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 --> 00:00:21,132 at ocw.mit.edu 8 00:00:21,132 --> 00:00:22,340 PROFESSOR: Hello and welcome. 9 00:00:24,839 --> 00:00:27,380 So today I want to start to talk about discrete time systems. 10 00:00:29,940 --> 00:00:33,200 But before beginning there, I want to say just a quick word 11 00:00:33,200 --> 00:00:35,479 about homework. 12 00:00:35,479 --> 00:00:37,520 I tried to stress this last time that the way you 13 00:00:37,520 --> 00:00:40,730 get to know this stuff is by doing the homework, 14 00:00:40,730 --> 00:00:44,780 and we'd like to provide positive feedback 15 00:00:44,780 --> 00:00:48,530 in a direction that will help you get 16 00:00:48,530 --> 00:00:50,390 the most out of the homework. 17 00:00:50,390 --> 00:00:52,700 So to that end, we've already talked about there's 18 00:00:52,700 --> 00:00:54,620 two kinds of problems. 19 00:00:54,620 --> 00:00:56,870 Each homework will have tutor problems that 20 00:00:56,870 --> 00:01:01,670 are intended to be problems like you will see on an exam, 21 00:01:01,670 --> 00:01:03,140 but they will also have the feature 22 00:01:03,140 --> 00:01:04,790 that they can be automatically graded 23 00:01:04,790 --> 00:01:08,570 so that you can get immediate feedback about whether you're 24 00:01:08,570 --> 00:01:11,170 doing it right or not. 25 00:01:11,170 --> 00:01:13,760 There will also be harder questions questions 26 00:01:13,760 --> 00:01:15,470 that look more like real world questions 27 00:01:15,470 --> 00:01:17,300 so you don't think that this class is only 28 00:01:17,300 --> 00:01:20,310 about toy problems. 29 00:01:20,310 --> 00:01:22,370 The idea is to give you problems that you 30 00:01:22,370 --> 00:01:25,520 can imagine running into. 31 00:01:25,520 --> 00:01:29,450 Those are harder to grade automatically so we won't. 32 00:01:29,450 --> 00:01:33,500 Instead they'll be graded by a human. 33 00:01:33,500 --> 00:01:36,710 Those problems often admit several different lines 34 00:01:36,710 --> 00:01:37,920 of reasoning. 35 00:01:37,920 --> 00:01:40,760 So there isn't necessarily a unique right 36 00:01:40,760 --> 00:01:43,460 way to think about it. 37 00:01:43,460 --> 00:01:47,710 So what we would like to do is encourage you to-- after you 38 00:01:47,710 --> 00:01:49,180 finish the homework-- 39 00:01:49,180 --> 00:01:52,014 read the solutions. 40 00:01:52,014 --> 00:01:54,430 Now we think it's kind of an impediment to wait for a week 41 00:01:54,430 --> 00:01:58,090 till you get your graded homework back because people-- 42 00:01:58,090 --> 00:02:00,760 in my experience-- just don't bother. 43 00:02:00,760 --> 00:02:03,880 So what we would like to do is post the solutions immediately 44 00:02:03,880 --> 00:02:05,980 after your solutions are due. 45 00:02:05,980 --> 00:02:08,880 Your solutions are due-- if you submit electronically-- 46 00:02:08,880 --> 00:02:10,160 at 5:00 p.m. 47 00:02:10,160 --> 00:02:11,170 on the due date. 48 00:02:11,170 --> 00:02:14,680 So homework one is due tomorrow, 5:00 p.m. 49 00:02:14,680 --> 00:02:16,976 electronic submission. 50 00:02:16,976 --> 00:02:18,100 You can also turn in paper. 51 00:02:18,100 --> 00:02:20,890 You have to do that in your recitation. 52 00:02:20,890 --> 00:02:23,320 Immediately after it was due at 5:00 p.m. 53 00:02:23,320 --> 00:02:26,110 we'll post the solutions, and we'll 54 00:02:26,110 --> 00:02:31,710 encourage you, in quotes, to read them, 55 00:02:31,710 --> 00:02:33,990 because it's good for you, right? 56 00:02:33,990 --> 00:02:36,720 We hope that that will help you find the errors that you made, 57 00:02:36,720 --> 00:02:40,080 but also expose you to other ways that maybe will be 58 00:02:40,080 --> 00:02:42,060 simpler, maybe will be harder. 59 00:02:42,060 --> 00:02:45,090 Your feedback on your coming up with a simpler way 60 00:02:45,090 --> 00:02:47,640 would be greatly appreciated, so please send me email anytime 61 00:02:47,640 --> 00:02:49,420 you think you have a better way of doing things, 62 00:02:49,420 --> 00:02:50,610 which is entirely possible. 63 00:02:53,130 --> 00:02:55,770 But as further encouragement, if you mark up 64 00:02:55,770 --> 00:02:59,470 all of your errors and resubmit, we'll 65 00:02:59,470 --> 00:03:03,190 give you back half of the points you lost. 66 00:03:03,190 --> 00:03:05,060 So you get to submit every homework twice. 67 00:03:05,060 --> 00:03:09,756 Once for the due date 5:00 p.m. on Wednesday. 68 00:03:09,756 --> 00:03:11,130 Immediately after that, you'll be 69 00:03:11,130 --> 00:03:13,650 able to see the published solutions. 70 00:03:13,650 --> 00:03:18,450 You can circle all the errors and say algebra, or conceptual, 71 00:03:18,450 --> 00:03:21,480 or I'm an idiot, or whatever is the right way of thinking 72 00:03:21,480 --> 00:03:24,670 about what you did. 73 00:03:24,670 --> 00:03:27,930 And if you get all of the errors, 74 00:03:27,930 --> 00:03:30,150 you'll get back half of the points you lost. 75 00:03:30,150 --> 00:03:32,950 If you say, I'm right, you're wrong, 76 00:03:32,950 --> 00:03:34,690 maybe you won't get all your points back. 77 00:03:34,690 --> 00:03:35,190 Yes? 78 00:03:35,190 --> 00:03:38,450 AUDIENCE: [INAUDIBLE] 79 00:03:38,450 --> 00:03:42,210 PROFESSOR: The resubmission is the same rules. 80 00:03:42,210 --> 00:03:43,710 If you want to submit by paper, you 81 00:03:43,710 --> 00:03:46,150 can turn them in on Friday's recitation. 82 00:03:46,150 --> 00:03:49,350 You have until Friday at 5:00 p.m. otherwise. 83 00:03:49,350 --> 00:03:52,810 Please notice that, if you're using paper, 84 00:03:52,810 --> 00:03:58,020 we will not have returned your solutions in time for you 85 00:03:58,020 --> 00:04:02,980 to write on them, so make a copy. 86 00:04:02,980 --> 00:04:04,650 If you want to do the paper route, 87 00:04:04,650 --> 00:04:08,010 make a photocopy before you turn it in the first time, 88 00:04:08,010 --> 00:04:11,310 so that you have something to write on for the second time. 89 00:04:11,310 --> 00:04:13,460 If you submit electronically, there's 90 00:04:13,460 --> 00:04:17,730 a feedback button on the tutor, so you can see the thing 91 00:04:17,730 --> 00:04:19,255 that you just submitted. 92 00:04:19,255 --> 00:04:20,630 So that's not a problem if you're 93 00:04:20,630 --> 00:04:22,880 doing the electronic thing, but if you're doing paper, 94 00:04:22,880 --> 00:04:27,210 please, make a photocopy before you submit. 95 00:04:27,210 --> 00:04:29,650 Otherwise, it will be hard to mark up your solutions. 96 00:04:29,650 --> 00:04:30,489 Yes? 97 00:04:30,489 --> 00:04:35,480 AUDIENCE: [INAUDIBLE] 98 00:04:35,480 --> 00:04:38,900 PROFESSOR: No, but you should probably tell us 99 00:04:38,900 --> 00:04:41,060 what you'd like us to do. 100 00:04:41,060 --> 00:04:44,920 We would normally try to make our best guess. 101 00:04:44,920 --> 00:04:48,450 So if you turn in everything in the paper copy, 102 00:04:48,450 --> 00:04:51,480 we will look at your previous submissions, 103 00:04:51,480 --> 00:04:55,320 but it would be best if you said some remark like, 104 00:04:55,320 --> 00:04:58,830 this overrides the electronic submission, 105 00:04:58,830 --> 00:05:03,780 or please ignore problems 1 to 4, just make sure we know. 106 00:05:03,780 --> 00:05:07,280 We're going to try to do what you want us to do. 107 00:05:07,280 --> 00:05:09,440 Other issues about logistics? 108 00:05:09,440 --> 00:05:10,063 Yes? 109 00:05:10,063 --> 00:05:13,525 AUDIENCE: So just to make sure, we 110 00:05:13,525 --> 00:05:17,118 don't have to submit homework both electronically 111 00:05:17,118 --> 00:05:18,555 and in paper? 112 00:05:18,555 --> 00:05:20,997 PROFESSOR: No, absolutely not. 113 00:05:20,997 --> 00:05:23,330 If you do the electronic route-- which is the preferred, 114 00:05:23,330 --> 00:05:26,680 we like electronic because it won't get lost-- 115 00:05:26,680 --> 00:05:29,240 the graders will absolutely have access to it 116 00:05:29,240 --> 00:05:31,620 because they're all in one place. 117 00:05:31,620 --> 00:05:34,241 We like the electronic route, but if you'd prefer paper, 118 00:05:34,241 --> 00:05:36,865 just make sure you give us paper and one or the other suffices. 119 00:05:40,070 --> 00:05:40,747 Yes? 120 00:05:40,747 --> 00:05:44,020 AUDIENCE: Did we ever submit the tutor questions on paper? 121 00:05:44,020 --> 00:05:46,860 PROFESSOR: You don't need to. 122 00:05:46,860 --> 00:05:49,200 If you submit the tutor problems online 123 00:05:49,200 --> 00:05:52,650 and it says you're 100% correct, you're done. 124 00:05:52,650 --> 00:05:54,540 We don't ever need to see anything else. 125 00:05:54,540 --> 00:05:55,500 You've got full credit. 126 00:05:58,560 --> 00:05:59,190 Other issues? 127 00:06:01,490 --> 00:06:01,990 Yes? 128 00:06:01,990 --> 00:06:03,798 AUDIENCE: If people are having technical problems 129 00:06:03,798 --> 00:06:05,154 with the tutor, please tell us. 130 00:06:05,154 --> 00:06:06,862 Don't just feel like you have to kick out 131 00:06:06,862 --> 00:06:08,380 and do the paper version. 132 00:06:08,380 --> 00:06:10,313 We're trying make sure that everything 133 00:06:10,313 --> 00:06:11,350 is in place this week. 134 00:06:11,350 --> 00:06:13,224 PROFESSOR: We will try to be very responsive. 135 00:06:13,224 --> 00:06:17,020 As I told you, this is a brand new tutor. 136 00:06:17,020 --> 00:06:21,550 We're very optimistic about it, but we're also realistic. 137 00:06:21,550 --> 00:06:23,429 Software has bugs, but don't tell 138 00:06:23,429 --> 00:06:25,720 any of my software people-- my software friends-- that. 139 00:06:29,650 --> 00:06:32,470 Software has bugs, so we're trying 140 00:06:32,470 --> 00:06:35,895 to get rid of all of ours. 141 00:06:35,895 --> 00:06:37,356 Yeah? 142 00:06:37,356 --> 00:06:38,820 AUDIENCE: [INAUDIBLE] 143 00:06:38,820 --> 00:06:39,810 PROFESSOR: I'm sorry? 144 00:06:39,810 --> 00:06:41,447 AUDIENCE: [INAUDIBLE] 145 00:06:41,447 --> 00:06:43,780 PROFESSOR: You have to shout because my hearing is gone. 146 00:06:43,780 --> 00:06:47,320 AUDIENCE: What happens if you run out of submits on tutor? 147 00:06:47,320 --> 00:06:51,220 PROFESSOR: If you run out of submits, that's bad. 148 00:06:51,220 --> 00:06:52,860 During this sort of ramp up phase, 149 00:06:52,860 --> 00:06:55,240 just send us an email, because the goal is not 150 00:06:55,240 --> 00:07:01,270 to have you run out of submits yet. 151 00:07:01,270 --> 00:07:02,921 Yes? 152 00:07:02,921 --> 00:07:04,504 AUDIENCE: So if you run out of submits 153 00:07:04,504 --> 00:07:06,504 and your last submission was an incorrect answer 154 00:07:06,504 --> 00:07:08,995 and [INAUDIBLE] that has the correct equation 155 00:07:08,995 --> 00:07:11,262 for that answer, will you still get full credit? 156 00:07:11,262 --> 00:07:12,970 PROFESSOR: If you continue to hit submit, 157 00:07:12,970 --> 00:07:16,540 we will continue to be aware of it. 158 00:07:16,540 --> 00:07:17,590 It's all in the grade. 159 00:07:17,590 --> 00:07:20,590 We know everything you've done. 160 00:07:20,590 --> 00:07:24,990 So just go ahead and hit submit some more times. 161 00:07:24,990 --> 00:07:27,130 You can always turn in paper to override it, 162 00:07:27,130 --> 00:07:28,960 or you can always send me an email 163 00:07:28,960 --> 00:07:32,971 and say, I ran out of submits, the answer is blah. 164 00:07:32,971 --> 00:07:34,720 We're going to try to be very cooperative. 165 00:07:34,720 --> 00:07:36,830 This is new. 166 00:07:36,830 --> 00:07:39,500 This is supposed to be helpful. 167 00:07:39,500 --> 00:07:40,215 Yes? 168 00:07:40,215 --> 00:07:43,872 AUDIENCE: [INAUDIBLE] 169 00:07:43,872 --> 00:07:46,080 PROFESSOR: That's not supposed to be on your website. 170 00:07:46,080 --> 00:07:48,270 That's supposed to be on my website. 171 00:07:48,270 --> 00:07:49,470 I'll have to look into that. 172 00:07:53,100 --> 00:07:55,580 If it works, use it. 173 00:07:59,730 --> 00:08:04,830 OK, other obvious errors that we've made? 174 00:08:04,830 --> 00:08:07,290 OK, let's go on and talk about 003. 175 00:08:07,290 --> 00:08:10,090 What I want to do today-- 176 00:08:10,090 --> 00:08:12,060 last time, I introduced what I call 177 00:08:12,060 --> 00:08:21,490 the 6.003 abstraction which is the idea that we represent 178 00:08:21,490 --> 00:08:22,060 a system-- 179 00:08:22,060 --> 00:08:30,850 whatever it is-- by the way it transforms an input signal 180 00:08:30,850 --> 00:08:31,780 to an output signal. 181 00:08:34,720 --> 00:08:37,720 The goal for today is to start to develop 182 00:08:37,720 --> 00:08:42,390 machinery that takes advantage of that representation. 183 00:08:42,390 --> 00:08:44,610 We'll start by thinking about DT. 184 00:08:44,610 --> 00:08:47,100 DT is conceptually easier than CT. 185 00:08:47,100 --> 00:08:48,740 That's why we start there. 186 00:08:48,740 --> 00:08:51,180 Algebra is slightly easier than calculus. 187 00:08:51,180 --> 00:08:51,950 That's the idea. 188 00:08:54,720 --> 00:08:57,900 What we learn in DT will transfer almost verbatim 189 00:08:57,900 --> 00:09:00,060 into CT. 190 00:09:00,060 --> 00:09:02,850 CT is like DT but a little harder 191 00:09:02,850 --> 00:09:06,470 so there is some extra stuff, but the things we learned in DT 192 00:09:06,470 --> 00:09:09,320 will still be applicable in CT. 193 00:09:09,320 --> 00:09:13,330 Discrete time, continuous time. 194 00:09:13,330 --> 00:09:15,860 Furthermore, DT is just important. 195 00:09:15,860 --> 00:09:19,310 So, increasingly, applications of this field 196 00:09:19,310 --> 00:09:21,350 are in discrete time. 197 00:09:21,350 --> 00:09:24,290 That wasn't necessarily true 20 years ago. 198 00:09:24,290 --> 00:09:26,647 It's still the case that there's lots of applications 199 00:09:26,647 --> 00:09:28,730 that are intrinsically married to the physics that 200 00:09:28,730 --> 00:09:30,290 underlie it. 201 00:09:30,290 --> 00:09:32,870 That makes it intrinsically CT, but a lot of stuff 202 00:09:32,870 --> 00:09:34,910 is done in DT. 203 00:09:34,910 --> 00:09:37,507 So for a lot of reasons, we're going to start with DT first, 204 00:09:37,507 --> 00:09:39,590 and we're going to look today at several different 205 00:09:39,590 --> 00:09:42,860 representations of DT. 206 00:09:42,860 --> 00:09:45,380 In particular, we're going to look at difference equations 207 00:09:45,380 --> 00:09:48,360 and block diagrams. 208 00:09:48,360 --> 00:09:52,170 And we're always interested in verbal descriptions, 209 00:09:52,170 --> 00:09:56,750 because that's the way the real world is described to you. 210 00:09:56,750 --> 00:09:59,334 So what we want to do is understand these. 211 00:09:59,334 --> 00:10:01,250 Understand there are strengths and weaknesses, 212 00:10:01,250 --> 00:10:04,249 so that we can capitalize on their strengths, 213 00:10:04,249 --> 00:10:05,540 and go around their weaknesses. 214 00:10:05,540 --> 00:10:07,950 That's the idea. 215 00:10:07,950 --> 00:10:10,439 So the easiest possible representation 216 00:10:10,439 --> 00:10:11,480 is a difference equation. 217 00:10:11,480 --> 00:10:15,050 Difference equations are good because they are absolutely 218 00:10:15,050 --> 00:10:19,610 precise, and they are the absolutely most concise 219 00:10:19,610 --> 00:10:22,330 representation we will have. 220 00:10:22,330 --> 00:10:23,710 Precise and concise. 221 00:10:23,710 --> 00:10:25,660 Those are good features. 222 00:10:25,660 --> 00:10:27,460 So here's an example. 223 00:10:27,460 --> 00:10:35,790 What if my output Y were my input X minus X, n minus 1. 224 00:10:35,790 --> 00:10:38,360 Let's say that the input is a unit sample. 225 00:10:38,360 --> 00:10:40,940 We will often ask for the unit sample response. 226 00:10:40,940 --> 00:10:47,780 The unit sample signal is the most simple non-trivial signal 227 00:10:47,780 --> 00:10:49,370 that we can have. 228 00:10:49,370 --> 00:10:53,900 It is 0 almost everywhere, and the only place it's not 0 229 00:10:53,900 --> 00:10:59,060 is at 0, and when it's not 0 it's 1. 230 00:10:59,060 --> 00:11:02,817 So that's sort of the easiest thing we can imagine. 231 00:11:02,817 --> 00:11:04,400 So what we're going to want to do then 232 00:11:04,400 --> 00:11:06,380 is think about what happens if we put 233 00:11:06,380 --> 00:11:09,530 this signal into that system. 234 00:11:09,530 --> 00:11:10,820 OK, that's completely trivial. 235 00:11:10,820 --> 00:11:12,650 You've done this a gazillion times before, 236 00:11:12,650 --> 00:11:14,780 and now is your chance to prove it. 237 00:11:14,780 --> 00:11:18,320 How many of those statements are true about the output 238 00:11:18,320 --> 00:11:20,784 of this system when that's the input? 239 00:11:20,784 --> 00:11:21,700 Talk to your neighbor. 240 00:11:21,700 --> 00:11:22,910 You've got 15 seconds. 241 00:11:22,910 --> 00:11:25,580 Is it too simple? 242 00:11:25,580 --> 00:11:28,100 Figure out how many of those statements are true. 243 00:12:30,076 --> 00:12:31,950 OK, you're either all bored out of your minds 244 00:12:31,950 --> 00:12:33,240 or paralyzed with fear. 245 00:12:33,240 --> 00:12:35,366 I can't tell. 246 00:12:35,366 --> 00:12:36,750 So everybody raise your hand. 247 00:12:36,750 --> 00:12:38,333 How many of these statements are true? 248 00:12:40,461 --> 00:12:41,710 More hands, more hands, c'mon. 249 00:12:48,550 --> 00:12:56,940 OK, less than half correct, so I'll just 250 00:12:56,940 --> 00:13:01,227 assume that you ran out of time, and I'll go on. 251 00:13:01,227 --> 00:13:02,310 So this is trivial, right? 252 00:13:02,310 --> 00:13:02,851 What do I do? 253 00:13:02,851 --> 00:13:04,190 How do I figure this out? 254 00:13:04,190 --> 00:13:04,690 Yeah? 255 00:13:04,690 --> 00:13:05,640 AUDIENCE: Step by step? 256 00:13:05,640 --> 00:13:07,098 PROFESSOR: Step by step, of course. 257 00:13:07,098 --> 00:13:11,730 Yes, so if I think about applying this equation 258 00:13:11,730 --> 00:13:16,550 to that input, then I can think about just substituting 259 00:13:16,550 --> 00:13:19,455 n equals minus 1, n equals 0, n equals 1, 260 00:13:19,455 --> 00:13:23,570 n equals 2, et cetera. 261 00:13:23,570 --> 00:13:25,550 And you can sort of see that I'm only 262 00:13:25,550 --> 00:13:28,790 looking at the input of two instances of time. 263 00:13:28,790 --> 00:13:32,649 So there's a finite range over which the input-- 264 00:13:32,649 --> 00:13:34,190 there's a finite range of inputs that 265 00:13:34,190 --> 00:13:36,600 can contribute to the output. 266 00:13:36,600 --> 00:13:38,780 And in particular, if I were to say-- 267 00:13:38,780 --> 00:13:41,380 if I were to substitute n equals minus 1, 268 00:13:41,380 --> 00:13:45,850 I would see that the output of time-- y minus 1-- 269 00:13:45,850 --> 00:13:50,000 depends on inputs before the delta function started. 270 00:13:52,880 --> 00:13:56,160 So therefore the answer is 0. 271 00:13:56,160 --> 00:14:03,700 If I look at the output of time 1, the time 0 input is 1, 272 00:14:03,700 --> 00:14:05,720 the time minus 1 input is 0. 273 00:14:05,720 --> 00:14:08,840 So my total answer is 1, et cetera. 274 00:14:08,840 --> 00:14:10,260 This is very straightforward. 275 00:14:10,260 --> 00:14:16,220 It's just plug and chug, and what you see is a waveform. 276 00:14:16,220 --> 00:14:18,410 From which is pretty easy to infer 277 00:14:18,410 --> 00:14:24,500 that Y2 is bigger than Y1, so Y2 was bigger than Y1-- 278 00:14:24,500 --> 00:14:27,540 bigger in a math sense. 279 00:14:27,540 --> 00:14:30,300 And Y3 is bigger than Y2-- 280 00:14:30,300 --> 00:14:31,510 not. 281 00:14:31,510 --> 00:14:38,500 Y3 and Y2 are identical, so that's not true. 282 00:14:38,500 --> 00:14:41,050 Y2 and, in fact, everything above that is 0. 283 00:14:41,050 --> 00:14:45,340 That's true, and if you do a little bit of manipulation, 284 00:14:45,340 --> 00:14:48,350 you can show that this must have also been true. 285 00:14:48,350 --> 00:14:49,230 Everyone see that? 286 00:14:52,414 --> 00:14:54,705 I'm assuming you've seen all this sort of stuff before, 287 00:14:54,705 --> 00:14:56,621 so I'm going to go through it kind of quickly, 288 00:14:56,621 --> 00:14:58,160 but the answer was 4. 289 00:14:58,160 --> 00:15:00,440 4 of those statements were true. 290 00:15:00,440 --> 00:15:02,110 It was just a refresher on how to think 291 00:15:02,110 --> 00:15:04,750 about difference equations. 292 00:15:04,750 --> 00:15:08,579 By comparison, I want to contrast a difference equation 293 00:15:08,579 --> 00:15:09,370 on a block diagram. 294 00:15:12,550 --> 00:15:16,840 This is the same system but represented graphically. 295 00:15:16,840 --> 00:15:18,310 Why would you want to do that? 296 00:15:18,310 --> 00:15:20,440 Well there's advantages and disadvantages. 297 00:15:20,440 --> 00:15:23,950 This is a direct way of expressing the system 298 00:15:23,950 --> 00:15:26,350 if you wanted to implement it in hardware. 299 00:15:26,350 --> 00:15:30,580 So you can imagine building a delay box, building an adder, 300 00:15:30,580 --> 00:15:34,620 building an inverter, and constructing 301 00:15:34,620 --> 00:15:37,800 a piece of hardware that does the computation. 302 00:15:37,800 --> 00:15:40,410 So this is more like a hardware realization. 303 00:15:40,410 --> 00:15:45,420 It's not as compact as the difference equation. 304 00:15:45,420 --> 00:15:48,630 It has another idiosyncrasy that we have to have, 305 00:15:48,630 --> 00:15:52,000 the notion of rest. 306 00:15:52,000 --> 00:15:54,550 When we say the system starts at rest-- which 307 00:15:54,550 --> 00:15:56,840 we will say all the time, and in fact, 308 00:15:56,840 --> 00:15:58,930 you can always assume that unless we tell you 309 00:15:58,930 --> 00:16:02,060 something to the contrary. 310 00:16:02,060 --> 00:16:04,280 When we say a system is at rest, what we mean 311 00:16:04,280 --> 00:16:09,320 is that all of the delays in the system start out being 0. 312 00:16:09,320 --> 00:16:13,220 So at rest means that this signal starts out at 0, 313 00:16:13,220 --> 00:16:14,720 because that's the only-- 314 00:16:14,720 --> 00:16:16,395 there's only one delay in the system. 315 00:16:19,020 --> 00:16:21,240 Its output is initially 0. 316 00:16:21,240 --> 00:16:23,550 So now we think about the input chugging along. 317 00:16:23,550 --> 00:16:25,860 So the input starts out being 1. 318 00:16:25,860 --> 00:16:27,990 That means the inverted input is minus 1. 319 00:16:27,990 --> 00:16:30,054 That doesn't propagate through the delay, 320 00:16:30,054 --> 00:16:31,470 because we're still at the instant 321 00:16:31,470 --> 00:16:35,970 when the input just became one for the first time. 322 00:16:35,970 --> 00:16:39,690 So this one goes through, and adds to this 0 to give us 1, 323 00:16:39,690 --> 00:16:40,570 and our answer is 1. 324 00:16:43,300 --> 00:16:47,760 Now the clock goes kachunk, and at that instant, 325 00:16:47,760 --> 00:16:50,790 the input changes from 1 to 0, because that's 326 00:16:50,790 --> 00:16:53,310 what happens here. 327 00:16:53,310 --> 00:16:55,860 And at precisely the same instant, 328 00:16:55,860 --> 00:16:58,920 the delay signal went from 0-- 329 00:16:58,920 --> 00:17:01,080 which it was because it started at rest-- 330 00:17:01,080 --> 00:17:05,940 to minus 1 which was its input just before kachunk. 331 00:17:08,839 --> 00:17:12,710 Then-- after the signal settle down-- 332 00:17:12,710 --> 00:17:18,950 this becomes 0, and this becomes minus 1, and that's our output. 333 00:17:18,950 --> 00:17:23,000 Then, again, kachunk. 334 00:17:23,000 --> 00:17:24,920 The input doesn't change, because it was zero. 335 00:17:24,920 --> 00:17:25,586 It's still zero. 336 00:17:28,440 --> 00:17:31,750 The output of the delay goes from minus 1 to 0, 337 00:17:31,750 --> 00:17:37,899 and after things settle down, we get 0 at the output, 338 00:17:37,899 --> 00:17:39,315 and that is a state that persists. 339 00:17:42,262 --> 00:17:44,220 So we have these two different representations. 340 00:17:44,220 --> 00:17:46,800 We have difference equations and block diagrams, 341 00:17:46,800 --> 00:17:47,890 so here's a question. 342 00:17:47,890 --> 00:17:49,150 How are they different? 343 00:17:52,492 --> 00:17:54,700 In particular, I've told you the difference equations 344 00:17:54,700 --> 00:17:56,880 are concise and precise. 345 00:18:00,370 --> 00:18:02,550 Are there advantages to a block diagram? 346 00:18:05,324 --> 00:18:06,240 Talk to your neighbor. 347 00:18:06,240 --> 00:18:10,390 Come up with a good hypothesis. 348 00:18:10,390 --> 00:18:12,250 What's good about a block diagram, anything? 349 00:18:48,184 --> 00:18:50,350 So can anybody think of something good about a block 350 00:18:50,350 --> 00:18:53,250 diagram? 351 00:18:53,250 --> 00:18:54,696 Yeah? 352 00:18:54,696 --> 00:18:57,588 AUDIENCE: It changes the [INAUDIBLE] 353 00:18:57,588 --> 00:18:59,516 and it's easy to see what's happening. 354 00:18:59,516 --> 00:19:01,690 PROFESSOR: So it's easy to see. 355 00:19:01,690 --> 00:19:05,420 So that could be something to do with a graphic representation, 356 00:19:05,420 --> 00:19:06,560 but it could be deeper too. 357 00:19:06,560 --> 00:19:09,332 Can somebody say a deeper reason why it might be easier to see? 358 00:19:09,332 --> 00:19:11,040 So that might be a very good-- that might 359 00:19:11,040 --> 00:19:13,140 be a deep comment too. 360 00:19:13,140 --> 00:19:15,630 Why might it be easier to use the block diagram 361 00:19:15,630 --> 00:19:17,421 than it is to use that difference equation? 362 00:19:20,830 --> 00:19:21,357 Yeah? 363 00:19:21,357 --> 00:19:24,160 AUDIENCE: Because equations are broken down into steps? 364 00:19:24,160 --> 00:19:25,220 PROFESSOR: How so? 365 00:19:25,220 --> 00:19:27,854 AUDIENCE: So there's [INAUDIBLE] and then 366 00:19:27,854 --> 00:19:31,926 there's the feedback [INAUDIBLE] difference equations are 367 00:19:31,926 --> 00:19:32,890 [INAUDIBLE] 368 00:19:32,890 --> 00:19:34,610 PROFESSOR: So that's kind of right. 369 00:19:34,610 --> 00:19:37,740 I'm wondering if you could say, I mean, 370 00:19:37,740 --> 00:19:39,330 you just do this, right? 371 00:19:39,330 --> 00:19:41,070 You do this, and then you do that. 372 00:19:41,070 --> 00:19:42,390 What's the difference? 373 00:19:44,421 --> 00:19:46,420 There's something very different about these two 374 00:19:46,420 --> 00:19:47,710 representations. 375 00:19:47,710 --> 00:19:49,900 There is more information in the bottom 376 00:19:49,900 --> 00:19:52,520 than there is in the top. 377 00:19:52,520 --> 00:19:54,020 Can somebody tell me the information 378 00:19:54,020 --> 00:19:55,561 that's available in the bottom that's 379 00:19:55,561 --> 00:19:56,870 not available in the top? 380 00:19:56,870 --> 00:19:59,066 Yes? 381 00:19:59,066 --> 00:20:00,533 AUDIENCE: Actually solving circuits 382 00:20:00,533 --> 00:20:05,423 it becomes a lot easier to use graphical representation. 383 00:20:05,423 --> 00:20:08,390 PROFESSOR: Circuits, graphical, but is there 384 00:20:08,390 --> 00:20:12,460 additional information here that wasn't available there? 385 00:20:12,460 --> 00:20:12,960 Yes? 386 00:20:12,960 --> 00:20:15,142 AUDIENCE: The number of signal paths? 387 00:20:15,142 --> 00:20:16,350 PROFESSOR: Number of signal-- 388 00:20:16,350 --> 00:20:17,160 AUDIENCE: Paths 389 00:20:17,160 --> 00:20:17,868 PROFESSOR: Paths. 390 00:20:17,868 --> 00:20:19,710 Signal paths is a good idea. 391 00:20:19,710 --> 00:20:20,552 Yes? 392 00:20:20,552 --> 00:20:23,444 AUDIENCE: Is it that X and negative 1 is 0? 393 00:20:23,444 --> 00:20:26,830 [INTERPOSING VOICES] 394 00:20:26,830 --> 00:20:30,940 PROFESSOR: Well that sort of goes with specifying X, yes? 395 00:20:30,940 --> 00:20:32,920 AUDIENCE: The structure is more emphasized, 396 00:20:32,920 --> 00:20:36,880 so you can just look at it, and [INAUDIBLE] or not, 397 00:20:36,880 --> 00:20:41,350 and it's more easy to see structure [INAUDIBLE] 398 00:20:41,350 --> 00:20:44,090 PROFESSOR: It is easier, but there's a very precise reason 399 00:20:44,090 --> 00:20:44,770 why it's easier. 400 00:20:44,770 --> 00:20:45,395 There's arrows. 401 00:20:49,210 --> 00:20:53,090 Arrows tell you what to do next. 402 00:20:53,090 --> 00:20:56,150 So the thing that's different is that. 403 00:20:56,150 --> 00:21:00,249 From the structure of the arrows, everything starts here, 404 00:21:00,249 --> 00:21:02,540 and you should worry about what goes that way, and what 405 00:21:02,540 --> 00:21:04,250 goes that way, and what goes that way, 406 00:21:04,250 --> 00:21:05,750 and how that comes together there. 407 00:21:05,750 --> 00:21:08,540 There's more information. 408 00:21:08,540 --> 00:21:11,960 In computer science terms, we would say that the difference 409 00:21:11,960 --> 00:21:15,110 equation is declarative. 410 00:21:15,110 --> 00:21:18,200 It tells you a statement of fact, 411 00:21:18,200 --> 00:21:22,990 not necessarily what you can do with it. 412 00:21:22,990 --> 00:21:26,240 The block diagram is imperative. 413 00:21:26,240 --> 00:21:29,480 It tells you how to do something, 414 00:21:29,480 --> 00:21:31,500 and those are very different. 415 00:21:31,500 --> 00:21:35,900 So the difference equation is concise, precise, 416 00:21:35,900 --> 00:21:38,270 and declarative. 417 00:21:38,270 --> 00:21:40,760 The block diagram is imperative. 418 00:21:40,760 --> 00:21:43,190 Do this, do this, do this. 419 00:21:43,190 --> 00:21:45,140 It shows you the signal flow path. 420 00:21:45,140 --> 00:21:49,310 There's no question about who causes what. 421 00:21:49,310 --> 00:21:54,230 In this, we could be computing X from Y or Y from X. 422 00:21:54,230 --> 00:21:57,620 There's no indication other than some convention-- some place-- 423 00:21:57,620 --> 00:21:59,480 that the x is the input. 424 00:21:59,480 --> 00:22:02,270 That make sense? 425 00:22:02,270 --> 00:22:05,110 So there's a very big difference. 426 00:22:05,110 --> 00:22:07,740 There's another step we want to take. 427 00:22:07,740 --> 00:22:10,070 So imperative and declarative is one step. 428 00:22:10,070 --> 00:22:13,630 Another step is lumping things in the same sense 429 00:22:13,630 --> 00:22:17,590 that we lump the coordinates of a three space point 430 00:22:17,590 --> 00:22:20,770 into a single thing called a point, or in the same way 431 00:22:20,770 --> 00:22:23,560 we make a vector in linear algebra. 432 00:22:23,560 --> 00:22:25,270 The idea is going to be that we're 433 00:22:25,270 --> 00:22:28,090 going to want to think about whole signals at a time. 434 00:22:33,320 --> 00:22:37,150 In the notation that we will use, we will think about-- 435 00:22:37,150 --> 00:22:42,360 this will be the X signal, this will be the Y signal, 436 00:22:42,360 --> 00:22:45,690 and we can re-interpret the block diagram 437 00:22:45,690 --> 00:22:51,090 to be operations on signals rather than operations 438 00:22:51,090 --> 00:22:54,030 on samples. 439 00:22:54,030 --> 00:22:58,320 So I'm trying to develop a level of abstraction that 440 00:22:58,320 --> 00:23:02,160 helps us predict the way systems will behave, 441 00:23:02,160 --> 00:23:05,100 so the next level of abstraction I want to think about 442 00:23:05,100 --> 00:23:08,770 is thinking about whole signals at once. 443 00:23:08,770 --> 00:23:11,580 So if we take this signal idea, we 444 00:23:11,580 --> 00:23:13,050 would say there's an X signal. 445 00:23:13,050 --> 00:23:14,730 It gets operated by this inverter 446 00:23:14,730 --> 00:23:19,070 to give an inverted signal, then this signal 447 00:23:19,070 --> 00:23:24,097 goes through a delay box to give us a delayed inverted signal. 448 00:23:24,097 --> 00:23:26,680 The delayed inverted signal gets added to the original signal, 449 00:23:26,680 --> 00:23:28,138 and that gives us an output signal. 450 00:23:28,138 --> 00:23:32,250 So we think about the nodes being whole signals, 451 00:23:32,250 --> 00:23:36,240 and the boxes being operators on signals, not samples. 452 00:23:36,240 --> 00:23:39,480 That's a huge difference, and to make 453 00:23:39,480 --> 00:23:42,960 sure to be explicit about that we'll develop a notation. 454 00:23:42,960 --> 00:23:45,390 We will say there's an operator R-- 455 00:23:45,390 --> 00:23:48,090 the right shift operator-- 456 00:23:48,090 --> 00:23:50,940 that operates on signals. 457 00:23:50,940 --> 00:23:55,500 So we'll think about if X is the whole input signal, not the nth 458 00:23:55,500 --> 00:23:58,720 sample, the whole thing. 459 00:23:58,720 --> 00:24:02,800 If X is a signal, we can operate on it by the R operator. 460 00:24:02,800 --> 00:24:07,000 We can apply R-- the right shift operator-- to the signal X, 461 00:24:07,000 --> 00:24:08,560 and generate a whole new signal which 462 00:24:08,560 --> 00:24:13,060 I'll call Y which will be a shifted version of the X 463 00:24:13,060 --> 00:24:14,350 signal. 464 00:24:14,350 --> 00:24:17,080 OK, you all got that? 465 00:24:17,080 --> 00:24:20,380 And that leads us then to a concise representation 466 00:24:20,380 --> 00:24:24,460 similar to the difference equation, 467 00:24:24,460 --> 00:24:28,360 but that has the features of the block diagram. 468 00:24:28,360 --> 00:24:32,750 Because it's imperative, R operating on X 469 00:24:32,750 --> 00:24:35,880 tells me start with X and apply R to it. 470 00:24:35,880 --> 00:24:37,750 There's a direction. 471 00:24:37,750 --> 00:24:40,300 R applied to X is imperative. 472 00:24:40,300 --> 00:24:42,740 It tells me what to do. 473 00:24:42,740 --> 00:24:48,310 So this is a way that I can make a notation that 474 00:24:48,310 --> 00:24:51,640 is as concise as difference equations, 475 00:24:51,640 --> 00:24:57,430 but has the same imperative feature as a block diagram. 476 00:24:57,430 --> 00:25:01,240 So it's an improved notation in that sense. 477 00:25:01,240 --> 00:25:03,040 Make sure that you know what it means. 478 00:25:03,040 --> 00:25:05,950 If I said that Y is RX, how many of the following is true? 479 00:25:09,360 --> 00:25:10,770 Now you have to redeem yourself. 480 00:25:10,770 --> 00:25:14,940 You were less than 50% right, so now I want 90% right. 481 00:25:25,630 --> 00:25:27,150 Oh, excuse me, not how many, which? 482 00:25:48,210 --> 00:25:51,520 OK, everybody raise your hands, c'mon. 483 00:25:51,520 --> 00:25:52,790 This is redemption time. 484 00:25:52,790 --> 00:25:54,170 Wonderful, you're redeemed. 485 00:25:54,170 --> 00:25:57,080 So you're about 100% correct. 486 00:25:57,080 --> 00:25:58,970 So to think about that. 487 00:25:58,970 --> 00:26:01,709 One easy way to think about it is by example. 488 00:26:01,709 --> 00:26:03,500 Examples can't prove that something's true, 489 00:26:03,500 --> 00:26:06,344 but they can quickly prove that something's false. 490 00:26:06,344 --> 00:26:08,510 So if you think about the simplest possible signal-- 491 00:26:08,510 --> 00:26:10,850 say you had a unit sample. 492 00:26:10,850 --> 00:26:13,370 If you applied R to that, it would shift it to the right. 493 00:26:15,890 --> 00:26:18,200 If it shifts it to the right, it's pretty clear 494 00:26:18,200 --> 00:26:25,670 that Y at time 1 is the same as X at time 0, 495 00:26:25,670 --> 00:26:30,100 and that's a feature of this equation 496 00:26:30,100 --> 00:26:33,050 and none of the others. 497 00:26:33,050 --> 00:26:35,175 So that proves the others are wrong, 498 00:26:35,175 --> 00:26:36,800 and if you think about it a little bit, 499 00:26:36,800 --> 00:26:38,508 you can generalize that argument to prove 500 00:26:38,508 --> 00:26:42,240 that the second one is actually true for all the cases. 501 00:26:45,700 --> 00:26:47,650 So the idea then is that we want to think 502 00:26:47,650 --> 00:26:51,669 about this hybrid notation, this operator notation, 503 00:26:51,669 --> 00:26:53,460 and we want to think about how to represent 504 00:26:53,460 --> 00:26:55,240 simple systems that way. 505 00:26:55,240 --> 00:26:56,460 So here's an example. 506 00:26:56,460 --> 00:26:59,640 Say I have this system. 507 00:26:59,640 --> 00:27:02,880 I can think about that as the cascade of a system that 508 00:27:02,880 --> 00:27:08,400 starts with X, X as a signal, and this whole thing 509 00:27:08,400 --> 00:27:12,170 gets turned into an operator. 510 00:27:12,170 --> 00:27:16,040 That operator can be written mathematically as 1. 511 00:27:16,040 --> 00:27:20,750 This path added to minus 1 times a delay 512 00:27:20,750 --> 00:27:23,930 minus 1 times a delay, so one minus R operating on X 513 00:27:23,930 --> 00:27:26,860 gives me Y1. 514 00:27:26,860 --> 00:27:34,100 Then a very similar system operates on Y1 to give me Y2 515 00:27:34,100 --> 00:27:37,510 and if the rules of math apply, I 516 00:27:37,510 --> 00:27:43,880 ought to be able to substitute 1 minus RX for Y1 517 00:27:43,880 --> 00:27:47,650 and get that expression. 518 00:27:47,650 --> 00:27:51,910 And if the world were nice and it is. 519 00:27:51,910 --> 00:27:54,790 If the world were nice, then this expression, 520 00:27:54,790 --> 00:27:58,100 since it looks like a polynomial in R, 521 00:27:58,100 --> 00:28:01,056 might behave like a polynomial in R, 522 00:28:01,056 --> 00:28:03,680 and it's pretty easy to convince yourself that that's, in fact, 523 00:28:03,680 --> 00:28:06,160 true. 524 00:28:06,160 --> 00:28:09,820 So think about the primitive definition. 525 00:28:09,820 --> 00:28:16,402 Sample by sample, difference equation, what should this do? 526 00:28:16,402 --> 00:28:18,110 Well, there's a difference representation 527 00:28:18,110 --> 00:28:20,342 for this, which is here. 528 00:28:20,342 --> 00:28:21,800 There's a difference representation 529 00:28:21,800 --> 00:28:26,520 for this, which I can substitute twice 530 00:28:26,520 --> 00:28:28,650 so I can generate an equivalent difference 531 00:28:28,650 --> 00:28:35,330 equation for the whole thing, and then I 532 00:28:35,330 --> 00:28:38,000 can think about term wise translating all of those 533 00:28:38,000 --> 00:28:40,580 into R expressions. 534 00:28:40,580 --> 00:28:43,570 And I see that I have proved the idea 535 00:28:43,570 --> 00:28:48,480 that this particular system at least 536 00:28:48,480 --> 00:28:53,150 behaves as though the operator expressions follow 537 00:28:53,150 --> 00:28:54,500 the rules of polynomials. 538 00:28:57,090 --> 00:29:01,080 So that's important because you know 539 00:29:01,080 --> 00:29:04,100 how to deal with polynomials. 540 00:29:04,100 --> 00:29:07,970 If there's an isomorphism between these kinds 541 00:29:07,970 --> 00:29:12,164 of discrete time signals and polynomials, 542 00:29:12,164 --> 00:29:13,580 the fact that you know how to deal 543 00:29:13,580 --> 00:29:17,030 with polynomials translates into you 544 00:29:17,030 --> 00:29:21,120 already know how to do systems, right? 545 00:29:21,120 --> 00:29:23,670 So it's a very powerful kind of thing 546 00:29:23,670 --> 00:29:26,859 to draw an isomorphism between two systems, 547 00:29:26,859 --> 00:29:29,150 especially when you already know how to do one of them. 548 00:29:33,602 --> 00:29:35,560 So let's think about equivalence a little more. 549 00:29:35,560 --> 00:29:38,180 Here's another example. 550 00:29:38,180 --> 00:29:41,080 So here is that same system that's 551 00:29:41,080 --> 00:29:46,170 composed of two cascaded difference engines, 552 00:29:46,170 --> 00:29:48,585 and the hypothesis is that this is the same. 553 00:29:53,230 --> 00:29:54,450 Two things I want you to see. 554 00:29:54,450 --> 00:29:57,250 First, it should be pretty clear that you 555 00:29:57,250 --> 00:29:59,410 can see how you would derive that thinking 556 00:29:59,410 --> 00:30:04,660 about the system as operators R. You should also 557 00:30:04,660 --> 00:30:07,630 be able to derive the equivalence by thinking 558 00:30:07,630 --> 00:30:12,510 about the individual blocks as difference equations, 559 00:30:12,510 --> 00:30:17,440 but more importantly, I'm hoping that you can see a caveat. 560 00:30:17,440 --> 00:30:19,689 Assuming both are initially at rest-- 561 00:30:19,689 --> 00:30:21,730 that will come up over and over and over and over 562 00:30:21,730 --> 00:30:25,980 again for exactly this reason-- 563 00:30:25,980 --> 00:30:30,250 the equivalence is only true if they started that rest. 564 00:30:30,250 --> 00:30:33,050 That's not quite the right way to say it. 565 00:30:33,050 --> 00:30:36,140 The equivalence is always true if they start at rest. 566 00:30:36,140 --> 00:30:39,060 That is precisely right. 567 00:30:39,060 --> 00:30:41,880 If they start at rest it will always be equivalent. 568 00:30:41,880 --> 00:30:44,070 Can you think of why they have to be at rest? 569 00:30:49,230 --> 00:30:50,790 Why do they have to be at rest? 570 00:30:59,990 --> 00:31:02,670 OK, smile. 571 00:31:02,670 --> 00:31:04,350 It's not torture. 572 00:31:04,350 --> 00:31:05,124 Yes? 573 00:31:05,124 --> 00:31:10,932 AUDIENCE: [INAUDIBLE] 574 00:31:10,932 --> 00:31:15,260 PROFESSOR: So if they weren't at rest, the outputs of the delays 575 00:31:15,260 --> 00:31:16,640 wouldn't necessarily be zero. 576 00:31:20,090 --> 00:31:22,100 There is no simple correspondence 577 00:31:22,100 --> 00:31:24,155 between these two delays and those two delays. 578 00:31:26,960 --> 00:31:30,260 If I told you what are these two delays, 579 00:31:30,260 --> 00:31:34,970 it may or may not be possible to make an equivalent system where 580 00:31:34,970 --> 00:31:39,110 there are two different numbers here. 581 00:31:39,110 --> 00:31:42,860 Say this delay starts at 1, and this delay starts at 7. 582 00:31:42,860 --> 00:31:46,310 There may or may not be a set of numbers 583 00:31:46,310 --> 00:31:48,800 for the bottom delays that will have 584 00:31:48,800 --> 00:31:56,030 the same behavior as those numbers did in a top system. 585 00:31:56,030 --> 00:31:58,610 But what is guaranteed is if they all start at 0, 586 00:31:58,610 --> 00:32:00,850 everything's fine. 587 00:32:00,850 --> 00:32:03,220 That's the reason we put in the caveat 588 00:32:03,220 --> 00:32:05,820 initially that both systems start at rest. 589 00:32:05,820 --> 00:32:08,350 The equivalence that we talk about 590 00:32:08,350 --> 00:32:10,180 is equivalence starting from rest. 591 00:32:13,930 --> 00:32:20,530 So think about these systems. 592 00:32:20,530 --> 00:32:23,594 They are equivalent. 593 00:32:23,594 --> 00:32:25,260 You can try to convince yourself they're 594 00:32:25,260 --> 00:32:32,100 equivalent by thinking about operations on block diagrams, 595 00:32:32,100 --> 00:32:34,380 but what I'd like you to think about is, 596 00:32:34,380 --> 00:32:40,060 they are a statement of what property of polynomials? 597 00:32:40,060 --> 00:32:41,860 The equivalence of those two systems 598 00:32:41,860 --> 00:32:47,000 is a statement about what property of polynomials? 599 00:32:47,000 --> 00:32:49,400 Is it the commutative property, the associative property, 600 00:32:49,400 --> 00:32:52,040 the distributive property, the transitive property, or none 601 00:32:52,040 --> 00:32:52,950 of the above? 602 00:33:10,870 --> 00:33:12,050 OK, you're far too quiet. 603 00:33:14,630 --> 00:33:16,170 Take a break. 604 00:33:16,170 --> 00:33:18,110 Turn to your neighbor. 605 00:33:18,110 --> 00:33:21,500 Tell your neighbor what living group you're in. 606 00:33:21,500 --> 00:33:24,320 Become accustomed to talking to your neighbor, 607 00:33:24,320 --> 00:33:26,608 and then figure this out. 608 00:33:26,608 --> 00:34:18,100 [INTERPOSING VOICES] 609 00:34:18,100 --> 00:34:21,121 PROFESSOR: a equals b means b equals a. 610 00:34:21,121 --> 00:34:24,940 AUDIENCE: [INAUDIBLE] 611 00:34:24,940 --> 00:34:25,630 PROFESSOR: Or 5. 612 00:34:25,630 --> 00:34:35,760 AUDIENCE: [INAUDIBLE] 613 00:34:35,760 --> 00:34:39,029 PROFESSOR: OK, what property is being described here. 614 00:34:39,029 --> 00:34:40,820 Everybody raise your hand with some number. 615 00:34:44,110 --> 00:34:45,520 Excellent, excellent. 616 00:34:45,520 --> 00:34:47,290 More than 90% correct. 617 00:34:47,290 --> 00:34:52,850 So the idea is that we write this as an R expression. 618 00:34:52,850 --> 00:34:55,719 So this system says, start with X, 619 00:34:55,719 --> 00:34:59,080 apply this operator to X. It's a little confusing that one 620 00:34:59,080 --> 00:35:01,000 goes left and one goes right. 621 00:35:01,000 --> 00:35:03,090 So start with X, apply this to X-- 622 00:35:03,090 --> 00:35:06,610 that's the 1 minus R operator-- then take whatever comes out, 623 00:35:06,610 --> 00:35:09,850 and apply R again. 624 00:35:09,850 --> 00:35:14,330 Similarly, down here, start with X. 625 00:35:14,330 --> 00:35:18,310 You've got the sum of two things, an R in the top, 626 00:35:18,310 --> 00:35:21,140 and an R squared minus 1 in the bottom, 627 00:35:21,140 --> 00:35:24,410 and the equivalence of those is the distributive principle. 628 00:35:24,410 --> 00:35:28,770 R distributes so multiplication over addition. 629 00:35:33,520 --> 00:35:35,850 OK, in the interest of time, let me skip that. 630 00:35:35,850 --> 00:35:39,000 That's an exercise that you can work on on your own. 631 00:35:39,000 --> 00:35:41,930 It's very straightforward. 632 00:35:41,930 --> 00:35:46,440 What I want to do is talk about another more slightly deeper 633 00:35:46,440 --> 00:35:53,670 issue, and that's the difference between recipes and constraints 634 00:35:53,670 --> 00:35:55,005 for whole block diagrams. 635 00:35:57,600 --> 00:36:03,180 So we can think about this as a recipe in the sense 636 00:36:03,180 --> 00:36:07,110 that we can take the R operators, 637 00:36:07,110 --> 00:36:11,680 and think about signals as the composition of other signals. 638 00:36:11,680 --> 00:36:17,250 So this says, add this signal to that signal. 639 00:36:17,250 --> 00:36:19,630 This signal is just the delay of that signal. 640 00:36:19,630 --> 00:36:22,230 This signal is just the minus 1 of that signal. 641 00:36:22,230 --> 00:36:26,400 So you can think about, there's a recipe start with X, 642 00:36:26,400 --> 00:36:32,030 invert it, delay it, add it to the original, that's 643 00:36:32,030 --> 00:36:33,810 the output. 644 00:36:33,810 --> 00:36:35,620 Let's contrast that to this. 645 00:36:35,620 --> 00:36:36,435 This is harder. 646 00:36:39,850 --> 00:36:42,800 So let's think about the R operators. 647 00:36:42,800 --> 00:36:45,750 What we can say here is that this output 648 00:36:45,750 --> 00:36:49,770 is composed of the sum of two things, the input 649 00:36:49,770 --> 00:36:51,395 and the delayed version of the output. 650 00:36:56,020 --> 00:36:59,165 That's a recipe for finding the input from the output. 651 00:37:01,960 --> 00:37:05,380 That's not what we want to do. 652 00:37:05,380 --> 00:37:11,620 So there's a recipe here that says, start with the output, 653 00:37:11,620 --> 00:37:14,650 shift it to the right, subtract that from the original, 654 00:37:14,650 --> 00:37:16,030 and that will tell you the input. 655 00:37:16,030 --> 00:37:22,030 But I don't know the output, I know the input. 656 00:37:22,030 --> 00:37:24,284 What that says is if you knew the output, 657 00:37:24,284 --> 00:37:25,450 it would have this property. 658 00:37:25,450 --> 00:37:26,410 It's declarative. 659 00:37:29,840 --> 00:37:34,480 Whatever the output is, it has to have that property 660 00:37:34,480 --> 00:37:38,290 but it didn't tell me how to get it. 661 00:37:38,290 --> 00:37:41,990 The R operator is imperative. 662 00:37:41,990 --> 00:37:46,270 Start with the input, apply R to it, you get the output. 663 00:37:46,270 --> 00:37:52,080 But you can compose systems using an imperative operator 664 00:37:52,080 --> 00:37:57,482 that is not-- where the overall system is not imperative, 665 00:37:57,482 --> 00:37:59,690 so what are you going to do with something like that? 666 00:37:59,690 --> 00:38:02,060 Well it's a constraint, so the question 667 00:38:02,060 --> 00:38:05,180 is how do you think about that? 668 00:38:05,180 --> 00:38:09,830 The thing to do in this case is the thing you should always 669 00:38:09,830 --> 00:38:11,060 do in this course. 670 00:38:11,060 --> 00:38:12,710 Fall back to a simpler method. 671 00:38:15,207 --> 00:38:17,040 The idea is we're going to start by teaching 672 00:38:17,040 --> 00:38:20,460 simple methods, then more abstract methods and more 673 00:38:20,460 --> 00:38:21,434 abstract methods. 674 00:38:21,434 --> 00:38:23,850 The reason being that as the methods become more abstract, 675 00:38:23,850 --> 00:38:24,933 they become more powerful. 676 00:38:24,933 --> 00:38:28,770 You can do things quicker, but if they fail, 677 00:38:28,770 --> 00:38:31,090 revert to the easier method. 678 00:38:31,090 --> 00:38:34,890 So here the easier method is sample by sample, 679 00:38:34,890 --> 00:38:37,140 so let's just think about how signals would 680 00:38:37,140 --> 00:38:40,120 propagate through this system. 681 00:38:40,120 --> 00:38:46,580 So think about this as a sample by sample block diagram system. 682 00:38:46,580 --> 00:38:49,890 Say that it starts at rest. 683 00:38:49,890 --> 00:38:52,250 So the output before any inputs come in 684 00:38:52,250 --> 00:38:54,530 have to be 0, because the initial condition was 685 00:38:54,530 --> 00:38:58,040 that delays were all 0. 686 00:38:58,040 --> 00:39:02,050 Now the input comes in at time 1. 687 00:39:02,050 --> 00:39:07,640 So X of 0 is 1, so the output becomes one. 688 00:39:07,640 --> 00:39:12,210 And now because of the way the feedback path works, 689 00:39:12,210 --> 00:39:14,370 the output as a second signal is-- 690 00:39:14,370 --> 00:39:16,090 the second sample is also one. 691 00:39:18,720 --> 00:39:21,190 Everybody see that? 692 00:39:21,190 --> 00:39:22,340 And in fact, it's stuck. 693 00:39:25,280 --> 00:39:28,740 So the idea in this system-- if you think about it 694 00:39:28,740 --> 00:39:33,810 from a sample by sample point of view, the signal at the output 695 00:39:33,810 --> 00:39:35,680 persists long. 696 00:39:35,680 --> 00:39:38,670 In fact, infinitely long after the signal as the input 697 00:39:38,670 --> 00:39:39,910 goes away. 698 00:39:39,910 --> 00:39:43,110 That's very different from what we saw in the simpler 699 00:39:43,110 --> 00:39:47,010 systems that were imperative. 700 00:39:47,010 --> 00:39:49,380 If in the previous system you had a finite length 701 00:39:49,380 --> 00:39:53,240 signal at the input, any number of operations 702 00:39:53,240 --> 00:39:55,670 that you did to it and then combined those 703 00:39:55,670 --> 00:39:57,380 would give you a finite length output. 704 00:39:57,380 --> 00:39:59,240 Maybe bigger than the last one because 705 00:39:59,240 --> 00:40:02,510 of extra delays in the system, but if it started finite, 706 00:40:02,510 --> 00:40:04,940 it's finite. 707 00:40:04,940 --> 00:40:06,130 This is different here. 708 00:40:06,130 --> 00:40:09,400 I'm having a persisting response persists 709 00:40:09,400 --> 00:40:14,900 to infinity from a transient input and that's because-- 710 00:40:14,900 --> 00:40:16,570 it's because of feedback. 711 00:40:16,570 --> 00:40:19,880 It's because there's a path that wraps back on itself. 712 00:40:22,950 --> 00:40:25,950 How do we think about that in terms of R operators? 713 00:40:25,950 --> 00:40:28,680 Well having figured out what the response looks like, 714 00:40:28,680 --> 00:40:34,050 there's a perfectly valid R expression for this signal. 715 00:40:34,050 --> 00:40:39,210 This is just the unit sample plus R times the unit sample 716 00:40:39,210 --> 00:40:41,700 plus R squared times the unit sample plus R cubed 717 00:40:41,700 --> 00:40:44,370 times the unit sample, ad infinitum. 718 00:40:44,370 --> 00:40:46,930 That's that system. 719 00:40:46,930 --> 00:40:50,870 So I've done something very powerful and not all that 720 00:40:50,870 --> 00:40:51,890 obvious. 721 00:40:51,890 --> 00:40:59,030 I've taken a system that was not imperative, 722 00:40:59,030 --> 00:41:01,628 and I've turned it into a system that is imperative. 723 00:41:04,440 --> 00:41:07,360 This system gives me a recipe for how to do things. 724 00:41:07,360 --> 00:41:09,525 Start with X, delay it by one, and add 725 00:41:09,525 --> 00:41:11,920 it delay by another one, and add it delay by another one, 726 00:41:11,920 --> 00:41:14,400 and add it delay it by another one, and add it. 727 00:41:14,400 --> 00:41:18,660 I've converted a declarative statement 728 00:41:18,660 --> 00:41:22,230 about how the system should work into an imperative statement 729 00:41:22,230 --> 00:41:24,390 about how the system should work, 730 00:41:24,390 --> 00:41:26,760 and I've got an equivalent system representation 731 00:41:26,760 --> 00:41:28,020 like this. 732 00:41:28,020 --> 00:41:32,400 And you should be saying to yourself, do I really buy that? 733 00:41:32,400 --> 00:41:34,770 So is it really going to be the case 734 00:41:34,770 --> 00:41:40,590 that a system that was previously described this way? 735 00:41:40,590 --> 00:41:44,280 Tell me the output operate on it by 1 minus R, 736 00:41:44,280 --> 00:41:45,990 and that'll be the input should that 737 00:41:45,990 --> 00:41:49,640 be the same as this system. 738 00:41:49,640 --> 00:41:54,710 If you think about that mathematically, here's a proof. 739 00:41:54,710 --> 00:41:57,290 It is OK to think about it that way. 740 00:41:57,290 --> 00:41:59,955 Start with the idea that X2 equals X1. 741 00:42:02,940 --> 00:42:06,450 This is a statement of the right hand side. 742 00:42:06,450 --> 00:42:07,930 Substitute X1 for X2. 743 00:42:10,830 --> 00:42:15,620 Substitute this equivalent expression for X1, 744 00:42:15,620 --> 00:42:16,797 and multiply it out. 745 00:42:19,600 --> 00:42:22,420 When you do that, you find Y2 equals Y1. 746 00:42:22,420 --> 00:42:26,440 It's true, and what we've done is 747 00:42:26,440 --> 00:42:28,690 showed a very important thing. 748 00:42:28,690 --> 00:42:31,690 This and this are reciprocals. 749 00:42:31,690 --> 00:42:33,710 When you multiply two polynomials together 750 00:42:33,710 --> 00:42:39,350 and the answer is one, they're reciprocals. 751 00:42:39,350 --> 00:42:43,130 So that implies that we can treat that first expression-- 752 00:42:43,130 --> 00:42:44,930 we can think about this-- 753 00:42:44,930 --> 00:42:46,320 let's see, backup. 754 00:42:46,320 --> 00:42:48,680 So here, we had an expression that said, 755 00:42:48,680 --> 00:42:54,690 start with the output, operate on it with this, get the input. 756 00:42:54,690 --> 00:43:02,630 We can think about that instead as operate on the input 757 00:43:02,630 --> 00:43:06,540 by the reciprocal operator. 758 00:43:06,540 --> 00:43:09,780 Now I've defined what this reciprocal mean. 759 00:43:12,530 --> 00:43:15,870 Reciprocal means that thing in the last slide, 760 00:43:15,870 --> 00:43:19,230 so now I know from the last slide 761 00:43:19,230 --> 00:43:22,790 the reciprocal of 1 minus R is 1 plus R plus R squared 762 00:43:22,790 --> 00:43:25,550 plus R cubed, blah, blah, blah, and that 763 00:43:25,550 --> 00:43:27,710 follows the normal rules of polynomials. 764 00:43:27,710 --> 00:43:29,180 This is not a surprise. 765 00:43:29,180 --> 00:43:30,290 You could have done that. 766 00:43:30,290 --> 00:43:32,081 You could have derived that same expression 767 00:43:32,081 --> 00:43:35,830 by thinking about synthetic division. 768 00:43:35,830 --> 00:43:38,890 You could have thought about it as a Taylor series. 769 00:43:38,890 --> 00:43:43,730 Anything you know about polynomials should have worked. 770 00:43:43,730 --> 00:43:48,110 So this idea of representing a signal by a system-- 771 00:43:48,110 --> 00:43:50,650 by an operator R-- 772 00:43:50,650 --> 00:43:53,300 is extraordinarily powerful. 773 00:43:53,300 --> 00:43:54,890 We've been able to take advantage 774 00:43:54,890 --> 00:43:59,180 of non-trivial properties of polynomials 775 00:43:59,180 --> 00:44:02,720 by having done that, and we'll continue that way. 776 00:44:02,720 --> 00:44:11,465 So the overriding feature of this system that was not-- 777 00:44:11,465 --> 00:44:15,710 that could not-- so this system was not a recipe. 778 00:44:15,710 --> 00:44:16,770 It was a constraint. 779 00:44:19,370 --> 00:44:21,650 The feature that made it that way is something 780 00:44:21,650 --> 00:44:25,190 we'll call feedback, and we'll be 781 00:44:25,190 --> 00:44:28,400 able to visualize feedback very simply by looking at the block 782 00:44:28,400 --> 00:44:29,240 diagram. 783 00:44:29,240 --> 00:44:32,060 Feedback means that there's a cycle somewhere. 784 00:44:34,680 --> 00:44:37,130 So if there's no cycles then it has 785 00:44:37,130 --> 00:44:38,990 to be-- so we call this the network that 786 00:44:38,990 --> 00:44:42,220 results when there's no cycles, we say it's acyclic. 787 00:44:42,220 --> 00:44:43,820 What a clever word, right? 788 00:44:43,820 --> 00:44:45,320 So no cycles. 789 00:44:45,320 --> 00:44:48,230 And when it's a acyclic, it will have the property 790 00:44:48,230 --> 00:44:52,380 that transient inputs give rise to transient outputs. 791 00:44:52,380 --> 00:44:56,400 If it has a cycle in it, then it has feedback and feedback has 792 00:44:56,400 --> 00:44:59,700 the potential-- doesn't always do it but has the potential-- 793 00:44:59,700 --> 00:45:04,930 to generate a persisting output to a transient input. 794 00:45:08,050 --> 00:45:14,650 So if we think about a slightly more complicated example where 795 00:45:14,650 --> 00:45:19,060 I've put a number p0 in the feedback loop that was not 796 00:45:19,060 --> 00:45:23,740 in the previous example, we can see that the way 797 00:45:23,740 --> 00:45:25,210 you can conceive of this-- 798 00:45:25,210 --> 00:45:27,042 since there's a single time when we're 799 00:45:27,042 --> 00:45:28,750 thinking about the unit sample response-- 800 00:45:28,750 --> 00:45:33,020 there's a single input that's not 0. 801 00:45:33,020 --> 00:45:35,450 And what you can do is think about the feedback-- 802 00:45:35,450 --> 00:45:37,220 the block diagram-- 803 00:45:37,220 --> 00:45:41,900 you can think about it separating times, 804 00:45:41,900 --> 00:45:46,500 because there's one unit of delay in every cycle. 805 00:45:46,500 --> 00:45:48,860 So you can think about that as the only way 806 00:45:48,860 --> 00:45:51,510 you can get from the input at times 0 to the output of times 807 00:45:51,510 --> 00:45:53,370 0 is through the straightforward path. 808 00:45:53,370 --> 00:45:55,610 The one highlighted by red. 809 00:45:55,610 --> 00:45:57,620 The only way you can get from the input of times 810 00:45:57,620 --> 00:46:02,480 0 to the output of times 1 is to make the circle at least once-- 811 00:46:02,480 --> 00:46:07,300 to make the circle exactly once there. 812 00:46:07,300 --> 00:46:11,350 If you make the circle exactly once then, 813 00:46:11,350 --> 00:46:12,790 instead of being height one, it's 814 00:46:12,790 --> 00:46:16,435 height p0, because you had to have gone through the p0 once. 815 00:46:19,320 --> 00:46:22,870 If you go around a third time, it's p-not squared, 816 00:46:22,870 --> 00:46:25,170 p-not to the fourth, et cetera. 817 00:46:25,170 --> 00:46:31,130 So there's a way of thinking about why the way is there 818 00:46:31,130 --> 00:46:36,210 this persistent response by looking at the block diagrams. 819 00:46:36,210 --> 00:46:38,520 What I'm trying to emphasize is that it's 820 00:46:38,520 --> 00:46:42,510 useful to be able to think about difference equations, block 821 00:46:42,510 --> 00:46:48,240 diagrams, R operators, they're all useful. 822 00:46:48,240 --> 00:46:50,790 What you want to know is the strengths and limitations 823 00:46:50,790 --> 00:46:51,420 of each. 824 00:46:51,420 --> 00:46:53,700 So here what I've showed is a very good way 825 00:46:53,700 --> 00:46:58,710 to think about feedback in terms of block diagrams. 826 00:47:03,230 --> 00:47:07,480 How many of these networks are cyclic? 827 00:47:07,480 --> 00:47:09,360 How many of these networks are acyclic? 828 00:47:13,190 --> 00:47:14,300 Is this cyclic? 829 00:47:14,300 --> 00:47:16,430 This network? 830 00:47:16,430 --> 00:47:21,250 No, so none of the paths wrap that back on itself. 831 00:47:21,250 --> 00:47:22,410 Here's a path. 832 00:47:22,410 --> 00:47:23,580 Here's a path. 833 00:47:23,580 --> 00:47:24,290 Here's a path. 834 00:47:24,290 --> 00:47:25,790 None of those wrap on itself. 835 00:47:25,790 --> 00:47:28,240 Is this network cyclic? 836 00:47:28,240 --> 00:47:31,810 Sure, this is cyclic over here. 837 00:47:31,810 --> 00:47:33,380 This one, Yeah. 838 00:47:33,380 --> 00:47:35,020 The input, this is cyclic. 839 00:47:35,020 --> 00:47:36,250 This one? 840 00:47:36,250 --> 00:47:38,140 Doubly cyclic. 841 00:47:38,140 --> 00:47:47,150 So that's the idea, and this gives rise to a notion 842 00:47:47,150 --> 00:47:48,380 that we will call a pole. 843 00:47:51,250 --> 00:47:54,250 A pole is the base of the geometric sequence 844 00:47:54,250 --> 00:47:58,400 that results in these simple sequences whenever 845 00:47:58,400 --> 00:48:02,620 you have feedback, and we'll be saying a whole lot more 846 00:48:02,620 --> 00:48:05,470 about those in upcoming times. 847 00:48:08,320 --> 00:48:11,540 In the interest of time, I have to stop now, 848 00:48:11,540 --> 00:48:13,960 but so the idea from today is supposed 849 00:48:13,960 --> 00:48:17,370 to have been lots of representations 850 00:48:17,370 --> 00:48:22,740 for DT signals, DT signals and systems, strengths 851 00:48:22,740 --> 00:48:26,370 and weaknesses of each, difference equations-- 852 00:48:26,370 --> 00:48:31,020 concise, precise-- block diagrams, visualizing signals 853 00:48:31,020 --> 00:48:35,910 flow paths, operators, thinking about things as polynomials, 854 00:48:35,910 --> 00:48:39,060 and what we'll do in the next few sessions 855 00:48:39,060 --> 00:48:41,640 is try to build on that polynomial representation 856 00:48:41,640 --> 00:48:45,990 to gain further insights into the way that systems will work. 857 00:48:45,990 --> 00:48:47,720 Thank you.