1 00:00:00,000 --> 00:00:02,360 The following content is provided under a Create 2 00:00:02,360 --> 00:00:03,630 Commons license. 3 00:00:03,630 --> 00:00:06,600 Your support will help MIT OpenCourseWare continue to 4 00:00:06,600 --> 00:00:09,970 offer high quality education resources for free. 5 00:00:09,970 --> 00:00:12,870 To make a donation or to view additional materials from 6 00:00:12,870 --> 00:00:15,280 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:15,280 --> 00:00:16,530 ocw.mit.edu. 8 00:00:21,070 --> 00:00:23,380 PROFESSOR: Let's get started then. 9 00:00:23,380 --> 00:00:26,260 We went through Rayleigh fading very, very 10 00:00:26,260 --> 00:00:29,200 quickly last time. 11 00:00:29,200 --> 00:00:34,110 I want to spend a little more time on it today since it's 12 00:00:34,110 --> 00:00:40,950 one of the sort of classical models of wireless channels. 13 00:00:40,950 --> 00:00:43,400 And it's good to understand how it works. 14 00:00:43,400 --> 00:00:46,070 And it's good to also understand what all the 15 00:00:46,070 --> 00:00:51,260 assumptions that are made when one assumes Rayleigh fading, 16 00:00:51,260 --> 00:00:54,220 because they're really quite a few of them. 17 00:00:54,220 --> 00:00:58,560 OK, so what we're doing is we're assuming flat fading. 18 00:00:58,560 --> 00:01:00,960 In other words when we talk about flat fading, we're 19 00:01:00,960 --> 00:01:06,830 talking about fading where if you generate a discrete model 20 00:01:06,830 --> 00:01:10,500 for the channel, that discrete model is just going to have 21 00:01:10,500 --> 00:01:12,280 one path in it. 22 00:01:12,280 --> 00:01:16,460 In other words, the output is going to look like a faded 23 00:01:16,460 --> 00:01:18,290 version of the input. 24 00:01:18,290 --> 00:01:21,700 It'll be shifted in phase because of the unknown phase 25 00:01:21,700 --> 00:01:22,760 in the channel. 26 00:01:22,760 --> 00:01:26,770 It'll be attenuated by some random amount. 27 00:01:26,770 --> 00:01:29,590 But if you look at the waveform, it'll look like the 28 00:01:29,590 --> 00:01:32,860 waveform that you transmitted except for the noise. 29 00:01:36,260 --> 00:01:39,070 And that's what really is represented by this one tap 30 00:01:39,070 --> 00:01:41,260 model that we've been looking at. 31 00:01:41,260 --> 00:01:45,980 In general we've said that you can model a pretty arbitrary 32 00:01:45,980 --> 00:01:49,590 channel for purposes of somewhat narrow band 33 00:01:49,590 --> 00:01:56,070 communication by using a sequence of taps where usually 34 00:01:56,070 --> 00:01:59,510 for want of something better to do, we model those taps as 35 00:01:59,510 --> 00:02:03,800 being Gaussian random variables, complex Gaussian 36 00:02:03,800 --> 00:02:13,630 random variables with zero mean, and variables which are 37 00:02:13,630 --> 00:02:15,340 circularly symmetric. 38 00:02:15,340 --> 00:02:19,350 And we assume for not any very good reason, that the taps are 39 00:02:19,350 --> 00:02:21,100 independent of each other. 40 00:02:21,100 --> 00:02:23,370 I mean we have to make some assumptions or we can't start 41 00:02:23,370 --> 00:02:27,490 to make any progress on trying to analyze these channels. 42 00:02:27,490 --> 00:02:30,830 But we should realize that all of these assumptions are 43 00:02:30,830 --> 00:02:33,940 subject to a certain amount of question. 44 00:02:33,940 --> 00:02:34,300 OK. 45 00:02:34,300 --> 00:02:42,760 When we assume a single tap model, and these tap models 46 00:02:42,760 --> 00:02:46,630 are always given with the number of the tap given first, 47 00:02:46,630 --> 00:02:48,340 and the time given second. 48 00:02:48,340 --> 00:02:50,760 So what we're assuming here is the only tap is 49 00:02:50,760 --> 00:02:52,580 the tap at time 0. 50 00:02:52,580 --> 00:02:56,270 And it's at time 0 because we're assuming the receiver 51 00:02:56,270 --> 00:02:59,080 timing is locked at transmitter timing. 52 00:02:59,080 --> 00:03:01,160 And we're just going to get rid of the zero because 53 00:03:01,160 --> 00:03:03,740 there's only one tap, and call this G sub m. 54 00:03:03,740 --> 00:03:08,210 We're also going to pretty much assume that G sub m stays 55 00:03:08,210 --> 00:03:10,270 relatively constant for a relatively 56 00:03:10,270 --> 00:03:11,560 long amount of time. 57 00:03:14,440 --> 00:03:17,680 Except as far as this analysis of Rayleigh fading goes, we 58 00:03:17,680 --> 00:03:19,890 don't have to assume that. 59 00:03:19,890 --> 00:03:23,860 Because in fact, when we're assuming Rayleigh fading, the 60 00:03:23,860 --> 00:03:28,180 analysis that we're going to follow, the receiver doesn't 61 00:03:28,180 --> 00:03:30,960 know anything about the channel at all, except that 62 00:03:30,960 --> 00:03:32,980 it's a single tap model. 63 00:03:32,980 --> 00:03:35,760 And therefore what the receiver does is it goes 64 00:03:35,760 --> 00:03:39,700 through maximum likelihood detection assuming that that 65 00:03:39,700 --> 00:03:45,010 single tap is just a complex Gaussian random variable. 66 00:03:45,010 --> 00:03:48,030 OK when you have a complex Gaussian random variable as 67 00:03:48,030 --> 00:03:52,060 you've seen in the problem sets and we've noted a number 68 00:03:52,060 --> 00:03:57,720 of times, the energy in that complex Gaussian random 69 00:03:57,720 --> 00:04:00,630 variable is exponential. 70 00:04:00,630 --> 00:04:08,620 And the magnitude is just a square root of the magnitude 71 00:04:08,620 --> 00:04:10,310 squared, namely the energy. 72 00:04:10,310 --> 00:04:13,460 And that has a Rayleigh distribution which looks like 73 00:04:13,460 --> 00:04:20,750 this, namely the probability density of how much 74 00:04:20,750 --> 00:04:22,210 response you get. 75 00:04:22,210 --> 00:04:23,980 We'll base this law here. 76 00:04:23,980 --> 00:04:28,810 And the phase of course, is equally likely to be anything. 77 00:04:28,810 --> 00:04:31,990 Namely the phase is uniform and random. 78 00:04:31,990 --> 00:04:34,610 This density looks like this. 79 00:04:34,610 --> 00:04:37,910 I wanted to draw this so it would emphasize the fact that 80 00:04:37,910 --> 00:04:41,290 the magnitude is in fact, always nonnegative. 81 00:04:41,290 --> 00:04:43,830 But also to emphasize the fact there's a whole lot of 82 00:04:43,830 --> 00:04:46,670 probability down here where there's 83 00:04:46,670 --> 00:04:48,990 very, very little channel. 84 00:04:48,990 --> 00:04:52,680 And this is in fact what gives rise to the fact that if you 85 00:04:52,680 --> 00:04:58,150 try to communicate over Rayleigh fading, and you don't 86 00:04:58,150 --> 00:05:00,860 make any use of diversity-- and we'll talk later about 87 00:05:00,860 --> 00:05:02,290 diversity-- 88 00:05:02,290 --> 00:05:05,660 in fact you can't communicate very well at all. 89 00:05:05,660 --> 00:05:09,270 And that's because of this very bad region here where the 90 00:05:09,270 --> 00:05:11,230 channel is very badly faded. 91 00:05:11,230 --> 00:05:16,210 You send a bit on this channel which is very badly faded, and 92 00:05:16,210 --> 00:05:18,010 there's nothing much the receiver can do. 93 00:05:18,010 --> 00:05:22,670 And that's the thing we want to try to get a 94 00:05:22,670 --> 00:05:24,000 real feeling for. 95 00:05:24,000 --> 00:05:26,970 OK so the output of the channel when you put in an 96 00:05:26,970 --> 00:05:31,230 input U sub m, and we'll think of this as being a binary 97 00:05:31,230 --> 00:05:33,470 digit for this time being. 98 00:05:33,470 --> 00:05:37,830 So the output is going to be the input times this tap 99 00:05:37,830 --> 00:05:40,500 variable, which is this complex Gaussian random 100 00:05:40,500 --> 00:05:44,800 variable plus a noise random variable, which we're also 101 00:05:44,800 --> 00:05:49,330 assuming is complex Gaussian and circularly symmetric. 102 00:05:49,330 --> 00:05:53,020 OK so what we have, if you're going to make two hypotheses 103 00:05:53,020 --> 00:05:56,780 about two possible values of U sub m, look at what this 104 00:05:56,780 --> 00:05:58,790 random phase does here. 105 00:05:58,790 --> 00:06:06,660 No matter what U sub m you transmit in one epoch of time, 106 00:06:06,660 --> 00:06:10,130 the channel is going to rotate this around by a completely 107 00:06:10,130 --> 00:06:11,420 random phase. 108 00:06:11,420 --> 00:06:14,020 It's going to add a noise to it which has a completely 109 00:06:14,020 --> 00:06:15,330 random phase. 110 00:06:15,330 --> 00:06:17,240 And the output is going to come out. 111 00:06:17,240 --> 00:06:21,530 And the output has a completely random phase. 112 00:06:21,530 --> 00:06:25,610 Namely the phase of the output cannot possibly tell you 113 00:06:25,610 --> 00:06:28,280 anything about what input you're 114 00:06:28,280 --> 00:06:29,980 putting into the channel. 115 00:06:29,980 --> 00:06:34,530 OK so in other words, in this model that we're using, the 116 00:06:34,530 --> 00:06:37,990 phase is completely useless. 117 00:06:37,990 --> 00:06:45,270 And if we want to talk about anything connected to using 118 00:06:45,270 --> 00:06:49,630 likelihoods, the only thing we can use is the magnitude of 119 00:06:49,630 --> 00:06:50,730 the output. 120 00:06:50,730 --> 00:06:52,110 OK. 121 00:06:52,110 --> 00:06:54,930 Now, why don't we just analyze it in terms of the magnitude 122 00:06:54,930 --> 00:06:56,060 of the output? 123 00:06:56,060 --> 00:07:00,360 Well when you analyze these problems, Gaussian things are 124 00:07:00,360 --> 00:07:05,680 usually much easier to analyze than things like this. 125 00:07:05,680 --> 00:07:07,590 Not always, I mean we have to get used to 126 00:07:07,590 --> 00:07:09,080 analyzing all of them. 127 00:07:09,080 --> 00:07:12,400 But this particular problem of Rayleigh fading is really 128 00:07:12,400 --> 00:07:14,310 easier to analyze in terms of these 129 00:07:14,310 --> 00:07:16,220 Gaussian random variables. 130 00:07:16,220 --> 00:07:19,510 But it's easier to understand in terms of recognizing that 131 00:07:19,510 --> 00:07:23,320 the only thing you can make any use is these magnitudes. 132 00:07:28,190 --> 00:07:32,040 OK if we only use one complex degree of freedom in a signal, 133 00:07:32,040 --> 00:07:37,540 namely if we try to send some signal and we only use one 134 00:07:37,540 --> 00:07:41,730 input to the channel, then we only get one output. 135 00:07:41,730 --> 00:07:45,800 Namely we sent U sub 0, we get V sub 0. 136 00:07:45,800 --> 00:07:49,330 And we try to decide from V sub 0 what was sent. 137 00:07:49,330 --> 00:07:52,850 We're really in a very bad pickle at that point. 138 00:07:52,850 --> 00:07:56,050 Because the only thing that makes any sense, since we can 139 00:07:56,050 --> 00:08:01,090 only use the magnitudes, is to send a very small magnitude or 140 00:08:01,090 --> 00:08:03,530 a positive magnitude. 141 00:08:03,530 --> 00:08:05,550 Magnitudes are positive anyway. 142 00:08:05,550 --> 00:08:08,170 So if you're going to send binary signals, this is your 143 00:08:08,170 --> 00:08:11,520 only choice-- if it makes any sense-- 144 00:08:11,520 --> 00:08:13,760 if you make this larger than zero you're 145 00:08:13,760 --> 00:08:15,980 just wasting energy. 146 00:08:15,980 --> 00:08:18,060 So you only have this choice. 147 00:08:18,060 --> 00:08:20,700 And you can choose the amplitude a that you're using. 148 00:08:20,700 --> 00:08:23,020 But that's the only thing that you can do. 149 00:08:23,020 --> 00:08:28,340 OK, this is a very nasty thing to analyze for one thing. 150 00:08:28,340 --> 00:08:29,820 It gives you a very large error 151 00:08:29,820 --> 00:08:31,780 probability for another thing. 152 00:08:31,780 --> 00:08:34,330 Nobody uses it for another thing. 153 00:08:34,330 --> 00:08:38,080 And therefore almost all systems of trying to transmit 154 00:08:38,080 --> 00:08:42,160 in this kind of Rayleigh fading, always use at least 155 00:08:42,160 --> 00:08:44,520 two sample values. 156 00:08:44,520 --> 00:08:47,180 In other words, instead of just putting one complex 157 00:08:47,180 --> 00:08:49,970 degree of freedom into the channel, you're going to put 158 00:08:49,970 --> 00:08:52,930 two complex degrees of freedom into the channel. 159 00:08:52,930 --> 00:08:55,860 And the thing that we're going to analyze, because it's the 160 00:08:55,860 --> 00:08:58,700 easiest thing to do in this discrete time model we've 161 00:08:58,700 --> 00:09:03,930 developed, is to think of modeling hypothesis 0 as 162 00:09:03,930 --> 00:09:08,320 sending two symbols U sub 0 and U sub 1 will make U sub 0 163 00:09:08,320 --> 00:09:12,380 equal to a, and U sub 1 equal to 0. 164 00:09:12,380 --> 00:09:15,530 And the alternative case, if we're going to try to send 165 00:09:15,530 --> 00:09:18,300 input 1, this is binary transmission. 166 00:09:18,300 --> 00:09:21,660 You can talk about more than binary transmission, but 167 00:09:21,660 --> 00:09:24,400 binary is awful enough. 168 00:09:24,400 --> 00:09:27,680 You get U sub 0 and U sub 1 is equal to 0 and a. 169 00:09:27,680 --> 00:09:31,480 So what you're going to be doing here in this 170 00:09:31,480 --> 00:09:35,030 pulse-position modulation, is choosing one of these two 171 00:09:35,030 --> 00:09:38,040 different epochs to put the data in. 172 00:09:38,040 --> 00:09:41,000 So in one case, you put all your energy in the first one. 173 00:09:41,000 --> 00:09:44,690 In the other case, you put all your energy in the second one. 174 00:09:44,690 --> 00:09:47,340 Mathematically, this is completely equivalent to 175 00:09:47,340 --> 00:09:50,190 frequency-shift keying, that's completely equivalent to 176 00:09:50,190 --> 00:09:52,320 phase-shift keying. 177 00:09:52,320 --> 00:09:54,740 And if we had a little more time, we 178 00:09:54,740 --> 00:09:55,890 could talk about that. 179 00:09:55,890 --> 00:09:58,750 And I'll probably put an appendix in which talks about 180 00:09:58,750 --> 00:10:00,020 those two systems. 181 00:10:00,020 --> 00:10:02,310 But in fact, it's completely the same thing. 182 00:10:02,310 --> 00:10:04,810 It's just that you're using different complex degrees of 183 00:10:04,810 --> 00:10:06,910 freedom than we're using here. 184 00:10:06,910 --> 00:10:10,500 So we're really analyzing FSK and PSK. 185 00:10:10,500 --> 00:10:13,160 And that's where people usually come up with these 186 00:10:13,160 --> 00:10:17,240 analyses of Rayleigh fading. 187 00:10:17,240 --> 00:10:25,000 OK when we have input 0, what we receive then, is the 0 is 188 00:10:25,000 --> 00:10:29,470 going to be the input a, times the magnitude of the channel 189 00:10:29,470 --> 00:10:32,060 of time 0, plus a noise variable. 190 00:10:32,060 --> 00:10:35,290 The noise is complex Gaussian, remember. 191 00:10:35,290 --> 00:10:39,110 The second input is just going to be the noise variable. 192 00:10:39,110 --> 00:10:42,120 Alternatively, if we're sending the second symbol, 193 00:10:42,120 --> 00:10:44,740 which means we put our energy into the second degree of 194 00:10:44,740 --> 00:10:48,440 freedom, it means that what we're going to get is the 0 195 00:10:48,440 --> 00:10:50,430 was just going to be the noise. 196 00:10:50,430 --> 00:10:53,840 And the second output is going to be the 197 00:10:53,840 --> 00:10:56,020 signal plus the noise. 198 00:10:56,020 --> 00:11:00,740 And remember, both this variable and this variable are 199 00:11:00,740 --> 00:11:02,790 both complex Gaussian. 200 00:11:02,790 --> 00:11:04,390 The phase doesn't mean anything. 201 00:11:04,390 --> 00:11:10,080 So what we can use is simply the magnitude. 202 00:11:10,080 --> 00:11:16,250 OK, so when we have hypothesis equal to 0, what comes out is 203 00:11:16,250 --> 00:11:20,710 going to be, the 0 is going to be a complex 204 00:11:20,710 --> 00:11:22,120 Gaussian random variable. 205 00:11:22,120 --> 00:11:24,920 Let me introduce a new piece of notation now. 206 00:11:24,920 --> 00:11:28,870 Because it gets to be a real mess to constantly talk about 207 00:11:28,870 --> 00:11:32,640 a Gaussian complex random variable, and talk about it's 208 00:11:32,640 --> 00:11:36,310 real part and imaginary part as being independent Gaussian. 209 00:11:36,310 --> 00:11:41,230 So I'll just call this normal complex. 210 00:11:41,230 --> 00:11:44,550 And this first thing is the mean, which is a real and 211 00:11:44,550 --> 00:11:47,510 imaginary part, but it's zero in most of the 212 00:11:47,510 --> 00:11:48,980 things we deal with. 213 00:11:48,980 --> 00:11:53,850 And the second one is the mean square value of this random 214 00:11:53,850 --> 00:11:55,660 variable V sub zero. 215 00:11:55,660 --> 00:12:01,640 So this quantity here is now twice the variance of the real 216 00:12:01,640 --> 00:12:05,635 part of V sub 0, and twice the imaginary part of 217 00:12:05,635 --> 00:12:09,910 the variance of v0. 218 00:12:09,910 --> 00:12:14,170 We scaled the noise in a peculiar way here. 219 00:12:14,170 --> 00:12:19,040 And I apologize for all of the mess that 220 00:12:19,040 --> 00:12:21,510 occurs when we do this. 221 00:12:21,510 --> 00:12:26,100 Because sometimes we think of the noise as having variance N 222 00:12:26,100 --> 00:12:29,720 sub 0 over 2 in each real and imaginary degree of freedom. 223 00:12:29,720 --> 00:12:33,310 And therefore N sub 0 in a complex degree of freedom. 224 00:12:33,310 --> 00:12:37,560 And sometimes we think of it as having variance N sub 0 W 225 00:12:37,560 --> 00:12:39,930 Where does that difference come from? 226 00:12:39,930 --> 00:12:44,060 It's this infernal problem of the sampling theorem being so 227 00:12:44,060 --> 00:12:47,810 critical in most of the models that we talk about. 228 00:12:47,810 --> 00:12:51,920 OK because when you use the sampling theorem, the sinc x 229 00:12:51,920 --> 00:12:56,080 over x waveforms that we use are not orthonormal, they're 230 00:12:56,080 --> 00:12:57,700 orthogonal. 231 00:12:57,700 --> 00:13:03,390 And this factor of W appears exactly because of that. 232 00:13:03,390 --> 00:13:08,240 They appear because the magnitude of the signal is a, 233 00:13:08,240 --> 00:13:10,690 and the energy and the power in the 234 00:13:10,690 --> 00:13:13,020 signal is then a-squared. 235 00:13:13,020 --> 00:13:13,910 OK. 236 00:13:13,910 --> 00:13:17,050 In this case the power in the signal is not quite a-squared 237 00:13:17,050 --> 00:13:21,950 because we only send energy in one or the other of alternate 238 00:13:21,950 --> 00:13:23,150 degrees of freedom. 239 00:13:23,150 --> 00:13:31,620 So therefore, if we look at a time one second, we get W 240 00:13:31,620 --> 00:13:33,810 complex degrees of freedom to use. 241 00:13:33,810 --> 00:13:41,990 We only send energy in half of those so that the actual power 242 00:13:41,990 --> 00:13:47,290 that we're sending is a-squared divided by 2. 243 00:13:47,290 --> 00:13:47,730 OK. 244 00:13:47,730 --> 00:13:50,810 Because of that, when we normalize the noise the same 245 00:13:50,810 --> 00:13:55,090 way the signal is normalized, we get this 246 00:13:55,090 --> 00:13:57,600 variance W N sub 0. 247 00:13:57,600 --> 00:14:00,910 If you're confused by that, everyone is confused by it. 248 00:14:00,910 --> 00:14:04,220 Everyone I know, when they go through calculations like 249 00:14:04,220 --> 00:14:07,410 this, they always start out with some arbitrary fudge 250 00:14:07,410 --> 00:14:08,840 factor like this. 251 00:14:08,840 --> 00:14:12,120 And after they get all done, they think it through or more 252 00:14:12,120 --> 00:14:14,785 likely they look it up in a book to see what somebody else 253 00:14:14,785 --> 00:14:16,390 has gotten. 254 00:14:16,390 --> 00:14:19,100 And then they sweat about it a little bit, and they finally 255 00:14:19,100 --> 00:14:21,430 decide what it ought to be. 256 00:14:21,430 --> 00:14:23,190 And that's just the way it is. 257 00:14:23,190 --> 00:14:27,710 It's the problem of having both the sampling theorem and 258 00:14:27,710 --> 00:14:29,800 orthonormal waveforms sitting around. 259 00:14:29,800 --> 00:14:34,400 It's also the problem of multiplying the power by 2 as 260 00:14:34,400 --> 00:14:36,110 soon as we go to passband. 261 00:14:36,110 --> 00:14:40,370 Because both of those things together generate all of this 262 00:14:40,370 --> 00:14:41,090 difficulty. 263 00:14:41,090 --> 00:14:43,710 But anyway, this is the way it is. 264 00:14:43,710 --> 00:14:47,820 And the important thing for us is that what we can have now 265 00:14:47,820 --> 00:14:49,950 is under these two hypotheses. 266 00:14:49,950 --> 00:14:53,300 We just have two Gaussian random variables, complex 267 00:14:53,300 --> 00:14:55,030 Gaussian random variables. 268 00:14:55,030 --> 00:14:59,690 And in one case, the larger mean square value is in one. 269 00:14:59,690 --> 00:15:01,500 And in the other case the larger mean square 270 00:15:01,500 --> 00:15:02,910 value is in the other. 271 00:15:09,900 --> 00:15:10,380 OK. 272 00:15:10,380 --> 00:15:12,470 So just reviewing that. 273 00:15:12,470 --> 00:15:17,450 If H is equal to zero, V sub 0 and V sub 1 are these complex 274 00:15:17,450 --> 00:15:19,250 Gaussian random variables. 275 00:15:19,250 --> 00:15:23,590 If H is equal to one, then we have this set of Gaussian 276 00:15:23,590 --> 00:15:25,190 random variables. 277 00:15:25,190 --> 00:15:29,560 The probability density of V sub 0 and V sub 1, and now 278 00:15:29,560 --> 00:15:34,460 it's more convenient to use the real and imaginary parts 279 00:15:34,460 --> 00:15:36,230 for the Gaussian density. 280 00:15:36,230 --> 00:15:40,260 Anytime you're working problems of this type, try 281 00:15:40,260 --> 00:15:45,300 both densities using real and imaginary parts, and using 282 00:15:45,300 --> 00:15:49,300 magnitude in phase, and see which one is easier. 283 00:15:49,300 --> 00:15:52,400 Here it turns out that the easiest thing is just to use 284 00:15:52,400 --> 00:15:56,230 the ordinary conventional density over real and 285 00:15:56,230 --> 00:15:58,060 imaginary parts. 286 00:15:58,060 --> 00:16:01,450 And what we wind up with is this Gaussian density. 287 00:16:04,130 --> 00:16:10,190 On V sub 0 the density is V sub 0 squared divided by the 288 00:16:10,190 --> 00:16:13,440 variance a-squared plus W N sub 0. 289 00:16:13,440 --> 00:16:18,760 And on V sub 1, it's this Gaussian density V sub 1 290 00:16:18,760 --> 00:16:21,250 squared divided by W N sub 0. 291 00:16:21,250 --> 00:16:23,920 Just because here we have this variance. 292 00:16:23,920 --> 00:16:26,330 Here we have this variance. 293 00:16:26,330 --> 00:16:29,910 OK, on the alternative hypothesis when H is equal to 294 00:16:29,910 --> 00:16:33,880 one, you have the same thing but the denominators are 295 00:16:33,880 --> 00:16:35,010 switched around. 296 00:16:35,010 --> 00:16:38,250 When you take the likelihood ratio, you want to take the 297 00:16:38,250 --> 00:16:41,660 ratio of this, to the ratio of this. 298 00:16:41,660 --> 00:16:44,420 If you look at it and you take the logarithm of that, you're 299 00:16:44,420 --> 00:16:46,650 taking the ratio of this to this. 300 00:16:46,650 --> 00:16:49,140 Incidentally the coefficient here, you could write it out 301 00:16:49,140 --> 00:16:50,320 if you want to. 302 00:16:50,320 --> 00:16:53,630 It's 1 over the square root of blah, times 1 over the square 303 00:16:53,630 --> 00:16:54,700 root of blah. 304 00:16:54,700 --> 00:16:58,350 But if you recognize that the coefficient here has to be the 305 00:16:58,350 --> 00:17:00,550 same as the coefficient here, you don't have 306 00:17:00,550 --> 00:17:01,850 to worry about it. 307 00:17:01,850 --> 00:17:05,320 So when you take the log likelihood ratio, you get this 308 00:17:05,320 --> 00:17:06,700 divided by this. 309 00:17:06,700 --> 00:17:08,790 You have the same form in both cases. 310 00:17:08,790 --> 00:17:13,050 In one case, you have this term minus this term and this 311 00:17:13,050 --> 00:17:14,840 term minus this term. 312 00:17:14,840 --> 00:17:24,340 And the other case well, for V sub 0, you have this term 313 00:17:24,340 --> 00:17:25,480 minus this term. 314 00:17:25,480 --> 00:17:28,830 And for V sub 1 you have this term minus this term. 315 00:17:28,830 --> 00:17:31,440 Because of the symmetry between the two, this just 316 00:17:31,440 --> 00:17:34,230 comes out to V sub 0 squared minus V sub 1 317 00:17:34,230 --> 00:17:36,010 squared times a-squared. 318 00:17:36,010 --> 00:17:39,040 And when you do the algebra, the denominator is a-squared 319 00:17:39,040 --> 00:17:43,330 plus W N sub 0 times W N sub 0. 320 00:17:43,330 --> 00:17:43,850 OK. 321 00:17:43,850 --> 00:17:48,600 What do we do for making a maximum likelihood decision? 322 00:17:48,600 --> 00:17:53,150 Maximum likelihood is map when the threshold is equal to 1, 323 00:17:53,150 --> 00:17:55,530 which is when the logarithm of the correct 324 00:17:55,530 --> 00:17:57,510 threshold is equal 0. 325 00:17:57,510 --> 00:18:02,040 Which says that you take this quantity, and if it's 326 00:18:02,040 --> 00:18:07,550 nonnegative, you choose H equals zero. 327 00:18:07,550 --> 00:18:10,490 And if it's negative, you choose H equals one. 328 00:18:10,490 --> 00:18:15,200 Which says you compare V sub 0 squared and V sub 1 squared. 329 00:18:15,200 --> 00:18:18,700 And whichever one is larger, that's the one you choose. 330 00:18:18,700 --> 00:18:22,660 And if you go back and look at the problem, it's pretty 331 00:18:22,660 --> 00:18:25,650 obvious that that's what you want to do anyway. 332 00:18:25,650 --> 00:18:27,770 I mean you'd be very, very surprised when you're 333 00:18:27,770 --> 00:18:31,280 comparing two Gaussian random variables where one of them 334 00:18:31,280 --> 00:18:33,410 has a larger variance than the other. 335 00:18:33,410 --> 00:18:35,490 And on the other hypothesis, the absolon 336 00:18:35,490 --> 00:18:38,650 has the larger variance. 337 00:18:38,650 --> 00:18:41,950 If you came up with any rule other than to take the 338 00:18:41,950 --> 00:18:46,360 magnitude squares and to then compare those two magnitude 339 00:18:46,360 --> 00:18:49,410 squares, you would go back and look at the problem again 340 00:18:49,410 --> 00:18:52,810 realizing you must have done something wrong. 341 00:18:52,810 --> 00:18:57,960 But anyway when you deal with problems like this, I advise 342 00:18:57,960 --> 00:19:01,430 you to take log likelihood ratio anyway. 343 00:19:01,430 --> 00:19:04,800 Because every once in awhile you find something which comes 344 00:19:04,800 --> 00:19:07,430 out in a somewhat peculiar way. 345 00:19:07,430 --> 00:19:10,370 But anyway, here there's nothing peculiar. 346 00:19:10,370 --> 00:19:12,950 So what we have to do now is to find the 347 00:19:12,950 --> 00:19:14,680 probability of error. 348 00:19:14,680 --> 00:19:16,480 Now what's the probability of error? 349 00:19:24,990 --> 00:19:34,250 OK if we actually transmit zero, then V sub 0 squared is 350 00:19:34,250 --> 00:19:36,160 exponential. 351 00:19:36,160 --> 00:19:42,600 It's exponential with this mean. 352 00:19:42,600 --> 00:19:45,830 Namely this is the mean of V sub 0 squared. 353 00:19:45,830 --> 00:19:48,710 And V sub 1 is exponential with this mean. 354 00:19:48,710 --> 00:19:52,340 In other words, this is a big exponential. 355 00:19:52,340 --> 00:19:54,310 And this is a little exponential. 356 00:19:54,310 --> 00:19:57,370 The two of them have probability densities that 357 00:19:57,370 --> 00:19:58,620 look like this. 358 00:20:01,170 --> 00:20:02,610 This is not going to work. 359 00:20:07,550 --> 00:20:11,620 The big one has a probability density that looks like this. 360 00:20:14,710 --> 00:20:16,680 And the little one-- 361 00:20:16,680 --> 00:20:19,220 this is big-- 362 00:20:19,220 --> 00:20:21,490 and the little one has a probability density 363 00:20:21,490 --> 00:20:23,140 that looks like this. 364 00:20:23,140 --> 00:20:28,190 And what you want to do is to subtract a random variable 365 00:20:28,190 --> 00:20:30,250 with this density from a random 366 00:20:30,250 --> 00:20:33,140 variable with this density. 367 00:20:33,140 --> 00:20:37,630 So you're convolving two exponential densities with 368 00:20:37,630 --> 00:20:39,300 each other. 369 00:20:39,300 --> 00:20:42,860 And unfortunately, you're taking the differences of two. 370 00:20:42,860 --> 00:20:47,040 So you're convolving the negatives of this with this. 371 00:20:47,040 --> 00:20:48,660 And then you have to integrate the thing. 372 00:20:48,660 --> 00:20:53,290 And it's just something you have to do. 373 00:20:53,290 --> 00:21:01,080 And the answer is, the probability of error is then 2 374 00:21:01,080 --> 00:21:05,570 plus a-squared over W N sub 0 to the minus 1. 375 00:21:05,570 --> 00:21:08,250 OK that is really an awful result. 376 00:21:08,250 --> 00:21:12,090 Because that says that if you increase the energy that 377 00:21:12,090 --> 00:21:15,910 you're using, the probability of error goes down very, very, 378 00:21:15,910 --> 00:21:17,390 very slowly. 379 00:21:17,390 --> 00:21:20,320 And if you look at this picture you think about it a 380 00:21:20,320 --> 00:21:23,490 little bit, it should be clear that that's the only thing 381 00:21:23,490 --> 00:21:26,050 that can happen. 382 00:21:26,050 --> 00:21:26,420 OK. 383 00:21:26,420 --> 00:21:30,660 Because if you increase a-squared a little bit, it's 384 00:21:30,660 --> 00:21:32,970 not going to save you much here. 385 00:21:32,970 --> 00:21:35,670 Because when you have a bigger a-squared, it's just going to 386 00:21:35,670 --> 00:21:38,570 move down the value of g bar that's 387 00:21:38,570 --> 00:21:39,960 going to give you trouble. 388 00:21:39,960 --> 00:21:44,900 Namely when you double a, the value of magnitude of g that 389 00:21:44,900 --> 00:21:48,130 gives you trouble just goes down by a factor of two. 390 00:21:48,130 --> 00:21:52,850 When that goes down by a factor two, this bad part of 391 00:21:52,850 --> 00:21:57,330 the curve just goes down in a quadratic way. 392 00:21:57,330 --> 00:22:01,400 Well that's what this is telling us. 393 00:22:01,400 --> 00:22:03,730 OK. 394 00:22:03,730 --> 00:22:08,010 I mean the thing that we see is a quadratic and a. 395 00:22:08,010 --> 00:22:10,590 So we're sort of assured that we're doing the right thing. 396 00:22:10,590 --> 00:22:13,750 And we're sort of also assured that the reason why this 397 00:22:13,750 --> 00:22:17,370 result is so awful, is just that sometimes the fading is 398 00:22:17,370 --> 00:22:21,640 so bad there's nothing you can do about it. 399 00:22:21,640 --> 00:22:23,790 OK now the signal power as we said before is 400 00:22:23,790 --> 00:22:25,150 a-squared over 2. 401 00:22:25,150 --> 00:22:27,040 Since half the inputs are zero. 402 00:22:27,040 --> 00:22:29,710 So we can put twice as much energy into the 403 00:22:29,710 --> 00:22:31,440 ones that are non-zero. 404 00:22:31,440 --> 00:22:35,520 And therefore when you put this in terms of the average 405 00:22:35,520 --> 00:22:39,240 signal energy that you're sending, what we get is E sub 406 00:22:39,240 --> 00:22:41,070 b over N sub 0. 407 00:22:41,070 --> 00:22:42,020 OK. 408 00:22:42,020 --> 00:22:46,550 So that again says exactly the same thing that this does. 409 00:22:46,550 --> 00:22:51,060 It's worthwhile keeping both of these notions around, 410 00:22:51,060 --> 00:22:55,460 because we have done something kind of peculiar here. 411 00:22:55,460 --> 00:22:57,590 I should mention it for you. 412 00:22:57,590 --> 00:23:02,580 As soon as you're looking at a fading channel, the power that 413 00:23:02,580 --> 00:23:05,380 you're talking about becomes a little peculiar. 414 00:23:05,380 --> 00:23:07,030 Because remember when we were looking 415 00:23:07,030 --> 00:23:09,020 at white noise channels? 416 00:23:09,020 --> 00:23:12,600 What we were looking at is the power at the receiver, the 417 00:23:12,600 --> 00:23:16,650 signal power as received at the receiver. 418 00:23:16,650 --> 00:23:19,890 Now at this point, we still want to talk about E sub b. 419 00:23:19,890 --> 00:23:23,320 We still want to isolate this problem from the attenuation 420 00:23:23,320 --> 00:23:26,590 that occurs just because of distance and things like. 421 00:23:26,590 --> 00:23:31,680 Because of that, when we model g, this original model that we 422 00:23:31,680 --> 00:23:39,220 use here was a model in which the magnitude 423 00:23:39,220 --> 00:23:41,840 of g had mean one. 424 00:23:41,840 --> 00:23:44,500 And we made it have mean one so that the energy 425 00:23:44,500 --> 00:23:45,560 would come out right. 426 00:23:45,560 --> 00:23:48,470 Which is another reason why you get confused with these E 427 00:23:48,470 --> 00:23:50,940 sub b over N sub 0 terms. 428 00:23:50,940 --> 00:23:57,720 OK so anyway, that's the answer. 429 00:23:57,720 --> 00:24:03,760 And E sub b is in terms of the received energy using the 430 00:24:03,760 --> 00:24:06,360 average value of fading. 431 00:24:06,360 --> 00:24:09,800 OK we next want to look at non-coherent detection. 432 00:24:09,800 --> 00:24:12,270 Non-coherent detection is another thing that 433 00:24:12,270 --> 00:24:15,370 communication engineers use all the time, talk 434 00:24:15,370 --> 00:24:16,760 about all the time. 435 00:24:16,760 --> 00:24:20,670 And you have to understand what the difference is between 436 00:24:20,670 --> 00:24:24,060 coherent transmission and incoherent transmission. 437 00:24:24,060 --> 00:24:27,850 The general idea is that when you're doing incoherent 438 00:24:27,850 --> 00:24:30,590 detection, you're assuming that you don't know what the 439 00:24:30,590 --> 00:24:32,590 phase of the channel is. 440 00:24:32,590 --> 00:24:35,960 And somehow you want to do your detection without knowing 441 00:24:35,960 --> 00:24:37,520 that phase. 442 00:24:37,520 --> 00:24:39,870 The difference between Rayleigh fading on this kind 443 00:24:39,870 --> 00:24:43,990 of channel and incoherent detection, is that with 444 00:24:43,990 --> 00:24:48,140 incoherent detection the receiver is assumed to know 445 00:24:48,140 --> 00:24:52,200 what the magnitude of the channel is, but not the phase. 446 00:24:52,200 --> 00:24:55,640 It's harder to measure the phase of the channel than it 447 00:24:55,640 --> 00:24:57,060 is the measure of the magnitude. 448 00:24:57,060 --> 00:25:01,010 Because the phase changes very, very fast. 449 00:25:01,010 --> 00:25:03,810 If you look at these equations we have for what the response 450 00:25:03,810 --> 00:25:08,610 of the channel is, you see the phase changing many, many 451 00:25:08,610 --> 00:25:13,240 times during the time where the amplitude of the fading 452 00:25:13,240 --> 00:25:15,560 changes by just a little bit. 453 00:25:15,560 --> 00:25:20,100 So a very common assumption that people make when trying 454 00:25:20,100 --> 00:25:23,910 to do detection is that it's incoherent. 455 00:25:23,910 --> 00:25:27,250 Partly, people get used to analyzing incoherent 456 00:25:27,250 --> 00:25:29,480 communication. 457 00:25:29,480 --> 00:25:31,450 And I've seen this so many times. 458 00:25:31,450 --> 00:25:34,740 And they insist on building communication systems using 459 00:25:34,740 --> 00:25:36,360 incoherent detection. 460 00:25:36,360 --> 00:25:38,300 They will swear up and down there's no way you 461 00:25:38,300 --> 00:25:39,740 can measure the phase. 462 00:25:39,740 --> 00:25:42,840 And what they're really saying is that's the only kind of 463 00:25:42,840 --> 00:25:45,250 communication they understand. 464 00:25:45,250 --> 00:25:48,090 And because that's the only thing they understand, they 465 00:25:48,090 --> 00:25:51,260 become very, very upset if anyone suggests that you ought 466 00:25:51,260 --> 00:25:53,050 to try to measure the phase. 467 00:25:53,050 --> 00:25:56,830 But that's a tale for another day. 468 00:26:00,310 --> 00:26:04,050 OK so now we want to look at the case where we're assuming 469 00:26:04,050 --> 00:26:07,560 that we know the magnitude of the channel. 470 00:26:07,560 --> 00:26:11,570 It's just some quantity that we'll call g tilde. 471 00:26:11,570 --> 00:26:16,390 We're assuming that the same magnitude occurs both on U sub 472 00:26:16,390 --> 00:26:17,810 0 and U sub 1. 473 00:26:17,810 --> 00:26:20,430 We're going to use the same transmission system that we 474 00:26:20,430 --> 00:26:24,190 used before, namely pulse-position modulation. 475 00:26:24,190 --> 00:26:28,340 We'll either put our energy in U sub 0 or we'll put our 476 00:26:28,340 --> 00:26:29,750 energy in U sub 1. 477 00:26:29,750 --> 00:26:31,830 We'll try to detect what's going on. 478 00:26:31,830 --> 00:26:34,770 But we just give the detector this little extra amount of 479 00:26:34,770 --> 00:26:37,920 ability of knowing what the channel is. 480 00:26:37,920 --> 00:26:42,050 I'm going to talk more later about how you can use this 481 00:26:42,050 --> 00:26:45,190 knowledge of what the channel is, and how you can measure 482 00:26:45,190 --> 00:26:46,050 what the channel is. 483 00:26:46,050 --> 00:26:48,710 But for now we just assume that we know it. 484 00:26:48,710 --> 00:26:51,450 So the phase is random and independent 485 00:26:51,450 --> 00:26:53,050 of everything else. 486 00:26:53,050 --> 00:27:00,400 So under hypothesis H equals zero, we have the output of 487 00:27:00,400 --> 00:27:01,060 the channel. 488 00:27:01,060 --> 00:27:07,690 And times 0 is whatever input level we put in a, times what 489 00:27:07,690 --> 00:27:12,190 the channel does to us, times e to this random phase. 490 00:27:12,190 --> 00:27:13,760 And V sub 1 it's just-- 491 00:27:19,650 --> 00:27:23,320 plus Z sub 0. 492 00:27:23,320 --> 00:27:28,050 And in the other case we have V sub 1 equals Z sub 1. 493 00:27:28,050 --> 00:27:32,355 And under the other hypothesis V sub 1 is this input with a 494 00:27:32,355 --> 00:27:37,820 random phase but a known magnitude and again, a 495 00:27:37,820 --> 00:27:39,040 Gaussian random variable. 496 00:27:39,040 --> 00:27:42,250 Phases are independent of the hypothesis. 497 00:27:42,250 --> 00:27:44,000 The phases are independent of the 498 00:27:44,000 --> 00:27:45,660 magnitudes which are known. 499 00:27:45,660 --> 00:27:48,470 The phases are independent of everything and therefore, we 500 00:27:48,470 --> 00:27:51,130 just want to forget about them. 501 00:27:51,130 --> 00:27:54,350 So the question is, how do we make a maximum likelihood 502 00:27:54,350 --> 00:27:57,510 decision on this problem? 503 00:27:57,510 --> 00:27:59,080 Well you look at the problem. 504 00:27:59,080 --> 00:28:03,510 And for the same reason as before you say, it's obvious 505 00:28:03,510 --> 00:28:06,070 how to make a maximum likelihood decision just from 506 00:28:06,070 --> 00:28:09,450 all the symmetry that you have. 507 00:28:09,450 --> 00:28:12,770 If the magnitude of V sub 0 is bigger than the magnitude of V 508 00:28:12,770 --> 00:28:16,000 sub 1, V sub 0 corresponds to this little bit of extra 509 00:28:16,000 --> 00:28:17,960 energy that you have. 510 00:28:17,960 --> 00:28:22,010 So if V sub 0, the magnitude of V sub 0 is positive, is 511 00:28:22,010 --> 00:28:24,790 bigger than the magnitude of V sub 1, you want to 512 00:28:24,790 --> 00:28:26,960 choose H equals 0. 513 00:28:26,960 --> 00:28:30,740 And alternatively you'll want to choose H equals 1. 514 00:28:30,740 --> 00:28:33,250 It's obvious right? 515 00:28:33,250 --> 00:28:37,040 I've tried for years to find a way to prove that. 516 00:28:37,040 --> 00:28:40,280 And the only way I can prove it is by going into Bessel 517 00:28:40,280 --> 00:28:43,340 functions which is the way that everybody else proves it. 518 00:28:43,340 --> 00:28:46,040 And this seems like absolute foolishness to me. 519 00:28:46,040 --> 00:28:48,930 And if any of you can find a way to do this, I would be 520 00:28:48,930 --> 00:28:51,000 delighted to hear it. 521 00:28:51,000 --> 00:28:55,120 I will be in great admiration of you. 522 00:28:55,120 --> 00:28:57,020 Because I'm sure there has to be an easy way to 523 00:28:57,020 --> 00:28:58,430 look at this problem. 524 00:28:58,430 --> 00:29:00,860 And I just can't find it. 525 00:29:00,860 --> 00:29:03,440 OK anyway, we're not going to worry about all these Bessel 526 00:29:03,440 --> 00:29:08,670 functions, because that's just arithmetic in a sense. 527 00:29:08,670 --> 00:29:12,040 So we're just going to say well it can be proven using 528 00:29:12,040 --> 00:29:13,380 all of this machinery. 529 00:29:13,380 --> 00:29:16,610 So what we really want to find is what is the probability of 530 00:29:16,610 --> 00:29:19,420 error when we make that decision. 531 00:29:19,420 --> 00:29:23,180 And when we make that decision, namely what we're 532 00:29:23,180 --> 00:29:26,740 looking for is the probability of this magnitude then, is 533 00:29:26,740 --> 00:29:34,370 bigger than this magnitude when H equals one is the 534 00:29:34,370 --> 00:29:35,420 correct hypothesis. 535 00:29:35,420 --> 00:29:38,440 Because that's the probability of error then. 536 00:29:38,440 --> 00:29:40,130 So you have these two different terms. 537 00:29:40,130 --> 00:29:43,230 You just go through all of the junk that's in the appendix to 538 00:29:43,230 --> 00:29:46,080 the notes we passed out last time. 539 00:29:46,080 --> 00:29:48,080 If you want to go through that, I think it's great. 540 00:29:48,080 --> 00:29:50,510 It's an interesting analysis. 541 00:29:50,510 --> 00:29:52,510 Certainly not going to do it now. 542 00:29:52,510 --> 00:29:55,490 When you get done doing that you find out the probability 543 00:29:55,490 --> 00:30:00,540 of error is exactly one half times e to the minus a-squared 544 00:30:00,540 --> 00:30:04,250 times this known magnitude of the channel. 545 00:30:04,250 --> 00:30:08,670 I mean, a-squared and g tilde have to appear together here. 546 00:30:08,670 --> 00:30:11,860 OK because what's coming out of the channel, the magnitude 547 00:30:11,860 --> 00:30:15,370 of what's coming out of the channel without noise is just 548 00:30:15,370 --> 00:30:17,510 a times g tilde. 549 00:30:17,510 --> 00:30:19,810 They both come together everywhere. 550 00:30:19,810 --> 00:30:24,560 And therefore, they have to come together anytime you're 551 00:30:24,560 --> 00:30:27,860 talking about optimal detection, probability of 552 00:30:27,860 --> 00:30:29,700 error, or anything else. 553 00:30:29,700 --> 00:30:31,030 So these two appear together. 554 00:30:31,030 --> 00:30:34,350 We have the same noise term down here as we had before. 555 00:30:34,350 --> 00:30:37,650 Because again we're using a sampling theorem analysis and 556 00:30:37,650 --> 00:30:42,510 the noise in each of these random variables is W N sub 0. 557 00:30:42,510 --> 00:30:44,630 OK so that's a little surprising that that's what 558 00:30:44,630 --> 00:30:47,270 the noise is. 559 00:30:47,270 --> 00:30:52,130 If you knew the phase also, if the detector knew both the 560 00:30:52,130 --> 00:30:55,200 magnitude and the phase of the channel, it would be the 561 00:30:55,200 --> 00:30:57,520 conventional Gaussian problem that we've 562 00:30:57,520 --> 00:31:00,000 analyzed many times before. 563 00:31:00,000 --> 00:31:04,770 And the solution would be that probability of error is equal 564 00:31:04,770 --> 00:31:08,420 to Q of a-squared times g tilde squared 565 00:31:08,420 --> 00:31:11,750 divided by W N sub 0. 566 00:31:11,750 --> 00:31:13,890 Now if you remember the estimates we've come up with 567 00:31:13,890 --> 00:31:17,350 and the bounds we've come up with on the Q function, the 568 00:31:17,350 --> 00:31:21,940 simplest bound that we came up with was this. 569 00:31:21,940 --> 00:31:25,490 Namely you take this thing, you take one half of it, which 570 00:31:25,490 --> 00:31:28,070 is the Gaussian density with the coefficient. 571 00:31:28,070 --> 00:31:29,970 You multiply it by one half. 572 00:31:29,970 --> 00:31:33,740 So this is the simplest estimate we can get of this. 573 00:31:33,740 --> 00:31:37,280 On the other hand when this quantity is large, a much 574 00:31:37,280 --> 00:31:40,930 better estimate of this is to have that estimate which has a 575 00:31:40,930 --> 00:31:47,030 1 over the square root of pi times W N sub 0 over a-squared 576 00:31:47,030 --> 00:31:48,560 g tilde squared in it. 577 00:31:48,560 --> 00:31:50,600 So we have that term extra. 578 00:31:50,600 --> 00:31:53,520 Which says that when this is large, whenever we're 579 00:31:53,520 --> 00:31:57,430 communicating at all reasonably, this probability 580 00:31:57,430 --> 00:32:03,230 of error is much smaller than this probability of error. 581 00:32:03,230 --> 00:32:06,345 However you talk to any communication engineer, and 582 00:32:06,345 --> 00:32:09,090 they'll say when you have a good signal noise ratio, 583 00:32:09,090 --> 00:32:11,130 incoherent detection is virtually as 584 00:32:11,130 --> 00:32:14,300 good as coherent detection. 585 00:32:14,300 --> 00:32:15,550 And why did they say that? 586 00:32:18,640 --> 00:32:22,920 Well it's because the probability of error goes down 587 00:32:22,920 --> 00:32:24,860 so quickly with energy here. 588 00:32:24,860 --> 00:32:27,330 It's going down as a square of an exponent. 589 00:32:27,330 --> 00:32:31,590 Well it's going down as an exponent in the energy. 590 00:32:31,590 --> 00:32:37,160 The question you want to ask is how much extra energy do I 591 00:32:37,160 --> 00:32:40,120 have to use? 592 00:32:40,120 --> 00:32:43,070 If I'm using coherent detection, how much more 593 00:32:43,070 --> 00:32:47,520 energy does an incoherent detector need at the input in 594 00:32:47,520 --> 00:32:49,930 order to get the same results? 595 00:32:49,930 --> 00:32:52,300 And then you see the question is very different. 596 00:32:52,300 --> 00:32:55,580 Because if I increase this quantity just a little bit, 597 00:32:55,580 --> 00:33:00,230 this probability of error goes down like a bat. 598 00:33:00,230 --> 00:33:01,590 OK. 599 00:33:01,590 --> 00:33:05,100 So what happens then, when you compare these two terms is 600 00:33:05,100 --> 00:33:09,220 that as the signal to noise ratio gets larger and larger, 601 00:33:09,220 --> 00:33:12,100 the amount of extra energy you need to make incoherent 602 00:33:12,100 --> 00:33:16,470 detection work as well as coherent detection goes down 603 00:33:16,470 --> 00:33:17,720 with 1 over a-squared. 604 00:33:21,250 --> 00:33:26,280 Which says that these communication engineers who 605 00:33:26,280 --> 00:33:30,770 swear that they like incoherent detection in fact, 606 00:33:30,770 --> 00:33:33,500 have something on their side. 607 00:33:33,500 --> 00:33:35,380 because they don't have to assume so 608 00:33:35,380 --> 00:33:36,830 much about the channel. 609 00:33:36,830 --> 00:33:39,620 They have something which is more robust. 610 00:33:39,620 --> 00:33:43,560 And in fact what's turning out here, is that even though this 611 00:33:43,560 --> 00:33:47,140 error probability is a little bigger than this error 612 00:33:47,140 --> 00:33:50,640 probability, there's only a very negligible amount of 613 00:33:50,640 --> 00:33:54,270 extra dB required to make the two the same. 614 00:33:54,270 --> 00:33:57,320 So it only costs a little bit of extra energy to be able to 615 00:33:57,320 --> 00:34:01,710 use incoherent detection instead of coherent detection. 616 00:34:01,710 --> 00:34:04,640 OK so this is very strange now. 617 00:34:04,640 --> 00:34:07,810 We have a nice error probability which is almost as 618 00:34:07,810 --> 00:34:11,140 good as the Gaussian error probability 619 00:34:11,140 --> 00:34:13,890 using incoherent detection. 620 00:34:13,890 --> 00:34:18,110 This is assuming that the channel, that the receiver 621 00:34:18,110 --> 00:34:22,440 knows what g tilde is. 622 00:34:22,440 --> 00:34:25,450 But now we go back and think about this, and look at our 623 00:34:25,450 --> 00:34:29,340 detection rule, which is the optimal detection rule. 624 00:34:29,340 --> 00:34:32,950 And the optimal detection rule is no matter what g tilde 625 00:34:32,950 --> 00:34:36,700 happens to be, we compare the magnitude of V sub 0 with the 626 00:34:36,700 --> 00:34:39,640 magnitude of V sub 1. 627 00:34:39,640 --> 00:34:42,020 In other words, we have analyzed this assuming that we 628 00:34:42,020 --> 00:34:43,350 know what g tilde is. 629 00:34:43,350 --> 00:34:45,660 We know what the gain of the channel is. 630 00:34:45,660 --> 00:34:47,900 But the receiver doesn't pay any attention to it. 631 00:34:50,420 --> 00:34:56,590 OK so now we have this very peculiar situation where 632 00:34:56,590 --> 00:35:01,910 incoherent detection with a known value channel is almost 633 00:35:01,910 --> 00:35:06,150 as good as coherent detection is. 634 00:35:06,150 --> 00:35:10,180 But at the same time Rayleigh fading gives this awful error 635 00:35:10,180 --> 00:35:12,800 probability. 636 00:35:12,800 --> 00:35:16,580 So now you have the final part of the argument take this 637 00:35:16,580 --> 00:35:22,270 probability of error, multiply it by the probability density 638 00:35:22,270 --> 00:35:27,690 of g tilde squared, integrate it to find out what the 639 00:35:27,690 --> 00:35:31,090 average error probability is when we average over the 640 00:35:31,090 --> 00:35:32,900 channel fading. 641 00:35:32,900 --> 00:35:34,340 And guess what answer you get? 642 00:35:37,830 --> 00:35:39,690 Well you ought to be able to guess it if you've looked at 643 00:35:39,690 --> 00:35:41,050 the homework already. 644 00:35:41,050 --> 00:35:43,190 Because in the homework you actually go through this 645 00:35:43,190 --> 00:35:44,260 integration. 646 00:35:44,260 --> 00:35:47,170 And bingo you get the Rayleigh fading result. 647 00:35:47,170 --> 00:35:50,510 Which says that the problem with Rayleigh fading is not 648 00:35:50,510 --> 00:35:52,960 any lack of knowledge about the channel. 649 00:35:52,960 --> 00:35:57,580 Knowing what the channel is would not give you, even 650 00:35:57,580 --> 00:36:00,010 knowing what the phase is of the channel would not give you 651 00:36:00,010 --> 00:36:01,020 a lot of extra help. 652 00:36:01,020 --> 00:36:03,790 The only help in knowing what the phase is, is to get this 653 00:36:03,790 --> 00:36:05,680 result instead of this result. 654 00:36:05,680 --> 00:36:07,870 And even that won't help you much. 655 00:36:07,870 --> 00:36:13,070 The problem is anytime you're dealing with Rayleigh fading, 656 00:36:13,070 --> 00:36:17,170 the channel has faded so badly a large fraction of the time, 657 00:36:17,170 --> 00:36:19,110 that you can't get an acceptable 658 00:36:19,110 --> 00:36:22,030 probability of error. 659 00:36:22,030 --> 00:36:23,730 OK so now we have to stop and think. 660 00:36:23,730 --> 00:36:25,080 What do you do about this? 661 00:36:34,190 --> 00:36:37,910 Well you have two general kinds of techniques to use at 662 00:36:37,910 --> 00:36:39,270 this point. 663 00:36:39,270 --> 00:36:42,330 OK and one of them is to try to measure the 664 00:36:42,330 --> 00:36:45,490 channel at the receiver. 665 00:36:45,490 --> 00:36:48,320 You take the measurement of the channel at the receiver. 666 00:36:48,320 --> 00:36:50,530 You send it to the transmitter. 667 00:36:50,530 --> 00:36:54,810 And the transmitter then does something to compensate for 668 00:36:54,810 --> 00:36:56,510 the amount of fading. 669 00:36:56,510 --> 00:37:00,210 One thing that the transmitter can do is anytime the channel 670 00:37:00,210 --> 00:37:03,080 is badly faded, it increases the amount of 671 00:37:03,080 --> 00:37:05,620 power that it's sending. 672 00:37:05,620 --> 00:37:09,220 That's what typical voice systems do. 673 00:37:09,220 --> 00:37:11,700 And the other thing that you can do is change the rate at 674 00:37:11,700 --> 00:37:13,830 which you're transmitting. 675 00:37:13,830 --> 00:37:16,700 You can do all sorts of things with the transmitter if you 676 00:37:16,700 --> 00:37:18,180 know what the channel is. 677 00:37:18,180 --> 00:37:21,300 You can respond to it in various ways. 678 00:37:21,300 --> 00:37:24,310 And all these different communication systems have 679 00:37:24,310 --> 00:37:28,250 various ways of dealing with that. 680 00:37:28,250 --> 00:37:31,780 And we'll talk a little about that on Wednesday when we talk 681 00:37:31,780 --> 00:37:33,030 about CDMA. 682 00:37:35,450 --> 00:37:38,270 The other thing you can do about it is use something 683 00:37:38,270 --> 00:37:40,050 called diversity. 684 00:37:40,050 --> 00:37:46,270 And the idea of diversity is that instead of sending this 685 00:37:46,270 --> 00:37:49,580 one bit, trying to use as few degrees of freedom as 686 00:37:49,580 --> 00:37:54,610 possible, you try to send your bits using as many degrees of 687 00:37:54,610 --> 00:37:56,320 freedom as possible. 688 00:37:56,320 --> 00:37:59,560 If you can use a large number of degrees of freedom, and if 689 00:37:59,560 --> 00:38:02,280 the fading is independent on these different degrees of 690 00:38:02,280 --> 00:38:05,790 freedom, then in fact you gain something. 691 00:38:05,790 --> 00:38:09,490 Because instead of having one random variable which can 692 00:38:09,490 --> 00:38:14,350 totally cripple you, you have lots of random variables. 693 00:38:14,350 --> 00:38:16,670 And if any one of them is good, you get through. 694 00:38:16,670 --> 00:38:20,960 So you get a benefit out of diversity. 695 00:38:20,960 --> 00:38:23,510 OK so that's our next topic. 696 00:38:27,460 --> 00:38:29,860 Namely, how do you measure the channel? 697 00:38:29,860 --> 00:38:33,000 Because if you're going to use diversity it's a help to know 698 00:38:33,000 --> 00:38:34,010 the channel. 699 00:38:34,010 --> 00:38:37,080 If you're going to use coding, coding is just another way to 700 00:38:37,080 --> 00:38:39,290 get diversity. 701 00:38:39,290 --> 00:38:42,180 Again your coding will work better if you know what the 702 00:38:42,180 --> 00:38:43,950 channel is. 703 00:38:43,950 --> 00:38:48,590 So somehow we would like to be able to measure the channel 704 00:38:48,590 --> 00:38:51,480 and send it back to the transmitter if we want to 705 00:38:51,480 --> 00:38:55,830 alter the power, the rate of the transmitter, and to let 706 00:38:55,830 --> 00:39:01,780 the receiver use it if the receiver is going to. 707 00:39:01,780 --> 00:39:06,430 Well we have seen that when you use one bit on just a 708 00:39:06,430 --> 00:39:10,060 couple of degrees of freedom, knowing what the channel is 709 00:39:10,060 --> 00:39:11,780 does not do you much help. 710 00:39:11,780 --> 00:39:15,890 If you use coding or if you use one bit and spread it over 711 00:39:15,890 --> 00:39:18,920 a large number of degrees of freedom, then knowing what the 712 00:39:18,920 --> 00:39:21,740 channel is gives you a great deal. 713 00:39:21,740 --> 00:39:25,240 This is one of the basic confusions that everyone has 714 00:39:25,240 --> 00:39:27,170 when they deal with Rayleigh fading. 715 00:39:27,170 --> 00:39:29,400 Because when you look at a Rayleigh faded channel, the 716 00:39:29,400 --> 00:39:32,580 first thing you analyze is this incredibly small number 717 00:39:32,580 --> 00:39:34,160 of degrees of freedom. 718 00:39:34,160 --> 00:39:35,770 And you say wow, that's awful. 719 00:39:35,770 --> 00:39:39,880 There's no way to deal with that. 720 00:39:39,880 --> 00:39:42,500 And then you start looking for something. 721 00:39:42,500 --> 00:39:45,510 And you say, well diversity helps me. 722 00:39:45,510 --> 00:39:48,140 But in general, this is the general scheme of things that 723 00:39:48,140 --> 00:39:50,110 we're going to use. 724 00:39:50,110 --> 00:39:51,020 OK. 725 00:39:51,020 --> 00:39:54,600 So as we said channel measurement helps if diversity 726 00:39:54,600 --> 00:39:55,820 is available. 727 00:39:55,820 --> 00:39:58,940 Why does that help when diversity is available? 728 00:39:58,940 --> 00:40:02,790 OK, think of sending this one bit. 729 00:40:02,790 --> 00:40:06,000 You get one reception, and then 730 00:40:06,000 --> 00:40:07,970 you get another reception. 731 00:40:07,970 --> 00:40:11,200 And on this reception, you get one amount of fading. 732 00:40:11,200 --> 00:40:15,470 On this reception you get another amount of fading. 733 00:40:15,470 --> 00:40:19,350 If I don't know how much fading there is it doesn't 734 00:40:19,350 --> 00:40:21,010 help me an awful lot. 735 00:40:21,010 --> 00:40:22,260 It helps me some. 736 00:40:22,260 --> 00:40:25,300 But if I know that this channel is faded badly and 737 00:40:25,300 --> 00:40:28,320 this channel is not faded, then I'm going to use what 738 00:40:28,320 --> 00:40:32,310 comes out here instead of what comes out here. 739 00:40:32,310 --> 00:40:35,990 And then my detector is going to work much, much better. 740 00:40:35,990 --> 00:40:40,100 When you look at diversity results, always ask yourself a 741 00:40:40,100 --> 00:40:42,360 couple of questions. 742 00:40:42,360 --> 00:40:45,620 Is the detector using knowledge of what the strength 743 00:40:45,620 --> 00:40:50,420 of the channel is on these two diversity outputs? 744 00:40:50,420 --> 00:40:53,360 Is the transmitter using it's knowledge of what those 745 00:40:53,360 --> 00:40:54,430 channels are? 746 00:40:54,430 --> 00:40:57,640 You get very different results for diversity depending on the 747 00:40:57,640 --> 00:41:02,670 answers to both of those questions. 748 00:41:02,670 --> 00:41:08,660 OK, so if you have a multi-tap model for a channel-- 749 00:41:08,660 --> 00:41:13,000 OK remember the multi-tap models that we came up with. 750 00:41:13,000 --> 00:41:18,970 We we're looking at transmission using multipath. 751 00:41:18,970 --> 00:41:22,470 And we had multipath in different ranges of a delay. 752 00:41:22,470 --> 00:41:29,950 We came up with a model which gave us multiple taps for a 753 00:41:29,950 --> 00:41:32,440 discrete model of the channel. 754 00:41:32,440 --> 00:41:35,560 You get a large number of taps if you're using broadband 755 00:41:35,560 --> 00:41:36,430 communication. 756 00:41:36,430 --> 00:41:40,210 Because using broadband communication 1 over W becomes 757 00:41:40,210 --> 00:41:41,250 very small. 758 00:41:41,250 --> 00:41:45,200 And therefore these ranges of delay become very small. 759 00:41:45,200 --> 00:41:48,030 And if you're using very narrow band communication, 760 00:41:48,030 --> 00:41:50,180 that's when you have the flat fading. 761 00:41:50,180 --> 00:41:54,080 Namely flat fading is not flat fading, it's fading which is 762 00:41:54,080 --> 00:41:56,740 flat over the bandwidth that you're using. 763 00:41:56,740 --> 00:41:59,370 So if you use a broader bandwidth and you have 764 00:41:59,370 --> 00:42:02,580 multiple taps, then these taps are going to be independent of 765 00:42:02,580 --> 00:42:03,580 each other. 766 00:42:03,580 --> 00:42:06,170 And you automatically have diversity. 767 00:42:06,170 --> 00:42:08,520 So the question is how do you use that. 768 00:42:08,520 --> 00:42:10,300 Well if you're going to use it, you better be able to 769 00:42:10,300 --> 00:42:12,270 measure it. 770 00:42:12,270 --> 00:42:16,310 OK so now we're going to try to figure out how to do that 771 00:42:16,310 --> 00:42:17,560 measurement. 772 00:42:20,170 --> 00:42:24,600 And the first thing to do is to assume the simplest 773 00:42:24,600 --> 00:42:26,180 possible thing. 774 00:42:26,180 --> 00:42:29,610 I mean, suppose you know how many taps the channel has. 775 00:42:29,610 --> 00:42:32,850 Suppose it has k sub 0 channel taps. 776 00:42:32,850 --> 00:42:35,400 So the channel looks like this, G sub 0, G sub 777 00:42:35,400 --> 00:42:38,120 1, and G sub 2. 778 00:42:38,120 --> 00:42:42,170 You're transmitting a sequence of inputs. 779 00:42:42,170 --> 00:42:45,960 OK remember all of this stuff came from trying to model a 780 00:42:45,960 --> 00:42:49,430 channel in terms of discrete inputs, where you're sending 781 00:42:49,430 --> 00:42:52,810 one input each one over W seconds. 782 00:42:52,810 --> 00:42:55,270 So you put in a sequence of inputs. 783 00:42:55,270 --> 00:42:58,510 You have these three different channel taps here. 784 00:42:58,510 --> 00:43:05,920 And what comes out when you put in a single bit here or a 785 00:43:05,920 --> 00:43:08,120 single symbol. 786 00:43:08,120 --> 00:43:11,110 You get something out when this tap right away. 787 00:43:11,110 --> 00:43:15,430 You get something out here one time unit later. 788 00:43:15,430 --> 00:43:19,720 You get something out here, one epoch still later. 789 00:43:19,720 --> 00:43:22,870 So all of these outputs get added up. 790 00:43:22,870 --> 00:43:32,290 And therefore the output here at time m, is the input at 791 00:43:32,290 --> 00:43:39,160 time m times this tap, plus the input at time m minus 1 792 00:43:39,160 --> 00:43:42,800 times this tap, plus the input at time m minus 793 00:43:42,800 --> 00:43:44,740 2 times this tap. 794 00:43:44,740 --> 00:43:48,370 Because it takes these inputs that long to go through there. 795 00:43:48,370 --> 00:43:52,360 All this is is just digital convolution, OK. 796 00:43:52,360 --> 00:43:54,370 I'm just drawing it out in the figure so you see 797 00:43:54,370 --> 00:43:55,090 what's going on. 798 00:43:55,090 --> 00:43:57,990 Because otherwise you tend to think everything happens at 799 00:43:57,990 --> 00:43:59,560 one instant of time. 800 00:43:59,560 --> 00:44:03,080 Then we're adding this white Gaussian noise. 801 00:44:03,080 --> 00:44:05,230 When we're talking about digital systems, white 802 00:44:05,230 --> 00:44:08,600 Gaussian noise just means that each of these random variables 803 00:44:08,600 --> 00:44:11,050 are independent at each other random variable. 804 00:44:11,050 --> 00:44:12,760 They all have the same variance. 805 00:44:12,760 --> 00:44:14,900 And the real parts and imaginary parts 806 00:44:14,900 --> 00:44:15,960 have the same variance. 807 00:44:15,960 --> 00:44:17,310 And they're independent of each other. 808 00:44:17,310 --> 00:44:20,660 Namely these are all normal random variables. 809 00:44:20,660 --> 00:44:27,270 Since we're sending a, or minus a, or something with 810 00:44:27,270 --> 00:44:30,960 magnitude a, we want to divide by a out here, if we want to 811 00:44:30,960 --> 00:44:34,940 figure out anything about what these taps are. 812 00:44:34,940 --> 00:44:39,550 OK so suppose that what we send now is a bunch of zeros, 813 00:44:39,550 --> 00:44:43,240 followed by a single input, followed by a bunch of zeros. 814 00:44:43,240 --> 00:44:44,730 What comes out? 815 00:44:44,730 --> 00:44:48,320 Well the thing that comes out is at the point that this big 816 00:44:48,320 --> 00:44:57,630 input comes in, we get a G sub 0 out at the time 817 00:44:57,630 --> 00:44:58,960 that you put in a. 818 00:44:58,960 --> 00:45:01,430 I mean we're leaving out propagation delay here. 819 00:45:01,430 --> 00:45:04,900 We got a times G sub 1, the next epoch. 820 00:45:04,900 --> 00:45:08,370 Then we get a times G sub 2, the next epoch. 821 00:45:08,370 --> 00:45:11,770 And by that time the input is completely out of the filter. 822 00:45:11,770 --> 00:45:14,270 And we get zeros after that. 823 00:45:14,270 --> 00:45:19,410 So if you put in a bunch of zeros and then a single a, you 824 00:45:19,410 --> 00:45:21,550 got a nice measurement of the channel. 825 00:45:21,550 --> 00:45:25,050 There's Gaussian noise added to each of these inputs. 826 00:45:25,050 --> 00:45:28,710 But in fact you do get a reading of 827 00:45:28,710 --> 00:45:29,960 each channel output. 828 00:45:29,960 --> 00:45:34,980 When you divide these by the a here, then you get something 829 00:45:34,980 --> 00:45:42,410 which is a measurement of the appropriate tap G plus 830 00:45:42,410 --> 00:45:44,690 Gaussian noise on it. 831 00:45:44,690 --> 00:45:47,930 OK now you try to make an estimation from this. 832 00:45:47,930 --> 00:45:50,040 And the trouble is we don't want to say much about 833 00:45:50,040 --> 00:45:52,020 estimation theory. 834 00:45:52,020 --> 00:45:55,430 But in fact the notes gives you a very brief introduction 835 00:45:55,430 --> 00:45:57,690 into estimation. 836 00:45:57,690 --> 00:46:00,080 There are two well-known kinds of estimation. 837 00:46:00,080 --> 00:46:03,550 One of them is maximum likelihood estimation. 838 00:46:03,550 --> 00:46:07,530 And the other one is minimum mean square error estimation. 839 00:46:07,530 --> 00:46:11,510 Maximum likelihood estimation is in fact exactly the same as 840 00:46:11,510 --> 00:46:13,630 maximum likelihood detection. 841 00:46:13,630 --> 00:46:16,000 Namely you look at the likelihoods which is the 842 00:46:16,000 --> 00:46:21,910 probabilities of the outputs given the inputs. 843 00:46:21,910 --> 00:46:23,630 And what's the input in this problem? 844 00:46:27,430 --> 00:46:30,140 The input is these channel variables. 845 00:46:30,140 --> 00:46:33,840 Because that's the thing we're trying to measure in this 846 00:46:33,840 --> 00:46:34,860 measurement problem. 847 00:46:34,860 --> 00:46:37,310 We assume that the probing signal is known. 848 00:46:37,310 --> 00:46:40,035 It's just a bunch of zeros, followed by a, followed by a 849 00:46:40,035 --> 00:46:41,100 bunch of zeros. 850 00:46:41,100 --> 00:46:42,220 So we know that. 851 00:46:42,220 --> 00:46:45,070 We're trying to estimate these things. 852 00:46:45,070 --> 00:46:48,020 So these are the variables that we're trying to estimate. 853 00:46:48,020 --> 00:46:50,880 So we try to find the probability density of the 854 00:46:50,880 --> 00:46:54,810 output conditional on the knowledge of G sub 0. 855 00:46:54,810 --> 00:46:58,560 Which is just the Gaussian density shifted to 856 00:46:58,560 --> 00:47:00,380 a times G sub 0. 857 00:47:04,130 --> 00:47:11,580 You then look at the maximum likelihood estimate of G. So 858 00:47:11,580 --> 00:47:17,120 you're looking at the value you can put in to maximize 859 00:47:17,120 --> 00:47:25,560 this estimate which comes out here as a times G sub 2. 860 00:47:25,560 --> 00:47:29,020 And then at this appropriate time, you're looking at G sub 861 00:47:29,020 --> 00:47:32,690 2 here, plus a noise random variable. 862 00:47:32,690 --> 00:47:36,160 And since the noise is zero mean, this quantity here is in 863 00:47:36,160 --> 00:47:39,570 fact the best estimate in terms of the maximum 864 00:47:39,570 --> 00:47:41,260 likelihood that you can get. 865 00:47:41,260 --> 00:47:44,395 If you assume that this is a Gaussian random variable and 866 00:47:44,395 --> 00:47:47,520 this is a Gaussian random variable, you can solve a 867 00:47:47,520 --> 00:47:50,310 minimum mean square error estimation problem. 868 00:47:50,310 --> 00:47:55,210 It's much like the map problem except these random variables 869 00:47:55,210 --> 00:47:57,770 are all continuous here. 870 00:47:57,770 --> 00:48:01,250 But it's a little different from the map problem in the 871 00:48:01,250 --> 00:48:05,400 sense that you can't have equally likely inputs where 872 00:48:05,400 --> 00:48:07,570 you have a continuous random variable. 873 00:48:07,570 --> 00:48:09,700 You make them all equally probable. 874 00:48:09,700 --> 00:48:11,990 The only possible value you can have is zero. 875 00:48:11,990 --> 00:48:13,840 Because it has to extend forever. 876 00:48:13,840 --> 00:48:19,040 So anyway, maximum likelihood detection is just normalize 877 00:48:19,040 --> 00:48:22,820 what you get so that in the absence of Gaussian noise, you 878 00:48:22,820 --> 00:48:25,120 would get the variable you're looking for. 879 00:48:25,120 --> 00:48:26,720 And then ignore the Gaussian noise, 880 00:48:26,720 --> 00:48:28,470 and that's your estimate. 881 00:48:28,470 --> 00:48:29,050 OK. 882 00:48:29,050 --> 00:48:32,630 If you want to do this and you want to use the strategy, it 883 00:48:32,630 --> 00:48:34,860 looks like a very nice strategy. 884 00:48:34,860 --> 00:48:38,350 But what's the problem in it? 885 00:48:38,350 --> 00:48:41,940 If this sequence is somewhat longer, you need a whole lot 886 00:48:41,940 --> 00:48:45,200 of zeros in between each probing signal. 887 00:48:45,200 --> 00:48:48,590 And what that means is you're going to be using your energy 888 00:48:48,590 --> 00:48:52,550 and clumping it all up into the small number 889 00:48:52,550 --> 00:48:53,920 of degrees of freedom. 890 00:48:53,920 --> 00:48:57,590 Which means you're going to be sending a lot of energy at one 891 00:48:57,590 --> 00:48:59,990 instant of time and then nothing for a long period of 892 00:48:59,990 --> 00:49:03,100 time, then a very big signal for awhile then nothing for a 893 00:49:03,100 --> 00:49:06,630 long period of time, and so forth. 894 00:49:06,630 --> 00:49:11,010 If you do that, the FTC is really going 895 00:49:11,010 --> 00:49:12,610 to be down on you. 896 00:49:12,610 --> 00:49:15,970 Because you're not supposed to send too much energy in any 897 00:49:15,970 --> 00:49:19,570 small amount of time or any small amount of frequency. 898 00:49:19,570 --> 00:49:21,950 So you're supposed to spread things out a little bit. 899 00:49:21,950 --> 00:49:24,470 You say OK, that doesn't work too well. 900 00:49:24,470 --> 00:49:25,960 What am I going to do? 901 00:49:25,960 --> 00:49:30,100 How can I choose a sequence of input so they have relatively 902 00:49:30,100 --> 00:49:34,850 constant amplitude, but at the same time so that when I go 903 00:49:34,850 --> 00:49:38,700 through this kind of filter, I can sort out what's coming 904 00:49:38,700 --> 00:49:41,290 from here, and what's coming from here, and 905 00:49:41,290 --> 00:49:43,480 what's coming from here. 906 00:49:43,480 --> 00:49:46,020 Well it turns out that the answer to that question is to 907 00:49:46,020 --> 00:49:48,880 use a pseudonoise sequence. 908 00:49:48,880 --> 00:49:52,540 And the next thing I want to do is to give you some idea 909 00:49:52,540 --> 00:49:55,530 about why these pseudonoise sequences work. 910 00:49:58,310 --> 00:50:00,410 OK so we'll think in terms of vectors now. 911 00:50:04,520 --> 00:50:09,240 OK so we have a vector input, u sub 1, u sub 2, up to u sub 912 00:50:09,240 --> 00:50:11,210 n, a vector of length n. 913 00:50:11,210 --> 00:50:15,670 So we're putting these discrete signals in 914 00:50:15,670 --> 00:50:17,610 one after the other. 915 00:50:17,610 --> 00:50:19,960 We're passing them through this, which is a digital 916 00:50:19,960 --> 00:50:21,140 filter now. 917 00:50:21,140 --> 00:50:26,350 So what comes out here V prime is just a convolution of u and 918 00:50:26,350 --> 00:50:30,370 G. We then add the noise to it. 919 00:50:30,370 --> 00:50:32,670 I claim that what we ought to do is use the matched 920 00:50:32,670 --> 00:50:35,140 filter here to u. 921 00:50:35,140 --> 00:50:39,950 And if I use a matched filter to u here, that matched 922 00:50:39,950 --> 00:50:43,830 filter, if I'm using a pseudonoise sequence, is going 923 00:50:43,830 --> 00:50:48,820 to bingo give me the filter that I started out with, plus 924 00:50:48,820 --> 00:50:49,910 some noise. 925 00:50:49,910 --> 00:50:51,780 OK why is that? 926 00:50:51,780 --> 00:50:55,480 The property that pseudonoise sequences have, if I choose 927 00:50:55,480 --> 00:51:01,070 each of the inputs to have the magnitude of a, and think of 928 00:51:01,070 --> 00:51:04,150 it as being real plus or minus a, which is what people 929 00:51:04,150 --> 00:51:05,620 usually do. 930 00:51:05,620 --> 00:51:09,090 If you look at the correlation of this sequence, namely the 931 00:51:09,090 --> 00:51:14,560 correlation of u sub m with the complex conjugate of u sub 932 00:51:14,560 --> 00:51:20,540 m spaced a little bit, PN sequences have the property 933 00:51:20,540 --> 00:51:24,720 that this correlation function looks like an impulse. 934 00:51:24,720 --> 00:51:25,610 OK. 935 00:51:25,610 --> 00:51:29,290 Now how you find sequences that have that property is 936 00:51:29,290 --> 00:51:31,060 another question. 937 00:51:31,060 --> 00:51:32,630 But in fact they do exist. 938 00:51:32,630 --> 00:51:34,020 There are lots of them. 939 00:51:34,020 --> 00:51:35,570 They're easy to find. 940 00:51:38,730 --> 00:51:41,910 And they have this very nice property. 941 00:51:41,910 --> 00:51:46,550 Another way to say this is that is that the vector u has 942 00:51:46,550 --> 00:51:49,260 to be orthogonal to all of its shifts. 943 00:51:49,260 --> 00:51:51,800 That's exactly what this is saying. 944 00:51:51,800 --> 00:51:55,850 And another way of saying it is that u, if you pass it 945 00:51:55,850 --> 00:51:58,440 through the matched filter to u-- now remember what a 946 00:51:58,440 --> 00:52:02,440 matched filter is on an analog waveform. 947 00:52:02,440 --> 00:52:06,850 You take a waveform, you switch it around in time. 948 00:52:06,850 --> 00:52:09,200 You take the complex conjugate of it, and 949 00:52:09,200 --> 00:52:10,900 that's the matched filter. 950 00:52:10,900 --> 00:52:14,200 And when you convolve u with this matched filter, what it's 951 00:52:14,200 --> 00:52:18,490 doing is just exactly the same operation of correlation. 952 00:52:18,490 --> 00:52:22,110 OK in other words, convolution with one of the sequences 953 00:52:22,110 --> 00:52:26,380 turned around this time, is the same as correlation. 954 00:52:26,380 --> 00:52:29,460 And most of you have seen that I'm sure. 955 00:52:29,460 --> 00:52:34,070 So that if we take this matched filter where u tilde 956 00:52:34,070 --> 00:52:38,180 sub j is equal to the complex conjugate of u at time minus 957 00:52:38,180 --> 00:52:43,940 j, then I pass u through the filter G. Forget about the 958 00:52:43,940 --> 00:52:45,480 noise for the time being. 959 00:52:45,480 --> 00:52:49,900 I then pass it through the matched filter u tilde. 960 00:52:49,900 --> 00:52:53,870 What I'm going to get out, I claim, is G. And I'll show you 961 00:52:53,870 --> 00:52:56,270 why that is in just a minute. 962 00:52:56,270 --> 00:52:59,120 Let make caution you about something. 963 00:52:59,120 --> 00:53:01,970 Because you can get very confused with this picture. 964 00:53:01,970 --> 00:53:07,180 Because as soon as I take this input, u 1 up to u sub m. 965 00:53:07,180 --> 00:53:11,610 This matched filter is going to start responding at time u 966 00:53:11,610 --> 00:53:12,740 sub minus m. 967 00:53:12,740 --> 00:53:14,650 And it's going to finish responding a 968 00:53:14,650 --> 00:53:16,750 time u sub minus 1. 969 00:53:16,750 --> 00:53:19,580 So it responds before it's hit. 970 00:53:19,580 --> 00:53:23,310 Which again is this business of thinking of timing at the 971 00:53:23,310 --> 00:53:27,030 receiver being very much delayed from timing at the 972 00:53:27,030 --> 00:53:29,020 transmitter. 973 00:53:29,020 --> 00:53:30,920 Which is a trick that we've always played. 974 00:53:30,920 --> 00:53:33,710 Which is why we don't have to think of filters as being 975 00:53:33,710 --> 00:53:34,880 realizable. 976 00:53:34,880 --> 00:53:37,820 Still in this example, this becomes confusing. 977 00:53:37,820 --> 00:53:39,070 And I'll show you why in a minute. 978 00:53:44,280 --> 00:53:49,100 OK so I'm going to assume that I picked a good PN sequence. 979 00:53:49,100 --> 00:53:53,150 So when I convolve it with its matched filter I essentially 980 00:53:53,150 --> 00:53:58,170 get an impulse function, namely a discrete impulse. 981 00:53:58,170 --> 00:54:02,130 Which is the same as saying that u is orthogonal to all of 982 00:54:02,130 --> 00:54:03,370 its shifts. 983 00:54:03,370 --> 00:54:04,900 And that's exactly what you what to do. 984 00:54:04,900 --> 00:54:06,320 You want to think of turning it around 985 00:54:06,320 --> 00:54:07,490 and passing it through. 986 00:54:07,490 --> 00:54:10,660 And that's exactly what this is doing. 987 00:54:10,660 --> 00:54:16,880 OK so we have the output of this filter, which is u 988 00:54:16,880 --> 00:54:21,660 convolved with G. We're then convolving that with this 989 00:54:21,660 --> 00:54:24,170 matched filter u tilde. 990 00:54:24,170 --> 00:54:29,860 And now we use the nice property of convolution, which 991 00:54:29,860 --> 00:54:32,630 you probably don't think of very often. 992 00:54:32,630 --> 00:54:38,860 But the nice property that convolution has, is that it's 993 00:54:38,860 --> 00:54:41,670 both associative and commutative. 994 00:54:41,670 --> 00:54:42,180 OK. 995 00:54:42,180 --> 00:54:47,010 And therefore when we look at V prime times u tilde, it's 996 00:54:47,010 --> 00:54:51,510 the convolution of u with G-- that's what the prime is-- all 997 00:54:51,510 --> 00:54:54,450 convolved with the matched filter u. 998 00:54:54,450 --> 00:54:57,600 Because of the associativity and the commutativity, you can 999 00:54:57,600 --> 00:55:00,760 reverse these two things so you're taking the convolution 1000 00:55:00,760 --> 00:55:03,080 of u with its matched filter. 1001 00:55:03,080 --> 00:55:05,870 When you take the convolution of u with its matched filter, 1002 00:55:05,870 --> 00:55:07,900 you get a delta function. 1003 00:55:07,900 --> 00:55:11,350 And you take a delta function and pass it through G. And 1004 00:55:11,350 --> 00:55:15,570 what comes out is a delta function weighted by a-squared 1005 00:55:15,570 --> 00:55:19,240 n, which is just the energy of what we're putting in. 1006 00:55:19,240 --> 00:55:23,050 OK so that says that if we can find pseudonoise sequences, 1007 00:55:23,050 --> 00:55:23,940 all of this works. 1008 00:55:23,940 --> 00:55:26,400 And it works just dandy. 1009 00:55:26,400 --> 00:55:29,340 If you put noise in, what happens there? 1010 00:55:29,340 --> 00:55:31,860 Well let's analyze the noise separately. 1011 00:55:31,860 --> 00:55:35,890 The noise is going through this matched filter. 1012 00:55:35,890 --> 00:55:41,030 Well if u is a pseudonoise sequence, if it has this nice 1013 00:55:41,030 --> 00:55:45,300 correlation property and you flip it around in time, it's 1014 00:55:45,300 --> 00:55:48,190 going to have the same nice correlation property. 1015 00:55:48,190 --> 00:55:57,020 So that in fact u tilde is going to have the same 1016 00:55:57,020 --> 00:56:01,240 property that it's orthogonal to all of its time shifts. 1017 00:56:01,240 --> 00:56:05,490 If you now look at what happens when you take Z and 1018 00:56:05,490 --> 00:56:10,290 send it through this filter, and you find the covariance 1019 00:56:10,290 --> 00:56:14,270 matrix for Z passed through this filter, what that 1020 00:56:14,270 --> 00:56:18,760 independence gives you is the correlation function is just 1021 00:56:18,760 --> 00:56:22,510 all diagonal, all terms, all the same. 1022 00:56:22,510 --> 00:56:27,770 Which says that all of these terms and this vector here are 1023 00:56:27,770 --> 00:56:29,770 all white Gaussian noise variables. 1024 00:56:29,770 --> 00:56:34,210 So what comes out is the filter plus white noise. 1025 00:56:34,210 --> 00:56:36,710 Which is the same thing that happened when we put in a 1026 00:56:36,710 --> 00:56:40,320 single input with zeros on both sides. 1027 00:56:40,320 --> 00:56:42,500 OK. 1028 00:56:42,500 --> 00:56:47,910 So using a PN sequence works in exactly the same way as 1029 00:56:47,910 --> 00:56:51,380 this very special pseudonoise sequence, which just has one 1030 00:56:51,380 --> 00:56:52,200 input in it. 1031 00:56:52,200 --> 00:56:54,960 Which happens to be a pseudonoise sequence 1032 00:56:54,960 --> 00:56:57,460 in this term also. 1033 00:56:57,460 --> 00:57:00,300 OK so the output then, is going to be a maximum 1034 00:57:00,300 --> 00:57:04,000 likelihood estimate of G. OK, this is the way that people 1035 00:57:04,000 --> 00:57:06,420 typically measure channels. 1036 00:57:06,420 --> 00:57:08,970 They use pseudonoise inputs. 1037 00:57:08,970 --> 00:57:11,890 And the output that comes out, namely the 1038 00:57:11,890 --> 00:57:14,360 output that comes out. 1039 00:57:14,360 --> 00:57:18,510 When we put in a finite duration pseudonoise sequence, 1040 00:57:18,510 --> 00:57:22,860 what we're going to look for is the output at the exact 1041 00:57:22,860 --> 00:57:26,580 instant of the last digit as the input goes in. 1042 00:57:26,580 --> 00:57:30,060 And the output then is G sub 0, followed by G sub 1, 1043 00:57:30,060 --> 00:57:33,600 followed by G sub 2, and then silence. 1044 00:57:33,600 --> 00:57:36,580 So you see nothing coming out until this big burst of 1045 00:57:36,580 --> 00:57:44,760 energy, which is all digits of G. 1046 00:57:44,760 --> 00:57:47,970 OK so now we want to put all of this together into 1047 00:57:47,970 --> 00:57:50,070 something called a rake receiver. 1048 00:57:50,070 --> 00:57:52,820 I wish I could spend more time on the rake receiver because 1049 00:57:52,820 --> 00:57:55,110 it's a really neat thing. 1050 00:57:55,110 --> 00:57:59,630 It was developed in the 50s about the same time that 1051 00:57:59,630 --> 00:58:01,570 information theory was getting developed. 1052 00:58:01,570 --> 00:58:06,770 But it was developed by people who were trying to do radar. 1053 00:58:06,770 --> 00:58:09,110 And at the same time trying to do a little communication 1054 00:58:09,110 --> 00:58:11,230 along with the radar. 1055 00:58:11,230 --> 00:58:14,170 And this was one of the things they came up with. 1056 00:58:14,170 --> 00:58:19,000 So they wanted to measure the channel and make decisions in 1057 00:58:19,000 --> 00:58:21,870 transmitting data both at the same time. 1058 00:58:21,870 --> 00:58:26,340 And the trick here is about the same as the trick we use 1059 00:58:26,340 --> 00:58:30,700 in trying to measure carrier frequency, and make decisions 1060 00:58:30,700 --> 00:58:31,930 at the same time. 1061 00:58:31,930 --> 00:58:34,480 Namely you use the decisions you make 1062 00:58:34,480 --> 00:58:36,000 to measure the frequency. 1063 00:58:36,000 --> 00:58:38,190 You use the frequency that you've measured 1064 00:58:38,190 --> 00:58:40,670 to make future decisions. 1065 00:58:40,670 --> 00:58:43,510 And here, we're going to do exactly the same thing. 1066 00:58:43,510 --> 00:58:44,770 We make decisions. 1067 00:58:44,770 --> 00:58:48,680 We use those decisions as a way of measuring the channel. 1068 00:58:48,680 --> 00:58:51,830 We then use the measurements of the channel to create this 1069 00:58:51,830 --> 00:58:55,210 matched filter G tilde. 1070 00:58:55,210 --> 00:58:58,940 And that's what we're going to use to make the decisions. 1071 00:58:58,940 --> 00:59:03,860 OK if you have two different inputs, I mean here we'll just 1072 00:59:03,860 --> 00:59:05,270 look at binary inputs. 1073 00:59:05,270 --> 00:59:09,370 You take u sub 0 and u sub 1, and you look at what happens 1074 00:59:09,370 --> 00:59:11,070 when you have those two inputs. 1075 00:59:11,070 --> 00:59:14,830 This is just a vector white Gaussian noise problem that we 1076 00:59:14,830 --> 00:59:17,570 looked at in quite a bit of detail when we were studying 1077 00:59:17,570 --> 00:59:19,580 decision theory. 1078 00:59:19,580 --> 00:59:24,460 What we want to do is to look at, I mean if these two 1079 00:59:24,460 --> 00:59:28,540 signals are not antipodal to each other you want to look at 1080 00:59:28,540 --> 00:59:30,650 the mean of them. 1081 00:59:30,650 --> 00:59:34,065 And you'll want to look at u sub 0 minus that mean, and u 1082 00:59:34,065 --> 00:59:37,280 sub 1 minus that mean as two antipodal signals. 1083 00:59:37,280 --> 00:59:41,160 When you go through all of that, you find that the 1084 00:59:41,160 --> 00:59:45,020 maximum likelihood decision is to take the real part of the 1085 00:59:45,020 --> 00:59:53,490 output, of the inner product of the output, with u sub 0 1086 00:59:53,490 --> 00:59:58,026 convolved with g, and the real part of v convolved with u sub 1087 00:59:58,026 --> 00:59:59,510 1 convolved with G. 1088 00:59:59,510 --> 01:00:02,520 OK in other words, what's happening here is that as far 1089 01:00:02,520 --> 01:00:07,340 as anybody is concerned, we're not using u sub 0 and u sub 1 1090 01:00:07,340 --> 01:00:09,330 in this making a decision. 1091 01:00:09,330 --> 01:00:11,420 We know what the channel is. 1092 01:00:11,420 --> 01:00:15,425 And therefore what exists right before the white noise 1093 01:00:15,425 --> 01:00:20,485 is added, is these two signals u sub 0 convolved with g and u 1094 01:00:20,485 --> 01:00:22,170 sub 1 convolved with g. 1095 01:00:22,170 --> 01:00:24,480 So we're doing binary detection on 1096 01:00:24,480 --> 01:00:26,750 those two known signals. 1097 01:00:26,750 --> 01:00:29,610 And we're using the output to try to make the best choice 1098 01:00:29,610 --> 01:00:30,500 between them. 1099 01:00:30,500 --> 01:00:32,670 So this is the thing that we do. 1100 01:00:32,670 --> 01:00:36,300 So we want to use a filter matched to u sub 0 1101 01:00:36,300 --> 01:00:38,040 convolved with g. 1102 01:00:38,040 --> 01:00:41,220 Now how do we build a filter matched to a convolution of 1103 01:00:41,220 --> 01:00:43,140 two things? 1104 01:00:43,140 --> 01:00:45,880 Well we convolve u sub 0 with g. 1105 01:00:45,880 --> 01:00:48,080 And then we turn the thing around. 1106 01:00:48,080 --> 01:00:50,340 And then we see that after turning it around what we've 1107 01:00:50,340 --> 01:00:53,890 gotten is the turned around version of u convolved with 1108 01:00:53,890 --> 01:00:56,460 the turned around version of g. 1109 01:00:56,460 --> 01:00:57,770 I mean write it down and you'll see that 1110 01:00:57,770 --> 01:00:59,850 that's what you have. 1111 01:00:59,850 --> 01:01:08,120 So what you wind up with is the following figure. 1112 01:01:08,120 --> 01:01:11,860 You either send u sub 0 or you send u sub 1. 1113 01:01:11,860 --> 01:01:14,450 This is a way to send one binary digit. 1114 01:01:14,450 --> 01:01:20,060 We're sending it by using these long PN sequences now. 1115 01:01:20,060 --> 01:01:28,190 If u sub 0 goes through g, we got a V prime out, which is 1116 01:01:28,190 --> 01:01:30,880 the output before noise is added. 1117 01:01:30,880 --> 01:01:34,980 We then add noise so we get. 1118 01:01:34,980 --> 01:01:38,610 And then we pass to try to detect whether 1119 01:01:38,610 --> 01:01:40,530 this or this is true. 1120 01:01:40,530 --> 01:01:48,000 We take this output V. We convolve it with u sub 1 1121 01:01:48,000 --> 01:01:52,060 convolved with g, and with u sub 0 convolved with g. 1122 01:01:52,060 --> 01:01:55,380 Now you'll all say I'm wasting stuff here. 1123 01:01:55,380 --> 01:01:59,490 Because I could just put the g over here and then follow it 1124 01:01:59,490 --> 01:02:02,850 with u sub 1 or u sub 0. 1125 01:02:02,850 --> 01:02:04,090 Be patient for a little bit. 1126 01:02:04,090 --> 01:02:05,390 I want to put both of them in. 1127 01:02:05,390 --> 01:02:07,280 And I want to put them in this order. 1128 01:02:07,280 --> 01:02:10,580 And then I make a decision here. 1129 01:02:10,580 --> 01:02:14,520 OK well here comes the clincher to the argument. 1130 01:02:14,520 --> 01:02:17,500 Look at what happens right there. 1131 01:02:17,500 --> 01:02:24,120 If I forget about this and I forget about this, what I get 1132 01:02:24,120 --> 01:02:26,440 here is u sub 0 coming in. 1133 01:02:26,440 --> 01:02:28,900 It's going through the filter g. 1134 01:02:28,900 --> 01:02:31,510 It has white noise added to it. 1135 01:02:31,510 --> 01:02:35,520 It goes through the matched filter to u sub 0. 1136 01:02:35,520 --> 01:02:38,770 And what comes out is a measurement of g. 1137 01:02:38,770 --> 01:02:39,650 That's what we showed before. 1138 01:02:39,650 --> 01:02:42,740 When we were trying to measure g, that was the way we did it. 1139 01:02:42,740 --> 01:02:44,970 We started out with a PN sequence, go 1140 01:02:44,970 --> 01:02:47,240 through g, add noise-- 1141 01:02:47,240 --> 01:02:49,970 we can't avoid the noise-- go through the matched filter. 1142 01:02:49,970 --> 01:02:53,260 That is a measurement of g at that point there. 1143 01:02:53,260 --> 01:02:57,540 And if we send u sub 1, that's a measurement of g at that 1144 01:02:57,540 --> 01:02:59,050 point there. 1145 01:02:59,050 --> 01:03:06,790 So finally we have the rake receiver which does both of 1146 01:03:06,790 --> 01:03:08,830 these things it once. 1147 01:03:08,830 --> 01:03:11,670 You either send u sub 0 or u sub 1. 1148 01:03:11,670 --> 01:03:12,810 You go through this filter. 1149 01:03:12,810 --> 01:03:14,510 You add white noise. 1150 01:03:17,240 --> 01:03:20,390 As far as making a decision is concerned, you do what we 1151 01:03:20,390 --> 01:03:22,310 talked about before. 1152 01:03:22,310 --> 01:03:25,070 You compare this output with this 1153 01:03:25,070 --> 01:03:27,360 output to make a decision. 1154 01:03:27,360 --> 01:03:31,270 After you make a decision you go forward in time, because 1155 01:03:31,270 --> 01:03:34,500 we've done everything backwards in time here. 1156 01:03:34,500 --> 01:03:39,290 And you take what is going to come out of here, which hasn't 1157 01:03:39,290 --> 01:03:40,460 come out yet. 1158 01:03:40,460 --> 01:03:44,550 And you use that to make a new estimate of g. 1159 01:03:44,550 --> 01:03:48,180 You use that estimate of g turned around in time, to 1160 01:03:48,180 --> 01:03:54,260 alter your estimate of the matched filter to g. 1161 01:03:54,260 --> 01:03:57,090 And if you read the notes, the notes explains what's going on 1162 01:03:57,090 --> 01:03:59,230 as far as the timing in here, a little bit 1163 01:03:59,230 --> 01:04:00,940 better than I can here. 1164 01:04:00,940 --> 01:04:02,970 But in fact, this is the kind of circuit that people 1165 01:04:02,970 --> 01:04:07,060 actually use to both measure channels, and to send data at 1166 01:04:07,060 --> 01:04:08,140 the same time. 1167 01:04:08,140 --> 01:04:11,400 I want stop here because we're supposed to 1168 01:04:11,400 --> 01:04:13,300 evaluate the class.