1
00:00:00,000 --> 00:00:05,604

2
00:00:05,604 --> 00:00:06,830
PROFESSOR: Hi.

3
00:00:06,830 --> 00:00:08,900
Today I'd like to talk to about
a new module called

4
00:00:08,900 --> 00:00:10,140
Signals and Systems.

5
00:00:10,140 --> 00:00:12,280
Our previous module on
programming, we focused on

6
00:00:12,280 --> 00:00:14,710
object-oriented programming
and state machines as

7
00:00:14,710 --> 00:00:16,350
different methods by which
we could model

8
00:00:16,350 --> 00:00:17,790
the physical world.

9
00:00:17,790 --> 00:00:20,100
In this module we're going to
talk about a new way to model

10
00:00:20,100 --> 00:00:22,070
the physical world called
the discrete linear

11
00:00:22,070 --> 00:00:24,140
time-invariant system.

12
00:00:24,140 --> 00:00:27,320
You might say, Kendra, why do
I need to do that, or why

13
00:00:27,320 --> 00:00:28,620
would I move away from
state machines.

14
00:00:28,620 --> 00:00:29,590
I seem to be able to model

15
00:00:29,590 --> 00:00:32,850
everything using state machines.

16
00:00:32,850 --> 00:00:35,200
The short answer is, you want
to be able to move away from

17
00:00:35,200 --> 00:00:36,450
state machines, because
you want to be able

18
00:00:36,450 --> 00:00:37,610
to predict the future.

19
00:00:37,610 --> 00:00:39,645
But I'm not really going to be
able to get to how you're

20
00:00:39,645 --> 00:00:42,250
going to be able to predict
the future until

21
00:00:42,250 --> 00:00:43,890
two videos from now.

22
00:00:43,890 --> 00:00:46,680
So for now we'll introduce
discrete linear time-invariant

23
00:00:46,680 --> 00:00:48,920
systems and also talk about
different knowledge

24
00:00:48,920 --> 00:00:51,000
representations you might
encounter them in, so that you

25
00:00:51,000 --> 00:00:55,200
recognize them when you see them
and can manipulate them

26
00:00:55,200 --> 00:00:58,670
as needed to talk to other
people about them.

27
00:00:58,670 --> 00:00:59,920
Let's go to the board.

28
00:00:59,920 --> 00:01:02,250

29
00:01:02,250 --> 00:01:03,920
Last time we talked about
state machines.

30
00:01:03,920 --> 00:01:06,560
We're able to use them to model
pretty much any system.

31
00:01:06,560 --> 00:01:09,140

32
00:01:09,140 --> 00:01:12,980
We can model the evolution of a
particular process over time

33
00:01:12,980 --> 00:01:17,010
using a record of all previous
inputs and outputs and

34
00:01:17,010 --> 00:01:20,780
possibly previous states, if we

35
00:01:20,780 --> 00:01:22,650
included that in our output.

36
00:01:22,650 --> 00:01:26,900
But the problem with state
machines is that they, like

37
00:01:26,900 --> 00:01:29,810
most programming, encounter
the halting problem.

38
00:01:29,810 --> 00:01:32,460
At some point your program can
reach a little complexity at

39
00:01:32,460 --> 00:01:35,660
which you cannot determine what
it's going to do without

40
00:01:35,660 --> 00:01:38,710
actually just running it and
seeing what happens.

41
00:01:38,710 --> 00:01:40,170
This is great for researchers.

42
00:01:40,170 --> 00:01:44,530
This is the method by which
people go out and have to find

43
00:01:44,530 --> 00:01:46,630
something or discover something
instead of being

44
00:01:46,630 --> 00:01:48,140
able to like, simulate
it using a machine.

45
00:01:48,140 --> 00:01:51,260
But if you want to be able to
promise things to other people

46
00:01:51,260 --> 00:01:54,730
or predict what's going to
happen, then it helps if you

47
00:01:54,730 --> 00:01:58,310
can model something that has a
high level of complexity by

48
00:01:58,310 --> 00:02:00,130
abstracting away the complexity
and interacting

49
00:02:00,130 --> 00:02:04,190
with it as though it has
features that indicate that

50
00:02:04,190 --> 00:02:06,580
it's going to behave in
a particular way.

51
00:02:06,580 --> 00:02:09,020
So if you want to make a control
system that does a

52
00:02:09,020 --> 00:02:13,110
particular thing, or look at
an existing system and say,

53
00:02:13,110 --> 00:02:17,790
well, I know that your power
plant's going to blow up in

54
00:02:17,790 --> 00:02:22,200
five years, because of this,
it helps if you have some

55
00:02:22,200 --> 00:02:28,660
understanding of particular
classes of systems and what

56
00:02:28,660 --> 00:02:30,700
kind of long term behaviors
those particular classes of

57
00:02:30,700 --> 00:02:33,090
systems generate.

58
00:02:33,090 --> 00:02:35,650
And that's why we look
at discrete LTI.

59
00:02:35,650 --> 00:02:37,480
It's a particular class
of system --

60
00:02:37,480 --> 00:02:40,150
in particular discrete LTI is
what we're going to look at in

61
00:02:40,150 --> 00:02:41,230
6.01 because it's
fairly simple.

62
00:02:41,230 --> 00:02:43,200
But once you've learned how to
deal with the system in that

63
00:02:43,200 --> 00:02:47,560
manner, you can use the skills
that you've learned there to

64
00:02:47,560 --> 00:02:50,590
approach more complex feedback
problems or more complex

65
00:02:50,590 --> 00:02:51,840
control problems.

66
00:02:51,840 --> 00:02:54,930

67
00:02:54,930 --> 00:02:57,430
First, an incredibly brief
review of linear

68
00:02:57,430 --> 00:02:58,680
time-invariant systems.

69
00:02:58,680 --> 00:03:01,380

70
00:03:01,380 --> 00:03:03,700
When you're talking about a
linear time-invariant system,

71
00:03:03,700 --> 00:03:05,590
both the inputs and the outputs
are going to be real.

72
00:03:05,590 --> 00:03:09,220
We're not going to deal with
the complex plane at all.

73
00:03:09,220 --> 00:03:14,090
If you were to model your LTI
system using a state machine,

74
00:03:14,090 --> 00:03:16,740
that state machine would be
dependent on a fixed amount of

75
00:03:16,740 --> 00:03:21,240
previous inputs, outputs
or states.

76
00:03:21,240 --> 00:03:23,730
You give some amount of leeway
for start state or the fixed

77
00:03:23,730 --> 00:03:26,900
number of states before you get
to your fixed amount of

78
00:03:26,900 --> 00:03:29,250
data structure that represents
your current state.

79
00:03:29,250 --> 00:03:32,330

80
00:03:32,330 --> 00:03:35,840
But if you're running a linear
time-invariant system, the

81
00:03:35,840 --> 00:03:38,200
amount of information that you
need to figure out your state

82
00:03:38,200 --> 00:03:42,260
in a long term sense is always
finite and fixed.

83
00:03:42,260 --> 00:03:46,850

84
00:03:46,850 --> 00:03:49,130
In terms of functional
expressions, you've probably

85
00:03:49,130 --> 00:03:53,180
seen these associated with
linear time-invariant systems.

86
00:03:53,180 --> 00:03:55,200
Know that this also means that
you can represent linear

87
00:03:55,200 --> 00:04:02,280
time-invariant systems as
additions or scalar

88
00:04:02,280 --> 00:04:04,440
multiplications of your
existing function.

89
00:04:04,440 --> 00:04:07,430

90
00:04:07,430 --> 00:04:09,800
So that's out of the way.

91
00:04:09,800 --> 00:04:11,310
These are all true of LTI.

92
00:04:11,310 --> 00:04:13,190
We're going to focus
on discrete LTI.

93
00:04:13,190 --> 00:04:15,760
One, because we can use
a discrete state

94
00:04:15,760 --> 00:04:17,220
machine to model it.

95
00:04:17,220 --> 00:04:19,180
And two, because it
makes it easier to

96
00:04:19,180 --> 00:04:21,720
represent using computers.

97
00:04:21,720 --> 00:04:23,550
Once you've got that digital
abstraction, you don't have to

98
00:04:23,550 --> 00:04:26,620
worry about having access to a
truly continuous function.

99
00:04:26,620 --> 00:04:32,900

100
00:04:32,900 --> 00:04:33,006
All right.

101
00:04:33,006 --> 00:04:34,460
Now that you know what we're
going to talk about, how do

102
00:04:34,460 --> 00:04:36,730
you talk about it
to other people?

103
00:04:36,730 --> 00:04:38,310
Here are the different
representations that you might

104
00:04:38,310 --> 00:04:41,150
see a discrete linear
time-invariant system in.

105
00:04:41,150 --> 00:04:42,630
One of them is called a
difference equation.

106
00:04:42,630 --> 00:04:44,640
And you've probably encountered
this in one of

107
00:04:44,640 --> 00:04:45,890
your math classes.

108
00:04:45,890 --> 00:04:47,940

109
00:04:47,940 --> 00:04:51,320
It says that you're going to
take a sample from a signal y

110
00:04:51,320 --> 00:04:54,530
at some sort of given time
step, typically n.

111
00:04:54,530 --> 00:04:56,030
And when you're writing out
a difference equation that

112
00:04:56,030 --> 00:04:58,760
represents an entire system, you
usually see y of n on one

113
00:04:58,760 --> 00:05:02,050
side on the left, and then
everything that represents the

114
00:05:02,050 --> 00:05:04,580
functional component on the
system on the right, which in

115
00:05:04,580 --> 00:05:09,760
this case determines what
determines the output

116
00:05:09,760 --> 00:05:11,280
at every time step.

117
00:05:11,280 --> 00:05:13,270
In this very particular case,
we're talking about an

118
00:05:13,270 --> 00:05:17,940
accumulator, which means that at
every time step the output

119
00:05:17,940 --> 00:05:23,520
is determined by the input
plus the previous output.

120
00:05:23,520 --> 00:05:29,020
So at every time step we take
the input and whatever it is

121
00:05:29,020 --> 00:05:32,280
we were outputting before
and output it again.

122
00:05:32,280 --> 00:05:36,720

123
00:05:36,720 --> 00:05:37,970
Pretty straightforward.

124
00:05:37,970 --> 00:05:41,160

125
00:05:41,160 --> 00:05:43,050
If somebody described to you a
difference equation in words,

126
00:05:43,050 --> 00:05:45,120
you could probably turn it into
a equation at this point.

127
00:05:45,120 --> 00:05:48,250

128
00:05:48,250 --> 00:05:50,880
Note that even though there are
variables associated with

129
00:05:50,880 --> 00:05:54,080
these samples, they are still
very particular samples.

130
00:05:54,080 --> 00:05:56,740
And the thing that you're
interested in is probably not

131
00:05:56,740 --> 00:05:57,760
even the samples at all.

132
00:05:57,760 --> 00:06:00,010
It's the relationship between
the samples, when they were

133
00:06:00,010 --> 00:06:03,130
sampled, and relative to one
another, and how that relates

134
00:06:03,130 --> 00:06:05,720
to the output at a
given time step.

135
00:06:05,720 --> 00:06:08,520
If you then want to talk about
an entire signal, then you can

136
00:06:08,520 --> 00:06:10,080
use an operator equation.

137
00:06:10,080 --> 00:06:12,730
And you'll probably see these
things when you do any sort of

138
00:06:12,730 --> 00:06:14,245
research on control theory
or feedback.

139
00:06:14,245 --> 00:06:17,160

140
00:06:17,160 --> 00:06:20,700
Instead of representing the
sample of the signal y at a

141
00:06:20,700 --> 00:06:24,220
given time n, we're just going
to talk about the overall

142
00:06:24,220 --> 00:06:28,715
signal capital Y. Likewise,
instead of talking about x at

143
00:06:28,715 --> 00:06:31,830
a given time n, we're going to
talk about the signal capital

144
00:06:31,830 --> 00:06:37,410
X.

145
00:06:37,410 --> 00:06:39,820
Remember that I said that the
most important part of this

146
00:06:39,820 --> 00:06:41,530
equation is the fact that
there's a different

147
00:06:41,530 --> 00:06:43,390
relationship between these two
signals and when you were

148
00:06:43,390 --> 00:06:44,640
sampling from them.

149
00:06:44,640 --> 00:06:47,340

150
00:06:47,340 --> 00:06:53,080
When we want to represent this
relative delay, we represent

151
00:06:53,080 --> 00:06:59,310
it using an R. And in
particular, I want to note the

152
00:06:59,310 --> 00:07:06,940
fact that the degree of R
represents the amount of delay

153
00:07:06,940 --> 00:07:10,250
associated with the sample
from a particular signal.

154
00:07:10,250 --> 00:07:19,020
So if I wanted to make this n
minus 2, I would reflect that

155
00:07:19,020 --> 00:07:28,137
change in my operator equation
by changing the degree of R.

156
00:07:28,137 --> 00:07:30,110
And because we're working with
a linear time-invariant

157
00:07:30,110 --> 00:07:34,900
system, if we wanted to scale
this by 2, it'd be the same as

158
00:07:34,900 --> 00:07:36,730
scaling this by 2, et cetera.

159
00:07:36,730 --> 00:07:43,770

160
00:07:43,770 --> 00:07:46,390
Now I can talk about signals.

161
00:07:46,390 --> 00:07:48,910
What if I want to talk about a
physical manifestation of the

162
00:07:48,910 --> 00:07:51,500
relationship between these
signals, and/or what if I want

163
00:07:51,500 --> 00:07:55,480
to use something kind of like
circuit diagrams to talk about

164
00:07:55,480 --> 00:07:57,630
what kind of signal manipulation
I'm going to do?

165
00:07:57,630 --> 00:08:00,600

166
00:08:00,600 --> 00:08:03,600
This is where block
diagrams come in.

167
00:08:03,600 --> 00:08:05,720
Block diagrams are incredibly
PowerPoint-friendly.

168
00:08:05,720 --> 00:08:08,370
Block diagrams mean that
everybody is on the same page,

169
00:08:08,370 --> 00:08:10,510
because all you have to do
is trace out the arrows.

170
00:08:10,510 --> 00:08:14,030
Block diagrams mean you're going
to do a combination of

171
00:08:14,030 --> 00:08:19,120
gains, delays and adders to
visually represent the

172
00:08:19,120 --> 00:08:21,160
relationship between your output
signal and any input

173
00:08:21,160 --> 00:08:24,280
signals that you have.

174
00:08:24,280 --> 00:08:26,830
Up here I have actually
drawn an accumulator.

175
00:08:26,830 --> 00:08:30,500
And it's got a box around
it, which will be

176
00:08:30,500 --> 00:08:33,500
relevant in two minutes.

177
00:08:33,500 --> 00:08:37,559
But right now you
can ignore it.

178
00:08:37,559 --> 00:08:39,820
I've got my input signal,
or my input signals, are

179
00:08:39,820 --> 00:08:41,590
typically on the left.

180
00:08:41,590 --> 00:08:44,370
And the progression of gains,
delays and adders associated

181
00:08:44,370 --> 00:08:47,320
with the block diagram represent
the relationship

182
00:08:47,320 --> 00:08:50,780
between the input signals
and the output signal.

183
00:08:50,780 --> 00:08:52,350
And then I'll have my output
signal on the right.

184
00:08:52,350 --> 00:08:57,320

185
00:08:57,320 --> 00:09:00,570
Most people indicate flow with
arrows, in part to make things

186
00:09:00,570 --> 00:09:01,590
easier to read.

187
00:09:01,590 --> 00:09:04,580
But typically, you only need
the arrow indicating where

188
00:09:04,580 --> 00:09:09,080
you're sampling from and
what your output's

189
00:09:09,080 --> 00:09:10,330
going to look like.

190
00:09:10,330 --> 00:09:13,280

191
00:09:13,280 --> 00:09:17,800
In this case in order to get Y,
and in the general sense, I

192
00:09:17,800 --> 00:09:21,670
can backtrace from the signal
that I'm interested in through

193
00:09:21,670 --> 00:09:29,820
my diagram and figure
out what values I'm

194
00:09:29,820 --> 00:09:31,640
actually interested in.

195
00:09:31,640 --> 00:09:37,990
So in this particular case, Y
is a linear combination of X

196
00:09:37,990 --> 00:09:42,710
and whatever this represents,
which is RY.

197
00:09:42,710 --> 00:09:50,700

198
00:09:50,700 --> 00:09:56,900
If I did want to put a 2 here,
I've been talking about gains,

199
00:09:56,900 --> 00:09:59,300
delays and adders, but
this diagram doesn't

200
00:09:59,300 --> 00:10:00,220
contain a gain yet.

201
00:10:00,220 --> 00:10:03,140
So let me show you how you
would include a gain.

202
00:10:03,140 --> 00:10:10,790

203
00:10:10,790 --> 00:10:13,770
Gains are typically represented
by the value of

204
00:10:13,770 --> 00:10:16,240
the gain inside an arrow.

205
00:10:16,240 --> 00:10:18,540
It looks like an op-amp from
schematic diagrams.

206
00:10:18,540 --> 00:10:19,790
We'll get to those later
in circuits.

207
00:10:19,790 --> 00:10:24,520

208
00:10:24,520 --> 00:10:26,330
It's not essential that the
arrow point in the direction

209
00:10:26,330 --> 00:10:29,480
of flow, but people might look
at you funny if you don't put

210
00:10:29,480 --> 00:10:32,140
it in the direction of flow.

211
00:10:32,140 --> 00:10:34,160
And the other thing that I think
I want to note here is

212
00:10:34,160 --> 00:10:43,750
that if you see a minus sign
here, it means X minus 2RY,

213
00:10:43,750 --> 00:10:48,800
which is the same as putting a
negative value on your gain.

214
00:10:48,800 --> 00:10:52,340
So right now, we're back
to X plus 2RY.

215
00:10:52,340 --> 00:10:55,908

216
00:10:55,908 --> 00:10:57,390
OK.

217
00:10:57,390 --> 00:10:59,440
We've got to block diagrams.

218
00:10:59,440 --> 00:11:01,400
You might be saying at this
point, Kendra, what are block

219
00:11:01,400 --> 00:11:03,900
diagrams good for other than
PowerPoint presentations and

220
00:11:03,900 --> 00:11:06,440
possibly arguing with my friends
over what's going on.

221
00:11:06,440 --> 00:11:09,250
And I say, they're really
good for abstraction.

222
00:11:09,250 --> 00:11:17,940

223
00:11:17,940 --> 00:11:22,050
I could draw a box around this
and abstract it away into a

224
00:11:22,050 --> 00:11:27,280
function and say, if I put
something into this box, and I

225
00:11:27,280 --> 00:11:32,480
want to get something out where
the action that happens

226
00:11:32,480 --> 00:11:37,450
in here is identical to this
schematic, then I can find an

227
00:11:37,450 --> 00:11:41,310
equation that actually
represents that operation.

228
00:11:41,310 --> 00:11:44,740
And the way I do that is I'm
going to take my operator

229
00:11:44,740 --> 00:11:55,840
equation and solve in particular
for Y over X. In

230
00:11:55,840 --> 00:12:01,430
this case, with the 2, Y
over X is going to be 1

231
00:12:01,430 --> 00:12:02,750
over (1 minus 2R).

232
00:12:02,750 --> 00:12:13,690

233
00:12:13,690 --> 00:12:19,080
Note that if H is equal to the
expression Y over X, and I

234
00:12:19,080 --> 00:12:27,380
multiply H by X, I get out Y.

235
00:12:27,380 --> 00:12:30,490
This concludes my coverage of
motivations for learning about

236
00:12:30,490 --> 00:12:33,410
discrete LTI systems and also
the different representations

237
00:12:33,410 --> 00:12:36,760
that we will want to use when
talking to other people about

238
00:12:36,760 --> 00:12:38,240
discrete LTI.

239
00:12:38,240 --> 00:12:40,720
At this point we can also start
talking about how we get

240
00:12:40,720 --> 00:12:42,780
to the point where we can start
predicting the future.

241
00:12:42,780 --> 00:12:45,260
And then in the video after
that I'll actually start

242
00:12:45,260 --> 00:12:46,510
talking about poles.

243
00:12:46,510 --> 00:12:47,615