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 00:00:23,242 --> 00:00:24,700 DENNIS FREEMAN: Hello, and welcome. 9 00:00:29,950 --> 00:00:32,320 So before starting today, there's 10 00:00:32,320 --> 00:00:36,010 one of those kind of required announcements for events 11 00:00:36,010 --> 00:00:37,640 coming up. 12 00:00:37,640 --> 00:00:40,690 So even though it seems like we just did exam 1, 13 00:00:40,690 --> 00:00:42,700 next week is exam 2. 14 00:00:42,700 --> 00:00:47,440 So next Wednesday evening, 7:30 to 9:30. 15 00:00:47,440 --> 00:00:51,940 Same as before other than it's Walker. 16 00:00:51,940 --> 00:00:54,850 So don't go to Building 26. 17 00:00:54,850 --> 00:00:58,010 Go to Walker. 18 00:00:58,010 --> 00:00:58,690 OK. 19 00:00:58,690 --> 00:01:01,420 Other than that, the rules are pretty much the same. 20 00:01:01,420 --> 00:01:04,900 No recitations on the day of the exam. 21 00:01:04,900 --> 00:01:08,680 Coverage will be up until the end of this week. 22 00:01:08,680 --> 00:01:11,860 That includes Lecture 12, Recitation 12. 23 00:01:11,860 --> 00:01:14,260 It includes Homework 7, but Homework 7 24 00:01:14,260 --> 00:01:18,640 will be handled the way Homework 4 was. 25 00:01:18,640 --> 00:01:20,500 So Homework 7 won't be collected. 26 00:01:20,500 --> 00:01:21,520 It won't be graded. 27 00:01:21,520 --> 00:01:24,700 There will be solutions posted. 28 00:01:24,700 --> 00:01:27,560 We'll post previous exams. 29 00:01:27,560 --> 00:01:30,790 You'll be allowed to use two pages of notes, presumably 30 00:01:30,790 --> 00:01:33,640 the page you used last time and one more. 31 00:01:33,640 --> 00:01:36,280 Although, we are not going to check whether it was the page 32 00:01:36,280 --> 00:01:37,250 you used last time. 33 00:01:37,250 --> 00:01:41,030 This is supposed to be a convenience. 34 00:01:41,030 --> 00:01:42,250 No calculators. 35 00:01:42,250 --> 00:01:44,470 No electronic devices. 36 00:01:44,470 --> 00:01:45,610 Just like last time-- 37 00:01:45,610 --> 00:01:47,440 I hope you all found it to be this way-- 38 00:01:47,440 --> 00:01:49,540 the exam was designed to be done in one hour. 39 00:01:49,540 --> 00:01:51,550 And you have two hours, so there's not supposed 40 00:01:51,550 --> 00:01:54,760 to be any time pressure. 41 00:01:54,760 --> 00:01:57,970 Review sessions during the normal open office hours. 42 00:01:57,970 --> 00:01:59,440 And conflicts, please let me know 43 00:01:59,440 --> 00:02:03,440 so that I can arrange to get somebody to proctor. 44 00:02:03,440 --> 00:02:05,340 And so that I can get a room. 45 00:02:05,340 --> 00:02:07,780 So for those two reasons, try to tell me 46 00:02:07,780 --> 00:02:09,384 about conflicts before Friday. 47 00:02:11,716 --> 00:02:13,090 I guess the other important thing 48 00:02:13,090 --> 00:02:15,850 is remember that the theory of the exams in this class 49 00:02:15,850 --> 00:02:17,590 is that they ramp up. 50 00:02:17,590 --> 00:02:20,050 The first exam only counted 10%, with the idea 51 00:02:20,050 --> 00:02:22,210 that that's to let you get acclimated to things. 52 00:02:22,210 --> 00:02:24,460 You're sort of walking in the shallow end of the pool 53 00:02:24,460 --> 00:02:26,080 is the idea. 54 00:02:26,080 --> 00:02:28,310 This one will count 15%. 55 00:02:28,310 --> 00:02:29,410 So it's coming up. 56 00:02:29,410 --> 00:02:31,210 The next one will count 20. 57 00:02:31,210 --> 00:02:33,760 And the final one will count 40. 58 00:02:33,760 --> 00:02:35,920 So this one counts slightly more than the last one, 59 00:02:35,920 --> 00:02:40,660 but you're still kind of in the not-so-deep end of the pool. 60 00:02:40,660 --> 00:02:41,240 Questions? 61 00:02:41,240 --> 00:02:41,740 Comments? 62 00:02:44,142 --> 00:02:45,100 Questions on the exams? 63 00:02:47,830 --> 00:02:49,000 Good. 64 00:02:49,000 --> 00:02:52,120 So I want to start talking about a new topic today, which 65 00:02:52,120 --> 00:02:54,250 is really an elaboration of what we talked about 66 00:02:54,250 --> 00:02:55,776 with frequency response. 67 00:02:55,776 --> 00:02:57,900 But I want to start by telling you the big picture, 68 00:02:57,900 --> 00:03:01,930 the sort of 30,000-foot view. 69 00:03:01,930 --> 00:03:03,460 We've been focusing in this class 70 00:03:03,460 --> 00:03:05,560 on multiple representations for thinking 71 00:03:05,560 --> 00:03:08,440 about how linear systems work. 72 00:03:08,440 --> 00:03:13,330 Differential equations, block diagrams, impulse responses, 73 00:03:13,330 --> 00:03:15,430 frequency response-- all kinds of different ways 74 00:03:15,430 --> 00:03:19,200 of thinking about it, with the idea that if you know all 75 00:03:19,200 --> 00:03:22,240 of them, then you can use the one that's the easiest to solve 76 00:03:22,240 --> 00:03:24,610 your particular problem. 77 00:03:24,610 --> 00:03:26,380 In almost all of the cases, we've 78 00:03:26,380 --> 00:03:29,740 tried to find representations that are 79 00:03:29,740 --> 00:03:32,465 small in conceptual complexity. 80 00:03:32,465 --> 00:03:34,090 So for example, differential equations. 81 00:03:34,090 --> 00:03:37,010 In principle, differential equations could be complicated. 82 00:03:37,010 --> 00:03:39,430 In fact, we're only interested in linear time invariant 83 00:03:39,430 --> 00:03:41,860 differential equations, which means that they always 84 00:03:41,860 --> 00:03:43,090 have a simple form. 85 00:03:43,090 --> 00:03:47,650 They always look something like y plus some coefficient times y 86 00:03:47,650 --> 00:03:50,750 dot plus some other coefficient times y dot. 87 00:03:50,750 --> 00:03:56,770 y double dot plus a whole bunch of things turns into say b0 88 00:03:56,770 --> 00:04:02,560 x plus b1 x dot plus b2 x dot-- double dot plus a whole bunch 89 00:04:02,560 --> 00:04:03,400 of things. 90 00:04:03,400 --> 00:04:06,490 Point being that although the differential equations could, 91 00:04:06,490 --> 00:04:08,140 in fact, be arbitrarily complicated, 92 00:04:08,140 --> 00:04:09,640 we only focus on the ones that have 93 00:04:09,640 --> 00:04:16,329 a simple representation in terms of a small number of constants. 94 00:04:16,329 --> 00:04:21,490 There they are-- a1, a2, a3, b0, b1, b2. 95 00:04:21,490 --> 00:04:23,960 Small number of constants. 96 00:04:23,960 --> 00:04:26,410 So that's the way of managing the complexity. 97 00:04:26,410 --> 00:04:28,360 There's only a small number of constants 98 00:04:28,360 --> 00:04:29,620 you need to worry about. 99 00:04:29,620 --> 00:04:32,320 Similarly, when we think about things like poles and zeros, 100 00:04:32,320 --> 00:04:36,610 the whole point of a pole-zero diagram is that you reduce. 101 00:04:36,610 --> 00:04:39,290 So say we're doing s-plane. 102 00:04:39,290 --> 00:04:42,190 So say we're doing some continuous kind of a problem. 103 00:04:42,190 --> 00:04:48,190 We reduced the whole system to thinking about a small number 104 00:04:48,190 --> 00:04:50,830 of singularities. 105 00:04:50,830 --> 00:04:53,740 Again, we're trying to think about the system in terms 106 00:04:53,740 --> 00:04:56,650 of a small number of numbers. 107 00:04:56,650 --> 00:05:03,820 We like this one because the numbers are in some sense more 108 00:05:03,820 --> 00:05:06,220 intuitive or help us to solve problems better 109 00:05:06,220 --> 00:05:08,290 than perhaps these numbers did. 110 00:05:08,290 --> 00:05:10,330 Both of these representations are characterized 111 00:05:10,330 --> 00:05:11,538 by a small number of numbers. 112 00:05:14,140 --> 00:05:17,070 But for certain purposes, one set of numbers 113 00:05:17,070 --> 00:05:20,730 is easier to work with than another set of numbers. 114 00:05:20,730 --> 00:05:25,230 Last week, we started to think about frequency responses. 115 00:05:25,230 --> 00:05:28,040 Frequency responses were really good 116 00:05:28,040 --> 00:05:30,210 for thinking about systems, like audio. 117 00:05:30,210 --> 00:05:32,370 I did some examples with audio. 118 00:05:32,370 --> 00:05:35,726 Or just other kinds of systems. 119 00:05:35,726 --> 00:05:37,350 It's very good to think about this kind 120 00:05:37,350 --> 00:05:39,030 of a system as a frequency response 121 00:05:39,030 --> 00:05:42,150 because there's a certain frequency at which 122 00:05:42,150 --> 00:05:44,010 a very small input-- 123 00:05:44,010 --> 00:05:45,480 you can barely see my hand moving. 124 00:05:45,480 --> 00:05:50,610 Or, if I weren't shaking, it wouldn't move so much. 125 00:05:50,610 --> 00:05:54,510 And in fact, the mass moves a lot. 126 00:05:54,510 --> 00:05:59,670 So that's some unique property of a particular frequency 127 00:05:59,670 --> 00:06:02,550 that's interesting for us to know about. 128 00:06:02,550 --> 00:06:04,140 The problem with frequency responses 129 00:06:04,140 --> 00:06:09,840 is that when we start to think about a frequency response, 130 00:06:09,840 --> 00:06:15,660 we're now thinking about, say, the magnitude of a system 131 00:06:15,660 --> 00:06:18,660 function as a function of omega, which 132 00:06:18,660 --> 00:06:24,030 might be horribly complicated, and an angle which 133 00:06:24,030 --> 00:06:26,400 might be horribly complicated. 134 00:06:26,400 --> 00:06:28,045 So we've kind of lost the modularity. 135 00:06:28,045 --> 00:06:29,670 We've kind of lost the ability to think 136 00:06:29,670 --> 00:06:31,086 about the whole frequency response 137 00:06:31,086 --> 00:06:32,340 by a handful of numbers. 138 00:06:32,340 --> 00:06:34,080 That's what Bode plots are. 139 00:06:34,080 --> 00:06:36,870 Bode plots are a way of thinking about the entire frequency 140 00:06:36,870 --> 00:06:39,810 response in terms of a handful of numbers. 141 00:06:39,810 --> 00:06:41,550 That's why we like it. 142 00:06:41,550 --> 00:06:44,820 We like frequency response because it's a very good way 143 00:06:44,820 --> 00:06:47,790 to think about certain systems. 144 00:06:47,790 --> 00:06:50,850 Audio because we like to think about bass differently 145 00:06:50,850 --> 00:06:54,040 from the way we think about treble. 146 00:06:54,040 --> 00:06:57,240 Mass-spring dashpots because there are certain frequencies 147 00:06:57,240 --> 00:07:00,150 at which the system goes crazy. 148 00:07:00,150 --> 00:07:04,290 Airplanes because you don't want it to do that, right? 149 00:07:04,290 --> 00:07:08,250 So there are certain reasons why frequency responses are good. 150 00:07:08,250 --> 00:07:11,100 And Bode plots are a way of getting back to this idea 151 00:07:11,100 --> 00:07:13,530 that we can think about a whole function of frequency 152 00:07:13,530 --> 00:07:15,390 with just a handful of numbers. 153 00:07:15,390 --> 00:07:16,720 That's what we're doing. 154 00:07:16,720 --> 00:07:17,220 OK. 155 00:07:17,220 --> 00:07:21,630 So just to get things going, remember 156 00:07:21,630 --> 00:07:23,310 where we're coming from. 157 00:07:23,310 --> 00:07:26,970 We think about frequency responses 158 00:07:26,970 --> 00:07:29,880 by thinking about eigenfunctions and eigenvalues. 159 00:07:29,880 --> 00:07:32,640 We think about how if your system is linear and time 160 00:07:32,640 --> 00:07:37,770 invariant, last week we showed that complex exponentials are 161 00:07:37,770 --> 00:07:40,730 eigenfunctions for all such systems. 162 00:07:40,730 --> 00:07:43,980 You put in any complex exponential and what comes out 163 00:07:43,980 --> 00:07:49,020 is a weighted version of the same complex exponential. 164 00:07:49,020 --> 00:07:52,740 Property of linear time invariant systems Furthermore, 165 00:07:52,740 --> 00:07:57,210 the eigenvalues are really easy to find from vector diagrams. 166 00:07:57,210 --> 00:08:01,620 The eigenvalue is the value of the system function evaluated 167 00:08:01,620 --> 00:08:04,080 at the frequency of the complex exponential. 168 00:08:07,360 --> 00:08:08,939 Really easy. 169 00:08:08,939 --> 00:08:11,230 And you can calculate that from a vector diagram, which 170 00:08:11,230 --> 00:08:15,780 capitalizes on the small number of numbers representation 171 00:08:15,780 --> 00:08:18,780 of the pole-zero plot. 172 00:08:18,780 --> 00:08:19,690 So it's all easy. 173 00:08:19,690 --> 00:08:21,540 We only do easy things. 174 00:08:21,540 --> 00:08:23,700 And then, we're really interested 175 00:08:23,700 --> 00:08:27,280 in these sinusoidal responses. 176 00:08:27,280 --> 00:08:31,350 So we think about a sinusoid by using Euler's formula 177 00:08:31,350 --> 00:08:34,380 as the sum of two complex exponentials. 178 00:08:34,380 --> 00:08:37,020 And that lets us see that all we need to do 179 00:08:37,020 --> 00:08:41,370 is evaluate the system function at j omega 0, 180 00:08:41,370 --> 00:08:44,610 j times the frequency of interest, 181 00:08:44,610 --> 00:08:46,560 the magnitude, and phase. 182 00:08:46,560 --> 00:08:48,870 And that lets us compute, how much 183 00:08:48,870 --> 00:08:50,850 is the amplification at that frequency 184 00:08:50,850 --> 00:08:57,150 and how much is the time delay associated with that frequency? 185 00:08:57,150 --> 00:09:01,290 And so that motivated us to look at, how 186 00:09:01,290 --> 00:09:04,560 does frequency response map to pole-zero? 187 00:09:04,560 --> 00:09:07,140 There's a simple mapping, the vector diagram. 188 00:09:07,140 --> 00:09:10,920 Think about for example, if you have an isolated zero, 189 00:09:10,920 --> 00:09:12,675 a single zero in your system. 190 00:09:12,675 --> 00:09:17,030 A Single zero here at minus 2. 191 00:09:17,030 --> 00:09:19,640 The magnitude can be determined by the magnitude of the-- 192 00:09:19,640 --> 00:09:22,880 by the length of the angle that goes from the 0 193 00:09:22,880 --> 00:09:24,380 to the frequency of interest. 194 00:09:24,380 --> 00:09:26,060 Frequencies are on the j omega axis. 195 00:09:28,670 --> 00:09:30,320 Angle can be found-- 196 00:09:30,320 --> 00:09:32,870 this angle can be computed from the angle 197 00:09:32,870 --> 00:09:34,760 that this vector makes with the x-axis. 198 00:09:38,250 --> 00:09:39,680 And that always works. 199 00:09:39,680 --> 00:09:42,101 Here it's illustrated for a single zero. 200 00:09:42,101 --> 00:09:43,850 If you think about the magnitude and angle 201 00:09:43,850 --> 00:09:47,090 as a function of frequency, you map out 202 00:09:47,090 --> 00:09:49,190 features that look like so. 203 00:09:49,190 --> 00:09:52,760 As the frequency goes from 0 to very big, 204 00:09:52,760 --> 00:09:55,610 the arrow goes from short to very long. 205 00:09:55,610 --> 00:09:58,790 That's manifested by this. 206 00:09:58,790 --> 00:10:00,410 And the angle changes systematically 207 00:10:00,410 --> 00:10:06,341 from being near 0 to being up around pi over 2. 208 00:10:06,341 --> 00:10:06,840 OK. 209 00:10:06,840 --> 00:10:08,510 And the same sort of thing works for a pole, 210 00:10:08,510 --> 00:10:09,968 except now a pole is in the bottom. 211 00:10:13,310 --> 00:10:17,690 So when it used to get big, now it gets small. 212 00:10:17,690 --> 00:10:20,510 The angle, which used to go positive, now goes negative. 213 00:10:20,510 --> 00:10:22,580 But it's the same idea. 214 00:10:22,580 --> 00:10:25,970 And you can compose more elaborate functions 215 00:10:25,970 --> 00:10:28,670 by thinking about the individual arrows that 216 00:10:28,670 --> 00:10:29,885 correspond to factors. 217 00:10:29,885 --> 00:10:33,020 That's using the factor theorem to reduce the system 218 00:10:33,020 --> 00:10:38,750 function, which is always a polynomial in s, into factors. 219 00:10:38,750 --> 00:10:39,250 OK. 220 00:10:39,250 --> 00:10:41,722 So that was what we talked about last time 221 00:10:41,722 --> 00:10:43,180 what I want to talk about today now 222 00:10:43,180 --> 00:10:45,550 is how to reduce the frequency response, which 223 00:10:45,550 --> 00:10:47,470 looks like a lot of numbers-- 224 00:10:47,470 --> 00:10:50,410 one for every frequency-- to thinking about it in terms 225 00:10:50,410 --> 00:10:52,840 of a small number of numbers. 226 00:10:52,840 --> 00:10:58,257 So the idea is that this magnitude response is simple 227 00:10:58,257 --> 00:10:59,590 if you look at it the right way. 228 00:11:02,390 --> 00:11:07,310 In particular, it has a simple value at low frequencies. 229 00:11:07,310 --> 00:11:09,860 If we think about the smallest frequencies you can have, 230 00:11:09,860 --> 00:11:11,840 that's omega equals 0. 231 00:11:11,840 --> 00:11:14,570 And the smallest frequencies, the magnitude 232 00:11:14,570 --> 00:11:19,700 response asymptotes becomes arbitrarily close to the line 233 00:11:19,700 --> 00:11:21,590 that you would get by substituting omega 234 00:11:21,590 --> 00:11:23,900 equals 0 here. 235 00:11:23,900 --> 00:11:29,070 If omega were 0, then h of j omega would be minus z1. 236 00:11:29,070 --> 00:11:34,910 The magnitude of minus z1 is z1. 237 00:11:34,910 --> 00:11:37,820 So at very low frequencies, the response-- so 238 00:11:37,820 --> 00:11:39,650 when you get to frequencies near 0, 239 00:11:39,650 --> 00:11:41,690 the response asymptotes to that line. 240 00:11:44,920 --> 00:11:49,080 And if you get to very high frequencies, then the fact 241 00:11:49,080 --> 00:11:51,330 that you're adding a small constant to a number that's 242 00:11:51,330 --> 00:11:55,260 already very big, if omega were large, then that little 243 00:11:55,260 --> 00:11:58,600 constant, z1, wouldn't matter. 244 00:11:58,600 --> 00:12:01,900 So the magnitude becomes very close to the magnitude of omega 245 00:12:01,900 --> 00:12:03,450 shown by the green dotted line. 246 00:12:05,980 --> 00:12:08,380 So you can think about the blue line 247 00:12:08,380 --> 00:12:10,615 being some kind of an interpolation between those two 248 00:12:10,615 --> 00:12:13,360 dash lines, the low-frequency behavior 249 00:12:13,360 --> 00:12:14,920 to the high-frequency behavior. 250 00:12:14,920 --> 00:12:19,060 And the relation looks even simpler if you plot it 251 00:12:19,060 --> 00:12:20,980 on a different kind of axis. 252 00:12:20,980 --> 00:12:24,280 If you make it log-log, then you can 253 00:12:24,280 --> 00:12:27,840 think about frequencies going-- so the log of frequency. 254 00:12:27,840 --> 00:12:29,590 Well, there's some-- if the frequency 255 00:12:29,590 --> 00:12:31,320 were 1, whatever that means. 256 00:12:31,320 --> 00:12:33,370 Or in fact here, it's important to note 257 00:12:33,370 --> 00:12:36,580 that I'm plotting here versus a normalized version 258 00:12:36,580 --> 00:12:39,160 of frequency. 259 00:12:39,160 --> 00:12:40,750 I'm normalizing by z1. 260 00:12:40,750 --> 00:12:46,090 What That has the effect that when omega is the same as z1, 261 00:12:46,090 --> 00:12:48,456 I get the log of 1, which is? 262 00:12:48,456 --> 00:12:49,170 AUDIENCE: 0. 263 00:12:49,170 --> 00:12:50,850 DENNIS FREEMAN: 0, right? 264 00:12:50,850 --> 00:12:54,030 That's a way of shifting the axis to make 265 00:12:54,030 --> 00:12:55,770 the interesting stuff happen at 0. 266 00:12:55,770 --> 00:12:56,700 That's all. 267 00:12:56,700 --> 00:13:02,130 So if you scale frequency by the singularity-- in this case, 268 00:13:02,130 --> 00:13:05,740 a 0-- 269 00:13:05,740 --> 00:13:09,490 then the critical frequency becomes the frequency zero. 270 00:13:09,490 --> 00:13:12,580 We all like 0, right? 271 00:13:12,580 --> 00:13:14,980 So what you can see then if you plot it this way-- 272 00:13:14,980 --> 00:13:19,060 and I also scaled the magnitude the same way. 273 00:13:19,060 --> 00:13:22,510 So if you realize that there's a simple asymptote, 274 00:13:22,510 --> 00:13:24,970 you can see very clearly on this log-log plot 275 00:13:24,970 --> 00:13:27,290 that you get a good approximation to the function, 276 00:13:27,290 --> 00:13:29,110 which is showed in blue. 277 00:13:29,110 --> 00:13:32,320 That one's calculated exactly. 278 00:13:32,320 --> 00:13:36,280 And you get a good approximation by the low- and high-frequency 279 00:13:36,280 --> 00:13:37,010 asymptotes. 280 00:13:37,010 --> 00:13:40,360 So instead of thinking about the whole function of frequency, 281 00:13:40,360 --> 00:13:44,570 what we do is we think about just the asymptotes. 282 00:13:44,570 --> 00:13:46,550 That's the simplification. 283 00:13:46,550 --> 00:13:49,340 We don't think about this complicated function. 284 00:13:49,340 --> 00:13:51,855 We only think about what happens at low frequencies 285 00:13:51,855 --> 00:13:52,730 and high frequencies. 286 00:13:52,730 --> 00:13:55,100 Of course, this was too simple, right? 287 00:13:55,100 --> 00:14:00,110 This is what happens with a single zero. 288 00:14:00,110 --> 00:14:01,580 The point is that a similar thing 289 00:14:01,580 --> 00:14:05,010 happens with a single pole or when you combine them. 290 00:14:05,010 --> 00:14:06,890 Let's look at the case of a pole. 291 00:14:06,890 --> 00:14:09,680 Here, the pole idea is a little bit more complicated. 292 00:14:09,680 --> 00:14:11,840 If you go to very low frequencies, 293 00:14:11,840 --> 00:14:14,630 think about s being j omega. 294 00:14:14,630 --> 00:14:18,110 The frequency response lives on the j omega axis. 295 00:14:18,110 --> 00:14:20,720 Always look at s equals j omega if you're 296 00:14:20,720 --> 00:14:23,030 interested in the frequency response. 297 00:14:23,030 --> 00:14:26,840 If you think about s equals j omega, then 298 00:14:26,840 --> 00:14:31,940 if you go to small omega, this term goes away. 299 00:14:31,940 --> 00:14:34,760 And we're left with, again, a constant at low 300 00:14:34,760 --> 00:14:36,220 frequency showed by the red line. 301 00:14:39,540 --> 00:14:41,820 If we look at high frequencies, now the curve's 302 00:14:41,820 --> 00:14:44,880 a little more funky. 303 00:14:44,880 --> 00:14:48,960 The curve at high frequencies-- the real answer 304 00:14:48,960 --> 00:14:52,290 is showed by this sort of bell-shaped curve. 305 00:14:52,290 --> 00:14:57,312 If you plot for different values of omega where s is j omega, 306 00:14:57,312 --> 00:14:59,020 plot out the magnitude, you get this sort 307 00:14:59,020 --> 00:15:01,510 of funky, bell-shaped curve. 308 00:15:01,510 --> 00:15:04,940 The point is that-- 309 00:15:04,940 --> 00:15:06,400 and the high-frequency asymptote. 310 00:15:06,400 --> 00:15:09,190 If you just say, what if j omega were so big 311 00:15:09,190 --> 00:15:11,800 that the p1 wouldn't matter? 312 00:15:11,800 --> 00:15:18,860 Then, you would get 9 over omega being the asymptote. 313 00:15:18,860 --> 00:15:21,410 So that's this hyperbolic sort of thing here. 314 00:15:21,410 --> 00:15:21,910 OK. 315 00:15:21,910 --> 00:15:23,740 Well, that's ugly. 316 00:15:23,740 --> 00:15:27,650 But if you go to log-log, it's very simple. 317 00:15:27,650 --> 00:15:30,890 That's why we like log-log. 318 00:15:30,890 --> 00:15:35,990 If you think about the function y' equals 1 over omega and plot 319 00:15:35,990 --> 00:15:40,190 that function on log-log axes, reciprocal 320 00:15:40,190 --> 00:15:41,630 turns into a straight line. 321 00:15:41,630 --> 00:15:43,560 We like straight lines. 322 00:15:43,560 --> 00:15:45,590 They're easy. 323 00:15:45,590 --> 00:15:48,260 So what happens then-- again, I've normalized things. 324 00:15:48,260 --> 00:15:52,100 I've normalized frequency by the pole. 325 00:15:52,100 --> 00:15:56,360 That makes the critical frequency be omega equals pole. 326 00:15:56,360 --> 00:15:58,850 The critical frequency comes out 1. 327 00:15:58,850 --> 00:16:00,890 The log of 1 is 0. 328 00:16:00,890 --> 00:16:05,860 So the interesting behavior happens at log equals 0. 329 00:16:08,710 --> 00:16:12,340 Below that frequency, the frequency response magnitude 330 00:16:12,340 --> 00:16:17,440 is well approximated by the constant whose log divided 331 00:16:17,440 --> 00:16:19,180 by some funny number that was put there 332 00:16:19,180 --> 00:16:20,430 for normalization is 0. 333 00:16:24,320 --> 00:16:27,140 And at high frequencies, it falls off 334 00:16:27,140 --> 00:16:30,300 with a slope of minus 1. 335 00:16:30,300 --> 00:16:32,090 So again, for the case of the pole, 336 00:16:32,090 --> 00:16:37,610 we get this simple behavior if we focus on the asymptotes. 337 00:16:37,610 --> 00:16:38,960 OK. 338 00:16:38,960 --> 00:16:42,140 So now, that's kind of the theory behind everything. 339 00:16:42,140 --> 00:16:44,702 Make sure you're all up to speed on the theory. 340 00:16:44,702 --> 00:16:47,380 So compare log-log plots of the frequency 341 00:16:47,380 --> 00:16:49,630 response magnitudes of the following system functions. 342 00:16:49,630 --> 00:16:52,810 H1, 1 over s plus 1. 343 00:16:52,810 --> 00:16:55,728 Where's the pole? 344 00:16:55,728 --> 00:16:57,100 AUDIENCE: [INAUDIBLE] 345 00:16:57,100 --> 00:16:58,860 DENNIS FREEMAN: Minus 1. 346 00:16:58,860 --> 00:17:01,080 And s2, 1 over s plus 10. 347 00:17:01,080 --> 00:17:02,750 Pole is? 348 00:17:02,750 --> 00:17:03,920 Minus 10. 349 00:17:03,920 --> 00:17:06,440 Compare the magnitude. 350 00:17:06,440 --> 00:17:07,880 Compare their magnitude functions 351 00:17:07,880 --> 00:17:11,300 when plotted on log-log and answer the following question, 352 00:17:11,300 --> 00:17:14,930 which of these best describes the transformation between H1 353 00:17:14,930 --> 00:17:16,250 and H2? 354 00:17:16,250 --> 00:17:17,540 You should shift horizontally. 355 00:17:17,540 --> 00:17:19,730 You should shift and scale horizontally. 356 00:17:19,730 --> 00:17:21,770 You should shift horizontally and vertically. 357 00:17:21,770 --> 00:17:24,410 You should shift and scale horizontally and vertically. 358 00:17:24,410 --> 00:17:28,050 Or, it's something completely different. 359 00:17:28,050 --> 00:17:29,600 OK, turn to your neighbor. 360 00:17:29,600 --> 00:17:30,740 Say hi. 361 00:17:30,740 --> 00:17:33,530 Right now, say hi. 362 00:17:33,530 --> 00:17:36,080 And now, figure out what answer best 363 00:17:36,080 --> 00:17:40,404 classifies that transformation. 364 00:17:40,404 --> 00:17:44,388 [SIDE CONVERSATIONS] 365 00:19:02,149 --> 00:19:02,940 DENNIS FREEMAN: OK. 366 00:19:02,940 --> 00:19:05,162 So which transformation do you like the best? 367 00:19:05,162 --> 00:19:06,870 Everybody raise your hand with the number 368 00:19:06,870 --> 00:19:08,845 of fingers between 1 and 5. 369 00:19:12,672 --> 00:19:14,130 Let me make sure I know the answer. 370 00:19:14,130 --> 00:19:15,960 I think I know the answer. 371 00:19:15,960 --> 00:19:16,770 OK. 372 00:19:16,770 --> 00:19:19,590 Greater audience participation. 373 00:19:19,590 --> 00:19:20,730 Blame it on your neighbor. 374 00:19:20,730 --> 00:19:23,625 My stupid partner thought it was-- 375 00:19:27,935 --> 00:19:28,435 OK. 376 00:19:28,435 --> 00:19:31,000 It's about 90% correct. 377 00:19:33,970 --> 00:19:38,590 If I plot the magnitude response for H1 on log-log axes, 378 00:19:38,590 --> 00:19:40,490 what will it look like? 379 00:19:40,490 --> 00:19:41,650 Sketch it in the air. 380 00:19:44,480 --> 00:19:47,190 Sketch it in the air is like that sort of thing. 381 00:19:50,290 --> 00:19:50,790 Go ahead. 382 00:19:50,790 --> 00:19:51,610 Go ahead. 383 00:19:51,610 --> 00:19:52,110 Sketch. 384 00:19:54,710 --> 00:19:57,060 OK. 385 00:19:57,060 --> 00:19:58,730 Maybe that didn't work. 386 00:19:58,730 --> 00:19:59,230 OK. 387 00:19:59,230 --> 00:20:05,160 What I want to do is plot the magnitude of H1 of j omega 388 00:20:05,160 --> 00:20:10,145 log versus the log of omega. 389 00:20:10,145 --> 00:20:13,590 But I only want to think about the asymptotes. 390 00:20:13,590 --> 00:20:16,680 Where should I draw the low-frequency asymptote 391 00:20:16,680 --> 00:20:18,600 for that function? 392 00:20:18,600 --> 00:20:20,550 What's the low-frequency asymptote 393 00:20:20,550 --> 00:20:23,360 for the log magnitude of H1? 394 00:20:33,940 --> 00:20:36,270 What's the low-frequency asymptote-- 395 00:20:36,270 --> 00:20:37,300 forget the log part. 396 00:20:37,300 --> 00:20:39,330 What's the low-frequency asymptote for H1? 397 00:20:41,720 --> 00:20:42,220 0. 398 00:20:46,290 --> 00:20:46,982 Yes. 399 00:20:46,982 --> 00:20:49,877 AUDIENCE: [INAUDIBLE] 400 00:20:49,877 --> 00:20:50,960 DENNIS FREEMAN: I'm sorry. 401 00:20:50,960 --> 00:20:51,380 I can't hear. 402 00:20:51,380 --> 00:20:52,796 AUDIENCE: Why is it not symmetric? 403 00:20:52,796 --> 00:21:00,165 So if you look at the amplitude of the system function, j 404 00:21:00,165 --> 00:21:01,516 omega, it's symmetric. 405 00:21:01,516 --> 00:21:05,432 But the Bode plots are not symmetric. 406 00:21:05,432 --> 00:21:07,390 DENNIS FREEMAN: The Bode plots are unsymmetric. 407 00:21:07,390 --> 00:21:09,556 Why are the Bode plots unsymmetric and the frequency 408 00:21:09,556 --> 00:21:10,930 response was symmetric? 409 00:21:10,930 --> 00:21:11,770 Somebody-- 410 00:21:11,770 --> 00:21:14,020 AUDIENCE: I get why, if you have a negative frequency. 411 00:21:14,020 --> 00:21:16,210 DENNIS FREEMAN: Negative frequency, exactly. 412 00:21:16,210 --> 00:21:18,875 Why do we have negative frequencies? 413 00:21:18,875 --> 00:21:19,750 AUDIENCE: [INAUDIBLE] 414 00:21:19,750 --> 00:21:23,696 DENNIS FREEMAN: Because time runs backwards. 415 00:21:23,696 --> 00:21:26,070 Negative frequency corresponds to running time backwards, 416 00:21:26,070 --> 00:21:26,850 right? 417 00:21:26,850 --> 00:21:30,150 The argument, it gets bigger when time gets smaller. 418 00:21:30,150 --> 00:21:30,780 OK. 419 00:21:30,780 --> 00:21:32,550 That's not the reason we do it, right? 420 00:21:32,550 --> 00:21:36,798 Why do we use negative frequencies? 421 00:21:36,798 --> 00:21:38,270 AUDIENCE: [INAUDIBLE] 422 00:21:38,270 --> 00:21:41,752 DENNIS FREEMAN: Why do we consider negative frequencies? 423 00:21:41,752 --> 00:21:42,700 AUDIENCE: [INAUDIBLE] 424 00:21:42,700 --> 00:21:44,033 DENNIS FREEMAN: To make it real. 425 00:21:44,033 --> 00:21:45,166 To make what real? 426 00:21:45,166 --> 00:21:46,812 AUDIENCE: A system. 427 00:21:46,812 --> 00:21:49,270 DENNIS FREEMAN: Why do we think about negative frequencies? 428 00:21:49,270 --> 00:21:49,770 Come on. 429 00:21:49,770 --> 00:21:50,348 Yeah. 430 00:21:50,348 --> 00:21:52,340 AUDIENCE: [INAUDIBLE] 431 00:21:52,340 --> 00:21:54,590 DENNIS FREEMAN: We want sines and cosines. 432 00:21:54,590 --> 00:21:57,320 It's easy to find complex exponentials. 433 00:21:57,320 --> 00:21:59,930 We like Euler, right? 434 00:21:59,930 --> 00:22:00,800 We like Euler. 435 00:22:00,800 --> 00:22:04,910 We like that cosine of omega t can 436 00:22:04,910 --> 00:22:08,115 be written as 1/2 e to the j omega t plus 1/2 437 00:22:08,115 --> 00:22:11,060 e to the minus j omega t. 438 00:22:11,060 --> 00:22:12,502 That's what we like. 439 00:22:12,502 --> 00:22:13,770 Whoops. 440 00:22:13,770 --> 00:22:17,050 I shouldn't put the parentheses there. 441 00:22:17,050 --> 00:22:19,410 That's what we like. 442 00:22:19,410 --> 00:22:21,357 We invent the negative frequencies. 443 00:22:21,357 --> 00:22:23,190 They are completely an invention of our own. 444 00:22:23,190 --> 00:22:26,940 When we do the Bode plots, we throw them away. 445 00:22:26,940 --> 00:22:30,610 We know that we can always figure them out 446 00:22:30,610 --> 00:22:32,980 because it was an invention of ours, 447 00:22:32,980 --> 00:22:34,480 so we don't need to write them down. 448 00:22:34,480 --> 00:22:36,370 If we ever needed them, we could figure them out. 449 00:22:36,370 --> 00:22:38,740 So we don't bother to write them down on the Bode plot. 450 00:22:38,740 --> 00:22:40,850 What's the low-frequency asymptote of H1? 451 00:22:43,610 --> 00:22:48,120 If omega 0 goes to 0, what's the value of H1? 452 00:22:48,120 --> 00:22:50,130 1. 453 00:22:50,130 --> 00:22:53,130 s equals j omega, omega equals 0. 454 00:22:53,130 --> 00:22:53,920 1. 455 00:22:53,920 --> 00:22:55,520 What's the log of 1? 456 00:22:55,520 --> 00:22:56,380 AUDIENCE: 0. 457 00:22:56,380 --> 00:22:57,280 DENNIS FREEMAN: 0. 458 00:22:57,280 --> 00:23:01,310 So the low-frequency asymptote for H1 is 0. 459 00:23:01,310 --> 00:23:03,150 What's the high-frequency asymptote of H1? 460 00:23:07,030 --> 00:23:11,168 If you put a high frequency- 461 00:23:11,168 --> 00:23:14,240 AUDIENCE: [INAUDIBLE] 462 00:23:14,240 --> 00:23:17,810 DENNIS FREEMAN: We're thinking about H1 of j omega. 463 00:23:17,810 --> 00:23:22,790 H1 of j omega is 1 over j omega plus 1. 464 00:23:22,790 --> 00:23:24,800 What happens when you make omega high? 465 00:23:24,800 --> 00:23:27,900 What happens to the magnitude? 466 00:23:27,900 --> 00:23:29,370 AUDIENCE: [INAUDIBLE] 467 00:23:29,370 --> 00:23:33,660 DENNIS FREEMAN: 1 over omega for the amplitude, right? 468 00:23:33,660 --> 00:23:35,159 You could say it approaches 0. 469 00:23:35,159 --> 00:23:37,200 That's kind of right, but we're interested in how 470 00:23:37,200 --> 00:23:39,120 it approaches 0. 471 00:23:39,120 --> 00:23:43,950 On a log plot, 0 is very far away. 472 00:23:43,950 --> 00:23:47,190 So we never think about being there. 473 00:23:47,190 --> 00:23:49,480 We think about getting there. 474 00:23:49,480 --> 00:23:52,380 So we get there with a slope of minus 1. 475 00:23:52,380 --> 00:23:53,250 So it's like this. 476 00:23:53,250 --> 00:23:55,770 What's the crossover point? 477 00:23:55,770 --> 00:23:56,400 Omega equals? 478 00:23:59,870 --> 00:24:02,960 What's the omega at the crossover point? 479 00:24:02,960 --> 00:24:04,190 AUDIENCE: 0. 480 00:24:04,190 --> 00:24:04,940 DENNIS FREEMAN: 0. 481 00:24:08,405 --> 00:24:12,860 AUDIENCE: We're plotting for negative [INAUDIBLE].. 482 00:24:12,860 --> 00:24:14,510 And I would think that since we're 483 00:24:14,510 --> 00:24:17,315 taking the absolute value [INAUDIBLE] 484 00:24:17,315 --> 00:24:20,780 and not having it just be 0. 485 00:24:20,780 --> 00:24:23,255 Because you're talking about as omega goes to 0, 486 00:24:23,255 --> 00:24:25,740 but not necessarily as omega goes to negative. 487 00:24:25,740 --> 00:24:28,190 DENNIS FREEMAN: We're thinking about the limit 488 00:24:28,190 --> 00:24:30,200 as we go towards 0, because we're 489 00:24:30,200 --> 00:24:32,420 plotting on the log of omega. 490 00:24:32,420 --> 00:24:39,170 0, omega equals 0, is way over there. 491 00:24:39,170 --> 00:24:41,442 Way, way, way off the blackboard. 492 00:24:41,442 --> 00:24:42,650 So we don't think about that. 493 00:24:42,650 --> 00:24:45,380 We think about how it got there. 494 00:24:45,380 --> 00:24:47,360 And it got there by being constant. 495 00:24:49,950 --> 00:24:52,770 The trend-- if you start at the critical frequency 496 00:24:52,770 --> 00:24:57,990 in the middle, the value becomes closer and closer to 1 497 00:24:57,990 --> 00:25:00,400 as you go to the left. 498 00:25:00,400 --> 00:25:05,280 That's why we say the asymptote is at log equals 0. 499 00:25:05,280 --> 00:25:06,450 Log of 1 is 0. 500 00:25:06,450 --> 00:25:09,500 The answer goes to 1. 501 00:25:09,500 --> 00:25:10,490 The log of 1 is 0. 502 00:25:10,490 --> 00:25:12,850 Does everybody know what I'm talking about? 503 00:25:12,850 --> 00:25:15,540 I'm getting a lot of blank stares. 504 00:25:15,540 --> 00:25:19,020 Then, if you try to go to a high frequency 505 00:25:19,020 --> 00:25:23,290 and think about the magnitude, you 506 00:25:23,290 --> 00:25:24,670 can go to a high enough frequency 507 00:25:24,670 --> 00:25:26,128 where the magnitude of this doesn't 508 00:25:26,128 --> 00:25:29,020 matter that that's there. 509 00:25:29,020 --> 00:25:31,900 The magnitude never matters that there's a j there. 510 00:25:31,900 --> 00:25:35,380 So the magnitude goes like 1 over omega. 511 00:25:35,380 --> 00:25:37,480 If you plot the function, 1 over omega, 512 00:25:37,480 --> 00:25:39,670 a reciprocal function on log coordinates, 513 00:25:39,670 --> 00:25:43,510 it turns into slope of minus 1. 514 00:25:43,510 --> 00:25:45,280 So that gives us this region where 515 00:25:45,280 --> 00:25:48,687 it has a slope of minus 1. 516 00:25:48,687 --> 00:25:50,395 What's the frequency at which they cross? 517 00:25:55,012 --> 00:25:56,446 AUDIENCE: [INAUDIBLE]. 518 00:25:56,446 --> 00:25:59,320 In log scale, it's only [INAUDIBLE].. 519 00:25:59,320 --> 00:26:01,630 DENNIS FREEMAN: On a log scale, it's omega equals 0. 520 00:26:01,630 --> 00:26:06,770 On a linear scale, it's omega equals 1. 521 00:26:06,770 --> 00:26:10,970 So the crossover point where 1 over omega-- 522 00:26:10,970 --> 00:26:13,460 the high-frequency asymptote is 1 ever omega. 523 00:26:13,460 --> 00:26:15,920 The low-frequency asymptote's 1. 524 00:26:15,920 --> 00:26:20,480 They are equal when 1 over omega equals 1. 525 00:26:20,480 --> 00:26:23,350 Omega equals 1, right? 526 00:26:23,350 --> 00:26:24,860 Log 0. 527 00:26:24,860 --> 00:26:29,600 So they cross at a point where the log is 0. 528 00:26:29,600 --> 00:26:31,640 Now, how about H2? 529 00:26:31,640 --> 00:26:34,040 What if I plot H2 on top of this? 530 00:26:34,040 --> 00:26:36,180 What's the low-frequency limit of H2? 531 00:26:39,568 --> 00:26:40,540 AUDIENCE: [INAUDIBLE] 532 00:26:40,540 --> 00:26:43,790 DENNIS FREEMAN: 1/10. 533 00:26:43,790 --> 00:26:46,760 What's the log of 1/10? 534 00:26:46,760 --> 00:26:48,000 Minus 1. 535 00:26:48,000 --> 00:26:57,160 So now, H2, if I do log of H2, it's going to-- 536 00:26:57,160 --> 00:26:59,800 so the low-frequency asymptote is 1/10. 537 00:26:59,800 --> 00:27:02,290 It's minus 1 on a log plot. 538 00:27:02,290 --> 00:27:04,510 What's the high-frequency asymptote for H2? 539 00:27:07,365 --> 00:27:10,850 AUDIENCE: [INAUDIBLE] 540 00:27:10,850 --> 00:27:11,870 DENNIS FREEMAN: 1/s. 541 00:27:11,870 --> 00:27:14,210 1 over omega. 542 00:27:14,210 --> 00:27:14,960 It's the same. 543 00:27:18,100 --> 00:27:20,710 If you go to a high enough frequency, 544 00:27:20,710 --> 00:27:23,470 the fact that there is divide by 10-- 545 00:27:23,470 --> 00:27:27,220 the fact that you've added 10 or added 1 doesn't matter. 546 00:27:27,220 --> 00:27:29,770 If omega gets sufficiently high, the thing you added to it 547 00:27:29,770 --> 00:27:31,810 has no relevance. 548 00:27:31,810 --> 00:27:33,880 You get the same high-frequency asymptote. 549 00:27:38,310 --> 00:27:43,830 So to transform the asymptotic view of the magnitude function 550 00:27:43,830 --> 00:27:48,375 from H1 to H2, what should you do to the curves? 551 00:27:48,375 --> 00:27:49,770 [INTERPOSING VOICES] 552 00:27:49,770 --> 00:27:51,660 DENNIS FREEMAN: Shift and shift. 553 00:27:51,660 --> 00:27:53,640 No scaling. 554 00:27:53,640 --> 00:27:54,930 That's why we like Bode. 555 00:27:54,930 --> 00:27:57,850 That's why we like log-log. 556 00:27:57,850 --> 00:28:01,770 Every pole looks the same. 557 00:28:01,770 --> 00:28:04,050 Every pole is straight across and down. 558 00:28:04,050 --> 00:28:09,382 Every zero looks the same, straight across and up. 559 00:28:09,382 --> 00:28:10,590 There's a critical frequency. 560 00:28:10,590 --> 00:28:13,920 That frequency is the frequency equal to the position 561 00:28:13,920 --> 00:28:15,780 of the pole or zero. 562 00:28:15,780 --> 00:28:18,941 That's all of the rules. 563 00:28:18,941 --> 00:28:19,440 OK. 564 00:28:19,440 --> 00:28:21,060 That's why we like this. 565 00:28:21,060 --> 00:28:21,940 Does that make sense? 566 00:28:21,940 --> 00:28:25,900 So the answer here was that we want 567 00:28:25,900 --> 00:28:27,900 to be able to shift horizontally and vertically. 568 00:28:27,900 --> 00:28:30,390 We don't need to scale. 569 00:28:30,390 --> 00:28:34,080 The shape of the curve is invariant to the position 570 00:28:34,080 --> 00:28:35,670 of the pole and zero. 571 00:28:35,670 --> 00:28:36,600 That's what we like. 572 00:28:36,600 --> 00:28:37,780 It's an invariant. 573 00:28:37,780 --> 00:28:40,820 We like invariant things. 574 00:28:40,820 --> 00:28:42,550 OK. 575 00:28:42,550 --> 00:28:48,290 Then, if you wanted to construct a more complicated system, 576 00:28:48,290 --> 00:28:50,205 it's also easy. 577 00:28:50,205 --> 00:28:52,330 If you want to construct a more complicated system, 578 00:28:52,330 --> 00:28:56,920 you can use the factor theorem to convert the system function, 579 00:28:56,920 --> 00:29:02,140 which for a system that's built out of integrators, adders, 580 00:29:02,140 --> 00:29:04,210 and gains. 581 00:29:04,210 --> 00:29:05,730 If you build a system out of that, 582 00:29:05,730 --> 00:29:10,600 the system function will always be a ratio of polynomials in s. 583 00:29:10,600 --> 00:29:13,180 We proved that a while back. 584 00:29:13,180 --> 00:29:16,780 You can always factor such things. 585 00:29:16,780 --> 00:29:19,030 We just found out a rule for how you think 586 00:29:19,030 --> 00:29:21,350 about the individual factors. 587 00:29:21,350 --> 00:29:23,290 And so all you need to worry about is 588 00:29:23,290 --> 00:29:25,870 the rule for combination. 589 00:29:25,870 --> 00:29:27,880 If you multiply a bunch of zeros and divide 590 00:29:27,880 --> 00:29:34,240 by the product of a bunch of poles, that's pretty easy. 591 00:29:34,240 --> 00:29:37,829 If you take the magnitude-- 592 00:29:37,829 --> 00:29:40,370 the magnitude of a product is the product of the magnitudes-- 593 00:29:40,370 --> 00:29:44,590 that's also pretty easy. 594 00:29:44,590 --> 00:29:47,160 Everybody's with me? 595 00:29:47,160 --> 00:29:48,910 And if you take the log, it's even easier. 596 00:29:51,650 --> 00:29:57,670 The log of a product is the sum of the logs. 597 00:29:57,670 --> 00:29:58,490 OK. 598 00:29:58,490 --> 00:30:01,150 Sum's easier to multiply. 599 00:30:01,150 --> 00:30:04,210 That's another reason we like this log-log thing. 600 00:30:04,210 --> 00:30:06,880 So all you need to do is think about each singularity. 601 00:30:06,880 --> 00:30:09,730 Each one of them can be represented by this and down 602 00:30:09,730 --> 00:30:11,770 or this and up. 603 00:30:11,770 --> 00:30:13,300 And compose them by adding. 604 00:30:13,300 --> 00:30:14,710 So it's really easy. 605 00:30:14,710 --> 00:30:18,840 So say I had this system function, a 0 at 0, 606 00:30:18,840 --> 00:30:21,655 a pole at minus 1, and a pole at minus 10. 607 00:30:25,430 --> 00:30:28,680 Think about the Bode representation. 608 00:30:28,680 --> 00:30:31,951 Bode's just a word that means think about the asymptotes. 609 00:30:31,951 --> 00:30:34,760 Think about the Bode representation for the top. 610 00:30:34,760 --> 00:30:39,060 Well, that grows linearly with omega. 611 00:30:39,060 --> 00:30:41,640 A function that grows linearly with omega 612 00:30:41,640 --> 00:30:45,150 has a log that grows with a slope of 1. 613 00:30:45,150 --> 00:30:48,440 So that's this. 614 00:30:48,440 --> 00:30:51,250 So that's the zero. 615 00:30:51,250 --> 00:30:55,240 That's the contribution to the magnitude function of the zero. 616 00:30:55,240 --> 00:30:57,430 The contribution to the magnitude function 617 00:30:57,430 --> 00:31:01,450 of the pole at minus 1 is this. 618 00:31:01,450 --> 00:31:04,090 The pole at minus 10 is this. 619 00:31:04,090 --> 00:31:07,770 All pole's look the same. 620 00:31:07,770 --> 00:31:09,880 So all you need to do is add them up. 621 00:31:09,880 --> 00:31:12,750 So you add the first two, you get the-- 622 00:31:12,750 --> 00:31:15,060 so backing up. 623 00:31:15,060 --> 00:31:16,020 0. 624 00:31:16,020 --> 00:31:17,350 Pole. 625 00:31:17,350 --> 00:31:18,900 Add them. 626 00:31:18,900 --> 00:31:21,690 In this region, the sum of a constant 627 00:31:21,690 --> 00:31:24,240 and a straight line sloping up is a straight line sloping up. 628 00:31:27,590 --> 00:31:30,890 In this region, the sum of a straight line sloping up 629 00:31:30,890 --> 00:31:35,576 with this sloping down is flat. 630 00:31:35,576 --> 00:31:36,700 So that's why you get that. 631 00:31:39,240 --> 00:31:41,880 Then, we add in this contribution. 632 00:31:41,880 --> 00:31:43,211 Add a constant to this. 633 00:31:43,211 --> 00:31:44,460 It just shifts it up and down. 634 00:31:44,460 --> 00:31:44,959 Who cares? 635 00:31:47,380 --> 00:31:48,510 That's easy. 636 00:31:48,510 --> 00:31:50,760 The important thing is that it breaks down. 637 00:31:50,760 --> 00:31:53,204 So the result of summing that is that it breaks down. 638 00:31:53,204 --> 00:31:55,889 Easy. 639 00:31:55,889 --> 00:31:57,930 So instead of thinking about a frequency response 640 00:31:57,930 --> 00:31:59,638 as something that's horribly complicated, 641 00:31:59,638 --> 00:32:02,100 we think about it as having parts that 642 00:32:02,100 --> 00:32:04,470 came from each pole and zero. 643 00:32:04,470 --> 00:32:06,990 So instead of thinking about it as a collection of arbitrary 644 00:32:06,990 --> 00:32:08,364 numbers at different frequencies, 645 00:32:08,364 --> 00:32:09,900 we think about it as a little part 646 00:32:09,900 --> 00:32:12,150 that comes from the first pole, another part that 647 00:32:12,150 --> 00:32:14,100 comes from the second pole, another part that 648 00:32:14,100 --> 00:32:15,450 comes from the first zero-- 649 00:32:15,450 --> 00:32:16,950 blah, blah, blah. 650 00:32:16,950 --> 00:32:19,290 It's a way of thinking about it, reducing 651 00:32:19,290 --> 00:32:23,672 the complexity, the conceptual complexity, 652 00:32:23,672 --> 00:32:24,755 of the frequency response. 653 00:32:28,560 --> 00:32:33,700 So the angles are the same. 654 00:32:33,700 --> 00:32:35,370 If we think about the low-frequency and 655 00:32:35,370 --> 00:32:39,660 high-frequency behavior of the angle starting with a 0. 656 00:32:39,660 --> 00:32:43,730 For low frequencies, the angle is 0. 657 00:32:43,730 --> 00:32:46,520 So that's the dot. 658 00:32:46,520 --> 00:32:48,004 For high frequencies, as frequency 659 00:32:48,004 --> 00:32:49,420 goes higher and higher and higher, 660 00:32:49,420 --> 00:32:51,710 the angle stands straighter and straighter up. 661 00:32:51,710 --> 00:32:53,060 The asymptotic value is? 662 00:32:59,310 --> 00:33:01,980 Pi over 2. 663 00:33:01,980 --> 00:33:05,040 So we have two asymptotes, 0 and pi over 2. 664 00:33:05,040 --> 00:33:07,710 If we plot that on log-log axes, we 665 00:33:07,710 --> 00:33:11,710 get something slightly more complicated. 666 00:33:11,710 --> 00:33:14,105 The blue line shows the calculated value. 667 00:33:17,010 --> 00:33:19,110 The red line shows a way of thinking 668 00:33:19,110 --> 00:33:22,560 about that from a straight line approximation point of view. 669 00:33:22,560 --> 00:33:26,100 Very nice construction is put a straight line 670 00:33:26,100 --> 00:33:29,700 starting one decade-- 671 00:33:29,700 --> 00:33:33,870 a factor of 10, minus 1 on a log scale. 672 00:33:33,870 --> 00:33:39,520 Draw a line from minus 1 to plus 1, a straight line. 673 00:33:39,520 --> 00:33:42,000 And you actually get a very good approximation 674 00:33:42,000 --> 00:33:45,004 to the angle function over the whole range. 675 00:33:45,004 --> 00:33:46,920 Notice that there are two critical frequencies 676 00:33:46,920 --> 00:33:48,572 associated with the phase. 677 00:33:48,572 --> 00:33:51,030 There was one critical frequency associated with magnitude. 678 00:33:54,270 --> 00:33:57,120 What was the crossing point of the low- and high-frequency 679 00:33:57,120 --> 00:33:59,000 asymptotes? 680 00:33:59,000 --> 00:34:01,130 In phase, we think about that same thing, 681 00:34:01,130 --> 00:34:02,830 but then we bump it up and down 1. 682 00:34:02,830 --> 00:34:05,400 So there are two critical frequencies. 683 00:34:05,400 --> 00:34:10,170 The same critical frequency we use with the magnitude 684 00:34:10,170 --> 00:34:13,380 plus or minus 1. 685 00:34:13,380 --> 00:34:15,560 The same sort of thing happens with a pole, 686 00:34:15,560 --> 00:34:18,260 but now it's upside down. 687 00:34:18,260 --> 00:34:19,850 Other than that, it's identical. 688 00:34:23,400 --> 00:34:25,139 Same thing. 689 00:34:25,139 --> 00:34:31,400 Having calculated the phase for a single pole or zero, 690 00:34:31,400 --> 00:34:33,679 it's easy to think about how you would generalize 691 00:34:33,679 --> 00:34:36,440 to multiple poles and zeros. 692 00:34:36,440 --> 00:34:41,420 The angle of a product is the sum of the angles. 693 00:34:41,420 --> 00:34:43,806 Don't need to take the log this way. 694 00:34:43,806 --> 00:34:45,389 This time, when we're doing the phase, 695 00:34:45,389 --> 00:34:46,330 we don't need to take the log. 696 00:34:46,330 --> 00:34:48,000 There's a way of thinking about phase. 697 00:34:48,000 --> 00:34:51,960 Because it's in the exponent, e to the j angle 698 00:34:51,960 --> 00:34:53,639 by Euler's equation. 699 00:34:53,639 --> 00:34:56,219 There's a way of thinking about the logs already 700 00:34:56,219 --> 00:34:58,170 in the angle function, right? 701 00:34:58,170 --> 00:35:02,650 The angle was in the exponent, e to the j omega. 702 00:35:02,650 --> 00:35:05,040 So since the angle is already up in the e, 703 00:35:05,040 --> 00:35:06,320 it's kind of like an-- 704 00:35:06,320 --> 00:35:09,650 it's already a logarithmic function. 705 00:35:09,650 --> 00:35:11,530 And that shows up here. 706 00:35:11,530 --> 00:35:16,080 The angle of a product is the sum of the angles. 707 00:35:16,080 --> 00:35:21,739 So same sort of thing happens if we had a complicated function. 708 00:35:21,739 --> 00:35:24,280 Just think about the angle that results for each of the poles 709 00:35:24,280 --> 00:35:26,830 and zeros and add them. 710 00:35:26,830 --> 00:35:37,180 The angle associated with the zero at 0 is pi over 2. 711 00:35:37,180 --> 00:35:39,900 j is the same as e to the j pi over 2. 712 00:35:39,900 --> 00:35:41,100 e to the j pi over 2. 713 00:35:41,100 --> 00:35:44,310 The angle's pi over 2. 714 00:35:44,310 --> 00:35:50,040 So the angle associated with this is always pi over 2. 715 00:35:50,040 --> 00:35:54,780 The angle associated with the pole at minus 1. 716 00:35:54,780 --> 00:35:58,980 Poles in the left-half plane cause the angle to start out 717 00:35:58,980 --> 00:36:01,560 at 0 and go negative. 718 00:36:01,560 --> 00:36:04,690 So we start out at 0 and go negative. 719 00:36:04,690 --> 00:36:07,845 We find the critical frequency labeled in this axis by 0. 720 00:36:07,845 --> 00:36:11,740 s equals 1 is the critical frequency. 721 00:36:11,740 --> 00:36:14,880 So we go up and down 1 unit and we 722 00:36:14,880 --> 00:36:17,950 draw the straight-line approximation. 723 00:36:17,950 --> 00:36:20,360 s equals 10 is the same thing, except now it's 724 00:36:20,360 --> 00:36:21,860 shifted to a higher frequency. 725 00:36:21,860 --> 00:36:26,910 Factor of 10, units shift on a log plot. 726 00:36:26,910 --> 00:36:30,240 And now all we do is sum them, add the first to the second, 727 00:36:30,240 --> 00:36:31,380 add the third. 728 00:36:31,380 --> 00:36:35,070 That's our angle approximation. 729 00:36:35,070 --> 00:36:36,780 Again, the idea is to take something 730 00:36:36,780 --> 00:36:38,647 that's conceptually hard-- 731 00:36:38,647 --> 00:36:40,230 what's the value of the angle function 732 00:36:40,230 --> 00:36:44,090 as a function of frequency-- and turn it into something simple. 733 00:36:44,090 --> 00:36:45,830 A few straight line segments associated 734 00:36:45,830 --> 00:36:46,940 with every pole and zero. 735 00:36:50,020 --> 00:36:53,820 So this is just a summary. 736 00:36:53,820 --> 00:36:57,090 Because we can represent a system that's 737 00:36:57,090 --> 00:37:03,430 composed of integrators, summers, and gains 738 00:37:03,430 --> 00:37:05,957 by a linear differential equation 739 00:37:05,957 --> 00:37:07,540 with constant coefficients, it follows 740 00:37:07,540 --> 00:37:18,490 that the system function is a quotient of polynomials in s. 741 00:37:18,490 --> 00:37:20,450 Because of the fundamental theorem in algebra, 742 00:37:20,450 --> 00:37:21,930 there's n roots. 743 00:37:21,930 --> 00:37:24,614 Because of factor theorem, you can break it up into factors. 744 00:37:24,614 --> 00:37:27,030 Because of all that, we can think about them one at a time 745 00:37:27,030 --> 00:37:28,950 and glue them together. 746 00:37:28,950 --> 00:37:31,101 Gluing them together in the case of the magnitude 747 00:37:31,101 --> 00:37:33,600 works best if you use the log because then the product turns 748 00:37:33,600 --> 00:37:36,330 into a sum. 749 00:37:36,330 --> 00:37:38,100 In the case of the angle, it's for free 750 00:37:38,100 --> 00:37:39,930 because the angle is, in some sense, 751 00:37:39,930 --> 00:37:43,672 already a logarithmic function. 752 00:37:43,672 --> 00:37:45,670 OK. 753 00:37:45,670 --> 00:37:47,710 OK, we'll see if you got it. 754 00:37:47,710 --> 00:37:49,544 So here is a complicated frequency response. 755 00:37:49,544 --> 00:37:51,876 This is the straight-line approximation to the frequency 756 00:37:51,876 --> 00:37:53,020 response of a system. 757 00:37:53,020 --> 00:37:53,850 Which system? 758 00:39:20,160 --> 00:39:21,660 So which of the system functions-- 759 00:39:21,660 --> 00:39:24,780 1, 2, 3, 4, or none of the above-- 760 00:39:24,780 --> 00:39:29,160 is represented by the Bode straight-line plot 761 00:39:29,160 --> 00:39:30,360 showed above? 762 00:39:30,360 --> 00:39:32,010 Raise your hands, which one is better-- 763 00:39:32,010 --> 00:39:33,755 1, 2, 3, 4, or 5? 764 00:39:37,395 --> 00:39:38,510 Ah, 100%. 765 00:39:38,510 --> 00:39:40,070 That's exactly the right answer. 766 00:39:40,070 --> 00:39:41,240 Wonderful. 767 00:39:41,240 --> 00:39:43,620 The idea is that Bode is easy. 768 00:39:43,620 --> 00:39:45,690 That's why we do it. 769 00:39:45,690 --> 00:39:46,280 It's easy. 770 00:39:46,280 --> 00:39:47,270 So tell me a rule. 771 00:39:47,270 --> 00:39:48,830 How do I think about this one? 772 00:39:48,830 --> 00:39:49,820 What would happen here? 773 00:39:49,820 --> 00:39:51,804 What's this saying? 774 00:39:51,804 --> 00:39:53,470 What would be the Bode plot of this one? 775 00:39:53,470 --> 00:39:54,344 Sketch it in the air. 776 00:39:57,660 --> 00:39:58,610 Exactly. 777 00:39:58,610 --> 00:40:02,841 You start at 0, right? 778 00:40:02,841 --> 00:40:03,340 OK. 779 00:40:03,340 --> 00:40:04,420 So start at 0. 780 00:40:04,420 --> 00:40:07,210 Then what? 781 00:40:07,210 --> 00:40:10,480 What do you run into first? 782 00:40:10,480 --> 00:40:11,590 You start at 0. 783 00:40:11,590 --> 00:40:13,840 What do you run into first when you're doing this guy? 784 00:40:18,730 --> 00:40:21,820 You run into this factor, or that factor, or that factor 785 00:40:21,820 --> 00:40:23,090 first? 786 00:40:23,090 --> 00:40:24,360 AUDIENCE: [INAUDIBLE] 787 00:40:24,360 --> 00:40:26,220 DENNIS FREEMAN: Yeah, the left one. 788 00:40:26,220 --> 00:40:28,440 So think about the order. 789 00:40:28,440 --> 00:40:32,070 By how close were they to the origin? 790 00:40:32,070 --> 00:40:34,710 Think about them in order from the origin. 791 00:40:34,710 --> 00:40:37,010 So you hit the first one first, the one at 10, 792 00:40:37,010 --> 00:40:38,350 then the one at 100. 793 00:40:38,350 --> 00:40:41,600 This would start flat, break down at 1, 794 00:40:41,600 --> 00:40:43,995 break down again at 10, break down again at 100. 795 00:40:43,995 --> 00:40:45,120 That's not the right shape. 796 00:40:48,150 --> 00:40:51,880 This one starts with a 0. 797 00:40:51,880 --> 00:40:54,600 That's a break up. 798 00:40:54,600 --> 00:40:59,870 The 0 happens at frequency 1. 799 00:40:59,870 --> 00:41:02,060 So that's log frequency 0. 800 00:41:02,060 --> 00:41:04,610 That's right. 801 00:41:04,610 --> 00:41:07,920 Then, you break down, break down. 802 00:41:07,920 --> 00:41:09,230 OK. 803 00:41:09,230 --> 00:41:11,840 So break up, break up, break down. 804 00:41:11,840 --> 00:41:14,850 Except that it would actually break down first. 805 00:41:14,850 --> 00:41:16,310 So the way to think about this one 806 00:41:16,310 --> 00:41:20,090 is break down, break up, break up 807 00:41:20,090 --> 00:41:23,090 ordered from going away from the origin. 808 00:41:23,090 --> 00:41:24,050 OK. 809 00:41:24,050 --> 00:41:26,900 So the point is it's very easy to take a system function 810 00:41:26,900 --> 00:41:31,060 and immediately draw the frequency response. 811 00:41:31,060 --> 00:41:36,010 That lets you take a pole-zero representation, which 812 00:41:36,010 --> 00:41:39,070 can be very concise in terms of the number of numbers 813 00:41:39,070 --> 00:41:41,890 you need to know, and quickly map out the frequency 814 00:41:41,890 --> 00:41:44,200 response, which can be very intuitive for thinking 815 00:41:44,200 --> 00:41:45,449 about how the system responds. 816 00:41:47,820 --> 00:41:49,260 OK. 817 00:41:49,260 --> 00:41:54,650 So one more issue. 818 00:41:54,650 --> 00:41:58,220 We don't really like log plots when it gets down 819 00:41:58,220 --> 00:42:03,560 to talking about the frequency at which the log was 7.129. 820 00:42:03,560 --> 00:42:06,590 We just don't like that because nobody measures frequency 821 00:42:06,590 --> 00:42:10,010 in the frequency whose log is x. 822 00:42:10,010 --> 00:42:12,920 So what we usually do, instead of plotting 823 00:42:12,920 --> 00:42:18,901 versus the log of omega, we plot versus omega on a log scale. 824 00:42:18,901 --> 00:42:19,400 OK. 825 00:42:19,400 --> 00:42:20,600 Very reasonable. 826 00:42:20,600 --> 00:42:25,710 We just put tick marks and we just label them exponentially. 827 00:42:25,710 --> 00:42:29,660 So the tick mark associated with omega is labeled 1, not 0. 828 00:42:32,270 --> 00:42:34,920 That's frequency 1. 829 00:42:34,920 --> 00:42:38,960 The tick mark associated with frequency 10 is labeled 10. 830 00:42:38,960 --> 00:42:42,350 What that means, though, is that the growth inside that interval 831 00:42:42,350 --> 00:42:45,074 is on a log scale. 832 00:42:45,074 --> 00:42:46,490 That's why we write log scale here 833 00:42:46,490 --> 00:42:50,030 to keep reminding ourselves that the ticks between 1 and 10-- 834 00:42:50,030 --> 00:42:53,330 1, 2, 3, 4, 5-- 835 00:42:53,330 --> 00:42:54,710 are not uniformly spaced. 836 00:42:54,710 --> 00:42:56,427 They're log spaced. 837 00:42:56,427 --> 00:42:58,760 The distance between 1 and 2 is bigger than the distance 838 00:42:58,760 --> 00:43:00,710 between 9 and 10. 839 00:43:00,710 --> 00:43:03,350 That's the way logs work. 840 00:43:03,350 --> 00:43:04,080 OK. 841 00:43:04,080 --> 00:43:07,230 So we think about frequency on a log scale rather than 842 00:43:07,230 --> 00:43:08,460 log frequency. 843 00:43:08,460 --> 00:43:12,510 Similarly, we think about amplitude on a dB scale 844 00:43:12,510 --> 00:43:14,330 rather than log amplitude. 845 00:43:14,330 --> 00:43:17,220 dB is for Alexander Graham Bell. 846 00:43:17,220 --> 00:43:19,970 It's decibel. 847 00:43:19,970 --> 00:43:24,510 A bell is a factor of 10. 848 00:43:24,510 --> 00:43:26,560 It's a little unintuitive, a factor of 10 849 00:43:26,560 --> 00:43:32,110 is labeled 20, because Bell was really thinking energy. 850 00:43:32,110 --> 00:43:33,520 And you have to square energy. 851 00:43:33,520 --> 00:43:36,260 You have to square voltage to get energy. 852 00:43:36,260 --> 00:43:39,000 Bell was thinking a factor of 10 in energy. 853 00:43:39,000 --> 00:43:40,990 We like to think amplitude. 854 00:43:40,990 --> 00:43:46,471 Therefore, a factor of 10 for us would be 20 decibels. 855 00:43:46,471 --> 00:43:46,970 20. 856 00:43:46,970 --> 00:43:48,720 OK, so that's a little weird. 857 00:43:48,720 --> 00:43:52,190 So we will think about this axis in decibels 858 00:43:52,190 --> 00:43:56,940 and we'll think about that one in decades. 859 00:43:56,940 --> 00:43:59,390 And so what that means is that the slopes are 860 00:43:59,390 --> 00:44:01,880 no longer minus 1 and 1. 861 00:44:01,880 --> 00:44:06,270 The slopes are 20 decibels per decade, 862 00:44:06,270 --> 00:44:08,344 or minus 20 decibels per decade. 863 00:44:08,344 --> 00:44:10,010 Now, this is all completely meaningless. 864 00:44:10,010 --> 00:44:13,170 It's a slope of 1. 865 00:44:13,170 --> 00:44:17,190 And you could equally have labeled this axis in dB 866 00:44:17,190 --> 00:44:19,110 and this axis on a log scale, and then you 867 00:44:19,110 --> 00:44:24,600 would have the slope of 1 decade per 20 dB. 868 00:44:24,600 --> 00:44:26,550 And that, too, would be 1. 869 00:44:26,550 --> 00:44:29,350 We just don't do it that way. 870 00:44:29,350 --> 00:44:29,850 OK. 871 00:44:29,850 --> 00:44:30,433 What do we do? 872 00:44:30,433 --> 00:44:34,680 What we do do is frequency on a log scale. 873 00:44:34,680 --> 00:44:40,440 So therefore, the unit of frequency is the decade. 874 00:44:40,440 --> 00:44:42,300 You have a frequency and a decade 875 00:44:42,300 --> 00:44:45,750 higher and a decade higher and a decade higher. 876 00:44:45,750 --> 00:44:49,050 The unit of frequency is decade or octave. 877 00:44:49,050 --> 00:44:51,780 Octave is a factor of 2. 878 00:44:51,780 --> 00:44:53,130 Octave higher, factor of 2. 879 00:44:53,130 --> 00:44:54,180 Octave higher. 880 00:44:54,180 --> 00:44:56,790 Octave makes more sense if you're a musician, right? 881 00:44:56,790 --> 00:45:01,220 The distance between two C's is an octave. 882 00:45:01,220 --> 00:45:02,300 So that's a dB scale. 883 00:45:05,150 --> 00:45:07,265 And that results in a little bit of funny math. 884 00:45:10,610 --> 00:45:13,520 So if you convert linear measures of amplitude 885 00:45:13,520 --> 00:45:19,390 to decibel measures, a factor of 1 in amplitude 886 00:45:19,390 --> 00:45:22,000 is a factor of 0 dB. 887 00:45:22,000 --> 00:45:24,602 A factor of 10 is 20 dB. 888 00:45:24,602 --> 00:45:25,810 We already talked about that. 889 00:45:25,810 --> 00:45:28,150 There are some convenient middle grounds. 890 00:45:28,150 --> 00:45:30,424 2 is 6 dB. 891 00:45:30,424 --> 00:45:31,880 That's a little weird. 892 00:45:31,880 --> 00:45:35,112 So we'll go around saying it's 6 dB and we mean 2. 893 00:45:35,112 --> 00:45:39,240 But every engineer in the world will call it 6 dB. 894 00:45:39,240 --> 00:45:43,670 Half of that is the square root of 2, which is 3 dB. 895 00:45:43,670 --> 00:45:46,280 It's just convenient, at least it 896 00:45:46,280 --> 00:45:48,710 is after you spend 30 years doing that. 897 00:45:48,710 --> 00:45:50,420 The other point from the slide is 898 00:45:50,420 --> 00:45:54,550 that the asymptotic responses are really quite good. 899 00:45:54,550 --> 00:45:57,490 The magnitude for a single pole deviates 900 00:45:57,490 --> 00:46:01,750 from the straight-line approximation by only 3 dB. 901 00:46:01,750 --> 00:46:05,260 I can hear sounds that range 120 dB. 902 00:46:05,260 --> 00:46:08,290 That's the sense in which 3 is small. 903 00:46:08,290 --> 00:46:11,080 The kinds of signals we work with every day 904 00:46:11,080 --> 00:46:14,110 have ranges that are big compared to 3. 905 00:46:14,110 --> 00:46:16,540 So we think of 3 as a small thing. 906 00:46:16,540 --> 00:46:18,890 The phases are within 6 degrees. 907 00:46:18,890 --> 00:46:24,130 And for a lot of applications, 6 degrees is a small number. 908 00:46:24,130 --> 00:46:26,950 Again, we're thinking 6 out of 180. 909 00:46:26,950 --> 00:46:30,240 So it's a small fraction of a cycle. 910 00:46:30,240 --> 00:46:33,320 OK, this is a good thing-- 911 00:46:33,320 --> 00:46:35,320 especially the fact that it's a trick question-- 912 00:46:35,320 --> 00:46:37,140 for you to practice for the exam. 913 00:46:37,140 --> 00:46:38,890 So let me skip it in the interest of time, 914 00:46:38,890 --> 00:46:41,730 because there's one more important thing. 915 00:46:41,730 --> 00:46:44,480 Don't look at the answer. 916 00:46:44,480 --> 00:46:47,470 Use this as practice. 917 00:46:47,470 --> 00:46:50,890 There's one more important thing in trying 918 00:46:50,890 --> 00:46:56,820 to reduce poles and zeros to a frequency response. 919 00:46:56,820 --> 00:46:59,610 And that is that when you use the fundamental theorem 920 00:46:59,610 --> 00:47:04,590 of algebra, even though the polynomial can 921 00:47:04,590 --> 00:47:08,980 have real-valued coefficients, that does not mean the roots 922 00:47:08,980 --> 00:47:11,250 are real. 923 00:47:11,250 --> 00:47:16,090 Right 924 00:47:16,090 --> 00:47:19,210 The roots to a polynomial with real value coefficients 925 00:47:19,210 --> 00:47:23,460 can have complex parts. 926 00:47:23,460 --> 00:47:25,650 So the remaining thing I want to talk about 927 00:47:25,650 --> 00:47:30,910 is, what do you do with these poles that have complex parts? 928 00:47:33,610 --> 00:47:37,740 The imaginary part has to do with oscillations. 929 00:47:37,740 --> 00:47:41,110 And so we want to think about, what's a Bode plot looks like? 930 00:47:41,110 --> 00:47:44,040 What's the asymptotes look like when 931 00:47:44,040 --> 00:47:46,000 you have a system like this? 932 00:47:46,000 --> 00:47:50,041 This was a system that has a mass spring, and dashpot. 933 00:47:50,041 --> 00:47:51,915 You should have done this by homework by now. 934 00:47:55,700 --> 00:47:58,160 The differential equation is second order. 935 00:47:58,160 --> 00:48:03,500 It has real-valued coefficients, but the holes are complex. 936 00:48:03,500 --> 00:48:06,290 What happens with complex roots? 937 00:48:06,290 --> 00:48:10,460 Well, if the com-- 938 00:48:10,460 --> 00:48:14,840 if the polynomial had real-valued coefficients, 939 00:48:14,840 --> 00:48:18,530 complex roots come in pairs. 940 00:48:18,530 --> 00:48:21,560 That's the only way to take a complex number 941 00:48:21,560 --> 00:48:23,870 and end up with a product that's all real. 942 00:48:23,870 --> 00:48:25,190 You take the complex number. 943 00:48:25,190 --> 00:48:27,410 It has to be paired with its complex conjugate. 944 00:48:27,410 --> 00:48:34,550 So that when you pair them, the result has real coefficients. 945 00:48:34,550 --> 00:48:36,920 So we only need to worry about the case 946 00:48:36,920 --> 00:48:42,830 when the poles or zeros come in complex pairs. 947 00:48:42,830 --> 00:48:45,800 And for that purpose, it's convenient to think about-- 948 00:48:45,800 --> 00:48:50,045 you might think that if you had mass-spring dashpot system, 949 00:48:50,045 --> 00:48:52,580 you might be expecting something like s 950 00:48:52,580 --> 00:48:57,350 squared m plus sb plus k, something like that. 951 00:48:57,350 --> 00:48:59,090 Mass-spring and dashpot. 952 00:48:59,090 --> 00:49:00,980 Three parameters. 953 00:49:00,980 --> 00:49:03,594 Well, you don't really need three. 954 00:49:03,594 --> 00:49:05,760 Three is nice because it has an association with how 955 00:49:05,760 --> 00:49:07,460 stiff is this thing and how massive 956 00:49:07,460 --> 00:49:08,862 is that thing, et cetera. 957 00:49:08,862 --> 00:49:10,820 But in terms of thinking about poles and zeros, 958 00:49:10,820 --> 00:49:12,650 you don't need to think about all three. 959 00:49:12,650 --> 00:49:18,140 First off, you could divide the top and bottom by k 960 00:49:18,140 --> 00:49:21,770 and the 1/k then is just a gain factor. 961 00:49:21,770 --> 00:49:22,830 Gains are easy. 962 00:49:22,830 --> 00:49:24,420 They don't affect shape. 963 00:49:24,420 --> 00:49:24,920 OK. 964 00:49:24,920 --> 00:49:26,370 We don't care about that one. 965 00:49:26,370 --> 00:49:29,330 So we went from 3 to 2. 966 00:49:29,330 --> 00:49:31,970 Now, there's another simplification 967 00:49:31,970 --> 00:49:35,970 because all of these are going to be oscillatory. 968 00:49:35,970 --> 00:49:39,330 Pole pairs work in an oscillatory fashion. 969 00:49:39,330 --> 00:49:43,170 This has a natural frequency that if I didn't shake it, 970 00:49:43,170 --> 00:49:45,240 it has a preferred frequency. 971 00:49:45,240 --> 00:49:50,250 If I divide by that preferred frequency, omega 0, 972 00:49:50,250 --> 00:49:52,500 I can get rid of another parameter. 973 00:49:52,500 --> 00:49:53,610 So don't think about s. 974 00:49:53,610 --> 00:49:56,940 Think about s over omega 0. 975 00:49:56,940 --> 00:50:00,591 Now, frequencies are normalized to 1. 976 00:50:00,591 --> 00:50:01,090 OK. 977 00:50:01,090 --> 00:50:03,010 So by dividing by omega 0, I turn 978 00:50:03,010 --> 00:50:06,220 every frequency-dependent system into something 979 00:50:06,220 --> 00:50:10,360 whose best frequency is near omega equals 1. 980 00:50:10,360 --> 00:50:13,990 Then finally, for my third parameter, 981 00:50:13,990 --> 00:50:16,406 if I write my third parameter as 1 over q, 982 00:50:16,406 --> 00:50:18,280 something very magical and wonderful happens. 983 00:50:21,460 --> 00:50:25,280 The roots fall on a circle. 984 00:50:25,280 --> 00:50:30,140 So what I want to show here then is as I change q, 985 00:50:30,140 --> 00:50:31,790 I don't need to think about omega. 986 00:50:31,790 --> 00:50:34,266 I can plot this on an s over omega 0 plane 987 00:50:34,266 --> 00:50:35,390 and it works for all omega. 988 00:50:39,110 --> 00:50:40,300 I do need to worry about q. 989 00:50:40,300 --> 00:50:47,630 So if I change q, here q is 1/2. 990 00:50:47,630 --> 00:50:48,470 So there's q of 1/2. 991 00:50:51,790 --> 00:50:53,970 1/4. 992 00:50:53,970 --> 00:50:54,936 1/8. 993 00:50:54,936 --> 00:50:58,570 Excuse me, I'm doing it backwards. 994 00:50:58,570 --> 00:51:02,190 I want 1/q to equal-- 995 00:51:02,190 --> 00:51:05,070 I want 1 over 2q to equal 1/2. 996 00:51:05,070 --> 00:51:06,750 I need q equals 1. 997 00:51:06,750 --> 00:51:07,600 OK, that's better. 998 00:51:07,600 --> 00:51:10,585 That's q equals 1. 999 00:51:10,585 --> 00:51:15,900 q equals 2 in order to half the distance to the origin-- 1000 00:51:15,900 --> 00:51:17,445 4, 8, 16. 1001 00:51:20,310 --> 00:51:23,390 Notice that as I change-- now watch this side. 1002 00:51:23,390 --> 00:51:26,294 So q equals 1. 1003 00:51:26,294 --> 00:51:28,660 So q equals 1, q equals 1. 1004 00:51:31,630 --> 00:51:33,380 Low-frequency magnitude is flat. 1005 00:51:33,380 --> 00:51:36,110 High-frequency is slope of 2. 1006 00:51:36,110 --> 00:51:37,590 Sloping down with minus 2. 1007 00:51:40,990 --> 00:51:51,060 As I change q from 1 to 2, 4, 8, 16, the peak gets bigger. 1008 00:51:51,060 --> 00:51:53,280 In fact, if you do a little bit of math, 1009 00:51:53,280 --> 00:51:59,310 you can show that the peak gets bigger with q. 1010 00:51:59,310 --> 00:52:02,800 The peak value, if you measure how big 1011 00:52:02,800 --> 00:52:06,430 is the peak compared to where is the crossover, 1012 00:52:06,430 --> 00:52:10,620 that distance is a factor of q. 1013 00:52:10,620 --> 00:52:13,480 Similarly-- and you can reason about that 1014 00:52:13,480 --> 00:52:14,470 with vector diagrams. 1015 00:52:14,470 --> 00:52:17,230 And we'll do homework problems to practice that. 1016 00:52:17,230 --> 00:52:19,900 Similarly, it got peakier. 1017 00:52:19,900 --> 00:52:22,120 It got sharper. 1018 00:52:22,120 --> 00:52:24,040 If you do the vector story to try 1019 00:52:24,040 --> 00:52:26,160 to figure out why it got peakier, 1020 00:52:26,160 --> 00:52:28,030 the width turns out to be 1/q. 1021 00:52:30,590 --> 00:52:32,130 So that's kind of impulse-y. 1022 00:52:32,130 --> 00:52:34,100 The height got bigger with q. 1023 00:52:34,100 --> 00:52:37,280 The width got bigger with 1/q. 1024 00:52:37,280 --> 00:52:39,599 The product always 1. 1025 00:52:39,599 --> 00:52:41,140 So that's a way of thinking about why 1026 00:52:41,140 --> 00:52:42,760 it got peaky that way. 1027 00:52:42,760 --> 00:52:45,940 And finally, if you think about the angle, the angle changes. 1028 00:52:45,940 --> 00:52:51,750 As you make q bigger and bigger, the angle changes very quickly. 1029 00:52:51,750 --> 00:52:55,300 And it turns out that the angle changes abruptly 1030 00:52:55,300 --> 00:52:56,950 over the same bandwidth. 1031 00:52:56,950 --> 00:52:59,080 Bandwidth is how many frequencies are there 1032 00:52:59,080 --> 00:53:03,430 between the low-frequency part and the high-frequency part. 1033 00:53:03,430 --> 00:53:07,480 The phase change over the bandwidth is always pi over 2. 1034 00:53:07,480 --> 00:53:08,020 OK. 1035 00:53:08,020 --> 00:53:09,460 So that's the whole story then. 1036 00:53:09,460 --> 00:53:12,070 Think about isolating the poles on the real axes. 1037 00:53:12,070 --> 00:53:13,090 They're just this. 1038 00:53:13,090 --> 00:53:16,000 Isolated zeros, they're just this. 1039 00:53:16,000 --> 00:53:19,400 Pairs can be more complicated because they can be peaky. 1040 00:53:19,400 --> 00:53:19,900 OK. 1041 00:53:19,900 --> 00:53:21,450 Thanks.