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DENNIS FREEMAN: So today's topic
is to talk about Z-transforms.

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00:00:27,960 --> 00:00:29,910
But before I start,
I want to mention

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00:00:29,910 --> 00:00:33,990
that we've already covered
a great deal of material.

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00:00:33,990 --> 00:00:39,030
Here, I've made a map just
of things we've done in DT,

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00:00:39,030 --> 00:00:42,390
and there's a fair
number of entries.

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00:00:42,390 --> 00:00:45,510
More importantly, there's a
fair number of connections

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00:00:45,510 --> 00:00:47,610
between those entries.

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00:00:47,610 --> 00:00:50,010
And if you're thinking
about systems,

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00:00:50,010 --> 00:00:51,900
you should be able
to think about,

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00:00:51,900 --> 00:00:55,950
what do each one of
those arrows stand for?

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00:00:55,950 --> 00:00:58,290
The reason that's
important is that we

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00:00:58,290 --> 00:01:01,480
like to have multiple
representations for the system

20
00:01:01,480 --> 00:01:03,480
so that we're always able
to choose the one that

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00:01:03,480 --> 00:01:06,800
lets us work the most simply.

22
00:01:06,800 --> 00:01:10,410
But sometimes that involves
moving between the squares.

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00:01:10,410 --> 00:01:12,240
So the way the problem
was posed to you

24
00:01:12,240 --> 00:01:14,220
may not be the simplest
way to solve it,

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00:01:14,220 --> 00:01:17,070
and that involves, then, going
across one of those arrows.

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00:01:17,070 --> 00:01:20,160
So for example, we
know that there's

27
00:01:20,160 --> 00:01:22,800
a simple relationship between
block diagrams and system

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00:01:22,800 --> 00:01:24,510
functionals.

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00:01:24,510 --> 00:01:26,560
All you need to do
is think about delays

30
00:01:26,560 --> 00:01:29,910
as being the right
shift operator.

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00:01:29,910 --> 00:01:33,150
That's a way of thinking about
the system functional is just

32
00:01:33,150 --> 00:01:40,600
a formula picture of
the block diagram.

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00:01:40,600 --> 00:01:43,270
Similarly, we can think about
moving between the block

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00:01:43,270 --> 00:01:44,980
diagram and the
difference equation

35
00:01:44,980 --> 00:01:47,875
by thinking about delay
being a shift of the index.

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00:01:50,450 --> 00:01:53,120
So you should be able to think
of ways of thinking about all

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00:01:53,120 --> 00:01:55,630
of the transformations
between the boxes.

38
00:01:55,630 --> 00:01:58,900
Over here, we thought
about system functionals.

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00:01:58,900 --> 00:02:03,190
How do you think about a system
as a sequence of operations

40
00:02:03,190 --> 00:02:05,920
that you do to a signal?

41
00:02:05,920 --> 00:02:07,720
We also thought about
system functions,

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00:02:07,720 --> 00:02:10,300
and there's a very simple
relationship between those.

43
00:02:10,300 --> 00:02:13,860
You think about the operator
expression that involves

44
00:02:13,860 --> 00:02:18,070
Rs and replace each R with 1/z.

45
00:02:20,830 --> 00:02:23,140
OK, that was all a big
set up, because now I

46
00:02:23,140 --> 00:02:25,300
want to ask a question.

47
00:02:25,300 --> 00:02:28,990
Using your vast knowledge of how
all these things interrelate,

48
00:02:28,990 --> 00:02:32,290
what's the relationship
between the system functional,

49
00:02:32,290 --> 00:02:37,180
a function of R, and the
unit sample response?

50
00:02:41,897 --> 00:02:44,480
As usual, I'd like you to talk
to your neighbor to figure this

51
00:02:44,480 --> 00:02:47,960
out, and as usual, you won't
do that unless I tell you

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00:02:47,960 --> 00:02:49,300
to do something trivial first.

53
00:02:49,300 --> 00:02:52,025
So look toward your
neighbor, say, "Hi."

54
00:02:54,980 --> 00:02:56,300
Wonderful, good.

55
00:02:56,300 --> 00:02:57,900
Now, figure out this problem.

56
00:04:44,160 --> 00:04:45,432
OK, how many of you--

57
00:04:45,432 --> 00:04:46,390
I want a show of hands.

58
00:04:46,390 --> 00:04:49,384
How many of you know a way of
getting between those boxes?

59
00:04:49,384 --> 00:04:50,050
Raise your hand.

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00:04:52,969 --> 00:04:54,760
How many of you are
sitting beside somebody

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00:04:54,760 --> 00:04:56,635
who knows the way to
get between those boxes?

62
00:04:59,270 --> 00:04:59,770
No.

63
00:04:59,770 --> 00:05:01,450
OK, that didn't work either.

64
00:05:01,450 --> 00:05:04,190
OK, can somebody tell me, if I
told you a system functional,

65
00:05:04,190 --> 00:05:07,810
say I told you that one,
how would I figure out

66
00:05:07,810 --> 00:05:10,230
the unit sample response
from the system function--

67
00:05:10,230 --> 00:05:11,880
functional?

68
00:05:11,880 --> 00:05:12,611
Yes.

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00:05:12,611 --> 00:05:14,770
AUDIENCE: Don't you take
the inverse Laplace of it?

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00:05:14,770 --> 00:05:16,630
DENNIS FREEMAN: Inverse Laplace.

71
00:05:16,630 --> 00:05:19,600
OK, that's kind of
right, but not quite.

72
00:05:22,366 --> 00:05:26,110
So usually, the
answer to the question

73
00:05:26,110 --> 00:05:28,570
is either the slide before
this one or the slide

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00:05:28,570 --> 00:05:30,290
after this one.

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00:05:30,290 --> 00:05:31,955
So that doesn't quite fit this.

76
00:05:31,955 --> 00:05:34,771
The slide before this one said
something about Z-transforms.

77
00:05:39,946 --> 00:05:41,320
Can somebody think
of a way, if I

78
00:05:41,320 --> 00:05:43,690
told you the system
functional is this thing,

79
00:05:43,690 --> 00:05:49,190
how would you derive the
unit sample response?

80
00:05:49,190 --> 00:05:49,928
Yes.

81
00:05:49,928 --> 00:05:51,920
AUDIENCE: You could convert
it into a difference equation

82
00:05:51,920 --> 00:05:53,420
and then just work
it out manually.

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00:05:53,420 --> 00:05:54,920
DENNIS FREEMAN: Convert it
into a difference equation.

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00:05:54,920 --> 00:05:55,590
That would work.

85
00:05:55,590 --> 00:06:01,012
So what you could do is go
from this box to that box.

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00:06:01,012 --> 00:06:02,470
Then, if you had
this box how would

87
00:06:02,470 --> 00:06:03,844
you get the unit
sample response?

88
00:06:08,610 --> 00:06:10,056
Yes.

89
00:06:10,056 --> 00:06:13,350
AUDIENCE: [INAUDIBLE] delta.

90
00:06:13,350 --> 00:06:15,350
DENNIS FREEMAN: Put a
delta into the difference.

91
00:06:15,350 --> 00:06:17,690
Think about the difference
equation being a system.

92
00:06:17,690 --> 00:06:19,970
Put a delta in and
see what comes out.

93
00:06:19,970 --> 00:06:21,316
Exactly.

94
00:06:21,316 --> 00:06:22,940
Can somebody think
of a more direct way

95
00:06:22,940 --> 00:06:25,250
that doesn't skirt through
the difference equation?

96
00:06:25,250 --> 00:06:25,990
Yes.

97
00:06:25,990 --> 00:06:29,070
AUDIENCE: Multiply the system
functional by a delta function.

98
00:06:29,070 --> 00:06:31,650
DENNIS FREEMAN: Multiply the
functional by a delta function.

99
00:06:31,650 --> 00:06:33,240
I'm not quite sure
what that means,

100
00:06:33,240 --> 00:06:35,250
but I'm willing to try anything.

101
00:06:35,250 --> 00:06:38,250
1 over 1 minus R
minus R squared.

102
00:06:38,250 --> 00:06:40,050
Multiply that times
a delta function.

103
00:06:44,050 --> 00:06:44,800
What do I do next?

104
00:06:50,040 --> 00:06:51,900
OK, as I said, the
answer to the question

105
00:06:51,900 --> 00:06:53,580
is usually something
that we just did

106
00:06:53,580 --> 00:06:55,584
or something we're
just about to do.

107
00:06:55,584 --> 00:06:57,500
Where did we spend the
last three weeks doing?

108
00:06:57,500 --> 00:06:58,630
No, it's not three weeks.

109
00:06:58,630 --> 00:07:02,990
It's only a week, but,
you know, I exaggerate.

110
00:07:02,990 --> 00:07:05,010
What have we been doing?

111
00:07:05,010 --> 00:07:06,274
Yes.

112
00:07:06,274 --> 00:07:08,990
AUDIENCE: We could use
long division [INAUDIBLE]

113
00:07:08,990 --> 00:07:11,127
DENNIS FREEMAN: One way
would be use long division.

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00:07:11,127 --> 00:07:11,960
How would that work?

115
00:07:11,960 --> 00:07:12,840
If we did 1/1--

116
00:07:12,840 --> 00:07:18,650
maybe I'd better use a
bigger blackboard space--

117
00:07:18,650 --> 00:07:24,309
if I tried to do 1 over 1 minus
R minus R squared, if I tried

118
00:07:24,309 --> 00:07:25,850
to think about that
by long division,

119
00:07:25,850 --> 00:07:30,110
I would do 1 minus R
minus R squared into 1.

120
00:07:30,110 --> 00:07:33,070
How would that work?

121
00:07:33,070 --> 00:07:33,800
Well, I'd go 1--

122
00:07:33,800 --> 00:07:36,290
I'd go 1 goes into 1 once.

123
00:07:36,290 --> 00:07:38,690
I get 1 minus R minus R squared.

124
00:07:38,690 --> 00:07:41,660
This minus this gives
me R plus R squared.

125
00:07:41,660 --> 00:07:48,690
This goes into this plus
R. That gives me whatever.

126
00:07:48,690 --> 00:07:52,790
Having done it in the
solitude of my own breakfast

127
00:07:52,790 --> 00:07:57,130
this morning, here's an
answer that I got there.

128
00:07:57,130 --> 00:08:00,050
So if I perform long
division on the functional,

129
00:08:00,050 --> 00:08:03,410
I get this kind of an answer.

130
00:08:03,410 --> 00:08:05,000
The question was,
how do I relate

131
00:08:05,000 --> 00:08:07,400
the functional to the
unit sample response?

132
00:08:07,400 --> 00:08:10,869
How do I-- how do I relate this
to the unit sample response?

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00:08:10,869 --> 00:08:12,410
AUDIENCE: The
coefficient [INAUDIBLE]

134
00:08:12,410 --> 00:08:12,710
DENNIS FREEMAN: Excuse me?

135
00:08:12,710 --> 00:08:14,720
AUDIENCE: The
coefficients [INAUDIBLE]

136
00:08:14,720 --> 00:08:17,130
DENNIS FREEMAN: The
coefficients of this expression

137
00:08:17,130 --> 00:08:19,745
are the unit sample response.

138
00:08:22,330 --> 00:08:23,440
OK?

139
00:08:23,440 --> 00:08:25,210
So there's a very
straightforward way.

140
00:08:25,210 --> 00:08:27,610
If I tell you the
system functional,

141
00:08:27,610 --> 00:08:29,360
if I tell you this
representation,

142
00:08:29,360 --> 00:08:32,679
there's a very simple way
that we've thought about

143
00:08:32,679 --> 00:08:36,500
by which you can
automatically calculate what

144
00:08:36,500 --> 00:08:39,160
is the unit sample response?

145
00:08:39,160 --> 00:08:40,090
OK?

146
00:08:40,090 --> 00:08:45,610
In fact, it's so simple that
we can write a formula for it.

147
00:08:45,610 --> 00:08:49,710
Think about the unit sample
response being h of n.

148
00:08:49,710 --> 00:08:52,340
Associate the h 0
term with a constant,

149
00:08:52,340 --> 00:08:56,550
the h 1 term with an R, with
the h 2 term with an R squared.

150
00:08:56,550 --> 00:09:01,270
Add them all together,
that's the answer.

151
00:09:01,270 --> 00:09:06,260
So the way you get between these
representations is very simple.

152
00:09:06,260 --> 00:09:08,950
Here's the unit sample response,
here's the system functional,

153
00:09:08,950 --> 00:09:11,920
and here's an equation
that relates the two

154
00:09:11,920 --> 00:09:12,910
representations.

155
00:09:12,910 --> 00:09:15,530
That all completely clear?

156
00:09:15,530 --> 00:09:16,030
Simple.

157
00:09:16,030 --> 00:09:18,380
Yes, hi.

158
00:09:18,380 --> 00:09:20,642
Everything clear?

159
00:09:20,642 --> 00:09:21,475
Questions, comments?

160
00:09:24,980 --> 00:09:27,800
OK, a follow-up question.

161
00:09:27,800 --> 00:09:30,990
So here's a relationship between
these two representations.

162
00:09:30,990 --> 00:09:33,765
How about a relationship between
these two representations?

163
00:09:38,710 --> 00:09:39,210
Yes.

164
00:09:39,210 --> 00:09:40,376
AUDIENCE: Same relationship.

165
00:09:40,376 --> 00:09:42,390
Just use one of
the place markers.

166
00:09:42,390 --> 00:09:44,020
DENNIS FREEMAN: Precisely.

167
00:09:44,020 --> 00:09:47,000
So it's really-- so we got
this one in the previous step.

168
00:09:47,000 --> 00:09:50,264
R is just 1/z, so we get this.

169
00:09:50,264 --> 00:09:52,180
So we get a relationship
that looks like that.

170
00:09:54,810 --> 00:09:58,600
OK, that relationship is the
basis of today's lecture.

171
00:09:58,600 --> 00:10:01,280
We call that relationship--

172
00:10:01,280 --> 00:10:04,680
so that relationship represents
a mathematical relationship

173
00:10:04,680 --> 00:10:12,300
between a function of
z and a function of n,

174
00:10:12,300 --> 00:10:16,460
and we call that
relationship the Z-transform.

175
00:10:16,460 --> 00:10:17,710
So that's the topic for today.

176
00:10:17,710 --> 00:10:22,140
We're going to think about
systems, not as an operator,

177
00:10:22,140 --> 00:10:26,160
but as a mathematical
function, h of z.

178
00:10:26,160 --> 00:10:29,820
So the first thing to realize
is that this transform then,

179
00:10:29,820 --> 00:10:32,070
this thing that we're going
to be worried about today

180
00:10:32,070 --> 00:10:37,810
is a map between a
discrete function

181
00:10:37,810 --> 00:10:42,180
and a continuous function of z.

182
00:10:42,180 --> 00:10:43,680
And even though
it's been motivated,

183
00:10:43,680 --> 00:10:45,210
I motivated it by
thinking about how

184
00:10:45,210 --> 00:10:48,780
it would relate to a
system, that relationship

185
00:10:48,780 --> 00:10:50,790
between a function of
z and a function of n

186
00:10:50,790 --> 00:10:55,830
is something that you could do
on any discrete time signal.

187
00:10:55,830 --> 00:10:58,380
So in fact, if you
have any signal x of n,

188
00:10:58,380 --> 00:11:00,270
we can think about
the Z-transform of it

189
00:11:00,270 --> 00:11:03,180
by simply thinking about what
that equation is telling us.

190
00:11:03,180 --> 00:11:09,040
How does it map from a function
of n to a function of z?

191
00:11:09,040 --> 00:11:10,990
I've written it in a little--

192
00:11:10,990 --> 00:11:12,890
in a way that you might
not have anticipated.

193
00:11:12,890 --> 00:11:15,487
You might have thought I
would have started at 0.

194
00:11:15,487 --> 00:11:17,320
That's just because
we're going to do what's

195
00:11:17,320 --> 00:11:19,690
called the bilateral transform.

196
00:11:19,690 --> 00:11:22,210
There are many kinds
of Z-transforms.

197
00:11:22,210 --> 00:11:23,920
We're going to do
this particular one.

198
00:11:23,920 --> 00:11:25,390
Other classes that
you might take

199
00:11:25,390 --> 00:11:28,360
may use something
called a unilateral.

200
00:11:28,360 --> 00:11:29,830
If you're not
taking such a class,

201
00:11:29,830 --> 00:11:31,663
there's no good reason
for you to know that.

202
00:11:31,663 --> 00:11:34,990
If you are taking such a
class, we're doing bilateral.

203
00:11:34,990 --> 00:11:36,700
That just means
two-sided, and we'll

204
00:11:36,700 --> 00:11:41,580
see by the end of the hour how
that makes some things simpler.

205
00:11:41,580 --> 00:11:44,040
The big picture is
identical whether you

206
00:11:44,040 --> 00:11:48,750
do unilateral or bilateral,
but there are differences.

207
00:11:48,750 --> 00:11:49,710
OK.

208
00:11:49,710 --> 00:11:50,790
Simple Z-transforms.

209
00:11:50,790 --> 00:11:53,130
What's the Z-transform
of the simplest

210
00:11:53,130 --> 00:11:54,630
signal that we can imagine?

211
00:11:54,630 --> 00:11:56,580
Simplest signal
that we can imagine

212
00:11:56,580 --> 00:11:58,170
is the unit sample signal.

213
00:11:58,170 --> 00:12:00,390
It's 0 everywhere
except n equals 0.

214
00:12:00,390 --> 00:12:03,720
At n equals 0, it's the simplest
possible non-zero answer,

215
00:12:03,720 --> 00:12:04,285
which is 1.

216
00:12:06,852 --> 00:12:07,810
What's the Z-transform?

217
00:12:07,810 --> 00:12:09,080
Well, that's trivial.

218
00:12:09,080 --> 00:12:11,970
You stick it in the formula.

219
00:12:11,970 --> 00:12:15,520
The only non-zero
answer is at n equals 0.

220
00:12:15,520 --> 00:12:17,870
Stick in n equals 0, and
we get that the answer

221
00:12:17,870 --> 00:12:20,960
is the Z-transform is 1.

222
00:12:20,960 --> 00:12:22,080
What could be easier?

223
00:12:22,080 --> 00:12:22,760
Good.

224
00:12:22,760 --> 00:12:28,860
Simple signal in time, simple
signal in the Z-transform.

225
00:12:28,860 --> 00:12:33,720
What's the Z-transform for a
delayed unit sample signal?

226
00:12:33,720 --> 00:12:37,050
What happens if the only
place that it's not 0

227
00:12:37,050 --> 00:12:39,910
is shifted to n equals 1?

228
00:12:39,910 --> 00:12:41,510
Not a big deal.

229
00:12:41,510 --> 00:12:43,950
Again, there's a
single non-zero answer,

230
00:12:43,950 --> 00:12:47,420
it's just that it's
now at n equals 1.

231
00:12:47,420 --> 00:12:49,870
So this answer is
z to the minus 1.

232
00:12:49,870 --> 00:12:53,750
We took a signal that
was a function of n,

233
00:12:53,750 --> 00:12:55,800
and we represent it
by a Z-transform,

234
00:12:55,800 --> 00:12:58,040
which is a function of z.

235
00:12:58,040 --> 00:13:00,590
Delta of n is 1.

236
00:13:00,590 --> 00:13:04,800
Delta of n minus 1
is z to the minus 1.

237
00:13:04,800 --> 00:13:08,460
OK, with that vast
knowledge of Z-transforms,

238
00:13:08,460 --> 00:13:11,850
figure out the Z-transform for
a slightly more complicated

239
00:13:11,850 --> 00:13:14,040
sequence of the
type-- of the type

240
00:13:14,040 --> 00:13:16,920
that we saw when we
were looking at systems.

241
00:13:16,920 --> 00:13:19,020
What if the signal that
we're interested in

242
00:13:19,020 --> 00:13:22,080
is 7/8 to the n, u of n,
where I'm using the u of n

243
00:13:22,080 --> 00:13:26,500
just to cut off the negative
parts of 7/8 to the n.

244
00:13:26,500 --> 00:13:28,350
So find the
Z-transform, and find

245
00:13:28,350 --> 00:13:30,630
if it looks like one
of those four answers

246
00:13:30,630 --> 00:13:32,080
or none, which is number five.

247
00:15:13,020 --> 00:15:17,850
So what answer best represents
the Z-transform of the signal

248
00:15:17,850 --> 00:15:19,020
x?

249
00:15:19,020 --> 00:15:21,135
Raise your hand, number
one through five.

250
00:15:25,020 --> 00:15:28,510
It's a participatory sport.

251
00:15:28,510 --> 00:15:32,260
Good 97% correct, I think.

252
00:15:32,260 --> 00:15:35,230
So OK.

253
00:15:35,230 --> 00:15:36,119
Everybody says two.

254
00:15:36,119 --> 00:15:36,910
How do you get two?

255
00:15:36,910 --> 00:15:38,000
What do you do?

256
00:15:38,000 --> 00:15:39,190
Plug in the formula, right?

257
00:15:39,190 --> 00:15:42,490
It's a very simple-minded thing.

258
00:15:42,490 --> 00:15:44,590
If you were to think
about this signal

259
00:15:44,590 --> 00:15:47,710
and think about the
definition of the Z-transform,

260
00:15:47,710 --> 00:15:51,910
just substitute this particular
signal, 7/8 to the n, u of n,

261
00:15:51,910 --> 00:15:57,440
in where there would
have been the x of n

262
00:15:57,440 --> 00:15:59,990
and try to close the sum, right?

263
00:15:59,990 --> 00:16:02,480
It's a geometric sequence.

264
00:16:02,480 --> 00:16:05,550
We've had lots of experience
with geometric sequences.

265
00:16:05,550 --> 00:16:07,250
It was in the
homework, and so we all

266
00:16:07,250 --> 00:16:10,000
know that the answer to
summing a geometric--

267
00:16:10,000 --> 00:16:13,400
a one-sided geometric sequence
is something of the form

268
00:16:13,400 --> 00:16:15,340
1 over 1 minus a.

269
00:16:15,340 --> 00:16:16,640
OK, easy, right?

270
00:16:19,360 --> 00:16:22,660
So the idea, then,
is that we understand

271
00:16:22,660 --> 00:16:25,570
a general rule by which we
can map a function of time

272
00:16:25,570 --> 00:16:28,420
into a function of z.

273
00:16:28,420 --> 00:16:33,030
So the function of time has
to make sense everywhere.

274
00:16:33,030 --> 00:16:36,180
That's-- we want
that to be true.

275
00:16:36,180 --> 00:16:39,300
We want to know what was
the system's unit sample

276
00:16:39,300 --> 00:16:42,400
response, for example, or what
was the input to the system,

277
00:16:42,400 --> 00:16:42,900
or whatever.

278
00:16:42,900 --> 00:16:46,050
We want to know that at
all possible values of n.

279
00:16:46,050 --> 00:16:47,310
How about the other case?

280
00:16:47,310 --> 00:16:49,140
Does it make sense?

281
00:16:49,140 --> 00:16:53,190
Is x of z makes sense for
all possible values of z?

282
00:16:55,710 --> 00:16:59,940
And since I ask the
question, the answer is no.

283
00:16:59,940 --> 00:17:01,120
Yes, of course.

284
00:17:01,120 --> 00:17:02,700
I wouldn't have
asked the question

285
00:17:02,700 --> 00:17:05,369
if the answer were yes, right?

286
00:17:05,369 --> 00:17:06,839
So by the theory
of lecturers, you

287
00:17:06,839 --> 00:17:09,660
can tell the answer
has to have been no.

288
00:17:09,660 --> 00:17:12,945
So the question is, what
values of z don't make sense?

289
00:17:17,300 --> 00:17:19,079
Shout.

290
00:17:19,079 --> 00:17:21,800
AUDIENCE: The summation
of the [INAUDIBLE]

291
00:17:21,800 --> 00:17:24,109
DENNIS FREEMAN: The
summation has to converge,

292
00:17:24,109 --> 00:17:29,540
so it's only going to be defined
if this sum, which happens

293
00:17:29,540 --> 00:17:34,230
to be an infinite sum and
therefore may not converge,

294
00:17:34,230 --> 00:17:36,620
it's going to have to be the
case that that infinite sum

295
00:17:36,620 --> 00:17:37,370
does converge.

296
00:17:37,370 --> 00:17:39,740
Otherwise, it won't make
sense to talk about it.

297
00:17:39,740 --> 00:17:41,930
So then the question
is, for what values of z

298
00:17:41,930 --> 00:17:43,910
will that converge?

299
00:17:43,910 --> 00:17:45,530
And again, we know
from our experience

300
00:17:45,530 --> 00:17:53,900
with geometric sequences that
the base of the geometric, a,

301
00:17:53,900 --> 00:17:56,720
when we did homework one,
the base of the geometric

302
00:17:56,720 --> 00:17:58,230
has to have what?

303
00:17:58,230 --> 00:17:59,972
What's the property
of the base of the--

304
00:17:59,972 --> 00:18:01,430
the geometric
that'll make it work?

305
00:18:04,650 --> 00:18:08,690
So if I want to sum some
series of the form n equals 0

306
00:18:08,690 --> 00:18:12,800
to infinity a to the
n, what's the values--

307
00:18:12,800 --> 00:18:16,425
what's the limitation
on a that makes it work?

308
00:18:16,425 --> 00:18:21,274
AUDIENCE: [INAUDIBLE]

309
00:18:21,274 --> 00:18:22,940
DENNIS FREEMAN: OK,
you're not saying it

310
00:18:22,940 --> 00:18:26,432
loud enough or clearly enough,
or I'm too dense or something.

311
00:18:26,432 --> 00:18:27,747
AUDIENCE: Abs less than 1?

312
00:18:27,747 --> 00:18:29,330
DENNIS FREEMAN: So
we need something--

313
00:18:29,330 --> 00:18:30,770
we need abs less than 1, right.

314
00:18:33,440 --> 00:18:34,950
So we do the same thing here.

315
00:18:34,950 --> 00:18:43,180
We're going to have to have
that the geometric base, 7/8 1

316
00:18:43,180 --> 00:18:47,660
over z has to be less-- has
to have a magnitude that

317
00:18:47,660 --> 00:18:49,950
is less than 1.

318
00:18:49,950 --> 00:18:53,150
And if we torture our minds
with inequalities and magnitude

319
00:18:53,150 --> 00:18:55,760
signs, that says that
the magnitude of z

320
00:18:55,760 --> 00:18:58,700
has to be bigger than 7/8.

321
00:18:58,700 --> 00:18:59,950
OK?

322
00:18:59,950 --> 00:19:04,110
The important idea is
that when we characterize

323
00:19:04,110 --> 00:19:08,380
the Z-transform, we
should be expecting

324
00:19:08,380 --> 00:19:10,840
it to work for all values
of n, but not necessarily

325
00:19:10,840 --> 00:19:13,030
all values of z.

326
00:19:13,030 --> 00:19:14,690
So we have to be
cognizant of that.

327
00:19:14,690 --> 00:19:18,640
We have to know which values
of z are you talking about.

328
00:19:18,640 --> 00:19:20,720
We call that idea the
region of convergence.

329
00:19:20,720 --> 00:19:24,430
We say the Z-transform converged
for all z inside the region

330
00:19:24,430 --> 00:19:25,790
of convergence.

331
00:19:25,790 --> 00:19:28,480
So when you specify
a Z-transform,

332
00:19:28,480 --> 00:19:32,650
generally, you have to tell me
not just some functional form,

333
00:19:32,650 --> 00:19:34,270
but you also have
to tell me what

334
00:19:34,270 --> 00:19:38,780
was the region for which that
functional form converged.

335
00:19:38,780 --> 00:19:41,956
So in general, you have to tell
me the region of convergence.

336
00:19:41,956 --> 00:19:43,330
In this particular
case, you have

337
00:19:43,330 --> 00:19:45,280
to tell me that
you should restrict

338
00:19:45,280 --> 00:19:48,730
your attention to values
of z whose magnitude is

339
00:19:48,730 --> 00:19:51,250
bigger than 7/8.

340
00:19:51,250 --> 00:19:53,380
OK?

341
00:19:53,380 --> 00:19:56,470
OK, that's completely the whole
definition of Z-transforms.

342
00:19:56,470 --> 00:20:00,090
We're done from a point of
view of mathematics, right?

343
00:20:00,090 --> 00:20:06,030
So all we need to know is that
the Z-transform of a signal

344
00:20:06,030 --> 00:20:14,280
is some sum of the signal z to
the minus n, and we're done.

345
00:20:14,280 --> 00:20:16,590
We need to know about the
region of convergence,

346
00:20:16,590 --> 00:20:18,630
but mathematically, we've
completely specified

347
00:20:18,630 --> 00:20:20,160
the problem at this point.

348
00:20:20,160 --> 00:20:22,410
To make it useful,
however, what we need to do

349
00:20:22,410 --> 00:20:25,740
is investigate properties
of that transformation.

350
00:20:25,740 --> 00:20:28,350
If the transformation were
not easy to manipulate,

351
00:20:28,350 --> 00:20:31,090
we wouldn't bother with it.

352
00:20:31,090 --> 00:20:35,810
So we want to know what's easy
to do with the Z-transform

353
00:20:35,810 --> 00:20:38,210
and what's not easy to
do with the Z-transform.

354
00:20:38,210 --> 00:20:40,940
When something's easy to do with
the Z-transform, we'll use it.

355
00:20:40,940 --> 00:20:42,920
When things are easier
to do some other way,

356
00:20:42,920 --> 00:20:44,420
we use the other way.

357
00:20:44,420 --> 00:20:45,980
That's the game plan.

358
00:20:45,980 --> 00:20:48,650
So there's a number of
properties of the Z-transform

359
00:20:48,650 --> 00:20:51,210
that make it easy or
hard to do things.

360
00:20:51,210 --> 00:20:53,420
So we'll talk about
the easy ones.

361
00:20:53,420 --> 00:20:58,610
The most fundamental
one is linearity.

362
00:20:58,610 --> 00:21:03,590
Basically, this entire subject
is about linear things.

363
00:21:03,590 --> 00:21:06,590
If it's not linear, largely,
we don't talk about it.

364
00:21:06,590 --> 00:21:08,840
The reason is we have such
powerful tools for thinking

365
00:21:08,840 --> 00:21:10,980
about things that are linear.

366
00:21:10,980 --> 00:21:13,090
This is one of them.

367
00:21:13,090 --> 00:21:16,100
If-- so we say that
the Z-transform is

368
00:21:16,100 --> 00:21:17,990
a linear operator,
and all that means

369
00:21:17,990 --> 00:21:23,120
is if you apply the Z-transform
to the sum of two things,

370
00:21:23,120 --> 00:21:24,932
you get the sum
of the things out.

371
00:21:24,932 --> 00:21:26,640
It's a little more
complicated than that.

372
00:21:26,640 --> 00:21:28,264
If I were being a
little more careful--

373
00:21:28,264 --> 00:21:32,289
in fact, I didn't quite say
a complete definition here.

374
00:21:32,289 --> 00:21:34,580
This is certainly true, but
there are additional things

375
00:21:34,580 --> 00:21:35,870
that are also true.

376
00:21:35,870 --> 00:21:37,550
We say that the
Z-transform is linear

377
00:21:37,550 --> 00:21:42,770
because if we knew the
z-transform for X 1,

378
00:21:42,770 --> 00:21:46,760
that includes a functional form
and a region of convergence,

379
00:21:46,760 --> 00:21:49,380
and if we knew the
Z-transform for X 2,

380
00:21:49,380 --> 00:21:52,580
again, a functional form
and a region of convergence,

381
00:21:52,580 --> 00:21:54,710
then by the linearity
of the operator,

382
00:21:54,710 --> 00:21:58,520
we can figure out just from
the two Z-transforms, what

383
00:21:58,520 --> 00:22:02,300
is the Z-transform of the sum.

384
00:22:02,300 --> 00:22:04,860
And that's trivial to see.

385
00:22:04,860 --> 00:22:07,340
It's easy to see why it
ought to be that way.

386
00:22:07,340 --> 00:22:09,680
Just look at--
let's think about y,

387
00:22:09,680 --> 00:22:12,890
which is the signal,
which is the sum.

388
00:22:12,890 --> 00:22:14,930
By the definition
of Z-transform,

389
00:22:14,930 --> 00:22:18,010
the Z-transform of the
sum is that formula.

390
00:22:21,500 --> 00:22:25,550
Y, by definition, is that thing.

391
00:22:25,550 --> 00:22:31,460
Z commutes-- distributes
over addition,

392
00:22:31,460 --> 00:22:35,080
and the sum separates.

393
00:22:35,080 --> 00:22:38,050
So we can always
reduce this to this,

394
00:22:38,050 --> 00:22:44,475
at least when the case that z is
in both regions of convergence.

395
00:22:44,475 --> 00:22:45,850
Though later in
the course, we'll

396
00:22:45,850 --> 00:22:47,830
see that that's a little
overly restrictive.

397
00:22:47,830 --> 00:22:52,791
Sometimes it works for more
zs than in the intersection.

398
00:22:52,791 --> 00:22:54,790
But it's guaranteed to
work in the intersection,

399
00:22:54,790 --> 00:23:00,880
because I know that I can do
this sum if z is in ROC 1.

400
00:23:00,880 --> 00:23:04,090
I know I can do this
sum if z is in ROC 2.

401
00:23:04,090 --> 00:23:06,640
So I know I can do both of them
if it's in the intersection.

402
00:23:06,640 --> 00:23:08,290
We'll see a little
later that sometimes you

403
00:23:08,290 --> 00:23:10,340
can do it even when it's
not in the intersection.

404
00:23:10,340 --> 00:23:12,214
But for the time being,
we know you can do it

405
00:23:12,214 --> 00:23:14,980
if it's in the intersection.

406
00:23:14,980 --> 00:23:21,610
So because of linearity, we know
that the Z-transform of a sum

407
00:23:21,610 --> 00:23:23,680
is the sum of the Z-transforms.

408
00:23:23,680 --> 00:23:26,190
I didn't realize when
I made this slide,

409
00:23:26,190 --> 00:23:28,240
linearity implies
more than that.

410
00:23:28,240 --> 00:23:29,640
I could have done
a weighted sum.

411
00:23:33,110 --> 00:23:38,570
I could have said if X 1 goes
to X1 and X 2 goes to X 2,

412
00:23:38,570 --> 00:23:42,650
then alpha 1, X 1
plus alpha 2, X 2

413
00:23:42,650 --> 00:23:46,190
goes to alpha 1, X
1 plus alpha 2, X 2.

414
00:23:46,190 --> 00:23:47,690
I should have put
that in the slide,

415
00:23:47,690 --> 00:23:50,660
and it just didn't occur to me.

416
00:23:50,660 --> 00:23:53,480
So the idea of linearity
is slightly more powerful

417
00:23:53,480 --> 00:23:56,300
than the example
that I gave here.

418
00:23:56,300 --> 00:23:58,760
But again, linearity is the
most fundamental property

419
00:23:58,760 --> 00:23:59,774
of Z-transforms.

420
00:23:59,774 --> 00:24:01,190
If the Z-transform
weren't linear,

421
00:24:01,190 --> 00:24:02,910
we wouldn't bother with it.

422
00:24:02,910 --> 00:24:06,080
But it is, so we do.

423
00:24:06,080 --> 00:24:08,870
Another important property
is the delay property.

424
00:24:12,930 --> 00:24:15,030
Think about what we do
with discrete signals.

425
00:24:15,030 --> 00:24:18,660
We put them through
discrete systems.

426
00:24:18,660 --> 00:24:21,000
Discrete systems of the
type we've looked at so far

427
00:24:21,000 --> 00:24:24,570
have adders, delays, gains.

428
00:24:24,570 --> 00:24:25,730
Delays.

429
00:24:25,730 --> 00:24:28,050
If the Z-transform
couldn't handle delays,

430
00:24:28,050 --> 00:24:30,690
again, we wouldn't do it, right?

431
00:24:30,690 --> 00:24:32,370
Delays are so
fundamental to the way

432
00:24:32,370 --> 00:24:34,590
we think about discrete
systems that it

433
00:24:34,590 --> 00:24:36,660
must be the case that
Z-transforms deal

434
00:24:36,660 --> 00:24:39,310
with delays well, and they do.

435
00:24:39,310 --> 00:24:40,900
We've already seen two examples.

436
00:24:40,900 --> 00:24:45,340
The first example I did
was the unit sample signal

437
00:24:45,340 --> 00:24:49,620
has a transform of 1, and the
delayed unit sample's signal

438
00:24:49,620 --> 00:24:52,044
has transform of
z to the minus 1.

439
00:24:52,044 --> 00:24:53,460
So there's a simple
relationship--

440
00:24:53,460 --> 00:24:57,060
if I knew the first, how
I could find the second.

441
00:24:57,060 --> 00:25:02,830
More generally, if I had
a signal X and I knew its

442
00:25:02,830 --> 00:25:09,710
transform X, then I could
readily compute the transform

443
00:25:09,710 --> 00:25:13,500
of the shifted version just from
the transform of the unshifted

444
00:25:13,500 --> 00:25:15,900
version, and that's-- you
can see how that would work

445
00:25:15,900 --> 00:25:16,950
mathematically.

446
00:25:16,950 --> 00:25:20,402
Let's call the
shifted version Y,

447
00:25:20,402 --> 00:25:22,110
then the transform of
the shifted version

448
00:25:22,110 --> 00:25:25,390
is given by this expression.

449
00:25:25,390 --> 00:25:27,670
By the definition
of shift, y of n

450
00:25:27,670 --> 00:25:30,862
is the same as x of n minus 1.

451
00:25:30,862 --> 00:25:35,350
And now all we need to
do is massage the math

452
00:25:35,350 --> 00:25:39,550
so it ends up looking like the
definition of a Z-transform.

453
00:25:39,550 --> 00:25:46,060
So the way to do that would be
to substitute m for n minus 1.

454
00:25:46,060 --> 00:25:48,730
We get something that looks
more like a Z-transform.

455
00:25:48,730 --> 00:25:50,950
And if we bring this
minus 1 out front,

456
00:25:50,950 --> 00:25:53,620
it looks exactly like a
Z-transform pre-multiplied by z

457
00:25:53,620 --> 00:25:56,070
to the minus 1.

458
00:25:56,070 --> 00:26:00,190
So the delay theorem
just says if I already

459
00:26:00,190 --> 00:26:02,020
know the Z-transform
of a signal,

460
00:26:02,020 --> 00:26:04,600
then to find the Z-transform
of the delayed version

461
00:26:04,600 --> 00:26:07,930
of that signal, simply multiply
the original Z-transform by z

462
00:26:07,930 --> 00:26:10,640
to the minus 1.

463
00:26:10,640 --> 00:26:11,140
OK?

464
00:26:11,140 --> 00:26:15,020
So so far, I've talked
about two properties

465
00:26:15,020 --> 00:26:18,740
of Z-transforms, linearity
and the delay property.

466
00:26:18,740 --> 00:26:24,530
And in combination, they
let me do a lot of stuff

467
00:26:24,530 --> 00:26:27,470
with discrete systems.

468
00:26:27,470 --> 00:26:29,295
So think, for
example, of a system,

469
00:26:29,295 --> 00:26:30,920
a discrete system
that can be described

470
00:26:30,920 --> 00:26:33,528
by a linear difference equation
with constant coefficients.

471
00:26:36,740 --> 00:26:39,290
If you can describe
a system that way,

472
00:26:39,290 --> 00:26:42,560
then you can write the
relationship between the input

473
00:26:42,560 --> 00:26:46,511
signal X and the output signal
Y that looks like a difference

474
00:26:46,511 --> 00:26:47,010
equation.

475
00:26:50,810 --> 00:26:53,900
Then, if you take into
account that the Z-transform

476
00:26:53,900 --> 00:27:00,460
is both linear and has a simple
representation for delays,

477
00:27:00,460 --> 00:27:03,910
I can take the Z-transform
of that difference equation

478
00:27:03,910 --> 00:27:08,030
and get a new expression.

479
00:27:08,030 --> 00:27:10,420
So the difference equation
represents an equality

480
00:27:10,420 --> 00:27:15,730
between two sums of
time domain signals.

481
00:27:15,730 --> 00:27:18,460
Taking the Z-transform
of that equality

482
00:27:18,460 --> 00:27:21,670
tells me some
equivalent relationship

483
00:27:21,670 --> 00:27:24,850
of the Z-transforms.

484
00:27:24,850 --> 00:27:29,890
So if I-- so I think about the
left-hand side by linearity,

485
00:27:29,890 --> 00:27:33,520
I can find the
Z-transform of this sum

486
00:27:33,520 --> 00:27:35,470
by finding the sum
of the Z-transforms.

487
00:27:38,620 --> 00:27:41,500
And so this one's easy, right?

488
00:27:41,500 --> 00:27:46,690
The Z-transform of y of n
is, by assumption, y of z.

489
00:27:46,690 --> 00:27:49,600
By linearity, the part
I didn't show you,

490
00:27:49,600 --> 00:27:51,310
if I pre-multiply
by b nought, it's

491
00:27:51,310 --> 00:27:52,450
pre-multiplied by b nought.

492
00:27:56,040 --> 00:27:58,830
Because it's linear, I can
just add the next term to it.

493
00:27:58,830 --> 00:28:00,990
The next term looks a lot
like the previous term

494
00:28:00,990 --> 00:28:03,150
except that it's shifted,
but shift is easy.

495
00:28:03,150 --> 00:28:06,420
You just put z to
the minus 1 in front.

496
00:28:06,420 --> 00:28:11,300
So this term just becomes
b one, z to the minus 1.

497
00:28:11,300 --> 00:28:14,040
This just becomes b
2, z to the minus 2,

498
00:28:14,040 --> 00:28:17,190
and the whole thing factors.

499
00:28:17,190 --> 00:28:19,467
Same sort of thing
happens on the X side,

500
00:28:19,467 --> 00:28:21,300
and you end up with a
very simple statement.

501
00:28:21,300 --> 00:28:25,750
The Z-transform of a system
that can be represented

502
00:28:25,750 --> 00:28:29,110
by a difference equation
with constant coefficients is

503
00:28:29,110 --> 00:28:33,730
the ratio of two polynomials
in z to the minus 1--

504
00:28:33,730 --> 00:28:36,054
this is a polynomial
in z to the minus 1.

505
00:28:36,054 --> 00:28:40,210
Or if I multiply top and
bottom by z to the k,

506
00:28:40,210 --> 00:28:43,742
I can make it look like a
ratio of polynomials in z.

507
00:28:43,742 --> 00:28:44,242
Yeah.

508
00:28:44,242 --> 00:28:49,274
AUDIENCE: Why is it y to
the x on the b 0 [INAUDIBLE]

509
00:28:49,274 --> 00:28:50,690
DENNIS FREEMAN:
Why is it y the x?

510
00:28:50,690 --> 00:28:53,390
Because of my poor typography.

511
00:28:53,390 --> 00:28:54,230
Thank you very much.

512
00:28:54,230 --> 00:28:57,500
That's completely wrong.

513
00:28:57,500 --> 00:29:00,970
OK, so I'll try to remember
that when I get to my office

514
00:29:00,970 --> 00:29:02,720
and change it before
I post it on the web.

515
00:29:02,720 --> 00:29:03,650
Thank you.

516
00:29:03,650 --> 00:29:07,370
So this is y of z, this
is y of z, this is y of z.

517
00:29:07,370 --> 00:29:12,020
That is-- and you can see I
used [? e-max, ?] so I copied

518
00:29:12,020 --> 00:29:17,660
the formula and I got the
single error to turn into two.

519
00:29:17,660 --> 00:29:20,780
It's always very good when
you can do that, right?

520
00:29:20,780 --> 00:29:22,160
Something like that.

521
00:29:22,160 --> 00:29:25,280
OK, so the point
then is that simply

522
00:29:25,280 --> 00:29:28,540
by knowing that the Z-transform
is linear and has a delay

523
00:29:28,540 --> 00:29:31,070
property, it results in
a very simple statement

524
00:29:31,070 --> 00:29:34,030
about the Z-transform for a
system that can be represented

525
00:29:34,030 --> 00:29:35,280
by such a difference equation.

526
00:29:38,120 --> 00:29:39,080
OK.

527
00:29:39,080 --> 00:29:43,040
Now we use a little bit of
knowledge about polynomials.

528
00:29:43,040 --> 00:29:45,785
We use the idea that
the fundamental theorem

529
00:29:45,785 --> 00:29:46,640
in algebra--

530
00:29:46,640 --> 00:29:48,890
anybody have any
idea what that is?

531
00:29:48,890 --> 00:29:49,889
Nah.

532
00:29:49,889 --> 00:29:51,430
Fundamental theorem
of algebra, what?

533
00:29:51,430 --> 00:29:52,700
AUDIENCE: That was
a long time ago.

534
00:29:52,700 --> 00:29:53,690
DENNIS FREEMAN: Long time ago.

535
00:29:53,690 --> 00:29:56,010
If it was long for you, think
about when it was for me.

536
00:29:56,010 --> 00:29:57,290
[LAUGHTER]

537
00:29:57,290 --> 00:29:59,450
No, you can't think that way.

538
00:29:59,450 --> 00:30:00,279
Yes?

539
00:30:00,279 --> 00:30:06,270
AUDIENCE: Are we [INAUDIBLE]

540
00:30:06,270 --> 00:30:07,460
DENNIS FREEMAN: Wonderful.

541
00:30:07,460 --> 00:30:08,310
Everybody hear that?

542
00:30:08,310 --> 00:30:09,393
Because I can't repeat it.

543
00:30:09,393 --> 00:30:11,750
But everybody hear that?

544
00:30:11,750 --> 00:30:15,800
An nth order
polynomial has n roots.

545
00:30:15,800 --> 00:30:17,690
Roughly speaking.

546
00:30:17,690 --> 00:30:19,040
I'm not a mathematician, right?

547
00:30:19,040 --> 00:30:20,120
I'm an engineer.

548
00:30:20,120 --> 00:30:20,930
Right?

549
00:30:20,930 --> 00:30:24,140
An nth order
polynomial has n roots.

550
00:30:24,140 --> 00:30:26,780
OK, and then the factor theorem.

551
00:30:26,780 --> 00:30:31,170
Surprisingly enough, that
says you can factor things.

552
00:30:31,170 --> 00:30:35,600
So the idea then is that if I
can represent the Z-transform

553
00:30:35,600 --> 00:30:40,530
as a ratio of polynomials and
z, there's a factored form.

554
00:30:40,530 --> 00:30:42,810
And that's the basis
of a decomposition

555
00:30:42,810 --> 00:30:45,150
that we will make
extensive use of.

556
00:30:45,150 --> 00:30:47,040
You've already
seen, extensively,

557
00:30:47,040 --> 00:30:48,300
the roots of the denominator.

558
00:30:48,300 --> 00:30:49,560
They're the poles.

559
00:30:49,560 --> 00:30:52,110
We will similarly define,
because of this manipulation,

560
00:30:52,110 --> 00:30:53,420
the roots of the numerator.

561
00:30:53,420 --> 00:30:55,661
They're the zeros.

562
00:30:55,661 --> 00:30:56,160
OK?

563
00:30:59,630 --> 00:31:02,960
So-- and it's pretty easy
to think through, then,

564
00:31:02,960 --> 00:31:05,120
how there is a relationship--

565
00:31:05,120 --> 00:31:06,974
it's all about
relationships, right?

566
00:31:06,974 --> 00:31:08,390
When I introduce
something, I want

567
00:31:08,390 --> 00:31:10,100
to think about how
the thing I just said

568
00:31:10,100 --> 00:31:13,130
relates to everything
else I've ever said.

569
00:31:13,130 --> 00:31:17,570
There's a simple relationship
between poles and regions

570
00:31:17,570 --> 00:31:19,700
of convergence.

571
00:31:19,700 --> 00:31:22,010
Turns out the regions
of convergence

572
00:31:22,010 --> 00:31:23,570
are always going to be circles.

573
00:31:23,570 --> 00:31:25,670
That has-- circles
in the z plane.

574
00:31:25,670 --> 00:31:27,830
That has to do with
things like convergence

575
00:31:27,830 --> 00:31:29,060
of geometric sequences.

576
00:31:32,490 --> 00:31:37,380
If I build my system out of
adders and delays and gains,

577
00:31:37,380 --> 00:31:39,570
then I have that
complex set of reasoning

578
00:31:39,570 --> 00:31:42,590
that gives rise to the idea
that I have polynomials.

579
00:31:42,590 --> 00:31:45,300
That's going to be-- that's
going to give rise to,

580
00:31:45,300 --> 00:31:51,180
if you think about
poles, partial fractions,

581
00:31:51,180 --> 00:31:52,920
each of those is going
to have some kind

582
00:31:52,920 --> 00:31:54,240
of a characteristic response.

583
00:31:54,240 --> 00:31:55,823
It's going to be a
geometric sequence.

584
00:31:55,823 --> 00:31:58,260
Each of those is going to have
a convergence property that

585
00:31:58,260 --> 00:32:00,370
has something to do with
a circle in the z plane.

586
00:32:03,170 --> 00:32:07,580
Each of those circles is
going to be bounded by a pole.

587
00:32:07,580 --> 00:32:10,670
So the upshot of
all that stuff is

588
00:32:10,670 --> 00:32:12,320
there's a relationship
between poles

589
00:32:12,320 --> 00:32:14,180
and regions of convergence.

590
00:32:14,180 --> 00:32:15,560
Regions of
convergence are always

591
00:32:15,560 --> 00:32:17,510
going to be circles
in the z plane,

592
00:32:17,510 --> 00:32:20,240
and they're always going
to be bounded by a pole.

593
00:32:20,240 --> 00:32:22,430
And we've seen an
example of that already.

594
00:32:22,430 --> 00:32:24,980
If we had a geometric
sequence that

595
00:32:24,980 --> 00:32:29,800
was defined only to the
right of n equals 0--

596
00:32:29,800 --> 00:32:34,190
if it is non-zero, only
at n equals 0 or bigger--

597
00:32:34,190 --> 00:32:39,530
then we get convergence
inside some region defined

598
00:32:39,530 --> 00:32:45,590
by when the base has an absolute
value that's less than 1.

599
00:32:45,590 --> 00:32:50,120
That's a circle in the z plane.

600
00:32:50,120 --> 00:32:56,000
And the pole, which is
alpha, turns into the edge

601
00:32:56,000 --> 00:32:59,100
of that circle of convergence.

602
00:32:59,100 --> 00:33:01,790
So the regions of convergence
for these kinds of systems

603
00:33:01,790 --> 00:33:04,970
will always be circles in
the z plane, always bounded

604
00:33:04,970 --> 00:33:05,500
by a pole.

605
00:33:09,750 --> 00:33:13,650
OK, enough of my talking.

606
00:33:13,650 --> 00:33:17,770
What DC-- what DT signal has
the following Z-transform?

607
00:33:23,120 --> 00:33:27,170
I want to know a DT signal
that has transform of the form

608
00:33:27,170 --> 00:33:31,310
z over z minus 7/8 with
a region of convergence

609
00:33:31,310 --> 00:33:37,410
inside absolute value
of z equals 7/8.

610
00:35:43,450 --> 00:35:48,032
So what's the DT signal
that has that Z-transform?

611
00:35:48,032 --> 00:35:49,490
AUDIENCE: It would
be Y to the n is

612
00:35:49,490 --> 00:35:53,055
equal to x to the n plus
7/8y to the n minus 1?

613
00:35:53,055 --> 00:35:55,180
DENNIS FREEMAN: Sounded
like a difference equation.

614
00:35:55,180 --> 00:35:55,580
Say it again.

615
00:35:55,580 --> 00:35:55,975
AUDIENCE: Oh yeah.

616
00:35:55,975 --> 00:35:57,370
Isn't that what
you're looking for?

617
00:35:57,370 --> 00:35:57,590
DENNIS FREEMAN: No.

618
00:35:57,590 --> 00:35:58,800
I wanted the signal.

619
00:35:58,800 --> 00:35:59,740
AUDIENCE: Oh, OK.

620
00:35:59,740 --> 00:36:01,150
DENNIS FREEMAN: So I
want to think about--

621
00:36:01,150 --> 00:36:02,274
this is a little confusing.

622
00:36:02,274 --> 00:36:04,220
I apologize if I didn't
say this clearly.

623
00:36:04,220 --> 00:36:05,920
We motivated the
idea of Z-transform

624
00:36:05,920 --> 00:36:11,740
by looking at systems, but
the result, the Z-transform's

625
00:36:11,740 --> 00:36:12,700
just a relationship.

626
00:36:12,700 --> 00:36:15,050
It's a map between a function
of n and a function of z.

627
00:36:15,050 --> 00:36:17,750
So we can do that
for every signal.

628
00:36:17,750 --> 00:36:19,570
So if I tell you
the function of z,

629
00:36:19,570 --> 00:36:21,550
you can figure out
the function of n.

630
00:36:21,550 --> 00:36:24,250
The question here is
intended-- what I intended was,

631
00:36:24,250 --> 00:36:27,910
what's the function
of n that corresponds

632
00:36:27,910 --> 00:36:30,220
to that function of z?

633
00:36:36,482 --> 00:36:36,982
Yes?

634
00:36:36,982 --> 00:36:40,150
AUDIENCE: [INAUDIBLE]

635
00:36:40,150 --> 00:36:41,990
DENNIS FREEMAN: 7/8
to the n, u of n.

636
00:36:41,990 --> 00:36:45,650
That sounds like
something we did before.

637
00:36:45,650 --> 00:36:49,166
7/8 to the n, u of n.

638
00:36:49,166 --> 00:36:51,290
That sounds like something
we did before, actually.

639
00:36:54,590 --> 00:36:55,422
Yes, no?

640
00:36:55,422 --> 00:36:57,380
Something we did before
is a good thing, right?

641
00:36:57,380 --> 00:36:59,005
That's one of the
general rules, right?

642
00:36:59,005 --> 00:37:01,250
When I ask a question,
look at what we did before.

643
00:37:01,250 --> 00:37:04,420
That's a good rule.

644
00:37:04,420 --> 00:37:07,090
Is that the same
Z-transform we did before?

645
00:37:07,090 --> 00:37:07,590
Yes?

646
00:37:07,590 --> 00:37:10,360
AUDIENCE: [INAUDIBLE]

647
00:37:10,360 --> 00:37:12,310
DENNIS FREEMAN: Yes, yes.

648
00:37:12,310 --> 00:37:18,966
The region of convergence that
we did before was outside 7/8,

649
00:37:18,966 --> 00:37:21,340
and the region of convergence
I asked for in this problem

650
00:37:21,340 --> 00:37:28,150
is inside 7/8 So what's
the effect of switching

651
00:37:28,150 --> 00:37:29,720
the region of convergence?

652
00:37:33,380 --> 00:37:36,490
What happens if I switch
the region of convergence?

653
00:37:40,490 --> 00:37:42,490
AUDIENCE: Does z change?

654
00:37:42,490 --> 00:37:44,990
Does the value of z change?

655
00:37:44,990 --> 00:37:48,930
DENNIS FREEMAN: Z. It's hard
to talk about the value of z,

656
00:37:48,930 --> 00:37:49,630
right?

657
00:37:49,630 --> 00:37:51,276
Z.

658
00:37:51,276 --> 00:37:52,687
AUDIENCE: [INAUDIBLE]

659
00:37:52,687 --> 00:37:53,770
DENNIS FREEMAN: Excuse me.

660
00:37:53,770 --> 00:37:55,390
So z is defined--

661
00:37:55,390 --> 00:37:57,940
the Z-transform is
defined this way.

662
00:37:57,940 --> 00:38:05,270
So I want to say that h of z
is always z over z minus 7/8.

663
00:38:07,910 --> 00:38:11,410
What happens if I say
the region switched?

664
00:38:11,410 --> 00:38:13,160
Well, it says something
about convergence.

665
00:38:13,160 --> 00:38:14,618
What's convergence
have to do with?

666
00:38:14,618 --> 00:38:20,840
Convergence has to do with,
well, I'm thinking about h

667
00:38:20,840 --> 00:38:29,030
of z is some sum over n of
h of n, z to the minus n.

668
00:38:29,030 --> 00:38:34,340
So convergent has to do
with which ones of those ns

669
00:38:34,340 --> 00:38:39,250
can be in the sum, because
some of these terms, z

670
00:38:39,250 --> 00:38:43,060
to the minus n, when
I switch the region,

671
00:38:43,060 --> 00:38:46,900
I consider a
different family of z.

672
00:38:46,900 --> 00:38:53,780
In the first one, that sum had
to converge outside the circle.

673
00:38:53,780 --> 00:38:56,590
And in the question of interest
now, it has to converge inside.

674
00:38:56,590 --> 00:38:59,410
So it's a different
set of zs for which

675
00:38:59,410 --> 00:39:03,290
the sum has to converge.

676
00:39:03,290 --> 00:39:04,930
There's a way you
can think about that.

677
00:39:04,930 --> 00:39:05,910
Think about that sum--

678
00:39:05,910 --> 00:39:09,389
think about exploding that sum.

679
00:39:09,389 --> 00:39:11,430
In general, we're going
to go from minus infinity

680
00:39:11,430 --> 00:39:13,020
to infinity, so explode that.

681
00:39:13,020 --> 00:39:16,290
So we get a whole
bunch of stuff.

682
00:39:16,290 --> 00:39:21,780
Then we get up to h
of minus 2, z squared.

683
00:39:21,780 --> 00:39:25,890
Then we have h of minus 1, z.

684
00:39:25,890 --> 00:39:30,790
Then we have h of 0, z
to the 0, which is 1.

685
00:39:30,790 --> 00:39:36,060
Then we have h of
1, z to the minus 1.

686
00:39:36,060 --> 00:39:40,840
Then we have h of 2, z to
the minus 2, et cetera.

687
00:39:40,840 --> 00:39:41,340
Right?

688
00:39:41,340 --> 00:39:44,160
That's what the Z-transform
always looks like.

689
00:39:44,160 --> 00:39:49,160
Which of those terms
are the most convergent

690
00:39:49,160 --> 00:39:51,760
when I have a z with
a large magnitude?

691
00:39:57,030 --> 00:40:00,020
So I'm thinking about
the n equals minus 2,

692
00:40:00,020 --> 00:40:02,410
n equals minus 1, n equals 0.

693
00:40:02,410 --> 00:40:05,500
I'm thinking about
all the terms and Ys.

694
00:40:05,500 --> 00:40:09,730
Which of those terms
is the most convergent

695
00:40:09,730 --> 00:40:14,800
if z has a large magnitude?

696
00:40:14,800 --> 00:40:17,275
AUDIENCE: I think it
would be z of negative n.

697
00:40:17,275 --> 00:40:19,227
Like, any negative.

698
00:40:19,227 --> 00:40:21,560
DENNIS FREEMAN: So they get
increasingly convergent as I

699
00:40:21,560 --> 00:40:24,360
go to the right.

700
00:40:24,360 --> 00:40:27,600
If I have z with
a big magnitude,

701
00:40:27,600 --> 00:40:30,420
each term is getting
increasingly convergent

702
00:40:30,420 --> 00:40:33,080
as I go to the right.

703
00:40:33,080 --> 00:40:39,760
That means that these
numbers h 0, h 1, h 2, h 3,

704
00:40:39,760 --> 00:40:42,250
they can keep being
some finite number.

705
00:40:42,250 --> 00:40:44,230
They don't need to be 0.

706
00:40:44,230 --> 00:40:47,650
And it will increasingly
come closer to 0

707
00:40:47,650 --> 00:40:50,180
as I keep going to the right.

708
00:40:50,180 --> 00:40:51,760
The implication of
that is something

709
00:40:51,760 --> 00:40:53,630
that's very important.

710
00:40:53,630 --> 00:40:59,485
So the implication of that is
that a right-sided signal--

711
00:41:03,730 --> 00:41:10,180
signal, not single-- has a--

712
00:41:10,180 --> 00:41:11,710
maps to an outside region.

713
00:41:17,130 --> 00:41:22,720
If I want the sum to get
increasingly convergent

714
00:41:22,720 --> 00:41:26,560
for large values of
z, outside regions--

715
00:41:26,560 --> 00:41:28,540
outside regions have
to take all the values,

716
00:41:28,540 --> 00:41:31,180
no matter how big they get.

717
00:41:31,180 --> 00:41:32,860
OK, I want things
to be on the right.

718
00:41:32,860 --> 00:41:35,710
I don't want things
to be on the left.

719
00:41:35,710 --> 00:41:40,650
Things on the left become
decreasingly convergent.

720
00:41:40,650 --> 00:41:49,270
And a corollary of
that, left-sided signals

721
00:41:49,270 --> 00:41:54,370
map to inside regions.

722
00:41:54,370 --> 00:41:58,000
So in fact, it's not a fluke
that this right-handed signal

723
00:41:58,000 --> 00:42:00,370
mapped to this outside region.

724
00:42:00,370 --> 00:42:02,740
That's where the convergence
for things to the right

725
00:42:02,740 --> 00:42:05,291
are the best.

726
00:42:05,291 --> 00:42:05,790
OK.

727
00:42:05,790 --> 00:42:07,800
Now I've just told you
everything you need to know.

728
00:42:07,800 --> 00:42:09,258
What's it going
to-- what do I need

729
00:42:09,258 --> 00:42:13,380
to have happen if I want it to
converge to the inside region?

730
00:42:16,540 --> 00:42:17,860
Flip the signal.

731
00:42:17,860 --> 00:42:20,860
More or less, I want
it to go to the left.

732
00:42:20,860 --> 00:42:23,880
I don't want it to
go to the right.

733
00:42:23,880 --> 00:42:28,400
So the way I can
think about that

734
00:42:28,400 --> 00:42:31,850
is by thinking about
the functional form

735
00:42:31,850 --> 00:42:38,550
is the same for the inside
and the outside region.

736
00:42:38,550 --> 00:42:41,947
That means they have the
same difference equation,

737
00:42:41,947 --> 00:42:43,780
because you can find
the difference equation

738
00:42:43,780 --> 00:42:46,210
from the functional form.

739
00:42:46,210 --> 00:42:47,970
But the way I want
to think about it

740
00:42:47,970 --> 00:42:50,800
is propagating the signal
that came in to the right

741
00:42:50,800 --> 00:42:52,450
or to the left.

742
00:42:52,450 --> 00:42:55,390
So rather then iterating
forward in time,

743
00:42:55,390 --> 00:42:58,900
which is what we did before,
one way I can think about it is,

744
00:42:58,900 --> 00:43:01,690
let's iterate backwards in time.

745
00:43:01,690 --> 00:43:07,270
Rather than solving for y of
n plus 1 in terms of y of n,

746
00:43:07,270 --> 00:43:11,050
solve instead for y of n
in terms of y of n plus 1.

747
00:43:11,050 --> 00:43:13,180
Run it backwards.

748
00:43:13,180 --> 00:43:15,914
Difference equation
doesn't care.

749
00:43:15,914 --> 00:43:17,580
If the difference
equation doesn't care,

750
00:43:17,580 --> 00:43:21,180
the functional form
will be the same.

751
00:43:21,180 --> 00:43:24,580
So run the difference
equation backwards.

752
00:43:24,580 --> 00:43:30,290
So if you think about doing
that, rest starts to say,

753
00:43:30,290 --> 00:43:32,670
the system starts--

754
00:43:32,670 --> 00:43:37,620
starts means future times,
OK, because we're flipped.

755
00:43:37,620 --> 00:43:39,000
So the signal starts at 0.

756
00:43:39,000 --> 00:43:39,960
That means it starts--

757
00:43:39,960 --> 00:43:45,390
that means at large values
of n, the signal y is 0.

758
00:43:45,390 --> 00:43:46,890
So I fill in this
table with a bunch

759
00:43:46,890 --> 00:43:51,420
of zeros for big values of n.

760
00:43:51,420 --> 00:43:53,640
So n's decreasing this way.

761
00:43:53,640 --> 00:43:55,140
I'm working from toward--

762
00:43:55,140 --> 00:43:58,340
0 toward the left.

763
00:43:58,340 --> 00:44:00,290
So I assume that
I start at rest.

764
00:44:00,290 --> 00:44:03,950
That means the output is
0 for times on the right.

765
00:44:06,880 --> 00:44:10,300
I want to find the unit sample
response, so x is 1 only at n

766
00:44:10,300 --> 00:44:13,070
equals 0.

767
00:44:13,070 --> 00:44:16,780
And now I just
compute each entry

768
00:44:16,780 --> 00:44:19,610
by sticking it into the
difference equation.

769
00:44:19,610 --> 00:44:20,970
And so I've done that.

770
00:44:20,970 --> 00:44:25,060
So you stick these values
into the difference equation,

771
00:44:25,060 --> 00:44:29,210
and that lets you
compute y of minus 1.

772
00:44:29,210 --> 00:44:31,300
So substitute y of minus 1.

773
00:44:31,300 --> 00:44:37,720
Y of minus 1, it depends
on y of 0 and x of 0.

774
00:44:37,720 --> 00:44:40,840
Similarly for all
the entries, and then

775
00:44:40,840 --> 00:44:42,910
I can make a plot of
that, and I get a function

776
00:44:42,910 --> 00:44:43,800
that looks like this.

777
00:44:47,250 --> 00:44:48,130
It's a geometric.

778
00:44:48,130 --> 00:44:50,590
I'm not surprised by that.

779
00:44:50,590 --> 00:44:53,230
The geometric is kind
of facing the wrong way.

780
00:44:53,230 --> 00:44:54,750
It was 7/8 to the end.

781
00:44:54,750 --> 00:44:57,130
It looked convergent.

782
00:44:57,130 --> 00:45:00,210
This one looks
divergent that way,

783
00:45:00,210 --> 00:45:04,240
but I'm multiplying
these h numbers, which

784
00:45:04,240 --> 00:45:08,110
are blowing up that
way, by numbers

785
00:45:08,110 --> 00:45:09,739
that look like z squared.

786
00:45:09,739 --> 00:45:12,155
So if I make z squared small
enough, it'll still converge.

787
00:45:15,320 --> 00:45:17,140
That's the idea.

788
00:45:17,140 --> 00:45:17,770
OK?

789
00:45:17,770 --> 00:45:20,290
By flipping the
region of convergence,

790
00:45:20,290 --> 00:45:25,000
I've changed the magnitudes
of these numbers,

791
00:45:25,000 --> 00:45:27,730
and I've changed which
side converges best.

792
00:45:27,730 --> 00:45:30,600
I've made something that--

793
00:45:30,600 --> 00:45:31,990
it's an inside region.

794
00:45:31,990 --> 00:45:36,280
It was converging for z
values inside some region,

795
00:45:36,280 --> 00:45:38,440
so it's become left-sided.

796
00:45:38,440 --> 00:45:44,590
So the idea then is that I have
two different kinds of time

797
00:45:44,590 --> 00:45:48,520
signals that are both associated
with the same functional form

798
00:45:48,520 --> 00:45:49,270
of z.

799
00:45:49,270 --> 00:45:51,580
They differ by the
region of convergence.

800
00:45:51,580 --> 00:45:54,220
That's why when you
tell me a Z-transform,

801
00:45:54,220 --> 00:45:57,250
you have to tell me the
region of convergence.

802
00:45:57,250 --> 00:45:57,750
OK?

803
00:46:00,770 --> 00:46:02,140
OK.

804
00:46:02,140 --> 00:46:04,480
So I've given some exercises.

805
00:46:04,480 --> 00:46:06,530
I'm running out of time.

806
00:46:06,530 --> 00:46:11,494
The best thing to do is to think
about the exercises offline.

807
00:46:11,494 --> 00:46:12,910
I could lead you
through the math.

808
00:46:12,910 --> 00:46:16,670
That's never very inspirational.

809
00:46:16,670 --> 00:46:20,880
So what I wanted to show you
is that there's many ways

810
00:46:20,880 --> 00:46:26,800
that you can go about
expressing a complicated form

811
00:46:26,800 --> 00:46:31,990
by partial fractions to get
a simpler form that can then

812
00:46:31,990 --> 00:46:36,160
be inverted this way.

813
00:46:36,160 --> 00:46:38,560
So that's the point
of this exercise.

814
00:46:38,560 --> 00:46:42,070
So I went through in the notes,
and the notes are online.

815
00:46:42,070 --> 00:46:44,580
So in the notes, I went
through three different ways

816
00:46:44,580 --> 00:46:48,000
you can think of the answer,
and all of those answers

817
00:46:48,000 --> 00:46:50,760
give you the same
functional form, right?

818
00:46:50,760 --> 00:46:53,370
So it was an exercise
in thinking through how

819
00:46:53,370 --> 00:46:55,680
you do partial fractions.

820
00:46:55,680 --> 00:46:58,320
But there's one more thing
that I want to talk about,

821
00:46:58,320 --> 00:47:01,860
and that is that a lot of these
problems-- one of the biggest

822
00:47:01,860 --> 00:47:05,700
uses of Z-transforms is to
solve difference equations.

823
00:47:08,290 --> 00:47:09,880
Z-transforms are great for that.

824
00:47:09,880 --> 00:47:11,500
I've already talked about that.

825
00:47:11,500 --> 00:47:14,170
With-- by using a Z-transform,
you can take a difference

826
00:47:14,170 --> 00:47:17,680
equation, think about
the difference equation,

827
00:47:17,680 --> 00:47:20,650
think about the input, take the
Laplace transform of everything

828
00:47:20,650 --> 00:47:22,260
you get--

829
00:47:22,260 --> 00:47:25,120
Laplace transform--
a Z-transform, sorry.

830
00:47:25,120 --> 00:47:25,810
Slipped.

831
00:47:25,810 --> 00:47:28,370
Next time it will be
Laplace transform.

832
00:47:28,370 --> 00:47:29,800
Take a Z-transform.

833
00:47:29,800 --> 00:47:32,500
You end up with a Z-transform,
and then the trick

834
00:47:32,500 --> 00:47:37,270
is to recognize the
inverse Z-transform.

835
00:47:37,270 --> 00:47:44,680
There are ways of thinking
about that as a mathematician.

836
00:47:44,680 --> 00:47:50,430
Those ways are not easy,
so by and large, we

837
00:47:50,430 --> 00:47:52,410
will never do this.

838
00:47:52,410 --> 00:47:56,730
If you would like to do that,
I highly recommend course 18.

839
00:47:56,730 --> 00:47:57,660
OK?

840
00:47:57,660 --> 00:47:59,940
It is certainly something
that people in course 18

841
00:47:59,940 --> 00:48:03,210
do all the time, but there
are always simpler ways

842
00:48:03,210 --> 00:48:06,270
that we will do it,
and those simpler ways

843
00:48:06,270 --> 00:48:10,440
derive from thinking about
properties of the z-transform.

844
00:48:10,440 --> 00:48:13,300
And we'll think more about those
as we go forward in the course.

845
00:48:13,300 --> 00:48:16,350
So the point of today
was to emphasize

846
00:48:16,350 --> 00:48:19,140
that there are lots of
different ways of thinking

847
00:48:19,140 --> 00:48:21,540
about DT systems.

848
00:48:21,540 --> 00:48:24,270
You should be able to think of
all the relations between them.

849
00:48:24,270 --> 00:48:25,800
And in particular,
today we talked

850
00:48:25,800 --> 00:48:28,560
about this thing, a
mathematical relationship

851
00:48:28,560 --> 00:48:31,320
for how you can go from
a unit sample response

852
00:48:31,320 --> 00:48:32,250
to a system function.

853
00:48:35,850 --> 00:48:38,000
See you next week.