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[MUSIC PLAYING]

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PROFESSOR: In the
last lecture, we

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00:00:53,340 --> 00:00:56,760
considered the frequency
response of linear shift

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00:00:56,760 --> 00:00:59,130
and variance systems.

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00:00:59,130 --> 00:01:01,350
In this lecture, what
I would like to do

14
00:01:01,350 --> 00:01:06,990
is extend some of those ideas
to the notion of the Fourier

15
00:01:06,990 --> 00:01:12,390
transform, which I'll refer to
as the discrete time Fourier

16
00:01:12,390 --> 00:01:14,280
transform.

17
00:01:14,280 --> 00:01:19,020
So let me begin by
reminding you of one

18
00:01:19,020 --> 00:01:22,360
of the key results
from last time,

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00:01:22,360 --> 00:01:24,720
namely, the notion
of the frequency

20
00:01:24,720 --> 00:01:28,590
response of a linear
shift invariant system.

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00:01:28,590 --> 00:01:31,440
For a linear shift
invariant system,

22
00:01:31,440 --> 00:01:36,090
we saw last time that
one of the key properties

23
00:01:36,090 --> 00:01:40,290
was that for a complex
exponential input,

24
00:01:40,290 --> 00:01:44,460
the output is a complex
exponential sequence

25
00:01:44,460 --> 00:01:50,100
with the same complex frequency,
but with a change in amplitude.

26
00:01:50,100 --> 00:01:52,470
And the change in
amplitude, that

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00:01:52,470 --> 00:01:55,890
is the function h
of e to the j omega,

28
00:01:55,890 --> 00:02:00,740
we refer to as the frequency
response of the system.

29
00:02:00,740 --> 00:02:04,070
The expression that we had
for the frequency response,

30
00:02:04,070 --> 00:02:06,890
in terms of the unit
sample response,

31
00:02:06,890 --> 00:02:11,570
was that the frequency
response h of e to the j omega,

32
00:02:11,570 --> 00:02:15,230
is given by the sum of
h of n, the unit sample

33
00:02:15,230 --> 00:02:20,355
response of the system,
multiplied by e to the minus j

34
00:02:20,355 --> 00:02:21,290
omega n.

35
00:02:21,290 --> 00:02:24,800
So this is a relationship
that tells us,

36
00:02:24,800 --> 00:02:28,040
in terms of the unit sample
response of the system,

37
00:02:28,040 --> 00:02:31,280
how to get the
frequency response.

38
00:02:31,280 --> 00:02:34,970
There were two properties
of the frequency response

39
00:02:34,970 --> 00:02:37,630
that I stressed last time.

40
00:02:37,630 --> 00:02:41,810
One of them was the fact
that the frequency response

41
00:02:41,810 --> 00:02:46,070
is a function of a
continuous variable, omega.

42
00:02:46,070 --> 00:02:50,300
That is, complex
exponential inputs

43
00:02:50,300 --> 00:02:55,790
can have the frequency variable
omega, vary continuously.

44
00:02:55,790 --> 00:02:59,090
Omega is a continuous
variable, whereas n

45
00:02:59,090 --> 00:03:02,160
is the discrete variable.

46
00:03:02,160 --> 00:03:05,840
So this was one
important property.

47
00:03:05,840 --> 00:03:10,220
The second important property is
that the frequency response is

48
00:03:10,220 --> 00:03:13,290
a periodic function of omega.

49
00:03:13,290 --> 00:03:17,510
And the period is equal 2 pi.

50
00:03:17,510 --> 00:03:21,730
H of e to the j omega
is equal to h of e

51
00:03:21,730 --> 00:03:28,810
to the j omega plus 2 pi
k, where k is any integer.

52
00:03:28,810 --> 00:03:34,540
Now one of the sort of heuristic
explanations or justifications

53
00:03:34,540 --> 00:03:37,780
for why the frequency
response is periodic,

54
00:03:37,780 --> 00:03:40,840
as I've tried to indicate
in several lectures,

55
00:03:40,840 --> 00:03:45,670
is basically tied to the fact
that complex exponentials

56
00:03:45,670 --> 00:03:53,170
or sinusoidal inputs, are very
periodically with frequency

57
00:03:53,170 --> 00:03:56,020
over an interval 0 to 2 pi.

58
00:03:56,020 --> 00:03:59,500
That is, once we've looked at
them for omega between 0 and 2

59
00:03:59,500 --> 00:04:02,395
pi, if we go further
than that in frequency,

60
00:04:02,395 --> 00:04:05,710
all we see are the same
complex exponentials

61
00:04:05,710 --> 00:04:09,100
or the same
sinusoids over again.

62
00:04:09,100 --> 00:04:11,650
So that, in essence,
is the reason

63
00:04:11,650 --> 00:04:16,690
why the frequency response is
a periodic function of omega.

64
00:04:16,690 --> 00:04:19,510
Well, one of the
things we'd like to do

65
00:04:19,510 --> 00:04:22,810
is develop an
inverse relationship,

66
00:04:22,810 --> 00:04:27,190
which permits us to get
the unit sample response

67
00:04:27,190 --> 00:04:29,950
from the frequency response.

68
00:04:29,950 --> 00:04:35,920
One heuristic notion
or style of developing

69
00:04:35,920 --> 00:04:38,380
such an inverse
relationship, we can

70
00:04:38,380 --> 00:04:44,980
get by observing that since
the frequency response is

71
00:04:44,980 --> 00:04:47,830
a function of a
continuous variable omega,

72
00:04:47,830 --> 00:04:51,890
and in fact, it's a
periodic function of omega,

73
00:04:51,890 --> 00:04:54,550
then it should have a Fourier
series representation.

74
00:04:54,550 --> 00:04:58,050
That is, it's a continuous
periodic function.

75
00:04:58,050 --> 00:05:00,670
A continuous periodic
function has a Fourier series

76
00:05:00,670 --> 00:05:02,810
representation.

77
00:05:02,810 --> 00:05:07,640
Well in fact, if we
look at this expression,

78
00:05:07,640 --> 00:05:11,620
we see that indeed what
this is is a Fourier series

79
00:05:11,620 --> 00:05:17,140
expansion of this
periodic function in terms

80
00:05:17,140 --> 00:05:19,630
of a complex Fourier series.

81
00:05:19,630 --> 00:05:24,460
The Fourier series expressed in
terms of complex exponentials.

82
00:05:24,460 --> 00:05:26,890
Well, what are the Fourier
series coefficients?

83
00:05:26,890 --> 00:05:31,120
The Fourier series
coefficients are the values

84
00:05:31,120 --> 00:05:35,830
of the unit sample response, in
other words, the values h of n.

85
00:05:35,830 --> 00:05:39,880
So in fact, this has the
form of a Fourier series.

86
00:05:39,880 --> 00:05:42,790
That means that these are
the Fourier coefficients.

87
00:05:42,790 --> 00:05:46,480
That means that we can get these
Fourier coefficients in terms

88
00:05:46,480 --> 00:05:49,630
of h of e to the j
omega, the way we always

89
00:05:49,630 --> 00:05:52,660
get Fourier series
coefficients in terms

90
00:05:52,660 --> 00:05:56,510
of a periodic
continuous function.

91
00:05:56,510 --> 00:05:59,440
Well, the resulting
expression then

92
00:05:59,440 --> 00:06:03,700
is that the Fourier series
coefficients, or equivalently

93
00:06:03,700 --> 00:06:08,080
the unit sample response,
is equal to 1 over 2 pi

94
00:06:08,080 --> 00:06:10,570
times the integral over
a period, which I've

95
00:06:10,570 --> 00:06:15,850
taken as minus pi to plus
pi, of h of e to the j omega,

96
00:06:15,850 --> 00:06:19,600
e to the omega n, d omega.

97
00:06:19,600 --> 00:06:21,860
This is just simply
the expression

98
00:06:21,860 --> 00:06:25,270
for Fourier series
coefficients in terms

99
00:06:25,270 --> 00:06:28,510
of the continuous
periodic function

100
00:06:28,510 --> 00:06:30,250
that we're dealing with.

101
00:06:30,250 --> 00:06:34,990
Well that's, in a sense,
a heuristic derivation.

102
00:06:34,990 --> 00:06:40,720
We can in fact verify that
this expression is valid simply

103
00:06:40,720 --> 00:06:46,450
by substituting in to the
integral, the expression for h

104
00:06:46,450 --> 00:06:48,950
of e to the j omega.

105
00:06:48,950 --> 00:06:51,910
And if we do that,
we end up with,

106
00:06:51,910 --> 00:06:55,060
for the expression for
h of e to the j omega,

107
00:06:55,060 --> 00:07:01,420
we have the sum of h of k,
e to the minus j omega k.

108
00:07:01,420 --> 00:07:07,810
Substituting in, the
result is this integral.

109
00:07:07,810 --> 00:07:13,300
If we interchange the order
of summation and integration,

110
00:07:13,300 --> 00:07:17,480
then the resulting
expression is shown here.

111
00:07:17,480 --> 00:07:22,480
And we can observe
that this integral--

112
00:07:22,480 --> 00:07:28,540
First of all, let's consider
it for n not equal to k.

113
00:07:28,540 --> 00:07:33,130
If n is not equal to k, this
is just simply the integral

114
00:07:33,130 --> 00:07:37,150
over a period of 2 pi
of a complex exponential

115
00:07:37,150 --> 00:07:40,155
whose period is 2 pi.

116
00:07:40,155 --> 00:07:42,030
The integral of the real
part is the integral

117
00:07:42,030 --> 00:07:45,780
of a cosine over an
integral number of periods.

118
00:07:45,780 --> 00:07:47,700
And the integral of
the imaginary part

119
00:07:47,700 --> 00:07:52,210
is the integral of the sine over
an integral number of periods.

120
00:07:52,210 --> 00:07:56,550
Obviously then, this integral,
if n is not equal to k

121
00:07:56,550 --> 00:07:57,825
is going to be 0.

122
00:07:57,825 --> 00:08:02,640
So for n not equal to k,
the value of this integral

123
00:08:02,640 --> 00:08:12,140
is 0, whereas for n equal
to k, this exponent is 0,

124
00:08:12,140 --> 00:08:17,360
the exponential is unity, unity
integrated from minus pi to pi

125
00:08:17,360 --> 00:08:18,920
is equal to 2 pi.

126
00:08:18,920 --> 00:08:22,190
So for n equal to k, the
value of the integral

127
00:08:22,190 --> 00:08:24,710
is equal to 2 pi.

128
00:08:24,710 --> 00:08:28,850
Well that means that the
only term in this sum that

129
00:08:28,850 --> 00:08:32,990
is going to contribute to
the answer is the term for k

130
00:08:32,990 --> 00:08:36,514
equal to n, because for all
the others the integral is 0.

131
00:08:36,514 --> 00:08:41,179
So for k equal to n, that's
the only non-zero term, for k

132
00:08:41,179 --> 00:08:43,280
equal to n, the value
of this integral

133
00:08:43,280 --> 00:08:47,300
is 2 pi, which will cancel
out this 1 over 2 pi

134
00:08:47,300 --> 00:08:50,976
and consequently we get h of n.

135
00:08:50,976 --> 00:08:57,200
All right, so this is, in a
sense, a formal justification

136
00:08:57,200 --> 00:09:02,330
for the fact that, indeed this
is the inverse relationship

137
00:09:02,330 --> 00:09:07,280
to express h of n terms
of the frequency response.

138
00:09:07,280 --> 00:09:11,300
Although, in fact it's more
useful in terms of insight

139
00:09:11,300 --> 00:09:14,090
to think of it as
the Fourier series

140
00:09:14,090 --> 00:09:18,080
coefficients of a function,
a periodic function,

141
00:09:18,080 --> 00:09:22,280
of the continuous
variable omega.

142
00:09:22,280 --> 00:09:30,410
Now this in essence, is a
Fourier domain representation

143
00:09:30,410 --> 00:09:33,020
for the unit sample
response of a system.

144
00:09:33,020 --> 00:09:37,310
And in fact, we can think of
this as a Fourier transform

145
00:09:37,310 --> 00:09:43,760
or a time domain, Fourier
domain transform pair,

146
00:09:43,760 --> 00:09:47,660
relating the unit sample
response of a system

147
00:09:47,660 --> 00:09:49,490
and the frequency
response of the system.

148
00:09:49,490 --> 00:09:51,840
And we can go back and forth.

149
00:09:51,840 --> 00:09:59,240
Now one of the obvious
facts is that any sequence,

150
00:09:59,240 --> 00:10:03,320
if we wanted to, we could
think of as the unit sample

151
00:10:03,320 --> 00:10:05,950
response of a system.

152
00:10:05,950 --> 00:10:11,170
Well, that suggests then
that this transform pair

153
00:10:11,170 --> 00:10:13,810
isn't restricted
to just sequences

154
00:10:13,810 --> 00:10:17,200
that we explicitly
identify as the unit sample

155
00:10:17,200 --> 00:10:18,490
response of a system.

156
00:10:18,490 --> 00:10:22,930
It in fact permits the
representation of an arbitrary,

157
00:10:22,930 --> 00:10:25,120
not quite arbitrary as
we'll see in a while,

158
00:10:25,120 --> 00:10:31,000
but the representation of a
more general class of sequences

159
00:10:31,000 --> 00:10:33,160
than just the ones
that we explicitly

160
00:10:33,160 --> 00:10:36,490
identify as the unit sample
response of a system.

161
00:10:36,490 --> 00:10:40,030
That is, we can generalize
this set of ideas

162
00:10:40,030 --> 00:10:46,480
to the Fourier transform,
which is a frequency domain

163
00:10:46,480 --> 00:10:51,760
representation of an arbitrary,
again not quite arbitrary,

164
00:10:51,760 --> 00:10:55,270
but for the time being we'll
consider arbitrary sequence,

165
00:10:55,270 --> 00:10:57,160
x of n.

166
00:10:57,160 --> 00:11:02,860
Well, the Fourier transform
then is defined as x of e

167
00:11:02,860 --> 00:11:06,910
to the j omega, which
is the sum of x of n, e

168
00:11:06,910 --> 00:11:12,110
to the minus j omega n, the
Fourier transform x of e

169
00:11:12,110 --> 00:11:17,760
to the j omega, of
a sequence, x of n,

170
00:11:17,760 --> 00:11:21,840
is essentially the frequency
response of a system whose

171
00:11:21,840 --> 00:11:24,960
unit sample response
would be x of n.

172
00:11:24,960 --> 00:11:30,120
So then obviously from what
we just finished discussing,

173
00:11:30,120 --> 00:11:34,170
we have an inverse relationship
that tells us what x of n

174
00:11:34,170 --> 00:11:37,260
is in terms of x of
e to the j omega.

175
00:11:37,260 --> 00:11:39,660
And that then is
this relationship.

176
00:11:42,420 --> 00:11:51,310
Well this then provides a
transform pair, or a frequency

177
00:11:51,310 --> 00:11:54,850
domain, time domain
relationship between sequences

178
00:11:54,850 --> 00:11:57,430
and frequency domain functions.

179
00:11:57,430 --> 00:12:03,550
And it's useful, in fact,
to interpret this expression

180
00:12:03,550 --> 00:12:08,920
somewhat heuristically again
as basically corresponding

181
00:12:08,920 --> 00:12:13,630
to a decomposition of
a sequence, x of n,

182
00:12:13,630 --> 00:12:20,590
in terms of complex exponentials
with incremental amplitude.

183
00:12:20,590 --> 00:12:25,660
This is basically an expression
that says that x of n is a sum,

184
00:12:25,660 --> 00:12:29,200
except that it's sum sort of
in the limit which corresponds

185
00:12:29,200 --> 00:12:34,060
to an integral, of a set
of complex exponentials

186
00:12:34,060 --> 00:12:36,340
with amplitudes
that are essentially

187
00:12:36,340 --> 00:12:40,720
given by the Fourier transform
x of e to the j omega.

188
00:12:40,720 --> 00:12:43,900
Well in fact, you can see
this a little more explicitly

189
00:12:43,900 --> 00:12:47,860
if we consider this integral
as the limiting form

190
00:12:47,860 --> 00:12:51,700
of a sum, the limit,
as delta omega goes

191
00:12:51,700 --> 00:12:57,940
to 0, of the sum of x of
e to the j k delta omega,

192
00:12:57,940 --> 00:13:04,630
times delta omega over 2 pi,
e to the j k delta omega n.

193
00:13:04,630 --> 00:13:10,090
This limit is by definition,
what this integral is.

194
00:13:10,090 --> 00:13:13,930
So an important
point to keep in mind

195
00:13:13,930 --> 00:13:18,670
is that basically the
Fourier transform corresponds

196
00:13:18,670 --> 00:13:22,750
to a decomposition of
a sequence in terms

197
00:13:22,750 --> 00:13:27,220
of a linear combination
of complex exponentials

198
00:13:27,220 --> 00:13:30,730
with incremental amplitudes.

199
00:13:30,730 --> 00:13:34,120
There are a number
of reasons why that's

200
00:13:34,120 --> 00:13:36,130
an important point of view.

201
00:13:36,130 --> 00:13:39,340
One of the reasons
is that it leads

202
00:13:39,340 --> 00:13:43,710
to a very important property
of linear shift invariance

203
00:13:43,710 --> 00:13:51,100
systems, which I refer to
as the convolution property,

204
00:13:51,100 --> 00:14:00,430
and which states that if I
have the convolution of two

205
00:14:00,430 --> 00:14:07,630
sequences, x of n and y
of n, then the Fourier

206
00:14:07,630 --> 00:14:12,340
transform of those is the
product of their Fourier

207
00:14:12,340 --> 00:14:13,780
transforms.

208
00:14:13,780 --> 00:14:19,940
And this of course, shouldn't
be an h, it should be a y.

209
00:14:19,940 --> 00:14:23,270
Or this shouldn't be a
y, it should be an h.

210
00:14:23,270 --> 00:14:26,960
The important property is
that the convolution of two

211
00:14:26,960 --> 00:14:30,620
sequences has, as its
Fourier transform,

212
00:14:30,620 --> 00:14:35,000
the product of the
Fourier transforms.

213
00:14:35,000 --> 00:14:39,800
Well let's look at again,
somewhat heuristically,

214
00:14:39,800 --> 00:14:41,840
an argument that
at least justifies

215
00:14:41,840 --> 00:14:45,920
this, keeping in mind that
we could go through this more

216
00:14:45,920 --> 00:14:48,650
formally, plugging sums
into integrals and integrals

217
00:14:48,650 --> 00:14:51,570
into sums, but
let's not do that.

218
00:14:51,570 --> 00:14:56,870
Let's look at this from a
somewhat heuristic point

219
00:14:56,870 --> 00:14:58,860
of view.

220
00:14:58,860 --> 00:15:03,350
First of all, we
have that again,

221
00:15:03,350 --> 00:15:06,320
a property that we
began the lecture with,

222
00:15:06,320 --> 00:15:10,450
that for a linear
shift invariant system,

223
00:15:10,450 --> 00:15:13,760
a complex exponential
input gives us

224
00:15:13,760 --> 00:15:18,140
at the output, the same
complex exponential, that

225
00:15:18,140 --> 00:15:20,480
is the same frequency,
and only one

226
00:15:20,480 --> 00:15:26,150
of them, multiplied by
h of e to the j omega 0.

227
00:15:26,150 --> 00:15:30,500
That's a consequence of
linearity and shift invariance.

228
00:15:30,500 --> 00:15:35,330
We know also, that because of
the fact that the system is

229
00:15:35,330 --> 00:15:40,190
linear, if we have
a linear combination

230
00:15:40,190 --> 00:15:45,530
of complex exponentials, then
the output of this system

231
00:15:45,530 --> 00:15:49,790
is going to be the
same linear combination

232
00:15:49,790 --> 00:15:54,260
of complex exponentials, with
the amplitudes multiplied

233
00:15:54,260 --> 00:15:57,410
by h of e to the j omega k.

234
00:15:57,410 --> 00:16:00,440
In other words, each
of these exponentials

235
00:16:00,440 --> 00:16:05,510
has as an output ase of k times
h of e to the j omegus of k, e

236
00:16:05,510 --> 00:16:08,900
to the j omegus of k n.

237
00:16:08,900 --> 00:16:11,090
A sum of those at
the input gives us

238
00:16:11,090 --> 00:16:13,340
a sum at the output
because of the fact

239
00:16:13,340 --> 00:16:16,010
that the system is linear.

240
00:16:16,010 --> 00:16:18,410
All right, so let's
take an input, which

241
00:16:18,410 --> 00:16:23,570
is an arbitrary input, more
or less arbitrary, an input

242
00:16:23,570 --> 00:16:32,590
x of n, which I can express in
terms of its Fourier transform,

243
00:16:32,590 --> 00:16:38,620
in terms of this inverse
Fourier transform relationship.

244
00:16:38,620 --> 00:16:44,500
All right, now m as I
emphasized just a minute ago,

245
00:16:44,500 --> 00:16:51,760
is a decomposition of x of
n as a linear combination

246
00:16:51,760 --> 00:16:54,190
of complex exponentials.

247
00:16:54,190 --> 00:16:57,430
This is a linear combination
of complex exponentials

248
00:16:57,430 --> 00:17:00,140
at the input, so
what's the output?

249
00:17:00,140 --> 00:17:04,329
Well, the output is then going
to be a linear combination

250
00:17:04,329 --> 00:17:08,560
of complex exponentials
with the amplitude

251
00:17:08,560 --> 00:17:11,560
of each one of the
input exponentials,

252
00:17:11,560 --> 00:17:15,670
multiplied by the frequency
response of the system

253
00:17:15,670 --> 00:17:18,050
at that frequency.

254
00:17:18,050 --> 00:17:23,020
So the result is that with
this as the input, what

255
00:17:23,020 --> 00:17:30,780
we have to get at
the output is this,

256
00:17:30,780 --> 00:17:34,260
the same linear
combination, the only change

257
00:17:34,260 --> 00:17:38,070
being that the complex
amplitudes of the input

258
00:17:38,070 --> 00:17:43,050
are multiplied by h
of e to the j omega.

259
00:17:43,050 --> 00:17:48,360
Well, this of course, is the
expression for the output.

260
00:17:48,360 --> 00:17:52,785
We put in x of n, we know
that we're getting out y of n.

261
00:17:55,430 --> 00:18:01,340
So obviously this then must
be the Fourier transform

262
00:18:01,340 --> 00:18:04,140
of the output of the system.

263
00:18:04,140 --> 00:18:07,460
So what this says then
is that the Fourier

264
00:18:07,460 --> 00:18:13,110
transform of the output y of e
to the j omega, has to be this,

265
00:18:13,110 --> 00:18:17,270
it has to be x of e to
the j omega times h of e

266
00:18:17,270 --> 00:18:20,150
to the j omega.

267
00:18:20,150 --> 00:18:26,180
And basically-- well, and
we know also that y of n

268
00:18:26,180 --> 00:18:30,650
is going to be equal to x
of n convolved with h of n.

269
00:18:30,650 --> 00:18:36,470
So basically, this justifies
the convolution property, that

270
00:18:36,470 --> 00:18:39,500
is the convolution
of two sequences

271
00:18:39,500 --> 00:18:43,430
has as its Fourier transform,
the product of the Fourier

272
00:18:43,430 --> 00:18:46,700
transform of each of the
individual sequences.

273
00:18:46,700 --> 00:18:48,230
Very important property.

274
00:18:48,230 --> 00:18:52,790
And although we can go through
a formal derivation of this,

275
00:18:52,790 --> 00:18:58,380
in fact the basic reason for
it is tied to the arguments

276
00:18:58,380 --> 00:18:59,670
that I've outlined here.

277
00:18:59,670 --> 00:19:02,430
And in terms of insight,
I feel that it's

278
00:19:02,430 --> 00:19:05,760
more important to understand
this way of looking at it

279
00:19:05,760 --> 00:19:10,380
than to understand
a formal derivation.

280
00:19:10,380 --> 00:19:13,410
Well, this is interesting,
also important,

281
00:19:13,410 --> 00:19:18,270
and also should be familiar
to you in terms of things

282
00:19:18,270 --> 00:19:19,930
that you're used
to thinking about

283
00:19:19,930 --> 00:19:22,740
for continuous time systems.

284
00:19:22,740 --> 00:19:25,290
Obviously, in
continuous time systems

285
00:19:25,290 --> 00:19:27,930
the same type of property holds.

286
00:19:27,930 --> 00:19:32,670
That is, that the
Fourier transform changes

287
00:19:32,670 --> 00:19:35,400
convolution to multiplication.

288
00:19:35,400 --> 00:19:39,690
And in fact, it permits the
description of a linear shift

289
00:19:39,690 --> 00:19:44,640
invariant system to be in
terms of multiplication

290
00:19:44,640 --> 00:19:47,062
rather than in terms
of convolution.

291
00:19:47,062 --> 00:19:48,770
And as we'll see in
a number of lectures,

292
00:19:48,770 --> 00:19:55,080
that basically the basis
for the notions of filtering

293
00:19:55,080 --> 00:19:57,130
and some other very
important notions,

294
00:19:57,130 --> 00:19:59,950
and notions of modulation etc.

295
00:19:59,950 --> 00:20:03,150
OK, key property,
this is a key property

296
00:20:03,150 --> 00:20:06,540
of linear shift invariant
systems and the Fourier

297
00:20:06,540 --> 00:20:08,130
transform.

298
00:20:08,130 --> 00:20:11,910
There are, of course,
lots of other properties

299
00:20:11,910 --> 00:20:14,910
of linear shift
invariant systems--

300
00:20:14,910 --> 00:20:17,040
sorry, other
properties, there are

301
00:20:17,040 --> 00:20:20,280
other properties, obviously, of
linear shift invariant systems.

302
00:20:20,280 --> 00:20:24,780
There are other properties
of Fourier transforms that

303
00:20:24,780 --> 00:20:30,000
are important, both in terms of
interpreting Fourier transforms

304
00:20:30,000 --> 00:20:34,500
and in terms of computing
Fourier transforms

305
00:20:34,500 --> 00:20:37,590
of a variety of sequences.

306
00:20:37,590 --> 00:20:42,210
A lot of these properties
will be developed in the text

307
00:20:42,210 --> 00:20:45,000
and also in the study
guide, so you'll

308
00:20:45,000 --> 00:20:47,340
have to do the work
rather than me.

309
00:20:47,340 --> 00:20:51,720
But let me just indicate one
class of properties that,

310
00:20:51,720 --> 00:20:55,020
again, should be
very familiar to you

311
00:20:55,020 --> 00:20:59,130
if you relate your thinking
back to continuous time

312
00:20:59,130 --> 00:21:00,990
Fourier transforms.

313
00:21:00,990 --> 00:21:04,890
And that is the class
of symmetry properties

314
00:21:04,890 --> 00:21:08,610
for the special case in which
the sequences that we're

315
00:21:08,610 --> 00:21:13,320
talking about are
real sequences.

316
00:21:13,320 --> 00:21:19,020
Well the basic symmetry
property is that for x of n

317
00:21:19,020 --> 00:21:25,680
real the Fourier transform is
a conjugate symmetric function,

318
00:21:25,680 --> 00:21:30,930
x of e to the j omega is
equal to x conjugate of e

319
00:21:30,930 --> 00:21:34,410
to the minus j omega.

320
00:21:34,410 --> 00:21:39,030
And we can see that,
essentially in a straightforward

321
00:21:39,030 --> 00:21:44,000
way, that is the derivation is
effectively straightforward.

322
00:21:44,000 --> 00:21:47,700
Here's x of e to the j
omega, the sum of x of n, e

323
00:21:47,700 --> 00:21:50,730
to the minus j omega n.

324
00:21:50,730 --> 00:21:54,150
Here is x of e to
the minus j omega.

325
00:21:54,150 --> 00:21:56,670
Well, the only difference
between that and that

326
00:21:56,670 --> 00:22:00,000
is that we replace
omega by minus omega,

327
00:22:00,000 --> 00:22:02,400
so this becomes a plus sign.

328
00:22:02,400 --> 00:22:07,110
Of course, it's not this that
we want from this expression,

329
00:22:07,110 --> 00:22:09,450
it's the conjugate of that.

330
00:22:09,450 --> 00:22:15,120
So we want to complex
conjugate this.

331
00:22:15,120 --> 00:22:18,940
Well, we will do the same
on the left--hand side--

332
00:22:18,940 --> 00:22:21,040
on the right-hand side.

333
00:22:21,040 --> 00:22:26,100
And so we would conjugate
that and conjugate

334
00:22:26,100 --> 00:22:33,910
this, which replaces this
plus sign with a minus sign.

335
00:22:33,910 --> 00:22:39,180
But we're talking about
a real sequence, x of n,

336
00:22:39,180 --> 00:22:41,920
so x conjugate of
n is just x of n.

337
00:22:41,920 --> 00:22:46,870
In other words, this is just
the x of an all over again.

338
00:22:46,870 --> 00:22:51,300
And so we have that x conjugate
of e to the minus j omega

339
00:22:51,300 --> 00:22:54,180
is the sum of x of
n, e to the minus j

340
00:22:54,180 --> 00:22:57,960
omega n, which is just
what we have up here.

341
00:22:57,960 --> 00:23:01,860
So obviously then,
these two are equal.

342
00:23:01,860 --> 00:23:08,070
So for a real sequence,
the Fourier transform

343
00:23:08,070 --> 00:23:10,660
is a conjugate
symmetric function.

344
00:23:10,660 --> 00:23:14,430
This is what we'll call
conjugate symmetric.

345
00:23:14,430 --> 00:23:18,390
Well let's press that
a little further.

346
00:23:18,390 --> 00:23:22,290
We have x of e to
the j omega, which

347
00:23:22,290 --> 00:23:26,430
I can represent in
terms of its real part

348
00:23:26,430 --> 00:23:31,810
and its imaginary part as
obviously x real, x sub r of e

349
00:23:31,810 --> 00:23:38,270
to the j omega, plus j times
x sub i of e to the j omega.

350
00:23:38,270 --> 00:23:42,870
And the conjugate
symmetric counterpart

351
00:23:42,870 --> 00:23:50,420
is this with omega replaced by
minus omega and the expression

352
00:23:50,420 --> 00:23:51,770
conjugated.

353
00:23:51,770 --> 00:23:56,390
So we have x sub r of
e to the minus j omega,

354
00:23:56,390 --> 00:24:02,090
minus j times x sub i of
e to the minus j omega.

355
00:24:02,090 --> 00:24:04,910
And we know that
these two are equal.

356
00:24:04,910 --> 00:24:07,460
Well, if these two are
equal, then these two

357
00:24:07,460 --> 00:24:11,430
are equal, and so are these.

358
00:24:11,430 --> 00:24:15,510
The real part of x
of e to the j omega

359
00:24:15,510 --> 00:24:21,300
must be equal to the real part
of x of e to the minus j omega.

360
00:24:21,300 --> 00:24:24,150
In other words,
the real part has

361
00:24:24,150 --> 00:24:30,480
to be the same if omega is
replaced by minus omega.

362
00:24:30,480 --> 00:24:34,380
That means then that the real
part of the Fourier transform

363
00:24:34,380 --> 00:24:38,790
is an even function of omega.

364
00:24:38,790 --> 00:24:43,230
Meaning that if we replace
omega by minus omega

365
00:24:43,230 --> 00:24:47,760
then the real part
doesn't change.

366
00:24:47,760 --> 00:24:50,310
The imaginary part, on
the other hand, does.

367
00:24:50,310 --> 00:24:54,960
In particular, on the basis
of what we're saying here

368
00:24:54,960 --> 00:24:59,970
and the equality of these, the
imaginary part x sub i of e

369
00:24:59,970 --> 00:25:03,900
to the j omega, must
be equal to minus,

370
00:25:03,900 --> 00:25:07,530
don't forget the minus
sign, minus x sub i of e

371
00:25:07,530 --> 00:25:09,630
to the minus j omega.

372
00:25:09,630 --> 00:25:15,810
And that says then, that if we
replace omega by minus omega,

373
00:25:15,810 --> 00:25:19,350
then the sign of the
imaginary part changes.

374
00:25:19,350 --> 00:25:24,720
So the imaginary part, in fact,
is an odd function of omega.

375
00:25:24,720 --> 00:25:28,590
The real part is an
even function of omega.

376
00:25:28,590 --> 00:25:32,550
Well, from this,
or from this, we

377
00:25:32,550 --> 00:25:37,650
can also show that the magnitude
of the Fourier transform

378
00:25:37,650 --> 00:25:42,520
is an even function of omega.

379
00:25:42,520 --> 00:25:47,220
And the angle of the
Fourier transform

380
00:25:47,220 --> 00:25:51,610
is an odd function of omega.

381
00:25:51,610 --> 00:25:56,460
And those, of course,
are identical to what

382
00:25:56,460 --> 00:25:59,640
we know is true for the
continuous time Fourier

383
00:25:59,640 --> 00:26:01,230
transform.

384
00:26:01,230 --> 00:26:04,530
Remember however, again, that
these are periodic functions,

385
00:26:04,530 --> 00:26:09,140
whereas in the continuous
time case, they're not.

386
00:26:09,140 --> 00:26:12,480
All right well, this
is an introduction

387
00:26:12,480 --> 00:26:17,820
to the Fourier transform.

388
00:26:17,820 --> 00:26:22,230
One of the things
that we've refrained

389
00:26:22,230 --> 00:26:25,020
from doing in all
of these lectures

390
00:26:25,020 --> 00:26:30,000
is tying our
development too closely

391
00:26:30,000 --> 00:26:35,940
to the notion of continuous
time signals, and in particular,

392
00:26:35,940 --> 00:26:38,640
avoiding to some
extent, the notion

393
00:26:38,640 --> 00:26:43,800
of interpreting discrete time
signals as just simply sampled

394
00:26:43,800 --> 00:26:47,100
replicas of continuous
time signals.

395
00:26:47,100 --> 00:26:49,650
And we'll continue to do
that throughout this set

396
00:26:49,650 --> 00:26:51,160
of lectures.

397
00:26:51,160 --> 00:26:54,960
But in particular
Fourier transform,

398
00:26:54,960 --> 00:27:01,980
I think that it's
instructive to tie together,

399
00:27:01,980 --> 00:27:04,170
at least in terms
of some insight

400
00:27:04,170 --> 00:27:07,980
into the relationship, the
continuous time Fourier

401
00:27:07,980 --> 00:27:12,090
transform of obviously
continuous time signal,

402
00:27:12,090 --> 00:27:17,820
and the discrete time Fourier
transform for a sequence that's

403
00:27:17,820 --> 00:27:20,970
obtained by periodic sampling.

404
00:27:20,970 --> 00:27:25,150
That is, equally spaced sampling
of the continuous time signal.

405
00:27:25,150 --> 00:27:29,460
So what I'd like to do now is
focus on that relationship,

406
00:27:29,460 --> 00:27:33,660
emphasizing again, that
not all sequences arise

407
00:27:33,660 --> 00:27:37,560
by periodic sampling of
continuous time signals.

408
00:27:37,560 --> 00:27:40,200
But for the cases
in which they do,

409
00:27:40,200 --> 00:27:43,440
the relationship between
the continuous time

410
00:27:43,440 --> 00:27:46,690
and discrete time Fourier
transform is instructive.

411
00:27:46,690 --> 00:27:55,170
So let's take a look
at the relationship

412
00:27:55,170 --> 00:28:02,280
between some continuous time
and discrete time Fourier

413
00:28:02,280 --> 00:28:07,860
transforms when we obtain
the discrete time signal

414
00:28:07,860 --> 00:28:11,740
by sampling a
continuous time signal.

415
00:28:11,740 --> 00:28:17,790
Well, we begin of course,
with a continuous time,

416
00:28:17,790 --> 00:28:22,470
time function which I denote by
x sub a of t, a sort of meaning

417
00:28:22,470 --> 00:28:24,030
analog.

418
00:28:24,030 --> 00:28:27,030
And the steps that
we would go through

419
00:28:27,030 --> 00:28:31,590
to convert that to a
sequence are first of all,

420
00:28:31,590 --> 00:28:36,550
to go through a sampler,
which I've indicated here.

421
00:28:36,550 --> 00:28:42,510
And the output of the sampler is
then a sampled continuous time

422
00:28:42,510 --> 00:28:46,500
version of this signal.

423
00:28:46,500 --> 00:28:49,740
This essentially
is the input signal

424
00:28:49,740 --> 00:28:53,810
multiplied by an impulse train.

425
00:28:53,810 --> 00:28:56,540
So I've indicated
that here, here's

426
00:28:56,540 --> 00:28:58,970
the continuous time input.

427
00:28:58,970 --> 00:29:02,630
Here is the
continuous time output

428
00:29:02,630 --> 00:29:05,030
of the sampler,
which is an impulse

429
00:29:05,030 --> 00:29:09,020
train with the envelope
of the impulse train

430
00:29:09,020 --> 00:29:14,430
being the continuous time
function x sub a of t.

431
00:29:14,430 --> 00:29:16,460
All right, well this
isn't the sequence.

432
00:29:16,460 --> 00:29:19,430
This is just simply
an impulse train.

433
00:29:19,430 --> 00:29:21,980
To turn this into
a sequence we need

434
00:29:21,980 --> 00:29:26,350
to go into a box,
which I've labeled

435
00:29:26,350 --> 00:29:30,470
c slash d, meaning continuous
time to discrete time

436
00:29:30,470 --> 00:29:32,180
converter.

437
00:29:32,180 --> 00:29:37,250
And the output then
is a sequence x of n.

438
00:29:37,250 --> 00:29:42,290
The sequence values being
samples of x sub a of t

439
00:29:42,290 --> 00:29:46,640
at the sampling
instances n times capital

440
00:29:46,640 --> 00:29:50,180
T. In other words,
it's converting

441
00:29:50,180 --> 00:29:55,500
the areas of these impulses
into sequence values.

442
00:29:55,500 --> 00:30:01,520
Now I've illustrated it here
for one choice for the sampling

443
00:30:01,520 --> 00:30:06,140
interval capital T.
Let's, down here,

444
00:30:06,140 --> 00:30:10,250
illustrate it for a sampling
interval that's twice as long.

445
00:30:10,250 --> 00:30:14,840
Well, of course we have the
same continuous time input.

446
00:30:14,840 --> 00:30:19,610
The sampled output,
x sub a, tilde of t,

447
00:30:19,610 --> 00:30:24,410
has the same envelope, but
the spacing of the impulses

448
00:30:24,410 --> 00:30:28,970
is twice what it is here,
and that I've indicated.

449
00:30:28,970 --> 00:30:32,150
The envelope, of
course, is the same.

450
00:30:32,150 --> 00:30:34,690
But at the output of
the continuous time

451
00:30:34,690 --> 00:30:38,190
to discrete time
converter, what do we have?

452
00:30:38,190 --> 00:30:43,970
Well, we have the areas
of these impulses lined up

453
00:30:43,970 --> 00:30:50,090
along this axis, again,
at integer values of n.

454
00:30:50,090 --> 00:30:54,020
That is, the spacing
of the lines,

455
00:30:54,020 --> 00:30:56,210
when we look at
the sequence here,

456
00:30:56,210 --> 00:31:00,290
must be exactly the same as
the spacing of the lines here,

457
00:31:00,290 --> 00:31:03,650
it's just that the values are
different because we picked out

458
00:31:03,650 --> 00:31:06,150
samples at different instance.

459
00:31:06,150 --> 00:31:11,420
So in fact, the
envelope here is indeed

460
00:31:11,420 --> 00:31:15,560
a compressed version of the
envelope that we had here.

461
00:31:15,560 --> 00:31:17,670
Very important point.

462
00:31:17,670 --> 00:31:22,760
The point being that no matter
what the sampling rate is

463
00:31:22,760 --> 00:31:26,600
the sequence values, when we
line them up as a sequence,

464
00:31:26,600 --> 00:31:31,760
are going to fall at integer
values along this argument,

465
00:31:31,760 --> 00:31:36,450
n, always at intervals
that correspond to 1.

466
00:31:36,450 --> 00:31:39,320
Whereas the output
of the sampler

467
00:31:39,320 --> 00:31:43,700
had impulses occurring at a
spacing of capital T, which

468
00:31:43,700 --> 00:31:46,610
in this case, was capital
T 1, and in this case

469
00:31:46,610 --> 00:31:49,520
is capital T 2.

470
00:31:49,520 --> 00:31:51,380
All right well,
this is essentially

471
00:31:51,380 --> 00:31:56,090
the sampling process plus the
conversion to a discrete time

472
00:31:56,090 --> 00:31:57,540
signal.

473
00:31:57,540 --> 00:32:00,050
And now let's take
a look at what this

474
00:32:00,050 --> 00:32:07,430
means in terms of the Fourier
transform of the discrete time

475
00:32:07,430 --> 00:32:11,210
signal as compared with
the Fourier transform

476
00:32:11,210 --> 00:32:15,090
of the continuous time signal.

477
00:32:15,090 --> 00:32:17,820
Let's do this in two steps.

478
00:32:17,820 --> 00:32:20,820
First of all, we
have the sampler.

479
00:32:20,820 --> 00:32:25,650
Here is the input x sub a of
t, continuous time function.

480
00:32:25,650 --> 00:32:29,390
Here is the output, x sub a
tilde of t, a continuous time

481
00:32:29,390 --> 00:32:30,800
function.

482
00:32:30,800 --> 00:32:34,520
And the output of the
sampler is the input

483
00:32:34,520 --> 00:32:40,220
to the sampler, multiplied
by an impulse train.

484
00:32:40,220 --> 00:32:44,990
Or equivalently,
it's an impulse train

485
00:32:44,990 --> 00:32:47,660
with the areas of
the impulses given

486
00:32:47,660 --> 00:32:52,190
by the values of x sub
a of t at the times

487
00:32:52,190 --> 00:32:54,560
that the impulses occur.

488
00:32:54,560 --> 00:32:57,230
Naught, by the way, is what
I'm using as the notation

489
00:32:57,230 --> 00:33:02,640
to designate a unit impulse,
unit continuous time impulse.

490
00:33:02,640 --> 00:33:05,810
Now in the Fourier
transformed domain then,

491
00:33:05,810 --> 00:33:09,500
the continuous time
Fourier transform,

492
00:33:09,500 --> 00:33:13,970
is the convolution of
the continuous time

493
00:33:13,970 --> 00:33:17,720
Fourier transform
of x sub a of t,

494
00:33:17,720 --> 00:33:21,770
convolved with the Fourier
transform of the impulse train,

495
00:33:21,770 --> 00:33:25,184
which is an impulse train
in the frequency domain.

496
00:33:25,184 --> 00:33:26,600
That's a result
that you should be

497
00:33:26,600 --> 00:33:30,830
familiar with for
continuous time signals.

498
00:33:30,830 --> 00:33:34,250
Or equivalently, it's
given by 1 over t,

499
00:33:34,250 --> 00:33:37,160
times the sum of x
sub a of j omega,

500
00:33:37,160 --> 00:33:42,650
plus j 2 pi r over capital
T. Basically, what that means

501
00:33:42,650 --> 00:33:48,430
is that the Fourier transform
of this continuous time signal

502
00:33:48,430 --> 00:33:50,590
is equal to the
Fourier transform

503
00:33:50,590 --> 00:33:55,000
of this continuous time signal,
but repeated over and over

504
00:33:55,000 --> 00:34:00,610
again in frequency at intervals
of 2 pi over capital T.

505
00:34:00,610 --> 00:34:06,100
So this is sort of a standard
sampling theorem kind of result

506
00:34:06,100 --> 00:34:08,650
in the continuous time
case, and a result

507
00:34:08,650 --> 00:34:11,770
that you should be more
or less familiar with.

508
00:34:11,770 --> 00:34:16,929
Let's look at this now from
a different point of view.

509
00:34:16,929 --> 00:34:21,310
Again looking at x sub
a tilde of j omega.

510
00:34:21,310 --> 00:34:25,600
Well x sub a tilde of
j omega is the integral

511
00:34:25,600 --> 00:34:32,050
of x sub a tilde of t, e
to the minus j omega t d t.

512
00:34:32,050 --> 00:34:35,889
Substitute in for
x sub a tilde of t,

513
00:34:35,889 --> 00:34:38,949
the relationship in terms
of an impulse train,

514
00:34:38,949 --> 00:34:42,010
and interchange summation
and integration,

515
00:34:42,010 --> 00:34:46,659
and I think you could verify
in a very straightforward way

516
00:34:46,659 --> 00:34:51,820
that what you end up with is
an expression for the Fourier

517
00:34:51,820 --> 00:34:55,600
transform at the
output of the sampler

518
00:34:55,600 --> 00:35:01,270
as given by the sampling
values times e to the minus j n

519
00:35:01,270 --> 00:35:05,470
capital omega T. Capital omega
is a continuous frequency

520
00:35:05,470 --> 00:35:07,490
variable.

521
00:35:07,490 --> 00:35:10,580
Now we want to look
at the relationship

522
00:35:10,580 --> 00:35:15,890
between the Fourier transform,
continuous time of x sub a of t

523
00:35:15,890 --> 00:35:20,150
and the discrete time
Fourier transform of x of n.

524
00:35:20,150 --> 00:35:23,420
So we have the
next step, which is

525
00:35:23,420 --> 00:35:28,280
to put this impulse train into
the continuous to discrete time

526
00:35:28,280 --> 00:35:30,240
converter.

527
00:35:30,240 --> 00:35:33,510
And I remind you that we
just developed two results.

528
00:35:33,510 --> 00:35:37,020
One result was that x
sub a tilde of j omega

529
00:35:37,020 --> 00:35:39,450
was given by this expression.

530
00:35:39,450 --> 00:35:44,410
Also we developed that it
was given by this expression.

531
00:35:44,410 --> 00:35:46,800
Well what's the
Fourier transform

532
00:35:46,800 --> 00:35:50,820
of the output of the continuous
to discrete time converter?

533
00:35:50,820 --> 00:35:55,850
Well, it's just simply
x of e to the j omega,

534
00:35:55,850 --> 00:35:59,380
our discrete time
Fourier transform,

535
00:35:59,380 --> 00:36:05,880
which is equal to the
sum on n of x of n.

536
00:36:05,880 --> 00:36:11,770
But x of n is x sub a of n
capital T. So we have x sub a

537
00:36:11,770 --> 00:36:20,850
of n capital T, times e
to the minus j omega n.

538
00:36:20,850 --> 00:36:24,810
Well let's compare
this with this.

539
00:36:24,810 --> 00:36:29,520
We see that they're exactly
the same except that for omega,

540
00:36:29,520 --> 00:36:33,270
little omega here we
have capital omega times

541
00:36:33,270 --> 00:36:35,700
capital T there.

542
00:36:35,700 --> 00:36:40,080
Consequently, we can say that
the discrete time Fourier

543
00:36:40,080 --> 00:36:43,920
transform is equal
to the Fourier

544
00:36:43,920 --> 00:36:47,910
transform of the impulse train,
the continuous time Fourier

545
00:36:47,910 --> 00:36:52,590
transform of the impulse train,
with omega times capital T

546
00:36:52,590 --> 00:36:53,790
equal to little omega.

547
00:36:56,380 --> 00:37:01,540
And we had another expression
for the Fourier transform

548
00:37:01,540 --> 00:37:05,380
of the impulse train,
which we derived here.

549
00:37:05,380 --> 00:37:08,770
Consequently, the final
result that we end up with

550
00:37:08,770 --> 00:37:14,980
is that the Fourier transform
of the discrete time sequence

551
00:37:14,980 --> 00:37:21,490
is equal to 1 over t times
the sum of x sub a of j omega,

552
00:37:21,490 --> 00:37:28,360
plus j 2 pi r over capital
T, with omega replaced

553
00:37:28,360 --> 00:37:34,550
by little omega, divided by
capital T. In other words,

554
00:37:34,550 --> 00:37:38,110
we had the expression that
capital omega times capital T

555
00:37:38,110 --> 00:37:40,180
is equal to little omega.

556
00:37:40,180 --> 00:37:44,470
And this then tells us
what the Fourier transform

557
00:37:44,470 --> 00:37:47,890
of the sequence is in
terms of the Fourier

558
00:37:47,890 --> 00:37:51,080
transform of the output.

559
00:37:51,080 --> 00:37:53,510
Well, this is just
the equations.

560
00:37:53,510 --> 00:37:57,690
Let's take a look at what
this looks like graphically.

561
00:38:06,760 --> 00:38:13,960
I've depicted here
the continuous time

562
00:38:13,960 --> 00:38:20,380
Fourier transform of
some time function,

563
00:38:20,380 --> 00:38:22,480
and I picked the
Fourier transform

564
00:38:22,480 --> 00:38:26,500
that looks like a triangle.

565
00:38:26,500 --> 00:38:29,440
Well, first of all
we derived the fact

566
00:38:29,440 --> 00:38:33,040
that the impulse
train that results

567
00:38:33,040 --> 00:38:35,980
from sampling
little x sub a of t

568
00:38:35,980 --> 00:38:41,110
has a Fourier transform, which
is this, periodically repeated

569
00:38:41,110 --> 00:38:47,830
in frequency with a period in
frequency equal to 2 pi divided

570
00:38:47,830 --> 00:38:53,990
by capital 1, where capital
1 is the sampling rate.

571
00:38:53,990 --> 00:38:57,110
And on the basis
of the expression

572
00:38:57,110 --> 00:39:02,060
that we derived relating the
discrete time Fourier transform

573
00:39:02,060 --> 00:39:05,430
and the continuous
time Fourier transform,

574
00:39:05,430 --> 00:39:07,970
the discrete time
Fourier transform

575
00:39:07,970 --> 00:39:12,830
looks exactly like this,
but with a re-normalization

576
00:39:12,830 --> 00:39:18,650
of the frequency axis, because
capital omega times capital T

577
00:39:18,650 --> 00:39:21,540
is equal to little omega.

578
00:39:21,540 --> 00:39:25,730
Yes, capital omega times capital
T is equal to little omega.

579
00:39:25,730 --> 00:39:33,170
So where capital omega is
equal to 2 pi over capital T,

580
00:39:33,170 --> 00:39:35,870
little omega is equal to 2 pi.

581
00:39:35,870 --> 00:39:40,760
So this point, which
was at pi over capital T

582
00:39:40,760 --> 00:39:44,930
ends up on the omega,
little omega axis at pi.

583
00:39:44,930 --> 00:39:50,180
So this picture just simply
gets scaled according

584
00:39:50,180 --> 00:39:53,690
to the relationship that
capital omega times capital T

585
00:39:53,690 --> 00:39:56,630
is equal to little omega.

586
00:39:56,630 --> 00:40:02,310
Well, this is, for one choice
of the sampling period.

587
00:40:02,310 --> 00:40:06,210
Obviously, if
capital T was large

588
00:40:06,210 --> 00:40:10,420
enough so that 2 pi over
t 1 was small enough,

589
00:40:10,420 --> 00:40:13,530
then each of these individual
replicas of the frequency

590
00:40:13,530 --> 00:40:16,200
response would interact.

591
00:40:16,200 --> 00:40:19,860
And we wouldn't have just
the simple separation

592
00:40:19,860 --> 00:40:22,380
of the spectra as we have here.

593
00:40:22,380 --> 00:40:26,940
We'd have an interaction,
which of course, is

594
00:40:26,940 --> 00:40:30,960
the interaction and the
relationship between when

595
00:40:30,960 --> 00:40:33,690
that interaction
occurs and capital T,

596
00:40:33,690 --> 00:40:37,740
is the basis for the well-known,
and hopefully, theorem

597
00:40:37,740 --> 00:40:41,550
that you're familiar with
namely, the sampling theorem.

598
00:40:41,550 --> 00:40:46,830
Well, to illustrate that if we
had a different sampling rate,

599
00:40:46,830 --> 00:40:50,580
say twice the sampling rate
that we have over here,

600
00:40:50,580 --> 00:40:53,700
so that capital T is
equal to 2 times capital

601
00:40:53,700 --> 00:41:01,060
T 1, then starting with the same
continuous time spectrum, what

602
00:41:01,060 --> 00:41:07,030
we have now is the spectra,
again periodically repeated,

603
00:41:07,030 --> 00:41:10,270
with a period again, which
is 2 pi over capital T,

604
00:41:10,270 --> 00:41:15,310
which in this case is
2 pi over capital T 2,

605
00:41:15,310 --> 00:41:17,740
but now they interact
and of course, we

606
00:41:17,740 --> 00:41:18,850
have to add these up.

607
00:41:18,850 --> 00:41:21,730
That was the expression
that we just derived.

608
00:41:21,730 --> 00:41:26,110
So in that case, we get some
interaction or aliasing,

609
00:41:26,110 --> 00:41:28,160
as it's referred to.

610
00:41:28,160 --> 00:41:32,320
And due to this
aliasing, the periodic,

611
00:41:32,320 --> 00:41:35,410
one period of this
periodic spectrum

612
00:41:35,410 --> 00:41:39,780
no longer resembles
the original spectrum.

613
00:41:39,780 --> 00:41:46,320
This is the Fourier transform
for the sampled continuous time

614
00:41:46,320 --> 00:41:46,980
function.

615
00:41:46,980 --> 00:41:49,230
That is, this is the
Fourier transform

616
00:41:49,230 --> 00:41:50,970
for the impulse train.

617
00:41:50,970 --> 00:41:54,330
And now if we renormalizing
the frequency axis

618
00:41:54,330 --> 00:41:58,050
so that we express this in terms
of the discrete time frequency

619
00:41:58,050 --> 00:42:04,560
variable, little omega, then
we simply scale this picture

620
00:42:04,560 --> 00:42:09,390
so that we end up with
2 pi over capital T,

621
00:42:09,390 --> 00:42:14,760
corresponding to 2
pi in little omega.

622
00:42:14,760 --> 00:42:19,080
Pi over capital T
corresponding to pi.

623
00:42:19,080 --> 00:42:20,910
And we see in fact, as--

624
00:42:20,910 --> 00:42:24,240
we better see, that is, it
better turn out this way,

625
00:42:24,240 --> 00:42:28,080
that the spectrum in
the discrete time case

626
00:42:28,080 --> 00:42:31,110
is a periodic function of omega.

627
00:42:31,110 --> 00:42:32,940
In other words, it's periodic.

628
00:42:32,940 --> 00:42:37,500
Furthermore, the period
is given by 2 pi,

629
00:42:37,500 --> 00:42:43,440
whereas here the period was
2 pi divided by capital T.

630
00:42:43,440 --> 00:42:46,290
All right, this perhaps
takes a little digesting.

631
00:42:46,290 --> 00:42:47,880
And you'll have
some chance to do

632
00:42:47,880 --> 00:42:53,040
that as we work some problems.

633
00:42:53,040 --> 00:42:55,980
We, meaning you, as
you work some problems

634
00:42:55,980 --> 00:42:59,580
in the study guide and
digest this a little

635
00:42:59,580 --> 00:43:01,540
while you're reading the text.

636
00:43:01,540 --> 00:43:04,310
But it's an important
relationship

637
00:43:04,310 --> 00:43:07,940
and it's important
to understand it.

638
00:43:07,940 --> 00:43:13,440
Now, this is basically
the Fourier transform,

639
00:43:13,440 --> 00:43:15,990
the relationship between
the discrete time

640
00:43:15,990 --> 00:43:18,990
and continuous time
Fourier transform.

641
00:43:18,990 --> 00:43:23,370
One of the difficulties
with the Fourier transform,

642
00:43:23,370 --> 00:43:28,440
which I've avoided illustrating
explicitly in this lecture,

643
00:43:28,440 --> 00:43:30,030
is that the Fourier
transform doesn't

644
00:43:30,030 --> 00:43:32,040
exist for all sequences.

645
00:43:32,040 --> 00:43:35,950
In particular, it doesn't
converge for all sequences.

646
00:43:35,950 --> 00:43:40,920
And this is a problem
which we can get around

647
00:43:40,920 --> 00:43:44,520
by generalizing the
notion of the Fourier

648
00:43:44,520 --> 00:43:49,260
transform to what we'll
call the z transform.

649
00:43:49,260 --> 00:43:53,010
And the z transform,
as you'll observe,

650
00:43:53,010 --> 00:43:57,660
is like, in the continuous time
case, the Laplace transform.

651
00:43:57,660 --> 00:44:01,000
And this is what we'll go
on to in the next lecture.

652
00:44:01,000 --> 00:44:02,500
Thanks

653
00:44:02,500 --> 00:44:04,650
[MUSIC PLAYING]