1 00:00:00,040 --> 00:00:02,470 The following content is provided under a Creative 2 00:00:02,470 --> 00:00:03,880 Commons license. 3 00:00:03,880 --> 00:00:06,920 Your support will help MIT OpenCourseWare continue to 4 00:00:06,920 --> 00:00:10,570 offer high quality educational resources for free. 5 00:00:10,570 --> 00:00:13,470 To make a donation or view additional materials from 6 00:00:13,470 --> 00:00:19,290 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:19,290 --> 00:00:20,540 ocw.mit.edu. 8 00:00:59,745 --> 00:01:02,330 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.