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[MUSIC PLAYING]

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ALAN V. OPPENHEIM:
In the last lecture,

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00:00:54,630 --> 00:00:59,600
we introduced the class
of discrete time systems,

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00:00:59,600 --> 00:01:03,980
and in particular, imposed the
conditions of linearity first

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00:01:03,980 --> 00:01:08,360
of all, and second, the
property or constraint of shift

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00:01:08,360 --> 00:01:10,880
invariance.

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00:01:10,880 --> 00:01:14,810
And those constraints led
us to the convolution sum

15
00:01:14,810 --> 00:01:16,880
representation.

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00:01:16,880 --> 00:01:20,420
In today's lecture,
there are several issues

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00:01:20,420 --> 00:01:22,470
that I'd like to focus on.

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00:01:22,470 --> 00:01:29,660
The first is the introduction
of two additional constraints

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00:01:29,660 --> 00:01:33,530
that it's sometimes useful to
impose, or at least consider,

20
00:01:33,530 --> 00:01:35,450
for discrete time systems--

21
00:01:35,450 --> 00:01:37,820
namely the constraints
or conditions

22
00:01:37,820 --> 00:01:41,630
of causality and stability.

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00:01:41,630 --> 00:01:46,770
Second of all, I would like to
talk about a particular class,

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00:01:46,770 --> 00:01:48,680
or at least introduce
a particular class

25
00:01:48,680 --> 00:01:51,800
of linear shift
invariant systems, namely

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00:01:51,800 --> 00:01:54,920
those representable by linear
constant coefficient difference

27
00:01:54,920 --> 00:01:56,280
equations.

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00:01:56,280 --> 00:01:59,990
And finally, I'd
like to introduce

29
00:01:59,990 --> 00:02:05,180
a representation of linear
shift invariant systems

30
00:02:05,180 --> 00:02:09,380
that is an alternative to the
convolution sum representation,

31
00:02:09,380 --> 00:02:11,270
and in particular,
that representation

32
00:02:11,270 --> 00:02:14,720
corresponds to the
representation in terms

33
00:02:14,720 --> 00:02:17,210
of a frequency response.

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00:02:17,210 --> 00:02:20,630
Well, let's begin
with the notions

35
00:02:20,630 --> 00:02:25,280
of stability and causality,
reminding you, first of all,

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00:02:25,280 --> 00:02:28,370
that, as we talked
about last time,

37
00:02:28,370 --> 00:02:31,610
we can consider a
general system--

38
00:02:31,610 --> 00:02:34,640
inputs and outputs
are sequences--

39
00:02:34,640 --> 00:02:37,820
and in general, the system
just simply corresponds

40
00:02:37,820 --> 00:02:40,730
to some transformation
from the input sequence

41
00:02:40,730 --> 00:02:43,310
to the output sequence.

42
00:02:43,310 --> 00:02:47,390
When we impose the conditions of
linearity and shift invariance,

43
00:02:47,390 --> 00:02:53,780
both conditions, then we can
express the output sequence

44
00:02:53,780 --> 00:02:58,580
in this form where h
of n is the response

45
00:02:58,580 --> 00:03:01,070
of the system to a unit sample.

46
00:03:01,070 --> 00:03:05,930
And this can also be
rearranged in the form

47
00:03:05,930 --> 00:03:08,900
that I've indicated here,
essentially interchanging

48
00:03:08,900 --> 00:03:12,620
the role of the unit sample
response and the input.

49
00:03:12,620 --> 00:03:16,430
The sum expressed in
either of these two forms,

50
00:03:16,430 --> 00:03:20,360
we referred to last time
as the convolution sum.

51
00:03:20,360 --> 00:03:24,080
So the convolution
sum comes out of--

52
00:03:24,080 --> 00:03:25,910
or is a consequence of--

53
00:03:25,910 --> 00:03:30,210
linearity and shift invariance.

54
00:03:30,210 --> 00:03:35,060
Two additional conditions
are stability and causality.

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00:03:35,060 --> 00:03:39,290
Stability of a
system, in general,

56
00:03:39,290 --> 00:03:46,070
corresponds to the statement
that if the input sequence

57
00:03:46,070 --> 00:03:47,780
is bounded--

58
00:03:47,780 --> 00:03:51,800
in other words, if x of
n, essentially if x of n

59
00:03:51,800 --> 00:03:57,290
is finite for all n, including
as n goes to infinity,

60
00:03:57,290 --> 00:04:00,260
then a system is
said to be stable

61
00:04:00,260 --> 00:04:06,170
if the output sequence,
y of n, is also bounded.

62
00:04:06,170 --> 00:04:11,900
In other words, the magnitude
of y of n is finite for all n.

63
00:04:11,900 --> 00:04:15,590
If that's true for any
bounded input sequence

64
00:04:15,590 --> 00:04:17,570
that the output
sequence is bounded,

65
00:04:17,570 --> 00:04:20,420
the system is said to be stable.

66
00:04:20,420 --> 00:04:22,460
Now, that's a
statement that applies

67
00:04:22,460 --> 00:04:24,920
to general discrete
time systems.

68
00:04:24,920 --> 00:04:28,220
For the specific class of
systems that we'll be dealing

69
00:04:28,220 --> 00:04:32,000
with in this set of lessons,
namely the class of linear

70
00:04:32,000 --> 00:04:37,130
shift invariant systems, you
can show that an equivalent

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00:04:37,130 --> 00:04:42,350
statement of stability-- or
a necessary and sufficient

72
00:04:42,350 --> 00:04:44,450
condition for this to be true--

73
00:04:44,450 --> 00:04:48,830
is simply that the unit
sample response of the system

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00:04:48,830 --> 00:04:50,990
be absolutely summable.

75
00:04:50,990 --> 00:04:53,690
In other words, that the
sum of the magnitudes

76
00:04:53,690 --> 00:04:58,620
of the values in the unit
sample response are finite.

77
00:04:58,620 --> 00:05:02,280
For example, if we had a
unit sample response which

78
00:05:02,280 --> 00:05:07,410
was 2 to the n times a unit
step, 2 to the n, of course,

79
00:05:07,410 --> 00:05:09,780
as n increases--

80
00:05:09,780 --> 00:05:12,500
as n goes from 0 to infinity--

81
00:05:12,500 --> 00:05:16,020
2 the n grows
exponentially, in fact.

82
00:05:16,020 --> 00:05:20,430
And so, obviously, this is
not absolutely summable.

83
00:05:20,430 --> 00:05:26,120
So this is an example of a
system that would be unstable.

84
00:05:29,310 --> 00:05:34,110
Whereas, if h of n were a
1/2 to the n times u of n,

85
00:05:34,110 --> 00:05:36,570
so that for n greater than 0--

86
00:05:36,570 --> 00:05:38,970
and less than 0, of course,
both of these are 0--

87
00:05:38,970 --> 00:05:42,600
for n greater than 0, this
were decaying exponentially,

88
00:05:42,600 --> 00:05:46,260
if you sum up these
values from 0 to infinity,

89
00:05:46,260 --> 00:05:48,610
it converges to a finite number.

90
00:05:48,610 --> 00:05:55,122
So that this, in fact, would
correspond to a stable system.

91
00:05:55,122 --> 00:05:56,580
Now, we have the
option, of course,

92
00:05:56,580 --> 00:06:01,030
of talking about unstable
systems or stable systems.

93
00:06:01,030 --> 00:06:05,610
Generally, it is
true that stability

94
00:06:05,610 --> 00:06:10,410
is a condition on a system
that it's useful to impose.

95
00:06:10,410 --> 00:06:13,320
In other words, generally,
we would like our systems

96
00:06:13,320 --> 00:06:15,900
to be stable, although
there are actually

97
00:06:15,900 --> 00:06:18,540
some cases where
unstable systems are

98
00:06:18,540 --> 00:06:21,670
useful to talk about.

99
00:06:21,670 --> 00:06:29,550
The second property or condition
that it is useful sometimes

100
00:06:29,550 --> 00:06:33,210
to consider, is the
condition of causality.

101
00:06:33,210 --> 00:06:38,190
And causality, first of
all, for a general system,

102
00:06:38,190 --> 00:06:42,240
is a statement basically
that the system doesn't

103
00:06:42,240 --> 00:06:44,950
respond before you kick it.

104
00:06:44,950 --> 00:06:52,200
In other words, if we have
an input, x of n, the output,

105
00:06:52,200 --> 00:06:58,250
y of n, for some value
of n-- let's say n 1--

106
00:06:58,250 --> 00:07:04,130
depends only on x of n
for previous values of n.

107
00:07:04,130 --> 00:07:09,830
So for any n 1, the
statement is that y of n

108
00:07:09,830 --> 00:07:16,100
only depends on x of n for
previous values of x of n.

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00:07:16,100 --> 00:07:19,610
In other words, the system
can't anticipate the values

110
00:07:19,610 --> 00:07:23,240
that are going to be coming in.

111
00:07:23,240 --> 00:07:26,330
For a linear shift
invariant system,

112
00:07:26,330 --> 00:07:30,800
we can show that a necessary
and sufficient condition

113
00:07:30,800 --> 00:07:36,080
for causality is that the unit
sample response of the system

114
00:07:36,080 --> 00:07:38,720
be 0 for n less than 0.

115
00:07:38,720 --> 00:07:41,930
In other words, if the unit
sample response of the system

116
00:07:41,930 --> 00:07:44,510
is 0 for n less
than 0, the system

117
00:07:44,510 --> 00:07:47,120
is guaranteed to be causal.

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00:07:47,120 --> 00:07:50,870
If it's not 0 for n less
than 0, then the system

119
00:07:50,870 --> 00:07:54,790
is guaranteed not to be causal.

120
00:07:54,790 --> 00:07:58,090
Just for example, if we had a
unit sample response which was

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00:07:58,090 --> 00:08:01,130
2 to the n times u of minus n--

122
00:08:01,130 --> 00:08:04,830
in other words, 0 for n
greater than 0, and 2 the n

123
00:08:04,830 --> 00:08:07,240
for n less than 0,
this, of course,

124
00:08:07,240 --> 00:08:13,780
would correspond to
a non-causal system,

125
00:08:13,780 --> 00:08:17,320
since the unit sample
response has non-zero values

126
00:08:17,320 --> 00:08:20,460
for negative values of n.

127
00:08:20,460 --> 00:08:23,610
And we could also
examine stability.

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00:08:23,610 --> 00:08:29,280
It would turn out that this
corresponds to a stable system,

129
00:08:29,280 --> 00:08:33,350
although in the previous
example, we had talked about 2

130
00:08:33,350 --> 00:08:36,210
to the n for n
positive corresponds

131
00:08:36,210 --> 00:08:37,750
to an unstable system.

132
00:08:37,750 --> 00:08:39,960
The point is that if
you look at 2 to the n

133
00:08:39,960 --> 00:08:45,380
as the index n runs
negative, then that is--

134
00:08:45,380 --> 00:08:48,240
as n runs negative,
it's an exponential

135
00:08:48,240 --> 00:08:51,090
that's decaying in
negative values of n.

136
00:08:51,090 --> 00:08:53,610
So then, in fact, this
is absolutely summable--

137
00:08:53,610 --> 00:08:57,870
that makes it stable, but
it's obviously non-causal.

138
00:08:57,870 --> 00:08:59,550
Now, I want to stress--

139
00:08:59,550 --> 00:09:02,790
it's a very important
point to stress--

140
00:09:02,790 --> 00:09:06,180
that causality is, of
course, a useful thing

141
00:09:06,180 --> 00:09:09,870
to inquire about about a
system, it's useful to ask,

142
00:09:09,870 --> 00:09:12,570
is the system causal
or is it not causal?

143
00:09:12,570 --> 00:09:17,340
But generally, it is
not necessarily true

144
00:09:17,340 --> 00:09:20,190
that causality is a
condition that we'll

145
00:09:20,190 --> 00:09:22,410
want to impose on the system.

146
00:09:22,410 --> 00:09:27,750
There are many examples of
useful, non-causal systems,

147
00:09:27,750 --> 00:09:30,510
and in many
instances, we'll want

148
00:09:30,510 --> 00:09:33,540
to talk about systems
which are non-causal.

149
00:09:33,540 --> 00:09:35,970
So again, it's useful
to inquire about

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00:09:35,970 --> 00:09:38,250
whether a system is
causal or not causal,

151
00:09:38,250 --> 00:09:42,990
but it is not generally useful
to constrain ourselves to talk

152
00:09:42,990 --> 00:09:46,120
only about causal systems.

153
00:09:46,120 --> 00:09:48,720
The story is somewhat different
for stability in the sense

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00:09:48,720 --> 00:09:53,220
that an unstable system
is somewhat less useful

155
00:09:53,220 --> 00:09:56,590
than a non-causal system.

156
00:09:56,590 --> 00:10:01,440
So these are two additional
conditions, or properties,

157
00:10:01,440 --> 00:10:05,160
that we'll sometimes want
to inquire about when

158
00:10:05,160 --> 00:10:08,670
we talk about a system.

159
00:10:08,670 --> 00:10:12,480
In general, for linear
shift invariant systems,

160
00:10:12,480 --> 00:10:16,500
there is a wide latitude
in terms of the description

161
00:10:16,500 --> 00:10:18,060
of those systems.

162
00:10:18,060 --> 00:10:21,360
And that is a similar
type of statement

163
00:10:21,360 --> 00:10:24,420
that applies in the
continuous time case, also.

164
00:10:24,420 --> 00:10:26,550
Just as in the
continuous time case,

165
00:10:26,550 --> 00:10:31,440
it is useful to
concentrate, in many cases,

166
00:10:31,440 --> 00:10:35,670
on systems that can
be implemented with

167
00:10:35,670 --> 00:10:36,960
r's, l's, and c's--

168
00:10:36,960 --> 00:10:38,580
and consequently
are representable

169
00:10:38,580 --> 00:10:41,940
by linear constant coefficient
differential equations.

170
00:10:41,940 --> 00:10:44,130
In the discrete time
case, it's often

171
00:10:44,130 --> 00:10:47,700
useful to concentrate
on systems that

172
00:10:47,700 --> 00:10:53,010
are describable by linear
constant coefficient difference

173
00:10:53,010 --> 00:10:54,540
equations.

174
00:10:54,540 --> 00:10:59,430
So I would like to
just briefly introduce

175
00:10:59,430 --> 00:11:03,450
the class of systems
which are representable

176
00:11:03,450 --> 00:11:07,230
by linear constant coefficient
difference equations.

177
00:11:07,230 --> 00:11:11,850
And the discussion
in this lecture

178
00:11:11,850 --> 00:11:14,700
is only a brief introduction,
and we'll, in fact,

179
00:11:14,700 --> 00:11:16,830
be returning to this
class of systems

180
00:11:16,830 --> 00:11:19,030
several times over
this set of lessons.

181
00:11:22,380 --> 00:11:25,290
We refer to an Nth order--

182
00:11:25,290 --> 00:11:28,290
linear constant coefficient
difference equation--

183
00:11:28,290 --> 00:11:31,990
as being in the form
that I've indicated here.

184
00:11:31,990 --> 00:11:37,470
And what it consists of
is a linear combination

185
00:11:37,470 --> 00:11:44,850
of the delayed output sequences
equal to a linear combination

186
00:11:44,850 --> 00:11:47,310
of delayed input sequences.

187
00:11:47,310 --> 00:11:49,980
A differential equation, of
course, in continuous time

188
00:11:49,980 --> 00:11:52,830
involves linear
combinations of derivatives.

189
00:11:52,830 --> 00:11:57,690
The corresponding situation
in the discrete time case,

190
00:11:57,690 --> 00:12:01,870
is a linear combination
of differences.

191
00:12:01,870 --> 00:12:08,520
So this is, then, a
general Nth order, linear--

192
00:12:08,520 --> 00:12:11,010
in the sense that it's
a linear combination--

193
00:12:11,010 --> 00:12:16,170
constant coefficient-- meaning
that these are constant

194
00:12:16,170 --> 00:12:18,780
numbers, as opposed to
being functions of n--

195
00:12:18,780 --> 00:12:21,450
difference equation--
meaning it involves

196
00:12:21,450 --> 00:12:23,505
differences of the input
and output sequence.

197
00:12:26,070 --> 00:12:28,620
The order of the
difference equation

198
00:12:28,620 --> 00:12:33,780
generally is used to refer to
the number of delays required

199
00:12:33,780 --> 00:12:35,790
in the output sequence.

200
00:12:35,790 --> 00:12:40,020
In general, the number of
delays in the input sequence, M,

201
00:12:40,020 --> 00:12:43,620
does not have to be equal to N.
But it's generally convenient

202
00:12:43,620 --> 00:12:45,960
to refer to the order
as corresponding

203
00:12:45,960 --> 00:12:48,855
to the number of delays
involved in the output sequence.

204
00:12:51,390 --> 00:12:56,550
For the 0th order
difference equation,

205
00:12:56,550 --> 00:13:00,450
the solution if it corresponds
to a representation

206
00:13:00,450 --> 00:13:02,710
of a linear shift
and invariant system,

207
00:13:02,710 --> 00:13:05,310
the solution is trivial--

208
00:13:05,310 --> 00:13:06,910
very straightforward.

209
00:13:06,910 --> 00:13:09,810
Particular, let's examine
what this difference equation

210
00:13:09,810 --> 00:13:13,620
reduces to, if N is equal to 0.

211
00:13:13,620 --> 00:13:16,800
And just for
convenience, we'll choose

212
00:13:16,800 --> 00:13:18,870
the coefficients
to be normalized

213
00:13:18,870 --> 00:13:21,240
so that a sub 0 is equal to 1.

214
00:13:21,240 --> 00:13:24,600
Obviously anything I
say is straightforwardly

215
00:13:24,600 --> 00:13:27,620
generalized for a
0 not equal to 1,

216
00:13:27,620 --> 00:13:33,520
but that's just a convenient
normalization to impose.

217
00:13:33,520 --> 00:13:36,970
All right, if N is equal to
0, and a 0 is equal to 1,

218
00:13:36,970 --> 00:13:39,120
then what this
equation looks like

219
00:13:39,120 --> 00:13:43,290
is just on the left hand side,
y of n, and the right hand side

220
00:13:43,290 --> 00:13:44,880
as it is.

221
00:13:44,880 --> 00:13:50,510
So we have y of n is
equal to this sum.

222
00:13:50,510 --> 00:13:54,960
And that looks suspiciously
like the convolution sum--

223
00:13:54,960 --> 00:14:00,040
it involves a linear combination
only of delayed input

224
00:14:00,040 --> 00:14:01,540
sequences.

225
00:14:01,540 --> 00:14:07,340
And so, in fact, this is
identical to the convolution

226
00:14:07,340 --> 00:14:08,110
sum.

227
00:14:08,110 --> 00:14:13,000
If we thought of this, say,
as h of r, h of r, then,

228
00:14:13,000 --> 00:14:14,920
is equal to b sub r.

229
00:14:14,920 --> 00:14:19,720
So the unit sample response
corresponding to the 0th order

230
00:14:19,720 --> 00:14:25,660
difference equation is just
this set of coefficients,

231
00:14:25,660 --> 00:14:31,120
and, of course, r
runs only from 0 to M,

232
00:14:31,120 --> 00:14:36,850
so it's the coefficients b
sub n for n between 0 and M.

233
00:14:36,850 --> 00:14:39,440
And it's equal to 0, otherwise.

234
00:14:39,440 --> 00:14:40,891
So that's very straightforward.

235
00:14:40,891 --> 00:14:42,640
We can pick off the
solution, essentially,

236
00:14:42,640 --> 00:14:46,090
by recognizing this as
the convolution sum.

237
00:14:46,090 --> 00:14:52,500
Obviously, another way of
obtaining the unit sample

238
00:14:52,500 --> 00:14:54,990
response corresponding
to this system

239
00:14:54,990 --> 00:14:58,500
is simply to plug in a
unit sample for x of n.

240
00:14:58,500 --> 00:15:00,810
And you'll see, in fact,
that what comes rolling out

241
00:15:00,810 --> 00:15:04,260
are these coefficients
b sub r, or b sub n.

242
00:15:04,260 --> 00:15:09,120
So if this is used to describe
a linear shift invariant system,

243
00:15:09,120 --> 00:15:11,970
the unit sample response
corresponds simply

244
00:15:11,970 --> 00:15:16,050
to the coefficients in
the difference equation.

245
00:15:16,050 --> 00:15:19,500
For n not equal to 0, it's
not quite as straightforward

246
00:15:19,500 --> 00:15:20,910
as that.

247
00:15:20,910 --> 00:15:27,150
First of all, let me point
out that for N not equal to 0,

248
00:15:27,150 --> 00:15:31,650
the left hand side of this
equation involves y of n,

249
00:15:31,650 --> 00:15:34,290
y of n minus 1, y of
minus 2, et cetera,

250
00:15:34,290 --> 00:15:38,070
up to y of little
n minus capital N.

251
00:15:38,070 --> 00:15:43,880
We can take all of the terms
except the one involving y of n

252
00:15:43,880 --> 00:15:47,600
over to the right hand
side of the equation,

253
00:15:47,600 --> 00:15:50,150
and end up with
an equation which

254
00:15:50,150 --> 00:15:55,640
expresses y of n as a linear
combination of delayed inputs,

255
00:15:55,640 --> 00:16:01,440
and a linear combination
of past output sequences.

256
00:16:01,440 --> 00:16:04,830
If we rewrite this
equation in this form--

257
00:16:04,830 --> 00:16:06,860
and again, just for
convenience, we'll

258
00:16:06,860 --> 00:16:13,250
choose a 0 equal to 1, simply
normalizing that coefficient.

259
00:16:13,250 --> 00:16:16,820
Then, the difference
equation that we end up

260
00:16:16,820 --> 00:16:22,070
with is in the form that
I've indicated here, y of n

261
00:16:22,070 --> 00:16:27,980
is a linear combination of
delayed input sequences minus--

262
00:16:27,980 --> 00:16:31,130
because we brought that from
the other side of the equation--

263
00:16:31,130 --> 00:16:33,920
minus a linear
combination of past--

264
00:16:33,920 --> 00:16:35,960
of the previous output values.

265
00:16:38,710 --> 00:16:43,960
What that says is
that we presumably

266
00:16:43,960 --> 00:16:48,610
can iterate this equation,
or equivalently, what it says

267
00:16:48,610 --> 00:16:51,430
is that the difference
equation corresponds

268
00:16:51,430 --> 00:16:56,320
to an explicit input/output
relationship for the system,

269
00:16:56,320 --> 00:17:00,940
because if somehow we could
get the equation going,

270
00:17:00,940 --> 00:17:05,319
then we can solve for y of
n-- we can solve for y of n

271
00:17:05,319 --> 00:17:09,880
if we have the right
previous values for y of n.

272
00:17:09,880 --> 00:17:12,671
And then we can solve
for y of n plus 1, y of n

273
00:17:12,671 --> 00:17:13,420
plus 2, et cetera.

274
00:17:13,420 --> 00:17:17,960
In other words, we can continue
to iterate the equation.

275
00:17:17,960 --> 00:17:21,010
What's required to
do that are that we

276
00:17:21,010 --> 00:17:22,140
have to know the input--

277
00:17:22,140 --> 00:17:24,339
and we assume, of course,
that we know that--

278
00:17:24,339 --> 00:17:28,750
and we have to know some
previous output values.

279
00:17:28,750 --> 00:17:32,960
And to know the
previous output values,

280
00:17:32,960 --> 00:17:37,010
then, means that there is some
additional information that we

281
00:17:37,010 --> 00:17:40,100
have to specify, and
those we'll generally

282
00:17:40,100 --> 00:17:41,960
refer to as the
boundary conditions,

283
00:17:41,960 --> 00:17:43,970
or the initial conditions.

284
00:17:43,970 --> 00:17:46,550
For example, let's
see how this equation

285
00:17:46,550 --> 00:17:51,950
would work if we focused
on the first order case.

286
00:17:51,950 --> 00:17:57,600
So the first order case,
capital N is equal to 1.

287
00:17:57,600 --> 00:18:00,740
We have the equation, y of
n minus a, y of n minus 1

288
00:18:00,740 --> 00:18:03,110
equals x of n.

289
00:18:03,110 --> 00:18:07,950
And let's find the unit sample
response for the system--

290
00:18:07,950 --> 00:18:12,110
in other words, choosing x
of n equal to a unit sample.

291
00:18:12,110 --> 00:18:15,590
And for the initial
conditions, let's assume

292
00:18:15,590 --> 00:18:19,640
that, for a unit sample
input, the output is

293
00:18:19,640 --> 00:18:22,730
equal to 0 for n less than 0.

294
00:18:22,730 --> 00:18:26,780
Now, when we do that, we're
making a specific assumption

295
00:18:26,780 --> 00:18:28,250
about causality--

296
00:18:28,250 --> 00:18:31,850
we're imposing the
boundary conditions

297
00:18:31,850 --> 00:18:35,810
that the unit sample response
has to be 0 for n less than 0--

298
00:18:35,810 --> 00:18:38,600
that's exactly the necessary
and sufficient condition

299
00:18:38,600 --> 00:18:40,410
that we need for causality.

300
00:18:40,410 --> 00:18:44,570
So basically, we're saying that
we're imposing on the solution

301
00:18:44,570 --> 00:18:47,720
that we're about to generate,
the additional condition

302
00:18:47,720 --> 00:18:49,880
of causality.

303
00:18:49,880 --> 00:18:53,440
All right, well now,
rewriting this equation

304
00:18:53,440 --> 00:18:55,990
by taking this term over
to the right hand side,

305
00:18:55,990 --> 00:18:59,050
and taking account of
the fact that we're

306
00:18:59,050 --> 00:19:01,660
considering the input
to be a unit sample,

307
00:19:01,660 --> 00:19:04,960
we have the output
sequences unit

308
00:19:04,960 --> 00:19:09,100
sample plus a times the
output sequence delayed.

309
00:19:09,100 --> 00:19:14,810
And now let's run through an
iteration of this equation,

310
00:19:14,810 --> 00:19:18,070
obtaining some
successive values.

311
00:19:18,070 --> 00:19:22,270
y of minus 1 we can
get simply by referring

312
00:19:22,270 --> 00:19:24,250
to the initial condition.

313
00:19:24,250 --> 00:19:27,400
We stated there y of n
is 0 for n less than 0,

314
00:19:27,400 --> 00:19:33,090
and so that means, of course,
that y of minus 1 is 0.

315
00:19:33,090 --> 00:19:38,510
y of 0 is equal to delta of
0, but that's equal to 1,

316
00:19:38,510 --> 00:19:42,360
plus a times y of
minus 1, which was 0.

317
00:19:42,360 --> 00:19:46,644
So y of 0 is equal to 1.

318
00:19:46,644 --> 00:19:50,720
y of 1 is equal to
delta of 1, which is 0--

319
00:19:50,720 --> 00:19:54,660
the unit sample is only
non-0 at n equals 0.

320
00:19:54,660 --> 00:19:57,940
So that 0 plus a times y of 0--

321
00:19:57,940 --> 00:19:59,880
y of 0 is equal to 1--

322
00:19:59,880 --> 00:20:06,090
so y of 1 is equal
to a times 1, or a.

323
00:20:06,090 --> 00:20:11,030
y of 2, if we follow that
through, is equal to a squared.

324
00:20:11,030 --> 00:20:13,100
And if you run through
a few more of those,

325
00:20:13,100 --> 00:20:17,410
you'll see rather quickly that
what we get for the unit sample

326
00:20:17,410 --> 00:20:22,750
response, imposing the
condition of causality,

327
00:20:22,750 --> 00:20:27,850
is that the unit sample
response is a to the N for n

328
00:20:27,850 --> 00:20:29,470
greater than 0.

329
00:20:29,470 --> 00:20:32,080
Of course, it's 0 for
n less than 0, because

330
00:20:32,080 --> 00:20:33,800
of our initial condition.

331
00:20:33,800 --> 00:20:39,220
So it's a to N times u of n.

332
00:20:39,220 --> 00:20:43,630
And it's obviously causal,
because that's what we imposed.

333
00:20:43,630 --> 00:20:47,360
It may or may not be stable,
depending on the value of a.

334
00:20:47,360 --> 00:20:52,850
In particular, if the
magnitude of a is less than 1,

335
00:20:52,850 --> 00:20:56,200
then this sequence will
be decaying exponentially

336
00:20:56,200 --> 00:20:57,890
as n increases.

337
00:20:57,890 --> 00:21:00,850
So for the magnitude
of a less than 1,

338
00:21:00,850 --> 00:21:06,590
this corresponds
to a stable system.

339
00:21:06,590 --> 00:21:09,920
Now this isn't the
only initial condition

340
00:21:09,920 --> 00:21:14,750
that we can impose on
the system and still

341
00:21:14,750 --> 00:21:18,500
have it correspond to a
linear shift invariant system.

342
00:21:18,500 --> 00:21:21,560
We could, alternatively,
impose a different set

343
00:21:21,560 --> 00:21:25,310
of initial conditions,
which is the--

344
00:21:25,310 --> 00:21:28,010
or boundary conditions,
really, because they're not,

345
00:21:28,010 --> 00:21:30,140
for this case,
initial conditions--

346
00:21:30,140 --> 00:21:33,500
boundary conditions
that state instead

347
00:21:33,500 --> 00:21:36,470
of the statement that
the system is causal,

348
00:21:36,470 --> 00:21:39,890
the statement that the
system is totally non-causal.

349
00:21:39,890 --> 00:21:42,890
What I mean is, let's
take the same example--

350
00:21:42,890 --> 00:21:45,830
the same difference equation--

351
00:21:45,830 --> 00:21:50,360
let's again choose the
input to be a unit sample,

352
00:21:50,360 --> 00:21:57,140
but let's impose another
condition on the solution,

353
00:21:57,140 --> 00:21:58,580
boundary condition.

354
00:21:58,580 --> 00:22:01,160
Namely that the
unit sample response

355
00:22:01,160 --> 00:22:04,700
is 0 for n greater
than 0, rather than

356
00:22:04,700 --> 00:22:10,750
the boundary condition that says
that it's 0 for n less than 0.

357
00:22:10,750 --> 00:22:14,950
In that case, it's convenient
to generate the solution

358
00:22:14,950 --> 00:22:19,450
iteratively by running the
difference equation backwards,

359
00:22:19,450 --> 00:22:23,270
in other words, by
expressing y of n minus 1,

360
00:22:23,270 --> 00:22:26,950
in terms of y of n and x
of n-- this is just simply

361
00:22:26,950 --> 00:22:30,260
another rearrangement of
the difference equation.

362
00:22:30,260 --> 00:22:36,580
And with this initial condition,
if we look at y of 1--

363
00:22:36,580 --> 00:22:40,210
of course, y of 1 is equal to
0 by virtue of the boundary

364
00:22:40,210 --> 00:22:45,130
condition that we've imposed,
so this is equal to 0--

365
00:22:45,130 --> 00:22:49,790
y of 0 corresponds to n
equal to 1 in this equation.

366
00:22:49,790 --> 00:22:55,990
So n equal to 1 here, we have
1 over a times y of 1, which is

367
00:22:55,990 --> 00:22:58,690
0 minus delta of 1 which is 0.

368
00:22:58,690 --> 00:23:01,970
So y of 0 is again equal to 0.

369
00:23:01,970 --> 00:23:07,030
y of minus 1 corresponds
to n equal to 0,

370
00:23:07,030 --> 00:23:11,140
so we have 1 over a times
y of 0, which was 0,

371
00:23:11,140 --> 00:23:13,790
minus delta of 0, which is 1.

372
00:23:13,790 --> 00:23:18,980
So we get minus a
minus to the minus 1.

373
00:23:18,980 --> 00:23:22,736
y of minus 2, we would
substitute n equals minus 1.

374
00:23:22,736 --> 00:23:28,430
y of minus 1 we had as
minus a to the minus 1.

375
00:23:28,430 --> 00:23:33,980
And delta of minus 1 is
0, so we have here minus a

376
00:23:33,980 --> 00:23:39,530
to the minus 2, and it
continues on that way.

377
00:23:39,530 --> 00:23:41,780
And if you generated
some more of these,

378
00:23:41,780 --> 00:23:44,270
what you'd see rather
quickly is that this

379
00:23:44,270 --> 00:23:51,480
is of the form minus a to the
n times u of minus n minus 1.

380
00:23:51,480 --> 00:23:56,150
In other words, it is an
exponential to form a to the n,

381
00:23:56,150 --> 00:23:59,600
as we found for the
causal solution, also.

382
00:23:59,600 --> 00:24:02,960
But it's 0 for n greater
than 0, and it's of the form

383
00:24:02,960 --> 00:24:05,600
a to the n for n less than 0.

384
00:24:05,600 --> 00:24:07,280
What that means,
in particular, is

385
00:24:07,280 --> 00:24:10,550
that if we again
inquired as to whether it

386
00:24:10,550 --> 00:24:15,160
was stable or unstable, for
the magnitude of a less than 1,

387
00:24:15,160 --> 00:24:17,330
what we would
find, in this case,

388
00:24:17,330 --> 00:24:22,530
is that it's unstable
rather than stable.

389
00:24:22,530 --> 00:24:25,110
So what we've
seen, then, is that

390
00:24:25,110 --> 00:24:29,940
a linear constant coefficient
difference equation, by itself,

391
00:24:29,940 --> 00:24:32,670
doesn't specify
uniquely a system--

392
00:24:32,670 --> 00:24:36,810
it requires a set of
initial conditions.

393
00:24:36,810 --> 00:24:39,780
And depending on the initial
conditions or boundary

394
00:24:39,780 --> 00:24:42,960
conditions that are
imposed, it may correspond

395
00:24:42,960 --> 00:24:45,030
to a causal system,
or it may correspond

396
00:24:45,030 --> 00:24:46,920
to a non-causal system.

397
00:24:46,920 --> 00:24:50,400
And in some situations,
we might want

398
00:24:50,400 --> 00:24:54,530
it to correspond to
either one of those two.

399
00:24:54,530 --> 00:24:57,890
Something that I haven't
stated explicitly,

400
00:24:57,890 --> 00:25:02,180
but there is some discussion
of this in the text,

401
00:25:02,180 --> 00:25:07,890
is the fact that it is not
for every set of boundary

402
00:25:07,890 --> 00:25:11,190
conditions that a linear
constant coefficient difference

403
00:25:11,190 --> 00:25:16,200
equation corresponds to a
linear shift invariant system,

404
00:25:16,200 --> 00:25:21,000
but it does in particular
for the two sets of boundary

405
00:25:21,000 --> 00:25:22,980
conditions that
are imposed here.

406
00:25:22,980 --> 00:25:26,090
More generally, what's required
of the boundary conditions,

407
00:25:26,090 --> 00:25:29,340
so that the difference equation
corresponds to a linear shift

408
00:25:29,340 --> 00:25:32,700
invariant system, is that
the boundary conditions

409
00:25:32,700 --> 00:25:36,330
have to be consistent
with the statement

410
00:25:36,330 --> 00:25:43,680
that if the input, x of
n were 0, 0 for all n,

411
00:25:43,680 --> 00:25:46,600
that the output would
also be 0 for all n.

412
00:25:46,600 --> 00:25:49,860
And there is some additional
discussion of this in the text.

413
00:25:49,860 --> 00:25:56,040
I indicate also that we'll be
returning from time to time

414
00:25:56,040 --> 00:25:59,730
to further discussion of linear
constant coefficient difference

415
00:25:59,730 --> 00:26:03,420
equations, and discussing,
in particular, other ways

416
00:26:03,420 --> 00:26:08,310
of solving this
class of equations.

417
00:26:08,310 --> 00:26:10,680
For the remainder
of the lecture,

418
00:26:10,680 --> 00:26:15,840
I'd like to focus on an
alternative representation

419
00:26:15,840 --> 00:26:18,180
of linear shift
invariant systems,

420
00:26:18,180 --> 00:26:23,430
alternative to the time
domain, or convolution sum

421
00:26:23,430 --> 00:26:28,350
representation, that we dealt
with in the last lecture.

422
00:26:28,350 --> 00:26:31,890
In particular, the
very useful alternative

423
00:26:31,890 --> 00:26:36,330
is a description of linear
shift invariant systems in terms

424
00:26:36,330 --> 00:26:38,850
of its frequency response.

425
00:26:38,850 --> 00:26:40,710
In other words, in
terms of its response

426
00:26:40,710 --> 00:26:43,530
either to sinusoidal
excitations,

427
00:26:43,530 --> 00:26:48,330
or to complex
exponential excitations.

428
00:26:48,330 --> 00:26:54,720
Well, the basic notion
behind the frequency response

429
00:26:54,720 --> 00:26:58,740
description of linear
shift invariant systems

430
00:26:58,740 --> 00:27:03,670
is the fact that
complex exponentials

431
00:27:03,670 --> 00:27:07,930
are eigenfunctions of linear
shift invariant systems.

432
00:27:07,930 --> 00:27:11,980
What I mean by that is that for
linear shift invariant systems,

433
00:27:11,980 --> 00:27:14,350
if you put in a
complex exponential,

434
00:27:14,350 --> 00:27:17,230
you get out a
complex exponential.

435
00:27:17,230 --> 00:27:21,970
And the only change is
in the complex amplitude

436
00:27:21,970 --> 00:27:25,180
of the complex exponential--
the functional form is

437
00:27:25,180 --> 00:27:27,730
the same, so complex
exponential in gives you

438
00:27:27,730 --> 00:27:30,370
a complex exponential out.

439
00:27:30,370 --> 00:27:34,750
We can see that rather
easily by referring

440
00:27:34,750 --> 00:27:39,370
to the convolution sum
description of linear shift

441
00:27:39,370 --> 00:27:42,280
invariant systems, as
I've indicated here.

442
00:27:42,280 --> 00:27:46,660
In particular, suppose that we
choose an input sequence which

443
00:27:46,660 --> 00:27:50,370
is a complex exponential.

444
00:27:50,370 --> 00:27:54,650
And let's substitute this
into this expression,

445
00:27:54,650 --> 00:27:58,500
so that the output is
then the sum of h of k

446
00:27:58,500 --> 00:28:02,140
e to the j omega n minus k.

447
00:28:02,140 --> 00:28:09,050
This term can be decomposed into
a product of two exponentials,

448
00:28:09,050 --> 00:28:17,920
e to the j omega n, and
e to the minus j omega k.

449
00:28:17,920 --> 00:28:22,960
And since e to the j omega n
doesn't depend on the index k,

450
00:28:22,960 --> 00:28:28,450
we can take that
piece outside the sum,

451
00:28:28,450 --> 00:28:31,750
and what we're
left with is y of n

452
00:28:31,750 --> 00:28:36,330
is e to the j omega n
times the sum of h of k

453
00:28:36,330 --> 00:28:41,770
e to the minus j omega k,
as I've indicated up here.

454
00:28:41,770 --> 00:28:45,220
So we started with a
complex exponential sequence

455
00:28:45,220 --> 00:28:47,710
going into the system.

456
00:28:47,710 --> 00:28:50,350
None of this stuff
over here depends on n,

457
00:28:50,350 --> 00:28:53,470
so when we sum all that
up, that's just a number--

458
00:28:53,470 --> 00:28:55,090
it's a function
of omega, depends

459
00:28:55,090 --> 00:28:57,850
on what complex
frequency we've put in.

460
00:28:57,850 --> 00:29:00,250
But it doesn't depend on n.

461
00:29:00,250 --> 00:29:05,860
And, in fact,
notationally, we'll

462
00:29:05,860 --> 00:29:09,790
refer to this as the
H of e to the j omega.

463
00:29:09,790 --> 00:29:14,680
So consequently,
the output sequence,

464
00:29:14,680 --> 00:29:17,560
due to a complex
exponential input,

465
00:29:17,560 --> 00:29:24,625
is this number, or
function of omega, times

466
00:29:24,625 --> 00:29:26,910
e of the j omega n.

467
00:29:26,910 --> 00:29:29,520
The change in the
complex amplitude, then,

468
00:29:29,520 --> 00:29:33,720
is H of e to the j omega,
but the functional form

469
00:29:33,720 --> 00:29:36,820
of the output is the same as the
functional form of the input.

470
00:29:36,820 --> 00:29:38,490
And that's what we
mean when we say

471
00:29:38,490 --> 00:29:41,730
that the complex exponential
is an eigenfunction

472
00:29:41,730 --> 00:29:44,860
of a linear shift
invariant system.

473
00:29:44,860 --> 00:29:49,080
So finally, H of e to the j
omega is given by this sum--

474
00:29:49,080 --> 00:29:54,810
I've just changed the
index of summation

475
00:29:54,810 --> 00:29:56,970
from what I had up here,
but obviously that's

476
00:29:56,970 --> 00:29:59,470
no important change.

477
00:29:59,470 --> 00:30:04,680
And we will refer to this,
or to H of e to the j omega,

478
00:30:04,680 --> 00:30:09,680
as the frequency
response of the system.

479
00:30:09,680 --> 00:30:14,150
One of the reasons why
the frequency response

480
00:30:14,150 --> 00:30:15,330
is important--

481
00:30:15,330 --> 00:30:17,720
of course, one of the facts
that you can see right away

482
00:30:17,720 --> 00:30:21,620
is that it's easily
obtained directly

483
00:30:21,620 --> 00:30:24,230
from the unit sample response.

484
00:30:24,230 --> 00:30:29,000
One of the reasons why the
frequency response is useful

485
00:30:29,000 --> 00:30:33,470
is because it's essentially
the frequency response that

486
00:30:33,470 --> 00:30:39,230
allows us to obtain quite easily
the response of the system

487
00:30:39,230 --> 00:30:43,340
to sinusoidal excitations.

488
00:30:43,340 --> 00:30:46,820
And, as we'll see
in later lectures,

489
00:30:46,820 --> 00:30:49,730
starting actually
with the next lecture,

490
00:30:49,730 --> 00:30:53,810
essentially arbitrary sequences
can be represented either

491
00:30:53,810 --> 00:30:57,110
as linear combinations
of complex exponentials,

492
00:30:57,110 --> 00:31:01,910
or as linear combinations
of sinusoidal sequences.

493
00:31:01,910 --> 00:31:04,820
And so, if we know
what the response is

494
00:31:04,820 --> 00:31:08,360
to a complex exponential
or to sinusoidal sequences,

495
00:31:08,360 --> 00:31:10,970
we can, in effect,
describe the response

496
00:31:10,970 --> 00:31:12,350
to arbitrary sequences.

497
00:31:12,350 --> 00:31:16,100
We'll see all of that coming
up in the next lecture.

498
00:31:16,100 --> 00:31:20,300
But to see how this
frequency response relates

499
00:31:20,300 --> 00:31:24,980
to the sinusoidal response,
the relation essentially

500
00:31:24,980 --> 00:31:28,580
pops out from the
fact that if we

501
00:31:28,580 --> 00:31:35,150
have a sinusoidal excitation,
a sinusoidal excitation

502
00:31:35,150 --> 00:31:39,290
can be represented as a
linear combination of two

503
00:31:39,290 --> 00:31:41,270
complex exponentials--

504
00:31:41,270 --> 00:31:43,420
as I've indicated here.

505
00:31:43,420 --> 00:31:47,720
So a over 2 to the j
phi, e to j omega 0 n,

506
00:31:47,720 --> 00:31:51,560
this is one complex exponential
with a complex frequency omega

507
00:31:51,560 --> 00:31:57,440
0, and a complex amplitude,
a over 2 e to j phi.

508
00:31:57,440 --> 00:32:00,380
Then its complex
conjugate term, those two

509
00:32:00,380 --> 00:32:04,980
added up give us a
complex exponential.

510
00:32:04,980 --> 00:32:11,790
We can find the response
to each of these simply

511
00:32:11,790 --> 00:32:13,200
from the frequency response.

512
00:32:13,200 --> 00:32:15,870
We know that all that happens
to a complex exponential

513
00:32:15,870 --> 00:32:19,260
is that its complex
amplitude gets multiplied

514
00:32:19,260 --> 00:32:21,610
by the frequency response.

515
00:32:21,610 --> 00:32:25,170
And so this is one complex
exponential, and another one.

516
00:32:25,170 --> 00:32:27,460
We're talking about
linear systems,

517
00:32:27,460 --> 00:32:31,330
so the response of
the sum of these two

518
00:32:31,330 --> 00:32:33,510
is the sum of the responses.

519
00:32:33,510 --> 00:32:40,320
And so if we express the
frequency response in a polar

520
00:32:40,320 --> 00:32:47,850
form, as I've indicated here,
in terms of magnitude and phase,

521
00:32:47,850 --> 00:32:52,050
and if you track through what
happens when you multiply each

522
00:32:52,050 --> 00:32:56,580
of these complex exponentials
by the appropriate frequency

523
00:32:56,580 --> 00:32:59,610
response-- this one at
a frequency omega 0,

524
00:32:59,610 --> 00:33:02,630
this one at a frequency
minus omega 0--

525
00:33:02,630 --> 00:33:05,310
and add the terms
back together again,

526
00:33:05,310 --> 00:33:08,340
the resulting
output sequence has

527
00:33:08,340 --> 00:33:14,130
a change in real amplitude
given by, or dictated

528
00:33:14,130 --> 00:33:18,030
by the magnitude of
the frequency response,

529
00:33:18,030 --> 00:33:24,780
and a change in phase dictated
by the angle of the frequency

530
00:33:24,780 --> 00:33:25,340
response.

531
00:33:25,340 --> 00:33:27,900
In other words, by
this theta of omega.

532
00:33:27,900 --> 00:33:30,840
So the frequency response,
when thought of in polar form--

533
00:33:30,840 --> 00:33:32,820
magnitude and angle--

534
00:33:32,820 --> 00:33:36,210
the magnitude represents the
change in the real amplitude

535
00:33:36,210 --> 00:33:39,600
of a sinusoidal excitation.

536
00:33:39,600 --> 00:33:43,110
And the angle, or
complex argument,

537
00:33:43,110 --> 00:33:45,190
represents the phase shift.

538
00:33:45,190 --> 00:33:47,250
And that is exactly
analogous, exactly

539
00:33:47,250 --> 00:33:52,260
identical to what we're used
to in the continuous time case.

540
00:33:52,260 --> 00:33:54,210
Well, let's just
look at an example

541
00:33:54,210 --> 00:33:56,850
of a simple linear
shift invariant system,

542
00:33:56,850 --> 00:33:59,795
and the resulting
frequency response.

543
00:34:02,600 --> 00:34:06,220
Let's return to the first
order difference equation

544
00:34:06,220 --> 00:34:09,969
that we talked about
just a short time ago.

545
00:34:09,969 --> 00:34:14,650
We saw that there
were two solutions

546
00:34:14,650 --> 00:34:16,725
that we could generate
for this, depending

547
00:34:16,725 --> 00:34:18,100
on whether we
assumed that it was

548
00:34:18,100 --> 00:34:22,030
a causal or non-causal system.

549
00:34:22,030 --> 00:34:24,790
Let's focus on the causal case.

550
00:34:24,790 --> 00:34:28,310
If the system was causal,
we generated, essentially,

551
00:34:28,310 --> 00:34:33,820
iteratively the solution that
the unit sample response is

552
00:34:33,820 --> 00:34:36,380
a to the n times u of n.

553
00:34:36,380 --> 00:34:39,580
And let's assume that we're
talking about a stable system,

554
00:34:39,580 --> 00:34:43,761
so that a is magnitude
of a is less than 1--

555
00:34:43,761 --> 00:34:45,969
of course, this could be
negative and still be stable

556
00:34:45,969 --> 00:34:47,320
between minus 1 and 0.

557
00:34:50,440 --> 00:34:54,230
I'll assume in the pictures
that I draw that a is, in fact,

558
00:34:54,230 --> 00:34:57,530
positive, and then
also less than 1.

559
00:35:00,500 --> 00:35:03,390
Now, the expression for
the frequency response

560
00:35:03,390 --> 00:35:07,410
of the system, from what
we derived just above,

561
00:35:07,410 --> 00:35:15,210
is the sum overall n of h of n
times e to the minus j omega n.

562
00:35:15,210 --> 00:35:20,010
Because of the unit step in
here, the limits on the sum

563
00:35:20,010 --> 00:35:23,940
change from 0 to infinity--
in other words, all of this

564
00:35:23,940 --> 00:35:28,260
is going to be 0 from minus
infinity up to 0, because

565
00:35:28,260 --> 00:35:30,100
of the unit step.

566
00:35:30,100 --> 00:35:32,980
For n greater than
0, we have a to the n

567
00:35:32,980 --> 00:35:39,610
times this exponential, so
we have a to the n times

568
00:35:39,610 --> 00:35:41,800
e to the--

569
00:35:41,800 --> 00:35:43,680
surely there's a
minus sign there--

570
00:35:43,680 --> 00:35:47,920
e to the minus j omega n.

571
00:35:47,920 --> 00:35:53,380
If we sum this, this is just
the sum of a geometric series.

572
00:35:53,380 --> 00:35:55,360
In other words,
it's of the form,

573
00:35:55,360 --> 00:35:57,610
the sum of alpha to the n--

574
00:35:57,610 --> 00:36:00,610
alpha equals 0 to infinity.

575
00:36:00,610 --> 00:36:05,990
Sums of this form are equal
to 1 over 1 minus alpha.

576
00:36:05,990 --> 00:36:10,180
And alpha, in this case, is
a e to the minus j omega.

577
00:36:10,180 --> 00:36:18,070
So that sum, then, is 1 over 1
minus a e to the minus j omega.

578
00:36:18,070 --> 00:36:19,540
Well, as we saw--

579
00:36:19,540 --> 00:36:21,640
at least for the
sinusoidal response--

580
00:36:21,640 --> 00:36:27,100
it's useful to look at the
magnitude and phase of this,

581
00:36:27,100 --> 00:36:29,570
in other words, to look
at it in polar form.

582
00:36:29,570 --> 00:36:35,170
So if we want the magnitude
of this frequency response,

583
00:36:35,170 --> 00:36:38,920
we can obtain that
by multiplying this

584
00:36:38,920 --> 00:36:41,560
by its complex conjugate.

585
00:36:41,560 --> 00:36:44,830
Consequently, the magnitude
of the frequency response

586
00:36:44,830 --> 00:36:48,220
is given by H of
e to the j omega--

587
00:36:48,220 --> 00:36:51,520
the minus sign is
conveniently there for us--

588
00:36:51,520 --> 00:36:53,980
multiplied by its
complex conjugate, which

589
00:36:53,980 --> 00:36:57,210
is 1 minus a e to the minus--

590
00:36:57,210 --> 00:37:02,530
I'm sorry, 1 over 1 minus
a 3 to the plus j omega.

591
00:37:02,530 --> 00:37:10,090
If we multiply these together,
then what we obtain is 1 over 1

592
00:37:10,090 --> 00:37:14,830
plus a squared minus 2
a times cosine omega.

593
00:37:14,830 --> 00:37:22,750
And the phase angle of this is
equal to the arctangent of--

594
00:37:22,750 --> 00:37:24,700
if you just simply
work this out--

595
00:37:24,700 --> 00:37:26,950
the arctangent of a
sine omega divided

596
00:37:26,950 --> 00:37:30,220
by 1 minus a cosine omega.

597
00:37:30,220 --> 00:37:33,490
I suspect, actually, that
because of that algebraic sign

598
00:37:33,490 --> 00:37:37,960
error I made, the
minus sign down here,

599
00:37:37,960 --> 00:37:41,060
actually I think that this
comes out with a minus sign.

600
00:37:41,060 --> 00:37:45,460
But I won't guarantee that
on the spot right now--

601
00:37:45,460 --> 00:37:47,440
you can simply
verify that, but I

602
00:37:47,440 --> 00:37:51,730
suspect that there should
be a minus sign there.

603
00:37:51,730 --> 00:37:56,590
OK, well this, then,
represents the phase shift

604
00:37:56,590 --> 00:38:00,520
that would be encountered
by a sinusoidal input going

605
00:38:00,520 --> 00:38:03,460
through the linear
shift invariant system.

606
00:38:03,460 --> 00:38:06,580
This would represent the
square of the magnitude

607
00:38:06,580 --> 00:38:11,030
change of a sinusoidal input.

608
00:38:11,030 --> 00:38:16,930
And if we were to sketch
this, then what we see

609
00:38:16,930 --> 00:38:21,205
is that at omega equal to
0, cosine omega of course

610
00:38:21,205 --> 00:38:23,620
at omega equal to 0 is 1.

611
00:38:23,620 --> 00:38:28,360
So this is 1 over 1 plus a
squared minus 2 a, or 1 over 1

612
00:38:28,360 --> 00:38:31,480
minus a quantity squared.

613
00:38:31,480 --> 00:38:34,220
We're assuming that
a is between 0 and 1,

614
00:38:34,220 --> 00:38:39,420
and so we're indicated
that by this value.

615
00:38:39,420 --> 00:38:43,410
When omega is equal
to pi, cosine omega

616
00:38:43,410 --> 00:38:45,300
is equal to minus 1.

617
00:38:45,300 --> 00:38:50,470
This, then, comes out to be 1
over 1 plus a quantity squared,

618
00:38:50,470 --> 00:38:54,030
which for a between 0 and
1 is less than this value.

619
00:38:54,030 --> 00:38:59,580
So the frequency response, then,
between minus pi and plus pi,

620
00:38:59,580 --> 00:39:03,700
would have the shape
that I've indicated here.

621
00:39:03,700 --> 00:39:07,880
Now, what happens if we
continue on in frequency-- omega

622
00:39:07,880 --> 00:39:12,900
can, of course, run from pi
to 2 pi, and on past that.

623
00:39:12,900 --> 00:39:15,660
You can verify in a
very straightforward way

624
00:39:15,660 --> 00:39:19,950
from this expression
that, from pi to 2 pi,

625
00:39:19,950 --> 00:39:23,250
the frequency response
will look exactly like it

626
00:39:23,250 --> 00:39:26,220
did from minus pi to pi.

627
00:39:26,220 --> 00:39:29,910
It will, then, look like that.

628
00:39:29,910 --> 00:39:33,070
And for minus pi
to minus 2 pi, it

629
00:39:33,070 --> 00:39:36,480
will look exactly as it
did from pi down to 0.

630
00:39:36,480 --> 00:39:39,850
So it will look like this.

631
00:39:39,850 --> 00:39:43,620
And in fact, it's
straightforward to verify

632
00:39:43,620 --> 00:39:48,090
for both the magnitude and
the phase that both of those

633
00:39:48,090 --> 00:39:53,970
are periodic in omega,
with a period of 2 pi.

634
00:39:53,970 --> 00:39:58,290
So this is one period,
say, from minus pi to pi,

635
00:39:58,290 --> 00:40:03,670
and then another period
would start from pi to 3 pi.

636
00:40:03,670 --> 00:40:06,550
And it would continue
on like that.

637
00:40:06,550 --> 00:40:08,980
So, in fact, if we
were to sketch this out

638
00:40:08,980 --> 00:40:11,350
over a wider range of
omega, we would see

639
00:40:11,350 --> 00:40:13,030
this periodically repeating.

640
00:40:13,030 --> 00:40:16,850
For this example, that's
straightforward to verify.

641
00:40:16,850 --> 00:40:23,630
So there are, then, two
properties of the frequency

642
00:40:23,630 --> 00:40:27,200
response which I would like
to call your attention to

643
00:40:27,200 --> 00:40:32,060
in preparation for our
discussion next time.

644
00:40:32,060 --> 00:40:34,220
One of them is the
fact, very important

645
00:40:34,220 --> 00:40:38,930
to keep in mind, that the
frequency response, as we're

646
00:40:38,930 --> 00:40:43,850
talking about it, is a function
of a continuous variable,

647
00:40:43,850 --> 00:40:44,960
omega.

648
00:40:44,960 --> 00:40:48,140
Omega, I hadn't stressed
this point previously,

649
00:40:48,140 --> 00:40:55,010
but omega is a variable
that changes continuously

650
00:40:55,010 --> 00:40:57,770
over whatever range
we're talking about,

651
00:40:57,770 --> 00:41:02,090
as opposed to sequences which
are functions, obviously,

652
00:41:02,090 --> 00:41:05,000
of a discrete variable.

653
00:41:05,000 --> 00:41:06,590
Here we're talking
about a function

654
00:41:06,590 --> 00:41:09,320
of a continuous variable, omega.

655
00:41:09,320 --> 00:41:14,300
As we saw, for our example,
the frequency response

656
00:41:14,300 --> 00:41:17,010
is a periodic function of omega.

657
00:41:17,010 --> 00:41:20,280
And the period is equal to 2 pi.

658
00:41:20,280 --> 00:41:22,970
Now, the reason that it's a
periodic function of omega

659
00:41:22,970 --> 00:41:27,080
is, in fact, somewhat obvious.

660
00:41:27,080 --> 00:41:31,490
Suppose that we take a complex
exponential, e to the j omega

661
00:41:31,490 --> 00:41:35,900
n, and inquire as to how
the complex exponential

662
00:41:35,900 --> 00:41:41,840
itself behaves if we change
omega over an interval of more

663
00:41:41,840 --> 00:41:43,130
than 2 pi.

664
00:41:43,130 --> 00:41:48,920
Suppose that we replace omega
by omega plus 2 pi times k,

665
00:41:48,920 --> 00:41:52,220
and now if we decompose
this into a product,

666
00:41:52,220 --> 00:41:55,910
we have e to the j omega
n, times e to the j

667
00:41:55,910 --> 00:41:59,540
2 pi k times n.

668
00:41:59,540 --> 00:42:02,990
This is an integer
multiple-- this exponent

669
00:42:02,990 --> 00:42:05,550
is an integer multiple of 2 pi.

670
00:42:05,550 --> 00:42:08,450
And so this is simply
equal to unity.

671
00:42:08,450 --> 00:42:13,460
Now what that says, is that
a complex exponential--

672
00:42:13,460 --> 00:42:19,130
once we've varied omega
over an interval of 2 pi,

673
00:42:19,130 --> 00:42:22,190
and we go past that,
there are no more

674
00:42:22,190 --> 00:42:24,170
no new complex
exponentials to see.

675
00:42:24,170 --> 00:42:26,990
We'll see the same ones over
and over and over again.

676
00:42:26,990 --> 00:42:30,260
And consequently,
the system response

677
00:42:30,260 --> 00:42:33,860
has to be periodic in
omega with period 2 pi,

678
00:42:33,860 --> 00:42:37,250
because we're putting in,
essentially, the same inputs

679
00:42:37,250 --> 00:42:39,810
over and over and over again.

680
00:42:39,810 --> 00:42:43,910
This is a point that I'll be
mentioning from time to time,

681
00:42:43,910 --> 00:42:49,100
and it's, in fact, somewhat
important to keep in mind.

682
00:42:49,100 --> 00:42:53,630
Also, it's discussed in some
detail again in the text.

683
00:42:53,630 --> 00:42:55,760
So these, then,
are some properties

684
00:42:55,760 --> 00:42:58,640
of the frequency response.

685
00:42:58,640 --> 00:43:02,720
There is a generalization of
the frequency response, which

686
00:43:02,720 --> 00:43:07,700
is, in fact, very important
for describing both signals

687
00:43:07,700 --> 00:43:08,900
and systems.

688
00:43:08,900 --> 00:43:13,370
The generalization is what
we'll refer to as the Fourier

689
00:43:13,370 --> 00:43:16,250
transform, which plays
the identical role

690
00:43:16,250 --> 00:43:19,820
in the discrete time case
that the Fourier transform did

691
00:43:19,820 --> 00:43:22,050
in the continuous time case.

692
00:43:22,050 --> 00:43:24,890
And so in the next
lecture, we'll

693
00:43:24,890 --> 00:43:29,240
be going on to a
discussion of the Fourier

694
00:43:29,240 --> 00:43:33,350
transform taking off
from the set of ideas

695
00:43:33,350 --> 00:43:38,540
that we've developed here, with
regard to frequency response.

696
00:43:38,540 --> 00:43:40,250
Thank you.

697
00:43:40,250 --> 00:43:42,400
[MUSIC PLAYING]