1 00:00:00,000 --> 00:00:02,620 SPEAKER: The following content is provided under a Creative 2 00:00:02,620 --> 00:00:03,640 Commons license. 3 00:00:03,640 --> 00:00:06,730 Your support will help MIT OpenCourseWare continue to 4 00:00:06,730 --> 00:00:10,030 offer high quality educational resources for free. 5 00:00:10,030 --> 00:00:12,780 To make a donation, or to view additional materials from 6 00:00:12,780 --> 00:00:16,900 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:16,900 --> 00:00:19,260 ocw.mit.edu. 8 00:00:19,260 --> 00:00:26,960 PROFESSOR: OK, last time we were talking about this thing 9 00:00:26,960 --> 00:00:30,200 called the theorem of irrelevance. 10 00:00:30,200 --> 00:00:33,780 At one level, the theorem of irrelevance is just another 11 00:00:33,780 --> 00:00:39,660 statement of, when you start looking at detection theory, 12 00:00:39,660 --> 00:00:43,050 there are things called sufficient statistics. 13 00:00:43,050 --> 00:00:45,480 And what the theorem of irrelevance says is you can 14 00:00:45,480 --> 00:00:50,560 ignore things that aren't part of a sufficient statistic. 15 00:00:50,560 --> 00:00:53,530 But more specifically, it says something about what those 16 00:00:53,530 --> 00:00:55,920 irrelevant things are. 17 00:00:55,920 --> 00:00:59,740 And in particular, let me repeat what we said last time 18 00:00:59,740 --> 00:01:01,780 but with a little less of the detail. 19 00:01:01,780 --> 00:01:03,990 You start out with a signal set. 20 00:01:03,990 --> 00:01:04,230 Ok? 21 00:01:04,230 --> 00:01:08,540 You're going to transmit one element of that single-- 22 00:01:08,540 --> 00:01:10,950 signal set-- 23 00:01:10,950 --> 00:01:15,000 and you're going to turn it into some waveform-- 24 00:01:15,000 --> 00:01:17,000 which we'll call X of t. 25 00:01:17,000 --> 00:01:20,680 And X of t is going to depend on which particular signal 26 00:01:20,680 --> 00:01:23,140 enters the transmitter at this point. 27 00:01:23,140 --> 00:01:25,870 You're going to receive something where the thing that 28 00:01:25,870 --> 00:01:29,060 you receive has noise added to it. 29 00:01:29,060 --> 00:01:32,170 And also if you'll notice I've written the thing that we 30 00:01:32,170 --> 00:01:36,600 receive that it's including a whole lot of other stuff. 31 00:01:36,600 --> 00:01:36,940 OK? 32 00:01:36,940 --> 00:01:40,910 In other words, the signal is constrained to j different 33 00:01:40,910 --> 00:01:43,420 degrees of freedom. 34 00:01:43,420 --> 00:01:49,340 And what you get out of the channel involves a much larger 35 00:01:49,340 --> 00:01:51,540 set of degrees of freedom. 36 00:01:51,540 --> 00:01:55,590 In other words, you're putting things in involving only a 37 00:01:55,590 --> 00:01:58,930 certain period of time, in a certain bandwidth, usually. 38 00:01:58,930 --> 00:02:02,220 And you can look at anything you want to at the output. 39 00:02:02,220 --> 00:02:05,055 The only thing is you can't peek at what the input was at 40 00:02:05,055 --> 00:02:08,380 the transmitter, because if you could there will be no 41 00:02:08,380 --> 00:02:11,410 sense in actually transmitting it. 42 00:02:11,410 --> 00:02:15,760 OK, so the point is, we receive something which, in 43 00:02:15,760 --> 00:02:21,030 the degrees of freedom that we know about, Yj is equal Xj 44 00:02:21,030 --> 00:02:23,670 plus the noise variables. 45 00:02:23,670 --> 00:02:26,680 And in the other degrees of freedom, Yj is 46 00:02:26,680 --> 00:02:28,770 just equal to Zj. 47 00:02:28,770 --> 00:02:33,780 And now the rules of the game are that these out of band 48 00:02:33,780 --> 00:02:42,980 things, all of these noise coefficients, which are not 49 00:02:42,980 --> 00:02:49,240 part of the phi 1 up to phi sub capital J. All of these 50 00:02:49,240 --> 00:02:55,060 things, the noise there, is independent of the noise in-- 51 00:02:55,060 --> 00:02:58,150 is independent of the noise and the signal in what we're 52 00:02:58,150 --> 00:02:59,740 actually sending. 53 00:02:59,740 --> 00:03:03,820 Because it's independent it's irrelevant that's relatively 54 00:03:03,820 --> 00:03:05,520 easy to show. 55 00:03:05,520 --> 00:03:10,440 But the broader way to look at this, and the way I want-- 56 00:03:10,440 --> 00:03:13,190 and the way I'd like to get you used to thinking about 57 00:03:13,190 --> 00:03:18,420 this-- is that this other stuff here can include signals 58 00:03:18,420 --> 00:03:20,260 sent by other users. 59 00:03:20,260 --> 00:03:23,390 It can include signals sent by you at other times. 60 00:03:23,390 --> 00:03:26,720 It can include anything else in the world. 61 00:03:26,720 --> 00:03:29,940 And what this theorem is saying is, if you're sending 62 00:03:29,940 --> 00:03:35,210 the signal in just these j degrees of freedom, then you 63 00:03:35,210 --> 00:03:37,620 don't have to look at anything else. 64 00:03:37,620 --> 00:03:40,370 So that all you have to look at is these received 65 00:03:40,370 --> 00:03:41,620 components, Y sub j. 66 00:03:43,970 --> 00:03:46,060 OK, it says something more than that. 67 00:03:46,060 --> 00:03:49,370 It says a lot more than that. 68 00:03:49,370 --> 00:03:52,710 These actual orthonormal functions that you use, phi 1 69 00:03:52,710 --> 00:03:55,950 up to phi sub capital J, don't appear in 70 00:03:55,950 --> 00:03:58,450 that solution at all. 71 00:03:58,450 --> 00:03:58,830 OK? 72 00:03:58,830 --> 00:04:00,640 In other words, you really have a vector 73 00:04:00,640 --> 00:04:02,650 problem at this point. 74 00:04:02,650 --> 00:04:05,500 You have a j dimensional vector that you send. 75 00:04:05,500 --> 00:04:09,180 You have a j dimensional vector that you receive. 76 00:04:09,180 --> 00:04:11,890 You can use orthonormal functions, which are anything 77 00:04:11,890 --> 00:04:16,560 in the world, so long as the noise in that region that 78 00:04:16,560 --> 00:04:20,150 you're looking at and sending things in does not vary. 79 00:04:20,150 --> 00:04:21,890 In other words, what you can send is 80 00:04:21,890 --> 00:04:23,570 very broadband signals. 81 00:04:23,570 --> 00:04:25,930 You can send very narrow band signals. 82 00:04:25,930 --> 00:04:30,980 You can send anything and it doesn't make any difference. 83 00:04:30,980 --> 00:04:33,660 What this says is when you're dealing with a white Gaussian 84 00:04:33,660 --> 00:04:37,020 noise channel, all signals are equivalent. 85 00:04:40,990 --> 00:04:41,340 OK? 86 00:04:41,340 --> 00:04:43,480 In other words, it doesn't make any difference what 87 00:04:43,480 --> 00:04:46,260 modulation system you use. 88 00:04:46,260 --> 00:04:47,700 They all behave the same way. 89 00:04:50,210 --> 00:04:54,000 The only difference between different modulation systems 90 00:04:54,000 --> 00:04:57,400 comes in these second order affects, which we haven't 91 00:04:57,400 --> 00:04:59,230 started to look at much yet. 92 00:04:59,230 --> 00:05:03,790 How hard is it to retrieve carrier? 93 00:05:03,790 --> 00:05:10,590 How hard is it to deal with things like time 94 00:05:10,590 --> 00:05:12,760 synchronization? 95 00:05:12,760 --> 00:05:14,580 How hard is it to build the filters? 96 00:05:14,580 --> 00:05:18,780 How hard is it to move from passband down to baseband if 97 00:05:18,780 --> 00:05:21,320 you want to do your operations at baseband? 98 00:05:21,320 --> 00:05:25,960 All of these questions become important, but the basic 99 00:05:25,960 --> 00:05:29,950 question of how to deal with the Gaussian noise, it doesn't 100 00:05:29,950 --> 00:05:32,370 make any difference. 101 00:05:32,370 --> 00:05:36,400 You can use whatever system you want to, and the analysis 102 00:05:36,400 --> 00:05:39,660 of the noise part of the problem is exactly the same at 103 00:05:39,660 --> 00:05:41,130 everything. 104 00:05:41,130 --> 00:05:41,910 OK? 105 00:05:41,910 --> 00:05:46,360 And that's fairly important because it says that in fact, 106 00:05:46,360 --> 00:05:48,750 this signal space there we're using-- 107 00:05:48,750 --> 00:05:50,640 I mean yes, we're all used to the fact that 108 00:05:50,640 --> 00:05:52,210 it's a vector space. 109 00:05:52,210 --> 00:05:56,820 L2 is a vector space, fine-- what's important here is 110 00:05:56,820 --> 00:06:01,330 you're using a finite part of that vector space, and you can 111 00:06:01,330 --> 00:06:04,690 deal with it just as if it's finite dimensional vectors. 112 00:06:04,690 --> 00:06:07,760 You can deal with vectors and matrices and all of that neat 113 00:06:07,760 --> 00:06:12,050 stuff and you can forget about all of the analog stuff. 114 00:06:12,050 --> 00:06:18,140 you look at 99 percent of the papers appear both in the 115 00:06:18,140 --> 00:06:22,310 information theory transactions and in the 116 00:06:22,310 --> 00:06:25,060 transactions of-- 117 00:06:25,060 --> 00:06:26,740 oh I guess it-- 118 00:06:26,740 --> 00:06:29,010 it's communication technology. 119 00:06:29,010 --> 00:06:32,480 You look at all of these things and 99 percent of the 120 00:06:32,480 --> 00:06:36,270 authors don't know anything about analog communication. 121 00:06:36,270 --> 00:06:38,550 All they've learned is how to deal with these vectors. 122 00:06:41,080 --> 00:06:41,600 OK? 123 00:06:41,600 --> 00:06:45,440 So in fact this is the key to that. 124 00:06:45,440 --> 00:06:48,260 And you can now play their game. 125 00:06:48,260 --> 00:06:51,450 But also when their game doesn't work you can go back 126 00:06:51,450 --> 00:06:54,290 to looking at the analog waveforms. 127 00:06:54,290 --> 00:06:56,350 But you now understand what that game is 128 00:06:56,350 --> 00:06:57,240 that they're playing. 129 00:06:57,240 --> 00:06:59,370 They're assuming white Gaussian noise. 130 00:06:59,370 --> 00:07:01,370 They don't have to deal with the fact that the white 131 00:07:01,370 --> 00:07:04,930 Gaussian noise is spread over all time and all frequency. 132 00:07:04,930 --> 00:07:07,930 Because part of this thoerem says it doesn't make any 133 00:07:07,930 --> 00:07:11,270 difference how the noise is characterized outside of this 134 00:07:11,270 --> 00:07:12,920 region that you're looking at. 135 00:07:12,920 --> 00:07:15,700 That's the other part of the argument that we've been 136 00:07:15,700 --> 00:07:16,910 dealing with all along. 137 00:07:16,910 --> 00:07:19,850 It doesn't matter how you model the noise anywhere 138 00:07:19,850 --> 00:07:21,910 outside of what you're looking at. 139 00:07:21,910 --> 00:07:25,530 The only thing you need is that the noise is independent 140 00:07:25,530 --> 00:07:27,500 outside of there. 141 00:07:27,500 --> 00:07:29,170 OK? 142 00:07:29,170 --> 00:07:30,420 So that's-- 143 00:07:32,770 --> 00:07:36,030 I mean this was sort of trivial analytically, but it's 144 00:07:36,030 --> 00:07:38,550 really an important aspect of what's going on. 145 00:07:42,440 --> 00:07:43,460 OK. 146 00:07:43,460 --> 00:07:46,040 Let's go back to QAM or PAM. 147 00:07:50,100 --> 00:07:53,300 The baseband input to a white Gaussian noise channel-- we're 148 00:07:53,300 --> 00:07:55,270 going to model it as u of t-- 149 00:07:55,270 --> 00:07:57,800 we're going to look at a succession 150 00:07:57,800 --> 00:07:59,240 of j different signals. 151 00:07:59,240 --> 00:08:03,400 In other words, when we study detection, we said we're going 152 00:08:03,400 --> 00:08:06,130 to build the system, we're going to send one signal, and 153 00:08:06,130 --> 00:08:09,410 then we're going to receive that signal and we're going to 154 00:08:09,410 --> 00:08:10,540 tear that system down. 155 00:08:10,540 --> 00:08:13,470 We're not going to send more than just this one signal. 156 00:08:13,470 --> 00:08:15,690 OK, now we're sending a big batch of signals. 157 00:08:15,690 --> 00:08:17,750 Were sending capital J of them. 158 00:08:17,750 --> 00:08:20,380 Where J can be as big as you please. 159 00:08:20,380 --> 00:08:24,200 Now we're going to look at two different alternatives. 160 00:08:24,200 --> 00:08:29,270 OK, one of these alternatives is sending all j signals and 161 00:08:29,270 --> 00:08:32,700 building a receiver which looks at all J of them 162 00:08:32,700 --> 00:08:37,430 together and makes a joint decision on everything. 163 00:08:37,430 --> 00:08:43,800 It makes a maximum likelihood decision on this sequence of J 164 00:08:43,800 --> 00:08:45,070 possible inputs. 165 00:08:45,070 --> 00:08:46,700 This is one of the things we've looked at. 166 00:08:46,700 --> 00:08:52,060 This is what happens when you have non binary detection. 167 00:08:52,060 --> 00:08:54,380 I mean, here you have an enormous number of things 168 00:08:54,380 --> 00:08:56,900 you're detecting between. 169 00:08:56,900 --> 00:08:59,800 And you do maximum likelihood detection on it. 170 00:08:59,800 --> 00:09:03,570 And the question is, do you get anything extra from that 171 00:09:03,570 --> 00:09:07,940 beyond what you get out of doing what we did before, just 172 00:09:07,940 --> 00:09:11,460 forgetting about the fact that other signals existed? 173 00:09:11,460 --> 00:09:14,660 Namely, which is better? 174 00:09:14,660 --> 00:09:17,620 What the notes prove, and what I'm going to sort of indicate 175 00:09:17,620 --> 00:09:20,670 here without any attempt to prove it-- 176 00:09:20,670 --> 00:09:23,240 I mean it's not a hard proof, it's just-- 177 00:09:23,240 --> 00:09:26,560 I mean the hardest part of this is realizing what the 178 00:09:26,560 --> 00:09:28,170 problem is. 179 00:09:28,170 --> 00:09:30,330 And the problem is you can detect things in 180 00:09:30,330 --> 00:09:31,270 two different ways. 181 00:09:31,270 --> 00:09:34,330 You can detect things one signal at a time. 182 00:09:34,330 --> 00:09:37,840 Or you can detect them the whole sequence at a time. 183 00:09:37,840 --> 00:09:41,550 And you do something different in each of these cases. 184 00:09:41,550 --> 00:09:45,830 OK, so we're going to assume again that these thetas are an 185 00:09:45,830 --> 00:09:47,290 orthonormal set. 186 00:09:47,290 --> 00:09:49,910 I'm going to assume that I've extended them so 187 00:09:49,910 --> 00:09:53,960 they span all of l2. 188 00:09:53,960 --> 00:09:58,430 I'm going to let v be a sample of this output 189 00:09:58,430 --> 00:10:01,950 vector, v1 to v sub j. 190 00:10:01,950 --> 00:10:04,620 You see, I use the theorem of irrelevance here because I 191 00:10:04,620 --> 00:10:07,980 don't care about anything beyond v sub j. 192 00:10:07,980 --> 00:10:11,590 Because all those other things are irrelevant. 193 00:10:11,590 --> 00:10:14,320 So I only look at v1 to v sub j. 194 00:10:14,320 --> 00:10:17,490 So the little v is going to be a sample value 195 00:10:17,490 --> 00:10:20,520 of this random vector. 196 00:10:20,520 --> 00:10:23,680 And the components of the random vector, the output, are 197 00:10:23,680 --> 00:10:26,250 going to be the input variables 198 00:10:26,250 --> 00:10:29,030 plus the noise variables. 199 00:10:29,030 --> 00:10:31,130 And the zj here are independent. 200 00:10:31,130 --> 00:10:32,370 I don't even have to assume that they're 201 00:10:32,370 --> 00:10:34,650 Gaussian for this argument. 202 00:10:34,650 --> 00:10:37,160 They can be anything at all, so long as they're 203 00:10:37,160 --> 00:10:39,010 independent. 204 00:10:39,010 --> 00:10:44,620 OK, now if I'm doing the signal by signal detection, in 205 00:10:44,620 --> 00:10:48,450 other words if what I'm doing is I'm saying, "OK, I want to 206 00:10:48,450 --> 00:10:56,140 take this little j signal, and I want to decide on what it is 207 00:10:56,140 --> 00:11:01,570 just from looking at the v sample of the output." OK, I 208 00:11:01,570 --> 00:11:02,760 can do that. 209 00:11:02,760 --> 00:11:05,000 My observation is v sub j. 210 00:11:05,000 --> 00:11:07,120 That's something we've talked about. 211 00:11:07,120 --> 00:11:09,570 I mean the fact that you might have other information 212 00:11:09,570 --> 00:11:11,760 available doesn't make any difference. 213 00:11:11,760 --> 00:11:13,720 You still can make a detection on the 214 00:11:13,720 --> 00:11:16,120 basis of this one variable. 215 00:11:16,120 --> 00:11:19,770 OK, so we do that. 216 00:11:19,770 --> 00:11:22,730 Now, we want to compare that with what happens when we make 217 00:11:22,730 --> 00:11:28,830 a detection for all capital J of these inputs, conditional 218 00:11:28,830 --> 00:11:30,440 on the whole sequence. 219 00:11:30,440 --> 00:11:33,150 OK, you write out the likelihood ratio. 220 00:11:33,150 --> 00:11:34,940 It factors. 221 00:11:34,940 --> 00:11:36,960 And the thing that happens when you do this-- and you 222 00:11:36,960 --> 00:11:40,000 have to read the notes for the details on this, 223 00:11:40,000 --> 00:11:41,920 and it's not hard-- 224 00:11:41,920 --> 00:11:46,350 what you find is that the maximum likelihood decision is 225 00:11:46,350 --> 00:11:49,260 exactly the same in both cases. 226 00:11:49,260 --> 00:11:53,300 OK, so it doesn't make any difference whether you detect 227 00:11:53,300 --> 00:11:58,910 little u sub j from the observation v sub j. 228 00:11:58,910 --> 00:12:03,530 Or whether you detect the vector, u sub capital j, from 229 00:12:03,530 --> 00:12:10,220 the whole set of outputs, v sub 1 up to v sub capital J. 230 00:12:10,220 --> 00:12:12,490 You get the same answer in both cases. 231 00:12:12,490 --> 00:12:14,920 Now, you might say, "What happens if some of these 232 00:12:14,920 --> 00:12:18,330 likelihood ratios come out to be right on the borderline, 233 00:12:18,330 --> 00:12:21,390 equal to one?" Well it doesn't make any difference because 234 00:12:21,390 --> 00:12:24,360 that's a zero probability thing. 235 00:12:24,360 --> 00:12:27,260 If you want to worry about that, you can worry about it 236 00:12:27,260 --> 00:12:29,510 and you get the same answer, you just have to be a little 237 00:12:29,510 --> 00:12:31,410 more careful. 238 00:12:31,410 --> 00:12:34,300 Now here is something the notes don't say, and it's also 239 00:12:34,300 --> 00:12:37,330 fairly important. 240 00:12:37,330 --> 00:12:40,240 Here we're talking about the case where all these signals 241 00:12:40,240 --> 00:12:42,320 are independent of each other. 242 00:12:42,320 --> 00:12:45,560 Which is sort of what we want to look at in communication 243 00:12:45,560 --> 00:12:50,820 because for the most part what we're doing is we're taking 244 00:12:50,820 --> 00:12:54,320 data of some sort, we're source processing it to make 245 00:12:54,320 --> 00:12:58,350 the bits coming up the channel be independent of each other, 246 00:12:58,350 --> 00:13:02,300 and then we're going to be sending them. 247 00:13:02,300 --> 00:13:04,925 Well except now we want to say, well suppose that these 248 00:13:04,925 --> 00:13:07,090 inputs are not independent. 249 00:13:07,090 --> 00:13:09,430 Suppose, for example, that we pass them through an error 250 00:13:09,430 --> 00:13:14,100 correction encoding device before transmitting them. 251 00:13:14,100 --> 00:13:17,020 And the question is, what happens then? 252 00:13:17,020 --> 00:13:20,900 Well the trouble is then you have dependence between 253 00:13:20,900 --> 00:13:23,720 u1 and u sub j. 254 00:13:23,720 --> 00:13:28,780 So you can still detect each of these just by looking at 255 00:13:28,780 --> 00:13:30,030 the single output. 256 00:13:32,330 --> 00:13:35,430 And it's a maximum likelihood detection based on that 257 00:13:35,430 --> 00:13:37,360 observation. 258 00:13:37,360 --> 00:13:40,784 But if you look at the entire observation, v1 up to v sub 259 00:13:40,784 --> 00:13:44,740 capital j, you get something better. 260 00:13:44,740 --> 00:13:46,350 How do I know you got something better? 261 00:13:49,420 --> 00:13:52,700 Well I know you get something better because in this case 262 00:13:52,700 --> 00:13:57,260 where u1 up to u sub j are dependent on each other, these 263 00:13:57,260 --> 00:14:01,440 output variables, v sub 1 up to v sub capital j, depends on 264 00:14:01,440 --> 00:14:02,130 each other. 265 00:14:02,130 --> 00:14:03,980 They aren't irrelevant. 266 00:14:03,980 --> 00:14:07,420 If you want to do the best job of maximum likelihood 267 00:14:07,420 --> 00:14:10,672 detection, given the observation of v sub 1 up to v 268 00:14:10,672 --> 00:14:14,160 sub capital j, then you're going to use all those 269 00:14:14,160 --> 00:14:16,660 variables in your detection and you're going to 270 00:14:16,660 --> 00:14:18,350 get a better -- 271 00:14:18,350 --> 00:14:20,190 And you're going to get a better decision. 272 00:14:20,190 --> 00:14:22,820 In other words, a smaller error probability, then you 273 00:14:22,820 --> 00:14:25,520 would have gotten otherwise. 274 00:14:25,520 --> 00:14:26,290 OK? 275 00:14:26,290 --> 00:14:29,380 But the important thing that comes out of here is that this 276 00:14:29,380 --> 00:14:31,380 simple minded decision-- 277 00:14:31,380 --> 00:14:36,430 where you make a decision on u sub 1 based just on v sub 1-- 278 00:14:36,430 --> 00:14:37,960 it's something you can do. 279 00:14:37,960 --> 00:14:41,780 You can do the maximum likelihood decision. 280 00:14:41,780 --> 00:14:43,020 And you know how it behaves. 281 00:14:43,020 --> 00:14:46,310 You can calculate what the error probability is. 282 00:14:46,310 --> 00:14:49,910 But you know now automatically that your error probability is 283 00:14:49,910 --> 00:14:53,040 going to be greater than or equal to the error probability 284 00:14:53,040 --> 00:14:56,300 that you would get if you base that decision one the whole 285 00:14:56,300 --> 00:14:57,950 set of inputs. 286 00:14:57,950 --> 00:15:00,680 OK, this is an argument that, it seems hard 287 00:15:00,680 --> 00:15:01,930 for most people to-- 288 00:15:04,890 --> 00:15:08,580 that it seems hard for most people to think of right at 289 00:15:08,580 --> 00:15:10,220 the beginning. 290 00:15:10,220 --> 00:15:13,350 Which is really to say, if you're doing an optimal 291 00:15:13,350 --> 00:15:15,820 detection, it is optimal. 292 00:15:15,820 --> 00:15:19,720 In other words, anything else that you do is worse. 293 00:15:19,720 --> 00:15:25,490 And you can always count on that to get bound between 294 00:15:25,490 --> 00:15:28,350 probabilities of error that you get doing something 295 00:15:28,350 --> 00:15:31,480 stupid, and probabilities of error that you get doing 296 00:15:31,480 --> 00:15:32,860 something intelligent. 297 00:15:32,860 --> 00:15:35,120 And the stupid thing is not always stupid-- 298 00:15:35,120 --> 00:15:37,540 I mean the stupid thing is sometimes better 299 00:15:37,540 --> 00:15:38,860 because it's cheaper-- 300 00:15:38,860 --> 00:15:42,120 but the error probability is always worse there then if you 301 00:15:42,120 --> 00:15:44,700 did the actual optimum thing. 302 00:15:44,700 --> 00:15:50,820 OK, so people in fact often do do detection where they have 303 00:15:50,820 --> 00:15:52,080 coded systems. 304 00:15:52,080 --> 00:15:55,950 They decode each received symbol separately based on the 305 00:15:55,950 --> 00:15:58,270 corresponding observation. 306 00:15:58,270 --> 00:15:59,810 They wind up with a larger error 307 00:15:59,810 --> 00:16:01,680 probability than they should. 308 00:16:01,680 --> 00:16:06,740 Then they pass this through some kind of, some kind of 309 00:16:06,740 --> 00:16:08,990 error correction device. 310 00:16:08,990 --> 00:16:14,040 And they wind up with a system that sort of performs 311 00:16:14,040 --> 00:16:15,090 reasonably. 312 00:16:15,090 --> 00:16:19,340 And you ask, "Well, would it have performed better if in 313 00:16:19,340 --> 00:16:23,470 fact what you did was to wait to make a final decision until 314 00:16:23,470 --> 00:16:25,850 you got all the data?" 315 00:16:25,850 --> 00:16:29,130 And we talk a little bit about various kinds of error control 316 00:16:29,130 --> 00:16:32,090 later when we get into wireless. 317 00:16:32,090 --> 00:16:36,680 We'll see that some kinds of coding systems can behave very 318 00:16:36,680 --> 00:16:40,260 easily and can make use of all this extra information. 319 00:16:40,260 --> 00:16:41,740 And other ones can't. 320 00:16:41,740 --> 00:16:44,610 Algebraic kinds of schemes can't seem to make use of the 321 00:16:44,610 --> 00:16:47,930 extra information. 322 00:16:47,930 --> 00:16:52,080 And various other kinds of schemes can make use of it. 323 00:16:52,080 --> 00:16:54,520 And if you want to understand what's happened in the error 324 00:16:54,520 --> 00:16:59,430 correction field over the last five years-- unfortunately 325 00:16:59,430 --> 00:17:01,980 6.451 won't be given, so you won't get all 326 00:17:01,980 --> 00:17:04,290 the details of this-- 327 00:17:04,290 --> 00:17:08,720 but the simplest one sentence statement you can say is that 328 00:17:08,720 --> 00:17:12,120 the world has changed from algebraic coding techniques to 329 00:17:12,120 --> 00:17:14,940 probablistic decoding techniques. 330 00:17:14,940 --> 00:17:19,510 And the primary reason for it is you want to make use of all 331 00:17:19,510 --> 00:17:22,260 that extra information you get from looking at the full 332 00:17:22,260 --> 00:17:27,770 observation, rather than just a partial observations. 333 00:17:27,770 --> 00:17:29,020 OK. 334 00:17:31,380 --> 00:17:35,630 Now, back to various signal sets. 335 00:17:35,630 --> 00:17:38,300 I put this slide up last time. 336 00:17:38,300 --> 00:17:43,790 I want to really talk about it this time because for each 337 00:17:43,790 --> 00:17:48,530 number of degrees of freedom, you can define what's called 338 00:17:48,530 --> 00:17:50,440 an orthogonal code. 339 00:17:50,440 --> 00:17:54,740 And the orthogonal codes for m equals 2 and for m equals 3 340 00:17:54,740 --> 00:17:56,120 are drawn here. 341 00:17:56,120 --> 00:17:59,620 For m equals 2, it's something that we've seen before. 342 00:17:59,620 --> 00:18:04,440 Namely, if you want to send a zero, you send a one in the 343 00:18:04,440 --> 00:18:07,130 first compound, the first degree of freedom, a zero in 344 00:18:07,130 --> 00:18:08,090 the second. 345 00:18:08,090 --> 00:18:13,310 And if you want to send a one, you send the opposite thing. 346 00:18:13,310 --> 00:18:16,490 We've all seen that this isn't a very sensible thing to do 347 00:18:16,490 --> 00:18:19,530 when we looked at binary detection, because when you 348 00:18:19,530 --> 00:18:23,760 use a scheme like this of course the thing that happens 349 00:18:23,760 --> 00:18:27,960 is we now know that we should look at this in terms of just 350 00:18:27,960 --> 00:18:30,630 looking at this line along here. 351 00:18:30,630 --> 00:18:35,300 Because what we're really transmitting is a pilot tone, 352 00:18:35,300 --> 00:18:38,240 so to speak, which is half way in the middle here, which 353 00:18:38,240 --> 00:18:39,490 sticks right here. 354 00:18:41,830 --> 00:18:45,760 Plus something that varies from that. 355 00:18:45,760 --> 00:18:51,020 So that when we take out this pilot tone, what we wind up 356 00:18:51,020 --> 00:18:53,540 with is a one dimensional system instead of a two 357 00:18:53,540 --> 00:18:55,660 dimensional system. 358 00:18:55,660 --> 00:19:00,280 Which we used to call antipodal communication-- and 359 00:19:00,280 --> 00:19:03,380 which everybody with any sense calls antipodal 360 00:19:03,380 --> 00:19:05,520 communication, even now-- 361 00:19:05,520 --> 00:19:08,300 but in terms of this, it's the simplest case 362 00:19:08,300 --> 00:19:09,980 of a simplex code. 363 00:19:09,980 --> 00:19:13,600 And a simplex code is simply an orthogonal code where 364 00:19:13,600 --> 00:19:16,070 you've taken the mean and moved it out. 365 00:19:16,070 --> 00:19:19,170 And as soon as you remove the mean from an orthogonal code, 366 00:19:19,170 --> 00:19:22,510 you get rid of one degree freedom, because one of the 367 00:19:22,510 --> 00:19:24,950 signals becomes dependent on the others. 368 00:19:24,950 --> 00:19:26,650 Which is exactly what's happened here. 369 00:19:26,650 --> 00:19:27,630 You've just-- 370 00:19:27,630 --> 00:19:30,640 if all your signals are in one degree of freedom here, when 371 00:19:30,640 --> 00:19:34,220 you do the same thing down here, well you get this. 372 00:19:34,220 --> 00:19:36,130 And I'll talk about that later. 373 00:19:36,130 --> 00:19:39,790 So, one thing you can do from an orthogonal code is go to a 374 00:19:39,790 --> 00:19:41,170 simplex code. 375 00:19:41,170 --> 00:19:44,560 The other thing you can do if you want to transmit one more 376 00:19:44,560 --> 00:19:49,300 bit out of this signal set is to go to a bi orthogonal code. 377 00:19:49,300 --> 00:19:52,640 Which says-- along with transmitting zero, one and 378 00:19:52,640 --> 00:19:56,910 one, zero-- you look this and you say, "Gee why don't I also 379 00:19:56,910 --> 00:20:02,940 put in zero minus one, minus one zero, and zero 380 00:20:02,940 --> 00:20:04,690 minus one down here. 381 00:20:04,690 --> 00:20:07,790 Which is exactly what the bi orthogonal code is. 382 00:20:07,790 --> 00:20:11,820 The bi orthogonal code simply says take your orthogonal code 383 00:20:11,820 --> 00:20:15,190 and every time you have a one, change it into a minus one and 384 00:20:15,190 --> 00:20:17,590 get an extra code word out of it. 385 00:20:17,590 --> 00:20:19,620 What's the difference between a set of code words and a 386 00:20:19,620 --> 00:20:21,860 signal set? 387 00:20:21,860 --> 00:20:23,110 Anybody have any idea? 388 00:20:26,660 --> 00:20:27,660 Absolutely none. 389 00:20:27,660 --> 00:20:31,070 They're both exactly the same thing. 390 00:20:31,070 --> 00:20:35,370 And you think of it as being a code, usually, if what you're 391 00:20:35,370 --> 00:20:38,570 doing is thinking of generating an error correcting 392 00:20:38,570 --> 00:20:41,910 code and then from that error correcting code you think of 393 00:20:41,910 --> 00:20:46,140 using QAM or PAM or something else out beyond that. 394 00:20:46,140 --> 00:20:48,540 You call it a signal set if you're just doing the whole 395 00:20:48,540 --> 00:20:50,420 thing as one unit. 396 00:20:50,420 --> 00:20:56,890 What a lot of systems now do is they start out with a code, 397 00:20:56,890 --> 00:21:01,410 then they turn this into an orthogonal signal set. 398 00:21:01,410 --> 00:21:02,120 I'll tell you in other words. 399 00:21:02,120 --> 00:21:04,000 A code produces bits. 400 00:21:04,000 --> 00:21:07,170 From the bits you group them together into sets of bits. 401 00:21:07,170 --> 00:21:11,520 From the sets of bits you go into a signal-- 402 00:21:11,520 --> 00:21:13,890 which is, for example something from one of these 403 00:21:13,890 --> 00:21:17,450 three possibilities here-- 404 00:21:17,450 --> 00:21:18,700 OK. 405 00:21:21,540 --> 00:21:24,760 The important thing to notice here-- and it's particularly 406 00:21:24,760 --> 00:21:28,800 important to think about it for a few minutes-- is because 407 00:21:28,800 --> 00:21:31,790 you do so many exercises. 408 00:21:31,790 --> 00:21:34,290 And you've done a number of them already and you will do a 409 00:21:34,290 --> 00:21:36,520 few more in this course where you deal with 410 00:21:36,520 --> 00:21:38,760 the m equals 2 case. 411 00:21:38,760 --> 00:21:41,700 And you can deal with this biorthogonal set here. 412 00:21:41,700 --> 00:21:46,260 You can shift the biorthogonal set around by 45 degrees. 413 00:21:46,260 --> 00:21:48,530 In which case it looks like this. 414 00:21:54,030 --> 00:21:56,420 OK, so that looks like two PAM sets. 415 00:21:56,420 --> 00:21:59,970 It looks like a standard QAM. 416 00:21:59,970 --> 00:22:02,970 It looks like standard 4QAM. 417 00:22:02,970 --> 00:22:06,270 This is exactly the same as this of course. 418 00:22:06,270 --> 00:22:10,680 When you do detection on this you do detection by saying, 419 00:22:10,680 --> 00:22:14,350 when you transmit this does the noise carry you across 420 00:22:14,350 --> 00:22:16,380 that boundary? 421 00:22:16,380 --> 00:22:19,610 And then does the noise carry you across this boundary? 422 00:22:19,610 --> 00:22:23,340 The noise in this direction is orthogonal from the noise in 423 00:22:23,340 --> 00:22:27,320 this direction and therefore, finding the probability of 424 00:22:27,320 --> 00:22:29,560 error is very, very simple. 425 00:22:29,560 --> 00:22:34,090 Because because you look at two separate orthogonal kinds 426 00:22:34,090 --> 00:22:37,190 of noise and you can just multiply these probabilities 427 00:22:37,190 --> 00:22:39,740 together in the appropriate way to 428 00:22:39,740 --> 00:22:41,890 find out what's happening. 429 00:22:41,890 --> 00:22:45,690 The important thing to have stick in your memory now is 430 00:22:45,690 --> 00:22:47,360 that as soon as you go to m equals 431 00:22:47,360 --> 00:22:50,590 three, life gets harder. 432 00:22:50,590 --> 00:22:54,470 In fact, if you look at this orthogonal set here and you 433 00:22:54,470 --> 00:22:57,780 try to find the error probability for it-- you try 434 00:22:57,780 --> 00:22:59,530 to find it exactly-- 435 00:22:59,530 --> 00:23:01,310 you can't do it like this. 436 00:23:01,310 --> 00:23:03,480 You can't just multiply three terms. 437 00:23:03,480 --> 00:23:06,610 You look at these regions in three dimensional space. 438 00:23:06,610 --> 00:23:09,300 If you want to visualize what they are, what do you do? 439 00:23:12,720 --> 00:23:15,760 What picture do you look at? 440 00:23:15,760 --> 00:23:17,960 You look at this picture. 441 00:23:17,960 --> 00:23:18,350 OK? 442 00:23:18,350 --> 00:23:22,350 Because this picture is just this with the mean taken away. 443 00:23:22,350 --> 00:23:25,590 So the error probability here is the same as the error 444 00:23:25,590 --> 00:23:27,050 probability here. 445 00:23:27,050 --> 00:23:33,420 When I send this point the regions that I'm looking at 446 00:23:33,420 --> 00:23:35,580 look like this. 447 00:23:40,900 --> 00:23:43,490 And they're not orthogonal to each other. 448 00:23:43,490 --> 00:23:47,470 So to find the probability that this point gets outside 449 00:23:47,470 --> 00:23:53,330 of this region looks just a little bit messy. 450 00:23:53,330 --> 00:23:57,930 And that happens for all m bigger than two. 451 00:23:57,930 --> 00:24:03,500 I never knew this because well, I think this is probably 452 00:24:03,500 --> 00:24:06,720 a disaster that happens more to teachers than to people 453 00:24:06,720 --> 00:24:08,460 working in the field. 454 00:24:08,460 --> 00:24:11,880 Because so often I have explained to people how these 455 00:24:11,880 --> 00:24:15,160 two dimensional pictures work that I just get used to 456 00:24:15,160 --> 00:24:17,630 thinking that this is an easy problem. 457 00:24:17,630 --> 00:24:20,510 And in fact when you start looking at the problem for m 458 00:24:20,510 --> 00:24:23,780 equals three, then the problem gets much more interesting and 459 00:24:23,780 --> 00:24:27,210 much more useful and much more practical when n becomes three 460 00:24:27,210 --> 00:24:30,250 or four or five or six or seven or eight. 461 00:24:30,250 --> 00:24:33,340 Beyond eight it becomes a little too hard to do. 462 00:24:33,340 --> 00:24:36,410 But up until there, it's very easy. 463 00:24:36,410 --> 00:24:38,930 OK, so we're going to make use of that in a little bit. 464 00:24:45,870 --> 00:24:49,030 As we said orthogonal codes and simplex codes-- if you 465 00:24:49,030 --> 00:24:51,970 scale the simplex code from the orthogonal code-- have 466 00:24:51,970 --> 00:24:54,560 exactly the same error probability. 467 00:24:54,560 --> 00:25:00,090 In other words, this code here where I've made the distances 468 00:25:00,090 --> 00:25:03,340 between the point square root of two over two, which 469 00:25:03,340 --> 00:25:07,270 corresponds to the distance between the points here. 470 00:25:07,270 --> 00:25:10,730 This and this have exactly the same error probability. 471 00:25:10,730 --> 00:25:14,380 So you can even, in fact, find the error probability for this 472 00:25:14,380 --> 00:25:17,050 or for this, whichever you find easier. 473 00:25:17,050 --> 00:25:18,790 You now think it's easier to find the error 474 00:25:18,790 --> 00:25:21,230 probability for this. 475 00:25:21,230 --> 00:25:24,590 I've lead you down the primrose path, because in fact 476 00:25:24,590 --> 00:25:28,180 this one is easier to find the error probability for. 477 00:25:28,180 --> 00:25:30,380 This one, again, you can find the error 478 00:25:30,380 --> 00:25:32,920 probability if you want. 479 00:25:32,920 --> 00:25:37,170 But the energy difference between this and this is 480 00:25:37,170 --> 00:25:44,020 simply the added energy that you have to use here to send 481 00:25:44,020 --> 00:25:48,570 the mean of these three signals. 482 00:25:48,570 --> 00:25:52,120 And one signal is sitting out here at 1,0,0 one is sitting 483 00:25:52,120 --> 00:25:56,850 at 0,1,0 one is sitting at 0,0,1. 484 00:25:56,850 --> 00:25:59,220 Even I can calculate the mean of those three. 485 00:25:59,220 --> 00:26:01,970 It's one third, one third, an one third. 486 00:26:01,970 --> 00:26:05,050 So you calculate the energy in one third, one third, and one 487 00:26:05,050 --> 00:26:09,490 third, and it's three times one ninth, or one third. 488 00:26:09,490 --> 00:26:11,800 Which is exactly what this says. 489 00:26:11,800 --> 00:26:16,630 The energy difference between orthogonal and simplex is 1 490 00:26:16,630 --> 00:26:17,930 minus 1 over m. 491 00:26:17,930 --> 00:26:21,730 In other words that's the factor in energy that you lose 492 00:26:21,730 --> 00:26:24,160 by using orthogonal codes instead of 493 00:26:24,160 --> 00:26:25,910 using simplex codes. 494 00:26:25,910 --> 00:26:29,090 Why do people ever use orthogonal codes? 495 00:26:29,090 --> 00:26:33,400 If it's just a pure loss in energy in doing so? 496 00:26:33,400 --> 00:26:37,220 Well, one reason is when n gets up to be 6 or 8, it 497 00:26:37,220 --> 00:26:40,140 doesn't amounts to a whole lot. 498 00:26:40,140 --> 00:26:48,290 And the other reason is if you look at if you look at 499 00:26:48,290 --> 00:26:51,510 modulating these things on to sine waves and things like 500 00:26:51,510 --> 00:26:58,140 that, you suddenly see that when you're using this it 501 00:26:58,140 --> 00:27:01,850 becomes easier to keep frequency lock and phase lock 502 00:27:01,850 --> 00:27:04,020 than it does when you use this. 503 00:27:04,020 --> 00:27:06,070 I mean you have to think about that argument a little bit to 504 00:27:06,070 --> 00:27:07,590 make sense out of it. 505 00:27:07,590 --> 00:27:11,170 But that, in fact, is why people often use orthogonal 506 00:27:11,170 --> 00:27:14,610 signals because, in fact, they can recover 507 00:27:14,610 --> 00:27:16,560 other things from it. 508 00:27:16,560 --> 00:27:19,400 Well because they're actually sending a mean, also. 509 00:27:19,400 --> 00:27:21,870 So the mean is the thing that lets them recover all these 510 00:27:21,870 --> 00:27:23,130 other neat things. 511 00:27:26,080 --> 00:27:29,670 So that sort of sometimes rules out this and it 512 00:27:29,670 --> 00:27:32,100 sometimes rules out that. 513 00:27:32,100 --> 00:27:32,600 OK. 514 00:27:32,600 --> 00:27:36,340 Orthogonal and biorthogonal codes have the same energy? 515 00:27:36,340 --> 00:27:38,560 Well, look at them. 516 00:27:38,560 --> 00:27:41,980 These two signal points each have the same energy. 517 00:27:41,980 --> 00:27:45,880 And these two have the same energy as these two. 518 00:27:45,880 --> 00:27:48,580 So the average energy is the same as the 519 00:27:48,580 --> 00:27:50,020 energy at each point. 520 00:27:50,020 --> 00:27:53,880 So this and this have the same energy. 521 00:27:53,880 --> 00:27:57,620 What happens to the probability of error? 522 00:27:57,620 --> 00:28:01,840 Well you can't evaluate it exactly. 523 00:28:01,840 --> 00:28:04,760 Except here is a case where just looking at the m equals 2 524 00:28:04,760 --> 00:28:07,030 case gives you the right answer. 525 00:28:07,030 --> 00:28:10,640 I mean here to make an error you have to go across that 526 00:28:10,640 --> 00:28:11,930 boundary there. 527 00:28:11,930 --> 00:28:15,030 Here to make an error you have to either go across that 528 00:28:15,030 --> 00:28:17,820 boundary or go across that boundary. 529 00:28:17,820 --> 00:28:19,880 The error of probability essentially goes up by a 530 00:28:19,880 --> 00:28:21,140 factor of two. 531 00:28:21,140 --> 00:28:23,000 Same thing happens here. 532 00:28:23,000 --> 00:28:24,860 You just get twice as many ways to make 533 00:28:24,860 --> 00:28:27,520 errors as you had before. 534 00:28:27,520 --> 00:28:30,910 And all of these ways to make errors are equally probable. 535 00:28:30,910 --> 00:28:33,910 All the points are equally distant from each other. 536 00:28:33,910 --> 00:28:37,480 So you essentially just double the number of likely ways you 537 00:28:37,480 --> 00:28:38,590 can make errors. 538 00:28:38,590 --> 00:28:42,490 And the error probability essentially goes up by two. 539 00:28:42,490 --> 00:28:43,740 Goes up a little-- 540 00:28:45,850 --> 00:28:48,240 well because you're using a union band it either goes up 541 00:28:48,240 --> 00:28:51,300 by a little more or a little less than two-- 542 00:28:51,300 --> 00:28:53,080 but it's almost two. 543 00:28:58,050 --> 00:29:03,210 OK, so I want to actually find the probability of error now. 544 00:29:03,210 --> 00:29:06,500 If you're sending an orthogonal code. 545 00:29:06,500 --> 00:29:10,480 Namely, we pick an orthogonal code where we pick as many 546 00:29:10,480 --> 00:29:13,010 code words as we want to. 547 00:29:13,010 --> 00:29:18,200 m might be 64, it might be 128, whatever. 548 00:29:18,200 --> 00:29:24,500 And we want to try to figure out how to evaluate the 549 00:29:24,500 --> 00:29:27,750 probability of error for this kind of code. 550 00:29:27,750 --> 00:29:33,360 After you face the fact that, in fact, these lines are not 551 00:29:33,360 --> 00:29:36,290 orthogonal to each other. 552 00:29:36,290 --> 00:29:42,190 OK, well the way you do this is you say, OK the, even 553 00:29:42,190 --> 00:29:47,530 though this is a slightly messy problem, it's clear from 554 00:29:47,530 --> 00:29:51,260 symmetry that you got the same error probability no matter 555 00:29:51,260 --> 00:29:53,100 which signal point you sent. 556 00:29:53,100 --> 00:29:56,370 Namely, every signal point is exactly the same as every 557 00:29:56,370 --> 00:29:57,820 other signal point. 558 00:29:57,820 --> 00:30:01,270 What you call the first signal depends only on which you 559 00:30:01,270 --> 00:30:04,620 happen to call the first orthonormal direction. 560 00:30:04,620 --> 00:30:08,140 Whether it's this, or this, or this. 561 00:30:08,140 --> 00:30:10,920 I can change it in anyway I want to and the problem is 562 00:30:10,920 --> 00:30:13,160 still exactly the same. 563 00:30:13,160 --> 00:30:16,210 OK, so all I'm going to do is try to find the error 564 00:30:16,210 --> 00:30:20,070 probability when I send this signal here. 565 00:30:20,070 --> 00:30:22,900 In other words, when I send 1,0,0-- 566 00:30:22,900 --> 00:30:26,980 actually I'm going to send the square root of e and 0,0-- 567 00:30:26,980 --> 00:30:29,150 because I want to talk about the energy here. 568 00:30:29,150 --> 00:30:33,100 Why am I torturing you with this? 569 00:30:33,100 --> 00:30:36,410 Well 50 years ago, 55 years ago, Shannon came out with 570 00:30:36,410 --> 00:30:40,370 this marvelous paper which says there's something called 571 00:30:40,370 --> 00:30:42,660 channel capacity. 572 00:30:42,660 --> 00:30:45,020 And what he said channel capacity was 573 00:30:45,020 --> 00:30:47,310 was the minimum rate-- 574 00:30:47,310 --> 00:30:50,560 the, was the maximum rate at which you could transmit on a 575 00:30:50,560 --> 00:30:55,190 channel and still get zero error probability. 576 00:30:55,190 --> 00:30:59,230 Sort of the simplest and most famous case of that is where 577 00:30:59,230 --> 00:31:02,120 you have white Gaussian noise to deal with. 578 00:31:02,120 --> 00:31:03,670 You're trying to transmit through this 579 00:31:03,670 --> 00:31:06,230 white Gaussian noise. 580 00:31:06,230 --> 00:31:09,380 And you can use as much bandwidth as you want. 581 00:31:09,380 --> 00:31:11,370 Namely, you can spread the signals out as 582 00:31:11,370 --> 00:31:12,710 much as you want to. 583 00:31:12,710 --> 00:31:14,920 You can sort of see from starting to look at this 584 00:31:14,920 --> 00:31:20,530 picture that you're going to be a little better off, for 585 00:31:20,530 --> 00:31:25,370 example, if you want to send one bit in these two 586 00:31:25,370 --> 00:31:28,040 dimensions here with orthogonal signals. 587 00:31:28,040 --> 00:31:30,890 If you can think of what happens down here for m equals 588 00:31:30,890 --> 00:31:35,770 4, you would wind up with four orthogonal signals. 589 00:31:35,770 --> 00:31:38,910 And if you wind up with four orthogonal signals, you're 590 00:31:38,910 --> 00:31:41,810 sending two bits, so you're going to use twice as much 591 00:31:41,810 --> 00:31:43,360 energy for each of them. 592 00:31:43,360 --> 00:31:49,570 So you can scale each of these up to be 2,0,0,0; 593 00:31:49,570 --> 00:31:55,240 0,2,0,0; and so forth. 594 00:31:55,240 --> 00:31:57,620 So you're sending twice as much energy. 595 00:31:57,620 --> 00:31:59,750 You're filling up more bandwidth because you need 596 00:31:59,750 --> 00:32:02,810 more degrees of freedom to send this signal. 597 00:32:02,810 --> 00:32:04,640 But who cares? 598 00:32:04,640 --> 00:32:06,920 Because we have all the bandwidth we want. 599 00:32:06,920 --> 00:32:09,020 Gets more complicated. 600 00:32:09,020 --> 00:32:12,090 But the question is, what happens if we go to a very 601 00:32:12,090 --> 00:32:14,580 large set of orthogonal signals? 602 00:32:14,580 --> 00:32:16,930 And what we're going to find is that when we go to a very 603 00:32:16,930 --> 00:32:20,550 large set of orthogonal signals, we can get an error 604 00:32:20,550 --> 00:32:24,820 probability which goes to zero as the number of orthogonal 605 00:32:24,820 --> 00:32:27,010 signals gets bigger. 606 00:32:27,010 --> 00:32:29,990 It goes to zero very fast as we send more 607 00:32:29,990 --> 00:32:32,220 bits with each signal. 608 00:32:32,220 --> 00:32:35,000 And the place where it goes to zero is 609 00:32:35,000 --> 00:32:37,390 exactly channel capacity. 610 00:32:37,390 --> 00:32:40,240 Now in your homework, you're going to work out a simpler 611 00:32:40,240 --> 00:32:42,740 version of all of this. 612 00:32:42,740 --> 00:32:44,280 And a simpler version is something 613 00:32:44,280 --> 00:32:46,330 called the union band. 614 00:32:46,330 --> 00:32:49,630 And in the union band you just assume that the probability of 615 00:32:49,630 --> 00:32:53,800 error when you send this is the sum of the probability of 616 00:32:53,800 --> 00:32:59,490 error, of making an error, to this and the possibility of 617 00:32:59,490 --> 00:33:01,500 making an error to this. 618 00:33:01,500 --> 00:33:03,520 The thing that we're going to add here-- 619 00:33:06,550 --> 00:33:09,280 and let me try to explain it from this picture. 620 00:33:09,280 --> 00:33:11,580 I'm sending this point here. 621 00:33:11,580 --> 00:33:12,830 OK? 622 00:33:17,950 --> 00:33:22,000 And I can find the probability of error over to that point. 623 00:33:22,000 --> 00:33:26,600 Which is the probability of going over that threshold. 624 00:33:26,600 --> 00:33:29,710 I can talk about the probability of error to over 625 00:33:29,710 --> 00:33:33,990 here, which is the probability of going over that threshold. 626 00:33:33,990 --> 00:33:36,920 These are not orthogonal to each other. 627 00:33:36,920 --> 00:33:40,150 And in fact they have a common component to them. 628 00:33:40,150 --> 00:33:44,120 And the common component is what happens in this first 629 00:33:44,120 --> 00:33:46,000 direction here. 630 00:33:46,000 --> 00:33:52,700 OK, in other words if you send 1,0,0 and the noise, and your 631 00:33:52,700 --> 00:33:54,920 own noise variable clobbers you. 632 00:33:54,920 --> 00:33:59,630 In other words, what you receive is something in this 633 00:33:59,630 --> 00:34:03,280 coordinate which is way down here and sort of arbitrary 634 00:34:03,280 --> 00:34:06,090 everywhere else-- 635 00:34:06,090 --> 00:34:10,690 conditional on the noise here being very, very large-- 636 00:34:10,690 --> 00:34:13,850 you're probably going to make an error. 637 00:34:13,850 --> 00:34:18,010 Now if you can imagine having a million orthogonal signals-- 638 00:34:18,010 --> 00:34:22,070 and the noise clobbering you on your own noise variable-- 639 00:34:22,070 --> 00:34:26,620 you're going to have a million ways to make errors. 640 00:34:26,620 --> 00:34:30,720 And they're all going to be kind of likely. 641 00:34:30,720 --> 00:34:34,440 If I go far enough down here, suppose what I receive in this 642 00:34:34,440 --> 00:34:37,040 coordinate is zero. 643 00:34:37,040 --> 00:34:40,520 Then there's a probability of one half that each one of 644 00:34:40,520 --> 00:34:45,100 these things is going to be greater than zero. 645 00:34:45,100 --> 00:34:48,990 If I use a union bound, adding up the probabilities of each 646 00:34:48,990 --> 00:34:52,600 of these, I'm going to add up a million one halves. 647 00:34:52,600 --> 00:34:56,820 Which is 500,000. 648 00:34:56,820 --> 00:35:00,310 As an upper bound to a probability. 649 00:35:00,310 --> 00:35:04,500 And that's going to clobber my bound pretty badly. 650 00:35:04,500 --> 00:35:09,020 Which says the thing I want to do here is, when I'm sending 651 00:35:09,020 --> 00:35:12,340 this I want to condition my whole argument on what the 652 00:35:12,340 --> 00:35:14,860 noise is in this direction. 653 00:35:14,860 --> 00:35:18,190 And given what the noise is in this direction, I will then 654 00:35:18,190 --> 00:35:21,440 try to evaluate the error probability, 655 00:35:21,440 --> 00:35:23,200 conditional on this. 656 00:35:23,200 --> 00:35:26,710 And conditional on a received value here. 657 00:35:26,710 --> 00:35:31,170 In fact, the noise in the direction, w2, is independent 658 00:35:31,170 --> 00:35:34,480 of the noise in direction, w3, independent of the noise in 659 00:35:34,480 --> 00:35:36,740 direction, w4, and so forth. 660 00:35:36,740 --> 00:35:42,350 So at that point, condition on w1 I'm dealing with m minus 1 661 00:35:42,350 --> 00:35:45,500 independent random variables. 662 00:35:45,500 --> 00:35:47,660 I can deal with independent random variables. 663 00:35:47,660 --> 00:35:50,740 You can deal with an independent random variables. 664 00:35:50,740 --> 00:35:52,500 Maybe some of you can integrate over 665 00:35:52,500 --> 00:35:55,000 these complex polycons. 666 00:35:55,000 --> 00:35:55,620 I can't. 667 00:35:55,620 --> 00:35:56,900 I don't want to. 668 00:35:56,900 --> 00:35:59,730 I don't want to write a program that does it. 669 00:35:59,730 --> 00:36:02,320 I don't want to be close to anybody who writes a program 670 00:36:02,320 --> 00:36:04,570 that does it. 671 00:36:04,570 --> 00:36:06,450 It offends me. 672 00:36:06,450 --> 00:36:09,550 OK, so here we go. 673 00:36:15,090 --> 00:36:17,480 Where am I? 674 00:36:17,480 --> 00:36:19,400 OK, so the first thing I'm going to do, which I didn't 675 00:36:19,400 --> 00:36:23,830 tell you, because I'm going to scale the 676 00:36:23,830 --> 00:36:25,080 problem a little bit. 677 00:36:29,600 --> 00:36:32,860 I did say that here. 678 00:36:32,860 --> 00:36:36,810 I'm going to normalize the whole problem by calling my 679 00:36:36,810 --> 00:36:40,670 output W sub j instead of Y sub j. 680 00:36:40,670 --> 00:36:44,540 And I'm going to normalize it by multiplying Y sub j by the 681 00:36:44,540 --> 00:36:47,530 square root of the noise variance. 682 00:36:47,530 --> 00:36:50,330 OK in other words, I'm going to scale the noise down so the 683 00:36:50,330 --> 00:36:52,520 noise has unit variance. 684 00:36:52,520 --> 00:36:55,950 And by scaling the noise down so it has unit variance, the 685 00:36:55,950 --> 00:36:58,770 signal will be scaled down in the same way. 686 00:36:58,770 --> 00:37:02,570 So the signal, now, instead of being the square root of e-- 687 00:37:02,570 --> 00:37:05,450 which is the energy I have available-- it's going to be 688 00:37:05,450 --> 00:37:11,160 the square root 2e divided by N0. 689 00:37:11,160 --> 00:37:14,690 Somehow this thing keeps creeping up everywhere. 690 00:37:14,690 --> 00:37:15,920 This 2e over N0. 691 00:37:15,920 --> 00:37:20,750 Well of course it's the difference between e, which is 692 00:37:20,750 --> 00:37:25,930 the energy we have to send the signal, and N0 over 2, which 693 00:37:25,930 --> 00:37:28,630 is the noise energy in each degree of freedom. 694 00:37:28,630 --> 00:37:30,810 So it's not surprising that it's sort of 695 00:37:30,810 --> 00:37:32,280 a fundamental quantity. 696 00:37:32,280 --> 00:37:35,180 And as soon as we normalize to make the noise variance equal 697 00:37:35,180 --> 00:37:38,860 to one, that's what the signal is. 698 00:37:38,860 --> 00:37:41,500 So I'm going to call alpha this so I don't have to write 699 00:37:41,500 --> 00:37:42,760 this all the time. 700 00:37:42,760 --> 00:37:45,670 Because it gets kind of messy on the slides. 701 00:37:45,670 --> 00:37:52,340 OK so given that I'm going to send input one, the received 702 00:37:52,340 --> 00:37:58,400 variable W1 is going to be normal with a mean alpha which 703 00:37:58,400 --> 00:38:03,530 is the square root of 2e over N0 and with a variance of one. 704 00:38:03,530 --> 00:38:05,730 All the other random variables are going to be 705 00:38:05,730 --> 00:38:07,200 normal random variables. 706 00:38:07,200 --> 00:38:09,690 Mean zero, variance one. 707 00:38:09,690 --> 00:38:11,050 OK? 708 00:38:11,050 --> 00:38:15,010 I'm going to make an error if any one of these other random 709 00:38:15,010 --> 00:38:20,810 variables happens to rise up and exceed W1. 710 00:38:20,810 --> 00:38:24,810 So the thing we have here is W1 is doing some crazy thing. 711 00:38:24,810 --> 00:38:28,860 I have this enormous sea of other code words in other 712 00:38:28,860 --> 00:38:30,760 directions. 713 00:38:30,760 --> 00:38:34,360 And then the question we ask is can the noise-- 714 00:38:34,360 --> 00:38:37,870 which is usually very small all over the place but it 715 00:38:37,870 --> 00:38:39,740 might rise up some place. 716 00:38:39,740 --> 00:38:43,210 And if it rises up someplace, we're asking what's the 717 00:38:43,210 --> 00:38:46,320 probability that it's going to rise up to be big enough that 718 00:38:46,320 --> 00:38:47,580 it's exceeds W1? 719 00:38:47,580 --> 00:38:55,920 So, I can at least write down what the error probability is 720 00:38:55,920 --> 00:38:58,810 exactly at that point. 721 00:38:58,810 --> 00:39:02,480 What I'm going to do is I'm going to write down the 722 00:39:02,480 --> 00:39:06,430 probability density for W1. 723 00:39:06,430 --> 00:39:10,300 And W1, remember, is a Gaussian random variable with 724 00:39:10,300 --> 00:39:12,720 mean alpha and with variance one. 725 00:39:12,720 --> 00:39:16,040 So we could actually write down what that density is. 726 00:39:16,040 --> 00:39:21,680 And for each W1 I'm going to make an error if the union of 727 00:39:21,680 --> 00:39:23,980 any one of these events occurs. 728 00:39:23,980 --> 00:39:26,630 Namely, if any one of the W sub j is bigger than 729 00:39:26,630 --> 00:39:30,140 W1, bingo I'm lost. 730 00:39:30,140 --> 00:39:31,060 OK? 731 00:39:31,060 --> 00:39:36,980 So I'm going to integrate that now over all values of W1. 732 00:39:36,980 --> 00:39:41,880 Now, as I said before if W1 is very small, then lots of other 733 00:39:41,880 --> 00:39:44,260 signals look more likely than W1. 734 00:39:44,260 --> 00:39:47,070 In other words I'm going to get clobbered no 735 00:39:47,070 --> 00:39:49,120 matter what I do. 736 00:39:49,120 --> 00:39:53,440 Whereas if W1 is large, it looks like the union bound 737 00:39:53,440 --> 00:39:53,950 might work here. 738 00:39:53,950 --> 00:39:57,000 And the union bound is what you're doing in the homework 739 00:39:57,000 --> 00:40:00,520 without paying any attention to W1. 740 00:40:00,520 --> 00:40:04,680 OK, so I'm going to use the union band-- and the union 741 00:40:04,680 --> 00:40:08,020 band says the probability that this union of all these 742 00:40:08,020 --> 00:40:11,390 different m minus 1 events-- 743 00:40:11,390 --> 00:40:15,020 exceeds some value, W1. 744 00:40:15,020 --> 00:40:18,690 W1 is just some arbitrary constant in here. 745 00:40:18,690 --> 00:40:20,270 The probability of this. 746 00:40:20,270 --> 00:40:22,200 I'm going to write it in two ways. 747 00:40:22,200 --> 00:40:24,400 It's upper bounded by one. 748 00:40:24,400 --> 00:40:27,310 Because all probabilities are upper bounded by one. 749 00:40:27,310 --> 00:40:30,210 This is not a probability density, this is a probability 750 00:40:30,210 --> 00:40:32,940 now so it's upper bounded by one. 751 00:40:32,940 --> 00:40:36,110 And it's also upper bounded by this union band. 752 00:40:36,110 --> 00:40:40,550 Mainly, the union of m minus 1 events. 753 00:40:40,550 --> 00:40:42,300 Each with probability. 754 00:40:42,300 --> 00:40:45,090 This is the tail of the Gaussian distribution 755 00:40:45,090 --> 00:40:46,850 evaluated at w1. 756 00:40:46,850 --> 00:40:48,710 So I have the union band here. 757 00:40:48,710 --> 00:40:50,070 I have one here. 758 00:40:50,070 --> 00:40:52,330 Going to pick some parameter gamma. 759 00:40:52,330 --> 00:40:55,840 I don't know what gamma should be yet, but whatever I pick 760 00:40:55,840 --> 00:40:58,420 gamma to be I still have a legitimate bound. 761 00:40:58,420 --> 00:41:01,980 This is less than or equal to this, and it's less than or 762 00:41:01,980 --> 00:41:03,390 equal to this. 763 00:41:03,390 --> 00:41:08,030 Just common sense dictates that I'm going to use this 764 00:41:08,030 --> 00:41:09,890 when this is less than one. 765 00:41:09,890 --> 00:41:13,650 And I'm going to use this when this is greater than one. 766 00:41:13,650 --> 00:41:15,360 That's what common sense says. 767 00:41:15,360 --> 00:41:19,050 As soon as I start to evaluate this, common sense goes out 768 00:41:19,050 --> 00:41:22,840 the window because then I start to deal with setting 769 00:41:22,840 --> 00:41:26,390 this quantity equal to one and dealing with the inverse of 770 00:41:26,390 --> 00:41:28,500 the Gaussian distribution function. 771 00:41:28,500 --> 00:41:30,350 Which is very painful. 772 00:41:30,350 --> 00:41:33,920 So I'm then going to bound what this is. 773 00:41:33,920 --> 00:41:36,260 And in terms of the bound on this, I'm going to affect 774 00:41:36,260 --> 00:41:37,530 gamma that way. 775 00:41:37,530 --> 00:41:40,130 Because all I'm interested in is an upper bound on error 776 00:41:40,130 --> 00:41:42,330 probability anyway. 777 00:41:42,330 --> 00:41:46,830 OK, so that's where we're going to go. 778 00:41:46,830 --> 00:41:51,630 So this probability of error then is then exceeded by using 779 00:41:51,630 --> 00:41:55,870 this bound for small W1. 780 00:41:55,870 --> 00:42:01,660 I get this integral over all W1 between minus 781 00:42:01,660 --> 00:42:03,280 infinity and gamma. 782 00:42:03,280 --> 00:42:05,650 And let's just look at what this becomes. 783 00:42:05,650 --> 00:42:08,940 This is just the tail of the Gaussian distribution. 784 00:42:08,940 --> 00:42:11,990 That's the lower tail instead of the upper tail, but that 785 00:42:11,990 --> 00:42:13,610 doesn't make any difference. 786 00:42:13,610 --> 00:42:18,460 So this is exactly Q of alpha minus gamma. 787 00:42:18,460 --> 00:42:24,080 Mainly, alpha is where the mean of this density is and 788 00:42:24,080 --> 00:42:26,760 gamma is where I integrate to. 789 00:42:26,760 --> 00:42:30,090 So if I shift the thing down, I get 790 00:42:30,090 --> 00:42:32,360 something that goes from-- 791 00:42:32,360 --> 00:42:35,190 well you might be surprised as to why this is minus gamma 792 00:42:35,190 --> 00:42:38,060 instead of plus gamma, and it's minus gamma because I'm 793 00:42:38,060 --> 00:42:42,060 looking at the lower tail rather than the upper tail and 794 00:42:42,060 --> 00:42:45,110 asking you to think this through in real time is 795 00:42:45,110 --> 00:42:47,970 unreasonable, but believe me if you sit down and think 796 00:42:47,970 --> 00:42:51,090 about it for a couple of seconds you'll realize that 797 00:42:51,090 --> 00:42:55,440 this integral is exactly this lowered tail of this Gaussian 798 00:42:55,440 --> 00:42:56,740 distribution. 799 00:42:56,740 --> 00:43:00,290 The other term is a little more complicated. 800 00:43:00,290 --> 00:43:05,950 It's m minus 1, which is that term there, times Q of w1, 801 00:43:05,950 --> 00:43:09,700 which is this term here, times the Gaussian density. 802 00:43:09,700 --> 00:43:14,440 Now this is the Gaussian density for W1, which is one 803 00:43:14,440 --> 00:43:15,860 over square root of 2pi. 804 00:43:15,860 --> 00:43:19,600 This is a normalized invariance, but it has a mean 805 00:43:19,600 --> 00:43:22,920 of alpha, so it's this. 806 00:43:22,920 --> 00:43:26,390 OK, well next thing we have to do is either fiddle around 807 00:43:26,390 --> 00:43:30,790 like mad or look at this. 808 00:43:30,790 --> 00:43:33,640 If you remember one of the things that you did-- 809 00:43:33,640 --> 00:43:37,400 I think in the previous homework that you passed in-- 810 00:43:37,400 --> 00:43:44,520 you found the bound on Q. Which looks like this. 811 00:43:51,650 --> 00:43:52,900 OK. 812 00:43:54,690 --> 00:43:57,120 That's just the tail of the Gaussian distribution. 813 00:43:57,120 --> 00:44:00,280 And the tail of the Gaussian distribution is upper banded 814 00:44:00,280 --> 00:44:05,330 by this for W1 greater than or equal to zero. 815 00:44:05,330 --> 00:44:08,480 It's upper bounded by a bunch of other things which you find 816 00:44:08,480 --> 00:44:10,630 in this problem. 817 00:44:10,630 --> 00:44:12,590 The other bounds are tighter. 818 00:44:12,590 --> 00:44:15,580 This is the most useful bound to the Gaussian distribution 819 00:44:15,580 --> 00:44:16,890 that there is. 820 00:44:16,890 --> 00:44:22,310 Because it works for W greater than or equal to zero. 821 00:44:22,310 --> 00:44:24,640 And it's exact when W1 is equal to zero. 822 00:44:24,640 --> 00:44:27,090 Because you're just integrating over half of the 823 00:44:27,090 --> 00:44:29,440 Gaussian density. 824 00:44:29,440 --> 00:44:33,500 And it's convenient and easy to work with. 825 00:44:33,500 --> 00:44:39,900 But what that says is the Q of W1, when W1 is anything 826 00:44:39,900 --> 00:44:43,200 greater than or equal to zero, looks very much like a 827 00:44:43,200 --> 00:44:45,740 Gaussian density. 828 00:44:45,740 --> 00:44:48,240 So the thing that you're doing here is taking one Gaussian 829 00:44:48,240 --> 00:44:50,670 density and you're multiplying it by 830 00:44:50,670 --> 00:44:53,000 another Gaussian density. 831 00:44:53,000 --> 00:44:58,590 And the one Gaussian density is sitting here looking like 832 00:44:58,590 --> 00:45:03,710 this with some scale factor on it at zero. 833 00:45:03,710 --> 00:45:08,290 The other Gaussian density is out here at alpha-- 834 00:45:08,290 --> 00:45:09,250 was alpha, wasn't it? 835 00:45:09,250 --> 00:45:10,540 Or was it gamma? 836 00:45:10,540 --> 00:45:14,450 Can't keep my gammas and alphas straight-- 837 00:45:14,450 --> 00:45:16,530 OK, and it looks like this. 838 00:45:20,260 --> 00:45:24,710 If you take the product of two Gaussian densities of the same 839 00:45:24,710 --> 00:45:25,890 amplitude and everything. 840 00:45:25,890 --> 00:45:27,250 And the same variance. 841 00:45:27,250 --> 00:45:29,790 What do you wind up with? 842 00:45:29,790 --> 00:45:32,320 Well you can go through and complete the square. 843 00:45:32,320 --> 00:45:33,910 And you can sort of see from looking at this from the 844 00:45:33,910 --> 00:45:36,190 symmetry of it that what you're going 845 00:45:36,190 --> 00:45:37,590 to get is the Gaussian. 846 00:45:37,590 --> 00:45:41,230 Which is right there. 847 00:45:41,230 --> 00:45:42,110 Alpha over 2. 848 00:45:42,110 --> 00:45:45,410 OK so these two things, when you multiply 849 00:45:45,410 --> 00:45:47,350 them, look like this. 850 00:45:51,440 --> 00:45:54,470 This has the same variance as these two things do. 851 00:45:54,470 --> 00:45:56,240 But it's centered on alpha over 2. 852 00:45:59,120 --> 00:46:01,740 If you want to take the Fourier transform and multiply 853 00:46:01,740 --> 00:46:03,210 the Fourier transform. 854 00:46:03,210 --> 00:46:05,330 I mean, you take the Fourier transform of the Gaussian and 855 00:46:05,330 --> 00:46:08,270 you got a Gaussian. 856 00:46:08,270 --> 00:46:11,400 So here we're multiplying. 857 00:46:11,400 --> 00:46:14,030 Up there in Fourier transform space you're convolving. 858 00:46:14,030 --> 00:46:16,770 And when you convolve a Gaussian with a Gaussian you 859 00:46:16,770 --> 00:46:17,920 got a Gaussian. 860 00:46:17,920 --> 00:46:20,680 When you multiply Gaussian by a Gaussian you got a Gaussian. 861 00:46:20,680 --> 00:46:23,320 When you do anything to a Gaussian, you got a Gaussian. 862 00:46:23,320 --> 00:46:24,570 OK? 863 00:46:26,630 --> 00:46:28,480 So this thing-- 864 00:46:28,480 --> 00:46:31,700 and if you don't believe me just actually take these two 865 00:46:31,700 --> 00:46:35,010 exponents and complete the square and see what you get-- 866 00:46:35,010 --> 00:46:38,070 so the mean is going to be alpha over 2. 867 00:46:38,070 --> 00:46:43,340 So, here you have a term which is one tail of the Gaussian 868 00:46:43,340 --> 00:46:48,300 distribution centered at alpha minus gamma. 869 00:46:48,300 --> 00:46:53,690 And here you have another one centered at alpha over 2. 870 00:46:53,690 --> 00:46:57,590 When you see these two terms, something clicks in your mind 871 00:46:57,590 --> 00:47:00,450 and says that sometimes this term is going to be the 872 00:47:00,450 --> 00:47:01,950 significant term. 873 00:47:01,950 --> 00:47:05,080 Sometimes this term is going to be significant term. 874 00:47:05,080 --> 00:47:07,490 And it depends on whether gamma-- 875 00:47:07,490 --> 00:47:10,410 alpha minus gamma- is greater than or less 876 00:47:10,410 --> 00:47:12,200 than alpha over 2. 877 00:47:12,200 --> 00:47:14,680 So you can sort of see what's going to happen right away. 878 00:47:18,660 --> 00:47:22,460 Well I hope you can see what's going to happen right away, 879 00:47:22,460 --> 00:47:23,590 because I'm not going to torture you 880 00:47:23,590 --> 00:47:24,560 with any more of this. 881 00:47:24,560 --> 00:47:29,360 And you can look at the notes to find the details. 882 00:47:29,360 --> 00:47:32,000 But you now sort of see what's happening. 883 00:47:32,000 --> 00:47:35,070 Because you have a sum of two terms. 884 00:47:35,070 --> 00:47:36,770 We're trying to upper bound this. 885 00:47:36,770 --> 00:47:40,360 We don't much care about factors of two, or factors of 886 00:47:40,360 --> 00:47:43,240 square root of 2pi or anything. 887 00:47:43,240 --> 00:47:46,590 We're trying to look at when this goes to zero. 888 00:47:46,590 --> 00:47:48,840 When you make m bigger and bigger. 889 00:47:48,840 --> 00:47:50,970 And when it doesn't go to zero. 890 00:47:50,970 --> 00:47:53,475 So when we do this, the probability of error is going 891 00:47:53,475 --> 00:47:56,660 to be less than or equal to either of these two terms. 892 00:47:56,660 --> 00:47:58,670 And here are these two alternatives 893 00:47:58,670 --> 00:48:00,170 that we spoke about. 894 00:48:00,170 --> 00:48:03,370 Namely when alpha over 2 is less than or equal to gamma, 895 00:48:03,370 --> 00:48:04,580 we get this. 896 00:48:04,580 --> 00:48:08,260 When alpha over 2 is greater than gamma, we get this. 897 00:48:08,260 --> 00:48:12,160 And this involves choosing gamma in the right way, so 898 00:48:12,160 --> 00:48:19,170 that that union bound is about equal to one at gamma. 899 00:48:19,170 --> 00:48:21,130 OK, well that doesn't tell us anything. 900 00:48:21,130 --> 00:48:25,540 So we say, "OK, what we're really interested in here is, 901 00:48:25,540 --> 00:48:28,870 as m is getting bigger and bigger, we're spending a 902 00:48:28,870 --> 00:48:32,410 certain amount of energy per input bit. 903 00:48:32,410 --> 00:48:34,240 And that's what we're interested in as far as 904 00:48:34,240 --> 00:48:35,870 Shannon's theorem is concerned. 905 00:48:35,870 --> 00:48:38,560 How much energy do you spend to send a bit 906 00:48:38,560 --> 00:48:41,110 through this channel? 907 00:48:41,110 --> 00:48:44,400 And this gives you the answer to that question. 908 00:48:44,400 --> 00:48:47,920 You let log m equal to b. m is the size of the signal 909 00:48:47,920 --> 00:48:52,120 alphabet, so b is the number of bits you're sending. 910 00:48:52,120 --> 00:48:57,400 So Eb, namely the energy per bit, is just the total energy 911 00:48:57,400 --> 00:49:00,960 in these orthogonal waveforms divided by b. 912 00:49:00,960 --> 00:49:03,310 So that's the energy per bit. 913 00:49:03,310 --> 00:49:07,110 OK, you substitute these two things into that equation and 914 00:49:07,110 --> 00:49:10,750 what you get is these two different terms. 915 00:49:10,750 --> 00:49:14,520 You got E to the minus b times this junk. 916 00:49:14,520 --> 00:49:18,240 And E to the minus b times this junk. 917 00:49:18,240 --> 00:49:20,330 What happens when m gets big? 918 00:49:20,330 --> 00:49:25,090 When m gets big, holding Eb fixed, so the game is we're 919 00:49:25,090 --> 00:49:29,930 going to keep doubling our orthogonal set, being able to 920 00:49:29,930 --> 00:49:31,740 transmit one more bit. 921 00:49:31,740 --> 00:49:34,850 And every time we transmit one more bit, we get a little more 922 00:49:34,850 --> 00:49:38,770 energy that we can use to transmit that one extra bit. 923 00:49:38,770 --> 00:49:42,980 So we used an orthogonal set, but using a little more energy 924 00:49:42,980 --> 00:49:44,790 in that orthogonal set. 925 00:49:44,790 --> 00:49:51,290 So what this says is that the probability of error goes down 926 00:49:51,290 --> 00:49:55,120 exponentially with b, if either one 927 00:49:55,120 --> 00:49:57,670 these terms are positive. 928 00:49:57,670 --> 00:50:00,890 And this, and looking at the biggest of these terms tells 929 00:50:00,890 --> 00:50:03,470 you which one you want to look at. 930 00:50:03,470 --> 00:50:09,140 So, anytime that Eb over 4N0 is less than or equal to log n 931 00:50:09,140 --> 00:50:15,100 is less than Eb over N0, you get this term. 932 00:50:15,100 --> 00:50:20,990 Any time Eb over 4N0 is greater than the natural log 933 00:50:20,990 --> 00:50:23,640 of 2, you get this term. 934 00:50:23,640 --> 00:50:26,870 Now when you go through the union bound in your homework, 935 00:50:26,870 --> 00:50:27,730 I'll give you a clue. 936 00:50:27,730 --> 00:50:29,710 This is the answer you ought to come up with 937 00:50:29,710 --> 00:50:31,330 when you're all done. 938 00:50:31,330 --> 00:50:33,950 Because that's what the union bound tells you. 939 00:50:33,950 --> 00:50:36,810 Here, remember we did something more sophisticated 940 00:50:36,810 --> 00:50:41,830 the union bound because we said the union bound is lousy 941 00:50:41,830 --> 00:50:44,330 when you get a lot of noise on W1. 942 00:50:44,330 --> 00:50:47,250 And therefore we did something separate for that case. 943 00:50:47,250 --> 00:50:50,800 And what we're finding now is depending on how much energy 944 00:50:50,800 --> 00:50:54,790 we're using, namely depending on whether we're trying to get 945 00:50:54,790 --> 00:50:57,580 very, very close to channel capacity or not. 946 00:50:57,580 --> 00:51:00,380 If you're trying to get very close to channel capacity 947 00:51:00,380 --> 00:51:02,330 you've got to use this answer here. 948 00:51:02,330 --> 00:51:04,690 Which comes from here. 949 00:51:04,690 --> 00:51:09,250 And in this case, this says that the probability of error 950 00:51:09,250 --> 00:51:12,620 goes to zero exponentially in the number of bits you're 951 00:51:12,620 --> 00:51:16,840 putting into this orthogonal code, at this rate. 952 00:51:16,840 --> 00:51:20,700 Eb over 2N0 minus log 2. 953 00:51:20,700 --> 00:51:23,390 Which is positive if Eb over 4N0, well-- 954 00:51:30,940 --> 00:51:33,880 I'm sorry. 955 00:51:33,880 --> 00:51:38,060 If you can, if you can trace back about three minutes, just 956 00:51:38,060 --> 00:51:42,730 reverse everything I said about this and about this. 957 00:51:42,730 --> 00:51:45,590 Somehow I wrote these inequalities in the wrong way 958 00:51:45,590 --> 00:51:47,410 and it sort of confused me. 959 00:51:50,990 --> 00:51:53,240 This is the answer you're going to get 960 00:51:53,240 --> 00:51:55,050 from the union bound. 961 00:51:55,050 --> 00:52:01,720 This is the answer that we get now because we use this more 962 00:52:01,720 --> 00:52:04,570 sophisticated way of looking at it. 963 00:52:26,470 --> 00:52:28,090 Sob. 964 00:52:28,090 --> 00:52:29,830 No. 965 00:52:29,830 --> 00:52:32,390 Erase what I just said in the last thirty seconds and go 966 00:52:32,390 --> 00:52:36,490 back to what I said before that. 967 00:52:36,490 --> 00:52:39,050 This is the answer you're going to get in the homework. 968 00:52:39,050 --> 00:52:42,020 This is the answer that we want to look at. 969 00:52:42,020 --> 00:52:44,500 This thing goes-- 970 00:52:44,500 --> 00:52:47,780 is valid-- anytime that Eb over N0 is 971 00:52:47,780 --> 00:52:50,200 greater than or equal-- 972 00:52:50,200 --> 00:52:52,650 is greater than natural log of 2. 973 00:52:52,650 --> 00:52:56,610 When Eb over N0 is equal to log 2, that's the capacity of 974 00:52:56,610 --> 00:52:57,470 this channel. 975 00:52:57,470 --> 00:53:08,190 Namely Eb equals N0 log 2. 976 00:53:08,190 --> 00:53:15,080 Well better to say it Eb over N0 equals 977 00:53:15,080 --> 00:53:18,300 natural log of 2 is capacity. 978 00:53:21,790 --> 00:53:25,810 And anytime Eb over N0 is greater than log 2 this term 979 00:53:25,810 --> 00:53:27,580 in here is positive. 980 00:53:27,580 --> 00:53:31,250 The error probability goes down exponentially as b gets 981 00:53:31,250 --> 00:53:34,370 large and eventually goes to zero. 982 00:53:34,370 --> 00:53:37,410 It doesn't get down as fast here as it does here. 983 00:53:37,410 --> 00:53:40,140 But this is where you really want to be because this is 984 00:53:40,140 --> 00:53:45,610 where you're transmitting with absolutely the smallest amount 985 00:53:45,610 --> 00:53:48,430 of energy possible. 986 00:53:48,430 --> 00:53:51,150 OK, so that's Shannon's formula. 987 00:53:51,150 --> 00:53:58,370 And at least we have caught up to 50 years behind what's 988 00:53:58,370 --> 00:54:01,720 going on in communication. 989 00:54:01,720 --> 00:54:05,840 And actually shown you something that's right there. 990 00:54:05,840 --> 00:54:09,950 And in fact, what we went through today is really the 991 00:54:09,950 --> 00:54:12,100 essence of that, of the proof of the 992 00:54:12,100 --> 00:54:15,000 channel capacity theorem. 993 00:54:15,000 --> 00:54:17,070 If you want to do it for finite bandwidth 994 00:54:17,070 --> 00:54:19,060 it gets much harder. 995 00:54:19,060 --> 00:54:21,980 But for this case, we really did it. 996 00:54:21,980 --> 00:54:22,660 It's all there. 997 00:54:22,660 --> 00:54:26,340 I mean you read the notes and in a little extra detail. 998 00:54:26,340 --> 00:54:29,310 But just with the grunge work left out, that's 999 00:54:29,310 --> 00:54:32,290 what's going on. 1000 00:54:32,290 --> 00:54:34,650 OK, wireless communication. 1001 00:54:34,650 --> 00:54:38,110 That's what we want to spend the rest of the term on. 1002 00:54:38,110 --> 00:54:40,600 We want to spend the rest of the term on that because 1003 00:54:40,600 --> 00:54:43,430 whether you realize it or not, we sort of said everything 1004 00:54:43,430 --> 00:54:48,170 there is to say about white Gaussian noise. 1005 00:54:48,170 --> 00:54:54,680 And when all of that sinks in, what you're left with is the 1006 00:54:54,680 --> 00:54:57,410 idea the white Gaussein noise. 1007 00:54:57,410 --> 00:55:01,190 You're really dealing with just a finite vector problem. 1008 00:55:01,190 --> 00:55:04,070 And you don't have to worry about anything else. 1009 00:55:04,070 --> 00:55:07,670 The QAM and the PAM, all that stuff, all disappear. 1010 00:55:07,670 --> 00:55:10,880 Doesn't matter whether you send things broadband, narrow 1011 00:55:10,880 --> 00:55:12,060 band, whatever. 1012 00:55:12,060 --> 00:55:14,220 It's all the same answer. 1013 00:55:14,220 --> 00:55:16,460 Wireless is different. 1014 00:55:16,460 --> 00:55:22,130 Wireless is different for a couple of reasons. 1015 00:55:22,130 --> 00:55:25,550 You're dealing with the radiation between antennas 1016 00:55:25,550 --> 00:55:28,500 because you're dealing with the radiation between antennas 1017 00:55:28,500 --> 00:55:31,180 rather than what's going on on a wire. 1018 00:55:31,180 --> 00:55:33,560 I mean, what's going on a wire, the wire is pretty much 1019 00:55:33,560 --> 00:55:35,530 shielded from the outside world. 1020 00:55:35,530 --> 00:55:39,240 So you send something, noise gets added, and you receive 1021 00:55:39,240 --> 00:55:41,010 signal plus noise. 1022 00:55:41,010 --> 00:55:43,270 There's not much fading, there's not much 1023 00:55:43,270 --> 00:55:44,760 awkward stuff going on. 1024 00:55:44,760 --> 00:55:46,580 Here all of the stuff goes on. 1025 00:55:50,340 --> 00:55:53,870 As soon as you start using wireless communication, you're 1026 00:55:53,870 --> 00:55:56,210 allowed to drive around in your car talking on 1027 00:55:56,210 --> 00:55:57,650 two phones at once. 1028 00:55:57,650 --> 00:56:00,530 With your ears and eyes shielded. 1029 00:56:00,530 --> 00:56:04,460 You can kill yourself much easier that way than you can 1030 00:56:04,460 --> 00:56:06,210 with ordinary telephony. 1031 00:56:06,210 --> 00:56:07,250 You can be in constant 1032 00:56:07,250 --> 00:56:14,210 communications with almost anyone. 1033 00:56:14,210 --> 00:56:18,670 OK, so you have motion, you have temporary locations. 1034 00:56:18,670 --> 00:56:20,410 You have all these neat things. 1035 00:56:20,410 --> 00:56:22,760 And if you look at what's happening in the world, the 1036 00:56:22,760 --> 00:56:26,670 less developed parts of the world have much more mobile 1037 00:56:26,670 --> 00:56:28,720 communication than we do. 1038 00:56:28,720 --> 00:56:31,130 Because in fact they don't have that much wire 1039 00:56:31,130 --> 00:56:31,960 communication. 1040 00:56:31,960 --> 00:56:33,760 It's not that good there. 1041 00:56:33,760 --> 00:56:37,080 So they find it's far cheaper to get a mobile phone. 1042 00:56:37,080 --> 00:56:40,830 Than like us where we have to pay for both a wire line phone 1043 00:56:40,830 --> 00:56:42,670 and a mobile phone. 1044 00:56:42,670 --> 00:56:48,050 So they have sort of the best of the two worlds there. 1045 00:56:48,050 --> 00:56:51,390 Except their mobile phones are like our mobile phones. 1046 00:56:51,390 --> 00:56:54,460 They only work three quarters of the time. 1047 00:56:54,460 --> 00:56:59,490 And all of the research that's going into sending video over 1048 00:56:59,490 --> 00:57:02,540 wireless phones, it seems that nobody's spending any time 1049 00:57:02,540 --> 00:57:06,160 trying to increase the amount of time you can use your 1050 00:57:06,160 --> 00:57:10,470 wireless phone from 75% to 90%. 1051 00:57:10,470 --> 00:57:13,670 And if any of you want to make a lot of money and also do 1052 00:57:13,670 --> 00:57:16,920 something worthwhile for the world, invent a wireless phone 1053 00:57:16,920 --> 00:57:18,990 the works 90% of the time. 1054 00:57:18,990 --> 00:57:21,790 And you'll clean up, believe me. 1055 00:57:21,790 --> 00:57:26,740 And you can even send video on it later if you want to. 1056 00:57:26,740 --> 00:57:29,430 OK that's another thing that wireless has turned out to be 1057 00:57:29,430 --> 00:57:30,710 very useful for. 1058 00:57:30,710 --> 00:57:32,930 And I'm sure you all know this. 1059 00:57:32,930 --> 00:57:34,490 It avoids mazes of wires. 1060 00:57:34,490 --> 00:57:37,520 I mean many people in their homes and offices and 1061 00:57:37,520 --> 00:57:41,780 everywhere are starting to use local area wireless networks 1062 00:57:41,780 --> 00:57:45,280 just as a way of getting rid of all of these maddening 1063 00:57:45,280 --> 00:57:47,460 wires that we have running all over the place. 1064 00:57:47,460 --> 00:57:52,570 As soon as we have a computer and a printer and a fax 1065 00:57:52,570 --> 00:57:56,620 machine and a blah blah blah, and a watch which is connected 1066 00:57:56,620 --> 00:58:00,480 to our, and a toaster which is connected to our computer. 1067 00:58:00,480 --> 00:58:01,800 Argh! 1068 00:58:01,800 --> 00:58:03,940 Pretty soon we're going to be connected to our computers. 1069 00:58:03,940 --> 00:58:06,650 We're going to have little things stuck in our head and 1070 00:58:06,650 --> 00:58:10,090 stuck our neck and all over the place. 1071 00:58:10,090 --> 00:58:13,590 So it'll be nice to have these, it'll be nice to not 1072 00:58:13,590 --> 00:58:15,790 have wires when we're doing that. 1073 00:58:15,790 --> 00:58:19,740 OK, but the new problem is that the channel, in fact, 1074 00:58:19,740 --> 00:58:21,100 changes with time. 1075 00:58:21,100 --> 00:58:24,210 It's very different from one time to another. 1076 00:58:24,210 --> 00:58:26,760 And you get a lot of interference between channels. 1077 00:58:26,760 --> 00:58:29,940 In other words, when you're dealing with wireless you 1078 00:58:29,940 --> 00:58:33,020 cannot think of just one transmitter and 1079 00:58:33,020 --> 00:58:34,900 one receiver anymore. 1080 00:58:34,900 --> 00:58:37,720 That's one of the problems you want to think about. 1081 00:58:37,720 --> 00:58:40,050 But you really have to think about what all the other 1082 00:58:40,050 --> 00:58:42,040 transmitters are doing and what all the other 1083 00:58:42,040 --> 00:58:43,290 receivers are doing. 1084 00:58:49,800 --> 00:58:53,880 It was started by Marconi in 1897. 1085 00:58:53,880 --> 00:58:56,490 It took him about three years to get transcontinental 1086 00:58:56,490 --> 00:58:58,720 communication. 1087 00:58:58,720 --> 00:59:01,590 I mean we think we're so great now-- 1088 00:59:01,590 --> 00:59:05,600 being able to have research move as quickly as it does-- 1089 00:59:05,600 --> 00:59:08,050 but if you think of the amount of time it takes to create a 1090 00:59:08,050 --> 00:59:11,420 new wireless system, it's a whole lot larger 1091 00:59:11,420 --> 00:59:13,440 now than it was then. 1092 00:59:13,440 --> 00:59:14,930 I mean he moved very fast. 1093 00:59:14,930 --> 00:59:18,550 I mean the technology was very primitive and very simple. 1094 00:59:18,550 --> 00:59:21,300 It was not a billion dollar business. 1095 00:59:21,300 --> 00:59:25,520 But in fact it was very, very rapid. 1096 00:59:25,520 --> 00:59:27,380 But what's happened since, with wireless, 1097 00:59:27,380 --> 00:59:30,470 has been very fitful. 1098 00:59:30,470 --> 00:59:31,660 Businesses have started. 1099 00:59:31,660 --> 00:59:32,740 Businesses have stopped. 1100 00:59:32,740 --> 00:59:34,300 People have tried to do one thing. 1101 00:59:34,300 --> 00:59:36,450 People have tried to do another thing. 1102 00:59:36,450 --> 00:59:40,230 They name things by something different all the time. 1103 00:59:40,230 --> 00:59:44,420 I mean one sort of amusing thing is back in the early 1104 00:59:44,420 --> 00:59:50,230 seventies the army was trying very hard to get wireless 1105 00:59:50,230 --> 00:59:52,140 communication in the field. 1106 00:59:52,140 --> 00:59:55,040 And they called this packet radio. 1107 00:59:55,040 --> 00:59:58,320 And they had all the universities in the country 1108 00:59:58,320 --> 01:00:01,480 spending enormous amounts of time developing packet radio. 1109 01:00:01,480 --> 01:00:03,400 Writing many papers about it. 1110 01:00:03,400 --> 01:00:07,360 They finally got disgusted because nothing was happening. 1111 01:00:07,360 --> 01:00:10,270 So they pulled all the funding for that. 1112 01:00:10,270 --> 01:00:13,550 And about five years later, when the people at DARPA and 1113 01:00:13,550 --> 01:00:16,420 NSF and all of that forgot about this unpleasant 1114 01:00:16,420 --> 01:00:20,340 experience, people started talking about ad hoc networks. 1115 01:00:20,340 --> 01:00:23,150 Guess what an ad hoc network is? 1116 01:00:23,150 --> 01:00:25,160 Same thing as packet radio. 1117 01:00:25,160 --> 01:00:29,400 Just a new name for an old system, and suddenly the money 1118 01:00:29,400 --> 01:00:33,060 started flowing in again. 1119 01:00:33,060 --> 01:00:35,340 We don't know whether it'll be any better this time than it 1120 01:00:35,340 --> 01:00:36,280 was last time. 1121 01:00:36,280 --> 01:00:41,240 But anyway that's the way funding goes. 1122 01:00:44,100 --> 01:00:47,960 OK, what we're going to talk about in this class is sort of 1123 01:00:47,960 --> 01:00:49,060 an old fashioned thing. 1124 01:00:49,060 --> 01:00:52,860 It's not as sexy as what all these other systems are. 1125 01:00:52,860 --> 01:00:55,820 It's just cellular networks. 1126 01:00:55,820 --> 01:00:59,900 It's probably because that's well understood by now, and 1127 01:00:59,900 --> 01:01:02,930 it's because we can talk about all of these fundamental 1128 01:01:02,930 --> 01:01:07,810 problems that occur in mobile communication just in the 1129 01:01:07,810 --> 01:01:12,800 context of this one kind of system that, by now, is 1130 01:01:12,800 --> 01:01:15,740 reasonably well understood. 1131 01:01:15,740 --> 01:01:20,000 When you're doing cellular communication, you wind up 1132 01:01:20,000 --> 01:01:24,430 with a large bunch of mobiles all communicating with one 1133 01:01:24,430 --> 01:01:26,090 base station. 1134 01:01:26,090 --> 01:01:28,570 OK, in other words you don't have the kind of thing you had 1135 01:01:28,570 --> 01:01:32,320 in the packet radio network or in the ad hoc network, where 1136 01:01:32,320 --> 01:01:35,470 you have a huge number of mobile telephones which are 1137 01:01:35,470 --> 01:01:37,640 all communicating to each other. 1138 01:01:37,640 --> 01:01:40,760 And where one phone has to relay things for others. 1139 01:01:40,760 --> 01:01:46,020 You wind up with a very complicated network problem. 1140 01:01:46,020 --> 01:01:49,690 Here, it's in a sense, a much simpler 1141 01:01:49,690 --> 01:01:52,050 and more sane structure. 1142 01:01:52,050 --> 01:01:54,910 Because you're using mobile for doing the things that 1143 01:01:54,910 --> 01:01:57,160 mobile does well. 1144 01:01:57,160 --> 01:01:59,070 And you're using wires for the things that 1145 01:01:59,070 --> 01:02:00,980 wires do very well. 1146 01:02:00,980 --> 01:02:03,250 Mainly you have lots of mobiles which are moving all 1147 01:02:03,250 --> 01:02:04,380 over the place. 1148 01:02:04,380 --> 01:02:08,090 You have these fixed base stations which are big and 1149 01:02:08,090 --> 01:02:12,470 expensive, and put up on hills or on buildings or on big 1150 01:02:12,470 --> 01:02:13,970 poles or something. 1151 01:02:13,970 --> 01:02:15,970 And you spend a lot of money on them. 1152 01:02:15,970 --> 01:02:19,350 You have optical fibers or cables or what have you 1153 01:02:19,350 --> 01:02:22,420 running between them or running from them to what's 1154 01:02:22,420 --> 01:02:25,400 called a MTSO. 1155 01:02:25,400 --> 01:02:30,470 Mobile Something Subscriber Office-- and I can never 1156 01:02:30,470 --> 01:02:33,720 remember what those letters stand for-- 1157 01:02:33,720 --> 01:02:37,750 Mobile Telephone Subscriber Office. 1158 01:02:37,750 --> 01:02:40,690 All I had to do to remember that was thank this was done 1159 01:02:40,690 --> 01:02:44,020 by telephone engineers. 1160 01:02:44,020 --> 01:02:48,770 No, Mobile Telephone Switching Office and telephone engineers 1161 01:02:48,770 --> 01:02:52,770 think in terms of switching and in terms of telephones. 1162 01:02:52,770 --> 01:02:56,630 And mobile and offices just follows along. 1163 01:02:56,630 --> 01:03:00,380 So the way these systems work is you go from a mobile to a 1164 01:03:00,380 --> 01:03:01,280 base station. 1165 01:03:01,280 --> 01:03:04,660 From the base station to one of these MTSOs, which is just 1166 01:03:04,660 --> 01:03:06,160 a big switching center. 1167 01:03:06,160 --> 01:03:09,140 From there you're in the wired network. 1168 01:03:09,140 --> 01:03:13,910 And from there you can either go back to a mobile or go back 1169 01:03:13,910 --> 01:03:20,370 to a wire line telephone or go anywhere you want to. 1170 01:03:20,370 --> 01:03:23,900 But but the point in that, and I think this is important to 1171 01:03:23,900 --> 01:03:28,590 remember, is that cellular networks are an appendage of 1172 01:03:28,590 --> 01:03:32,020 the wire line network. 1173 01:03:32,020 --> 01:03:35,650 And you always have this wire line network in the middle. 1174 01:03:35,650 --> 01:03:37,990 You probably always will. 1175 01:03:37,990 --> 01:03:41,270 Because wire line networks have things like fiber which 1176 01:03:41,270 --> 01:03:45,700 carries enormous amounts of data very, very cheaply. 1177 01:03:45,700 --> 01:03:50,010 And mobile is very limited as far as capacity goes. 1178 01:03:50,010 --> 01:03:51,870 And it's very noisy. 1179 01:03:51,870 --> 01:03:56,250 OK, so that lets us avoid the question of 1180 01:03:56,250 --> 01:03:58,620 how do you do relaying. 1181 01:03:58,620 --> 01:04:01,930 When you see pictures of this, people draw pictures of 1182 01:04:01,930 --> 01:04:03,610 hexagon cells. 1183 01:04:03,610 --> 01:04:05,470 AUDIENCE: [UNINTELLIGIBLE] turn around. 1184 01:04:05,470 --> 01:04:11,760 PROFESSOR: Oh, when I hit this, and it uh, OK. 1185 01:04:19,950 --> 01:04:21,760 I mean there's not much information on this picture 1186 01:04:21,760 --> 01:04:25,810 anyway but, [LAUGHTER] 1187 01:04:25,810 --> 01:04:30,230 OK, but people think in terms of base stations put down 1188 01:04:30,230 --> 01:04:33,470 uniformly with nice hexagons around them. 1189 01:04:33,470 --> 01:04:37,400 And any time a mobile within one hexagon it communicates 1190 01:04:37,400 --> 01:04:44,180 with the base station which is at the center of that hexagon. 1191 01:04:44,180 --> 01:04:47,030 And in reality what happens is that the base stations are 1192 01:04:47,030 --> 01:04:51,100 spread all over the place in a very haphazard way. 1193 01:04:51,100 --> 01:04:54,030 I shouldn't say haphazard, because people worked very 1194 01:04:54,030 --> 01:04:56,760 hard to find places to put these base stations. 1195 01:04:56,760 --> 01:05:00,150 Because you need to rent real estate, or buy real estate to 1196 01:05:00,150 --> 01:05:01,670 put them in. 1197 01:05:01,670 --> 01:05:04,200 You have to find out somehow what kind of 1198 01:05:04,200 --> 01:05:06,300 coverage they have. 1199 01:05:06,300 --> 01:05:08,930 And it's a very fascinating and very difficult problem. 1200 01:05:08,930 --> 01:05:12,610 One thing I'm going to try to convince you of in the next 1201 01:05:12,610 --> 01:05:16,310 lecture or so is that the problems of choosing base 1202 01:05:16,310 --> 01:05:20,020 stations are very heavily electromagnetic in nature. 1203 01:05:20,020 --> 01:05:23,730 You really have to understand electromagnetism very well. 1204 01:05:23,730 --> 01:05:28,970 And and you have to understand the modeling of these physical 1205 01:05:28,970 --> 01:05:33,130 communication links very, very well in order to try to sort 1206 01:05:33,130 --> 01:05:35,360 out where base stations should go and where 1207 01:05:35,360 --> 01:05:37,340 they shouldn't go. 1208 01:05:37,340 --> 01:05:40,910 The other part of the problem is the part of the problem 1209 01:05:40,910 --> 01:05:44,850 dealing with how do you design the mobile phone itself? 1210 01:05:44,850 --> 01:05:50,840 How do you design the base station itself? 1211 01:05:50,840 --> 01:05:54,450 And these are questions which don't depend so much on the 1212 01:05:54,450 --> 01:05:57,780 exact modeling of the electromagnetic channel. 1213 01:05:57,780 --> 01:06:02,540 They only depend on very coarse characteristics of it. 1214 01:06:02,540 --> 01:06:06,150 And very often, when you start to study mobile, you will 1215 01:06:06,150 --> 01:06:10,450 spend an inordinate amount of time studying all of the 1216 01:06:10,450 --> 01:06:14,220 details of these electromagnetic channels. 1217 01:06:14,220 --> 01:06:17,480 Which in fact are very important as far as choosing 1218 01:06:17,480 --> 01:06:19,410 base stations are concerned. 1219 01:06:19,410 --> 01:06:22,130 And have relatively little to do with the questions of how 1220 01:06:22,130 --> 01:06:23,520 do you design mobiles. 1221 01:06:23,520 --> 01:06:27,450 How do you design base stations? 1222 01:06:27,450 --> 01:06:29,300 It has enough to do with it that you have to know 1223 01:06:29,300 --> 01:06:33,980 something about it, but it's not central anymore. 1224 01:06:33,980 --> 01:06:40,740 OK, so let's look at what the problems are. 1225 01:06:43,780 --> 01:06:47,110 As I said the cellular network is really an appendage to the 1226 01:06:47,110 --> 01:06:48,920 wire network. 1227 01:06:48,920 --> 01:06:53,870 The problems we're going to have to deal with is when 1228 01:06:53,870 --> 01:06:58,370 you're outgoing from your own cell phone, there's some kind 1229 01:06:58,370 --> 01:07:01,700 of strategy that has to be used for you to find the best 1230 01:07:01,700 --> 01:07:03,410 base station to use. 1231 01:07:03,410 --> 01:07:06,140 And it's a difficult question because you're trying to find 1232 01:07:06,140 --> 01:07:11,460 a base station you can communicate with and one 1233 01:07:11,460 --> 01:07:15,380 that's not so overcrowded that you can't talk to it. 1234 01:07:15,380 --> 01:07:17,470 So that's one big problem. 1235 01:07:17,470 --> 01:07:19,340 We won't talk about that much. 1236 01:07:19,340 --> 01:07:21,710 Another is the ingoing problem. 1237 01:07:21,710 --> 01:07:23,530 Finding a mobile. 1238 01:07:23,530 --> 01:07:27,180 If you think about that, it's really a very tricky problem. 1239 01:07:27,180 --> 01:07:30,160 Because I run around in my car with my cellphone turned off 1240 01:07:30,160 --> 01:07:31,230 all the time. 1241 01:07:31,230 --> 01:07:33,860 And I only turn it on if I want to talk to somebody. 1242 01:07:33,860 --> 01:07:37,470 So I turn it on and somehow the whole network has to 1243 01:07:37,470 --> 01:07:39,550 suddenly realize where I am. 1244 01:07:39,550 --> 01:07:43,120 And you know that happens with all of these cellular networks 1245 01:07:43,120 --> 01:07:44,510 all over the place. 1246 01:07:44,510 --> 01:07:47,310 And every time somebody turns on their cell phone there's a 1247 01:07:47,310 --> 01:07:49,080 lot of stuff going back and forth that 1248 01:07:49,080 --> 01:07:50,430 says who is this guy? 1249 01:07:50,430 --> 01:07:52,420 Does he have the right to talk? 1250 01:07:52,420 --> 01:07:55,670 Has he paid is bill? 1251 01:07:55,670 --> 01:08:00,880 And how do I actually find a base station for him to use? 1252 01:08:00,880 --> 01:08:02,900 So this is kind of, both these 1253 01:08:02,900 --> 01:08:06,740 questions are kind of difficult. 1254 01:08:06,740 --> 01:08:11,330 And the even worse question is if somebody's calling me and I 1255 01:08:11,330 --> 01:08:15,740 live say, in Boston, or close to Boston, and I'm out in San 1256 01:08:15,740 --> 01:08:20,460 Francisco and somebody calls me on my cell phone, the call 1257 01:08:20,460 --> 01:08:22,160 gets to me. 1258 01:08:22,160 --> 01:08:24,850 And if you just imagine a little bit what has to go on 1259 01:08:24,850 --> 01:08:28,920 in this cellular network in order for the cellular network 1260 01:08:28,920 --> 01:08:32,250 to realize that I'm in San Francisco instead of Boston. 1261 01:08:32,250 --> 01:08:36,640 And then realize how to get calls to me in San Francisco. 1262 01:08:36,640 --> 01:08:37,820 I mean there's a lot of interesting 1263 01:08:37,820 --> 01:08:38,870 stuff going on here. 1264 01:08:38,870 --> 01:08:41,090 But we're not going to talk about any of that because 1265 01:08:41,090 --> 01:08:47,530 that's really sort of an organizational question as 1266 01:08:47,530 --> 01:08:52,150 opposed to a physical communication question, which 1267 01:08:52,150 --> 01:08:55,440 is the kind of thing we're interested in here. 1268 01:08:55,440 --> 01:08:58,700 OK, when you have these multiple mobiles which are 1269 01:08:58,700 --> 01:09:02,590 sending to the same base station. 1270 01:09:02,590 --> 01:09:06,100 People who are working on mobile communication, sort of 1271 01:09:06,100 --> 01:09:10,290 the practical side of it, call this the reverse channel. 1272 01:09:10,290 --> 01:09:12,860 Why they call this the reverse channel and the other one the 1273 01:09:12,860 --> 01:09:14,500 forward channel, I don't know. 1274 01:09:14,500 --> 01:09:17,580 Forward channel goes from the base station to the mobile. 1275 01:09:17,580 --> 01:09:21,160 Reverse station, reverse channel goes from the mobile 1276 01:09:21,160 --> 01:09:22,560 to the base station. 1277 01:09:22,560 --> 01:09:25,690 And what it says is the terminology was chosen by the 1278 01:09:25,690 --> 01:09:28,870 people designing the base stations. 1279 01:09:28,870 --> 01:09:30,700 That's sort of clear. 1280 01:09:30,700 --> 01:09:35,040 But if you read about this in any more technical 1281 01:09:35,040 --> 01:09:38,820 publication, you will see this thing being called a multi 1282 01:09:38,820 --> 01:09:40,180 access channel. 1283 01:09:40,180 --> 01:09:43,610 It's the multi access channel because many, many users are 1284 01:09:43,610 --> 01:09:47,400 all trying to get into the same base station. 1285 01:09:47,400 --> 01:09:51,690 And this one electromagnetic wave-- which is impinging on 1286 01:09:51,690 --> 01:09:56,540 the various space station antennas-- is carrying all of 1287 01:09:56,540 --> 01:09:59,820 that stuff all multiplexed together in some way. 1288 01:09:59,820 --> 01:10:02,720 And it's not multiplexed together in a sensible way, 1289 01:10:02,720 --> 01:10:05,290 because it's multiplexed together just by all of these 1290 01:10:05,290 --> 01:10:08,750 waveforms randomly adding to each other. 1291 01:10:08,750 --> 01:10:12,030 So information theorists call these things 1292 01:10:12,030 --> 01:10:14,140 multi access channels. 1293 01:10:14,140 --> 01:10:17,840 When you're going the other way, base station to mobiles, 1294 01:10:17,840 --> 01:10:20,230 it's called the forward channel by 1295 01:10:20,230 --> 01:10:22,030 the telephone engineers. 1296 01:10:22,030 --> 01:10:24,020 It's called the broadcast channel 1297 01:10:24,020 --> 01:10:26,470 by information theorists. 1298 01:10:26,470 --> 01:10:30,380 For those of you who think about broadcast in terms of TV 1299 01:10:30,380 --> 01:10:32,890 and FM and all that sort of stuff, this is 1300 01:10:32,890 --> 01:10:34,010 a little bit confusing. 1301 01:10:34,010 --> 01:10:38,250 Because this is not the same kind of broadcast that you're 1302 01:10:38,250 --> 01:10:39,770 usually thinking about. 1303 01:10:39,770 --> 01:10:43,180 I mean the usual kind of broadcast is where everybody 1304 01:10:43,180 --> 01:10:46,090 gets the same thing whether you want it or not. 1305 01:10:46,090 --> 01:10:49,720 But that whole signal is there, and you all get the 1306 01:10:49,720 --> 01:10:50,530 whole thing. 1307 01:10:50,530 --> 01:10:54,290 Here what it is, is you're sending a 1308 01:10:54,290 --> 01:10:56,190 different message to everyone. 1309 01:10:56,190 --> 01:10:59,020 You don't want everyone to be able to tune in and receive 1310 01:10:59,020 --> 01:11:00,420 what anybody else is getting. 1311 01:11:00,420 --> 01:11:01,840 You want a little privacy here. 1312 01:11:01,840 --> 01:11:05,070 So it's really broadcasting separate messages and trying 1313 01:11:05,070 --> 01:11:06,320 to keep them separate. 1314 01:11:13,890 --> 01:11:16,250 While the systems, almost all of them, are 1315 01:11:16,250 --> 01:11:18,080 now digital I think. 1316 01:11:18,080 --> 01:11:20,680 In the sense of having a binary interface, this is the 1317 01:11:20,680 --> 01:11:23,130 same issue we've been talking about all along. 1318 01:11:23,130 --> 01:11:25,840 You say something is digital if there's the binary 1319 01:11:25,840 --> 01:11:27,790 interface on it. 1320 01:11:27,790 --> 01:11:30,440 The source is either analog or digital. 1321 01:11:30,440 --> 01:11:34,070 Cellular communication was really designed for voice. 1322 01:11:34,070 --> 01:11:36,890 Now all the research is concerned with how do you make 1323 01:11:36,890 --> 01:11:39,810 it work for data also. 1324 01:11:39,810 --> 01:11:43,020 One of the things we're going to talk about a little bit is 1325 01:11:43,020 --> 01:11:47,120 why the problems are so very different. 1326 01:11:47,120 --> 01:11:50,030 I mean you would think they're both the same problem. 1327 01:11:50,030 --> 01:11:51,610 Because in both cases you're just 1328 01:11:51,610 --> 01:11:52,950 transmitting a string of bits. 1329 01:11:52,950 --> 01:11:54,600 That's all that's going on. 1330 01:11:54,600 --> 01:11:58,700 But the big difference is that in voice, you can't tolerate 1331 01:11:58,700 --> 01:11:59,090 delay in voice. 1332 01:11:59,090 --> 01:12:01,270 In data you can tolerate delay. 1333 01:12:01,270 --> 01:12:05,220 You can tolerate a lot of delay in data. 1334 01:12:05,220 --> 01:12:07,840 And therefore you can do lots of things with data that you 1335 01:12:07,840 --> 01:12:09,300 can't do with voice. 1336 01:12:09,300 --> 01:12:12,020 If you want to have a system that deals with both voice and 1337 01:12:12,020 --> 01:12:15,330 data, it's got to be able to get the voice 1338 01:12:15,330 --> 01:12:17,280 through without delay. 1339 01:12:17,280 --> 01:12:20,600 And you have to find some way of solving that problem. 1340 01:12:20,600 --> 01:12:28,600 OK, let me just say this quickly. 1341 01:12:28,600 --> 01:12:30,850 Let me just see if there's anything else of content here. 1342 01:12:36,740 --> 01:12:39,520 This is all just boiler plate stuff. 1343 01:12:39,520 --> 01:12:41,270 Let me skip that. 1344 01:12:41,270 --> 01:12:42,520 And skip this. 1345 01:12:51,060 --> 01:12:55,770 The thing we're going to be concerned about here is really 1346 01:12:55,770 --> 01:12:57,600 these physical modeling issues. 1347 01:13:01,220 --> 01:13:04,770 And where we wind up with that is we're typically talking 1348 01:13:04,770 --> 01:13:11,050 about bandwidths that are maybe a few megahertz wide. 1349 01:13:11,050 --> 01:13:12,070 Maybe a few kilohertz wide. 1350 01:13:12,070 --> 01:13:15,330 Or maybe a few megahertz. 1351 01:13:15,330 --> 01:13:18,980 But we're talking about carrier frequencies which are 1352 01:13:18,980 --> 01:13:23,310 usually up in the gigahertz range. 1353 01:13:23,310 --> 01:13:26,120 And they keep varying depending on which new range 1354 01:13:26,120 --> 01:13:28,320 of frequencies gets opened up. 1355 01:13:28,320 --> 01:13:30,510 They started out a little below a gigahertz. 1356 01:13:30,510 --> 01:13:33,470 They only went up to 2.4 and now they're up 1357 01:13:33,470 --> 01:13:35,100 around five or six. 1358 01:13:35,100 --> 01:13:37,370 And things like that. 1359 01:13:37,370 --> 01:13:39,530 When we talk about physical modeling, we want to 1360 01:13:39,530 --> 01:13:41,730 understand what difference it makes what carrier 1361 01:13:41,730 --> 01:13:43,220 frequency you're at. 1362 01:13:43,220 --> 01:13:45,540 And it does make a difference because we'll talk about 1363 01:13:45,540 --> 01:13:46,730 Doppler shift. 1364 01:13:46,730 --> 01:13:50,490 And Doppler shift changes a lot as you go from one range 1365 01:13:50,490 --> 01:13:52,050 to another. 1366 01:13:52,050 --> 01:13:54,910 But for the most part these systems are narrow band. 1367 01:13:54,910 --> 01:13:58,750 There's a lot of work now on wide band systems. 1368 01:13:58,750 --> 01:14:00,110 And what's does a wide band mean? 1369 01:14:00,110 --> 01:14:02,600 Does it mean more than a megahertz? 1370 01:14:02,600 --> 01:14:06,080 No, it means a system where the bandwidth that you're 1371 01:14:06,080 --> 01:14:09,350 communicating over is a significant fraction of the 1372 01:14:09,350 --> 01:14:10,430 carrier frequency. 1373 01:14:10,430 --> 01:14:12,230 If there is a carrier frequency. 1374 01:14:12,230 --> 01:14:15,520 Many of these wide band systems are not even done in 1375 01:14:15,520 --> 01:14:17,110 terms of the carrier frequency. 1376 01:14:17,110 --> 01:14:20,220 They're just done in terms of an arbitrary waveform which 1377 01:14:20,220 --> 01:14:24,830 takes over an enormous amount of bandwidth. 1378 01:14:24,830 --> 01:14:27,290 If you're dealing with the narrow band problems, white 1379 01:14:27,290 --> 01:14:31,540 Gaussian noise is a good assumption for the noise. 1380 01:14:31,540 --> 01:14:33,480 But now, along with the noise you have 1381 01:14:33,480 --> 01:14:34,910 all these other effects. 1382 01:14:34,910 --> 01:14:38,830 You have a channel, where the channel is not just a pass 1383 01:14:38,830 --> 01:14:42,270 through wire with a little attenuation on it. 1384 01:14:42,270 --> 01:14:44,520 I mean, remember what we've done all along. 1385 01:14:44,520 --> 01:14:48,620 We have absolutely ignored the question of attenuation. 1386 01:14:48,620 --> 01:14:51,170 We've just gotten rid of it and say what you send is what 1387 01:14:51,170 --> 01:14:52,870 you receive. 1388 01:14:52,870 --> 01:14:54,490 We've gotten rid of the problem of 1389 01:14:54,490 --> 01:14:55,700 filtering on the channel. 1390 01:14:55,700 --> 01:14:58,870 We've said a little bit about it, but essentially we've 1391 01:14:58,870 --> 01:14:59,970 avoided it. 1392 01:14:59,970 --> 01:15:02,630 Now the problem that you have is this channel that you're 1393 01:15:02,630 --> 01:15:06,970 transmitting over really comes and goes. 1394 01:15:06,970 --> 01:15:09,140 Sometimes it's there, sometimes it's not. 1395 01:15:09,140 --> 01:15:12,540 So it's a time varying channel. 1396 01:15:12,540 --> 01:15:15,290 It's a time varying channel which depends on the frequency 1397 01:15:15,290 --> 01:15:16,880 band that we're using. 1398 01:15:16,880 --> 01:15:19,340 And one of the things that we have to talk about in order to 1399 01:15:19,340 --> 01:15:22,850 come to grips with this is questions about how quickly 1400 01:15:22,850 --> 01:15:26,240 does it change and why does it change. 1401 01:15:26,240 --> 01:15:28,930 And how much do you have to change the frequency before 1402 01:15:28,930 --> 01:15:31,480 you got something that looks like an independently 1403 01:15:31,480 --> 01:15:33,890 different channel? 1404 01:15:33,890 --> 01:15:36,160 So we have to deal with both of those and we're going to do 1405 01:15:36,160 --> 01:15:38,580 that next time. 1406 01:15:38,580 --> 01:15:42,380 And in trying to come to grips with these questions, the 1407 01:15:42,380 --> 01:15:45,400 first thing we're going to do is to look at very, very 1408 01:15:45,400 --> 01:15:49,220 idealized models of what goes on in communication. 1409 01:15:49,220 --> 01:15:50,340 Like, we're going to look at a point 1410 01:15:50,340 --> 01:15:54,130 source radiating outwards. 1411 01:15:54,130 --> 01:15:57,470 We're going to look at a point source radiating outwords, 1412 01:15:57,470 --> 01:16:00,100 hitting a barrier, and coming back. 1413 01:16:00,100 --> 01:16:02,960 Interesting problem to look at and you ought to read the 1414 01:16:02,960 --> 01:16:05,220 notes about this. 1415 01:16:05,220 --> 01:16:09,200 What happens when you're in a car and you're driving at 60 1416 01:16:09,200 --> 01:16:12,390 miles an hour towards the reflecting wall. 1417 01:16:12,390 --> 01:16:14,680 And right before you hit the wall, what's the 1418 01:16:14,680 --> 01:16:17,920 communication look like? 1419 01:16:17,920 --> 01:16:20,750 OK, that's a very neat and very simple problem. 1420 01:16:20,750 --> 01:16:24,890 You can't do it many times, but we will talk 1421 01:16:24,890 --> 01:16:26,900 about that next time.