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 00:00:26,660 --> 00:00:29,640 DENNIS FREEMAN: So hello and welcome. 9 00:00:29,640 --> 00:00:32,100 No particular announcements today other than, 10 00:00:32,100 --> 00:00:34,186 in the unrelenting tradition of MIT, 11 00:00:34,186 --> 00:00:36,060 we're going to forge straight ahead, and keep 12 00:00:36,060 --> 00:00:39,054 going with new material. 13 00:00:39,054 --> 00:00:41,220 Today, we're going to talk about frequency response. 14 00:00:41,220 --> 00:00:44,760 And just to set the context, remember that last time we 15 00:00:44,760 --> 00:00:49,170 talked about a way to characterize a system, 16 00:00:49,170 --> 00:00:52,980 in terms of a single signal. 17 00:00:52,980 --> 00:00:55,230 By knowing the impulse response of a CTE system, 18 00:00:55,230 --> 00:00:59,010 you could characterize the response to any system 19 00:00:59,010 --> 00:01:02,820 by thinking about the response to any signal 20 00:01:02,820 --> 00:01:05,519 by thinking about the output as the convolution of the impulse 21 00:01:05,519 --> 00:01:09,510 response with the input signal. 22 00:01:09,510 --> 00:01:12,030 And that gave rise to a very compact way of thinking 23 00:01:12,030 --> 00:01:14,550 about the output of a system. 24 00:01:14,550 --> 00:01:16,380 We think about the output y as being 25 00:01:16,380 --> 00:01:20,940 x convolved with h, regardless of whether it's dt or ct. 26 00:01:20,940 --> 00:01:24,210 And the way that you carry out that convolution 27 00:01:24,210 --> 00:01:27,420 is, in fact, very similar in the two systems. 28 00:01:27,420 --> 00:01:28,980 That way of thinking about a system 29 00:01:28,980 --> 00:01:34,830 is particularly useful for certain kinds of systems. 30 00:01:34,830 --> 00:01:38,070 The examples we looked at last time were optical. 31 00:01:38,070 --> 00:01:40,170 In the case of an optical microscope, 32 00:01:40,170 --> 00:01:42,480 it's very convenient to think about the effect 33 00:01:42,480 --> 00:01:48,070 of the microscope is to blur the target. 34 00:01:48,070 --> 00:01:51,960 So whatever the target was, it always comes out blurrier, 35 00:01:51,960 --> 00:01:54,210 and that blur function is something that in optics we 36 00:01:54,210 --> 00:01:55,501 call the point spread function. 37 00:01:55,501 --> 00:01:59,520 The point spread function in 6003 terms 38 00:01:59,520 --> 00:02:03,960 is simply the three dimensional impulse response. 39 00:02:03,960 --> 00:02:06,060 We can think about three dimensions very similar 40 00:02:06,060 --> 00:02:07,726 to the way we think about one dimension. 41 00:02:07,726 --> 00:02:09,900 And in fact, in the next homework assignment-- 42 00:02:09,900 --> 00:02:12,840 which amazingly enough is already issued. 43 00:02:12,840 --> 00:02:15,390 So in the next homework assignment 44 00:02:15,390 --> 00:02:19,147 we'll think about two dimensional signal processing. 45 00:02:19,147 --> 00:02:21,480 And the difference between one and two is not very much, 46 00:02:21,480 --> 00:02:23,146 and the difference between two and three 47 00:02:23,146 --> 00:02:27,910 is even less than the difference between one and two. 48 00:02:27,910 --> 00:02:31,620 So we'll be able to think about an optical system 49 00:02:31,620 --> 00:02:35,790 purely in terms of the impulse response, 50 00:02:35,790 --> 00:02:39,180 and that's particularly useful, because it's so intuitive. 51 00:02:39,180 --> 00:02:42,210 Since the effect of the optical system is to blur, 52 00:02:42,210 --> 00:02:47,197 the impulse response is a direct measure of how much it blurs. 53 00:02:47,197 --> 00:02:48,780 That's true, not only for microscopes, 54 00:02:48,780 --> 00:02:50,520 but it's true for optics, in general. 55 00:02:50,520 --> 00:02:53,640 Imaging with light, it's a fundamental principle 56 00:02:53,640 --> 00:02:56,880 that derives from the wavelength, the coherent nature 57 00:02:56,880 --> 00:02:58,320 of light. 58 00:02:58,320 --> 00:03:01,650 And here's an example taken from the Hubble Space Telescope, 59 00:03:01,650 --> 00:03:04,050 where we see that there is much more blurring 60 00:03:04,050 --> 00:03:08,340 from a ground based microscope than there is from the Hubble, 61 00:03:08,340 --> 00:03:12,150 simply because the blurring is the result of two things. 62 00:03:12,150 --> 00:03:14,910 Generally speaking, if you have a ground based system then 63 00:03:14,910 --> 00:03:18,030 there's two kinds of blurring that we usually think about, 64 00:03:18,030 --> 00:03:21,510 blurring to the atmosphere and bouncing off 65 00:03:21,510 --> 00:03:24,600 particles in the atmosphere, and blurring 66 00:03:24,600 --> 00:03:27,589 due to the optics, the same as it was in the microscope. 67 00:03:27,589 --> 00:03:29,880 By going to space, we completely eliminate the blurring 68 00:03:29,880 --> 00:03:34,350 in the atmosphere, and so we're left 69 00:03:34,350 --> 00:03:36,960 with what can be a much sharper point spread function. 70 00:03:39,247 --> 00:03:41,580 Today, we're going to look at a completely different way 71 00:03:41,580 --> 00:03:44,010 of thinking about a system. 72 00:03:44,010 --> 00:03:49,210 Rather than thinking, what we will call in the time domain, 73 00:03:49,210 --> 00:03:51,067 h of t was a function of time. 74 00:03:51,067 --> 00:03:52,900 The impulse response was a function of time. 75 00:03:52,900 --> 00:03:55,540 We think about that as time domain signal processing. 76 00:03:55,540 --> 00:03:57,290 Rather than thinking about the time today, 77 00:03:57,290 --> 00:03:59,110 we'll think about the frequency domain. 78 00:03:59,110 --> 00:04:01,310 Frequency domain, just like time domain, 79 00:04:01,310 --> 00:04:04,750 is very convenient for certain kinds of signal processing 80 00:04:04,750 --> 00:04:06,040 tasks. 81 00:04:06,040 --> 00:04:09,370 One very natural example is audio. 82 00:04:09,370 --> 00:04:11,710 You all have lots of familiarity with this. 83 00:04:11,710 --> 00:04:14,200 What I'm going to do now, to prove to you 84 00:04:14,200 --> 00:04:16,769 that you have great intuition, is 85 00:04:16,769 --> 00:04:20,470 I'm going to play a clip followed 86 00:04:20,470 --> 00:04:23,890 by the same clip processed in four different-- in two 87 00:04:23,890 --> 00:04:25,450 different ways. 88 00:04:25,450 --> 00:04:26,680 All right. 89 00:04:26,680 --> 00:04:31,060 Clip-- so original clip 2. 90 00:04:31,060 --> 00:04:37,090 And clip 1 and clip 2 will have some transformation 91 00:04:37,090 --> 00:04:38,650 applied to it. 92 00:04:38,650 --> 00:04:41,770 Either the high frequencies HF or the low frequencies LF 93 00:04:41,770 --> 00:04:44,550 will be increased up arrow or decreased down arrow. 94 00:04:47,180 --> 00:04:47,820 OK? 95 00:04:47,820 --> 00:04:49,740 Is that clear? 96 00:04:49,740 --> 00:04:56,190 So I'm going to first play of the original, then play clip 1. 97 00:04:56,190 --> 00:04:57,832 You should decide whether the clip 1 98 00:04:57,832 --> 00:04:59,790 sounds like the high frequencies are increased, 99 00:04:59,790 --> 00:05:01,140 the high frequencies are decreased, 100 00:05:01,140 --> 00:05:02,556 the low frequencies are increased, 101 00:05:02,556 --> 00:05:06,990 the low frequencies are decreased, or none of those. 102 00:05:06,990 --> 00:05:11,325 Then I'll play the original clip 1, original clip 2. 103 00:05:11,325 --> 00:05:13,700 So there's going to be two answers, what happened to clip 104 00:05:13,700 --> 00:05:15,750 1, what happened to clip 2. 105 00:05:15,750 --> 00:05:16,250 OK. 106 00:05:16,250 --> 00:05:20,120 Nod your head yes, everybody knows what's going to happen. 107 00:05:20,120 --> 00:05:22,250 OK, now everybody listen. 108 00:05:22,250 --> 00:05:25,160 So you're supposed figure out two different answers, what 109 00:05:25,160 --> 00:05:27,579 happened to clip 1, and what happened to clip 2. 110 00:05:27,579 --> 00:05:35,563 [MUSIC PLAYING] 111 00:05:35,563 --> 00:05:44,046 [MUSIC PLAYING] 112 00:05:44,046 --> 00:05:52,529 [MUSIC PLAYING] 113 00:05:52,529 --> 00:06:00,513 [MUSIC PLAYING] 114 00:06:00,513 --> 00:06:08,996 [MUSIC PLAYING] 115 00:06:08,996 --> 00:06:16,980 [MUSIC PLAYING] 116 00:06:16,980 --> 00:06:24,964 [MUSIC PLAYING] 117 00:06:24,964 --> 00:06:34,730 [MUSIC PLAYING] 118 00:06:34,730 --> 00:06:35,230 OK. 119 00:06:35,230 --> 00:06:36,830 What happened? 120 00:06:36,830 --> 00:06:39,404 Answer 1, answer 2, answer 3, answer 4, answer 5, 121 00:06:39,404 --> 00:06:40,070 raise your hand. 122 00:06:43,517 --> 00:06:44,600 Come on, come on, come on. 123 00:06:44,600 --> 00:06:46,641 This is the part that you blame on your neighbor. 124 00:06:46,641 --> 00:06:47,222 OK, no don't. 125 00:06:47,222 --> 00:06:48,680 Talk to your neighbor, that way you 126 00:06:48,680 --> 00:06:49,580 can blame it on your neighbor. 127 00:06:49,580 --> 00:06:50,621 Yeah, I forgot that part. 128 00:06:50,621 --> 00:06:53,267 OK, talk to your neighbor. 129 00:06:53,267 --> 00:07:35,730 [SIDE CONVERSATIONS] 130 00:07:35,730 --> 00:07:38,465 OK, everybody raise your hands what happened, 131 00:07:38,465 --> 00:07:40,590 and if you're wrong just blame it on your neighbor. 132 00:07:40,590 --> 00:07:43,254 Just point to your neighbor as you raise your hand, right, 133 00:07:43,254 --> 00:07:44,670 and I'll understand what you mean. 134 00:07:48,170 --> 00:07:50,622 OK, you're only about a half right, 135 00:07:50,622 --> 00:07:52,080 you're supposed to be young people. 136 00:07:52,080 --> 00:07:56,230 I'm old, I'm not supposed to understand these things. 137 00:07:56,230 --> 00:07:57,420 So you're about half right. 138 00:07:57,420 --> 00:08:00,030 So now let's see maybe I can do it again. 139 00:08:00,030 --> 00:08:00,750 So listen again. 140 00:08:04,310 --> 00:08:12,418 [MUSIC PLAYING] 141 00:08:12,418 --> 00:08:20,901 [MUSIC PLAYING] 142 00:08:20,901 --> 00:08:28,885 [MUSIC PLAYING] 143 00:08:28,885 --> 00:08:36,869 [MUSIC PLAYING] 144 00:08:36,869 --> 00:08:44,853 [MUSIC PLAYING] 145 00:08:44,853 --> 00:08:53,336 [MUSIC PLAYING] 146 00:08:53,336 --> 00:09:01,320 [MUSIC PLAYING] 147 00:09:01,320 --> 00:09:12,880 [MUSIC PLAYING] 148 00:09:12,880 --> 00:09:14,490 OK, now the answer is perfectly clear. 149 00:09:14,490 --> 00:09:16,081 Everybody raise your hand. 150 00:09:16,081 --> 00:09:16,830 What's the answer? 151 00:09:19,807 --> 00:09:20,306 OK. 152 00:09:23,410 --> 00:09:24,410 I know the problem. 153 00:09:24,410 --> 00:09:26,789 The problem is-- does anybody know the title 154 00:09:26,789 --> 00:09:27,580 through this piece? 155 00:09:27,580 --> 00:09:28,972 AUDIENCE: No. 156 00:09:28,972 --> 00:09:30,430 DENNIS FREEMAN: That's the problem. 157 00:09:30,430 --> 00:09:34,410 It's 1970s, I wasn't thinking. 158 00:09:34,410 --> 00:09:36,160 This was new in 1970. 159 00:09:36,160 --> 00:09:37,960 I heard it when I was in-- 160 00:09:37,960 --> 00:09:38,495 but anyway. 161 00:09:38,495 --> 00:09:38,995 OK. 162 00:09:38,995 --> 00:09:39,828 AUDIENCE: [LAUGHING] 163 00:09:39,828 --> 00:09:42,690 DENNIS FREEMAN: The answer was. 164 00:09:42,690 --> 00:09:44,800 The answer is that one. 165 00:09:44,800 --> 00:09:47,950 OK, so the high frequencies in clip one were increased. 166 00:09:47,950 --> 00:09:50,470 If you heard sort of a tapping in the background, 167 00:09:50,470 --> 00:09:53,500 kind of a symbol thing that was louder, 168 00:09:53,500 --> 00:09:56,560 because the high frequencies were enhanced. 169 00:09:56,560 --> 00:09:58,990 In the second part, it sounded sort of similar, 170 00:09:58,990 --> 00:10:00,370 except the volume was lower. 171 00:10:00,370 --> 00:10:04,600 That's because I killed the low frequencies. 172 00:10:04,600 --> 00:10:05,570 OK. 173 00:10:05,570 --> 00:10:08,090 So in the first part the high frequencies were increased. 174 00:10:08,090 --> 00:10:11,240 In the second part the low frequencies were decreased. 175 00:10:11,240 --> 00:10:13,500 And I'll assume that if I had done the updated music, 176 00:10:13,500 --> 00:10:14,499 you would have got that. 177 00:10:14,499 --> 00:10:19,740 So I assume that you all got 100% correct, and we'll go on. 178 00:10:19,740 --> 00:10:23,778 So the-- yes, please. 179 00:10:23,778 --> 00:10:25,694 AUDIENCE: I have a question about when you say 180 00:10:25,694 --> 00:10:27,044 increasing degrees [INAUDIBLE]? 181 00:10:27,044 --> 00:10:28,210 DENNIS FREEMAN: Yes, you do. 182 00:10:28,210 --> 00:10:29,683 AUDIENCE: And what that include the amplitude 183 00:10:29,683 --> 00:10:31,046 when you say increasing degrees? 184 00:10:31,046 --> 00:10:32,920 DENNIS FREEMAN: Will be a little more clear-- 185 00:10:32,920 --> 00:10:35,253 So the question was, when I say increasing degrees, what 186 00:10:35,253 --> 00:10:36,620 exactly do I mean by that. 187 00:10:36,620 --> 00:10:39,550 What I meant was the magnitude of the frequency 188 00:10:39,550 --> 00:10:41,440 components was increased. 189 00:10:41,440 --> 00:10:44,740 We'll be saying-- we'll be developing some language, 190 00:10:44,740 --> 00:10:48,040 in the next 40 minutes, that will make that statement more 191 00:10:48,040 --> 00:10:51,040 precise, OK. 192 00:10:51,040 --> 00:10:53,449 So the idea here was to give you an intuitive feeling 193 00:10:53,449 --> 00:10:55,240 for something that will make mathematically 194 00:10:55,240 --> 00:10:56,950 more rigorous as we go along. 195 00:10:56,950 --> 00:10:59,950 By the end of the hour it should be completely clear 196 00:10:59,950 --> 00:11:02,740 what I mean by high frequencies increased or low frequencies 197 00:11:02,740 --> 00:11:03,460 decreased. 198 00:11:03,460 --> 00:11:06,610 If it's not tell me. 199 00:11:06,610 --> 00:11:08,440 So the idea then, in frequency analysis, 200 00:11:08,440 --> 00:11:12,504 is to think about the input as frequencies. 201 00:11:12,504 --> 00:11:14,170 Think about the input being cos omega t, 202 00:11:14,170 --> 00:11:17,260 and then think about what the system 203 00:11:17,260 --> 00:11:21,910 would do to a signal that was of the form cos omega t. 204 00:11:21,910 --> 00:11:23,680 We'll find out, as we go through the hour, 205 00:11:23,680 --> 00:11:26,740 that the signal that comes out of a linear time invariance 206 00:11:26,740 --> 00:11:31,330 system, when the input is cos omega t, 207 00:11:31,330 --> 00:11:34,180 is also cos omega t of frequent-- 208 00:11:34,180 --> 00:11:36,740 it's a signal with the same frequency. 209 00:11:36,740 --> 00:11:39,940 However, the amplitude and the phase-- 210 00:11:39,940 --> 00:11:42,010 that which sort of addresses your question. 211 00:11:42,010 --> 00:11:44,000 The amplitude and the phase can be different. 212 00:11:44,000 --> 00:11:45,375 Linear time invariant systems can 213 00:11:45,375 --> 00:11:49,360 change the amplitude and phase, but not the frequency 214 00:11:49,360 --> 00:11:52,060 of a pure sinusoid. 215 00:11:52,060 --> 00:11:55,810 So then the trick in thinking about what 216 00:11:55,810 --> 00:12:00,010 linear time the invariant system does, to a sinusoid, 217 00:12:00,010 --> 00:12:03,690 is thinking about how's the magnitude to change, 218 00:12:03,690 --> 00:12:05,910 and how's the frequency change. 219 00:12:05,910 --> 00:12:07,620 So motivate that a little more, well, I 220 00:12:07,620 --> 00:12:10,500 want to think about mass, spring, dashpot system. 221 00:12:10,500 --> 00:12:12,164 We talked about this before. 222 00:12:12,164 --> 00:12:13,830 If I think about the input of the system 223 00:12:13,830 --> 00:12:18,232 being the position of the top of the spring, 224 00:12:18,232 --> 00:12:20,190 and if I think about the response of the system 225 00:12:20,190 --> 00:12:25,050 being the position of the mass, then I 226 00:12:25,050 --> 00:12:29,230 can characterize that system as a linear time invariant system. 227 00:12:29,230 --> 00:12:31,830 And I can think about how does the amplitude change 228 00:12:31,830 --> 00:12:36,160 and how does the phase change, as a function of frequency 229 00:12:36,160 --> 00:12:39,060 of the sinusoid. 230 00:12:39,060 --> 00:12:40,860 OK, this is just like the music example, 231 00:12:40,860 --> 00:12:42,420 but this time it's mechanical. 232 00:12:42,420 --> 00:12:46,050 And what I want to do first is do a demonstration 233 00:12:46,050 --> 00:12:51,000 of figuring out the frequency response for a physical system. 234 00:12:51,000 --> 00:12:53,910 OK, so here's my mass, spring, dashpot. 235 00:12:53,910 --> 00:12:55,800 Hopefully, you can all see that. 236 00:12:55,800 --> 00:13:01,080 It is 10 loops of a slinky connected up to a bolt. 237 00:13:01,080 --> 00:13:06,540 And the idea is that I want to characterize 238 00:13:06,540 --> 00:13:08,910 how does the magnitude of the response change, 239 00:13:08,910 --> 00:13:10,740 as a function of frequency, and how 240 00:13:10,740 --> 00:13:12,870 does the phase of the response change, 241 00:13:12,870 --> 00:13:15,970 as a function of frequency. 242 00:13:15,970 --> 00:13:20,560 So if I turn it on to a low frequency-- 243 00:13:20,560 --> 00:13:27,700 [MOTOR BUZZING] 244 00:13:27,700 --> 00:13:30,060 OK, so I've got a low voltage going to the motor. 245 00:13:30,060 --> 00:13:31,340 The motor has got a cam on it. 246 00:13:31,340 --> 00:13:33,600 The cam's turning, that's giving the sinusoidal motion 247 00:13:33,600 --> 00:13:34,500 up and down. 248 00:13:34,500 --> 00:13:35,550 If you have very good eyes, you should 249 00:13:35,550 --> 00:13:37,508 be able to see this a little knot in the string 250 00:13:37,508 --> 00:13:40,440 is going up and down, that's my input. 251 00:13:40,440 --> 00:13:43,830 Because the cam is not changing physically, 252 00:13:43,830 --> 00:13:45,510 as I change the speed of the motor, 253 00:13:45,510 --> 00:13:48,510 the amplitude of the sinusoid won't change, 254 00:13:48,510 --> 00:13:49,710 but the frequency will. 255 00:13:49,710 --> 00:13:51,540 That's the idea. 256 00:13:51,540 --> 00:13:53,070 OK, so what you're supposed to do 257 00:13:53,070 --> 00:13:56,910 is this is x and that's y, how would you characterize 258 00:13:56,910 --> 00:14:00,609 the magnitude of the response as the ratio of the magnitude 259 00:14:00,609 --> 00:14:01,650 of the input, the output? 260 00:14:04,940 --> 00:14:06,540 They're the same. 261 00:14:06,540 --> 00:14:11,190 So the magnitude for this low frequency, the magnitude is 1. 262 00:14:11,190 --> 00:14:14,550 How about the phase relationship between the two? 263 00:14:14,550 --> 00:14:16,319 AUDIENCE: [INAUDIBLE]. 264 00:14:16,319 --> 00:14:18,360 DENNIS FREEMAN: They're in phase with each other. 265 00:14:18,360 --> 00:14:20,730 Right, when one goes up, the other one goes up. 266 00:14:20,730 --> 00:14:25,500 So at these low frequencies, the magnitude starts out at 1, 267 00:14:25,500 --> 00:14:29,100 and the phase starts out at 0. 268 00:14:29,100 --> 00:14:31,760 Now I'll increase the speed. 269 00:14:31,760 --> 00:14:45,150 [MOTOR BUZZING] 270 00:14:45,150 --> 00:14:46,270 Now what? 271 00:14:46,270 --> 00:14:47,476 What's the magnitude? 272 00:14:54,916 --> 00:15:02,260 Up, up, down, don't have a clue, don't care. 273 00:15:02,260 --> 00:15:03,260 AUDIENCE: [LAUGHING] 274 00:15:03,260 --> 00:15:04,470 DENNIS FREEMAN: OK, it's up a little bit. 275 00:15:04,470 --> 00:15:05,750 So the magnitude is up a little bit. 276 00:15:05,750 --> 00:15:06,583 How about the phase? 277 00:15:12,880 --> 00:15:14,979 How about the phase? 278 00:15:14,979 --> 00:15:17,300 AUDIENCE: [INAUDIBLE]. 279 00:15:17,300 --> 00:15:20,710 DENNIS FREEMAN: It's kind of similar. 280 00:15:20,710 --> 00:15:23,720 I'll just put a big dot. 281 00:15:23,720 --> 00:15:27,808 OK, so now I'll turn up the speed a little more. 282 00:15:27,808 --> 00:15:36,925 [MOTOR BUZZING] 283 00:15:36,925 --> 00:15:39,050 AUDIENCE: I think the magnitude might have changed. 284 00:15:39,050 --> 00:15:41,630 DENNIS FREEMAN: The magnitude might have changed. 285 00:15:41,630 --> 00:15:44,414 Up or down? 286 00:15:44,414 --> 00:15:45,330 AUDIENCE: [INAUDIBLE]. 287 00:15:45,330 --> 00:15:46,760 DENNIS FREEMAN: Yeah, went way up. 288 00:15:46,760 --> 00:15:49,840 So it came up here someplace, right. 289 00:15:49,840 --> 00:15:51,888 How about the phase? 290 00:15:51,888 --> 00:15:53,320 AUDIENCE: [INAUDIBLE]. 291 00:15:53,320 --> 00:15:54,035 DENNIS FREEMAN: It's totally-- it's 292 00:15:54,035 --> 00:15:55,284 going completely out of phase. 293 00:15:55,284 --> 00:15:58,190 So the phase-- in fact, we will call that a lag, 294 00:15:58,190 --> 00:15:59,870 but that's not perfectly clear now. 295 00:15:59,870 --> 00:16:02,810 We'll see later why I'm going to call it that. 296 00:16:02,810 --> 00:16:06,320 What do you think will happen if I go further higher? 297 00:16:06,320 --> 00:16:07,730 Hit the roof, right. 298 00:16:07,730 --> 00:16:10,670 So if I go-- so this is 4 1/2 volts. 299 00:16:10,670 --> 00:16:12,140 If I go to a higher frequency-- 300 00:16:12,140 --> 00:16:19,580 [MOTOR BUZZING] 301 00:16:19,580 --> 00:16:20,580 Ooh, funky. 302 00:16:20,580 --> 00:16:22,690 Why is it funky? 303 00:16:22,690 --> 00:16:24,190 AUDIENCE: Out of sync. 304 00:16:24,190 --> 00:16:26,200 DENNIS FREEMAN: Out of sync of what? 305 00:16:26,200 --> 00:16:27,220 It's only got one input. 306 00:16:27,220 --> 00:16:28,618 How can it be out of sync? 307 00:16:28,618 --> 00:16:30,052 AUDIENCE: [INAUDIBLE]. 308 00:16:35,352 --> 00:16:36,810 DENNIS FREEMAN: So it's remembering 309 00:16:36,810 --> 00:16:40,200 some of the response that it had before. 310 00:16:40,200 --> 00:16:42,854 Right, so I really have to wait for the old response to die. 311 00:16:42,854 --> 00:16:44,520 If I were to kill the input, it wouldn't 312 00:16:44,520 --> 00:16:47,700 stop moving right away. 313 00:16:47,700 --> 00:16:51,810 Right, so it's-- so the response to the previous input is 314 00:16:51,810 --> 00:16:54,960 interfering with the response to the current input. 315 00:16:54,960 --> 00:16:58,040 If I wait long enough, it ought to settle down. 316 00:16:58,040 --> 00:16:59,670 OK, magnitude up or down? 317 00:16:59,670 --> 00:17:01,110 AUDIENCE: [INAUDIBLE]. 318 00:17:01,110 --> 00:17:03,780 DENNIS FREEMAN: Compared to one? 319 00:17:03,780 --> 00:17:04,589 It's still down. 320 00:17:04,589 --> 00:17:07,849 It's down even compared to one. 321 00:17:07,849 --> 00:17:08,574 How about phase? 322 00:17:11,779 --> 00:17:15,210 AUDIENCE: [INAUDIBLE]. 323 00:17:15,210 --> 00:17:17,700 DENNIS FREEMAN: So it's almost out of phase. 324 00:17:17,700 --> 00:17:20,760 Out of phase would be the top is going up, 325 00:17:20,760 --> 00:17:22,720 while the bottom is going down. 326 00:17:22,720 --> 00:17:26,670 So it's almost out of phase, so there's even more delay. 327 00:17:26,670 --> 00:17:28,200 OK, so that's the idea. 328 00:17:28,200 --> 00:17:31,500 So that's-- what this is supposed to motivate is why we 329 00:17:31,500 --> 00:17:34,530 like to think about systems in terms of frequency response. 330 00:17:34,530 --> 00:17:38,760 It's a natural way to think of certain kinds of systems. 331 00:17:38,760 --> 00:17:42,870 Just like the time response, the impulse response, 332 00:17:42,870 --> 00:17:45,360 the time domain thinking convolution, 333 00:17:45,360 --> 00:17:48,450 was a convenient way to think about some kinds of systems, 334 00:17:48,450 --> 00:17:50,040 the optical system. 335 00:17:50,040 --> 00:17:51,780 Frequency response is a good way to think 336 00:17:51,780 --> 00:17:53,580 about some kinds of systems. 337 00:17:53,580 --> 00:17:54,687 Here's one. 338 00:17:54,687 --> 00:17:56,520 The reason it's a good way to think about it 339 00:17:56,520 --> 00:18:00,060 is that this particular system had a big response 340 00:18:00,060 --> 00:18:02,640 at a certain frequency. 341 00:18:02,640 --> 00:18:05,390 OK, now for something that has nothing to do with 003. 342 00:18:05,390 --> 00:18:07,520 Why is the CD there? 343 00:18:12,400 --> 00:18:13,376 Yes. 344 00:18:13,376 --> 00:18:14,850 AUDIENCE: [INAUDIBLE]. 345 00:18:14,850 --> 00:18:16,641 DENNIS FREEMAN: It keep it-- so it somewhat 346 00:18:16,641 --> 00:18:17,895 stabilizes it, that's correct. 347 00:18:17,895 --> 00:18:19,110 AUDIENCE: See it better? 348 00:18:19,110 --> 00:18:20,860 DENNIS FREEMAN: See it better that's true, 349 00:18:20,860 --> 00:18:22,720 but then I should've put one up here too. 350 00:18:22,720 --> 00:18:26,020 But of course, who knows, I make mistakes. 351 00:18:26,020 --> 00:18:26,590 Yes. 352 00:18:26,590 --> 00:18:27,390 AUDIENCE: [INAUDIBLE]. 353 00:18:27,390 --> 00:18:28,320 DENNIS FREEMAN: Damping. 354 00:18:28,320 --> 00:18:28,945 What's damping? 355 00:18:34,690 --> 00:18:37,710 So damping controls how big the amplitude 356 00:18:37,710 --> 00:18:40,020 got at the resonance frequency. 357 00:18:40,020 --> 00:18:42,850 Without the CD, it went crazy. 358 00:18:42,850 --> 00:18:43,350 Right. 359 00:18:43,350 --> 00:18:46,650 In order to get the response to be less 360 00:18:46,650 --> 00:18:50,340 than the elongation of the spring, 361 00:18:50,340 --> 00:18:52,980 I had to make the displacements tiny, tiny, tiny 362 00:18:52,980 --> 00:18:53,910 without the CD. 363 00:18:53,910 --> 00:18:56,040 So the CD was putting some damping-- 364 00:18:56,040 --> 00:18:57,950 air resistance. 365 00:18:57,950 --> 00:18:58,980 OK. 366 00:18:58,980 --> 00:19:03,210 Loss-- So in physics terms, the mass and the spring 367 00:19:03,210 --> 00:19:06,420 are lossless, no energy goes away. 368 00:19:06,420 --> 00:19:09,360 I'm continuously pumping energy into it from the motor. 369 00:19:09,360 --> 00:19:12,180 The response gets bigger and bigger and bigger and bigger. 370 00:19:12,180 --> 00:19:17,620 With the CD, there's loss, because of viscosity loss 371 00:19:17,620 --> 00:19:19,210 to the air. 372 00:19:19,210 --> 00:19:21,990 And so that means that it doesn't reach as high a peak, 373 00:19:21,990 --> 00:19:23,800 as it would have otherwise. 374 00:19:23,800 --> 00:19:25,080 OK? 375 00:19:25,080 --> 00:19:25,860 OK. 376 00:19:25,860 --> 00:19:28,080 So now what I want to do is think 377 00:19:28,080 --> 00:19:33,470 about having measured it, let's think about calculating it. 378 00:19:33,470 --> 00:19:35,184 Let's calculate the frequency response. 379 00:19:35,184 --> 00:19:36,850 We have a number of ways we could do it, 380 00:19:36,850 --> 00:19:38,455 because we have all those boxes, we 381 00:19:38,455 --> 00:19:39,830 have all the different methods we 382 00:19:39,830 --> 00:19:43,120 thought about for characterizing systems. 383 00:19:43,120 --> 00:19:45,990 So for example, we could figure out the differential equation 384 00:19:45,990 --> 00:19:49,590 for the system, and we could solve it 385 00:19:49,590 --> 00:19:54,480 with a particular input, cos omega t. 386 00:19:54,480 --> 00:19:55,570 That's one way to do it. 387 00:19:55,570 --> 00:19:57,486 Another way we could do it is find the impulse 388 00:19:57,486 --> 00:19:58,222 to the system-- 389 00:19:58,222 --> 00:20:00,180 the impulse response of the system, and then we 390 00:20:00,180 --> 00:20:01,740 could convolve. 391 00:20:01,740 --> 00:20:04,600 Right, we're all very familiar, we could do all those things. 392 00:20:04,600 --> 00:20:06,810 So rather than doing something we know how to do, what I'll do 393 00:20:06,810 --> 00:20:08,610 is something that we don't know how to do. 394 00:20:08,610 --> 00:20:11,280 I'll use a different method, which 395 00:20:11,280 --> 00:20:13,802 has to do with eigenfunctions and eigenvalues. 396 00:20:13,802 --> 00:20:15,510 And we'll see why that's a convenient way 397 00:20:15,510 --> 00:20:19,002 of thinking about frequency response in just a minute. 398 00:20:19,002 --> 00:20:20,460 So we'll be define in eigenfunction 399 00:20:20,460 --> 00:20:24,450 to be sort of the same thing as we do in linear algebra. 400 00:20:24,450 --> 00:20:27,240 So it's very much the same concept. 401 00:20:27,240 --> 00:20:29,050 We think about if we have a system 402 00:20:29,050 --> 00:20:31,440 and if we put in an input, and the output 403 00:20:31,440 --> 00:20:37,470 has the same shape as the input, just changed in amplitude, 404 00:20:37,470 --> 00:20:41,580 then we will say the input was an eigenfunction. 405 00:20:41,580 --> 00:20:44,880 And the change in amplitude was the eigenvalue. 406 00:20:44,880 --> 00:20:48,720 So you put in x of t, you get out then x of t. 407 00:20:48,720 --> 00:20:53,490 Same idea that we have in linear algebra for eigenvalues. 408 00:20:53,490 --> 00:20:54,720 OK? 409 00:20:54,720 --> 00:20:57,540 So that's the idea. 410 00:20:57,540 --> 00:21:00,030 And it turns out that certain very common functions 411 00:21:00,030 --> 00:21:05,190 are eigenfunctions of linear time invariant systems. 412 00:21:05,190 --> 00:21:09,210 Consider this system, y dot plus 2 y is x. 413 00:21:09,210 --> 00:21:12,470 Figure out if any of these functions, e to the minus c, 414 00:21:12,470 --> 00:21:16,020 e to the t, e to the jt, cos t, or u of t. 415 00:21:16,020 --> 00:21:20,945 Are any of those functions eigenfunctions of that system? 416 00:21:29,770 --> 00:21:31,440 And the answer is yes, and so the answer 417 00:21:31,440 --> 00:21:32,314 is really which ones? 418 00:23:12,300 --> 00:23:17,267 So how do I think about is e to the minus t 419 00:23:17,267 --> 00:23:18,600 an eigenfunction of that system? 420 00:23:18,600 --> 00:23:19,308 What should I do? 421 00:23:22,862 --> 00:23:24,570 Let's think about-- how many think it is? 422 00:23:27,600 --> 00:23:29,330 How many think it isn't? 423 00:23:29,330 --> 00:23:30,940 OK, it must be, right. 424 00:23:30,940 --> 00:23:34,360 Crowdsourcing, it has to be true. 425 00:23:34,360 --> 00:23:37,740 So how do I think about that? 426 00:23:37,740 --> 00:23:41,320 How could I set that up to convince somebody else that it 427 00:23:41,320 --> 00:23:42,790 is or is not an eigenfunction? 428 00:23:42,790 --> 00:23:43,611 What would I do? 429 00:23:43,611 --> 00:23:44,110 Yes. 430 00:23:44,110 --> 00:23:45,110 AUDIENCE: [INAUDIBLE]. 431 00:23:49,194 --> 00:23:50,110 DENNIS FREEMAN: Close. 432 00:23:50,110 --> 00:23:52,525 So what should I make x be? 433 00:23:52,525 --> 00:23:55,171 AUDIENCE: So you make x be either negative t 434 00:23:55,171 --> 00:23:59,610 or make y be [INAUDIBLE] negative t. 435 00:23:59,610 --> 00:24:00,610 DENNIS FREEMAN: Exactly. 436 00:24:00,610 --> 00:24:03,688 So then y dot would be-- 437 00:24:03,688 --> 00:24:07,090 AUDIENCE: [INAUDIBLE]. 438 00:24:07,090 --> 00:24:08,810 DENNIS FREEMAN: So now if I plug that in, 439 00:24:08,810 --> 00:24:11,610 I get y dot minus lambda e to the minus-- 440 00:24:11,610 --> 00:24:17,740 e to the minus t plus 2 y plus 2 lambda e to the minus t, 441 00:24:17,740 --> 00:24:19,450 should be e to the minus t. 442 00:24:19,450 --> 00:24:21,520 Is that true for any lambda? 443 00:24:21,520 --> 00:24:23,290 What should lambda be? 444 00:24:23,290 --> 00:24:24,040 1. 445 00:24:24,040 --> 00:24:27,070 So the answer is yes, this is true, 446 00:24:27,070 --> 00:24:30,440 and lambda would have to be 1 for that to be true. 447 00:24:30,440 --> 00:24:32,380 How about this guy? 448 00:24:32,380 --> 00:24:34,000 Same thing, lambda comes out the same? 449 00:24:37,330 --> 00:24:41,050 What's lambda for the second line? 450 00:24:41,050 --> 00:24:41,950 AUDIENCE: 1/3. 451 00:24:41,950 --> 00:24:43,510 DENNIS FREEMAN: 1/3, yeah. 452 00:24:43,510 --> 00:24:46,600 So now we have to use x is e to the t lambda 453 00:24:46,600 --> 00:24:51,757 e to the t lambda e to the t. 454 00:24:51,757 --> 00:24:52,590 Did I do that right? 455 00:24:52,590 --> 00:24:53,090 Yeah. 456 00:24:53,090 --> 00:24:57,040 So now we have lambda plus 2 lambda e to the t, 457 00:24:57,040 --> 00:24:58,870 should be e to the t. 458 00:24:58,870 --> 00:25:03,030 So lambda's going to have to be 1/3 this time. 459 00:25:03,030 --> 00:25:04,480 OK. 460 00:25:04,480 --> 00:25:07,160 How about this guy? 461 00:25:07,160 --> 00:25:07,760 What's lambda? 462 00:25:12,657 --> 00:25:13,740 AUDIENCE: 1 over 2 plus j? 463 00:25:13,740 --> 00:25:16,980 DENNIS FREEMAN: 1 over 2 plus j, exactly. 464 00:25:16,980 --> 00:25:19,167 Everybody see that? 465 00:25:19,167 --> 00:25:19,875 How about cosine? 466 00:25:23,429 --> 00:25:32,200 Yes, no, I'd like to break the tie. 467 00:25:32,200 --> 00:25:35,480 So there's a tie, the yes's and the no's are equal. 468 00:25:35,480 --> 00:25:36,220 So what do I do? 469 00:25:36,220 --> 00:25:40,942 You do the same thing, right, no difference. 470 00:25:40,942 --> 00:25:42,810 Oops, a little too hard. 471 00:25:42,810 --> 00:25:49,290 So I want to say that x is cos right. 472 00:25:49,290 --> 00:25:54,270 So y should be lambda cos. 473 00:25:54,270 --> 00:25:55,590 So y dot should be-- 474 00:26:01,260 --> 00:26:05,680 lambda sine minus, right. 475 00:26:05,680 --> 00:26:08,250 So then I need y dot minus lambda sine 476 00:26:08,250 --> 00:26:13,060 t plus twice, this one, lambda cosine t, 477 00:26:13,060 --> 00:26:16,290 should be that one, cosine t. 478 00:26:16,290 --> 00:26:17,414 What should lambda be? 479 00:26:20,318 --> 00:26:22,260 AUDIENCE: [INAUDIBLE]. 480 00:26:22,260 --> 00:26:24,100 DENNIS FREEMAN: Complex exponentials. 481 00:26:24,100 --> 00:26:27,720 So if you're willing to somehow think about something over here 482 00:26:27,720 --> 00:26:30,540 is imaginary, maybe you can do that. 483 00:26:30,540 --> 00:26:32,910 If you think about these as purely real functions, 484 00:26:32,910 --> 00:26:34,410 there's no real-- 485 00:26:34,410 --> 00:26:36,780 there's no way to take a cosine and turn it into a sine. 486 00:26:36,780 --> 00:26:40,320 That's actually why we like the complex numbers, right. 487 00:26:40,320 --> 00:26:44,820 Had we thought about the cosine as the sum of these, 488 00:26:44,820 --> 00:26:49,905 if we had thought about cosine t as e to the j t plus-- 489 00:26:53,280 --> 00:26:56,490 minus j t by 2, if we had thought about it that way, 490 00:26:56,490 --> 00:26:57,900 it would have actually turned out 491 00:26:57,900 --> 00:27:00,900 that we could think about the complex exponentials 492 00:27:00,900 --> 00:27:03,120 as being eigenfunctions. 493 00:27:03,120 --> 00:27:05,570 But the cosine function itself, thought of it 494 00:27:05,570 --> 00:27:07,290 as a strictly real function, there's 495 00:27:07,290 --> 00:27:12,090 no way you could choose the constants to get rid 496 00:27:12,090 --> 00:27:16,180 of the phase shift that would be introduced by the sine. 497 00:27:16,180 --> 00:27:18,920 OK, if we call the phase of this 0, 498 00:27:18,920 --> 00:27:21,574 this is phase shifted 90 degrees relative to that, 499 00:27:21,574 --> 00:27:23,740 and there's no way that I could choose a real number 500 00:27:23,740 --> 00:27:28,750 lambda to make that come out with 0 phase. 501 00:27:28,750 --> 00:27:31,160 OK. 502 00:27:31,160 --> 00:27:33,716 How about u of t? 503 00:27:33,716 --> 00:27:34,550 Im function? 504 00:27:38,140 --> 00:27:39,720 Same thing, right. 505 00:27:39,720 --> 00:27:42,060 I would think of the input is u of t, 506 00:27:42,060 --> 00:27:47,360 the output is lambda u of t, the derivative of the output. 507 00:27:47,360 --> 00:27:53,440 If the output is lambda u of t, what's the derivative? 508 00:27:53,440 --> 00:27:56,330 Lambda delta. 509 00:27:56,330 --> 00:27:58,380 So I'm left with trying to make a lambda-- 510 00:27:58,380 --> 00:28:01,000 left with trying to make a u function out 511 00:28:01,000 --> 00:28:03,730 of the sum of a delta and a u, and there's 512 00:28:03,730 --> 00:28:05,710 no way you can do that. 513 00:28:05,710 --> 00:28:11,620 So the answer is that the first three are eigenfunctions, 514 00:28:11,620 --> 00:28:15,510 and the last two are not. 515 00:28:15,510 --> 00:28:16,870 OK. 516 00:28:16,870 --> 00:28:18,419 So now what's this have to do-- 517 00:28:18,419 --> 00:28:19,960 I mean, I set this all up by thinking 518 00:28:19,960 --> 00:28:24,400 about frequency response. 519 00:28:24,400 --> 00:28:28,660 So what we want to do is make the connection between the two. 520 00:28:28,660 --> 00:28:31,390 The first step in the connection is complex exponentials, 521 00:28:31,390 --> 00:28:34,600 just like I'm motivated here, right. 522 00:28:34,600 --> 00:28:36,970 So the complex exponentials-- 523 00:28:36,970 --> 00:28:41,840 so it turns out that all complex exponentials 524 00:28:41,840 --> 00:28:45,110 are eigenfunctions to all linear time invariant systems. 525 00:28:45,110 --> 00:28:47,780 That's kind of amazing. 526 00:28:47,780 --> 00:28:50,720 That follows very directly from what we did last lecture. 527 00:28:50,720 --> 00:28:53,270 If we think about a linear time invariant system 528 00:28:53,270 --> 00:28:55,790 as having an impulse response, then we 529 00:28:55,790 --> 00:28:59,840 can find the response to the system, 530 00:28:59,840 --> 00:29:03,290 with the impulse response h of t, by convolving h of t 531 00:29:03,290 --> 00:29:05,900 with the input, which is a complex exponential. 532 00:29:05,900 --> 00:29:08,480 So if I say that my input is e to the st, 533 00:29:08,480 --> 00:29:10,460 and that my response is characterized 534 00:29:10,460 --> 00:29:14,650 by the impulse response h of t, all I need to do is convolve. 535 00:29:14,650 --> 00:29:18,630 So I convolve h, whatever it is, with x, 536 00:29:18,630 --> 00:29:23,060 which is e to the minus st. 537 00:29:23,060 --> 00:29:24,560 You remember that when you convolve, 538 00:29:24,560 --> 00:29:26,690 you do the funny things with the axes. 539 00:29:26,690 --> 00:29:30,230 So instead of thinking about e to the s t as a function of t, 540 00:29:30,230 --> 00:29:32,714 I think about it as the function of t minus tau. 541 00:29:32,714 --> 00:29:34,880 Instead of thinking about h of t as a function of t, 542 00:29:34,880 --> 00:29:39,200 I think of it as h of tau, and I run an integral. 543 00:29:39,200 --> 00:29:43,710 And because of the special form of the exponential function-- 544 00:29:43,710 --> 00:29:45,984 which is no coincidence. 545 00:29:45,984 --> 00:29:47,900 Because of the special form of the exponential 546 00:29:47,900 --> 00:29:50,650 function the e to the st factors out. 547 00:29:53,760 --> 00:29:57,240 So then I'm left with something that the integral, the function 548 00:29:57,240 --> 00:29:59,550 of tau, the taus all go away. 549 00:29:59,550 --> 00:30:03,000 I'm left with purely a function of s. 550 00:30:03,000 --> 00:30:05,460 And in fact, it's extraordinarily friendly. 551 00:30:05,460 --> 00:30:08,310 If I put in e to the st into a linear time invariant system, 552 00:30:08,310 --> 00:30:11,550 characterized by h of t, then the output 553 00:30:11,550 --> 00:30:17,340 has the shape e to the st, and the eigenvalue is h of s. 554 00:30:17,340 --> 00:30:18,890 Amazing. 555 00:30:18,890 --> 00:30:22,080 It's the value of the system function, the same thing we've 556 00:30:22,080 --> 00:30:26,520 been doing since week two, evaluated 557 00:30:26,520 --> 00:30:31,890 at the s that is the exponent in the complex exponential 558 00:30:31,890 --> 00:30:33,990 of interest. 559 00:30:33,990 --> 00:30:37,710 In my opinion, that's wholly remarkable. 560 00:30:37,710 --> 00:30:42,150 Then, knowing that we can form sinusoids out 561 00:30:42,150 --> 00:30:46,080 of complex exponentials, the problem is done. 562 00:30:46,080 --> 00:30:48,210 All complex exponentials are eigenfunctions 563 00:30:48,210 --> 00:30:51,180 of all LTI systems. 564 00:30:51,180 --> 00:30:52,510 And I can write-- 565 00:30:52,510 --> 00:30:54,540 I can always write an eternal sine wave 566 00:30:54,540 --> 00:30:56,970 in terms of a complex exponential, 567 00:30:56,970 --> 00:30:59,430 by using Euler's expression. 568 00:30:59,430 --> 00:31:01,320 And so I'm done. 569 00:31:01,320 --> 00:31:03,990 Furthermore, the eigenvalues that I need to do this 570 00:31:03,990 --> 00:31:08,400 are trivial, they're the value of the system function. 571 00:31:08,400 --> 00:31:15,690 That's even easier in the case that the LTI system 572 00:31:15,690 --> 00:31:20,250 can be written as a system of partial differential equations 573 00:31:20,250 --> 00:31:24,160 with constant coefficients. 574 00:31:24,160 --> 00:31:26,350 When that is true-- that is not all-- that is not 575 00:31:26,350 --> 00:31:28,230 true for all LTI systems. 576 00:31:28,230 --> 00:31:30,520 What I've said with the convolved part, 577 00:31:30,520 --> 00:31:33,850 that's true for all LTI systems. 578 00:31:33,850 --> 00:31:36,790 If I specialize it to the case that the system can 579 00:31:36,790 --> 00:31:38,680 be represented by a linear differential 580 00:31:38,680 --> 00:31:42,520 equation with constant coefficients, 581 00:31:42,520 --> 00:31:44,830 then the system function is always 582 00:31:44,830 --> 00:31:47,690 a rational polynomial in s. 583 00:31:47,690 --> 00:31:52,950 It's always the ratio of two polynomials in s. 584 00:31:52,950 --> 00:31:57,280 The reason that's interesting is that we can factor it. 585 00:31:57,280 --> 00:32:01,750 The reason that's interesting is that each of those factors 586 00:32:01,750 --> 00:32:07,700 has a very simple geometric interpretation. 587 00:32:07,700 --> 00:32:11,140 So what-- so if I can represent the LTI system in terms 588 00:32:11,140 --> 00:32:15,160 of a system of linear differential equations 589 00:32:15,160 --> 00:32:18,580 with constant coefficients, then the transfer-- then 590 00:32:18,580 --> 00:32:20,760 the system function has to shape, 591 00:32:20,760 --> 00:32:23,512 has the form of a rational polynomial in s. 592 00:32:23,512 --> 00:32:25,720 By fundamental theorem in algebra, and by the factors 593 00:32:25,720 --> 00:32:28,710 here, I can factor it. 594 00:32:28,710 --> 00:32:31,980 Makes it look like that, and then each of these terms 595 00:32:31,980 --> 00:32:34,215 looks like a vector in the s plane. 596 00:32:37,187 --> 00:32:39,520 Difference between two complex numbers, that's a vector. 597 00:32:43,380 --> 00:32:46,820 So for example, if I wanted to think about-- 598 00:32:46,820 --> 00:32:53,210 here's a system, a single pole at minus 2. 599 00:32:53,210 --> 00:32:58,870 Say I wanted to find the output when the input is e to 2 j t. 600 00:32:58,870 --> 00:33:02,120 E to the 2 jt is a complex exponential. 601 00:33:02,120 --> 00:33:05,840 The system is linear in time invariant, 602 00:33:05,840 --> 00:33:08,930 therefore I know that the complex exponential 603 00:33:08,930 --> 00:33:09,840 is an eigenfunction. 604 00:33:09,840 --> 00:33:11,990 Therefore, all I need to do is find the eigenvalue. 605 00:33:11,990 --> 00:33:14,120 The eigenvalue is the value of the system 606 00:33:14,120 --> 00:33:17,930 function at the s in question. 607 00:33:17,930 --> 00:33:20,330 So all I need to do is look at this diagram, 608 00:33:20,330 --> 00:33:22,940 and I have the entire picture. 609 00:33:22,940 --> 00:33:25,880 The system is a single pole at minus 2. 610 00:33:25,880 --> 00:33:28,830 That's the x. 611 00:33:28,830 --> 00:33:35,500 I want to know the response when the input is e to the j 2 t. 612 00:33:35,500 --> 00:33:41,020 So the s in question is s equals 2 j. 613 00:33:41,020 --> 00:33:42,680 So it's that point right there. 614 00:33:45,320 --> 00:33:49,060 So all I need to know is the length and direction 615 00:33:49,060 --> 00:33:51,070 of the vector that connects the pole 616 00:33:51,070 --> 00:33:53,380 to the frequency of interest. 617 00:33:53,380 --> 00:33:58,090 The length of that vector is 2 route 2, the angle is plus 45. 618 00:33:58,090 --> 00:34:00,430 It's a pole, so that contributions 619 00:34:00,430 --> 00:34:03,560 in the denominator. 620 00:34:03,560 --> 00:34:07,010 So the eigenvalue is 1 over the length 621 00:34:07,010 --> 00:34:11,690 of the vector, because it's a pole, and minus the angle, 622 00:34:11,690 --> 00:34:13,699 because it's the denominator. 623 00:34:13,699 --> 00:34:15,800 Is that clear? 624 00:34:15,800 --> 00:34:18,570 Point is it's really easy to do, that's why we do it. 625 00:34:18,570 --> 00:34:22,790 We only do things in this class that are easy to do. 626 00:34:22,790 --> 00:34:25,219 And you can then divide and conquer. 627 00:34:27,759 --> 00:34:30,050 If you have a system that's linear time invariant, that 628 00:34:30,050 --> 00:34:32,750 can be represented by a system of linear differential 629 00:34:32,750 --> 00:34:38,370 equations constant coefficients, rational polynomial s factor, 630 00:34:38,370 --> 00:34:40,230 do it for each factor. 631 00:34:40,230 --> 00:34:42,537 Do the same thing I just did for each factor. 632 00:34:42,537 --> 00:34:45,120 The magnitude of the response is the product of the magnitudes 633 00:34:45,120 --> 00:34:46,078 of each of those parts. 634 00:34:48,550 --> 00:34:51,159 The angle of the response is the sum 635 00:34:51,159 --> 00:34:55,260 of the angles of all of those parts. 636 00:34:55,260 --> 00:34:59,100 So the idea is that it's very simple. 637 00:34:59,100 --> 00:35:04,140 One last step-- if I'm interested in eternal 638 00:35:04,140 --> 00:35:08,840 of sine waves, like I was for the motor example, then 639 00:35:08,840 --> 00:35:12,210 I'll always be interested, according to Euler, 640 00:35:12,210 --> 00:35:17,320 in two complex exponentials. 641 00:35:17,320 --> 00:35:19,257 So if I were interested in cos omeganaut t, 642 00:35:19,257 --> 00:35:21,340 I would be interested in the complex exponential e 643 00:35:21,340 --> 00:35:25,660 to the j, omeganaut t, and e to the minus 644 00:35:25,660 --> 00:35:31,180 j omeganaut t, because I need both of those in order to sum-- 645 00:35:31,180 --> 00:35:33,195 to get the purely real function cos omeganaut t. 646 00:35:36,270 --> 00:35:39,030 So that means then, I can write the response 647 00:35:39,030 --> 00:35:46,260 to this eternal cosine wave as simply the sum of the system 648 00:35:46,260 --> 00:35:51,450 function evaluated at j omegnaut times e to the j omeganaut t, 649 00:35:51,450 --> 00:35:54,170 and the system of function evaluated at s 650 00:35:54,170 --> 00:35:58,045 equals minus j omeganaut times e to the minus j omeganaut t. 651 00:35:58,045 --> 00:35:59,130 Done, I have the answer. 652 00:36:04,040 --> 00:36:11,020 It's even easier, because the system function 653 00:36:11,020 --> 00:36:13,780 has a symmetry that we call conjugate symmetry. 654 00:36:16,330 --> 00:36:18,820 That's easy to show also by convolution, 655 00:36:18,820 --> 00:36:20,470 that's why we did convolution before we 656 00:36:20,470 --> 00:36:21,880 did frequency response. 657 00:36:21,880 --> 00:36:24,640 You can prove the properties of a frequency response 658 00:36:24,640 --> 00:36:29,350 by resorting back to convolution. 659 00:36:29,350 --> 00:36:32,620 Convolution is a very powerful thought tool. 660 00:36:32,620 --> 00:36:34,300 Less powerful as a computational tool, 661 00:36:34,300 --> 00:36:36,560 very powerful thought tool. 662 00:36:36,560 --> 00:36:39,490 So here again, if we think about the system function 663 00:36:39,490 --> 00:36:41,870 is the Laplace transform of h of t-- 664 00:36:41,870 --> 00:36:44,170 you remember that from two lectures ago. 665 00:36:44,170 --> 00:36:48,850 System function is always the Laplace transform of h of t. 666 00:36:48,850 --> 00:36:50,980 If I have a physical system, like the mass spring 667 00:36:50,980 --> 00:36:56,560 dashpot, h of t is real, how could it be complex. 668 00:36:56,560 --> 00:36:59,600 H of t is the response when I have a particular input. 669 00:36:59,600 --> 00:37:02,930 The response of a real system is real. 670 00:37:02,930 --> 00:37:06,830 So if I have a system that is real, then h of t is real. 671 00:37:10,500 --> 00:37:12,720 If I'm interested in frequencies that 672 00:37:12,720 --> 00:37:14,910 are the negative of each other, plus or minus 673 00:37:14,910 --> 00:37:20,970 j omeganaut in order to form cos omeganaut t, 674 00:37:20,970 --> 00:37:23,640 then I'm interested in two different system functions, 675 00:37:23,640 --> 00:37:27,810 h of j omega and h of minus j omega, which 676 00:37:27,810 --> 00:37:30,560 happen to be complex conjugates of each other. 677 00:37:30,560 --> 00:37:33,550 Since the h of t is real-- 678 00:37:33,550 --> 00:37:36,930 I just argued that for a physical system h of t 679 00:37:36,930 --> 00:37:38,610 has to be real. 680 00:37:38,610 --> 00:37:43,340 If h of t is real, then the only j in this equation is that one. 681 00:37:43,340 --> 00:37:47,460 It's negative here, so therefore negating j 682 00:37:47,460 --> 00:37:51,820 is the same as taking the complex conjugate. 683 00:37:51,820 --> 00:37:54,990 So I don't really have to compute two different system 684 00:37:54,990 --> 00:37:57,900 functions, two different eigenvalues. 685 00:37:57,900 --> 00:38:00,170 One is the complex conjugate of the other. 686 00:38:03,170 --> 00:38:08,300 That means that the response to the sum simplifies. 687 00:38:12,460 --> 00:38:15,490 So if I want to think about the input 688 00:38:15,490 --> 00:38:18,060 for-- if I want to think about the input as an eternal cosine 689 00:38:18,060 --> 00:38:20,800 wave, cos omeganaut t for all time, 690 00:38:20,800 --> 00:38:24,340 I write that by Euler, this way. 691 00:38:24,340 --> 00:38:28,060 I know from this eigenvalue eigenfunction idea 692 00:38:28,060 --> 00:38:31,320 that the response to this guy can be written this way. 693 00:38:34,290 --> 00:38:35,760 It's a complex exponential. 694 00:38:35,760 --> 00:38:38,219 Complex exponentials are eigenfunctions of LTI systems, 695 00:38:38,219 --> 00:38:40,135 so it has the same shape in an eigenfunction-- 696 00:38:40,135 --> 00:38:43,610 in an eigenvalues, sorry. 697 00:38:43,610 --> 00:38:45,350 This one can be written similarly, 698 00:38:45,350 --> 00:38:48,850 except there's a minus sign. 699 00:38:48,850 --> 00:38:51,890 Because this is the complex conjugate of this, 700 00:38:51,890 --> 00:38:53,900 and this is the complex conjugate of that, 701 00:38:53,900 --> 00:38:58,550 I've got the sum of a number and its complex conjugate. 702 00:38:58,550 --> 00:39:00,620 The sum of a number and its complex conjugate 703 00:39:00,620 --> 00:39:02,610 is just the real part of that number. 704 00:39:05,820 --> 00:39:11,190 Then this h, this eigenvalue, the system function 705 00:39:11,190 --> 00:39:15,030 evaluated at j omeganaut, has a magnitude and then angle. 706 00:39:17,800 --> 00:39:20,440 If I write the system function in terms of its magnitude 707 00:39:20,440 --> 00:39:25,300 and angle, I can factor the magnitude out of the real part, 708 00:39:25,300 --> 00:39:31,270 and combine the angle by Euler with the angle that generates 709 00:39:31,270 --> 00:39:35,730 the cosine wave, j omeganaut t. 710 00:39:35,730 --> 00:39:37,750 And I'm left with my final answer, 711 00:39:37,750 --> 00:39:42,070 that you can compute the output as the magnitude of a system 712 00:39:42,070 --> 00:39:45,980 function evaluated as j omeganaut, 713 00:39:45,980 --> 00:39:50,270 with a phase shift on the cosine that's equal to the angle of h 714 00:39:50,270 --> 00:39:52,490 of j omeganaut. 715 00:39:52,490 --> 00:39:54,980 So this is the final answer. 716 00:39:54,980 --> 00:39:59,160 If I want to think about the frequency response-- 717 00:39:59,160 --> 00:40:02,760 if I want to think about how this system has a response that 718 00:40:02,760 --> 00:40:08,820 depends on frequency, I think about the cos omega t going in, 719 00:40:08,820 --> 00:40:13,260 and what comes out is also cos omega t-- 720 00:40:13,260 --> 00:40:15,480 cos omega t. 721 00:40:15,480 --> 00:40:18,240 Except the magnitudes change by the magnitude of the system 722 00:40:18,240 --> 00:40:22,290 function, and the phase is changed by the phase 723 00:40:22,290 --> 00:40:23,820 of the system function. 724 00:40:23,820 --> 00:40:27,390 That's this thing I said back on slide 2. 725 00:40:27,390 --> 00:40:32,430 Sine in, sine out, however the magnitude can possibly change, 726 00:40:32,430 --> 00:40:35,280 and the angle, the phase can possibly change, 727 00:40:35,280 --> 00:40:37,560 but the frequency cannot change. 728 00:40:37,560 --> 00:40:42,060 Omega went in, omega comes out. 729 00:40:42,060 --> 00:40:43,920 OK? 730 00:40:43,920 --> 00:40:46,680 So that leads to a very easy way of thinking 731 00:40:46,680 --> 00:40:51,135 about frequency responses, in terms of pole-zero diagrams. 732 00:40:54,740 --> 00:40:58,610 The idea is if you think about the pole-zero diagram-- 733 00:40:58,610 --> 00:41:02,060 if the system can be represented by a linear differential 734 00:41:02,060 --> 00:41:04,610 equation with constant coefficients, 735 00:41:04,610 --> 00:41:06,590 then the system can be represented 736 00:41:06,590 --> 00:41:10,290 by a collection of poles and 0s, a handful of numbers. 737 00:41:10,290 --> 00:41:12,290 How many poles are there, how many 0s are there, 738 00:41:12,290 --> 00:41:15,680 and where are they? 739 00:41:15,680 --> 00:41:18,040 So if I have that representation, 740 00:41:18,040 --> 00:41:20,080 how many poles are there, how many 0s are there, 741 00:41:20,080 --> 00:41:23,560 where are they, I can think about the frequency response 742 00:41:23,560 --> 00:41:26,800 just by thinking about the vector that 743 00:41:26,800 --> 00:41:31,480 connects each pole and each 0 in turn, from the pole 744 00:41:31,480 --> 00:41:34,150 and 0 location to the point on the j omega 745 00:41:34,150 --> 00:41:38,470 axis at the frequency of interest. 746 00:41:38,470 --> 00:41:41,950 So if I have a system that has a single 0, 747 00:41:41,950 --> 00:41:45,670 here I have a 0 at z 1, at s equals z 1. 748 00:41:45,670 --> 00:41:52,210 Let's say that 0 is at the point z 1 equals minus 2. 749 00:41:52,210 --> 00:41:58,190 Single zero at the point s equals minus 2. 750 00:41:58,190 --> 00:42:02,180 Then I only need to think about one vector. 751 00:42:02,180 --> 00:42:04,430 If I'm interested in the response for very 752 00:42:04,430 --> 00:42:07,250 low frequencies, very low frequencies 753 00:42:07,250 --> 00:42:11,330 are at omega equals 0, I only need 754 00:42:11,330 --> 00:42:15,260 to think about the vector that connects the 0 to the point j 755 00:42:15,260 --> 00:42:16,940 omega equals 0. 756 00:42:16,940 --> 00:42:20,720 So j 0, which you say is at zero. 757 00:42:20,720 --> 00:42:24,380 So the eigenvalue is the length of this vector, that's 758 00:42:24,380 --> 00:42:28,950 the magnitude, and the angle of this vector-- we always measure 759 00:42:28,950 --> 00:42:31,370 angle relative to the x-axis. 760 00:42:31,370 --> 00:42:35,030 So the angle of this vector is 0. 761 00:42:35,030 --> 00:42:39,320 So I get a magnitude which is that big, 762 00:42:39,320 --> 00:42:40,830 and an angle which is 0. 763 00:42:40,830 --> 00:42:44,590 And that's plotted over here. 764 00:42:44,590 --> 00:42:46,940 That's my result for a low frequency, 765 00:42:46,940 --> 00:42:50,240 a frequency close to 0. 766 00:42:50,240 --> 00:42:54,342 Then if I think about a slightly higher frequency, what happens 767 00:42:54,342 --> 00:42:55,425 to the magnitude of the 0? 768 00:42:57,950 --> 00:42:59,630 Bigger. 769 00:42:59,630 --> 00:43:04,900 Bigger, it's the magnitude of a 0, the 0s are in the top. 770 00:43:04,900 --> 00:43:08,200 Bigger is bigger, so the bigger arrow 771 00:43:08,200 --> 00:43:12,340 translates to a slightly bigger magnitude. 772 00:43:12,340 --> 00:43:14,740 The angle that's made with the x-axis 773 00:43:14,740 --> 00:43:17,200 is now slightly positive, so that's illustrated by the fact 774 00:43:17,200 --> 00:43:20,350 that the angle is deviating from 0. 775 00:43:20,350 --> 00:43:23,860 And in general, I can think about the frequency response 776 00:43:23,860 --> 00:43:26,620 is just how the length of this vector 777 00:43:26,620 --> 00:43:28,330 changes as I change omega. 778 00:43:31,330 --> 00:43:33,400 The change in length tells me the magnitude, 779 00:43:33,400 --> 00:43:35,260 the change in angle tells me the phase. 780 00:43:38,090 --> 00:43:41,690 Same thing happens if I do minus omega. 781 00:43:41,690 --> 00:43:45,780 What on earth is minus omega? 782 00:43:45,780 --> 00:43:48,580 So omega was cos omega t. 783 00:43:48,580 --> 00:43:52,320 Omega, right, it had to do with how fast the motor was going. 784 00:43:52,320 --> 00:43:53,210 What's minus omega? 785 00:43:53,210 --> 00:43:54,980 Why am I drawing minus omega here? 786 00:44:02,270 --> 00:44:04,950 Yes? 787 00:44:04,950 --> 00:44:07,476 Why am I interested in these negative frequencies 788 00:44:07,476 --> 00:44:08,475 that don't really exist? 789 00:44:11,730 --> 00:44:14,610 Because I have an hour to fill, and I don't have enough 790 00:44:14,610 --> 00:44:16,980 stuff to fill the hour. 791 00:44:16,980 --> 00:44:17,480 No. 792 00:44:20,310 --> 00:44:22,680 Why am I interested in negative frequencies? 793 00:44:25,950 --> 00:44:28,320 Why am I interested in negative frequencies? 794 00:44:32,100 --> 00:44:35,834 I'm interested in negative frequencies because-- 795 00:44:35,834 --> 00:44:37,329 AUDIENCE: [INAUDIBLE]. 796 00:44:37,329 --> 00:44:39,620 DENNIS FREEMAN: --they let me create sines and cosines. 797 00:44:39,620 --> 00:44:43,970 Euler-- I invent negative frequencies, 798 00:44:43,970 --> 00:44:47,520 because I like complex numbers. 799 00:44:47,520 --> 00:44:49,820 I like complex numbers, because they're eigenfunction-- 800 00:44:49,820 --> 00:44:53,300 because the corresponding complex exponentials are 801 00:44:53,300 --> 00:44:55,390 eigenfunctions. 802 00:44:55,390 --> 00:44:59,630 Cos omega t was not an eigenfunction, I don't like it. 803 00:44:59,630 --> 00:45:03,990 E to the j omega t is an eigenfunction, I like it. 804 00:45:03,990 --> 00:45:09,410 I invent negative frequencies so that I can construct cos out 805 00:45:09,410 --> 00:45:12,410 of the sum of two complex things, one of them 806 00:45:12,410 --> 00:45:15,680 happens to be this imaginary frequency thing. 807 00:45:15,680 --> 00:45:19,830 Who cares, it's just a number, it's just math. 808 00:45:19,830 --> 00:45:22,580 So in general, we will think about positive and negative 809 00:45:22,580 --> 00:45:23,480 frequencies. 810 00:45:23,480 --> 00:45:26,330 Negative frequencies don't exist, 811 00:45:26,330 --> 00:45:31,380 we simply invent them to make the math easy. 812 00:45:31,380 --> 00:45:33,950 OK, so generally speaking, because 813 00:45:33,950 --> 00:45:37,350 of the complex conjugate symmetry, 814 00:45:37,350 --> 00:45:40,370 you can predict what the negative frequency 815 00:45:40,370 --> 00:45:42,890 part of the system function will look like, 816 00:45:42,890 --> 00:45:44,501 if you know the positive part. 817 00:45:44,501 --> 00:45:47,000 So generally speaking, we'll only look at the positive parts 818 00:45:47,000 --> 00:45:50,317 when we know the system is real, right. 819 00:45:50,317 --> 00:45:52,650 This symmetry only comes about when the system was real. 820 00:45:52,650 --> 00:45:57,624 I proved it, because I knew the h of t was real value. 821 00:45:57,624 --> 00:45:59,790 OK, so now I can think about the same sort of thing, 822 00:45:59,790 --> 00:46:02,150 but this time with a pole. 823 00:46:02,150 --> 00:46:03,800 The vector diagram looks the same. 824 00:46:03,800 --> 00:46:05,390 The shape of the curve's upside down, 825 00:46:05,390 --> 00:46:08,840 because the pole's in the bottom. 826 00:46:08,840 --> 00:46:13,310 Longer arrows now make smaller eigenvalues. 827 00:46:13,310 --> 00:46:16,480 So now as you go to higher frequencies, 828 00:46:16,480 --> 00:46:18,990 the idea is that a single pole gives you 829 00:46:18,990 --> 00:46:21,670 a frequency response, whose magnitude 830 00:46:21,670 --> 00:46:26,890 decays as you deviate from 0. 831 00:46:26,890 --> 00:46:29,080 And you get increasing phase lag. 832 00:46:29,080 --> 00:46:32,290 You get increasing phase lag, because this angle increases 833 00:46:32,290 --> 00:46:36,869 with omega, but it's in the bottom. 834 00:46:36,869 --> 00:46:38,910 Increases in the bottom, that means it decreases. 835 00:46:38,910 --> 00:46:41,100 That's why the angle goes down, in this case. 836 00:46:41,100 --> 00:46:44,670 And if you have a pole and a 0, the answer 837 00:46:44,670 --> 00:46:48,450 is easily derived from the previous two answers. 838 00:46:48,450 --> 00:46:52,710 You think about two vectors, one connecting the 0 to the point 839 00:46:52,710 --> 00:46:56,990 at the frequency of interest, the other connecting the pole. 840 00:46:56,990 --> 00:46:59,490 And the magnitude is the product to the lengths of those two 841 00:46:59,490 --> 00:47:02,400 vectors, and the angle is the sum of the angles of those two 842 00:47:02,400 --> 00:47:04,660 vectors. 843 00:47:04,660 --> 00:47:07,440 So you can see that the 0 is slightly shorter 844 00:47:07,440 --> 00:47:12,270 at low frequencies, and becomes asymptotically the same length. 845 00:47:12,270 --> 00:47:13,860 That means that at high frequencies 846 00:47:13,860 --> 00:47:16,860 the magnitude goes to 1, except, of course, 847 00:47:16,860 --> 00:47:20,730 I've got a constant here, so it goes to 3. 848 00:47:20,730 --> 00:47:23,520 And at low frequencies, this length 849 00:47:23,520 --> 00:47:25,800 is shorter than that one, so it's slightly smaller. 850 00:47:25,800 --> 00:47:29,010 The magnitude smaller lower frequencies. 851 00:47:29,010 --> 00:47:32,250 Similarly for the angles, the angles both start out at 0, 852 00:47:32,250 --> 00:47:34,939 so the angle starts out at 0. 853 00:47:34,939 --> 00:47:36,480 If you go to a high enough frequency, 854 00:47:36,480 --> 00:47:40,470 both angles go 2 pi over 2. 855 00:47:40,470 --> 00:47:42,390 So they both angle-- so the difference 856 00:47:42,390 --> 00:47:45,421 goes to 0 at high frequencies, as well. 857 00:47:45,421 --> 00:47:46,920 There's a little blip in the middle, 858 00:47:46,920 --> 00:47:51,210 because the angle of the zero increases more rapidly 859 00:47:51,210 --> 00:47:53,560 than the angle at the pole. 860 00:47:53,560 --> 00:47:57,780 0's in the top, so there's a little blip for positive phase. 861 00:47:57,780 --> 00:48:01,680 Same sort of thing works if I think about the mass, spring, 862 00:48:01,680 --> 00:48:02,400 dashpot system. 863 00:48:05,490 --> 00:48:09,510 Now if I write the-- so if I just write the differential 864 00:48:09,510 --> 00:48:11,340 equation, figure out the system function, 865 00:48:11,340 --> 00:48:15,720 I get a simple form that looks like so, second order. 866 00:48:15,720 --> 00:48:19,320 If I imagine very low damping, you 867 00:48:19,320 --> 00:48:22,810 can see that the poles are going to have the form s 868 00:48:22,810 --> 00:48:26,810 squared m plus k must be 0. 869 00:48:26,810 --> 00:48:30,240 So they are complex. 870 00:48:30,240 --> 00:48:31,810 OK? 871 00:48:31,810 --> 00:48:36,220 So if you factor this, if you set this equal to 0, 872 00:48:36,220 --> 00:48:39,160 you're going to get that squared m plus k must be zero 873 00:48:39,160 --> 00:48:41,590 if b were small. 874 00:48:41,590 --> 00:48:44,740 So the poles are going to be complex. 875 00:48:44,740 --> 00:48:46,300 So for example, if the poles were, 876 00:48:46,300 --> 00:48:51,610 as indicated here, about minus 1 plus or minus j 3, 877 00:48:51,610 --> 00:48:53,860 all you do is the same thing we did before, 878 00:48:53,860 --> 00:48:55,350 consider the length of the vectors, 879 00:48:55,350 --> 00:48:57,160 consider the angle of the vectors. 880 00:48:57,160 --> 00:48:59,320 Here you get sum product, and you 881 00:48:59,320 --> 00:49:01,570 get angles that are complimentary, so 882 00:49:01,570 --> 00:49:04,330 plus and minus. 883 00:49:04,330 --> 00:49:07,630 So the angle starts out at 0. 884 00:49:07,630 --> 00:49:11,110 As you increase the frequency, one of the pole-- 885 00:49:11,110 --> 00:49:13,990 one of the arrows gets longer and longer 886 00:49:13,990 --> 00:49:18,610 by about a factor of 2, and the other gets shorter-- 887 00:49:18,610 --> 00:49:22,540 gets shorter by a lot more than a factor of 2. 888 00:49:22,540 --> 00:49:28,680 So that means that the product of the lankes becomes small, 889 00:49:28,680 --> 00:49:33,070 but they're in the bottom, so the eigenvalue becomes large. 890 00:49:33,070 --> 00:49:36,150 So that's what's going on here. 891 00:49:36,150 --> 00:49:38,730 So that's the idea. 892 00:49:38,730 --> 00:49:40,960 I've written two check yourself questions. 893 00:49:40,960 --> 00:49:45,660 Those are intended to just be practice with complex numbers, 894 00:49:45,660 --> 00:49:48,210 because you need to know how to do complex numbers. 895 00:49:48,210 --> 00:49:54,480 But the point-- so the point of today 896 00:49:54,480 --> 00:49:56,970 is just that there's a very natural way-- that's 897 00:49:56,970 --> 00:49:59,370 completely complementary to thinking about convolution 898 00:49:59,370 --> 00:50:00,570 in time. 899 00:50:00,570 --> 00:50:03,450 There's a different natural way to think about some systems, 900 00:50:03,450 --> 00:50:05,430 in terms of the frequency response. 901 00:50:05,430 --> 00:50:08,850 How does the system respond to an eternal sine wave? 902 00:50:08,850 --> 00:50:11,130 It's a very easy thing to compute. 903 00:50:11,130 --> 00:50:12,960 It follows very naturally from the things 904 00:50:12,960 --> 00:50:14,730 that we thought about with convolution, 905 00:50:14,730 --> 00:50:17,160 and the results quite amazing. 906 00:50:17,160 --> 00:50:19,680 The result is that the magnitude of the angle 907 00:50:19,680 --> 00:50:23,850 can be computed from the Laplace transform of the impulse 908 00:50:23,850 --> 00:50:27,130 response, which just happens to be the system function. 909 00:50:27,130 --> 00:50:29,010 So the point-- so hopefully, what you see 910 00:50:29,010 --> 00:50:30,870 is that we have lots of representations, 911 00:50:30,870 --> 00:50:34,724 and now we're seeing even more connections between them. 912 00:50:34,724 --> 00:50:35,580 OK. 913 00:50:35,580 --> 00:50:37,400 Have a good day.