1 00:00:01,040 --> 00:00:03,460 The following content is provided under a Creative 2 00:00:03,460 --> 00:00:04,870 Commons license. 3 00:00:04,870 --> 00:00:07,910 Your support will help MIT OpenCourseWare continue to 4 00:00:07,910 --> 00:00:11,560 offer high quality educational resources for free. 5 00:00:11,560 --> 00:00:14,460 To make a donation or view additional materials from 6 00:00:14,460 --> 00:00:18,390 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:18,390 --> 00:00:19,640 ocw.mit.edu. 8 00:00:24,320 --> 00:00:29,520 PROFESSOR: Last time, which is now two weeks ago, we started 9 00:00:29,520 --> 00:00:33,050 the talk about the signals and systems approach, by which I 10 00:00:33,050 --> 00:00:36,890 mean, think about a system by the way it transforms its 11 00:00:36,890 --> 00:00:41,090 input signal into an output signal. 12 00:00:41,090 --> 00:00:43,520 That's kind of a bizarre way of thinking about systems. 13 00:00:43,520 --> 00:00:45,690 I demonstrated that last time by thinking about a mass and a 14 00:00:45,690 --> 00:00:49,760 spring, something you have a lot of experience with, but 15 00:00:49,760 --> 00:00:53,680 you probably didn't use this kind of approach for 16 00:00:53,680 --> 00:00:55,060 thinking about it. 17 00:00:55,060 --> 00:00:57,020 Over this lecture and over the next lecture, what I'd like to 18 00:00:57,020 --> 00:01:00,455 do is show you the advantages of this kind of approach. 19 00:01:00,455 --> 00:01:04,920 And so for today, what I'd like to do is talk about how 20 00:01:04,920 --> 00:01:09,900 to think about feedback within this structure, and I'd like 21 00:01:09,900 --> 00:01:13,260 to also think about how you can use this structure to 22 00:01:13,260 --> 00:01:15,760 characterize the performance of a system in a 23 00:01:15,760 --> 00:01:18,900 quantitative fashion. 24 00:01:18,900 --> 00:01:21,400 So first off, I want to just think about feedback. 25 00:01:21,400 --> 00:01:25,370 Feedback is so pervasive that you don't notice 26 00:01:25,370 --> 00:01:27,470 it most of the time. 27 00:01:27,470 --> 00:01:31,110 You use feedback in virtually everything that you do. 28 00:01:31,110 --> 00:01:33,680 Here is a very simple example of driving a car. 29 00:01:33,680 --> 00:01:36,050 If you want to keep the car in the center of your lane, 30 00:01:36,050 --> 00:01:38,630 something that many people outside of Boston, at least, 31 00:01:38,630 --> 00:01:43,880 think is a good idea, then you are mentally doing feedback. 32 00:01:43,880 --> 00:01:46,370 You're constantly comparing where you are to where you'd 33 00:01:46,370 --> 00:01:49,950 like to be and making small adjustments to the system 34 00:01:49,950 --> 00:01:52,760 based on that. 35 00:01:52,760 --> 00:01:54,900 When you think of even the most simple of systems, like 36 00:01:54,900 --> 00:01:55,920 the thermostat in a house-- 37 00:01:55,920 --> 00:01:58,110 I'm not talking about a cheap motel. 38 00:01:58,110 --> 00:01:59,550 The cheap motels don't do this. 39 00:01:59,550 --> 00:02:03,540 But in a real house, there's a thermostat, which regulates 40 00:02:03,540 --> 00:02:06,060 the temperature. 41 00:02:06,060 --> 00:02:08,160 That's important, because if the temperature suddenly 42 00:02:08,160 --> 00:02:11,550 drops, it would never do that of course, but if the 43 00:02:11,550 --> 00:02:16,790 temperature ever dropped, it could compensate for it. 44 00:02:16,790 --> 00:02:20,070 Here's one of my favorite examples. 45 00:02:20,070 --> 00:02:23,280 Feedback is enormously pervasive in biology. 46 00:02:23,280 --> 00:02:27,770 There's no general rules in biology, but to a first cut, 47 00:02:27,770 --> 00:02:31,250 everything is regulated. 48 00:02:31,250 --> 00:02:35,550 And in many cases, the regulation is amazing. 49 00:02:35,550 --> 00:02:39,930 Here's an example from 6.021, where it's illustrating the 50 00:02:39,930 --> 00:02:44,540 system that your body uses to regulate glucose delivery from 51 00:02:44,540 --> 00:02:48,330 food sources to every cell in your body, which is crucial to 52 00:02:48,330 --> 00:02:50,950 your being cognizant and mobile. 53 00:02:53,600 --> 00:03:00,580 And the idea is that it does that with amazing precision 54 00:03:00,580 --> 00:03:04,690 despite the fact that eating and exercise 55 00:03:04,690 --> 00:03:07,880 are enormously episodic. 56 00:03:07,880 --> 00:03:10,970 In order for you to remain healthy and functional, you 57 00:03:10,970 --> 00:03:14,220 need to have something between approximately two and ten 58 00:03:14,220 --> 00:03:16,510 millimoles per liter of glucose in your 59 00:03:16,510 --> 00:03:18,260 blood at all times. 60 00:03:18,260 --> 00:03:20,380 Were it to go higher than that, systematically, you 61 00:03:20,380 --> 00:03:25,050 would develop cardiac problems and could lead to even 62 00:03:25,050 --> 00:03:28,170 congestive heart failure. 63 00:03:28,170 --> 00:03:32,310 If you were to have lower than two millimoles per liter, you 64 00:03:32,310 --> 00:03:34,420 would go comatose. 65 00:03:34,420 --> 00:03:37,470 That's a very narrow range, two to five, especially 66 00:03:37,470 --> 00:03:43,260 because what we eat is so episodic, when we exercise is 67 00:03:43,260 --> 00:03:44,430 so episodic. 68 00:03:44,430 --> 00:03:49,440 It's amazing, and just to dramatize how amazing it is, 69 00:03:49,440 --> 00:03:52,140 how much sugar do you think circulates in 70 00:03:52,140 --> 00:03:53,390 your blood right now? 71 00:03:56,970 --> 00:03:59,430 Well, if you convert five millimoles per liter, and if 72 00:03:59,430 --> 00:04:01,540 you assume the average person has about three liters of 73 00:04:01,540 --> 00:04:05,290 blood, which is true, and if you calculate it, that comes 74 00:04:05,290 --> 00:04:08,920 out to 2.7 grams. 75 00:04:08,920 --> 00:04:12,150 That's this much. 76 00:04:12,150 --> 00:04:14,690 This is 2.7 grams of table sugar. 77 00:04:19,800 --> 00:04:21,740 So this is how much sugar is in your blood, now. 78 00:04:21,740 --> 00:04:22,800 This is what's keeping you healthy. 79 00:04:22,800 --> 00:04:26,730 This is what's keeping you from becoming comatose. 80 00:04:26,730 --> 00:04:27,980 Or not. 81 00:04:30,310 --> 00:04:34,288 How much sugar do you think is in this? 82 00:04:34,288 --> 00:04:35,650 AUDIENCE: The amount in that other cup. 83 00:04:35,650 --> 00:04:38,110 PROFESSOR: Exactly. 84 00:04:38,110 --> 00:04:40,690 We call that the Theory of Lectures. 85 00:04:40,690 --> 00:04:43,800 I don't want to look like an idiot, so my 86 00:04:43,800 --> 00:04:45,220 problems make sense. 87 00:04:45,220 --> 00:04:47,550 This is the amount of sugar in a can of soda. 88 00:04:50,160 --> 00:04:51,970 Numerically, this is 39 grams. 89 00:04:57,140 --> 00:05:00,350 That's more than 13 times this much. 90 00:05:00,350 --> 00:05:03,080 So when you down one of these, it's very important that the 91 00:05:03,080 --> 00:05:07,400 sugar gets taken out of the blood quickly, and it does. 92 00:05:07,400 --> 00:05:10,710 And that happens by way of a feedback system, and the 93 00:05:10,710 --> 00:05:12,850 feedback system is illustrated here. 94 00:05:12,850 --> 00:05:16,510 Basically, the feedback involves the hormone insulin, 95 00:05:16,510 --> 00:05:19,370 and that's why insulin deficiency is such a 96 00:05:19,370 --> 00:05:20,620 devastating disease. 97 00:05:22,890 --> 00:05:26,690 Finally, everything we do has feedback in it. 98 00:05:26,690 --> 00:05:28,400 Think about how your life would be 99 00:05:28,400 --> 00:05:30,520 different if it didn't. 100 00:05:30,520 --> 00:05:33,150 Even a simple task, like removing a light bulb, would 101 00:05:33,150 --> 00:05:36,300 be virtually impossible except for the fact that you get 102 00:05:36,300 --> 00:05:37,610 feedback from everything. 103 00:05:37,610 --> 00:05:39,170 Your hand's amazing. 104 00:05:39,170 --> 00:05:42,240 You have touch sensors, you have proprioceptive sensors, 105 00:05:42,240 --> 00:05:45,160 you have stress sensors on the muscles and ligaments, and 106 00:05:45,160 --> 00:05:48,520 they all coordinate to tell you when to stop squeezing on 107 00:05:48,520 --> 00:05:51,310 the light bulb so you don't break it. 108 00:05:51,310 --> 00:05:53,800 That's all completely amazing, and what we'd like to do, 109 00:05:53,800 --> 00:05:58,070 then, is think about that kind of a system, a feedback 110 00:05:58,070 --> 00:06:03,450 system, within the signals and systems construct. 111 00:06:03,450 --> 00:06:06,330 As an example, I want to think through the WallFinder problem 112 00:06:06,330 --> 00:06:09,820 that you did last week in Design Lab. 113 00:06:09,820 --> 00:06:12,670 I'm sure you all remember that, in that problem, we were 114 00:06:12,670 --> 00:06:15,300 trying to move the robot to a fixed distance away from the 115 00:06:15,300 --> 00:06:19,220 wall, and we thought about that as a feedback system 116 00:06:19,220 --> 00:06:21,820 comprised of three parts, a controller, a 117 00:06:21,820 --> 00:06:24,180 plant, and a sensor. 118 00:06:24,180 --> 00:06:26,800 We wrote difference equations to characterize each of those 119 00:06:26,800 --> 00:06:32,130 parts, and then we figured out how to solve those difference 120 00:06:32,130 --> 00:06:35,100 equations to make some meaningful prediction about 121 00:06:35,100 --> 00:06:36,350 how the robot would work. 122 00:06:38,690 --> 00:06:41,905 So just to make sure you're all on board with me, now, 123 00:06:41,905 --> 00:06:44,210 here's a question for you. 124 00:06:44,210 --> 00:06:45,990 These are the equations that describe 125 00:06:45,990 --> 00:06:48,760 the WallFinder problem. 126 00:06:48,760 --> 00:06:50,560 How many equations and how many unknowns are there? 127 00:06:54,860 --> 00:06:56,870 Take 20 seconds, talk to your neighbor, figure out an answer 128 00:06:56,870 --> 00:06:58,120 between (1) and (5). 129 00:07:09,920 --> 00:07:12,490 Unlike during the exam next week, you are allowed to talk. 130 00:07:19,915 --> 00:07:39,220 [CLASSROOM SIDE CONVERSATIONS] 131 00:07:39,220 --> 00:07:40,705 PROFESSOR: T and K are no. 132 00:07:40,705 --> 00:08:09,010 [CLASSROOM SIDE CONVERSATIONS] 133 00:08:09,010 --> 00:08:10,614 PROFESSOR: OK, so what's the answer, number (1), (2), (3), 134 00:08:10,614 --> 00:08:12,060 (4), or (5)? 135 00:08:12,060 --> 00:08:13,160 Everybody raise your hands. 136 00:08:13,160 --> 00:08:15,010 Let me see if I got the answer. 137 00:08:15,010 --> 00:08:15,920 OK. 138 00:08:15,920 --> 00:08:16,890 Come on, raise your hands. 139 00:08:16,890 --> 00:08:18,050 Come on, everybody vote. 140 00:08:18,050 --> 00:08:20,010 If you're wrong, just blame it on your neighbor. 141 00:08:20,010 --> 00:08:21,240 You had a poor partner. 142 00:08:21,240 --> 00:08:22,020 That's the idea. 143 00:08:22,020 --> 00:08:23,730 Right? 144 00:08:23,730 --> 00:08:25,530 So you can all vote, and you don't need to worry about 145 00:08:25,530 --> 00:08:26,540 being wrong. 146 00:08:26,540 --> 00:08:31,520 And you're all wrong, so that worked out well. 147 00:08:31,520 --> 00:08:34,960 OK, so I don't like the-- so the predominant answer is 148 00:08:34,960 --> 00:08:37,190 number (2). 149 00:08:37,190 --> 00:08:38,200 I don't like number (2). 150 00:08:38,200 --> 00:08:40,740 Can somebody think of a reason, now that you know the 151 00:08:40,740 --> 00:08:43,909 answer by the Theory of Lectures, the 152 00:08:43,909 --> 00:08:45,159 answer is not (2)? 153 00:08:49,060 --> 00:08:50,170 Why isn't the answer (2)? 154 00:08:50,170 --> 00:08:50,840 Yeah? 155 00:08:50,840 --> 00:08:53,110 AUDIENCE: Oh, I was saying (5). 156 00:08:53,110 --> 00:08:53,590 PROFESSOR: You were saying (5). 157 00:08:53,590 --> 00:08:55,682 Why did you say (5)? 158 00:08:55,682 --> 00:08:59,852 AUDIENCE: I don't remember for sure, but can you substitute n 159 00:08:59,852 --> 00:09:01,092 for different things? 160 00:09:01,092 --> 00:09:05,804 Like Df of n, you could substitute n for, and if you 161 00:09:05,804 --> 00:09:07,788 did that you would have two equations, which is 162 00:09:07,788 --> 00:09:08,780 [INAUDIBLE]. 163 00:09:08,780 --> 00:09:11,010 PROFESSOR: [UNINTELLIGIBLE] kind of thing, you count 164 00:09:11,010 --> 00:09:13,502 before or after doing substitution and 165 00:09:13,502 --> 00:09:15,240 simplification. 166 00:09:15,240 --> 00:09:18,940 And I mean to count before you do simplification. 167 00:09:18,940 --> 00:09:22,065 Any other issues? 168 00:09:22,065 --> 00:09:22,540 Yeah? 169 00:09:22,540 --> 00:09:25,865 AUDIENCE: Is D0 [UNINTELLIGIBLE] 170 00:09:28,730 --> 00:09:29,422 PROFESSOR: Say again? 171 00:09:29,422 --> 00:09:32,185 AUDIENCE: D0 [UNINTELLIGIBLE] 172 00:09:32,185 --> 00:09:36,000 PROFESSOR: Is D0_n different or the same from D0_n-1? 173 00:09:36,000 --> 00:09:38,310 That's the key question. 174 00:09:38,310 --> 00:09:41,330 So I want to think about this as a system of algebraic 175 00:09:41,330 --> 00:09:46,615 equations, and if I do that, then there's a lot of them. 176 00:09:50,545 --> 00:09:52,790 So it looks like there's three equations. 177 00:09:52,790 --> 00:09:55,620 The problem with that approach is that there's actually three 178 00:09:55,620 --> 00:09:59,670 equations for every value of n. 179 00:09:59,670 --> 00:10:02,000 If you think about a system of equations that you could solve 180 00:10:02,000 --> 00:10:05,940 with an algebra solver, you would have to treat all the 181 00:10:05,940 --> 00:10:07,050 n's separately. 182 00:10:07,050 --> 00:10:10,600 That's what we call the samples approach. 183 00:10:10,600 --> 00:10:13,660 So here's a way you could solve them. 184 00:10:13,660 --> 00:10:18,660 You could think about what if k and t are parameters, so 185 00:10:18,660 --> 00:10:21,730 they're known, they're given? 186 00:10:21,730 --> 00:10:25,500 What if my input signal is known, say it's a unit sample 187 00:10:25,500 --> 00:10:28,220 signal, for example? 188 00:10:28,220 --> 00:10:30,650 What would I need to solve this system? 189 00:10:30,650 --> 00:10:35,400 Well, I'd need to tell you the initial conditions. 190 00:10:35,400 --> 00:10:39,810 So in some sense, I want to consider those to be known. 191 00:10:39,810 --> 00:10:46,290 So my knowns kind of comprise t and k, the initial 192 00:10:46,290 --> 00:10:52,680 conditions for the output of the robot, and the sensor, all 193 00:10:52,680 --> 00:10:55,650 of the input signals because I'm telling you the input and 194 00:10:55,650 --> 00:10:58,230 asking you to calculate the output. 195 00:10:58,230 --> 00:11:01,700 My unknowns are all of the different velocities for all 196 00:11:01,700 --> 00:11:06,260 values of n bigger than or equal to 0 because I didn't 197 00:11:06,260 --> 00:11:08,360 tell you those. 198 00:11:08,360 --> 00:11:10,990 All of the values of robot's output at samples 199 00:11:10,990 --> 00:11:13,100 n bigger than 0. 200 00:11:13,100 --> 00:11:16,510 All the values of the sensor output for values 201 00:11:16,510 --> 00:11:18,750 n bigger than 0. 202 00:11:18,750 --> 00:11:23,030 So I get a lot of unknowns, infinitely many, and I get a 203 00:11:23,030 --> 00:11:26,380 lot of equations, also infinitely many. 204 00:11:26,380 --> 00:11:28,670 So the thing I want you think about is, if you're thinking 205 00:11:28,670 --> 00:11:31,050 about solving difference equations using algebra, 206 00:11:31,050 --> 00:11:32,565 that's a big system of equations. 207 00:11:35,320 --> 00:11:40,320 By contrast, what if you were to try to solve the system 208 00:11:40,320 --> 00:11:41,570 using operators? 209 00:11:43,990 --> 00:11:46,470 Now, how many equations and unknowns do you see? 210 00:11:50,990 --> 00:11:52,515 By the Theory of Lectures. 211 00:11:56,610 --> 00:11:59,430 OK, raise your hand. 212 00:11:59,430 --> 00:12:02,736 Or talk to your neighbor so you can blame your neighbor. 213 00:12:02,736 --> 00:12:05,196 Talk to your neighbor. 214 00:12:05,196 --> 00:12:06,180 Get a good alibi. 215 00:12:06,180 --> 00:12:25,880 [CLASSROOM SIDE CONVERSATIONS] 216 00:12:25,880 --> 00:12:28,830 PROFESSOR: So the idea here is that if you think about 217 00:12:28,830 --> 00:12:32,030 operators instead, so that you look at a whole signal at a 218 00:12:32,030 --> 00:12:38,620 time, then each equation only specifies one relationship 219 00:12:38,620 --> 00:12:45,070 among signals, and there's a small number of signals. 220 00:12:45,070 --> 00:12:48,970 So if I think about the knowns being k, t the parameters and 221 00:12:48,970 --> 00:12:54,040 the signal d(i), and if I think about the unknowns being 222 00:12:54,040 --> 00:12:57,520 the velocity, the output, and the sensor signal, then I get 223 00:12:57,520 --> 00:13:00,160 three equations and three unknowns. 224 00:13:00,160 --> 00:13:03,190 So one of the values of thinking about the operator 225 00:13:03,190 --> 00:13:05,480 approach is that it just simply reduces the amount of 226 00:13:05,480 --> 00:13:06,530 things you need to think about. 227 00:13:06,530 --> 00:13:08,020 It reduces complexity. 228 00:13:08,020 --> 00:13:10,030 That's what we're trying to do in this course. 229 00:13:10,030 --> 00:13:12,460 We're trying to think of methods that allow you to 230 00:13:12,460 --> 00:13:16,180 solve problems by reducing complexity. 231 00:13:16,180 --> 00:13:18,470 We would like, ultimately, to solve very complicated 232 00:13:18,470 --> 00:13:21,740 problems, and this operator approach is an approach that 233 00:13:21,740 --> 00:13:24,380 lets you do that. 234 00:13:24,380 --> 00:13:26,360 It does a lot more than that, too. 235 00:13:26,360 --> 00:13:30,900 It also generates new kinds of insights. 236 00:13:30,900 --> 00:13:36,840 So it lets you focus on the relations, but the relation is 237 00:13:36,840 --> 00:13:38,800 now not quite the same as we would have 238 00:13:38,800 --> 00:13:40,140 expected from algebra. 239 00:13:40,140 --> 00:13:43,290 Now, the relation between the input signal and the output 240 00:13:43,290 --> 00:13:45,390 signal is an operator. 241 00:13:45,390 --> 00:13:49,580 We're going to represent the operation that transforms the 242 00:13:49,580 --> 00:13:52,080 input to the output by this symbol, h. 243 00:13:52,080 --> 00:13:54,300 We'll call that the system functional. 244 00:13:54,300 --> 00:13:56,690 It's an operator. 245 00:13:56,690 --> 00:14:00,255 It's a thing that, when operated on x, gives you y. 246 00:14:06,160 --> 00:14:08,890 And this is one of the main purposes of today's lecture, 247 00:14:08,890 --> 00:14:12,860 it's also convenient to think about h as a ratio. 248 00:14:12,860 --> 00:14:16,250 We like to think of it that way because, as we'll see, 249 00:14:16,250 --> 00:14:20,310 there's a way of thinking about h as a ratio of 250 00:14:20,310 --> 00:14:24,790 polynomials in R. So two ways of thinking about it. 251 00:14:24,790 --> 00:14:27,190 We're trying to develop a signals and systems approach 252 00:14:27,190 --> 00:14:28,900 for thinking about feedback. 253 00:14:28,900 --> 00:14:32,530 We want to think about the input goes into a box. 254 00:14:32,530 --> 00:14:35,210 The box represents an operation. 255 00:14:35,210 --> 00:14:36,950 We will characterize that by a functional. 256 00:14:36,950 --> 00:14:39,610 We'll call the functional h. 257 00:14:39,610 --> 00:14:41,890 The functional, when applied to the input, produces the 258 00:14:41,890 --> 00:14:45,660 output, and what we'd like to do is infer what is the nature 259 00:14:45,660 --> 00:14:48,400 of that functional, and what are the properties of the 260 00:14:48,400 --> 00:14:49,920 system that functional represents. 261 00:14:52,970 --> 00:14:56,450 OK, so I see that you're all with me. 262 00:14:56,450 --> 00:14:59,530 Think about the WallFinder system. 263 00:14:59,530 --> 00:15:02,790 Think about the equations for the various components of that 264 00:15:02,790 --> 00:15:06,000 system when expressed an operator form. 265 00:15:06,000 --> 00:15:09,930 And figure out the system functional for that system. 266 00:15:09,930 --> 00:15:14,280 figure out the ratio of polynomials that can be in r 267 00:15:14,280 --> 00:15:17,420 that is represented by h. 268 00:15:17,420 --> 00:15:19,500 Take 30 seconds, talk to your neighbor, figure out whether 269 00:15:19,500 --> 00:15:20,750 the answer is (1), (2), (3), (4), or (5). 270 00:17:45,980 --> 00:17:47,230 So what's the answer -- (1), (2), (3), (4), or (5)? 271 00:17:51,670 --> 00:17:53,910 Come on, more voter participation. 272 00:17:53,910 --> 00:17:54,600 All right? 273 00:17:54,600 --> 00:17:55,850 Blame it on your partner. 274 00:17:58,630 --> 00:18:01,990 OK, virtually 100% correct. 275 00:18:01,990 --> 00:18:05,690 So the idea is algebra. 276 00:18:05,690 --> 00:18:09,180 You solve the operator equations exactly as though 277 00:18:09,180 --> 00:18:10,670 they were algebraic. 278 00:18:10,670 --> 00:18:13,640 Here, I've started with the second equation and just done 279 00:18:13,640 --> 00:18:15,970 substitutions until I got rid of everything other 280 00:18:15,970 --> 00:18:17,220 than d(0) and d(i). 281 00:18:20,280 --> 00:18:25,730 So I express v in terms of ke, then I express e in terms of 282 00:18:25,730 --> 00:18:28,640 d(i) minus rd(0). 283 00:18:28,640 --> 00:18:32,110 Then I'm left with one equation that relates d(0) and 284 00:18:32,110 --> 00:18:34,140 d(i), which I can solve for the ratio. 285 00:18:36,830 --> 00:18:41,830 And the answer comes out there, which was number (3). 286 00:18:41,830 --> 00:18:47,240 Point is that you can treat the operator just as though it 287 00:18:47,240 --> 00:18:52,103 were algebra, so that results in enormous implications. 288 00:18:54,760 --> 00:18:57,660 But what we want to understand is what's the relationship 289 00:18:57,660 --> 00:19:00,230 between that functional, that thing that we just calculated, 290 00:19:00,230 --> 00:19:01,040 and the behaviors. 291 00:19:01,040 --> 00:19:03,710 These are the kinds of behaviors that you observed 292 00:19:03,710 --> 00:19:05,850 with the WallFinder system. 293 00:19:05,850 --> 00:19:08,328 When you built the WallFinder system, depending on what you 294 00:19:08,328 --> 00:19:13,280 made k, you could get behaviors that were monotonic 295 00:19:13,280 --> 00:19:20,360 and slow, faster and oscillatory, or even faster 296 00:19:20,360 --> 00:19:22,250 and even more oscillatory. 297 00:19:22,250 --> 00:19:27,120 And what we'd like to know is, before we build it, how should 298 00:19:27,120 --> 00:19:29,640 we have constructed the system so that it 299 00:19:29,640 --> 00:19:31,770 has a desirable behavior? 300 00:19:31,770 --> 00:19:36,010 And, incidentally, are these the best you can do? 301 00:19:36,010 --> 00:19:38,790 Or is there some other set of parameters that's lurking 302 00:19:38,790 --> 00:19:42,700 behind some door that we just don't know about, and if we 303 00:19:42,700 --> 00:19:45,150 could discover it, it would work a lot better? 304 00:19:45,150 --> 00:19:48,300 So the question is, given the structure of our problem, 305 00:19:48,300 --> 00:19:51,860 what's the most general kind of answer that we can expect? 306 00:19:51,860 --> 00:19:55,670 And how do we choose the best behavior out of that set of 307 00:19:55,670 --> 00:19:58,280 possible behaviors? 308 00:19:58,280 --> 00:20:00,596 So that's what I want to think about for the rest of the hour 309 00:20:00,596 --> 00:20:05,020 and a half, and I want to begin by taking a step 310 00:20:05,020 --> 00:20:08,180 backwards and look at something simpler. 311 00:20:08,180 --> 00:20:10,440 The idea's going to be the same as the idea that we used 312 00:20:10,440 --> 00:20:11,400 when we studied Python. 313 00:20:11,400 --> 00:20:16,760 I want to find simple behaviors, think about that as 314 00:20:16,760 --> 00:20:22,070 a primitive, and combine primitives to get a more 315 00:20:22,070 --> 00:20:24,710 complicated behavior, so I want to use an approach that's 316 00:20:24,710 --> 00:20:26,240 very much PCAP. 317 00:20:26,240 --> 00:20:29,770 Find the most simple behavior, and then see if I can leverage 318 00:20:29,770 --> 00:20:32,310 that simple behavior to somehow understand more 319 00:20:32,310 --> 00:20:34,270 complicated things. 320 00:20:34,270 --> 00:20:36,940 So let's think about this very simple system that has a 321 00:20:36,940 --> 00:20:43,170 feedback loop that has a delay in it and a gain of P0. 322 00:20:43,170 --> 00:20:45,250 What I want to do is think about what would be the 323 00:20:45,250 --> 00:20:50,130 response of that very simple system if the input were a 324 00:20:50,130 --> 00:20:51,550 unit sample. 325 00:20:51,550 --> 00:20:55,355 So find y, given that the input x is delta. 326 00:20:58,060 --> 00:21:01,630 In order to do that, I have to start the system somehow. 327 00:21:01,630 --> 00:21:03,160 I will start it at rest. 328 00:21:03,160 --> 00:21:05,890 You've all seen already, I'm sure, that rest is the 329 00:21:05,890 --> 00:21:07,340 simplest assumption I can make. 330 00:21:07,340 --> 00:21:10,520 I'll say something at the end of the hour about how you deal 331 00:21:10,520 --> 00:21:13,950 with things that are not at rest. 332 00:21:13,950 --> 00:21:16,140 For the time being, we'll just assume rest 333 00:21:16,140 --> 00:21:17,600 because that's simple. 334 00:21:17,600 --> 00:21:19,950 Assume that the system is at rest, that means that the 335 00:21:19,950 --> 00:21:23,280 output of every delay box is 0. 336 00:21:23,280 --> 00:21:25,360 That's what rest means. 337 00:21:25,360 --> 00:21:27,820 If the output of this starts at 0, then the output of the 338 00:21:27,820 --> 00:21:31,970 scale by P0 is also 0. 339 00:21:31,970 --> 00:21:34,530 And if that's 0, and if the input is at 0 because I'm at 340 00:21:34,530 --> 00:21:40,070 time before 0, then the output is 0, indicated here. 341 00:21:40,070 --> 00:21:47,320 So now if I step, then the input becomes 1 because 342 00:21:47,320 --> 00:21:49,920 delta of 0 is 1. 343 00:21:49,920 --> 00:21:54,390 The output of the delay box is still 0, so the first answer 344 00:21:54,390 --> 00:21:57,010 is 1, the 1 just propagates straight through 345 00:21:57,010 --> 00:21:58,260 [UNINTELLIGIBLE] box. 346 00:22:00,490 --> 00:22:05,990 Then I step, and the 1 that was here goes through the r 347 00:22:05,990 --> 00:22:12,150 and becomes 1, the 1 goes through P0 and becomes P0, but 348 00:22:12,150 --> 00:22:15,210 at the same time, the 1 that was at the input goes to 0 349 00:22:15,210 --> 00:22:19,870 because the input is 1 only at time equals 0. 350 00:22:19,870 --> 00:22:24,030 So the result, then, is that after one step, the 351 00:22:24,030 --> 00:22:25,950 output has become P0. 352 00:22:25,950 --> 00:22:29,180 This propagated to 1, that became P0, add it to 0, and it 353 00:22:29,180 --> 00:22:33,250 became P0, so now the answer, which had been 1, is P0. 354 00:22:35,940 --> 00:22:38,250 On the next step, a very similar thing happens. 355 00:22:38,250 --> 00:22:42,190 The P0 that was here becomes the output of the delay, gets 356 00:22:42,190 --> 00:22:46,370 multiplied by P0 to give you P0 squared, gets added to 0 to 357 00:22:46,370 --> 00:22:49,140 give you P0 squared, et cetera. 358 00:22:55,560 --> 00:22:59,460 The thing that I want you to see is that the output was in 359 00:22:59,460 --> 00:23:00,710 some sense simple. 360 00:23:03,530 --> 00:23:08,930 The value simply increased as P0 to the n, geometrically. 361 00:23:08,930 --> 00:23:11,370 There's another way we can think about that. 362 00:23:11,370 --> 00:23:13,510 I just did the sample by sample approach, but the whole 363 00:23:13,510 --> 00:23:15,270 theme of this part of the course 364 00:23:15,270 --> 00:23:17,810 is the signals approach. 365 00:23:17,810 --> 00:23:21,630 If I think about the whole signal in one fell swoop, then 366 00:23:21,630 --> 00:23:23,570 I can develop an operator expression to 367 00:23:23,570 --> 00:23:25,400 characterize the system. 368 00:23:25,400 --> 00:23:28,760 The operator expression says the signal y is constructed by 369 00:23:28,760 --> 00:23:32,090 adding the signal x to the signal P0RY. 370 00:23:35,610 --> 00:23:39,660 If I solve that for the ratio of the output to input, I get 371 00:23:39,660 --> 00:23:42,300 1 over (1 minus P0R). 372 00:23:42,300 --> 00:23:44,702 Again, going back to the idea that I started with, that 373 00:23:44,702 --> 00:23:49,060 we're going to get ratios of polynomials in R now the R is 374 00:23:49,060 --> 00:23:55,390 in the bottom, and now I can expand that just as though it 375 00:23:55,390 --> 00:23:57,760 were an algebraic expression. 376 00:23:57,760 --> 00:24:04,730 I can expand 1 over P0R in a power series by using 377 00:24:04,730 --> 00:24:07,630 synthetic division. 378 00:24:07,630 --> 00:24:11,820 The result is very similar in structure to the result we saw 379 00:24:11,820 --> 00:24:14,230 in sample by sample. 380 00:24:14,230 --> 00:24:17,410 It consists of an ascending series in R, which means an 381 00:24:17,410 --> 00:24:21,320 ascending number of delays. 382 00:24:21,320 --> 00:24:24,300 Every time you increase the number of delays by 1, you 383 00:24:24,300 --> 00:24:29,770 also multiply the amplitude by P0, so this is, in fact, the 384 00:24:29,770 --> 00:24:32,320 same kind of result, but viewed from a 385 00:24:32,320 --> 00:24:35,490 signal point of view. 386 00:24:35,490 --> 00:24:39,000 Finally, I want to think about it in terms of block diagrams. 387 00:24:39,000 --> 00:24:42,570 Same idea, I've got the same feedback system, but now I 388 00:24:42,570 --> 00:24:45,870 want to take advantage of this ascending series expansion 389 00:24:45,870 --> 00:24:49,870 that I did and think about each of the terms in that 390 00:24:49,870 --> 00:24:55,720 series as a signal flow path through the feedback system. 391 00:24:55,720 --> 00:24:59,970 So one, the first term in the ascending series, represents 392 00:24:59,970 --> 00:25:02,730 the path that goes directly from the input to the output, 393 00:25:02,730 --> 00:25:03,980 passing through no delays. 394 00:25:06,840 --> 00:25:10,370 The second term in the series, P0R, represents the path that 395 00:25:10,370 --> 00:25:13,900 goes to the output, loops around, comes back through the 396 00:25:13,900 --> 00:25:16,890 adder, and then comes out. 397 00:25:16,890 --> 00:25:19,490 In traversing that more complicated path, you picked 398 00:25:19,490 --> 00:25:25,120 up 1 delay and 1 multiply by P0. 399 00:25:25,120 --> 00:25:27,990 Second term, two loops. 400 00:25:27,990 --> 00:25:29,300 Third term, three loops. 401 00:25:29,300 --> 00:25:32,560 Fourth term, four loops. 402 00:25:32,560 --> 00:25:37,440 The idea is that the block diagram gives us a way to 403 00:25:37,440 --> 00:25:39,930 visualize how the answer came about. 404 00:25:39,930 --> 00:25:43,620 It came about by all the possible paths that lead from 405 00:25:43,620 --> 00:25:45,820 the input to the output. 406 00:25:45,820 --> 00:25:48,660 Those possible paths all differed by a delay, and 407 00:25:48,660 --> 00:25:53,090 that's why the decomposition was so simple, each path 408 00:25:53,090 --> 00:25:54,610 corresponding to a different number of 409 00:25:54,610 --> 00:25:55,620 delays through the system. 410 00:25:55,620 --> 00:25:58,660 That won't always be true from more complicated systems, but 411 00:25:58,660 --> 00:26:01,760 it is true for this one. 412 00:26:01,760 --> 00:26:04,170 This flow diagram also lets you see something that's 413 00:26:04,170 --> 00:26:05,420 extremely interesting. 414 00:26:08,170 --> 00:26:12,540 Cyclical flow paths, which are characteristic of feedback-- 415 00:26:12,540 --> 00:26:15,780 feedback means the signal comes back. 416 00:26:15,780 --> 00:26:24,720 Cyclical flow paths require that transient inputs generate 417 00:26:24,720 --> 00:26:28,240 persistent outputs. 418 00:26:28,240 --> 00:26:30,670 They generate persistent outputs because the output at 419 00:26:30,670 --> 00:26:34,450 time n is not triggered by the input at time n. 420 00:26:34,450 --> 00:26:37,830 It's triggered by the output at time n minus 1. 421 00:26:37,830 --> 00:26:41,880 It keeps going on itself. 422 00:26:41,880 --> 00:26:43,550 That's fundamental to feedback. 423 00:26:43,550 --> 00:26:45,720 There's no way of getting around that. 424 00:26:45,720 --> 00:26:49,690 That's what feedback is. 425 00:26:49,690 --> 00:26:53,150 And it also shows why you got that funny oscillatory 426 00:26:53,150 --> 00:26:55,000 behavior in WallFinder. 427 00:26:55,000 --> 00:26:58,670 There wasn't any way around that. 428 00:26:58,670 --> 00:27:01,760 Feedback meant that you were looping back. 429 00:27:01,760 --> 00:27:04,790 That meant that there was a cycle in 430 00:27:04,790 --> 00:27:07,230 the signal flow paths. 431 00:27:07,230 --> 00:27:11,060 That means that even transient signals, signals that go away 432 00:27:11,060 --> 00:27:13,650 very quickly like the [INAUDIBLE] sample, generate 433 00:27:13,650 --> 00:27:15,070 responses that go on forever. 434 00:27:18,540 --> 00:27:21,680 So that's a fundamental way of thinking about systems. 435 00:27:21,680 --> 00:27:26,460 Systems are either feedforward or feedback. 436 00:27:26,460 --> 00:27:30,230 Feedforward means that there are no cyclic 437 00:27:30,230 --> 00:27:33,320 paths in the system. 438 00:27:36,830 --> 00:27:39,390 No path in the system that take you from the input to the 439 00:27:39,390 --> 00:27:41,650 output has a cycle in it. 440 00:27:41,650 --> 00:27:42,820 That's what acyclic means. 441 00:27:42,820 --> 00:27:45,760 That's what feedforward means. 442 00:27:45,760 --> 00:27:51,350 Acyclic, feedforward, those all have responses to 443 00:27:51,350 --> 00:27:55,880 transient inputs that are transient. 444 00:27:55,880 --> 00:27:59,600 That contrasts with cyclic systems. 445 00:27:59,600 --> 00:28:04,160 A cyclic system has feedback and will have the property 446 00:28:04,160 --> 00:28:06,590 that transient signals can generate 447 00:28:06,590 --> 00:28:10,200 outputs that go on forever. 448 00:28:10,200 --> 00:28:14,830 OK, how many of these systems are cyclic? 449 00:28:14,830 --> 00:28:15,560 Easy questions. 450 00:28:15,560 --> 00:28:16,815 15 seconds, talk to your neighbor. 451 00:29:46,420 --> 00:29:48,080 OK, so what's the answer? 452 00:29:48,080 --> 00:29:50,900 How many? 453 00:29:50,900 --> 00:29:52,310 OK, virtually 100%. 454 00:29:52,310 --> 00:29:53,720 Correct, the answer's (3). 455 00:29:53,720 --> 00:29:57,610 I've illustrated the cycles in red, so there's a cycle here, 456 00:29:57,610 --> 00:30:00,730 there's two cycles in this one, and there's a cycle here. 457 00:30:00,730 --> 00:30:05,340 So the idea is that, when you see a block diagram, one of 458 00:30:05,340 --> 00:30:07,120 the first things you want to characterize, because it's 459 00:30:07,120 --> 00:30:10,790 such a big difference between systems, is whether or not 460 00:30:10,790 --> 00:30:11,580 there's a cycle in it. 461 00:30:11,580 --> 00:30:13,710 If there's a cycle, then you know there's feedback. 462 00:30:13,710 --> 00:30:16,460 If there's feedback, then you know you have the potential to 463 00:30:16,460 --> 00:30:19,715 have a persistent response to even a transient signal. 464 00:30:22,870 --> 00:30:26,860 OK, so if you only have one loop of the type that I 465 00:30:26,860 --> 00:30:29,340 started with, where we had just one loop with an R and a 466 00:30:29,340 --> 00:30:36,680 P0, then the question is, as you go around the loop, do the 467 00:30:36,680 --> 00:30:38,750 samples get bigger, or smaller, or do 468 00:30:38,750 --> 00:30:40,570 they stay the same? 469 00:30:40,570 --> 00:30:43,130 That's a fundamental characterization of how the 470 00:30:43,130 --> 00:30:45,030 simple feedback system works. 471 00:30:45,030 --> 00:30:48,810 So here, if on every cycle the amplitude of the signal 472 00:30:48,810 --> 00:30:54,650 diminishes by multiplication by half, that means that the 473 00:30:54,650 --> 00:30:57,730 response ultimately decays. 474 00:30:57,730 --> 00:31:00,370 Mathematically, it goes on forever, just like I said 475 00:31:00,370 --> 00:31:05,440 previously, but the amplitude is decaying, so practically it 476 00:31:05,440 --> 00:31:07,410 stops after a while. 477 00:31:07,410 --> 00:31:11,090 It becomes small enough that you lose track of it. 478 00:31:11,090 --> 00:31:14,410 By contrast, if every time you go around the loop, you pick 479 00:31:14,410 --> 00:31:18,990 up amplitude, if the amplitude here were multiplied by 1.2, 480 00:31:18,990 --> 00:31:21,330 then it gets bigger. 481 00:31:21,330 --> 00:31:24,900 So the idea, then, is that you can characterize this kind of 482 00:31:24,900 --> 00:31:28,490 a feedback by one number. 483 00:31:28,490 --> 00:31:31,460 We call that number the pole. 484 00:31:31,460 --> 00:31:33,350 Very mysterious word. 485 00:31:33,350 --> 00:31:36,270 I won't go into the origins of the word. 486 00:31:36,270 --> 00:31:41,120 For our purposes, it just simply means the base of the 487 00:31:41,120 --> 00:31:45,270 geometric sequence that characterizes the response of 488 00:31:45,270 --> 00:31:49,440 a system to the unit sample signal. 489 00:31:49,440 --> 00:31:52,140 So here, I've showed an illustration of what can 490 00:31:52,140 --> 00:31:57,270 happen if p is 1/2, p is one, p is 1.2, which you can see 491 00:31:57,270 --> 00:32:00,485 decay, persistence, divergence. 492 00:32:04,780 --> 00:32:07,840 Can you characterize this system by P0? 493 00:32:07,840 --> 00:32:09,231 And if so, what is P0? 494 00:32:21,950 --> 00:32:22,210 Yes? 495 00:32:22,210 --> 00:32:23,317 No? 496 00:32:23,317 --> 00:32:30,772 [CLASSROOM SIDE CONVERSATIONS] 497 00:32:30,772 --> 00:32:33,008 PROFESSOR: Yeah, and virtually everybody's 498 00:32:33,008 --> 00:32:33,754 getting the right answer. 499 00:32:33,754 --> 00:32:35,710 The right answer's (2). 500 00:32:35,710 --> 00:32:37,960 So we like algebra. 501 00:32:37,960 --> 00:32:43,170 We like negative numbers, so we're allowed to think about 502 00:32:43,170 --> 00:32:44,310 poles being negative. 503 00:32:44,310 --> 00:32:46,000 In fact, by the end of the hour, we'll even think about 504 00:32:46,000 --> 00:32:48,820 poles having imaginary parts, but for the time 505 00:32:48,820 --> 00:32:50,440 being, this is fine. 506 00:32:50,440 --> 00:32:53,400 If the pole were negative, what that means is the 507 00:32:53,400 --> 00:32:57,370 consecutive terms in the unit sample response, the response 508 00:32:57,370 --> 00:33:00,960 of the system to a unit sample signal, the unit sample 509 00:33:00,960 --> 00:33:03,800 response, the unit sample response can 510 00:33:03,800 --> 00:33:05,050 alternate in sine. 511 00:33:07,420 --> 00:33:12,320 OK, so this then represents all the possible behaviors 512 00:33:12,320 --> 00:33:17,000 that you could get from a feedback system 513 00:33:17,000 --> 00:33:20,020 with a single pole. 514 00:33:20,020 --> 00:33:24,360 If a feedback system has a single pole, the only 515 00:33:24,360 --> 00:33:27,110 behaviors that you can get are represented 516 00:33:27,110 --> 00:33:28,480 by these three cartoons. 517 00:33:34,310 --> 00:33:38,895 So here, this z-axis contains all possible values of P0. 518 00:33:38,895 --> 00:33:44,540 If P0 is bigger than 1, then the magnitude diverges, and 519 00:33:44,540 --> 00:33:48,010 the signal grows monotonically. 520 00:33:48,010 --> 00:33:53,340 If the pole is between 0 and 1, the response is also 521 00:33:53,340 --> 00:34:00,870 monotonic, but now it converges towards 0. 522 00:34:00,870 --> 00:34:04,350 If you flip the sign, the relations are still the same, 523 00:34:04,350 --> 00:34:08,360 except that you now get sign alternation. 524 00:34:08,360 --> 00:34:14,159 So if the P0 is between 0 and minus 1, which is here, the 525 00:34:14,159 --> 00:34:17,989 output still converges because the magnitude of the pole is 526 00:34:17,989 --> 00:34:20,429 less than 1. 527 00:34:20,429 --> 00:34:22,659 But now the sign flips. 528 00:34:22,659 --> 00:34:28,139 And if the pole is below minus 1, then you get alternation, 529 00:34:28,139 --> 00:34:30,460 but you also get divergence. 530 00:34:30,460 --> 00:34:33,710 The important thing is we started with a simple system, 531 00:34:33,710 --> 00:34:35,699 and we ended up with an absolutely complete 532 00:34:35,699 --> 00:34:36,610 characterization of it. 533 00:34:36,610 --> 00:34:40,090 This is everything that can happen. 534 00:34:40,090 --> 00:34:41,860 That's a powerful statement. 535 00:34:41,860 --> 00:34:45,050 When I can analyze a system, even if it's simple, and find 536 00:34:45,050 --> 00:34:46,974 all the possible behaviors, I have something. 537 00:34:50,360 --> 00:34:52,820 If you have a simple system with a single pole, this is 538 00:34:52,820 --> 00:34:55,460 all that can happen. 539 00:34:55,460 --> 00:34:56,350 There might be offsets. 540 00:34:56,350 --> 00:34:57,770 There might be delays. 541 00:34:57,770 --> 00:35:02,810 The signal may not start until the fifth sample, but the 542 00:35:02,810 --> 00:35:06,660 persistent signal will either grow without bounds, the k to 543 00:35:06,660 --> 00:35:11,250 0, or do one of those two with alternating sign. 544 00:35:11,250 --> 00:35:16,970 That's the only things that can happen, which of course, 545 00:35:16,970 --> 00:35:19,130 begs the question, well, what if the system's more 546 00:35:19,130 --> 00:35:21,640 complicated. 547 00:35:21,640 --> 00:35:23,730 OK, so here's a more complicated system. 548 00:35:23,730 --> 00:35:27,260 This system cannot be represented by just one pole. 549 00:35:27,260 --> 00:35:31,280 In fact, the system's complicated enough you should 550 00:35:31,280 --> 00:35:34,400 think through how you would solve it. 551 00:35:34,400 --> 00:35:35,570 You should all be very comfortable with 552 00:35:35,570 --> 00:35:36,590 this sort of thing. 553 00:35:36,590 --> 00:35:47,210 So if you were to think about what if I had a system like 554 00:35:47,210 --> 00:36:02,670 so, and I want it to be 1.6 minus 0.63. 555 00:36:02,670 --> 00:36:09,080 What would be the output signal at time 2 if the input 556 00:36:09,080 --> 00:36:12,830 were a unit sample signal? 557 00:36:12,830 --> 00:36:17,050 OK, as with all systems we're going to think about, we have 558 00:36:17,050 --> 00:36:18,430 to specify initial conditions. 559 00:36:18,430 --> 00:36:20,240 The simplest kind of initial conditions we could think 560 00:36:20,240 --> 00:36:22,280 about would be rest. 561 00:36:22,280 --> 00:36:25,070 If I thought about this system at rest, then the initial 562 00:36:25,070 --> 00:36:27,770 outputs of the R's would be 0. 563 00:36:40,650 --> 00:36:42,560 That's at rest. 564 00:36:42,560 --> 00:36:47,880 For times less than 0, the input would be 0. 565 00:36:47,880 --> 00:36:56,530 0 times 1.6 plus 0 times -0.63 plus 0 would give me 0. 566 00:36:56,530 --> 00:36:59,240 Now, the clock ticks. 567 00:36:59,240 --> 00:37:04,840 When the clock ticks, it becomes times 0. 568 00:37:04,840 --> 00:37:06,495 At times 0, the input is 1. 569 00:37:09,820 --> 00:37:12,770 This 0 just propagated down to here, but this was 0, so 570 00:37:12,770 --> 00:37:14,325 nothing interesting happens at the R's. 571 00:37:17,420 --> 00:37:20,560 But now my output is 1. 572 00:37:23,790 --> 00:37:26,100 Now, the clock ticks. 573 00:37:26,100 --> 00:37:27,350 What happens? 574 00:37:31,390 --> 00:37:34,315 When the clock ticks, this 1 propagates down here. 575 00:37:38,120 --> 00:37:40,300 This 0 propagates down here, but that was 0. 576 00:37:43,360 --> 00:37:49,650 This 1 goes to 0 because the input's only 1 at times 0. 577 00:37:49,650 --> 00:37:50,900 So what's the output? 578 00:37:53,540 --> 00:37:58,490 1.6. 579 00:37:58,490 --> 00:38:01,660 Now, the clock ticks. 580 00:38:01,660 --> 00:38:02,910 What happens? 581 00:38:08,170 --> 00:38:12,690 Well, this 1 comes down here. 582 00:38:12,690 --> 00:38:14,440 This 1.6 comes down here. 583 00:38:19,170 --> 00:38:23,240 This 0 becomes another 0 because the input has an 584 00:38:23,240 --> 00:38:27,010 infinite stream of 0's after the initial time. 585 00:38:27,010 --> 00:38:28,260 So what's the output? 586 00:38:30,880 --> 00:38:37,650 Well, it's 1.6 times 1.6 plus 1 times -0.63, so the answer 587 00:38:37,650 --> 00:38:38,900 is number (3). 588 00:38:42,700 --> 00:38:43,020 Yeah? 589 00:38:43,020 --> 00:38:44,270 OK. 590 00:38:47,350 --> 00:38:48,750 I forgot to write it up there, so the 591 00:38:48,750 --> 00:38:50,720 answer's in red down here. 592 00:38:50,720 --> 00:38:53,260 1.6 squared minus 0.63. 593 00:38:53,260 --> 00:38:53,940 OK? 594 00:38:53,940 --> 00:38:55,850 The point is that it's slightly more complicated to 595 00:38:55,850 --> 00:39:00,610 think about than the case with a single pole, and in fact, if 596 00:39:00,610 --> 00:39:02,910 you use that logic to simply step through all the 597 00:39:02,910 --> 00:39:08,740 responses, you get a response that doesn't look geometric. 598 00:39:12,080 --> 00:39:15,060 The geometric sequences that we looked at previously either 599 00:39:15,060 --> 00:39:17,240 monotonically increased, monotonically decreased 600 00:39:17,240 --> 00:39:19,640 towards 0, or give one of those two things and 601 00:39:19,640 --> 00:39:20,410 alternated. 602 00:39:20,410 --> 00:39:22,030 This does none of those behaviors. 603 00:39:22,030 --> 00:39:25,430 So the point is Freeman's an idiot. 604 00:39:25,430 --> 00:39:28,010 He spent all that time telling us what one pole does, and now 605 00:39:28,010 --> 00:39:30,100 two poles does something completely different. 606 00:39:30,100 --> 00:39:31,900 Right? 607 00:39:31,900 --> 00:39:34,750 So the response is not geometric. 608 00:39:34,750 --> 00:39:37,110 The response grows and then decays. 609 00:39:37,110 --> 00:39:38,350 It never changes sign. 610 00:39:38,350 --> 00:39:41,430 It does something completely different from what we would 611 00:39:41,430 --> 00:39:45,020 have expected from a single pole system. 612 00:39:45,020 --> 00:39:47,750 As you might expect from the Theory of Lectures, that's not 613 00:39:47,750 --> 00:39:49,980 the end of the story. 614 00:39:49,980 --> 00:39:57,610 So the idea is to now capitalize on this notion that 615 00:39:57,610 --> 00:40:02,720 we can think about operators as algebra. 616 00:40:02,720 --> 00:40:06,950 If our expressions behaved like I told you they did last 617 00:40:06,950 --> 00:40:12,280 lecture, if they behaved as entities upon which-- 618 00:40:12,280 --> 00:40:17,300 if they are isomorphic with polynomials, as I said, then 619 00:40:17,300 --> 00:40:20,760 there's a very cute thing we can do with this system to 620 00:40:20,760 --> 00:40:23,800 make it a lot simpler. 621 00:40:23,800 --> 00:40:27,070 The thing we can do is factor. 622 00:40:27,070 --> 00:40:30,510 If we think about the operator expression to characterize 623 00:40:30,510 --> 00:40:34,690 this system, the thing that's different is that 624 00:40:34,690 --> 00:40:37,850 there's an R squared. 625 00:40:37,850 --> 00:40:40,990 But if R operators work just like polynomials-- 626 00:40:40,990 --> 00:40:42,690 you can factor polynomials. 627 00:40:42,690 --> 00:40:47,410 That's the factor theorem from algebra. 628 00:40:47,410 --> 00:40:51,120 And if I factor it, I get two things that look like 629 00:40:51,120 --> 00:40:52,770 first-order systems. 630 00:40:52,770 --> 00:40:54,020 Well, that's good. 631 00:40:56,260 --> 00:41:01,080 The factored form means that I can think about this more 632 00:41:01,080 --> 00:41:04,590 complicated system as the cascade of 633 00:41:04,590 --> 00:41:07,140 two first-order systems. 634 00:41:07,140 --> 00:41:09,860 Well, that's pretty good. 635 00:41:09,860 --> 00:41:11,910 In fact, it doesn't even matter what order I put them 636 00:41:11,910 --> 00:41:16,210 in because, as we've seen previously, if the system 637 00:41:16,210 --> 00:41:20,350 started at initial rest, then you can swap things because 638 00:41:20,350 --> 00:41:24,390 they obey all the principles of polynomials, which include 639 00:41:24,390 --> 00:41:25,640 commutation. 640 00:41:27,830 --> 00:41:30,520 So what we've done, then, is transform this more 641 00:41:30,520 --> 00:41:35,280 complicated system into the cascade of two simple systems, 642 00:41:35,280 --> 00:41:36,530 and that's very good. 643 00:41:39,270 --> 00:41:44,880 Even better, we can think about the complicated system 644 00:41:44,880 --> 00:41:51,940 as the sum of simpler parts, and that uses more intuition 645 00:41:51,940 --> 00:41:53,970 from polynomials. 646 00:41:53,970 --> 00:41:57,750 If we have one over a second-order polynomial, we 647 00:41:57,750 --> 00:42:01,680 can write it in a factored form here, but we can expand 648 00:42:01,680 --> 00:42:04,870 it in what we call partial fractions. 649 00:42:04,870 --> 00:42:09,110 We can expand this thing in this sum, and if you think 650 00:42:09,110 --> 00:42:12,140 about putting this over a common denominator and working 651 00:42:12,140 --> 00:42:17,100 out the relationship, this difference, 4.5 over 1 minus 652 00:42:17,100 --> 00:42:22,280 0.9R minus 3.5 over 1 minus 0.7R. 653 00:42:22,280 --> 00:42:25,020 That's precisely the same using the normal rules for 654 00:42:25,020 --> 00:42:26,110 polynomials. 655 00:42:26,110 --> 00:42:29,650 That's precisely the same as that expression. 656 00:42:29,650 --> 00:42:31,400 But the difference, from the point of view of thinking 657 00:42:31,400 --> 00:42:33,550 about systems, is enormous. 658 00:42:33,550 --> 00:42:37,310 We know the answer to that one. 659 00:42:37,310 --> 00:42:39,670 That's the sum of the responses to two first-order 660 00:42:39,670 --> 00:42:45,350 systems, so we can write that symbolically this way. 661 00:42:45,350 --> 00:42:47,830 We can think about having a sum system that 662 00:42:47,830 --> 00:42:51,190 generates this term. 663 00:42:51,190 --> 00:42:55,110 This term is a simple system of the type of that we looked 664 00:42:55,110 --> 00:43:01,670 at previously that, then, gets multiplied by 4.5. 665 00:43:01,670 --> 00:43:03,540 I'm just factoring again. 666 00:43:03,540 --> 00:43:07,690 I'm saying I've got something over something, which means 667 00:43:07,690 --> 00:43:10,950 that I can put something in each of two different parts of 668 00:43:10,950 --> 00:43:13,220 two things that I multiply together. 669 00:43:13,220 --> 00:43:16,230 And I can think about this as having been generated by this 670 00:43:16,230 --> 00:43:18,040 system, and you just add them together. 671 00:43:21,430 --> 00:43:24,790 The amazing thing is that that says that, despite the fact 672 00:43:24,790 --> 00:43:27,630 that the response looked complicated, it was in fact 673 00:43:27,630 --> 00:43:30,720 the sum of two geometrics. 674 00:43:30,720 --> 00:43:33,850 So it wasn't very different from the answer 675 00:43:33,850 --> 00:43:36,550 for a single pole. 676 00:43:36,550 --> 00:43:40,050 What I've just done is amazing. 677 00:43:40,050 --> 00:43:43,210 I've just taken something that, had you studied the 678 00:43:43,210 --> 00:43:45,410 difference equations and had you studied the block 679 00:43:45,410 --> 00:43:49,880 diagrams, it would have been very hard for you to conclude 680 00:43:49,880 --> 00:43:52,200 that something this complicated has a response 681 00:43:52,200 --> 00:43:55,930 that can be written as the sum of two geometrics. 682 00:43:55,930 --> 00:43:59,470 By thinking about the system as a polynomial in R, it's 683 00:43:59,470 --> 00:44:00,650 completely trivial. 684 00:44:00,650 --> 00:44:05,290 It's a simple application of the rules for polynomials that 685 00:44:05,290 --> 00:44:07,950 you all know. 686 00:44:07,950 --> 00:44:11,190 So what we've shown, then, is that this complicated system 687 00:44:11,190 --> 00:44:15,305 has a way of thinking about as just two 688 00:44:15,305 --> 00:44:18,190 of the simpler systems. 689 00:44:18,190 --> 00:44:22,860 The complicated response that grew and decayed, that's just 690 00:44:22,860 --> 00:44:28,750 the difference, really, 4.5 minus 3.5. 691 00:44:28,750 --> 00:44:32,560 It's the weighted difference of a part that goes 0.7 to the 692 00:44:32,560 --> 00:44:34,963 n, then a different part that goes 0.9 to the n. 693 00:44:42,640 --> 00:44:46,250 So far, we've got to results, the n equals 1 case, the 694 00:44:46,250 --> 00:44:48,620 first-order polynomial in our case, the one pole case, 695 00:44:48,620 --> 00:44:49,550 that's trivial. 696 00:44:49,550 --> 00:44:51,490 It's just a geometric sequence. 697 00:44:51,490 --> 00:44:54,010 The response is just a geometric sequence. 698 00:44:54,010 --> 00:44:56,160 If it happens to be second-order, this is 699 00:44:56,160 --> 00:45:03,240 second-order because when you write the operator expression, 700 00:45:03,240 --> 00:45:07,260 the polynomial in the bottom is second-order, second-order 701 00:45:07,260 --> 00:45:08,740 polynomial in R. 702 00:45:08,740 --> 00:45:12,270 This second-order system has a response that 703 00:45:12,270 --> 00:45:15,300 looks like two pieces. 704 00:45:15,300 --> 00:45:17,750 Each piece looks like a piece that was from 705 00:45:17,750 --> 00:45:19,410 a first-order system. 706 00:45:19,410 --> 00:45:21,675 And in fact, that idea generalizes. 707 00:45:25,470 --> 00:45:29,790 If we have a system that can be represented by linear 708 00:45:29,790 --> 00:45:34,500 difference equation with constant coefficients that 709 00:45:34,500 --> 00:45:37,410 will always be true if the system was constructed out of 710 00:45:37,410 --> 00:45:43,270 the parts that we talked about, adders, gains, delays. 711 00:45:43,270 --> 00:45:46,060 If the system is constructed out of adders, gains, and 712 00:45:46,060 --> 00:45:49,290 delays, then it will be possible to express the system 713 00:45:49,290 --> 00:45:51,830 in terms of one difference equation. 714 00:45:51,830 --> 00:45:53,850 General form is showed here. 715 00:45:53,850 --> 00:45:58,200 Y then can be constructed out of parts that are delayed 716 00:45:58,200 --> 00:46:04,210 versions of Y and delayed versions of X. If you do that, 717 00:46:04,210 --> 00:46:08,120 then you can always write the operator that expresses the 718 00:46:08,120 --> 00:46:11,630 ratio between the output and the input as the ratio of two 719 00:46:11,630 --> 00:46:13,000 polynomials. 720 00:46:13,000 --> 00:46:15,880 That will always be true. 721 00:46:15,880 --> 00:46:18,590 So this, now, is the generalization step. 722 00:46:18,590 --> 00:46:23,310 We did the n equals 1 case, we did the n equals 2 case, and 723 00:46:23,310 --> 00:46:24,730 now we're generalizing. 724 00:46:24,730 --> 00:46:28,220 We will always get, for any system that can be represented 725 00:46:28,220 --> 00:46:31,080 by a linear difference equation with constant 726 00:46:31,080 --> 00:46:34,170 coefficients, we can always represent the system 727 00:46:34,170 --> 00:46:37,750 functional in this form. 728 00:46:37,750 --> 00:46:42,060 Then just like we did in the second-order case, we can use 729 00:46:42,060 --> 00:46:46,420 the factor theorem to break this polynomial in the 730 00:46:46,420 --> 00:46:49,940 denominator into factors. 731 00:46:49,940 --> 00:46:51,945 That comes from the factor theorem in algebra. 732 00:46:54,640 --> 00:46:59,480 Then we can re-express that in terms of partial fractions. 733 00:46:59,480 --> 00:47:02,820 And what I've just showed is that, in the general case, 734 00:47:02,820 --> 00:47:05,910 regardless of how many delays are in the system, if the 735 00:47:05,910 --> 00:47:09,430 system only has adders, gains, and delays, I can always 736 00:47:09,430 --> 00:47:12,020 express the answer as a sum of geometrics. 737 00:47:12,020 --> 00:47:14,710 That's interesting. 738 00:47:14,710 --> 00:47:19,130 That means that if I knew the bases for all of those 739 00:47:19,130 --> 00:47:24,100 geometric sequences, I know something about the response. 740 00:47:24,100 --> 00:47:26,050 The bases are things we call poles. 741 00:47:26,050 --> 00:47:31,120 If you knew all the poles, you'd know something very 742 00:47:31,120 --> 00:47:32,370 powerful about the system. 743 00:47:36,330 --> 00:47:40,590 So every one of the factors corresponds to a pole, and by 744 00:47:40,590 --> 00:47:44,770 partial fractions, you'll get one response for each pole. 745 00:47:44,770 --> 00:47:48,411 The response for each pole goes like pole to the n. 746 00:47:48,411 --> 00:47:49,750 You know the basic shape. 747 00:47:49,750 --> 00:47:53,470 You don't know the constants, but you know the basic shape 748 00:47:53,470 --> 00:47:58,880 of the response just by knowing the poles. 749 00:47:58,880 --> 00:48:01,110 We can go one more step, which makes the 750 00:48:01,110 --> 00:48:03,790 computation somewhat simpler. 751 00:48:03,790 --> 00:48:06,600 I used the factor theorem. 752 00:48:06,600 --> 00:48:08,160 Here, I'm using the fundamental theorem of 753 00:48:08,160 --> 00:48:12,960 algebra, which says that if I have a polynomial of order n, 754 00:48:12,960 --> 00:48:15,740 I have n roots. 755 00:48:15,740 --> 00:48:19,530 The poles are related to the roots of the R polynomial. 756 00:48:22,510 --> 00:48:28,020 The relationship is take the functional, 757 00:48:28,020 --> 00:48:29,250 substitute for R -- 758 00:48:29,250 --> 00:48:38,500 1 over Z. Re-express the functional as a ratio of 759 00:48:38,500 --> 00:48:43,240 polynomials in Z. The poles are the roots of the 760 00:48:43,240 --> 00:48:44,490 denominator. 761 00:48:47,780 --> 00:48:50,350 So recapping, I started with a first-order system. 762 00:48:50,350 --> 00:48:52,440 I showed you how to get a second-order system. 763 00:48:52,440 --> 00:48:56,760 I showed that, in general, you can use the factor theorem to 764 00:48:56,760 --> 00:49:00,730 break down the response of a higher-order system into a sum 765 00:49:00,730 --> 00:49:03,880 of responses of first-order systems. 766 00:49:03,880 --> 00:49:06,720 Now, I've shown that you can use the fundamental theorem of 767 00:49:06,720 --> 00:49:11,590 algebra to find the poles directly, and then by knowing 768 00:49:11,590 --> 00:49:16,030 the poles, you know each of the behaviors, monotonic 769 00:49:16,030 --> 00:49:18,280 divergence, monotonic convergence, 770 00:49:18,280 --> 00:49:20,870 or alternating signs. 771 00:49:20,870 --> 00:49:23,770 And so here is this same example that I started with, 772 00:49:23,770 --> 00:49:26,260 worked out by thinking about what are the poles. 773 00:49:26,260 --> 00:49:29,280 The poles are 0.7 and 0.9, which we see by a simple 774 00:49:29,280 --> 00:49:32,300 application of the fundamental theorem of algebra. 775 00:49:32,300 --> 00:49:38,210 OK, we got a long way by just thinking about operators as 776 00:49:38,210 --> 00:49:39,080 polynomials. 777 00:49:39,080 --> 00:49:40,880 We haven't done anything that you haven't 778 00:49:40,880 --> 00:49:42,300 done in high school. 779 00:49:42,300 --> 00:49:46,090 Polynomials are very familiar and we've made an isomorphism 780 00:49:46,090 --> 00:49:50,100 between systems and polynomials. 781 00:49:53,210 --> 00:49:55,920 OK, make sure you're all with me. 782 00:49:55,920 --> 00:49:57,880 Here's a higher order system. 783 00:49:57,880 --> 00:50:00,140 How many of these statements are true -- 784 00:50:00,140 --> 00:50:02,030 0, 1, 2, 3, 4, or 5? 785 00:50:04,740 --> 00:50:06,060 Talk to your neighbor, get an answer. 786 00:52:42,920 --> 00:52:44,650 So how many are true -- 787 00:52:44,650 --> 00:52:46,660 0, 1, 2, 3, 4, or 5? 788 00:52:49,180 --> 00:52:50,290 Oh, come on. 789 00:52:50,290 --> 00:52:51,900 Blame it on your neighbor. 790 00:52:51,900 --> 00:52:53,595 You weren't talking, but I didn't hear you not talking. 791 00:52:56,490 --> 00:52:57,385 How many are true -- 792 00:52:57,385 --> 00:52:59,080 0, 1, 2, 3, 4, or 5? 793 00:52:59,080 --> 00:53:00,511 Raise your hands. 794 00:53:00,511 --> 00:53:02,420 AUDIENCE: They can't all be true. 795 00:53:02,420 --> 00:53:03,555 PROFESSOR: They can't all be true. 796 00:53:03,555 --> 00:53:05,270 Are they mutually contradictory? 797 00:53:05,270 --> 00:53:06,180 AUDIENCE: Well, yeah. 798 00:53:06,180 --> 00:53:07,090 5 -- 799 00:53:07,090 --> 00:53:08,390 PROFESSOR: N1 of the above, that sounds like-- 800 00:53:08,390 --> 00:53:10,650 OK, so you've eliminated 1. 801 00:53:13,310 --> 00:53:14,786 Which one's true? 802 00:53:14,786 --> 00:53:18,132 How many statements are true? 803 00:53:18,132 --> 00:53:21,700 Looks like about 75%. 804 00:53:21,700 --> 00:53:23,130 Correct? 805 00:53:23,130 --> 00:53:25,261 What should I do? 806 00:53:25,261 --> 00:53:26,390 How do I figure it out? 807 00:53:26,390 --> 00:53:27,120 What's my first step? 808 00:53:27,120 --> 00:53:27,878 What do I do? 809 00:53:27,878 --> 00:53:30,270 AUDIENCE: Operators. 810 00:53:30,270 --> 00:53:32,080 PROFESSOR: Operators, absolutely. 811 00:53:32,080 --> 00:53:34,690 So turn it into operators. 812 00:53:34,690 --> 00:53:36,060 So take the difference equation, 813 00:53:36,060 --> 00:53:39,110 turn it into operators. 814 00:53:39,110 --> 00:53:44,500 The important thing to see is that there are three Y terms. 815 00:53:44,500 --> 00:53:47,950 Take them all to the same side, and I get an operator 816 00:53:47,950 --> 00:53:51,420 expression like that. 817 00:53:51,420 --> 00:53:54,200 The ones that depend on X, there are two of them, that's 818 00:53:54,200 --> 00:53:57,410 represented here. 819 00:53:57,410 --> 00:54:01,210 The thing this is critical for determining poles is figuring 820 00:54:01,210 --> 00:54:02,770 out the denominator. 821 00:54:02,770 --> 00:54:04,225 The poles are going to come from this one. 822 00:54:06,920 --> 00:54:16,080 After I get the ratio of two polynomials in R, I substitute 823 00:54:16,080 --> 00:54:21,790 1 over Z for each R. So for this R, I get 1 over Z. For 824 00:54:21,790 --> 00:54:25,350 this R squared, I get 1 over Z squared. 825 00:54:25,350 --> 00:54:29,020 Then I want to turn it back into a ratio of polynomials in 826 00:54:29,020 --> 00:54:34,710 Z, so I have to multiply top and bottom by Z squared. 827 00:54:34,710 --> 00:54:45,700 And when I do that, I get this ratio of polynomials in Z. 828 00:54:45,700 --> 00:54:51,400 The poles are the roots of the denominator polynomial in Z. 829 00:54:51,400 --> 00:54:57,820 The poles are minus 1/2 and plus 1/4. 830 00:54:57,820 --> 00:55:00,095 So the unit sample response converges to 0. 831 00:55:04,680 --> 00:55:06,690 What would be the condition that that represents? 832 00:55:10,050 --> 00:55:12,990 AUDIENCE: Take the polynomial on the bottom. 833 00:55:12,990 --> 00:55:15,780 PROFESSOR: Something about the polynomial on the bottom. 834 00:55:15,780 --> 00:55:18,030 Would all second-order systems have that property that the 835 00:55:18,030 --> 00:55:20,774 unit sample response would converge to 0? 836 00:55:20,774 --> 00:55:22,024 AUDIENCE: [INAUDIBLE] 837 00:55:24,150 --> 00:55:25,520 PROFESSOR: Louder? 838 00:55:25,520 --> 00:55:27,820 AUDIENCE: Absolute value of the poles? 839 00:55:27,820 --> 00:55:31,526 PROFESSOR: Absolute value of the poles has to be? 840 00:55:31,526 --> 00:55:32,990 AUDIENCE: Less than 1. 841 00:55:32,990 --> 00:55:35,110 PROFESSOR: Less than 1. 842 00:55:35,110 --> 00:55:37,380 If the magnitude of the poles is less than 1, then the 843 00:55:37,380 --> 00:55:40,370 response magnitude will decay with time. 844 00:55:40,370 --> 00:55:44,080 So that's true here, and it would be true so long as none 845 00:55:44,080 --> 00:55:47,930 of the poles have a magnitude exceeding 1. 846 00:55:47,930 --> 00:55:50,030 There are poles at 1/2 and 1/4. 847 00:55:50,030 --> 00:55:51,050 No, that's not right. 848 00:55:51,050 --> 00:55:53,640 It's -1/2 or 1/4. 849 00:55:53,640 --> 00:55:54,830 There's a pole at 1/2. 850 00:55:54,830 --> 00:55:55,640 No, that's not right. 851 00:55:55,640 --> 00:55:57,450 There's a pole at -1/2. 852 00:55:57,450 --> 00:55:58,260 There are two poles. 853 00:55:58,260 --> 00:56:00,870 Yes, that's true. 854 00:56:00,870 --> 00:56:01,520 None of the above. 855 00:56:01,520 --> 00:56:02,340 No, that's not true. 856 00:56:02,340 --> 00:56:05,210 So the answer was (2). 857 00:56:05,210 --> 00:56:07,410 Everybody's comfortable? 858 00:56:07,410 --> 00:56:08,970 We've done something very astonishing. 859 00:56:08,970 --> 00:56:13,410 We took an arbitrary system, and we've figured out a rule 860 00:56:13,410 --> 00:56:15,820 that let's us break it into the sum 861 00:56:15,820 --> 00:56:17,820 of geometric sequences. 862 00:56:17,820 --> 00:56:20,750 We can always write the response to a unit sample 863 00:56:20,750 --> 00:56:25,250 signal, we can always write, as a weighted sum of geometric 864 00:56:25,250 --> 00:56:30,440 sequences, and the number of geometric sequences in the sum 865 00:56:30,440 --> 00:56:35,030 is the number of poles, which is the order of the operator 866 00:56:35,030 --> 00:56:38,920 that operates on Y. OK, so we've done 867 00:56:38,920 --> 00:56:41,720 something very powerful. 868 00:56:41,720 --> 00:56:44,490 There's one more thing that we have to think about, and then 869 00:56:44,490 --> 00:56:48,340 we have a complete picture of what's going on. 870 00:56:48,340 --> 00:56:49,800 Think about when you learned polynomials. 871 00:56:52,450 --> 00:56:57,180 One of the big shocks was that roots can be complex. 872 00:56:57,180 --> 00:56:58,940 What would that mean? 873 00:56:58,940 --> 00:57:05,660 What would it mean if we had a system whose poles were 874 00:57:05,660 --> 00:57:07,510 complex valued? 875 00:57:07,510 --> 00:57:09,480 So first off, does such a system exist? 876 00:57:09,480 --> 00:57:11,050 Well, here's one. 877 00:57:11,050 --> 00:57:12,720 I just pulled that out of the air. 878 00:57:12,720 --> 00:57:16,230 If I think about the functional, 1 over (1 minus R 879 00:57:16,230 --> 00:57:18,490 plus R squared) -- 880 00:57:18,490 --> 00:57:23,480 if I convert that into our ratio of polynomials in Z and 881 00:57:23,480 --> 00:57:25,640 then find the roots, I find that the roots 882 00:57:25,640 --> 00:57:27,070 have a complex part. 883 00:57:27,070 --> 00:57:30,195 The roots are 1/2 plus or minus root 3 over 2j. 884 00:57:33,020 --> 00:57:35,970 There's an imaginary part. 885 00:57:35,970 --> 00:57:40,330 So the question is what would that mean? 886 00:57:40,330 --> 00:57:42,370 Or is perhaps that system just meaningless? 887 00:57:45,970 --> 00:57:51,530 Well, complex numbers work in algebra, and complex numbers 888 00:57:51,530 --> 00:57:54,000 work here, too. 889 00:57:54,000 --> 00:57:59,280 So the fact that a pole has a complex value in the context 890 00:57:59,280 --> 00:58:02,770 of signals and systems simply means that the pole is 891 00:58:02,770 --> 00:58:10,520 complex, that the base of the geometric sequence, that base 892 00:58:10,520 --> 00:58:12,200 is complex. 893 00:58:12,200 --> 00:58:17,500 So that means that we can still rewrite the denominator, 894 00:58:17,500 --> 00:58:20,960 which was 1 minus R plus R squared, we can rewrite that 895 00:58:20,960 --> 00:58:25,830 denominator in terms of a product of two first-order R 896 00:58:25,830 --> 00:58:28,460 polynomials. 897 00:58:28,460 --> 00:58:33,580 The coefficients are now complex, but it still works. 898 00:58:33,580 --> 00:58:34,950 The algebra still works right. 899 00:58:34,950 --> 00:58:36,560 That has to work because that's just polynomials. 900 00:58:36,560 --> 00:58:39,020 That's the way polynomials behave. 901 00:58:39,020 --> 00:58:40,610 Now, we can still factor it. 902 00:58:40,610 --> 00:58:41,930 We can still use the factor theorem. 903 00:58:41,930 --> 00:58:44,760 In fact, we can still use the fundamental theorem of algebra 904 00:58:44,760 --> 00:58:47,820 to find the poles by the Z trick. 905 00:58:47,820 --> 00:58:48,950 That's fine. 906 00:58:48,950 --> 00:58:50,890 We can still use partial fractions. 907 00:58:50,890 --> 00:58:54,050 All of these numbers are complex, but 908 00:58:54,050 --> 00:58:56,250 the math still works. 909 00:58:56,250 --> 00:59:00,120 The funny thing is that it implies that 910 00:59:00,120 --> 00:59:02,620 the fundamental modes-- 911 00:59:02,620 --> 00:59:06,440 by fundamental mode, I mean the simple geometric for the 912 00:59:06,440 --> 00:59:09,380 case of a first-order system, more complex behaviors for 913 00:59:09,380 --> 00:59:10,630 higher order systems. 914 00:59:10,630 --> 00:59:15,130 The mode is the time response associated with a pole. 915 00:59:15,130 --> 00:59:20,006 So the modes are, in this case, complexes sequences. 916 00:59:24,140 --> 00:59:32,040 So in general, the modes look like P0 to the n. 917 00:59:32,040 --> 00:59:35,260 Here, my modes are simply have a complex value. 918 00:59:35,260 --> 00:59:36,740 So what did I say they were? 919 00:59:36,740 --> 00:59:40,210 The poles were 1/2 plus or minus-- 920 00:59:47,320 --> 00:59:48,940 so my modes simply look that. 921 00:59:48,940 --> 00:59:51,010 Same thing. 922 00:59:51,010 --> 00:59:55,625 The strange thing that happened was that those modes, 923 00:59:55,625 --> 01:00:00,860 those geometric sequences, are now have complex values. 924 01:00:00,860 --> 01:00:05,680 The first one up here, if I just look at the denominator, 925 01:00:05,680 --> 01:00:08,680 these coefficients mean that it's proportionate to the mode 926 01:00:08,680 --> 01:00:13,190 associated with this, which is that, which has a real part, 927 01:00:13,190 --> 01:00:15,490 which is the blue part, and the imaginary part, 928 01:00:15,490 --> 01:00:16,740 which is a red part. 929 01:00:19,260 --> 01:00:22,330 There were two poles, plus and minus. 930 01:00:22,330 --> 01:00:25,420 The other pole just flips the imaginary part. 931 01:00:28,480 --> 01:00:34,740 So if I have imaginary poles, all I get is complex modes, 932 01:00:34,740 --> 01:00:36,660 complex geometric sequences. 933 01:00:42,620 --> 01:00:45,840 An easier way of thinking about that is thinking about-- 934 01:00:45,840 --> 01:00:51,020 so when we had a simple, real pole, we just had P0 to the n. 935 01:00:51,020 --> 01:00:53,900 That's easy to visualize because we just think about 936 01:00:53,900 --> 01:00:57,960 each time you go from 0 to 1 to 2 to 3, it goes from 1 to 937 01:00:57,960 --> 01:01:00,220 P0 to P0 squared to P0 cubed. 938 01:01:00,220 --> 01:01:02,990 Here, when you're multiplying complex numbers, it's easier 939 01:01:02,990 --> 01:01:06,800 to imagine that on the complex plane. 940 01:01:06,800 --> 01:01:10,020 Think about the location of the point 1, think about the 941 01:01:10,020 --> 01:01:14,270 location of the point P0, think about the location of 942 01:01:14,270 --> 01:01:18,180 the point P0 squared, and in this particular case, where 943 01:01:18,180 --> 01:01:22,400 the pole was 1/2 plus or minus the square root of 3 over 2 944 01:01:22,400 --> 01:01:28,700 times j, this would be pole to the 0. 945 01:01:28,700 --> 01:01:30,510 This is pole to the 1. 946 01:01:30,510 --> 01:01:33,330 This is pole squared, pole cubed. 947 01:01:33,330 --> 01:01:36,870 As you can see when you have a complex number, the trajectory 948 01:01:36,870 --> 01:01:39,280 in complex space can be complicated. 949 01:01:39,280 --> 01:01:42,560 In this case, it's circular. 950 01:01:42,560 --> 01:01:46,680 The circular trajectory in the complex plane corresponds to 951 01:01:46,680 --> 01:01:50,070 the sinusoidal behavior in time. 952 01:01:50,070 --> 01:01:51,850 So there's a correlation between the way you think 953 01:01:51,850 --> 01:01:55,300 about the modes evolving on the complex plane and the way 954 01:01:55,300 --> 01:01:56,720 you think about the real and imaginary 955 01:01:56,720 --> 01:01:58,210 parts evolving in time. 956 01:02:02,560 --> 01:02:03,950 It seems a little weird that the 957 01:02:03,950 --> 01:02:08,570 response should be complex. 958 01:02:08,570 --> 01:02:12,040 We're studying this kind of system theory primarily 959 01:02:12,040 --> 01:02:14,290 because we're trying to gain insight into real systems. 960 01:02:14,290 --> 01:02:16,690 We want to know how things like robots work. 961 01:02:16,690 --> 01:02:17,900 How does the WallFinder work? 962 01:02:17,900 --> 01:02:21,190 What would it mean if the WallFinder went to position 963 01:02:21,190 --> 01:02:26,040 one plus the square root of 3 over 2j? 964 01:02:26,040 --> 01:02:29,100 That doesn't make sense. 965 01:02:29,100 --> 01:02:31,930 So there's a little bit of a strange thing going on here. 966 01:02:31,930 --> 01:02:35,550 How is it that we need complex numbers to model real things? 967 01:02:35,550 --> 01:02:37,690 That doesn't seem to sound right. 968 01:02:37,690 --> 01:02:42,710 But the answer is that, if the difference equation had real 969 01:02:42,710 --> 01:02:48,270 coefficients, as they will for a real system-- 970 01:02:48,270 --> 01:02:50,770 if you think about a real system, like a bank account, 971 01:02:50,770 --> 01:02:53,850 the coefficients in the difference equation are real 972 01:02:53,850 --> 01:02:55,940 numbers, not complex numbers. 973 01:02:55,940 --> 01:02:58,320 If you think about the WallFinder system, the 974 01:02:58,320 --> 01:03:03,140 coefficients in the WallFinder system, the coefficients of 975 01:03:03,140 --> 01:03:05,940 the difference equations-- 976 01:03:05,940 --> 01:03:07,720 the coefficients of the different equations describe 977 01:03:07,720 --> 01:03:11,080 the WallFinder behavior were all real numbers. 978 01:03:16,050 --> 01:03:18,620 Here, I'm thinking about the denominator polynomial. 979 01:03:18,620 --> 01:03:23,230 If we try to find the roots of a polynomial, and if we find a 980 01:03:23,230 --> 01:03:28,430 complex root, if the coefficients were all real, it 981 01:03:28,430 --> 01:03:33,150 follows that the complex conjugate of the original root 982 01:03:33,150 --> 01:03:36,390 is also a root. 983 01:03:36,390 --> 01:03:38,210 That's pretty simple, if you think about what it means to 984 01:03:38,210 --> 01:03:41,460 be a polynomial. 985 01:03:41,460 --> 01:03:45,020 If you think about a polynomial is whatever-- 986 01:03:45,020 --> 01:03:51,380 so I've got 1 plus Z plus Z squared plus blah, blah, blah. 987 01:03:54,850 --> 01:04:04,230 2, 3 minus 16, if all of those coefficients real, then the 988 01:04:04,230 --> 01:04:05,480 only way the-- 989 01:04:10,530 --> 01:04:16,880 If P were a root of this polynomial, then P* would have 990 01:04:16,880 --> 01:04:23,340 to be a root, too, because if you complex conjugate each of 991 01:04:23,340 --> 01:04:25,790 the Z's, it's the same thing as complex conjugating the 992 01:04:25,790 --> 01:04:30,970 whole thing because the coefficients are real valued. 993 01:04:30,970 --> 01:04:36,780 So the idea then is that if I happen to get a complex root 994 01:04:36,780 --> 01:04:39,530 for my system that can be described by real value 995 01:04:39,530 --> 01:04:43,380 coefficients, it must also be true that it's complex 996 01:04:43,380 --> 01:04:45,790 conjugate is a root. 997 01:04:45,790 --> 01:04:51,720 If that happens, the two roots co-conspire so that the modes 998 01:04:51,720 --> 01:04:54,240 have canceling imaginary parts. 999 01:04:54,240 --> 01:04:54,920 You can prove that. 1000 01:04:54,920 --> 01:04:57,000 I'm not worried about you being able to prove that. 1001 01:04:57,000 --> 01:05:00,800 I just want you to understand that if you have two roots 1002 01:05:00,800 --> 01:05:06,080 that are complex conjugates, they can conspire to have 1003 01:05:06,080 --> 01:05:08,980 their imaginary parts cancel, and that's 1004 01:05:08,980 --> 01:05:10,120 exactly what happens. 1005 01:05:10,120 --> 01:05:15,190 Here, in this example, the example that I started with, 1006 01:05:15,190 --> 01:05:18,860 where the system was 1 over (1 minus R plus R squared), the 1007 01:05:18,860 --> 01:05:20,510 unit sample responses showed here. 1008 01:05:23,170 --> 01:05:29,350 You can write as the sum of sinusoidal and cosinusoidal 1009 01:05:29,350 --> 01:05:35,780 signals, and the sum that falls out has the property 1010 01:05:35,780 --> 01:05:39,490 that the imaginary parts cancel. 1011 01:05:39,490 --> 01:05:42,560 It's still useful to look at the imaginary parts the same 1012 01:05:42,560 --> 01:05:45,560 as it is when you're trying to solve polynomials. 1013 01:05:45,560 --> 01:05:49,230 It's useful because the period of all of these 1014 01:05:49,230 --> 01:05:50,490 signals is the same. 1015 01:05:53,500 --> 01:05:56,900 If I think about the period of the individual modes, if I 1016 01:05:56,900 --> 01:06:00,380 think about the period of P0 to the n, 1/2 plus or minus 1017 01:06:00,380 --> 01:06:04,740 root 3 over 2j to the n, the period of that signal -- 1018 01:06:04,740 --> 01:06:07,810 I can see in the complex plane, this is the n equals 0, 1019 01:06:07,810 --> 01:06:13,000 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 -- 1020 01:06:13,000 --> 01:06:17,310 the period is 6. 1021 01:06:17,310 --> 01:06:20,940 If I were to take the minus 1, I would go around the circle 1022 01:06:20,940 --> 01:06:22,220 the other way. 1023 01:06:22,220 --> 01:06:26,850 This would be zero term 1, 2, 3, 4, 5, 6, et cetera. 1024 01:06:26,850 --> 01:06:33,620 Both of the modes, the geometric sequence associated 1025 01:06:33,620 --> 01:06:37,670 with the two complex conjugate poles, both of those modes had 1026 01:06:37,670 --> 01:06:40,590 the same period, they're both 6, as does the 1027 01:06:40,590 --> 01:06:43,670 response to the real part. 1028 01:06:43,670 --> 01:06:50,690 So you can deduce the period of the real signal by looking 1029 01:06:50,690 --> 01:06:54,055 at the periods of the two complex signals. 1030 01:06:57,440 --> 01:06:58,980 So think about this system. 1031 01:06:58,980 --> 01:07:04,420 Here, I've got a system whose responses showed here. 1032 01:07:04,420 --> 01:07:07,740 I'm going to tell you that the response was generated by a 1033 01:07:07,740 --> 01:07:10,420 second-order system, that is to say a system whose 1034 01:07:10,420 --> 01:07:13,720 polynomial was second-order, whose polynomial in R was 1035 01:07:13,720 --> 01:07:15,440 second-order. 1036 01:07:15,440 --> 01:07:17,375 Which of these statements is true? 1037 01:07:21,830 --> 01:07:25,730 I want to think about the pole as being a complex number. 1038 01:07:25,730 --> 01:07:29,640 Here, I'm showing you the complex number in polar form. 1039 01:07:29,640 --> 01:07:34,370 It's got a magnitude and an angle, and I'd like you to 1040 01:07:34,370 --> 01:07:38,210 figure how what must the magnitude have been, and what 1041 01:07:38,210 --> 01:07:39,997 must the angle have been. 1042 01:08:40,140 --> 01:08:43,290 So what's the utility? 1043 01:08:43,290 --> 01:08:46,553 Why did I tell you the pole in terms of it's magnitude and 1044 01:08:46,553 --> 01:08:51,310 angle rather than telling it to you in it's Cartesian form 1045 01:08:51,310 --> 01:08:54,100 as a real and imaginary part? 1046 01:08:54,100 --> 01:08:58,266 What's good about thinking about magnitude and angle? 1047 01:09:02,577 --> 01:09:06,140 Or if I break the pole into magnitude and angle, and if I 1048 01:09:06,140 --> 01:09:07,390 think about the mode-- 1049 01:09:10,370 --> 01:09:14,870 the modes are always of the form P0 to the n. 1050 01:09:14,870 --> 01:09:20,010 If I think about modes as having a magnitude and an 1051 01:09:20,010 --> 01:09:25,680 angle, when I raise it to the n, something 1052 01:09:25,680 --> 01:09:29,090 very special happens. 1053 01:09:29,090 --> 01:09:30,340 What happens? 1054 01:09:32,830 --> 01:09:35,420 You can separate it. 1055 01:09:35,420 --> 01:09:39,250 What is R e to the j omega raised to the n? 1056 01:09:39,250 --> 01:09:49,260 That's the same as R to the n, e to the j n omega. 1057 01:09:49,260 --> 01:09:55,040 It's the product of a real thing times a very simple 1058 01:09:55,040 --> 01:09:56,950 complex thing. 1059 01:09:56,950 --> 01:09:58,660 What's simple about the complex thing? 1060 01:10:01,330 --> 01:10:05,050 The magnitude is everywhere 1 e to the j, the 1061 01:10:05,050 --> 01:10:08,730 magnitude of that term. 1062 01:10:08,730 --> 01:10:11,180 So all of the magnitude is here, none of the 1063 01:10:11,180 --> 01:10:12,210 magnitude is here. 1064 01:10:12,210 --> 01:10:14,920 All of the angle is here, none of the angle is here. 1065 01:10:14,920 --> 01:10:18,270 I've separated the magnitude and the angle. 1066 01:10:18,270 --> 01:10:23,300 So it's very insightful to think about poles in terms of 1067 01:10:23,300 --> 01:10:28,910 magnitude and angle because it decouples the 1068 01:10:28,910 --> 01:10:31,080 parts of the mode. 1069 01:10:31,080 --> 01:10:34,050 So I can think, then, of this complicated signal that I gave 1070 01:10:34,050 --> 01:10:39,350 you as being the product of a magnitude part and a pure 1071 01:10:39,350 --> 01:10:41,180 angle part. 1072 01:10:41,180 --> 01:10:48,180 From the magnitude part, I can infer something about R. R is 1073 01:10:48,180 --> 01:10:54,480 the ratio of the nth 1 to the n minus 1-th one, so R is a 1074 01:10:54,480 --> 01:10:57,120 lot bigger than 1/2. 1075 01:10:57,120 --> 01:11:04,390 In fact, R in this case is 0.97. 1076 01:11:04,390 --> 01:11:06,930 And this is pure angle. 1077 01:11:06,930 --> 01:11:11,600 This lets me infer something about the oscillations. 1078 01:11:11,600 --> 01:11:13,530 In fact, I can say something about the period. 1079 01:11:13,530 --> 01:11:17,130 The period is, here's a peak, here's a peak, 1, 2, 3, 4, 5, 1080 01:11:17,130 --> 01:11:18,380 6, 7, 8, 9, 10, 11, 12. 1081 01:11:21,460 --> 01:11:24,270 The period's 12. 1082 01:11:24,270 --> 01:11:26,410 If the period is 12, what is omega? 1083 01:11:42,330 --> 01:11:47,350 Omega is a number, such that by the time I got up to 12 1084 01:11:47,350 --> 01:11:51,860 times it, I got up to 2 pi. 1085 01:11:55,420 --> 01:11:56,070 What's omega? 1086 01:11:56,070 --> 01:11:58,290 AUDIENCE: Pi over 6. 1087 01:11:58,290 --> 01:11:59,540 PROFESSOR: 2 pi over 12. 1088 01:12:02,180 --> 01:12:03,430 So it's about 1/2. 1089 01:12:05,510 --> 01:12:10,710 So that's a way that I can infer the form of the answer. 1090 01:12:10,710 --> 01:12:15,660 I ask you what pole corresponds to this behavior. 1091 01:12:15,660 --> 01:12:18,570 Well, the decay that comes from the real part, that comes 1092 01:12:18,570 --> 01:12:21,580 from R to the n. 1093 01:12:21,580 --> 01:12:26,100 The oscillation, that comes from the imaginary part, and I 1094 01:12:26,100 --> 01:12:28,400 can figure that out by thinking about the period and 1095 01:12:28,400 --> 01:12:29,650 thinking about that relationship. 1096 01:12:40,260 --> 01:12:43,720 So the answer, then, is this one, R is between 1/2 and 1, 1097 01:12:43,720 --> 01:12:47,040 and omega is about 1/2. 1098 01:12:47,040 --> 01:12:50,150 OK, one last-- whoops, wrong button. 1099 01:12:50,150 --> 01:12:51,400 One last example. 1100 01:12:54,330 --> 01:12:57,110 So what we've seen is a very powerful way of decomposing 1101 01:12:57,110 --> 01:13:01,650 systems, so that we can always think about them in terms of 1102 01:13:01,650 --> 01:13:06,420 poles and complex geometrics, which we will call modes, 1103 01:13:06,420 --> 01:13:09,070 poles and modes. 1104 01:13:09,070 --> 01:13:12,530 And that behavior works for any system in an enormous 1105 01:13:12,530 --> 01:13:16,080 class of systems, so I want to think about one last one, 1106 01:13:16,080 --> 01:13:17,620 which is Fibonacci sequence. 1107 01:13:17,620 --> 01:13:19,080 You all know the Fibonacci sequence. 1108 01:13:19,080 --> 01:13:20,260 You've all programmed it. 1109 01:13:20,260 --> 01:13:22,640 We started with that when we were doing Python, and we made 1110 01:13:22,640 --> 01:13:24,800 some illustrations about the recursion and all that sort of 1111 01:13:24,800 --> 01:13:26,120 thing by thinking about it. 1112 01:13:26,120 --> 01:13:27,190 Now, I'm going to use signals and 1113 01:13:27,190 --> 01:13:30,270 systems do the same problem. 1114 01:13:30,270 --> 01:13:33,370 So Fibonacci was interested in population growth. 1115 01:13:33,370 --> 01:13:36,720 How many pairs of rabbits can be produced from a single pair 1116 01:13:36,720 --> 01:13:40,140 in a year if it is suppose that every month each pair 1117 01:13:40,140 --> 01:13:44,400 begets a new pair, from which the second month it becomes 1118 01:13:44,400 --> 01:13:45,040 productive? 1119 01:13:45,040 --> 01:13:49,500 OK, it's not quite the same English I would have used. 1120 01:13:49,500 --> 01:13:54,470 From this statement, you can infer a difference equation. 1121 01:13:54,470 --> 01:13:56,910 I've written it in terms of X. What do you think X is? 1122 01:14:00,700 --> 01:14:05,740 X is the input signal, and here, I'm thinking about X as 1123 01:14:05,740 --> 01:14:09,960 something that's specifies the initial condition. 1124 01:14:09,960 --> 01:14:12,170 This is the thing I alluded to earlier. 1125 01:14:12,170 --> 01:14:16,780 One trick that we use to make it easy to think about initial 1126 01:14:16,780 --> 01:14:21,660 conditions is that we embed them in the input. 1127 01:14:21,660 --> 01:14:24,540 So in this particular case, I'll think about the initial 1128 01:14:24,540 --> 01:14:28,060 condition arising from a delta function. 1129 01:14:28,060 --> 01:14:33,630 If I think about X as a delta, then the sequence of results 1130 01:14:33,630 --> 01:14:37,590 Y0, Y1, Y2, Y3 from this difference equation, is the 1131 01:14:37,590 --> 01:14:39,860 conventional Fibonacci sequence. 1132 01:14:39,860 --> 01:14:44,590 It would correspond to what if you had a baby rabbit, one 1133 01:14:44,590 --> 01:14:48,330 baby rabbit, that's the one, in generation 1134 01:14:48,330 --> 01:14:52,030 0, that's the delta. 1135 01:14:52,030 --> 01:14:54,960 So the input is a way that I can specify initial 1136 01:14:54,960 --> 01:14:57,530 conditions, and that's a very powerful way of thinking about 1137 01:14:57,530 --> 01:14:59,100 initial conditions. 1138 01:14:59,100 --> 01:15:00,350 So here's the problem. 1139 01:15:00,350 --> 01:15:04,370 I've got one set of baby rabbits at times 0. 1140 01:15:04,370 --> 01:15:07,370 They grow up. 1141 01:15:07,370 --> 01:15:13,750 They have baby rabbits, which grow up at the same time the 1142 01:15:13,750 --> 01:15:19,980 parents had more baby rabbits, at which point more babies 1143 01:15:19,980 --> 01:15:22,110 grow into bigger rabbits. 1144 01:15:22,110 --> 01:15:26,460 And big rabbits have more babies, et cetera, et cetera, 1145 01:15:26,460 --> 01:15:28,640 et cetera, et cetera, et cetera. 1146 01:15:33,520 --> 01:15:34,850 So you all know that. 1147 01:15:34,850 --> 01:15:36,710 You all know about Fibonacci sequence. 1148 01:15:36,710 --> 01:15:38,390 It blows up very quickly. 1149 01:15:38,390 --> 01:15:40,945 What are the poles of the Fibonacci sequence? 1150 01:15:49,920 --> 01:15:51,970 The difference equation looks just like the difference 1151 01:15:51,970 --> 01:15:53,560 equations we've looked at throughout 1152 01:15:53,560 --> 01:15:56,470 this hour and a half. 1153 01:15:56,470 --> 01:16:00,320 We can do it just like we did all the other problems. 1154 01:16:00,320 --> 01:16:04,260 We write the difference equation in terms of R. We 1155 01:16:04,260 --> 01:16:08,340 rewrite the system functional in terms of a ratio of two 1156 01:16:08,340 --> 01:16:13,180 polynomials in R. We substitute R goes to 1 over Z, 1157 01:16:13,180 --> 01:16:18,340 deduce a ratio of polynomials in Z, factor the denominator, 1158 01:16:18,340 --> 01:16:20,810 find the roots of the denominator, and find that 1159 01:16:20,810 --> 01:16:24,050 there are two poles. 1160 01:16:24,050 --> 01:16:27,830 The poles for the Fibonacci sequence are plus or minus the 1161 01:16:27,830 --> 01:16:30,400 root of 5 over 2. 1162 01:16:30,400 --> 01:16:31,290 That's curious. 1163 01:16:31,290 --> 01:16:33,330 There's no recursion there. 1164 01:16:33,330 --> 01:16:36,310 It's a different way of thinking about things. 1165 01:16:36,310 --> 01:16:38,920 There are two poles. 1166 01:16:38,920 --> 01:16:41,300 The first pole, the plus one -- 1167 01:16:41,300 --> 01:16:46,540 1 plus the root of 5 over 1, corresponds to a pole whose 1168 01:16:46,540 --> 01:16:48,180 magnitude is bigger than 1. 1169 01:16:48,180 --> 01:16:49,710 It explodes. 1170 01:16:49,710 --> 01:16:51,970 There it is. 1171 01:16:51,970 --> 01:16:55,170 The second one is a negative number. 1172 01:16:55,170 --> 01:17:00,770 So the first one is the golden ratio, 1.618... 1173 01:17:00,770 --> 01:17:03,070 The second one is the negative reciprocal of the golden 1174 01:17:03,070 --> 01:17:08,590 ratio, which is -0.618... 1175 01:17:08,590 --> 01:17:14,710 And those two numbers, amazingly, conspire so that 1176 01:17:14,710 --> 01:17:21,240 their sum, horrendous as they are, is an integer. 1177 01:17:21,240 --> 01:17:25,330 And in fact, that's the integer that we computed here. 1178 01:17:25,330 --> 01:17:28,210 So we've used Fibonacci before to think about the way you 1179 01:17:28,210 --> 01:17:33,190 structure programs, iteration, recursion, that sort of thing. 1180 01:17:33,190 --> 01:17:35,030 Here, by thinking about signals and systems, we can 1181 01:17:35,030 --> 01:17:40,370 think about exactly the same problem as poles. 1182 01:17:40,370 --> 01:17:42,190 There's no complexity in this problem. 1183 01:17:42,190 --> 01:17:44,780 It doesn't take N or N squared or N log N 1184 01:17:44,780 --> 01:17:46,190 or anything to compute. 1185 01:17:46,190 --> 01:17:48,470 It's closed form. 1186 01:17:48,470 --> 01:17:53,100 The answer is (pole one) to the N plus (pole two) to the 1187 01:17:53,100 --> 01:17:54,400 N. That's it. 1188 01:17:54,400 --> 01:17:55,500 That's the answer. 1189 01:17:55,500 --> 01:17:57,810 So what we've done is we found a whole new way of thinking 1190 01:17:57,810 --> 01:18:02,160 about the Fibonacci sequence in terms of poles, and more 1191 01:18:02,160 --> 01:18:05,380 than that, we found that that way of thinking about poles 1192 01:18:05,380 --> 01:18:10,150 works for any difference equation of this type. 1193 01:18:10,150 --> 01:18:14,160 And we found that poles, this way of thinking about systems 1194 01:18:14,160 --> 01:18:18,190 in terms of polynomials, is a powerful abstraction that's 1195 01:18:18,190 --> 01:18:21,110 exactly the same kind of PCAP abstraction 1196 01:18:21,110 --> 01:18:22,360 that we used for Python.