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PROFESSOR: in the last
lecture, we discussed
9
00:01:02,330 --> 00:01:06,370
sinusoidal and real and complex
exponential signals
10
00:01:06,370 --> 00:01:10,110
both for continuous time
and discrete time.
11
00:01:10,110 --> 00:01:14,520
And those signals will form very
important building blocks
12
00:01:14,520 --> 00:01:19,100
when we return to a discussion
of Fourier
13
00:01:19,100 --> 00:01:22,540
analysis in a later lecture.
14
00:01:22,540 --> 00:01:25,830
In today's lecture, I'd like to
introduce some additional
15
00:01:25,830 --> 00:01:29,480
basic signals, specifically
the unit step and
16
00:01:29,480 --> 00:01:31,830
unit impulse signal.
17
00:01:31,830 --> 00:01:35,760
Let's begin with discrete time
and the discrete-time unit
18
00:01:35,760 --> 00:01:38,820
step and unit impulse.
19
00:01:38,820 --> 00:01:42,740
The discrete-time unit step is
a sequence as I've indicated
20
00:01:42,740 --> 00:01:50,570
here, specifically a sequence
which is 0 for negative values
21
00:01:50,570 --> 00:01:56,520
of its argument, and equal to
1 for positive values of its
22
00:01:56,520 --> 00:01:58,970
argument and 0.
23
00:01:58,970 --> 00:02:06,230
So mathematically, the unit
step sequence is 1 for n
24
00:02:06,230 --> 00:02:10,460
greater than or equal to 0
and 0 for n less than 0.
25
00:02:13,660 --> 00:02:17,730
The unit impulse sequence,
likewise, is defined in a
26
00:02:17,730 --> 00:02:19,430
straightforward way.
27
00:02:19,430 --> 00:02:25,630
The unit impulse sequence is a
sequence which is 0 for all
28
00:02:25,630 --> 00:02:31,210
values of its argument
except for n = 0.
29
00:02:31,210 --> 00:02:35,520
So the unit step and unit
impulse sequence are defined
30
00:02:35,520 --> 00:02:38,210
in a straightforward
way mathematically.
31
00:02:38,210 --> 00:02:42,430
And in fact, they are also
related to each other in a
32
00:02:42,430 --> 00:02:45,170
straightforward way
mathematically.
33
00:02:45,170 --> 00:02:50,050
Specifically, the unit impulse
can be related to the unit
34
00:02:50,050 --> 00:02:55,590
step through the relationship
that I've indicated here--
35
00:02:55,590 --> 00:03:01,000
delta of n, the unit impulse,
equal to a unit step minus the
36
00:03:01,000 --> 00:03:02,740
unit step delayed.
37
00:03:02,740 --> 00:03:06,930
So mathematically, the
relationship is what is
38
00:03:06,930 --> 00:03:11,075
referred to as a first
difference.
39
00:03:14,180 --> 00:03:19,330
And to see the validity of this
expression, we can simply
40
00:03:19,330 --> 00:03:24,380
look at the unit step and
its delayed version.
41
00:03:24,380 --> 00:03:29,400
So here, we show the
unit step, u[n].
42
00:03:29,400 --> 00:03:34,060
Here, we show the unit
step delayed by 1.
43
00:03:34,060 --> 00:03:37,890
So it's 0 for n less
than or equal to 0.
44
00:03:37,890 --> 00:03:42,700
And clearly, if we subtract
the delayed step from the
45
00:03:42,700 --> 00:03:49,350
original unit step, everything
subtracts out except at n = 0,
46
00:03:49,350 --> 00:03:52,070
at which point the difference
is equal to 1.
47
00:03:52,070 --> 00:03:55,540
And so the difference
between u[n]
48
00:03:55,540 --> 00:03:57,430
and u[n-1]
49
00:03:57,430 --> 00:04:02,180
is simply the unit impulse,
sometimes incidentally also
50
00:04:02,180 --> 00:04:03,880
referred to as the
unit sample.
51
00:04:06,490 --> 00:04:12,500
Now, in a similar way, we can
express the unit step in terms
52
00:04:12,500 --> 00:04:13,550
of the unit impulse.
53
00:04:13,550 --> 00:04:16,579
And there are several
ways of doing this.
54
00:04:16,579 --> 00:04:21,110
One way is through a
relationship referred to as a
55
00:04:21,110 --> 00:04:23,080
running sum.
56
00:04:23,080 --> 00:04:25,740
What I mean by that is the
following expression.
57
00:04:28,260 --> 00:04:37,960
If we think of forming the sum
from minus infinity up to some
58
00:04:37,960 --> 00:04:45,620
value n of a unit impulse or
unit sample, then this running
59
00:04:45,620 --> 00:04:50,480
sum, in fact, is equal
to the unit step.
60
00:04:50,480 --> 00:04:55,000
And we can see that in a fairly
straightforward way, by
61
00:04:55,000 --> 00:05:01,260
simply observing that in this
expression, when n is less
62
00:05:01,260 --> 00:05:05,750
than 0, there's nothing
accumulated in the sum.
63
00:05:05,750 --> 00:05:07,530
And we can see that graphically
64
00:05:07,530 --> 00:05:09,880
as I've shown here.
65
00:05:09,880 --> 00:05:12,530
So for n less than 0,
so we accumulate no
66
00:05:12,530 --> 00:05:14,340
terms in the sum.
67
00:05:14,340 --> 00:05:20,800
Whereas for n greater than 0, we
accumulate 1 non-zero value
68
00:05:20,800 --> 00:05:26,900
in the sum, namely the value of
the unit sample at n = 0.
69
00:05:26,900 --> 00:05:30,520
So we have, then, one
expression for the
70
00:05:30,520 --> 00:05:34,700
relationship between the unit
step and the unit sample.
71
00:05:34,700 --> 00:05:41,810
We can also develop another
relationship by observing in
72
00:05:41,810 --> 00:05:46,460
essence that if we look at the
unit step sequence, as I've
73
00:05:46,460 --> 00:05:51,850
returned to here, we can, in
effect, think of the unit step
74
00:05:51,850 --> 00:05:56,890
sequence as a succession
of unit impulses,
75
00:05:56,890 --> 00:06:00,320
one following another.
76
00:06:00,320 --> 00:06:14,270
So if we consider forming a sum
of delayed impulses, as I
77
00:06:14,270 --> 00:06:18,430
indicate mathematically here,
and as I indicate graphically
78
00:06:18,430 --> 00:06:24,680
down below, we have an impulse
here at n = 0 and an impulse
79
00:06:24,680 --> 00:06:30,160
here at n = 1, an impulse here
at n = 2, et cetera.
80
00:06:30,160 --> 00:06:34,930
And when we continue to add
these up, then what they add
81
00:06:34,930 --> 00:06:39,740
up to is the unit
step sequence.
82
00:06:39,740 --> 00:06:43,840
And so mathematically, then,
that would correspond to an
83
00:06:43,840 --> 00:06:48,640
impulse at n = 0 plus an impulse
at n = 1 plus an
84
00:06:48,640 --> 00:06:50,960
impulse at n = 2, et cetera.
85
00:06:53,770 --> 00:06:57,980
Now, in continuous time, we
have a very more or less
86
00:06:57,980 --> 00:07:00,270
similar situation.
87
00:07:00,270 --> 00:07:05,330
We will find it equally useful
to talk about a unit step
88
00:07:05,330 --> 00:07:08,540
continuous-time signal
and a unit impulse
89
00:07:08,540 --> 00:07:10,550
continuous-time signal.
90
00:07:10,550 --> 00:07:16,670
Let's begin with the
continuous-time unit step.
91
00:07:16,670 --> 00:07:22,090
The continuous-time unit step
function is graphically
92
00:07:22,090 --> 00:07:24,870
indicated as I've shown here.
93
00:07:24,870 --> 00:07:30,850
It's a time function which
is 0 for t less than 0.
94
00:07:30,850 --> 00:07:34,360
And it's 1 for t
greater than 0.
95
00:07:34,360 --> 00:07:38,970
And so mathematically, what
it corresponds to is u[t]
96
00:07:38,970 --> 00:07:44,720
t defined as a time function
which is 0 for t less than 0,
97
00:07:44,720 --> 00:07:48,250
1 for t greater than 0.
98
00:07:48,250 --> 00:07:53,710
Now, an obvious question is,
what happens at t = 0?
99
00:07:53,710 --> 00:07:56,890
And the difficulty here-- which
is not a difficulty that
100
00:07:56,890 --> 00:07:59,360
arises in the discrete-time
case--
101
00:07:59,360 --> 00:08:03,550
is that at t = 0, the units
step function is in fact
102
00:08:03,550 --> 00:08:06,770
discontinuous, which generates
a variety of
103
00:08:06,770 --> 00:08:08,460
mathematical problems.
104
00:08:08,460 --> 00:08:14,350
And one can define the unit step
at t = 0 in a variety of
105
00:08:14,350 --> 00:08:19,020
ways, but the essential point is
that the unit step function
106
00:08:19,020 --> 00:08:21,220
is discontinuous.
107
00:08:21,220 --> 00:08:25,190
So in effect, what we need to
do is think of the unit step
108
00:08:25,190 --> 00:08:29,980
function as the limit of
a continuous function.
109
00:08:29,980 --> 00:08:34,549
And so we can define a function,
which I specify here
110
00:08:34,549 --> 00:08:38,480
as u_delta(t) (u sub
delta of t).
111
00:08:38,480 --> 00:08:44,420
And u_delta(t) is a time
function which is 0 for t less
112
00:08:44,420 --> 00:08:49,950
than 0, linearly increases
to time delta which would
113
00:08:49,950 --> 00:08:52,720
correspond to this
break point, and
114
00:08:52,720 --> 00:08:55,450
then 1 following that.
115
00:08:55,450 --> 00:09:01,620
And so we can think then of the
discontinuous unit step as
116
00:09:01,620 --> 00:09:08,350
the limiting form of u_delta(t)
as delta goes to 0.
117
00:09:08,350 --> 00:09:13,880
Now, we also want to define
a unit impulse function.
118
00:09:13,880 --> 00:09:16,880
And it had a fairly
straightforward definition in
119
00:09:16,880 --> 00:09:17,840
discrete time.
120
00:09:17,840 --> 00:09:22,130
In continuous time, things get
slightly more difficult.
121
00:09:22,130 --> 00:09:25,370
And to motivate the definition,
let me return to
122
00:09:25,370 --> 00:09:28,980
the discrete-time definition--
123
00:09:28,980 --> 00:09:33,130
or rather, the discrete-time
relationship between the unit
124
00:09:33,130 --> 00:09:37,010
step and the unit impulse
function.
125
00:09:37,010 --> 00:09:44,660
In discrete time, we saw that
the unit impulse function is
126
00:09:44,660 --> 00:09:50,790
the first difference of the
unit step function.
127
00:09:50,790 --> 00:09:56,400
Well, similarly in continuous
time, we can talk about an
128
00:09:56,400 --> 00:10:02,560
impulse function, which is the
first derivative of a unit
129
00:10:02,560 --> 00:10:05,030
step function.
130
00:10:05,030 --> 00:10:09,320
So the unit impulse, as we
want to define it, is the
131
00:10:09,320 --> 00:10:11,850
derivative of the unit step.
132
00:10:11,850 --> 00:10:15,640
Of course, we just finished
discussing the fact that the
133
00:10:15,640 --> 00:10:21,710
unit step function is, in fact,
discontinuous at t = 0.
134
00:10:21,710 --> 00:10:26,960
But we can think of its
derivative as related to this
135
00:10:26,960 --> 00:10:29,840
approximation to
the unit step.
136
00:10:29,840 --> 00:10:34,310
And specifically, we will think
of the continuous-time
137
00:10:34,310 --> 00:10:42,540
impulse as the derivative of
u_delta(t) as delta goes to 0.
138
00:10:42,540 --> 00:10:47,590
So to define the unit impulse,
we think of the derivative of
139
00:10:47,590 --> 00:10:51,950
this approximation to the unit
step and then observe what
140
00:10:51,950 --> 00:10:55,440
happens as delta goes to 0.
141
00:10:55,440 --> 00:11:04,270
We have, then, the definition
of the unit impulse function
142
00:11:04,270 --> 00:11:09,100
more or less formally defined as
the first derivative of the
143
00:11:09,100 --> 00:11:17,320
unit step, or thought of as
the limiting form of the
144
00:11:17,320 --> 00:11:22,360
derivative of the approximation
to the unit step
145
00:11:22,360 --> 00:11:27,380
in the limit, as delta,
the duration of the
146
00:11:27,380 --> 00:11:29,210
discontinuity, goes to 0.
147
00:11:32,500 --> 00:11:35,780
Well, let's look at that.
148
00:11:35,780 --> 00:11:42,820
If we think about the derivative
of u_delta(t), the
149
00:11:42,820 --> 00:11:47,230
derivative, of course, is
0 for t less than 0.
150
00:11:47,230 --> 00:11:51,270
It's equal to a constant during
this linear slope, and
151
00:11:51,270 --> 00:11:55,220
then, it's 0 for t greater
than delta.
152
00:11:55,220 --> 00:12:01,070
So the derivative of u_delta(t)
will then be as
153
00:12:01,070 --> 00:12:04,000
I've indicated here.
154
00:12:04,000 --> 00:12:09,110
And it's simply a rectangle with
a height, which is 1 /
155
00:12:09,110 --> 00:12:13,260
delta, and a width, which
is equal to delta.
156
00:12:13,260 --> 00:12:17,460
And observe that no matter what
the value of delta is,
157
00:12:17,460 --> 00:12:22,260
the area is always equal to 1.
158
00:12:22,260 --> 00:12:27,340
Now, as we let delta go to 0,
what happens is that the width
159
00:12:27,340 --> 00:12:30,270
of the rectangle gets smaller,
the height of the rectangle
160
00:12:30,270 --> 00:12:33,900
gets bigger, the area
still remains 1.
161
00:12:33,900 --> 00:12:38,210
As delta goes to 0, of course,
the width goes to 0, and the
162
00:12:38,210 --> 00:12:40,470
height goes to infinity.
163
00:12:40,470 --> 00:12:48,550
And graphically, we choose to
depict that as an arrow, where
164
00:12:48,550 --> 00:12:53,240
the arrow indicates the fact
that we have an impulse
165
00:12:53,240 --> 00:12:58,890
occurring at t = 0, and the
height of the impulse is used
166
00:12:58,890 --> 00:13:03,380
to represent what the area
of the impulse is.
167
00:13:03,380 --> 00:13:06,350
And in this case, since we took
the derivative of a unit
168
00:13:06,350 --> 00:13:10,010
step, the height
is equal to 1.
169
00:13:13,300 --> 00:13:19,360
So the impulse has 0 width,
infinite height, area 1.
170
00:13:19,360 --> 00:13:24,060
It's mathematically not terribly
comfortable because
171
00:13:24,060 --> 00:13:27,310
what we've done is taken the
derivative of the unit step,
172
00:13:27,310 --> 00:13:30,970
which has a discontinuity at the
origin, and there's some
173
00:13:30,970 --> 00:13:33,780
mathematical difficulties
in doing that.
174
00:13:33,780 --> 00:13:37,700
We'll in fact return to another
interpretation of the
175
00:13:37,700 --> 00:13:43,990
impulse later to emphasize some
of the discomfort with
176
00:13:43,990 --> 00:13:45,460
the impulse.
177
00:13:45,460 --> 00:13:49,820
I remember something that Sam
Mason used to say that his
178
00:13:49,820 --> 00:13:52,990
students said about
the unit impulse.
179
00:13:52,990 --> 00:13:56,810
His definition was the unit
impulse is something that's so
180
00:13:56,810 --> 00:14:00,470
small every place I can't see
it except at one point where
181
00:14:00,470 --> 00:14:01,870
it's so big I can't see it.
182
00:14:01,870 --> 00:14:04,090
In other words, I
can't see it at.
183
00:14:04,090 --> 00:14:10,440
Well, accept some informality
with the unit impulse function
184
00:14:10,440 --> 00:14:15,410
in the continuous time case, and
generally, we'll see that
185
00:14:15,410 --> 00:14:17,505
we won't get into particular
difficulty.
186
00:14:20,040 --> 00:14:25,230
OK, now the impulse is the
derivative of the step.
187
00:14:25,230 --> 00:14:29,510
We saw in the discrete-time case
that the step could be
188
00:14:29,510 --> 00:14:32,952
recovered from the impulse
through a running sum.
189
00:14:32,952 --> 00:14:36,250
In continuous time, if the
impulse is the derivative of a
190
00:14:36,250 --> 00:14:39,900
step, we would more or less
reasonably expect that the
191
00:14:39,900 --> 00:14:43,420
step would be like an integral
of the impulse.
192
00:14:43,420 --> 00:14:45,370
And indeed, that's true.
193
00:14:45,370 --> 00:14:52,350
In fact, mathematically the
relationship is that the unit
194
00:14:52,350 --> 00:14:59,340
step function in continuous time
is the running integral
195
00:14:59,340 --> 00:15:02,230
of the unit impulse function.
196
00:15:02,230 --> 00:15:07,330
And so it's the integral from
minus infinity up to time t,
197
00:15:07,330 --> 00:15:13,900
where t is the argument at which
we're examining u(t).
198
00:15:13,900 --> 00:15:20,270
And so, just as with the
discrete-time case, if we are
199
00:15:20,270 --> 00:15:24,580
looking at t less than 0,
there's no area accumulated in
200
00:15:24,580 --> 00:15:25,780
the integral.
201
00:15:25,780 --> 00:15:29,620
If we're looking at t greater
than 0, then we accumulate the
202
00:15:29,620 --> 00:15:31,255
area under the impulse.
203
00:15:35,760 --> 00:15:43,160
OK, now we'll shortly be
returning to a further
204
00:15:43,160 --> 00:15:48,390
discussion on use of impulses
and step functions.
205
00:15:48,390 --> 00:15:52,580
And in particular, what we'll
see is that they provide a
206
00:15:52,580 --> 00:15:58,300
very convenient and powerful
mechanism for describing a
207
00:15:58,300 --> 00:16:01,900
particular class of systems
referred to as linear
208
00:16:01,900 --> 00:16:04,650
time-invariant systems.
209
00:16:04,650 --> 00:16:07,710
To lead up to that discussion,
which will be the principal
210
00:16:07,710 --> 00:16:12,470
focus of the next lecture, let's
for the remainder of
211
00:16:12,470 --> 00:16:15,980
this lecture talk about systems
in general and then
212
00:16:15,980 --> 00:16:17,860
some properties of systems.
213
00:16:17,860 --> 00:16:23,380
And as the lecture proceeds, the
specific properties that I
214
00:16:23,380 --> 00:16:26,085
want to get to are properties
of linearity and time
215
00:16:26,085 --> 00:16:28,010
invariance.
216
00:16:28,010 --> 00:16:31,820
So let's first talk about
systems in general.
217
00:16:31,820 --> 00:16:36,850
And a system in general, in its
most general definition,
218
00:16:36,850 --> 00:16:41,770
is simply a transformation from
an input signal to an
219
00:16:41,770 --> 00:16:43,840
output signal.
220
00:16:43,840 --> 00:16:48,360
So in a continuous-time
case, we would have a
221
00:16:48,360 --> 00:16:53,740
continuous-time system,
the input, x[t],
222
00:16:53,740 --> 00:16:56,140
and the output, y[t].
223
00:16:56,140 --> 00:17:00,410
And the box, in essence,
is used to denote a
224
00:17:00,410 --> 00:17:02,600
transformation from x[t]
225
00:17:02,600 --> 00:17:04,740
to y[t].
226
00:17:04,740 --> 00:17:09,079
And sometimes, we'll also use a
shorthand notation along the
227
00:17:09,079 --> 00:17:14,240
lines of indicating that the
input, x[t], is transformed to
228
00:17:14,240 --> 00:17:17,700
the output, y[t].
229
00:17:17,700 --> 00:17:21,540
Now, we have exactly the same
kind of definition in the
230
00:17:21,540 --> 00:17:23,349
discrete-time case.
231
00:17:23,349 --> 00:17:27,020
In the discrete-time case,
of course, the inputs are
232
00:17:27,020 --> 00:17:31,000
sequences, and the outputs
are sequences.
233
00:17:31,000 --> 00:17:33,450
And our shorthand notation
is similar.
234
00:17:36,250 --> 00:17:39,280
Often when we talk about
systems, we'll want to talk
235
00:17:39,280 --> 00:17:41,380
about interconnections
of systems.
236
00:17:41,380 --> 00:17:45,080
And we'll see again in later
lectures that interconnections
237
00:17:45,080 --> 00:17:48,080
become very important
and powerful.
238
00:17:48,080 --> 00:17:53,660
And in the way of introducing
terminology, let me introduce
239
00:17:53,660 --> 00:17:56,490
the terminology for a few
basic and important
240
00:17:56,490 --> 00:17:59,840
interconnections of systems.
241
00:17:59,840 --> 00:18:07,230
The first is what's referred to
as a cascade of systems, or
242
00:18:07,230 --> 00:18:11,110
sometimes as a series
interconnection of systems.
243
00:18:11,110 --> 00:18:14,720
And putting two systems in
cascade, as I've indicated
244
00:18:14,720 --> 00:18:19,860
here, means taking the output
of one system--
245
00:18:19,860 --> 00:18:21,780
let's say system 1--
246
00:18:21,780 --> 00:18:25,890
and using that as the input to
the second system, which I've
247
00:18:25,890 --> 00:18:27,880
denoted as system 2.
248
00:18:27,880 --> 00:18:34,420
So in this cascade with system
1 first and system 2 second,
249
00:18:34,420 --> 00:18:37,740
we have the output of system
1 going into the
250
00:18:37,740 --> 00:18:40,710
input of system 2.
251
00:18:40,710 --> 00:18:46,000
Now, we could, of course, take
those two systems and cascade
252
00:18:46,000 --> 00:18:48,570
them in the reverse order.
253
00:18:48,570 --> 00:18:52,630
Namely, take the output of
system 2 and put it into the
254
00:18:52,630 --> 00:18:54,650
input of system 1.
255
00:18:54,650 --> 00:18:56,995
And I've indicated that here.
256
00:18:56,995 --> 00:19:00,330
And we have system 2 first
in the cascade
257
00:19:00,330 --> 00:19:04,440
followed by system 1.
258
00:19:04,440 --> 00:19:09,370
It's important to keep in mind
that in general, for general
259
00:19:09,370 --> 00:19:13,480
systems, the order in which
you cascade the systems is
260
00:19:13,480 --> 00:19:15,440
very important.
261
00:19:15,440 --> 00:19:19,640
And in fact, the overall system
transformation will be
262
00:19:19,640 --> 00:19:22,710
different depending on which
system came first and which
263
00:19:22,710 --> 00:19:24,840
system came second.
264
00:19:24,840 --> 00:19:29,130
For example, if system 1 was,
let's say, a system for which
265
00:19:29,130 --> 00:19:35,390
the output doubles the input,
and system 2, the output is
266
00:19:35,390 --> 00:19:39,550
the square root of the input,
clearly doubling first and
267
00:19:39,550 --> 00:19:43,080
then taking the square root is
different than taking the
268
00:19:43,080 --> 00:19:46,680
square root first and
then doubling.
269
00:19:46,680 --> 00:19:52,570
Now, kind of amazingly, what
we'll see again when we get to
270
00:19:52,570 --> 00:19:55,140
this issue of linearity
and time invariance--
271
00:19:55,140 --> 00:19:58,340
which are a class of systems
that we'll focus in on--
272
00:19:58,340 --> 00:20:02,640
somewhat amazingly, it turns
out that for that specific
273
00:20:02,640 --> 00:20:07,480
class of systems, the overall
system transformation is
274
00:20:07,480 --> 00:20:09,680
independent of the
order in which
275
00:20:09,680 --> 00:20:12,280
the systems are cascaded.
276
00:20:12,280 --> 00:20:16,280
And we'll see that, of course,
in more detail later on.
277
00:20:16,280 --> 00:20:19,110
All right, well, that's
a cascade or a series
278
00:20:19,110 --> 00:20:20,970
interconnection.
279
00:20:20,970 --> 00:20:24,240
Let's look at another
interconnection, which is a
280
00:20:24,240 --> 00:20:27,270
parallel interconnection
of systems.
281
00:20:27,270 --> 00:20:33,110
And here, what's meant by the
interconnection is that the
282
00:20:33,110 --> 00:20:39,220
input is fed simultaneously
into system 1
283
00:20:39,220 --> 00:20:42,260
and into system 2.
284
00:20:42,260 --> 00:20:47,410
And then, the outputs of these
two systems are added to give
285
00:20:47,410 --> 00:20:50,900
the overall system output.
286
00:20:50,900 --> 00:20:57,670
Now, in this particular case,
in contrast to a cascade, no
287
00:20:57,670 --> 00:21:02,710
matter what the systems are, it
turns out that a parallel
288
00:21:02,710 --> 00:21:06,520
combination, the order
in which they're
289
00:21:06,520 --> 00:21:09,740
put in parallel is--
290
00:21:09,740 --> 00:21:13,660
the overall transformation is
independent of the order.
291
00:21:13,660 --> 00:21:16,230
And of course, that follows
from the fact that we're
292
00:21:16,230 --> 00:21:18,460
simply adding up outputs.
293
00:21:18,460 --> 00:21:22,120
And the outputs can be added
in any order because of the
294
00:21:22,120 --> 00:21:25,000
fact that addition doesn't care
in which order you're
295
00:21:25,000 --> 00:21:27,840
adding things up.
296
00:21:27,840 --> 00:21:30,880
OK, so that's the parallel
interconnection.
297
00:21:30,880 --> 00:21:35,460
And let's look at one more
interconnection, which is
298
00:21:35,460 --> 00:21:39,320
what's referred to as a feedback
interconnection.
299
00:21:39,320 --> 00:21:43,530
And this, again, is an
interconnection that will
300
00:21:43,530 --> 00:21:47,320
become a very important topic
much later in the course.
301
00:21:47,320 --> 00:21:51,940
But let me just indicate at this
point what I mean by it.
302
00:21:51,940 --> 00:21:55,000
What a feedback interconnection
means is that
303
00:21:55,000 --> 00:22:05,090
we have one system, system 1,
with an input and an output.
304
00:22:05,090 --> 00:22:09,710
The output of system 1 is the
output of the overall system.
305
00:22:09,710 --> 00:22:16,120
And that output is fed
into system 2.
306
00:22:16,120 --> 00:22:22,260
The output of system 2 is then
added to the input to the
307
00:22:22,260 --> 00:22:23,860
overall system.
308
00:22:23,860 --> 00:22:27,790
So in essence, what happens in
a feedback interconnection is
309
00:22:27,790 --> 00:22:29,910
we have one system.
310
00:22:29,910 --> 00:22:35,410
The output of that system is
fed back through system 2,
311
00:22:35,410 --> 00:22:37,290
added to the overall input.
312
00:22:37,290 --> 00:22:40,770
And then, that sum is what forms
the input of system 1.
313
00:22:40,770 --> 00:22:45,870
And we'll see that there are
lots of uses for feedback and,
314
00:22:45,870 --> 00:22:49,680
of course, also lots of ways
that feedback gets in the way.
315
00:22:49,680 --> 00:22:53,700
In fact, probably some of you
are already familiar with some
316
00:22:53,700 --> 00:22:55,580
of the issues in feedback--
317
00:22:55,580 --> 00:22:57,910
for example, in audio
systems or whatever.
318
00:22:57,910 --> 00:23:01,750
But this will be a topic that
we'll devote a considerable
319
00:23:01,750 --> 00:23:05,580
amount of time to later
in the course.
320
00:23:05,580 --> 00:23:10,170
And then, of course, there are
lots of other interconnections
321
00:23:10,170 --> 00:23:12,640
of systems.
322
00:23:12,640 --> 00:23:15,900
And as the course progresses,
we'll see lots of ways in
323
00:23:15,900 --> 00:23:19,270
which systems get interconnected
both in series
324
00:23:19,270 --> 00:23:22,700
and in parallel and feedback
interconnections, et cetera,
325
00:23:22,700 --> 00:23:28,040
to achieve a wide variety
of things.
326
00:23:28,040 --> 00:23:33,720
Now, what I've said so far
relates to systems in general.
327
00:23:33,720 --> 00:23:37,240
And you can't say much about
systems when you try to treat
328
00:23:37,240 --> 00:23:39,470
them in their most
general form.
329
00:23:39,470 --> 00:23:44,845
So it's useful and important to
focus in on properties that
330
00:23:44,845 --> 00:23:47,160
a system may or may not have.
331
00:23:47,160 --> 00:23:51,730
So what I'd like to do now is
turn our attention to system
332
00:23:51,730 --> 00:23:53,000
properties.
333
00:23:53,000 --> 00:23:56,120
And we'll define a
number of them.
334
00:23:56,120 --> 00:23:59,220
Some of them, we'll want
to impose on a system.
335
00:23:59,220 --> 00:24:02,460
Some of them, we may not want
to impose on a system.
336
00:24:02,460 --> 00:24:08,810
But as things progress, we'll
tend to find it useful to ask
337
00:24:08,810 --> 00:24:10,540
whether a system
does or doesn't
338
00:24:10,540 --> 00:24:13,200
have a certain property.
339
00:24:13,200 --> 00:24:18,640
Well, let's begin with a
property which I refer to here
340
00:24:18,640 --> 00:24:20,690
as memoryless.
341
00:24:20,690 --> 00:24:26,540
And what I mean by a system
being memoryless is that the
342
00:24:26,540 --> 00:24:35,590
output at any given time, t_0,
depends only on the input at
343
00:24:35,590 --> 00:24:36,550
the same time.
344
00:24:36,550 --> 00:24:41,560
And so what I'm suggesting is
that a memoryless system is
345
00:24:41,560 --> 00:24:48,320
one for which the output at a
specific time depends only on
346
00:24:48,320 --> 00:24:50,520
the input at that time.
347
00:24:50,520 --> 00:24:56,080
And that statement is true, or
that definition applies, both
348
00:24:56,080 --> 00:25:03,130
for continuous time, as I've
indicated here, and also for
349
00:25:03,130 --> 00:25:05,390
discrete time, as I've
indicated here.
350
00:25:05,390 --> 00:25:10,670
And so we have a similar
definition in discrete time,
351
00:25:10,670 --> 00:25:16,650
that the system is memoryless
if the output at any given
352
00:25:16,650 --> 00:25:20,750
time depends only on the
input at that time.
353
00:25:23,490 --> 00:25:27,000
Well, I have a number
of examples here.
354
00:25:27,000 --> 00:25:31,760
Let's look at the first example
in which the output is
355
00:25:31,760 --> 00:25:33,520
the square of the input.
356
00:25:33,520 --> 00:25:39,170
And that is a possible system
either in continuous time or
357
00:25:39,170 --> 00:25:40,680
discrete time.
358
00:25:40,680 --> 00:25:45,410
And that system, as you would
expect, is commonly referred
359
00:25:45,410 --> 00:25:48,670
to a squarer.
360
00:25:48,670 --> 00:25:53,480
And since the square of a signal
at any time depends
361
00:25:53,480 --> 00:25:58,220
only on the value of the signal
at that time, clearly a
362
00:25:58,220 --> 00:26:01,190
squarer is a memoryless
system.
363
00:26:01,190 --> 00:26:06,870
So in fact, we can indicate
here that this system is
364
00:26:06,870 --> 00:26:08,120
memoryless.
365
00:26:10,850 --> 00:26:15,980
Another important system is
what is referred to as an
366
00:26:15,980 --> 00:26:17,230
integrator.
367
00:26:18,990 --> 00:26:26,460
The output is equal to the
integral of the input.
368
00:26:26,460 --> 00:26:31,240
Here, as I've indicated it,
it's not just integrating
369
00:26:31,240 --> 00:26:35,100
x(t), but it's integrating
the square of x(t).
370
00:26:35,100 --> 00:26:40,610
And whether we square before
we integrate or not, the
371
00:26:40,610 --> 00:26:46,060
essential point is that since
we're integrating the input,
372
00:26:46,060 --> 00:26:50,800
the value of the output at any
time is an accumulation of
373
00:26:50,800 --> 00:26:52,900
past history of the input.
374
00:26:52,900 --> 00:26:55,690
Well, an accumulation
of past history in
375
00:26:55,690 --> 00:26:57,580
essence implies memory.
376
00:26:57,580 --> 00:27:04,580
To get the output at a given
time requires the input over
377
00:27:04,580 --> 00:27:08,330
an interval, specifically for
longer than that time.
378
00:27:08,330 --> 00:27:12,340
So this system, in fact,
is not memoryless.
379
00:27:16,350 --> 00:27:21,970
And now, I have a third
system defined here.
380
00:27:21,970 --> 00:27:26,530
The third system is a system,
a discrete-time system, in
381
00:27:26,530 --> 00:27:34,210
which the output is equal to
the input but not quite.
382
00:27:34,210 --> 00:27:36,190
The output at some time--
let's say for
383
00:27:36,190 --> 00:27:38,690
example at n = 0--
384
00:27:38,690 --> 00:27:44,260
is equal to the input at one
time sample, or instant, or
385
00:27:44,260 --> 00:27:47,260
value of the index
before that.
386
00:27:47,260 --> 00:27:52,930
And so this, in fact, is a
system for which the output is
387
00:27:52,930 --> 00:27:57,930
simply the input delayed or
shifted by one sample.
388
00:27:57,930 --> 00:28:05,610
So this system is referred
to as a unit delay.
389
00:28:05,610 --> 00:28:10,190
And now, the question is, is
that system memoryless?
390
00:28:12,830 --> 00:28:16,530
Well, the output depends
only on the input--
391
00:28:16,530 --> 00:28:20,530
the output at any instant
depends only on the input at
392
00:28:20,530 --> 00:28:22,130
one instant.
393
00:28:22,130 --> 00:28:27,960
But since it depends on an
instant prior to the time at
394
00:28:27,960 --> 00:28:30,040
which we're looking, or
different than the time at
395
00:28:30,040 --> 00:28:33,310
which we're looking, it violates
the definition of
396
00:28:33,310 --> 00:28:35,950
memoryless that we introduced.
397
00:28:35,950 --> 00:28:40,350
And so, in fact, the unit delay
is a system that has
398
00:28:40,350 --> 00:28:43,990
memory, and so let's
indicate that here.
399
00:28:43,990 --> 00:28:50,120
So this is not a memoryless
system because of the fact
400
00:28:50,120 --> 00:28:52,540
that there is 1 unit of delay.
401
00:28:52,540 --> 00:28:54,800
And in essence, delay
requires memory.
402
00:28:58,260 --> 00:29:04,500
OK, so that's the issue of
memory and memoryless systems.
403
00:29:04,500 --> 00:29:10,080
Let's now turn to another
property which a system may or
404
00:29:10,080 --> 00:29:14,155
may not have, the property
referred to as invertibility.
405
00:29:16,700 --> 00:29:21,330
Now, essentially what
invertibility means is that
406
00:29:21,330 --> 00:29:25,120
given the output of the system,
you can figure out
407
00:29:25,120 --> 00:29:27,140
uniquely what the input was.
408
00:29:27,140 --> 00:29:30,740
That's one definition
for invertibility.
409
00:29:30,740 --> 00:29:34,640
Said another way, invertibility
means that given
410
00:29:34,640 --> 00:29:37,100
the output, there's only
one input that
411
00:29:37,100 --> 00:29:38,180
could have caused it.
412
00:29:38,180 --> 00:29:43,180
That's another common definition
for invertibility.
413
00:29:43,180 --> 00:29:46,700
Another way of looking at it is,
in fact, to look at it in
414
00:29:46,700 --> 00:29:50,275
the context of a cascade
of systems.
415
00:29:53,040 --> 00:29:55,810
So let's consider a system.
416
00:29:55,810 --> 00:30:03,490
And here is a system which I
refer to as system A. And this
417
00:30:03,490 --> 00:30:06,260
could be continuous-time
or discrete-time.
418
00:30:06,260 --> 00:30:11,270
It has an input, x_1(t) or
x_1[n], depending on whether
419
00:30:11,270 --> 00:30:13,200
it's continuous-time or
discrete-time that we're
420
00:30:13,200 --> 00:30:18,760
talking about, and an
associated output.
421
00:30:18,760 --> 00:30:24,970
And here, we have system
B with its
422
00:30:24,970 --> 00:30:29,510
associated input and output.
423
00:30:29,510 --> 00:30:34,490
And now, let's put these
two systems in cascade.
424
00:30:34,490 --> 00:30:40,040
So we'll take the output of
system 1 and feed it into the
425
00:30:40,040 --> 00:30:43,280
input of system 2--
426
00:30:43,280 --> 00:30:48,590
or system B. So the output of
system A goes into the input
427
00:30:48,590 --> 00:30:56,625
of system B. And if system A is
invertible and system B is
428
00:30:56,625 --> 00:31:04,030
its inverse, then the
consequence is that the output
429
00:31:04,030 --> 00:31:10,730
of system B is equal to
the input of system A.
430
00:31:10,730 --> 00:31:15,560
Now, I know there are a lot
of inputs and outputs and
431
00:31:15,560 --> 00:31:17,080
inverses in there.
432
00:31:17,080 --> 00:31:22,050
But essentially, what we mean by
what I've just said is that
433
00:31:22,050 --> 00:31:30,160
if we have system A and it's
invertible, and if we cascade
434
00:31:30,160 --> 00:31:37,650
it with its inverse, system B,
then the overall cascade of
435
00:31:37,650 --> 00:31:43,710
these two systems is simply
what's referred to as the
436
00:31:43,710 --> 00:31:45,110
identity system.
437
00:31:48,180 --> 00:31:52,280
And the identity system is
simply a system which if you
438
00:31:52,280 --> 00:31:56,280
put a signal into it, you get
the same signal out of it.
439
00:31:56,280 --> 00:31:59,200
In other words, the overall
system is no
440
00:31:59,200 --> 00:32:00,830
transformation at all.
441
00:32:00,830 --> 00:32:07,330
And clearly, of course, for
a system to be able to be
442
00:32:07,330 --> 00:32:11,400
cascaded with another system
to generate the identity
443
00:32:11,400 --> 00:32:15,920
system requires that the first
system, system A, be
444
00:32:15,920 --> 00:32:18,250
invertible.
445
00:32:18,250 --> 00:32:23,730
So let's look at
some examples.
446
00:32:23,730 --> 00:32:30,090
If we had system A as I've
indicated here, where now the
447
00:32:30,090 --> 00:32:36,980
output is the running integral
of the input--
448
00:32:36,980 --> 00:32:39,420
and remember that we saw the
running integral when we
449
00:32:39,420 --> 00:32:42,990
talked about the relationship
between steps and impulses--
450
00:32:42,990 --> 00:32:45,970
if this happened to be
an impulse, then the
451
00:32:45,970 --> 00:32:48,640
output would be a step.
452
00:32:48,640 --> 00:32:54,790
This system is referred to, of
course, as an integrator since
453
00:32:54,790 --> 00:33:01,180
the output is the running
integral of the input.
454
00:33:01,180 --> 00:33:09,570
And the integrator, in fact,
is an invertible system.
455
00:33:09,570 --> 00:33:14,450
And its inverse is a system for
which the output is the
456
00:33:14,450 --> 00:33:16,350
derivative of the input.
457
00:33:16,350 --> 00:33:22,700
So the inverse of system A, if
system A is an integrator, is
458
00:33:22,700 --> 00:33:27,970
a system for which the output is
equal to the derivative of
459
00:33:27,970 --> 00:33:32,360
the input which, not
surprisingly, is referred to
460
00:33:32,360 --> 00:33:35,515
as a differentiator.
461
00:33:40,870 --> 00:33:46,690
So an integrator
is invertible.
462
00:33:46,690 --> 00:33:50,270
Its inverse is a
differentiator.
463
00:33:50,270 --> 00:33:52,810
What you might want to think
about is the question of
464
00:33:52,810 --> 00:33:56,550
whether a differentiator
is invertible.
465
00:33:56,550 --> 00:34:00,130
Now, to answer that, what you
would ask yourself is, if you
466
00:34:00,130 --> 00:34:02,390
always knew what the derivative
of the signal is,
467
00:34:02,390 --> 00:34:06,410
would you necessarily know
what the signal was?
468
00:34:06,410 --> 00:34:10,090
In other words, if you have a
differentiator and you have
469
00:34:10,090 --> 00:34:12,060
the output of the
differentiator, could you
470
00:34:12,060 --> 00:34:14,550
always figure out what
the input was?
471
00:34:14,550 --> 00:34:17,429
If you could, the system
would be invertible.
472
00:34:17,429 --> 00:34:21,639
If you couldn't, the system
would not be invertible.
473
00:34:21,639 --> 00:34:24,060
So you might just want
to think about that.
474
00:34:24,060 --> 00:34:26,630
I guess I won't tell
you right now.
475
00:34:26,630 --> 00:34:32,260
But I'm sure that you'll think
about that more, particularly
476
00:34:32,260 --> 00:34:36,760
with the guidance of
the video manual.
477
00:34:36,760 --> 00:34:41,639
OK, well let's look at
one last system.
478
00:34:41,639 --> 00:34:48,000
I've indicated here a system
for which the output is
479
00:34:48,000 --> 00:34:50,620
related to the input
through this curve.
480
00:34:50,620 --> 00:34:56,800
And what I mean by this curve,
which wasn't done quite as
481
00:34:56,800 --> 00:35:04,210
well as I might have, is that
the output is equal to the
482
00:35:04,210 --> 00:35:06,530
square of the input.
483
00:35:06,530 --> 00:35:13,070
So this system is our squarer
as we talked about before.
484
00:35:13,070 --> 00:35:17,510
We saw previously, or discussed
the fact previously,
485
00:35:17,510 --> 00:35:22,315
that the squarer is a
memoryless system.
486
00:35:25,500 --> 00:35:30,440
And now the question is, is a
squarer an invertible system?
487
00:35:30,440 --> 00:35:35,020
Well, the question then is, if
you're given the square of a
488
00:35:35,020 --> 00:35:37,990
signal, can you figure out
what the signal is?
489
00:35:37,990 --> 00:35:42,800
And as I'm sure you've already
suspected, the answer to that
490
00:35:42,800 --> 00:35:47,030
is no, because obviously if
the signal was negative or
491
00:35:47,030 --> 00:35:49,540
positive, you wouldn't be able
to figure that out after
492
00:35:49,540 --> 00:35:50,820
you've squared.
493
00:35:50,820 --> 00:35:55,085
So in fact, the squarer
is not invertible.
494
00:35:57,670 --> 00:36:00,650
All right, so we've introduced
several properties.
495
00:36:00,650 --> 00:36:04,260
And by the way, as we've gone
through it, also introduced
496
00:36:04,260 --> 00:36:07,190
some systems that will
turn out to be useful
497
00:36:07,190 --> 00:36:09,100
and important systems.
498
00:36:09,100 --> 00:36:13,930
And now, let's continue with
some other properties.
499
00:36:13,930 --> 00:36:18,720
A property that we'll find
useful to make reference to,
500
00:36:18,720 --> 00:36:21,610
from time to time, and will,
in fact, play a fairly
501
00:36:21,610 --> 00:36:26,130
important role in a variety of
discussions during the course,
502
00:36:26,130 --> 00:36:30,830
is a property which is referred
to as causality.
503
00:36:30,830 --> 00:36:36,410
Now, in essence what causality
means is the following.
504
00:36:36,410 --> 00:36:41,930
A system is set to be causal if,
as one way of saying it,
505
00:36:41,930 --> 00:36:45,340
it only responds when
you kick it.
506
00:36:45,340 --> 00:36:49,450
Is another way of saying it, its
response at any time only
507
00:36:49,450 --> 00:36:54,020
depends on values of the input
prior to that time.
508
00:36:54,020 --> 00:36:58,110
So a causal system, both
continuous time and discrete
509
00:36:58,110 --> 00:37:04,320
time, is defined sometimes
as a system which has the
510
00:37:04,320 --> 00:37:10,470
property that the output at any
time depends only on the
511
00:37:10,470 --> 00:37:15,520
input prior or equal
to that time.
512
00:37:15,520 --> 00:37:19,220
Essentially what we're saying
is that the system can't
513
00:37:19,220 --> 00:37:22,550
anticipate future inputs.
514
00:37:22,550 --> 00:37:27,450
And so that, in fact, is another
possible definition
515
00:37:27,450 --> 00:37:32,930
for causality, that the system
is causal if it can't
516
00:37:32,930 --> 00:37:36,490
anticipate future inputs.
517
00:37:36,490 --> 00:37:39,110
Finally, an alternative
way of saying it
518
00:37:39,110 --> 00:37:42,630
mathematically is as follows.
519
00:37:42,630 --> 00:37:50,920
If I have two signals, x_1(t)
and x_2(t), with their
520
00:37:50,920 --> 00:37:58,070
associated outputs, y_1(t) and
y_2(t), then a system is said
521
00:37:58,070 --> 00:38:03,870
to be causal if and only if it
has the property that if those
522
00:38:03,870 --> 00:38:08,520
two inputs are identical up
until some time, then the
523
00:38:08,520 --> 00:38:12,300
outputs are identical up
until the same time.
524
00:38:12,300 --> 00:38:18,060
So if we have two signals that
are exactly the same up until
525
00:38:18,060 --> 00:38:22,050
some time and perhaps do
something different later on,
526
00:38:22,050 --> 00:38:26,980
causality requires that the
outputs not anticipate the
527
00:38:26,980 --> 00:38:31,150
fact that those inputs are at
some future time going to do
528
00:38:31,150 --> 00:38:32,330
something different.
529
00:38:32,330 --> 00:38:36,140
And that, in fact, is the most
useful mathematical definition
530
00:38:36,140 --> 00:38:37,620
of causality.
531
00:38:37,620 --> 00:38:43,830
And of course, I've written that
here for continuous time.
532
00:38:43,830 --> 00:38:52,050
And the same definition of
causality also applies for
533
00:38:52,050 --> 00:38:53,300
discrete time.
534
00:38:55,520 --> 00:39:00,560
OK, well let's look
at an example.
535
00:39:00,560 --> 00:39:06,290
Let's take an example which
is a system which is the
536
00:39:06,290 --> 00:39:09,380
following discrete-time
system.
537
00:39:09,380 --> 00:39:17,980
The output at any given time
is the sum of x[n], x[n]
538
00:39:17,980 --> 00:39:20,270
delayed, and x[n]
539
00:39:20,270 --> 00:39:22,800
anticipated.
540
00:39:22,800 --> 00:39:27,570
And this is, in fact, a system
that's very useful and
541
00:39:27,570 --> 00:39:30,735
referred to as a
moving average.
542
00:39:34,210 --> 00:39:41,720
And so if we think of a moving
average, if we have here a
543
00:39:41,720 --> 00:39:44,510
sequence, x[n]
544
00:39:44,510 --> 00:39:45,880
and x[n]
545
00:39:45,880 --> 00:39:53,770
with other values going off in
both directions, for any
546
00:39:53,770 --> 00:39:57,680
value, n_0, at which we're
computing the output--
547
00:39:57,680 --> 00:40:01,050
and this is y[n], the output--
548
00:40:01,050 --> 00:40:10,040
we take x[n 0], x[n 0-1],
and x[n 0+1].
549
00:40:10,040 --> 00:40:14,590
And so to form that moving
average, we would take these
550
00:40:14,590 --> 00:40:19,580
three values and combine them
together, adding them and then
551
00:40:19,580 --> 00:40:23,540
dividing by 3 to get that.
552
00:40:23,540 --> 00:40:27,100
Well, is the system causal?
553
00:40:27,100 --> 00:40:31,360
One way to answer that is to
determine whether the output
554
00:40:31,360 --> 00:40:34,890
at any given time depends on
future values of the input.
555
00:40:34,890 --> 00:40:38,370
And clearly, if you look at
this, what you see is that the
556
00:40:38,370 --> 00:40:43,940
output at time n_0 depends both
on past values and on
557
00:40:43,940 --> 00:40:46,290
future values.
558
00:40:46,290 --> 00:40:50,890
As opposed to another moving
average, which I've indicated
559
00:40:50,890 --> 00:40:55,100
here, where I simply
shifted the values
560
00:40:55,100 --> 00:40:57,210
that I combined together.
561
00:40:57,210 --> 00:41:01,490
And in this case, because of the
way in which I picked the
562
00:41:01,490 --> 00:41:09,270
values, the output depends on
the value at n_0, the value at
563
00:41:09,270 --> 00:41:14,380
n_0 - 1, and the value
at n_0 - 2.
564
00:41:14,380 --> 00:41:16,160
So y[n 0]
565
00:41:16,160 --> 00:41:23,430
would depend here on x[n 0],
x[n 0-1], and x[n 0-2].
566
00:41:23,430 --> 00:41:31,210
And so in this case, the
system is not causal.
567
00:41:31,210 --> 00:41:36,660
And in this case, the
system is causal.
568
00:41:39,250 --> 00:41:42,900
All right, now let's turn to
another system property, the
569
00:41:42,900 --> 00:41:45,975
property referred
to as stability.
570
00:41:48,650 --> 00:41:53,730
Now, there are lots of
definitions of stability, and
571
00:41:53,730 --> 00:41:57,720
some of them get very
mathematical and formal.
572
00:41:57,720 --> 00:42:01,180
But we've chosen, and what we'll
use as our definition of
573
00:42:01,180 --> 00:42:04,100
stability, is what's called
bounded-input
574
00:42:04,100 --> 00:42:07,000
bounded-output stability.
575
00:42:07,000 --> 00:42:11,610
And essentially, the definition
is that a system is
576
00:42:11,610 --> 00:42:16,880
stable if and only if for
every bounded input, the
577
00:42:16,880 --> 00:42:18,650
output is bounded.
578
00:42:18,650 --> 00:42:23,180
So the notion is if you have a
system and the input never
579
00:42:23,180 --> 00:42:29,520
gets above some finite value,
then stability requires that
580
00:42:29,520 --> 00:42:33,600
the output also stay within
some bounded values.
581
00:42:33,600 --> 00:42:36,150
And I'm sure that stability
and instability are things
582
00:42:36,150 --> 00:42:40,030
that you're kind of informally
familiar with.
583
00:42:40,030 --> 00:42:45,840
Let me just emphasize the point
with something that I
584
00:42:45,840 --> 00:42:50,410
borrowed actually from
my son with some
585
00:42:50,410 --> 00:42:53,360
reluctance on his part.
586
00:42:53,360 --> 00:42:57,170
If we, for example, take a
system like this, which is in
587
00:42:57,170 --> 00:43:01,140
essence a pendulum, this system
as I'm holding it here
588
00:43:01,140 --> 00:43:05,040
is stable because if I put in
a bounded input, which is a
589
00:43:05,040 --> 00:43:09,170
displacement, the output, which
is the movement of it,
590
00:43:09,170 --> 00:43:11,680
remains bounded.
591
00:43:11,680 --> 00:43:15,510
Now on the other hand, if I
put the system like this,
592
00:43:15,510 --> 00:43:18,200
which is, in fact, what's
referred to as an inverted
593
00:43:18,200 --> 00:43:22,990
pendulum, although we could
conceivably get this to
594
00:43:22,990 --> 00:43:27,550
balance, just a slight
displacement because of the
595
00:43:27,550 --> 00:43:30,050
fact that the pendulum
is inverted, a slight
596
00:43:30,050 --> 00:43:35,130
displacement and the output
becomes unbounded.
597
00:43:35,130 --> 00:43:37,630
Now, an interesting thing with
the inverted pendulum, by the
598
00:43:37,630 --> 00:43:42,560
way, which I'm sure all of you,
if you were anything like
599
00:43:42,560 --> 00:43:46,150
me, were intrigued with as a
kid, was the notion that you
600
00:43:46,150 --> 00:43:50,920
could take an inverted pendulum
and in effect turn it
601
00:43:50,920 --> 00:43:55,360
back into a stable system by
using what I'm doing right
602
00:43:55,360 --> 00:43:58,730
now, which is feedback.
603
00:43:58,730 --> 00:44:03,200
What I've done in that case is
I've stabilized the system by
604
00:44:03,200 --> 00:44:06,390
using feedback, visual
feedback, from
605
00:44:06,390 --> 00:44:08,830
my eye to my hand.
606
00:44:08,830 --> 00:44:11,070
And in fact, one of the very
important things that we'll
607
00:44:11,070 --> 00:44:13,850
see about feedback when we talk
about feedback systems
608
00:44:13,850 --> 00:44:16,670
much later in the course is
that one of their very
609
00:44:16,670 --> 00:44:22,330
important applications is in
stabilizing unstable systems.
610
00:44:22,330 --> 00:44:25,960
By the way, one of their
problems is that if not used
611
00:44:25,960 --> 00:44:30,370
correctly, it can destabilize
stable systems.
612
00:44:30,370 --> 00:44:32,720
OK, well let's continue on with
613
00:44:32,720 --> 00:44:36,160
our property of stability.
614
00:44:36,160 --> 00:44:40,950
I have here, again, the example
of an integrator.
615
00:44:40,950 --> 00:44:46,380
And as I indicate here, if we
have an integrator and we put
616
00:44:46,380 --> 00:44:50,900
a step function into it or a
step signal into it, the
617
00:44:50,900 --> 00:44:55,330
output is what's referred
to as a ramp signal.
618
00:44:55,330 --> 00:44:58,240
It linearly increases.
619
00:44:58,240 --> 00:45:01,860
Now, the question is,
is a ramp unbounded?
620
00:45:01,860 --> 00:45:03,680
The input is bounded.
621
00:45:03,680 --> 00:45:05,580
The step input is bounded.
622
00:45:05,580 --> 00:45:10,590
The ramp is unbounded because
if you try to establish any
623
00:45:10,590 --> 00:45:15,820
bound on it, you can always go
out far enough in time so that
624
00:45:15,820 --> 00:45:18,430
the output will exceed
that bound.
625
00:45:18,430 --> 00:45:24,320
So in fact, the integrator
is not stable.
626
00:45:29,680 --> 00:45:34,960
OK, now finally, I'd like to
turn to two properties that
627
00:45:34,960 --> 00:45:39,180
we'll make considerable use of
as the course goes on, the
628
00:45:39,180 --> 00:45:42,630
properties of time invariance
and linearity.
629
00:45:45,210 --> 00:45:52,630
Time invariance, in essence,
says that the system doesn't
630
00:45:52,630 --> 00:45:56,360
really care what you
call the origin.
631
00:45:56,360 --> 00:46:00,430
In other words, it says if you
take the input and you shift
632
00:46:00,430 --> 00:46:04,370
it in time, all that you've done
is taken the output and
633
00:46:04,370 --> 00:46:07,630
shifted it in time by
the same amount.
634
00:46:07,630 --> 00:46:12,680
Somewhat more formally as I've
indicated here, if in
635
00:46:12,680 --> 00:46:16,270
continuous time we have an
input, x(t), which generate an
636
00:46:16,270 --> 00:46:22,990
output, y(t), then time
invariance requires that if
637
00:46:22,990 --> 00:46:26,820
the input is shifted by any
amount of time, the output is
638
00:46:26,820 --> 00:46:29,610
shifted by the same
amount of time.
639
00:46:29,610 --> 00:46:33,560
And exactly the same applies
in discrete time.
640
00:46:36,370 --> 00:46:41,340
For example, if we have a system
which is a system I've
641
00:46:41,340 --> 00:46:45,170
shown here, which by the way is
the system that we talked
642
00:46:45,170 --> 00:46:48,930
about previously to go from
a step sequence--
643
00:46:48,930 --> 00:46:51,940
I'm sorry, from an impulse
sequence to a step sequence.
644
00:46:51,940 --> 00:46:53,880
We called it a running sum.
645
00:46:53,880 --> 00:46:57,890
And actually, what it's also
often called is an
646
00:46:57,890 --> 00:46:59,450
accumulator.
647
00:46:59,450 --> 00:47:04,580
What it does is accumulate
past values of the input.
648
00:47:04,580 --> 00:47:10,590
Well, is the accumulator
time-invariant?
649
00:47:10,590 --> 00:47:13,080
The best way to establish that
is to work through the
650
00:47:13,080 --> 00:47:17,880
equations and verify that it
either does or doesn't satisfy
651
00:47:17,880 --> 00:47:20,920
the formal definition
of time invariance.
652
00:47:20,920 --> 00:47:26,130
Informally, if you think about
it, it makes intuitive sense
653
00:47:26,130 --> 00:47:30,640
that the accumulator is
time-invariant because if
654
00:47:30,640 --> 00:47:35,620
you're accumulating values and
if you delay the values that
655
00:47:35,620 --> 00:47:40,010
you're putting into the
accumulator, then the
656
00:47:40,010 --> 00:47:42,880
associated values that come
out will be delayed by the
657
00:47:42,880 --> 00:47:43,550
same amount.
658
00:47:43,550 --> 00:47:47,930
The accumulator doesn't care
really if you shift the input.
659
00:47:47,930 --> 00:47:51,730
It'll just simply shift
the associated output.
660
00:47:51,730 --> 00:47:55,630
But more generally, if you're
trying to test time
661
00:47:55,630 --> 00:47:58,940
invariance, it's important to
return to the definition.
662
00:47:58,940 --> 00:48:01,690
And that's what you're required
to do in the examples
663
00:48:01,690 --> 00:48:06,140
in the video manual.
664
00:48:06,140 --> 00:48:10,700
OK, well, I indicate
another example.
665
00:48:10,700 --> 00:48:14,130
We had the example of
an accumulator.
666
00:48:14,130 --> 00:48:20,430
Here's another example which, in
fact, as we'll see later is
667
00:48:20,430 --> 00:48:23,500
a system which is a modulator.
668
00:48:23,500 --> 00:48:26,820
The output is the input,
modulated.
669
00:48:26,820 --> 00:48:32,840
And although you might think at
first that this system is
670
00:48:32,840 --> 00:48:40,330
time invariant, in fact it is
not, because the input shifted
671
00:48:40,330 --> 00:48:46,240
generates an output which is
the input shifted times the
672
00:48:46,240 --> 00:48:49,180
same modulation function.
673
00:48:49,180 --> 00:48:55,170
Whereas if we were to take the
output of the system, we have
674
00:48:55,170 --> 00:49:03,220
x(t) is the input, then what
that would correspond to is
675
00:49:03,220 --> 00:49:10,220
sin(t-t_0) * x(t-t_0).
676
00:49:10,220 --> 00:49:13,770
And since these two are
not equal, this
677
00:49:13,770 --> 00:49:18,190
system is not time invariant.
678
00:49:18,190 --> 00:49:22,000
And this is an example that
often causes a slight amount
679
00:49:22,000 --> 00:49:27,030
of difficulty because it seems
like when you look
680
00:49:27,030 --> 00:49:29,180
at it ought to be.
681
00:49:29,180 --> 00:49:33,480
And so I strongly encourage you,
in the context of working
682
00:49:33,480 --> 00:49:37,860
problems in the manual, that you
think very carefully about
683
00:49:37,860 --> 00:49:42,030
this and at least believe
that what I told you
684
00:49:42,030 --> 00:49:44,690
is the right answer.
685
00:49:44,690 --> 00:49:48,500
OK, now the final property that
I want to introduce today
686
00:49:48,500 --> 00:49:52,580
is the property of linearity.
687
00:49:52,580 --> 00:49:58,290
And linearity is defined in a
manner similar for continuous
688
00:49:58,290 --> 00:50:00,790
time and discrete time.
689
00:50:00,790 --> 00:50:07,090
And what it says is that if
we have some inputs with
690
00:50:07,090 --> 00:50:13,300
associated outputs, let's say
x_1(t) and x_2(t), then a
691
00:50:13,300 --> 00:50:19,930
system is linear if it has the
property that the output to a
692
00:50:19,930 --> 00:50:24,000
linear combination of those
inputs is the same linear
693
00:50:24,000 --> 00:50:27,250
combination of the associated
outputs.
694
00:50:27,250 --> 00:50:32,650
And so that's what I've
indicated here, that if we now
695
00:50:32,650 --> 00:50:36,290
put into the system a linear
combination of those inputs,
696
00:50:36,290 --> 00:50:40,610
then for linearity, we require
that the output is the same
697
00:50:40,610 --> 00:50:42,320
linear combination.
698
00:50:42,320 --> 00:50:47,110
And exactly the same applies
in discrete time.
699
00:50:47,110 --> 00:50:50,960
And you can show from this
definition that if a system is
700
00:50:50,960 --> 00:50:54,930
linear with two inputs, then
it's linear in terms of an
701
00:50:54,930 --> 00:50:58,330
arbitrary number of inputs.
702
00:50:58,330 --> 00:51:01,280
I have a number of examples.
703
00:51:01,280 --> 00:51:07,410
And these are examples that,
again, I ask you to think
704
00:51:07,410 --> 00:51:11,250
about as you look at
the video manual.
705
00:51:11,250 --> 00:51:12,940
Just to suggest the answer--
706
00:51:12,940 --> 00:51:16,420
well, not to suggest but to
tell you the answer, the
707
00:51:16,420 --> 00:51:19,400
integrator as we have
here is linear.
708
00:51:23,550 --> 00:51:26,890
This system in which the output
is double the input
709
00:51:26,890 --> 00:51:30,570
plus a constant, you would
kind of think it's linear
710
00:51:30,570 --> 00:51:33,030
because it's a straight line.
711
00:51:33,030 --> 00:51:35,780
But one has to be careful.
712
00:51:35,780 --> 00:51:40,630
And in fact, as it turns out,
this is not linear.
713
00:51:40,630 --> 00:51:43,950
There is a qualifier attached
to it because it has a
714
00:51:43,950 --> 00:51:47,470
property referred to as
incrementally linear, which is
715
00:51:47,470 --> 00:51:51,590
discussed somewhat
more in the text.
716
00:51:51,590 --> 00:51:57,340
And finally, we have a system
which is the squarer that I've
717
00:51:57,340 --> 00:51:59,980
indicated again here.
718
00:51:59,980 --> 00:52:03,830
And squaring it is definitely
not a linear operation.
719
00:52:07,080 --> 00:52:12,200
OK, so what we've done, then,
is to introduce a number of
720
00:52:12,200 --> 00:52:15,570
properties of systems.
721
00:52:15,570 --> 00:52:19,370
And we've also, by the way, as
I've stressed previously, as
722
00:52:19,370 --> 00:52:23,090
we've gone along introduced also
a number of important and
723
00:52:23,090 --> 00:52:25,730
useful systems, like the
accumulator, the integrator,
724
00:52:25,730 --> 00:52:29,550
the differentiator, et cetera.
725
00:52:29,550 --> 00:52:33,110
What we'll see is that the
properties of linearity and
726
00:52:33,110 --> 00:52:38,640
time invariance in particular
become central and important
727
00:52:38,640 --> 00:52:40,260
properties.
728
00:52:40,260 --> 00:52:45,930
And in the next lecture, what
we'll show is that with
729
00:52:45,930 --> 00:52:51,890
systems that are linear and
time-invariant, the use of the
730
00:52:51,890 --> 00:52:55,310
impulse function, both in
continuous time and discrete
731
00:52:55,310 --> 00:53:00,150
time, provides an
extraordinarily important and
732
00:53:00,150 --> 00:53:04,300
useful mechanism for
characterizing those systems.
733
00:53:04,300 --> 00:53:05,550
Thank you.