such
a way that the area is

487
00:32:56,240 --> 00:33:01,890
constant, even as we make delta
get arbitrarily small.

488
00:33:01,890 --> 00:33:07,820
So we can just modify our
argument so that what we're

489
00:33:07,820 --> 00:33:12,890
referring to is a modulated
pulse train, which is a pulse

490
00:33:12,890 --> 00:33:15,360
train with pulses of
width delta and

491
00:33:15,360 --> 00:33:18,900
height, 1 over delta.

492
00:33:18,900 --> 00:33:24,160
In that case, as delta gets
arbitrarily small, then, in

493
00:33:24,160 --> 00:33:29,570
fact, what these rectangles
become are impulses, in which

494
00:33:29,570 --> 00:33:36,290
case, what we're talking about
is a carrier signal which, in

495
00:33:36,290 --> 00:33:39,550
fact, is an impulse train.

496
00:33:39,550 --> 00:33:45,200
And the resulting modulated
signal is an impulse train for

497
00:33:45,200 --> 00:33:50,630
which the amplitudes of the
impulses are proportional to

498
00:33:50,630 --> 00:33:56,290
the original input waveform at
the times at which these

499
00:33:56,290 --> 00:33:59,050
impulses occur.

500
00:33:59,050 --> 00:34:03,510
OK well, let's look at
the analysis of that.

501
00:34:03,510 --> 00:34:15,010
And so now, what we're talking
about, is a spectrum that

502
00:34:15,010 --> 00:34:20,840
consists of the result of the
spectrum we talked about

503
00:34:20,840 --> 00:34:24,900
before with the sine x over x
envelope, except that, now, as

504
00:34:24,900 --> 00:34:28,260
delta goes to zero that
becomes flat.

505
00:34:28,260 --> 00:34:32,000
In other words, the modulated
signal is an impulse train.

506
00:34:32,000 --> 00:34:36,210
And so as we look at the
spectrum of the modulated

507
00:34:36,210 --> 00:34:39,460
signal, that is, then,
an impulse train in

508
00:34:39,460 --> 00:34:41,380
the frequency domain.

509
00:34:41,380 --> 00:34:46,389
The height is proportional to
the frequency of the impulse

510
00:34:46,389 --> 00:34:50,659
train and omega sub s now
denotes the frequency of the

511
00:34:50,659 --> 00:34:52,170
impulse train.

512
00:34:52,170 --> 00:34:59,090
And the resulting output of the
modulator has a spectrum

513
00:34:59,090 --> 00:35:04,060
which is this original spectrum,
again, replicated

514
00:35:04,060 --> 00:35:08,800
around each of these impulses,
in other words, replicated in

515
00:35:08,800 --> 00:35:12,410
multiples of the sampling
frequency

516
00:35:12,410 --> 00:35:16,410
Now this is very much
identical to the

517
00:35:16,410 --> 00:35:17,460
more general case.

518
00:35:17,460 --> 00:35:21,130
We have this replication
of the spectra.

519
00:35:21,130 --> 00:35:30,220
And as long as the frequency of
the impulse train is large

520
00:35:30,220 --> 00:35:33,840
enough, compared with the
bandwidth of the signal so

521
00:35:33,840 --> 00:35:40,670
that these triangles don't
overlap, I can extract this

522
00:35:40,670 --> 00:35:45,070
portion of the spectrum by low
pass filtering, in fact, would

523
00:35:45,070 --> 00:35:49,340
then give us back the
original signal.

524
00:35:49,340 --> 00:35:54,380
Now if, instead, this frequency
omega sub m is

525
00:35:54,380 --> 00:36:00,060
greater than omega sub s minus
omega sub m, we would have a

526
00:36:00,060 --> 00:36:04,140
spectrum that looked something
more like this.

527
00:36:04,140 --> 00:36:08,640
And what's happened, in this
case, is that, because we have

528
00:36:08,640 --> 00:36:12,980
an overlap here, we've destroyed
the ability to

529
00:36:12,980 --> 00:36:16,430
recover the original signal
from the impulse train.

530
00:36:16,430 --> 00:36:20,750
And that would be true, also
in a more general case, of

531
00:36:20,750 --> 00:36:26,820
pulse amplitude modulation with
pulses of non-zero width.

532
00:36:26,820 --> 00:36:32,520
This effect by the way, is one
that we'll be exploring in

533
00:36:32,520 --> 00:36:35,100
considerably more detail
in the next lecture.

534
00:36:35,100 --> 00:36:38,440
And it's a phenomenon or
distortion refer to as

535
00:36:38,440 --> 00:36:40,540
aliasing which, in fact,
is an important

536
00:36:40,540 --> 00:36:42,090
and interesting topic.

537
00:36:42,090 --> 00:36:45,620
But going back to the case
in which we've chosen the

538
00:36:45,620 --> 00:36:50,070
frequency of the impulse train
high enough, then we would

539
00:36:50,070 --> 00:36:57,300
recover the original signal by
processing it through a low

540
00:36:57,300 --> 00:37:00,050
pass filter.

541
00:37:00,050 --> 00:37:06,480
And in that case, what this
says is, that if we have a

542
00:37:06,480 --> 00:37:13,030
signal, and we modulate it with
an impulse train, if we

543
00:37:13,030 --> 00:37:16,960
then process that impulse train
through an idea low pass

544
00:37:16,960 --> 00:37:20,950
filter, given the right
conditions on the frequency of

545
00:37:20,950 --> 00:37:23,760
impulse train and the bandwidth
of the signal, we

546
00:37:23,760 --> 00:37:27,050
can recover the original
signal back again.

547
00:37:27,050 --> 00:37:32,770
Now let me stress, just going
back to the picture in which

548
00:37:32,770 --> 00:37:37,560
we had done this modulation,
that this process, where the

549
00:37:37,560 --> 00:37:40,880
modulation, where the carrier
signal involves an impulse

550
00:37:40,880 --> 00:37:45,380
train, is often referred
to as sampling.

551
00:37:45,380 --> 00:37:50,290
And what that means,
specifically, is that, if we

552
00:37:50,290 --> 00:37:58,510
notice, this resulting impulse
train is, in fact, a sequence

553
00:37:58,510 --> 00:38:04,190
of samples of the original
continuous time signal.

554
00:38:04,190 --> 00:38:07,930
In other words, what we've
done, in effect, is taken

555
00:38:07,930 --> 00:38:10,940
instantaneous sample
of this wave form.

556
00:38:10,940 --> 00:38:14,560
And the implication is that,
if we do that at a rapid

557
00:38:14,560 --> 00:38:21,190
enough rate in relation to the
bandwidth of the signal, then

558
00:38:21,190 --> 00:38:25,280
we can, in fact, recover the
original signal back again.

559
00:38:25,280 --> 00:38:31,960
And, finally to remind you of
the argument once more, we

560
00:38:31,960 --> 00:38:37,910
have an original signal and
we have its spectrum.

561
00:38:37,910 --> 00:38:40,800
When we've sampled it, and
this is now the sampled

562
00:38:40,800 --> 00:38:45,400
signal, it's an impulse train
whose instantaneous values are

563
00:38:45,400 --> 00:38:49,840
samples of the original
waveform, the spectrum of that

564
00:38:49,840 --> 00:38:53,890
is the original one
replicated.

565
00:38:53,890 --> 00:38:59,110
And when that is processed,
through a low pass filter, to

566
00:38:59,110 --> 00:39:03,500
extract this part of the
spectrum, then, after the low

567
00:39:03,500 --> 00:39:06,530
pass filter, we can recover the

568
00:39:06,530 --> 00:39:07,910
original signal back again.

569
00:39:11,390 --> 00:39:18,650
OK well, in fact, although if
you follow through the spectra

570
00:39:18,650 --> 00:39:22,610
and the wave forms, this all
seems fairly straightforward

571
00:39:22,610 --> 00:39:29,210
and, perhaps or perhaps not,
obvious, it's really worth

572
00:39:29,210 --> 00:39:34,510
reflecting on how amazing
the result really is.

573
00:39:34,510 --> 00:39:39,040
We began this discussion by
talking about modulation.

574
00:39:39,040 --> 00:39:42,030
And in fact modulation and
sinusoidal of modulation is

575
00:39:42,030 --> 00:39:44,410
important in its own right.

576
00:39:44,410 --> 00:39:49,890
We ended the discussion by
talking about first pulse

577
00:39:49,890 --> 00:39:55,210
amplitude modulation, and then
showing how, under the right

578
00:39:55,210 --> 00:40:01,320
set of conditions, you can, in
fact, take a wave form and

579
00:40:01,320 --> 00:40:05,520
sample it with a set of
instantaneous samples.

580
00:40:05,520 --> 00:40:08,820
And that set of instantaneous
samples, in fact, are

581
00:40:08,820 --> 00:40:11,690
sufficient to totally
represent and

582
00:40:11,690 --> 00:40:14,280
reconstruct the signal.

583
00:40:14,280 --> 00:40:19,070
What in fact, the formal
statement that is, is refer to

584
00:40:19,070 --> 00:40:24,030
as the sampling theorem, a very
powerful theorem that

585
00:40:24,030 --> 00:40:28,190
says, if we're given equally
spaced samples of a time

586
00:40:28,190 --> 00:40:36,950
function, and if that time
function is band limited, and

587
00:40:36,950 --> 00:40:45,110
if the bandwidth and if the
sampling frequency is chosen

588
00:40:45,110 --> 00:40:54,100
in the right way, in relation
to the bandwidth, then, in

589
00:40:54,100 --> 00:40:58,990
fact, the original time
function is uniquely

590
00:40:58,990 --> 00:41:02,320
recoverable with a
low pass filter.

591
00:41:05,150 --> 00:41:11,110
Now the sampling theorem is,
I would say, a watershed or

592
00:41:11,110 --> 00:41:14,520
cornerstone of a lot of the
discussion that we've been

593
00:41:14,520 --> 00:41:17,450
having for a whole variety
of reasons.

594
00:41:17,450 --> 00:41:21,620
It, first of all, drops out
almost as a straightforward

595
00:41:21,620 --> 00:41:22,920
obvious statement.

596
00:41:22,920 --> 00:41:28,700
But more importantly what
it says is, if I have a

597
00:41:28,700 --> 00:41:32,160
continuous time signal which
satisfies the right set of

598
00:41:32,160 --> 00:41:38,300
conditions, I could represent it
by what it does at sampling

599
00:41:38,300 --> 00:41:42,200
instance or, equivalently, at
discrete instance of time.

600
00:41:44,950 --> 00:41:49,920
Now what that leads to is
a whole host of things.

601
00:41:49,920 --> 00:41:53,630
One of which is this statement
that says, if we have a

602
00:41:53,630 --> 00:41:58,880
continuous time signal, I could
in fact, represent it as

603
00:41:58,880 --> 00:42:01,090
a discrete time signal.

604
00:42:01,090 --> 00:42:04,890
And I could even think of
processing a continuous time

605
00:42:04,890 --> 00:42:09,520
signal using discrete
time concepts.

606
00:42:09,520 --> 00:42:13,010
And when I'm all done converting
back, through the

607
00:42:13,010 --> 00:42:15,230
power of the sampling theorem,
converting back to a

608
00:42:15,230 --> 00:42:16,870
continuous time signal.

609
00:42:16,870 --> 00:42:21,790
So the sampling theorem provides
us with a very major

610
00:42:21,790 --> 00:42:25,240
important bridge between
continuous time and discrete

611
00:42:25,240 --> 00:42:28,930
time implementations
and ideas.

612
00:42:28,930 --> 00:42:31,950
In the next several lectures,
we will be exploring some of

613
00:42:31,950 --> 00:42:34,140
this in considerable detail.

614
00:42:34,140 --> 00:42:38,860
First, to focus in more, next
time, on some of the specific

615
00:42:38,860 --> 00:42:42,620
issues and distortions
associated with sampling.

616
00:42:42,620 --> 00:42:47,520
And following that, a discussion
of what is referred

617
00:42:47,520 --> 00:42:52,400
to discrete time processing of
continuous time signals.

618
00:42:52,400 --> 00:42:53,650
Thank you.