1 00:00:00,090 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,720 continue to offer high quality educational resources for free. 5 00:00:10,720 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,280 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,280 --> 00:00:18,450 at ocw.mit.edu. 8 00:00:21,029 --> 00:00:22,445 DENNIS FREEMAN: Hello and welcome. 9 00:00:27,730 --> 00:00:29,540 So as I said last time, the important part 10 00:00:29,540 --> 00:00:33,510 of doing this class is doing the homework. 11 00:00:33,510 --> 00:00:35,820 We have a relatively complicated new system 12 00:00:35,820 --> 00:00:38,130 for how to do the homework, so I apologize if there's 13 00:00:38,130 --> 00:00:39,150 a couple of kinks in it. 14 00:00:39,150 --> 00:00:42,060 I appreciate the feedback pointing out the kinks. 15 00:00:42,060 --> 00:00:43,140 We will try to fix them. 16 00:00:43,140 --> 00:00:47,970 And we won't hold anybody responsible for our kinks, 17 00:00:47,970 --> 00:00:49,620 just so you know. 18 00:00:49,620 --> 00:00:54,340 So the kinks include there's two parts. 19 00:00:54,340 --> 00:00:57,840 There's the tutor problems that are automatically graded. 20 00:00:57,840 --> 00:01:01,500 And if you think the tutor didn't give you a just answer, 21 00:01:01,500 --> 00:01:03,870 please just let us know and we will 22 00:01:03,870 --> 00:01:06,120 take care of it, provided of course 23 00:01:06,120 --> 00:01:10,090 that your answer is just. 24 00:01:10,090 --> 00:01:14,360 And for the problems that will be graded by a human, 25 00:01:14,360 --> 00:01:17,660 we would like you to read the solution so that you understand 26 00:01:17,660 --> 00:01:24,620 what went wrong, if anything, in your solution and perhaps 27 00:01:24,620 --> 00:01:25,800 an alternative approach. 28 00:01:25,800 --> 00:01:27,680 Most of the engineering design problems 29 00:01:27,680 --> 00:01:31,760 will be solvable in multiple ways. 30 00:01:31,760 --> 00:01:34,840 So, it's good for you to see other approaches. 31 00:01:34,840 --> 00:01:36,340 To encourage you to do that, we're 32 00:01:36,340 --> 00:01:40,960 asking you to identify any errors that were 33 00:01:40,960 --> 00:01:44,140 in your original submission. 34 00:01:44,140 --> 00:01:46,390 So the importance of that is that Homework One was due 35 00:01:46,390 --> 00:01:49,040 yesterday at 5:00 PM. 36 00:01:49,040 --> 00:01:53,600 And if you want to recoup points that you lost, 37 00:01:53,600 --> 00:01:58,760 you must submit marked up version of the same thing 38 00:01:58,760 --> 00:02:04,860 that you submitted yesterday by Friday 5 PM, by tomorrow 5 PM. 39 00:02:04,860 --> 00:02:08,590 You are welcome to mark up a photocopy of the paper 40 00:02:08,590 --> 00:02:11,440 and give us a paper mark-up. 41 00:02:11,440 --> 00:02:15,010 If you do that, it's due in at the beginning of recitation. 42 00:02:15,010 --> 00:02:17,950 If you want to scan it, you can send us a scan anytime up 43 00:02:17,950 --> 00:02:20,750 to 5:00 PM. 44 00:02:20,750 --> 00:02:21,270 Questions? 45 00:02:21,270 --> 00:02:21,874 Yes. 46 00:02:21,874 --> 00:02:23,540 AUDIENCE: Can I just email you the scan? 47 00:02:23,540 --> 00:02:24,331 DENNIS FREEMAN: No. 48 00:02:24,331 --> 00:02:25,610 Enter it into the web site. 49 00:02:25,610 --> 00:02:27,320 It's much easier for me. 50 00:02:27,320 --> 00:02:29,900 The graders will all be able to see it that way. 51 00:02:29,900 --> 00:02:33,012 If you send the email it's much harder. 52 00:02:33,012 --> 00:02:34,970 It's easier if we have everything in one place. 53 00:02:34,970 --> 00:02:36,469 If there's a problem with the tutor, 54 00:02:36,469 --> 00:02:40,340 then send it to me by email and we'll fix it. 55 00:02:40,340 --> 00:02:41,570 Other comments or questions? 56 00:02:44,965 --> 00:02:46,905 Yeah? 57 00:02:46,905 --> 00:02:51,280 AUDIENCE: [INAUDIBLE] Do you ever get them [INAUDIBLE]? 58 00:02:51,280 --> 00:02:53,580 DENNIS FREEMAN: Yes, eventually. 59 00:02:53,580 --> 00:02:56,550 Now, what hasn't been completely worked out yet is whether what 60 00:02:56,550 --> 00:03:02,120 you get back will be paper or [? bits. ?] Because 61 00:03:02,120 --> 00:03:05,271 we'll probably end up doing all the grading via [? bits, ?] 62 00:03:05,271 --> 00:03:07,270 because it's just easier to communicate with all 63 00:03:07,270 --> 00:03:10,300 the graders via [? bits. ?] So that's not been resolved yet. 64 00:03:10,300 --> 00:03:12,508 We're still going to work out with the graders what's 65 00:03:12,508 --> 00:03:16,060 the easiest way to synchronize grading the electronic 66 00:03:16,060 --> 00:03:17,710 and dealing with it. 67 00:03:17,710 --> 00:03:20,050 So you will get feedback but you won't necessarily 68 00:03:20,050 --> 00:03:21,490 get paper feedback. 69 00:03:24,115 --> 00:03:24,740 Other comments? 70 00:03:28,380 --> 00:03:29,170 OK. 71 00:03:29,170 --> 00:03:30,580 So last time, we started to think 72 00:03:30,580 --> 00:03:35,495 about how to think about discrete time systems. 73 00:03:35,495 --> 00:03:37,620 The thing you were supposed to get out of last time 74 00:03:37,620 --> 00:03:39,480 is that there's different representations 75 00:03:39,480 --> 00:03:40,710 for discrete time systems. 76 00:03:40,710 --> 00:03:43,740 We're interested in all of them. 77 00:03:43,740 --> 00:03:46,200 We're interested in all of them because of the strengths 78 00:03:46,200 --> 00:03:47,820 and weaknesses of each. 79 00:03:47,820 --> 00:03:50,770 We looked particularly at difference equations, 80 00:03:50,770 --> 00:03:54,270 which are mathematically precise and concise. 81 00:03:54,270 --> 00:03:56,180 That's very good. 82 00:03:56,180 --> 00:03:57,930 We looked at block diagrams, that was good 83 00:03:57,930 --> 00:04:02,100 because we could trace the flow of information 84 00:04:02,100 --> 00:04:05,270 through the network. 85 00:04:05,270 --> 00:04:07,670 And we looked at operator representations 86 00:04:07,670 --> 00:04:08,600 and they were nice. 87 00:04:08,600 --> 00:04:12,170 And we will see further today how that's nice. 88 00:04:12,170 --> 00:04:13,670 They are nice because they allow us 89 00:04:13,670 --> 00:04:19,230 to think about a system in terms of polynomial mathematics. 90 00:04:19,230 --> 00:04:21,390 So what I want to do today primarily 91 00:04:21,390 --> 00:04:22,945 is develop that latter theme. 92 00:04:25,950 --> 00:04:30,300 Keep in mind that one of the important features of the block 93 00:04:30,300 --> 00:04:32,470 diagram was the arrow notation. 94 00:04:32,470 --> 00:04:34,020 It told you what causes what. 95 00:04:34,020 --> 00:04:36,885 It let you make a hardware realization of a system. 96 00:04:39,620 --> 00:04:42,290 An important point from last time, which we just ended on 97 00:04:42,290 --> 00:04:45,580 at the end of the hour, an important point 98 00:04:45,580 --> 00:04:49,270 was that when you're thinking about the signals, 99 00:04:49,270 --> 00:04:51,190 there's a higher level abstraction that 100 00:04:51,190 --> 00:04:53,980 is also important. 101 00:04:53,980 --> 00:04:56,830 In addition to knowing which is the input to the R operator 102 00:04:56,830 --> 00:04:59,290 and which is the output of the R operator, 103 00:04:59,290 --> 00:05:01,120 there are some kinds of systems that 104 00:05:01,120 --> 00:05:03,610 just behave categorically different from other kinds 105 00:05:03,610 --> 00:05:04,540 of systems. 106 00:05:04,540 --> 00:05:06,580 And they are indicated here. 107 00:05:06,580 --> 00:05:09,400 This system has the property that 108 00:05:09,400 --> 00:05:14,320 is easy to write a recipe for how to compute y given x. 109 00:05:14,320 --> 00:05:15,670 How do you compute y given x? 110 00:05:15,670 --> 00:05:21,700 Well, you take x, copy it, invert it, 111 00:05:21,700 --> 00:05:23,470 shift it to the right. 112 00:05:23,470 --> 00:05:27,580 Add the copy to the inverted shifted. 113 00:05:27,580 --> 00:05:29,560 Easy to make a recipe to do that. 114 00:05:29,560 --> 00:05:31,780 It's not so easy to make a recipe here. 115 00:05:35,610 --> 00:05:36,110 Right? 116 00:05:36,110 --> 00:05:38,750 So there, so if you think about how 117 00:05:38,750 --> 00:05:42,770 you would try to write that in an operator, you could say, y. 118 00:05:42,770 --> 00:05:43,640 What is y? 119 00:05:43,640 --> 00:05:47,160 y is going to be x plus a right-shift of y. 120 00:05:50,170 --> 00:05:53,920 So you could say, well that's 1 minus a right-shift of y equals 121 00:05:53,920 --> 00:05:54,850 x. 122 00:05:54,850 --> 00:05:59,380 That's not a recipe for computing y from x. 123 00:05:59,380 --> 00:06:03,330 That's a recipe for computing x from y. 124 00:06:03,330 --> 00:06:05,487 But we don't know y. 125 00:06:05,487 --> 00:06:06,070 Everybody see? 126 00:06:06,070 --> 00:06:08,420 That's completely different. 127 00:06:08,420 --> 00:06:11,090 And so it's that difference that makes 128 00:06:11,090 --> 00:06:13,130 this kind of a network, this kind 129 00:06:13,130 --> 00:06:18,770 of a system interesting by comparison to that one. 130 00:06:18,770 --> 00:06:23,510 And the distinguishing feature that makes that behavior occur 131 00:06:23,510 --> 00:06:26,600 is feedback. 132 00:06:26,600 --> 00:06:30,230 When you have feedback, feedback means, 133 00:06:30,230 --> 00:06:32,600 for some signal in the system, we 134 00:06:32,600 --> 00:06:34,160 would say the system has feedback 135 00:06:34,160 --> 00:06:38,600 if there is some signal somewhere whose output at time 136 00:06:38,600 --> 00:06:44,780 n depends on it's same, the output of that same signal 137 00:06:44,780 --> 00:06:47,810 at a previous time. 138 00:06:47,810 --> 00:06:52,040 If that's true for some signal in the system, 139 00:06:52,040 --> 00:06:56,240 then we say the system has feedback. 140 00:06:56,240 --> 00:07:00,150 The implications of feedback are profound. 141 00:07:00,150 --> 00:07:03,950 They mean, for example, that a transient input can give rise 142 00:07:03,950 --> 00:07:05,735 to a persisting output. 143 00:07:05,735 --> 00:07:07,610 And we took a look at how to think about that 144 00:07:07,610 --> 00:07:09,920 in terms of the block diagrams. 145 00:07:09,920 --> 00:07:13,190 You can see that if you had even a signal that only lasts 146 00:07:13,190 --> 00:07:16,430 for one sample, a signal-- 147 00:07:16,430 --> 00:07:19,250 a system that has this property of feedback 148 00:07:19,250 --> 00:07:24,050 could generate a response that lasts for a very long time. 149 00:07:24,050 --> 00:07:28,000 In fact, the response at time 0, the only way that could happen 150 00:07:28,000 --> 00:07:29,770 is through the straight through path. 151 00:07:29,770 --> 00:07:32,560 The only way you could get an output at time 0, given 152 00:07:32,560 --> 00:07:35,700 that the input is non-0 only at 0 is through the straight 153 00:07:35,700 --> 00:07:37,240 through path. 154 00:07:37,240 --> 00:07:39,550 But then you could get an output at time 1 155 00:07:39,550 --> 00:07:42,170 if you cycled around once. 156 00:07:42,170 --> 00:07:44,570 This is the idea that a signal somewhere depends 157 00:07:44,570 --> 00:07:48,160 on a prior value of itself. 158 00:07:48,160 --> 00:07:50,940 And you could get a response at time 2 159 00:07:50,940 --> 00:07:53,940 by going around the loop twice, three, 3 times, 4, 160 00:07:53,940 --> 00:07:55,970 four times, et cetera. 161 00:07:55,970 --> 00:08:00,300 And in general, the response could persist forever. 162 00:08:00,300 --> 00:08:02,220 That's such an interesting behavior. 163 00:08:02,220 --> 00:08:04,950 It's such a fundamental property of these systems 164 00:08:04,950 --> 00:08:07,320 that we want to characterize systems 165 00:08:07,320 --> 00:08:10,550 by thinking about them that way. 166 00:08:10,550 --> 00:08:14,410 So in this particular case, if we 167 00:08:14,410 --> 00:08:16,570 think about sort of the most simple form 168 00:08:16,570 --> 00:08:19,060 that feedback could take, this is 169 00:08:19,060 --> 00:08:22,420 sort of the simplest way you could imagine. 170 00:08:22,420 --> 00:08:23,950 This is kind of the simplest system 171 00:08:23,950 --> 00:08:25,658 that you could imagine that has feedback. 172 00:08:28,310 --> 00:08:31,900 And you can see that the kinds of responses that you can get 173 00:08:31,900 --> 00:08:34,720 depend on this number p-naught. 174 00:08:34,720 --> 00:08:38,080 And that p-naught characterizes the family of responses 175 00:08:38,080 --> 00:08:39,919 that you could get. 176 00:08:39,919 --> 00:08:44,900 So for example, if p-naught were a number less than 1, 177 00:08:44,900 --> 00:08:49,250 you could get a decaying geometric sequence. 178 00:08:49,250 --> 00:08:53,060 If p-naught were 1 exactly, you would get a response 179 00:08:53,060 --> 00:08:56,870 that never decayed or grew. 180 00:08:56,870 --> 00:08:59,360 And if p-naught were a number bigger than 1, 181 00:08:59,360 --> 00:09:01,557 then the response could grow infinitely. 182 00:09:01,557 --> 00:09:03,390 So that's a very fundamental characteristic. 183 00:09:03,390 --> 00:09:07,711 And we call that number p-naught the pole. 184 00:09:07,711 --> 00:09:08,210 OK. 185 00:09:08,210 --> 00:09:09,376 Everybody's happy with that? 186 00:09:09,376 --> 00:09:11,170 That's very simple-minded. 187 00:09:11,170 --> 00:09:14,800 So we're going to characterize-- so the point is, 188 00:09:14,800 --> 00:09:19,690 there's a higher level of structure in systems 189 00:09:19,690 --> 00:09:20,940 that we would like to capture. 190 00:09:20,940 --> 00:09:23,550 We're going to call that higher level feedback. 191 00:09:23,550 --> 00:09:26,640 It means there's a cycle in the block diagram. 192 00:09:26,640 --> 00:09:31,140 And it means that you can get characteristically different 193 00:09:31,140 --> 00:09:35,370 kinds of responses then result when you don't have feedback. 194 00:09:35,370 --> 00:09:37,350 In particular, simple systems will 195 00:09:37,350 --> 00:09:39,660 have this behavior of a geometric sequence 196 00:09:39,660 --> 00:09:41,380 characterized by a pole. 197 00:09:41,380 --> 00:09:43,800 OK, just to make sure that everybody 198 00:09:43,800 --> 00:09:48,480 is with me, how many of the following unit-sample responses 199 00:09:48,480 --> 00:09:50,247 can be represented by a single pole? 200 00:09:52,840 --> 00:09:53,340 OK. 201 00:09:53,340 --> 00:09:56,340 Before you answer that question, talk to your neighbor. 202 00:09:56,340 --> 00:09:58,590 Don't even bother trying to think about the question. 203 00:09:58,590 --> 00:10:00,300 Look at your neighbor. 204 00:10:00,300 --> 00:10:01,710 Say hi. 205 00:10:01,710 --> 00:10:03,837 And then try to answer this question. 206 00:10:03,837 --> 00:10:05,825 [AUDIENCE CHATTER] 207 00:11:26,409 --> 00:11:27,200 DENNIS FREEMAN: OK. 208 00:11:27,200 --> 00:11:27,950 What's the answer? 209 00:11:27,950 --> 00:11:29,150 Everybody raise your hand. 210 00:11:29,150 --> 00:11:31,310 Raise some number of fingers equal to the answer 211 00:11:31,310 --> 00:11:34,244 for how many of those responses could result from-- 212 00:11:34,244 --> 00:11:35,660 how many of those systems could be 213 00:11:35,660 --> 00:11:38,100 represented by a single pole? 214 00:11:41,990 --> 00:11:42,490 OK. 215 00:11:42,490 --> 00:11:43,870 A little variety. 216 00:11:43,870 --> 00:11:47,430 There's about 80% correct. 217 00:11:47,430 --> 00:11:50,184 Tell me a simple one. 218 00:11:50,184 --> 00:11:51,600 So tell me one that can definitely 219 00:11:51,600 --> 00:11:53,111 be represented by a pole. 220 00:11:53,111 --> 00:11:53,610 OK. 221 00:11:53,610 --> 00:11:54,757 Upper right. 222 00:11:54,757 --> 00:11:57,090 Because it looks just like one that I showed you before. 223 00:11:57,090 --> 00:11:58,965 What's another simple one that can definitely 224 00:11:58,965 --> 00:12:01,860 be represented by a pole? 225 00:12:01,860 --> 00:12:02,385 Excuse me? 226 00:12:02,385 --> 00:12:03,510 AUDIENCE: Right underneath. 227 00:12:03,510 --> 00:12:04,170 DENNIS FREEMAN: Underneath. 228 00:12:04,170 --> 00:12:06,690 OK, so these are things that were on the previous slide. 229 00:12:06,690 --> 00:12:08,910 That should be pretty easy. 230 00:12:08,910 --> 00:12:09,750 Any others? 231 00:12:09,750 --> 00:12:10,485 Yes? 232 00:12:10,485 --> 00:12:11,700 AUDIENCE: Top left. 233 00:12:11,700 --> 00:12:12,180 DENNIS FREEMAN: Top left. 234 00:12:12,180 --> 00:12:12,680 Why? 235 00:12:12,680 --> 00:12:15,216 There's none like that on the other side? 236 00:12:15,216 --> 00:12:16,692 AUDIENCE: Because it's oscillating. 237 00:12:16,692 --> 00:12:19,644 It's changing sides and it's pole is negative 238 00:12:19,644 --> 00:12:22,596 but because it's converging to 0, the pole, it's magnitude 239 00:12:22,596 --> 00:12:24,090 is less than 1. 240 00:12:24,090 --> 00:12:27,400 DENNIS FREEMAN: So what would be the pole for the upper left? 241 00:12:27,400 --> 00:12:29,732 AUDIENCE: Some p-naught-- so that 242 00:12:29,732 --> 00:12:33,259 is like negative 1 p-naught plus p-naught less than 0. 243 00:12:33,259 --> 00:12:35,550 DENNIS FREEMAN: So can you figure out what p-naught is? 244 00:12:35,550 --> 00:12:38,101 Back in the back? 245 00:12:38,101 --> 00:12:38,600 Any? 246 00:12:38,600 --> 00:12:40,960 And so what's p-naught? 247 00:12:40,960 --> 00:12:44,139 For the upper left, what is the value of p-naught? 248 00:12:44,139 --> 00:12:46,460 AUDIENCE: 1/2. 249 00:12:46,460 --> 00:12:47,590 DENNIS FREEMAN: Minus 1/2. 250 00:12:47,590 --> 00:12:48,090 OK. 251 00:12:48,090 --> 00:12:48,590 Maybe. 252 00:12:48,590 --> 00:12:49,470 Not really. 253 00:12:49,470 --> 00:12:50,030 But yeah. 254 00:12:50,030 --> 00:12:53,190 So if it were, if p-naught were minus 1/2, 255 00:12:53,190 --> 00:12:56,741 let's think about p-naught is minus 1/2, what would that do? 256 00:12:56,741 --> 00:12:57,240 OK. 257 00:12:57,240 --> 00:12:59,250 If the first answer were 1, then the second one 258 00:12:59,250 --> 00:13:00,390 should be minus 1/2? 259 00:13:00,390 --> 00:13:01,598 What should the third one be? 260 00:13:03,760 --> 00:13:04,450 Plus 1/4. 261 00:13:04,450 --> 00:13:06,890 What should the next one be? 262 00:13:06,890 --> 00:13:07,700 Minus 1/8. 263 00:13:07,700 --> 00:13:10,070 So it doesn't quite look like a minus 1/2. 264 00:13:10,070 --> 00:13:16,020 It looks more like minus 3/4. 265 00:13:16,020 --> 00:13:16,680 Yep. 266 00:13:16,680 --> 00:13:18,320 Having made the graph myself, I know 267 00:13:18,320 --> 00:13:20,660 that the answer was minus 0.8. 268 00:13:20,660 --> 00:13:22,320 OK. 269 00:13:22,320 --> 00:13:25,900 You wouldn't necessarily have any way of knowing that. 270 00:13:25,900 --> 00:13:28,140 OK so three of them are, how about this one? 271 00:13:33,804 --> 00:13:36,001 Yes? 272 00:13:36,001 --> 00:13:36,500 No? 273 00:13:39,790 --> 00:13:42,650 Completely uncommitted? 274 00:13:42,650 --> 00:13:44,210 My neighbor didn't know? 275 00:13:44,210 --> 00:13:45,390 My partner's an idiot? 276 00:13:49,890 --> 00:13:52,780 So this is geometric, isn't it? 277 00:13:52,780 --> 00:13:53,280 Right? 278 00:13:53,280 --> 00:13:55,785 The only thing you see there is like 1 and 1/2 and 1/4 279 00:13:55,785 --> 00:13:56,910 and stuff like that, right? 280 00:14:01,040 --> 00:14:02,030 Blank stares. 281 00:14:02,030 --> 00:14:03,306 Yes? 282 00:14:03,306 --> 00:14:05,258 AUDIENCE: So it's not [INAUDIBLE], 283 00:14:05,258 --> 00:14:10,989 so it's not diverting, so it can't be [INAUDIBLE] unit step. 284 00:14:10,989 --> 00:14:13,280 DENNIS FREEMAN: If it were responding to the unit step, 285 00:14:13,280 --> 00:14:14,850 that's absolutely true. 286 00:14:14,850 --> 00:14:18,640 So if I'm thinking about the unit-sample response, 287 00:14:18,640 --> 00:14:22,920 then the unit-sample response always looks like p-naught 288 00:14:22,920 --> 00:14:26,740 to the n, n bigger than or equal to 0. 289 00:14:30,030 --> 00:14:34,290 And there's no way you can choose a p-naught that 290 00:14:34,290 --> 00:14:37,030 fits that lower left hand. 291 00:14:37,030 --> 00:14:39,270 How about this one? 292 00:14:39,270 --> 00:14:40,090 Same problem. 293 00:14:40,090 --> 00:14:42,660 It's alternating with the same problem. 294 00:14:42,660 --> 00:14:43,650 Right? 295 00:14:43,650 --> 00:14:46,050 So the only response-- so you can't do those two. 296 00:14:46,050 --> 00:14:47,790 The answer was three. 297 00:14:47,790 --> 00:14:54,210 The only kinds of responses you can model with a single pole 298 00:14:54,210 --> 00:14:56,180 can be shown here. 299 00:14:59,450 --> 00:15:00,710 That's not quite true. 300 00:15:00,710 --> 00:15:02,030 Let me retract that slightly. 301 00:15:02,030 --> 00:15:05,540 Here are some-- and I will make this more complicated 302 00:15:05,540 --> 00:15:08,780 and about two pages of notes. 303 00:15:08,780 --> 00:15:14,040 Here's all the real values of p-naught and the associated 304 00:15:14,040 --> 00:15:15,140 behaviors. 305 00:15:15,140 --> 00:15:18,500 In two pages, we'll get to the complex values for p-naught. 306 00:15:18,500 --> 00:15:20,970 So here's all of that the responses 307 00:15:20,970 --> 00:15:28,840 that can be associated with a single real valued pole. 308 00:15:28,840 --> 00:15:31,210 So you can get, as somebody has already said, 309 00:15:31,210 --> 00:15:36,880 you can get the idea of diverging with alternation, 310 00:15:36,880 --> 00:15:41,140 converging with alternation, converging monotonically, 311 00:15:41,140 --> 00:15:42,600 diverging monotonically. 312 00:15:42,600 --> 00:15:45,431 Those are the behaviors you can get with a single real value 313 00:15:45,431 --> 00:15:45,930 pole. 314 00:15:48,810 --> 00:15:50,110 OK. 315 00:15:50,110 --> 00:15:52,710 That's what can happen with a single real valued pole 316 00:15:52,710 --> 00:15:56,327 but systems, there are more complicated systems. 317 00:15:56,327 --> 00:15:58,160 And when you have a more complicated system, 318 00:15:58,160 --> 00:16:00,110 you can get a more complicated response. 319 00:16:00,110 --> 00:16:01,940 And one example is shown here. 320 00:16:01,940 --> 00:16:03,930 So here a system. 321 00:16:03,930 --> 00:16:06,320 Here's the unit-sample response. 322 00:16:06,320 --> 00:16:10,910 Is this something that can be represented by a single pole? 323 00:16:10,910 --> 00:16:14,350 No, because it doesn't fall into one of those canonical shapes. 324 00:16:14,350 --> 00:16:15,080 Right. 325 00:16:15,080 --> 00:16:17,940 In fact, this response has the property 326 00:16:17,940 --> 00:16:21,670 that it grows and then decays. 327 00:16:21,670 --> 00:16:24,070 But it only does the grow and decay once, 328 00:16:24,070 --> 00:16:25,950 so it's just not one of the canonical forms. 329 00:16:25,950 --> 00:16:28,450 So now there's something different going on here 330 00:16:28,450 --> 00:16:30,380 and we'd like to understand what that is. 331 00:16:30,380 --> 00:16:34,280 So we cannot represent that by a single pole. 332 00:16:34,280 --> 00:16:38,890 However, this is where the polynomial stuff becomes useful 333 00:16:38,890 --> 00:16:41,200 because I can think about tricks I 334 00:16:41,200 --> 00:16:43,060 can do with the polynomials, which you all 335 00:16:43,060 --> 00:16:45,672 learned in high school so you all know how to do that. 336 00:16:45,672 --> 00:16:50,461 That we can think about the system in terms of, 337 00:16:50,461 --> 00:16:52,210 we can think about this complicated system 338 00:16:52,210 --> 00:16:56,140 in terms of simpler systems that can be represented by one pole. 339 00:16:56,140 --> 00:16:59,200 So for example, one of the more powerful things 340 00:16:59,200 --> 00:17:01,180 you can do with a polynomial is factor. 341 00:17:01,180 --> 00:17:04,599 If you factor this polynomial-- 342 00:17:04,599 --> 00:17:06,849 OK, so this is a simple representation for that right? 343 00:17:06,849 --> 00:17:10,520 I can just read that straight off. 344 00:17:10,520 --> 00:17:12,619 So I can make this representation just reading it 345 00:17:12,619 --> 00:17:14,510 straight off the block diagram. 346 00:17:14,510 --> 00:17:18,240 I can turn that into an operator type representation here. 347 00:17:18,240 --> 00:17:22,109 So I operate on y with this mass to get x. 348 00:17:22,109 --> 00:17:26,450 It's backwards feedback, that's what happens. 349 00:17:26,450 --> 00:17:28,560 But the second order polynomial in R 350 00:17:28,560 --> 00:17:32,850 can be factored into two first order polynomials in R. 351 00:17:32,850 --> 00:17:34,800 And not surprisingly, that then leads 352 00:17:34,800 --> 00:17:38,430 to a representation of the more complex system in terms 353 00:17:38,430 --> 00:17:40,900 of two simpler systems. 354 00:17:40,900 --> 00:17:44,460 So in fact, I can write the original system. 355 00:17:44,460 --> 00:17:47,550 If I factor it, there's an equivalent representation. 356 00:17:47,550 --> 00:17:52,390 Equivalent means provided that all the systems 357 00:17:52,390 --> 00:17:53,830 start initial rest. 358 00:17:53,830 --> 00:17:59,320 Initial rest means provided all the delays started out at 0. 359 00:17:59,320 --> 00:18:01,210 Putting in the normal caveats, there 360 00:18:01,210 --> 00:18:03,710 is an equivalent representation to the complex system. 361 00:18:03,710 --> 00:18:07,680 In fact, there's two equivalent systems shown here. 362 00:18:07,680 --> 00:18:10,500 Each corresponds to factoring. 363 00:18:10,500 --> 00:18:13,440 And it corresponds to doing which factor first. 364 00:18:13,440 --> 00:18:15,030 So in the first realization, it says 365 00:18:15,030 --> 00:18:17,610 that generate an intermediate signal 366 00:18:17,610 --> 00:18:21,204 by thinking about the factor that has a base of 0.7. 367 00:18:21,204 --> 00:18:25,650 Then cascade that with another system that 368 00:18:25,650 --> 00:18:31,210 has a base that looks just like a canonical first order system, 369 00:18:31,210 --> 00:18:33,670 except that the base now is 0.9. 370 00:18:33,670 --> 00:18:37,060 And presumably, if you cascade those two things, 371 00:18:37,060 --> 00:18:40,250 you will get an equivalent response. 372 00:18:40,250 --> 00:18:42,235 And the order didn't make any difference. 373 00:18:44,890 --> 00:18:48,550 So that's a way we can think about the response 374 00:18:48,550 --> 00:18:52,220 of a more complex system in terms of first order responses. 375 00:18:52,220 --> 00:18:55,430 So that's nice. 376 00:18:55,430 --> 00:18:58,020 And that gives us then-- 377 00:18:58,020 --> 00:19:03,560 so we can then use the idea of the Taylor series expansion 378 00:19:03,560 --> 00:19:06,980 to think about the response of each of those first order 379 00:19:06,980 --> 00:19:11,600 systems, so we can replace the response of the canonical first 380 00:19:11,600 --> 00:19:17,030 order system with the infinite feedforward realization-- 381 00:19:17,030 --> 00:19:21,110 remember from last time, the first order feedback system can 382 00:19:21,110 --> 00:19:24,120 be equivalently represented by an infinite order 383 00:19:24,120 --> 00:19:27,540 of feedforward system-- 384 00:19:27,540 --> 00:19:30,880 so that the unit-sample response here. 385 00:19:30,880 --> 00:19:34,800 So this system, excuse me, this first order feedback system 386 00:19:34,800 --> 00:19:40,350 can be rewritten as that infinite order feedforward 387 00:19:40,350 --> 00:19:41,280 system. 388 00:19:41,280 --> 00:19:43,740 This one can be written that way. 389 00:19:43,740 --> 00:19:48,210 And then again, we can use polynomial math 390 00:19:48,210 --> 00:19:52,210 to figure out the response of that more complicated system. 391 00:19:52,210 --> 00:19:55,500 All we need to do is multiply and then collect terms. 392 00:19:55,500 --> 00:19:57,510 You all know how to multiply polynomials, 393 00:19:57,510 --> 00:20:01,160 therefore you all know how to solve second order systems. 394 00:20:01,160 --> 00:20:01,660 OK. 395 00:20:01,660 --> 00:20:04,630 That's the advantage of converting one representation, 396 00:20:04,630 --> 00:20:07,112 a system representation, into a polynomial representation 397 00:20:07,112 --> 00:20:09,070 because you already know all about polynomials. 398 00:20:09,070 --> 00:20:11,440 Since you already know how to multiply polynomials, 399 00:20:11,440 --> 00:20:13,420 you already know how to cascade systems. 400 00:20:17,150 --> 00:20:22,450 The only difficulty here is bookkeeping, 401 00:20:22,450 --> 00:20:23,560 not missing a term. 402 00:20:23,560 --> 00:20:25,900 So there are graphic aids to that. 403 00:20:25,900 --> 00:20:27,670 Think about the block diagram that 404 00:20:27,670 --> 00:20:32,200 would correspond to the infinite feedforward realization 405 00:20:32,200 --> 00:20:37,910 of the left part and the infinite feedforward 406 00:20:37,910 --> 00:20:40,540 realization of the right part. 407 00:20:40,540 --> 00:20:45,290 So I'm thinking about this system and that system being 408 00:20:45,290 --> 00:20:46,940 represented by this infinite system 409 00:20:46,940 --> 00:20:48,740 and that infinite system. 410 00:20:48,740 --> 00:20:52,430 And now just making a block diagram realization of this 411 00:20:52,430 --> 00:20:54,920 and a separate block diagram realization of that 412 00:20:54,920 --> 00:20:56,365 and cascading them to get this. 413 00:20:59,540 --> 00:21:01,740 And now the bookkeeping is trivial. 414 00:21:01,740 --> 00:21:05,460 What's the only way you can get a response if the input is 415 00:21:05,460 --> 00:21:07,500 a unit-sample signal? 416 00:21:07,500 --> 00:21:10,440 What's the only way you can get a response at time n equals 0? 417 00:21:13,950 --> 00:21:16,740 What's the only signal path that generates a response of n 418 00:21:16,740 --> 00:21:17,250 equals 0? 419 00:21:22,940 --> 00:21:23,846 Say it louder? 420 00:21:23,846 --> 00:21:25,370 AUDIENCE: The top right left. 421 00:21:25,370 --> 00:21:26,870 DENNIS FREEMAN: The only signal path 422 00:21:26,870 --> 00:21:29,600 is to go through here, because if I go through here, I 423 00:21:29,600 --> 00:21:30,470 introduce a delay. 424 00:21:30,470 --> 00:21:31,970 This introduces even more delay. 425 00:21:31,970 --> 00:21:33,639 This introduces even more delay. 426 00:21:33,639 --> 00:21:35,180 The only way I can get from the input 427 00:21:35,180 --> 00:21:39,260 to the output with no delay is to go through the one 428 00:21:39,260 --> 00:21:42,320 here and go through the one here. 429 00:21:42,320 --> 00:21:44,769 So that that's where that term comes from. 430 00:21:44,769 --> 00:21:46,310 The only way I can get from the input 431 00:21:46,310 --> 00:21:51,000 to the output with no delay, no R, is through that top path. 432 00:21:51,000 --> 00:21:53,200 If I want to get one level of delay, 433 00:21:53,200 --> 00:21:56,250 the only way I can do that is to pick it up here and then not 434 00:21:56,250 --> 00:21:57,690 pick it up there or vice versa. 435 00:22:00,550 --> 00:22:05,050 There's more ways that I could get two delays. 436 00:22:05,050 --> 00:22:06,670 I can pick up two delays all at once 437 00:22:06,670 --> 00:22:10,390 on the left with none on the right or vice versa 438 00:22:10,390 --> 00:22:12,310 or one in each. 439 00:22:12,310 --> 00:22:14,260 That's just bookkeeping, right? 440 00:22:14,260 --> 00:22:15,760 It's just showing you how you can 441 00:22:15,760 --> 00:22:17,950 think about polynomial multiplication 442 00:22:17,950 --> 00:22:21,130 in different ways. 443 00:22:21,130 --> 00:22:25,589 Alternatively, you could represent it in a table. 444 00:22:25,589 --> 00:22:27,380 So if you think about multiplying these two 445 00:22:27,380 --> 00:22:30,200 polynomials where I've represented the first pole as a 446 00:22:30,200 --> 00:22:35,460 in the second pole as b, you can think 447 00:22:35,460 --> 00:22:38,940 about polynomial multiplication in a tabular form. 448 00:22:38,940 --> 00:22:41,610 So you have to think about the distributive property, 449 00:22:41,610 --> 00:22:43,630 the 1 distributes over all of these, 450 00:22:43,630 --> 00:22:45,570 then the a distributes over all, the aR 451 00:22:45,570 --> 00:22:48,000 distributes over all of these and the a-squared R-squared 452 00:22:48,000 --> 00:22:50,430 distributes over all of these. 453 00:22:50,430 --> 00:22:53,760 And you can represent each of those distribution operations 454 00:22:53,760 --> 00:22:55,320 by a row in the table. 455 00:22:55,320 --> 00:22:58,230 And then you can see that the only entries 456 00:22:58,230 --> 00:23:02,370 with no R, with R to the 0 is along that diagonal. 457 00:23:02,370 --> 00:23:04,790 And then the R terms are here, the R-squared terms, 458 00:23:04,790 --> 00:23:07,434 the R-cubed terms, et cetera. 459 00:23:07,434 --> 00:23:08,850 And you can see how that would all 460 00:23:08,850 --> 00:23:13,140 sum to give you something that's not a geometric. 461 00:23:13,140 --> 00:23:15,470 OK, all those things, just bookkeeping. 462 00:23:15,470 --> 00:23:16,040 Right? 463 00:23:16,040 --> 00:23:17,600 All I'm trying to say is that there is a way 464 00:23:17,600 --> 00:23:19,940 that you can think about the second order system which 465 00:23:19,940 --> 00:23:24,170 does not generate a simple geometric sequence by thinking 466 00:23:24,170 --> 00:23:27,290 about the cascade of two systems that each 467 00:23:27,290 --> 00:23:34,540 do generate a unit-sample response that is geometric. 468 00:23:34,540 --> 00:23:36,970 Nevertheless, all of this is bookkeeping. 469 00:23:36,970 --> 00:23:39,254 It's still the case that this is a complicated shape. 470 00:23:39,254 --> 00:23:41,170 And you still have to do a fair amount of work 471 00:23:41,170 --> 00:23:43,870 to figure out what that complicated shape is. 472 00:23:43,870 --> 00:23:46,840 We can use polynomials one more time 473 00:23:46,840 --> 00:23:48,490 to make it conceptually simpler. 474 00:23:51,940 --> 00:23:56,100 We can do something that we call partial fractions. 475 00:23:56,100 --> 00:23:58,050 You can take the expression that we 476 00:23:58,050 --> 00:24:01,450 found for this complicated system, 477 00:24:01,450 --> 00:24:04,520 factor it just like we did before. 478 00:24:04,520 --> 00:24:07,880 But now, according to the method of partial fractions, 479 00:24:07,880 --> 00:24:11,780 you can represent the reciprocal of a quadratic 480 00:24:11,780 --> 00:24:15,260 as the sum of two terms, each the reciprocal 481 00:24:15,260 --> 00:24:20,220 of a linear function in R. So you can always do that. 482 00:24:20,220 --> 00:24:24,750 That's a property of polynomials, right? 483 00:24:24,750 --> 00:24:27,750 So what that says is that there's an equivalent 484 00:24:27,750 --> 00:24:34,206 representation for this complicated system that 485 00:24:34,206 --> 00:24:37,670 looks like this. 486 00:24:37,670 --> 00:24:43,970 So backing up, so we showed that by factoring, we 487 00:24:43,970 --> 00:24:46,070 take this and turn it into this. 488 00:24:46,070 --> 00:24:49,460 By partial fractions we turn it into a sum difference. 489 00:24:49,460 --> 00:24:51,600 What's the difference? 490 00:24:51,600 --> 00:24:54,120 So now we think about that difference here. 491 00:24:56,730 --> 00:25:00,980 This term can be generated by multiplying 4.5 times 492 00:25:00,980 --> 00:25:03,385 a canonic first order term. 493 00:25:06,180 --> 00:25:10,380 This can be generated by multiplying minus 3.5 494 00:25:10,380 --> 00:25:13,610 times a canonic first order term. 495 00:25:13,610 --> 00:25:16,590 And the response adds. 496 00:25:16,590 --> 00:25:20,960 OK, well that's completely new insight. 497 00:25:20,960 --> 00:25:24,170 We've taken what looks like a very complicated response 498 00:25:24,170 --> 00:25:27,360 from a system that is not a canonical form. 499 00:25:27,360 --> 00:25:28,940 And we've turned it into something 500 00:25:28,940 --> 00:25:33,224 that does look like several canonical forms, 501 00:25:33,224 --> 00:25:34,640 just like we did with the cascade, 502 00:25:34,640 --> 00:25:37,940 but now we can see that the answer is 503 00:25:37,940 --> 00:25:40,100 the sum of two geometrics. 504 00:25:44,157 --> 00:25:44,990 Everybody's with it? 505 00:25:44,990 --> 00:25:50,270 So it's even better than being able to cascade, 506 00:25:50,270 --> 00:25:54,340 wherein the cascade, we have to do some complicated bookkeeping 507 00:25:54,340 --> 00:25:56,470 in order to figure out the responses. 508 00:25:56,470 --> 00:25:59,260 By thinking about it with partial fractions, 509 00:25:59,260 --> 00:26:02,621 it's the simple sum of two simple things. 510 00:26:02,621 --> 00:26:04,870 So then if you know properties of each individual one, 511 00:26:04,870 --> 00:26:07,120 life it rattles for a long time or it 512 00:26:07,120 --> 00:26:11,470 decays quickly, you can instantly say something 513 00:26:11,470 --> 00:26:14,670 about the combined response. 514 00:26:14,670 --> 00:26:17,150 Does that follow? 515 00:26:17,150 --> 00:26:20,540 So then this response, which I introduced 516 00:26:20,540 --> 00:26:23,090 by saying it's complicated, it's not really complicated. 517 00:26:23,090 --> 00:26:24,680 It's the sum of two geometrics. 518 00:26:28,320 --> 00:26:31,030 It goes up and then down simply because the decay constants 519 00:26:31,030 --> 00:26:33,570 for the two different geometrics are different, that's all. 520 00:26:36,680 --> 00:26:41,660 The point is that this is a very different kind 521 00:26:41,660 --> 00:26:43,820 of representation. 522 00:26:43,820 --> 00:26:46,670 Many things-- so you could do a lot 523 00:26:46,670 --> 00:26:49,400 by just taking a difference equation 524 00:26:49,400 --> 00:26:52,340 and just factoring it and moving terms around and figuring out 525 00:26:52,340 --> 00:26:57,170 different ways to do the sums and so forth to try 526 00:26:57,170 --> 00:26:58,924 to simplify your life. 527 00:26:58,924 --> 00:27:00,590 You can do the same with block diagrams. 528 00:27:00,590 --> 00:27:02,048 You could shuffle around the blocks 529 00:27:02,048 --> 00:27:04,130 and try to make things simpler. 530 00:27:04,130 --> 00:27:09,720 This idea of turning the top system into this 531 00:27:09,720 --> 00:27:13,320 is trivial to see with polynomial manipulations 532 00:27:13,320 --> 00:27:16,650 and relatively and not at all straightforward to do 533 00:27:16,650 --> 00:27:19,230 by simple block diagram manipulations. 534 00:27:19,230 --> 00:27:22,200 It follows directly from the polynomial realization. 535 00:27:22,200 --> 00:27:24,750 That's what we mean by a more powerful representation. 536 00:27:24,750 --> 00:27:28,150 It's very easy to see how we got there if you represent 537 00:27:28,150 --> 00:27:29,340 the system by a polynomial. 538 00:27:29,340 --> 00:27:32,006 It's much less easy to see how we got there 539 00:27:32,006 --> 00:27:33,630 by thinking about it as a block diagram 540 00:27:33,630 --> 00:27:36,500 or as a difference equation. 541 00:27:36,500 --> 00:27:37,436 OK? 542 00:27:37,436 --> 00:27:37,935 Makes sense? 543 00:27:42,560 --> 00:27:45,680 Because they're so important, we would like shorthand ways 544 00:27:45,680 --> 00:27:46,630 of finding poles. 545 00:27:49,092 --> 00:27:51,050 As soon as you know the poles of a system, then 546 00:27:51,050 --> 00:27:53,540 you know the geometric sequences that characterize 547 00:27:53,540 --> 00:27:55,610 unit-sample response. 548 00:27:55,610 --> 00:27:58,200 And for at least a broad class of systems, 549 00:27:58,200 --> 00:28:01,490 we're going to be able to think about the combined 550 00:28:01,490 --> 00:28:06,200 unit-sample responses of the system as some weighted sum 551 00:28:06,200 --> 00:28:10,750 of individual contributions. 552 00:28:10,750 --> 00:28:12,310 So that makes the idea of finding 553 00:28:12,310 --> 00:28:14,374 the pose very important. 554 00:28:14,374 --> 00:28:15,790 There's a very straightforward way 555 00:28:15,790 --> 00:28:17,960 that we've sort of already alluded to. 556 00:28:17,960 --> 00:28:19,360 If you had this complicated thing 557 00:28:19,360 --> 00:28:23,170 and you wanted to find the poles, what you should do 558 00:28:23,170 --> 00:28:25,930 is try to coerce it into the canonical form. 559 00:28:25,930 --> 00:28:32,500 You remember the canonic form for a single pole system 560 00:28:32,500 --> 00:28:34,790 was this sort of representation. 561 00:28:40,880 --> 00:28:42,360 So you have x coming in. 562 00:28:42,360 --> 00:28:45,780 Coming out as y and feeding around that way. 563 00:28:45,780 --> 00:28:50,880 That's the canonical form for one pole system. 564 00:28:50,880 --> 00:28:53,610 And that had a system function of the form y 565 00:28:53,610 --> 00:29:01,730 over x is 1 over 1 minus p-naught R. 566 00:29:01,730 --> 00:29:07,220 So if you can coerce all the parts to look like 1 over 1 567 00:29:07,220 --> 00:29:11,480 minus p-naught R then according to that, here's a pole 568 00:29:11,480 --> 00:29:14,210 and there's a pole just by canonic forms 569 00:29:14,210 --> 00:29:16,730 and matching up the terms in a canonic form, right? 570 00:29:20,570 --> 00:29:24,830 That requires some sort of graphic memory or something 571 00:29:24,830 --> 00:29:29,120 graphic ability to take one form and turn it into another form. 572 00:29:29,120 --> 00:29:33,680 A simpler rule is to say, look, just replace 573 00:29:33,680 --> 00:29:38,970 all the Rs by 1 over z, rewrite things, 574 00:29:38,970 --> 00:29:41,420 and the z's turn out to be-- 575 00:29:41,420 --> 00:29:43,850 the values of z that are the roots of the denominator 576 00:29:43,850 --> 00:29:46,320 turn out to be the values of the poles. 577 00:29:46,320 --> 00:29:48,230 That's very straightforward to see just 578 00:29:48,230 --> 00:29:50,360 by looking at the canonical form. 579 00:29:50,360 --> 00:29:51,890 If I have the canonical form where 580 00:29:51,890 --> 00:29:58,540 I know that the pole is p-naught and if I replaced the R by 1 581 00:29:58,540 --> 00:30:02,080 over z and cleared the fraction, I 582 00:30:02,080 --> 00:30:04,940 would end up with z over z minus p-naught. 583 00:30:04,940 --> 00:30:08,290 And now if I ask what's the poles at the bottom, what's 584 00:30:08,290 --> 00:30:09,670 the roots of the bottom? 585 00:30:09,670 --> 00:30:14,170 For what values of z is the denominator equal to 0? 586 00:30:14,170 --> 00:30:17,650 The answer is clearly z equals p-naught. 587 00:30:17,650 --> 00:30:23,840 It's just a trick to turn a system function in terms of R 588 00:30:23,840 --> 00:30:29,400 into a root-finding problem. 589 00:30:29,400 --> 00:30:31,510 So rather than having some program that 590 00:30:31,510 --> 00:30:33,940 knows how to write canonical forms, 591 00:30:33,940 --> 00:30:40,480 you can use any root finder to find the values of the poles. 592 00:30:40,480 --> 00:30:43,880 OK so make sure everybody is with me. 593 00:30:43,880 --> 00:30:48,260 Consider the following system, blah, blah, blah. 594 00:30:48,260 --> 00:30:50,690 How many of the following are true? 595 00:30:50,690 --> 00:30:53,860 Answer looks like it ought to be a number between 0 and 5 596 00:30:53,860 --> 00:30:54,920 probably. 597 00:30:54,920 --> 00:30:57,050 So talk to your neighbor and figure out 598 00:30:57,050 --> 00:30:58,450 a number between 0 and 5. 599 00:33:23,635 --> 00:33:25,260 So how many of the statements are true? 600 00:33:28,672 --> 00:33:29,172 Hands? 601 00:33:35,530 --> 00:33:37,300 OK, participation is a little small, 602 00:33:37,300 --> 00:33:42,654 but it's like 95, 93, something like that percent, 603 00:33:42,654 --> 00:33:44,820 except people keep changing their number of fingers. 604 00:33:44,820 --> 00:33:47,230 That makes it confusing. 605 00:33:47,230 --> 00:33:48,149 OK. 606 00:33:48,149 --> 00:33:49,690 How do you think about the first one? 607 00:33:49,690 --> 00:33:51,940 The unit-sample response converges to 0, yes or no? 608 00:33:51,940 --> 00:33:53,023 How do you think about it? 609 00:33:55,660 --> 00:33:58,980 Yes because they all do. 610 00:33:58,980 --> 00:34:00,960 True or false. 611 00:34:00,960 --> 00:34:01,460 Yes. 612 00:34:01,460 --> 00:34:05,310 Because there's something special about this one. 613 00:34:05,310 --> 00:34:07,797 What is special about this one that convinces you 614 00:34:07,797 --> 00:34:08,880 that that's true or false? 615 00:34:08,880 --> 00:34:09,643 Yes. 616 00:34:09,643 --> 00:34:11,499 AUDIENCE: It has two poles both of which 617 00:34:11,499 --> 00:34:12,681 are magnitude less than 1. 618 00:34:12,681 --> 00:34:14,639 DENNIS FREEMAN: It has two poles, both of which 619 00:34:14,639 --> 00:34:15,722 are magnitude less than 1. 620 00:34:15,722 --> 00:34:17,310 That's precisely right. 621 00:34:17,310 --> 00:34:21,409 So if we simply-- 622 00:34:21,409 --> 00:34:25,199 if we just do one blackboard full of math. 623 00:34:25,199 --> 00:34:26,670 So we take this. 624 00:34:26,670 --> 00:34:30,150 Turn it into a polynomial form. 625 00:34:30,150 --> 00:34:32,790 Turn it into a functional form. 626 00:34:32,790 --> 00:34:33,645 Turn it into z's. 627 00:34:36,960 --> 00:34:39,880 Everyone's happy with that? 628 00:34:39,880 --> 00:34:42,170 Now you may have made slight algebra errors 629 00:34:42,170 --> 00:34:45,690 in what you were doing, but the procedure is very easy, right? 630 00:34:45,690 --> 00:34:48,167 We see that we get two poles. 631 00:34:48,167 --> 00:34:49,000 Where are the poles? 632 00:34:51,739 --> 00:34:54,060 Negative 1/2 and plus 1/4. 633 00:34:54,060 --> 00:34:58,950 They're at the roots of the denominator. 634 00:34:58,950 --> 00:35:03,210 So is the unit-sample response converge to 0? 635 00:35:03,210 --> 00:35:05,000 It's the unit-sample response. 636 00:35:05,000 --> 00:35:06,870 Unit-sample responses are going to be 637 00:35:06,870 --> 00:35:11,940 the sum of two pieces from the partial fraction argument. 638 00:35:11,940 --> 00:35:15,937 And so it will depend on the positions of the poles. 639 00:35:15,937 --> 00:35:17,520 Both poles are inside the unit circle. 640 00:35:17,520 --> 00:35:20,050 Both poles have magnitude less than 1. 641 00:35:20,050 --> 00:35:23,000 Therefore, they converge, therefore the first one 642 00:35:23,000 --> 00:35:24,990 is right. 643 00:35:24,990 --> 00:35:30,310 Poles at a 1/2 and 1/4, obviously wrong, right? 644 00:35:30,310 --> 00:35:33,040 Obviously wrong, right? 645 00:35:33,040 --> 00:35:35,450 There are two poles. 646 00:35:35,450 --> 00:35:36,200 None of the above. 647 00:35:36,200 --> 00:35:36,870 Obviously wrong. 648 00:35:36,870 --> 00:35:37,370 OK. 649 00:35:37,370 --> 00:35:38,300 So the answer is 2. 650 00:35:41,252 --> 00:35:43,220 OK? 651 00:35:43,220 --> 00:35:45,200 OK. 652 00:35:45,200 --> 00:35:47,420 One more example problem. 653 00:35:47,420 --> 00:35:49,580 Favorite computer science problems 654 00:35:49,580 --> 00:35:51,160 is the Fibonacci problem. 655 00:35:51,160 --> 00:35:54,080 Fibonacci was thinking about population growth 656 00:35:54,080 --> 00:35:55,670 when he formulated the problem. 657 00:35:55,670 --> 00:36:00,470 The idea was what if you have a pair of rabbits that 658 00:36:00,470 --> 00:36:03,770 takes one cycle to grow up. 659 00:36:03,770 --> 00:36:08,780 And every cycle thereafter, has a pair of baby rabbits. 660 00:36:08,780 --> 00:36:12,950 Then the next cycle, there is a pair of baby rabbits. 661 00:36:12,950 --> 00:36:14,380 Then they grow up. 662 00:36:14,380 --> 00:36:18,620 But meanwhile mom and dad had another pair of rabbits. 663 00:36:18,620 --> 00:36:19,890 But then those grow up. 664 00:36:19,890 --> 00:36:21,120 And now we have more. 665 00:36:21,120 --> 00:36:22,060 And then they grow up. 666 00:36:22,060 --> 00:36:22,976 And then they grow up. 667 00:36:22,976 --> 00:36:24,360 And then they grow up. 668 00:36:24,360 --> 00:36:26,050 And you get lots of rabbits. 669 00:36:26,050 --> 00:36:26,550 OK. 670 00:36:26,550 --> 00:36:27,633 So that's the idea, right? 671 00:36:32,400 --> 00:36:33,549 Question? 672 00:36:33,549 --> 00:36:35,340 What are the poles of the Fibonacci system? 673 00:37:49,542 --> 00:37:51,534 AUDIENCE: The poles, why isn't that-- 674 00:37:51,534 --> 00:37:54,522 [INAUDIBLE] to find the poles [INAUDIBLE] 675 00:37:54,522 --> 00:38:00,327 to solve for a denominator [INAUDIBLE] 676 00:38:00,327 --> 00:38:02,410 DENNIS FREEMAN: Why is a hard philosophy question. 677 00:38:02,410 --> 00:38:05,130 I'm not sure I can answer philosophy questions. 678 00:38:05,130 --> 00:38:09,060 I guess I have a PhD, maybe I can. 679 00:38:09,060 --> 00:38:11,560 So it has to do with partial fractions, right? 680 00:38:11,560 --> 00:38:12,450 It's the bottom. 681 00:38:12,450 --> 00:38:15,050 The bottoms are special. 682 00:38:15,050 --> 00:38:16,550 And in fact, in about a week, we'll 683 00:38:16,550 --> 00:38:19,430 talk about what's the tops. 684 00:38:19,430 --> 00:38:23,330 Not surprisingly, the tops are like an inverse bottom. 685 00:38:23,330 --> 00:38:25,280 What a deep insight. 686 00:38:25,280 --> 00:38:29,190 So if you were to run signals through the system backwards, 687 00:38:29,190 --> 00:38:32,310 the tops look like bottoms and the bottoms look like tops. 688 00:38:32,310 --> 00:38:34,050 And there's some really useful things 689 00:38:34,050 --> 00:38:36,600 that you can derive from that. 690 00:38:36,600 --> 00:38:38,229 OK what's the answer to the question? 691 00:38:38,229 --> 00:38:40,020 What are the poles in the Fibonacci system? 692 00:38:40,020 --> 00:38:41,990 Give me a number between 0 and-- 693 00:38:41,990 --> 00:38:43,644 1 and 5. 694 00:38:43,644 --> 00:38:44,310 Raise your hand. 695 00:38:48,787 --> 00:38:49,995 OK, how many are you guessed? 696 00:38:52,521 --> 00:38:53,020 OK. 697 00:38:53,020 --> 00:38:54,340 That's what I thought. 698 00:38:54,340 --> 00:38:59,146 OK so, are there poles in a Fibonacci system? 699 00:39:03,784 --> 00:39:05,950 How would you convince yourself there are or are not 700 00:39:05,950 --> 00:39:08,020 poles in a Fibonacci system? 701 00:39:08,020 --> 00:39:10,370 Yeah? 702 00:39:10,370 --> 00:39:13,180 AUDIENCE: You can [INAUDIBLE] as a difference equation, 703 00:39:13,180 --> 00:39:16,636 so [INAUDIBLE] some visual representation. 704 00:39:16,636 --> 00:39:18,510 DENNIS FREEMAN: So you have to start somehow. 705 00:39:18,510 --> 00:39:21,450 Somehow you want to translate that graphical problem 706 00:39:21,450 --> 00:39:25,350 into a difference equation or an R expression or something. 707 00:39:25,350 --> 00:39:27,720 So one way to think about it is a difference equation 708 00:39:27,720 --> 00:39:29,090 of this form. 709 00:39:29,090 --> 00:39:33,060 At time n, add some number of rabbits 710 00:39:33,060 --> 00:39:35,670 that somebody just plunked in. 711 00:39:35,670 --> 00:39:37,500 That's my input. 712 00:39:37,500 --> 00:39:42,780 In the example I showed by magic, by spontaneous creation, 713 00:39:42,780 --> 00:39:45,840 there was a pair of rabbits at the beginning that 714 00:39:45,840 --> 00:39:51,570 didn't obey the rules for that were true for the rest of time. 715 00:39:51,570 --> 00:39:54,210 That was x. 716 00:39:54,210 --> 00:39:58,720 And y of n, also you have to add in the two previous values. 717 00:39:58,720 --> 00:40:03,490 That's kind of a rule for the Fibonacci system. 718 00:40:03,490 --> 00:40:06,640 And that has the form that we've been talking about. 719 00:40:06,640 --> 00:40:09,490 It's something that has just adder's and delays and stuff 720 00:40:09,490 --> 00:40:10,510 like that in it. 721 00:40:10,510 --> 00:40:13,210 So we can write that as a functional. 722 00:40:13,210 --> 00:40:14,950 And we get a second order denominator 723 00:40:14,950 --> 00:40:17,950 so we see that there are two poles. 724 00:40:17,950 --> 00:40:22,170 And if you find the roots by the z method, 725 00:40:22,170 --> 00:40:23,690 you get these funny numbers. 726 00:40:26,840 --> 00:40:28,970 And they're indeed funny. 727 00:40:28,970 --> 00:40:31,490 You get the golden ratio and you get the reciprocal 728 00:40:31,490 --> 00:40:33,500 of the golden ratio, the negative reciprocal 729 00:40:33,500 --> 00:40:35,670 of the golden ratio. 730 00:40:35,670 --> 00:40:37,520 So that's really bizarre because we end up 731 00:40:37,520 --> 00:40:40,622 with this whole pole, 1.618. 732 00:40:40,622 --> 00:40:42,830 Tell me something about the response of a system that 733 00:40:42,830 --> 00:40:48,140 has a pole at 1.618. 734 00:40:48,140 --> 00:40:49,730 AUDIENCE: It [INAUDIBLE]. 735 00:40:49,730 --> 00:40:50,870 DENNIS FREEMAN: Diverges. 736 00:40:50,870 --> 00:40:54,749 What about this one at minus 0.618? 737 00:40:54,749 --> 00:40:56,050 AUDIENCE: Oscillate. 738 00:40:56,050 --> 00:40:57,460 DENNIS FREEMAN: It oscillates. 739 00:40:57,460 --> 00:41:00,617 So you take diverging plus oscillating 740 00:41:00,617 --> 00:41:01,450 and what do you get? 741 00:41:04,609 --> 00:41:05,900 You get a sequence of integers. 742 00:41:10,500 --> 00:41:11,000 OK. 743 00:41:11,000 --> 00:41:12,208 That's pretty bizarre, right? 744 00:41:12,208 --> 00:41:13,790 You take diverging and oscillating 745 00:41:13,790 --> 00:41:17,150 and you get 1, 1, 2, 3, 5, 8, 13. 746 00:41:17,150 --> 00:41:18,110 That's pretty weird. 747 00:41:21,960 --> 00:41:23,946 Also notice that it doesn't behave 748 00:41:23,946 --> 00:41:25,320 the way we think about it when we 749 00:41:25,320 --> 00:41:27,370 think about computer science. 750 00:41:27,370 --> 00:41:30,270 There's no r-squared growth of anything. 751 00:41:30,270 --> 00:41:32,190 How many operations is required to compute 752 00:41:32,190 --> 00:41:33,690 the output at time 119? 753 00:41:36,987 --> 00:41:39,350 AUDIENCE: [INAUDIBLE] operations. 754 00:41:39,350 --> 00:41:41,641 DENNIS FREEMAN: Depends on what you mean by operations, 755 00:41:41,641 --> 00:41:42,540 that's a good point. 756 00:41:42,540 --> 00:41:47,801 So if you used a modern laptop, how many steps 757 00:41:47,801 --> 00:41:49,050 would the laptop have to take? 758 00:41:52,800 --> 00:41:55,680 You have to know something about what's inside the laptop. 759 00:41:55,680 --> 00:41:58,065 These days, exponentiation is pretty simple. 760 00:42:00,760 --> 00:42:01,930 So you can think about-- 761 00:42:01,930 --> 00:42:04,440 so all you need to do is raise phi 762 00:42:04,440 --> 00:42:11,080 to the n an add minus phi the minus n 763 00:42:11,080 --> 00:42:12,820 and scale them and add them. 764 00:42:12,820 --> 00:42:15,070 So it's a different way of thinking about things. 765 00:42:15,070 --> 00:42:16,120 That's the point. 766 00:42:16,120 --> 00:42:19,690 So you can think about the Fibonacci system as a recursion 767 00:42:19,690 --> 00:42:21,460 problem like we did in 601 or you 768 00:42:21,460 --> 00:42:23,380 can think about it as the response 769 00:42:23,380 --> 00:42:26,380 to an LTI system, a linear time invariant system 770 00:42:26,380 --> 00:42:30,440 and you get this kind of a way of thinking about things. 771 00:42:30,440 --> 00:42:33,660 OK one more thing to talk about today. 772 00:42:33,660 --> 00:42:37,200 And that is, what happens when things are complex? 773 00:42:37,200 --> 00:42:39,420 You can make a terribly small change to the system 774 00:42:39,420 --> 00:42:40,770 functional. 775 00:42:40,770 --> 00:42:45,240 Here all I've done is change one sign from the Fibonacci system 776 00:42:45,240 --> 00:42:49,330 and the result is complex roots. 777 00:42:49,330 --> 00:42:54,950 So what's it mean to have a system that has complex poles? 778 00:42:54,950 --> 00:43:01,270 So complex poles, one of the interesting things 779 00:43:01,270 --> 00:43:04,390 about polynomial math, complex numbers 780 00:43:04,390 --> 00:43:06,990 work like any other number. 781 00:43:06,990 --> 00:43:10,500 Not surprisingly then, complex poles in a linear system 782 00:43:10,500 --> 00:43:13,410 work sort of like any other pole. 783 00:43:13,410 --> 00:43:16,170 The fundamental mode associated with any pole 784 00:43:16,170 --> 00:43:19,320 is p-naught to the n. 785 00:43:19,320 --> 00:43:23,340 The fundamental modes corresponding to a complex pole 786 00:43:23,340 --> 00:43:24,960 have complex values. 787 00:43:24,960 --> 00:43:27,190 And that's shown here. 788 00:43:27,190 --> 00:43:30,810 We can write-- so we can think about that system polynomial 789 00:43:30,810 --> 00:43:32,310 that I showed in the previous slide. 790 00:43:32,310 --> 00:43:34,080 This one. 791 00:43:34,080 --> 00:43:36,085 We can write that in a factored form. 792 00:43:38,970 --> 00:43:41,345 But now, the roots are complex numbers. 793 00:43:43,799 --> 00:43:44,840 This is a complex number. 794 00:43:44,840 --> 00:43:46,280 There's complex numbers all over the place. 795 00:43:46,280 --> 00:43:46,780 Who cares? 796 00:43:46,780 --> 00:43:48,740 Complex numbers, they're just numbers. 797 00:43:48,740 --> 00:43:51,120 That's the nice part about algebra. 798 00:43:51,120 --> 00:43:53,980 So we don't need to worry about that. 799 00:43:53,980 --> 00:43:56,430 Other than the modes are complex. 800 00:43:56,430 --> 00:43:59,240 So if you try to plot the mode associated 801 00:43:59,240 --> 00:44:03,550 with one of the roots and the other root, 802 00:44:03,550 --> 00:44:06,160 they have real and imaginary parts. 803 00:44:06,160 --> 00:44:08,530 So here I'm showing the real part in blue 804 00:44:08,530 --> 00:44:12,150 and the imaginary part in red. 805 00:44:12,150 --> 00:44:14,960 So each of the fundamental modes has a real and imaginary part. 806 00:44:14,960 --> 00:44:16,683 That's the only complicated thing. 807 00:44:19,930 --> 00:44:26,240 It's sometimes easier to visualize the modes 808 00:44:26,240 --> 00:44:29,600 on a polar plot. 809 00:44:29,600 --> 00:44:31,370 And that's illustrated here. 810 00:44:31,370 --> 00:44:34,200 These plots look complicated. 811 00:44:34,200 --> 00:44:37,850 If you think about what they look like in polar coordinates. 812 00:44:37,850 --> 00:44:39,950 If you think about this p-naught to the n, 813 00:44:39,950 --> 00:44:43,340 but now p-naught is complex, just 814 00:44:43,340 --> 00:44:48,260 plot the number p-naught to the n on a complex plane, 815 00:44:48,260 --> 00:44:49,610 it's pretty simple. 816 00:44:49,610 --> 00:44:55,175 So in this first case, we have this pole at e 817 00:44:55,175 --> 00:44:57,590 to the j and pi over 3. 818 00:44:57,590 --> 00:45:00,380 Gets raised to the n-th power. 819 00:45:00,380 --> 00:45:01,970 And we use trig and we can come up 820 00:45:01,970 --> 00:45:04,040 with some expression for that. 821 00:45:04,040 --> 00:45:08,330 But forget trig, just remember the Euler relationship. 822 00:45:08,330 --> 00:45:09,530 And that's equivalent. 823 00:45:09,530 --> 00:45:12,350 We can just use this form to say well, if you had 0, 824 00:45:12,350 --> 00:45:13,740 e to the 0 is 1. 825 00:45:13,740 --> 00:45:15,750 So I'm there. 826 00:45:15,750 --> 00:45:18,670 If n were 1, then it would be e to the j pi over 3. 827 00:45:18,670 --> 00:45:19,850 Well, that's there. 828 00:45:19,850 --> 00:45:22,540 You go up by 60 degrees. 829 00:45:22,540 --> 00:45:23,920 If n is 2, you're over here. 830 00:45:23,920 --> 00:45:26,770 3, 4, 5, 6, 7, 8, 9, 10. 831 00:45:26,770 --> 00:45:29,510 So you can visualize how the root is behaving, 832 00:45:29,510 --> 00:45:32,320 the complex root, by looking at a complex plane. 833 00:45:32,320 --> 00:45:34,310 That other root works just the same way, 834 00:45:34,310 --> 00:45:37,610 except it's negative so it spins around the other way. 835 00:45:37,610 --> 00:45:40,330 So it's convenient to watch the behavior of complex roots 836 00:45:40,330 --> 00:45:41,230 on a complex plane. 837 00:45:41,230 --> 00:45:44,800 That's not too surprising. 838 00:45:44,800 --> 00:45:48,340 What's slightly more surprising is the idea 839 00:45:48,340 --> 00:45:51,880 of a real system that's made out of adders and delays 840 00:45:51,880 --> 00:45:56,680 and that kind of stuff generating a complex response. 841 00:45:56,680 --> 00:45:57,700 That's a little weird. 842 00:45:57,700 --> 00:45:58,960 How would that happen? 843 00:45:58,960 --> 00:46:01,270 Does that mean every time I build a delay, 844 00:46:01,270 --> 00:46:03,689 I need to have an output that is imaginary? 845 00:46:03,689 --> 00:46:05,230 Or does it sort of just squirt it out 846 00:46:05,230 --> 00:46:07,510 into some orthogonal space? 847 00:46:07,510 --> 00:46:10,260 Or how do you think about that? 848 00:46:10,260 --> 00:46:12,690 So it's a little weird to think about the response 849 00:46:12,690 --> 00:46:15,510 of a real system coming out complex. 850 00:46:15,510 --> 00:46:16,350 And they don't. 851 00:46:19,020 --> 00:46:22,650 Systems with real value coefficients 852 00:46:22,650 --> 00:46:24,930 of the type that could be represented by a difference 853 00:46:24,930 --> 00:46:27,270 equation, think about a difference equation. 854 00:46:27,270 --> 00:46:30,602 If you require that the coefficients are real valued, 855 00:46:30,602 --> 00:46:33,060 that puts a constraint on the kinds of difference equations 856 00:46:33,060 --> 00:46:34,920 that you can have. 857 00:46:34,920 --> 00:46:38,610 Even though the roots may come out complex, 858 00:46:38,610 --> 00:46:41,755 if you started with a system that 859 00:46:41,755 --> 00:46:43,880 could be expressed with a difference equation whose 860 00:46:43,880 --> 00:46:48,590 coefficients were all real, then algebra 861 00:46:48,590 --> 00:46:57,060 will conspire to make the sum of the modes be real. 862 00:46:57,060 --> 00:46:59,190 It sort of all that could happen. 863 00:46:59,190 --> 00:47:00,060 Right? 864 00:47:00,060 --> 00:47:03,315 So we're expecting that if it were a physical system, if it 865 00:47:03,315 --> 00:47:05,190 could be represented by a difference equation 866 00:47:05,190 --> 00:47:08,940 with real value coefficients, how on earth 867 00:47:08,940 --> 00:47:11,260 could a real valued input generate a complex valued 868 00:47:11,260 --> 00:47:11,760 output? 869 00:47:11,760 --> 00:47:13,410 That doesn't make any sense. 870 00:47:13,410 --> 00:47:15,010 And it doesn't happen. 871 00:47:15,010 --> 00:47:17,200 And that's because things conspire. 872 00:47:17,200 --> 00:47:19,590 So this is just showing long-winded math 873 00:47:19,590 --> 00:47:22,710 for those of you who like long-winded math. 874 00:47:22,710 --> 00:47:27,360 The idea is that these coefficients conspire 875 00:47:27,360 --> 00:47:33,340 as complex numbers so that for every integer value of n, 876 00:47:33,340 --> 00:47:35,490 the answer will be real valued. 877 00:47:35,490 --> 00:47:38,820 And what's showed in the bottom is the unit-sample response 878 00:47:38,820 --> 00:47:40,150 for that system. 879 00:47:40,150 --> 00:47:44,050 And you can see everything's real valued now. 880 00:47:44,050 --> 00:47:47,050 So last question. 881 00:47:47,050 --> 00:47:51,010 Here is the response of a system with poles, two 882 00:47:51,010 --> 00:47:53,370 poles at those complex numbers. 883 00:47:57,520 --> 00:48:03,470 How do you think about the values R and omega? 884 00:48:03,470 --> 00:48:07,850 So assume that the poles are at the polar form of the pole 885 00:48:07,850 --> 00:48:13,340 location is magnitude R, angle plus or minus j omega. 886 00:48:13,340 --> 00:48:16,400 How do you think about R and omega? 887 00:48:16,400 --> 00:48:19,580 So let's say that I plotted the response here, 888 00:48:19,580 --> 00:48:23,180 can you figure out by looking at that, what is R? 889 00:48:23,180 --> 00:48:25,750 Can you estimate R? 890 00:48:25,750 --> 00:48:39,050 R is approximately 1/9 0.9, approximately 0.9. 891 00:48:39,050 --> 00:48:41,890 So if it were 0.9, how's it go up? 892 00:48:44,866 --> 00:48:47,850 Hmm. 893 00:48:47,850 --> 00:48:50,630 So it's going around in that circle right? 894 00:48:50,630 --> 00:48:55,270 So as it's spinning around in a circle, it's going up and down. 895 00:48:55,270 --> 00:48:57,160 As it's spinning around the circle, 896 00:48:57,160 --> 00:49:00,190 the R is also spiraling in. 897 00:49:00,190 --> 00:49:02,900 So the R, R to the n. 898 00:49:02,900 --> 00:49:03,400 Right? 899 00:49:03,400 --> 00:49:05,430 If we think about pole to the n. 900 00:49:05,430 --> 00:49:08,070 The key is always pole to the n. 901 00:49:08,070 --> 00:49:10,480 If we think about pole to the n, then it's R to the n, 902 00:49:10,480 --> 00:49:13,280 e to the j omega n. 903 00:49:13,280 --> 00:49:16,940 The R part, R the n is making it spiral in or spiral 904 00:49:16,940 --> 00:49:19,850 out, one of those. 905 00:49:19,850 --> 00:49:23,390 And the omega part is making it go up and down and up and down. 906 00:49:23,390 --> 00:49:28,910 So the R here is actually 0.97, because I made the plot 907 00:49:28,910 --> 00:49:31,250 so I know what it is. 908 00:49:31,250 --> 00:49:32,540 So R is 0.97. 909 00:49:32,540 --> 00:49:37,444 That has to do with sort of a gentle decay across cycles. 910 00:49:37,444 --> 00:49:39,110 So you could figure that out for example 911 00:49:39,110 --> 00:49:41,234 by finding the ratio of that height to that height. 912 00:49:43,960 --> 00:49:47,760 Sort of, how much did the decay when it came around once? 913 00:49:47,760 --> 00:49:50,540 So R is 0.97, what do you think omega is? 914 00:49:54,725 --> 00:49:56,120 Yeah? 915 00:49:56,120 --> 00:49:57,050 AUDIENCE: [INAUDIBLE] 916 00:49:57,050 --> 00:49:58,250 DENNIS FREEMAN: Around 0.5. 917 00:49:58,250 --> 00:50:03,380 So what you need to think about in omega is if you go n steps 918 00:50:03,380 --> 00:50:06,590 and each advances the angle by omega, 919 00:50:06,590 --> 00:50:08,330 how long does it take to get to 2pi? 920 00:50:12,840 --> 00:50:13,340 Right? 921 00:50:13,340 --> 00:50:16,300 It will go around the circle once when the radiant angle, 922 00:50:16,300 --> 00:50:20,080 when the angle goes increments by 2 pi. 923 00:50:20,080 --> 00:50:24,870 Every time, every step, the angle is incremented by omega. 924 00:50:24,870 --> 00:50:30,645 How many steps does it take to get to 2 pi? 925 00:50:34,769 --> 00:50:36,810 That takes-- if I started here and went to there, 926 00:50:36,810 --> 00:50:42,290 I get 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. 927 00:50:42,290 --> 00:50:43,570 Takes about 12. 928 00:50:43,570 --> 00:50:48,650 So 12 omega is 2 pi. 929 00:50:48,650 --> 00:50:51,682 And so omega is approximately, well it's exactly, 930 00:50:51,682 --> 00:50:53,390 well, no it's approximately-- this should 931 00:50:53,390 --> 00:50:59,720 be approximately 2 pi over 12. 932 00:50:59,720 --> 00:51:05,360 6 over 12 by blackboard math is 1/2. 933 00:51:05,360 --> 00:51:08,120 OK. 934 00:51:08,120 --> 00:51:09,890 OK there's one more slide. 935 00:51:09,890 --> 00:51:13,970 But that will turn out to be a good question for you to review 936 00:51:13,970 --> 00:51:16,810 as preparation for the exam.