1 00:00:00,000 --> 00:00:02,210 NARRATOR: The following content is provided under a 2 00:00:02,210 --> 00:00:03,640 Creative Commons license. 3 00:00:03,640 --> 00:00:06,540 Your support will help MIT OpenCourseWare continue to 4 00:00:06,540 --> 00:00:09,970 offer high quality educational resources for free. 5 00:00:09,970 --> 00:00:12,810 To make a donation or to view additional materials from 6 00:00:12,810 --> 00:00:16,830 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:16,830 --> 00:00:19,260 ocw.mit.edu. 8 00:00:19,260 --> 00:00:24,870 PROFESSOR: Today, I want to spend a fair amount of time, 9 00:00:24,870 --> 00:00:28,750 to start out with, talking about this business of, how do 10 00:00:28,750 --> 00:00:33,300 you go from pass band to a baseband in a wireless system 11 00:00:33,300 --> 00:00:39,970 and some of these questions about time reference and all 12 00:00:39,970 --> 00:00:40,730 of these things. 13 00:00:40,730 --> 00:00:45,400 When we were dealing with ordinary channels that didn't 14 00:00:45,400 --> 00:00:49,190 have fading in them and didn't have motion or anything like 15 00:00:49,190 --> 00:00:52,370 that, it was a fairly straightforward thing to think 16 00:00:52,370 --> 00:00:57,000 of a transmitter having its time reference and the 17 00:00:57,000 --> 00:01:01,740 receiver locking in on its time reference, which had a 18 00:01:01,740 --> 00:01:03,310 certain amount of delay from what the 19 00:01:03,310 --> 00:01:05,610 transmitter was doing. 20 00:01:05,610 --> 00:01:08,390 As soon as you start having multiple paths, each with 21 00:01:08,390 --> 00:01:12,610 different delays in them, this starts to get confusing. 22 00:01:12,610 --> 00:01:15,060 If you try to write it down carefully, 23 00:01:15,060 --> 00:01:18,090 it gets doubly confusing. 24 00:01:18,090 --> 00:01:21,760 If these path lanes are changing dynamically, it gets 25 00:01:21,760 --> 00:01:23,620 even more confusing. 26 00:01:23,620 --> 00:01:26,170 So I wanted to spend a little bit of time at the beginning 27 00:01:26,170 --> 00:01:29,560 of the hour trying to sort that out. 28 00:01:29,560 --> 00:01:33,040 It's not sorted out that well in the notes. 29 00:01:33,040 --> 00:01:36,050 As you'll see, it doesn't get sorted out too well here 30 00:01:36,050 --> 00:01:40,600 either because the problem is, whenever you write everything 31 00:01:40,600 --> 00:01:42,990 down, it becomes a nightmare. 32 00:01:42,990 --> 00:01:49,040 So this is just another effort at trying to do this. 33 00:01:49,040 --> 00:01:52,420 Let's start out by just looking at a single path. 34 00:01:52,420 --> 00:01:55,880 This single path, we're going to assume, has a time varying 35 00:01:55,880 --> 00:02:00,990 attenuation, which we'll call beta of t and a time varying 36 00:02:00,990 --> 00:02:03,590 delay, which we'll call tau of t. 37 00:02:03,590 --> 00:02:06,290 Namely, this is the same thing that we did before. 38 00:02:06,290 --> 00:02:08,460 We started out with a fixed 39 00:02:08,460 --> 00:02:11,190 transmitter and a fixed receiver. 40 00:02:11,190 --> 00:02:15,810 Now, since we're allowing both the attenuation and the delay 41 00:02:15,810 --> 00:02:19,510 to vary, but only having one path. 42 00:02:19,510 --> 00:02:22,760 What we're essentially thinking is, that is the 43 00:02:22,760 --> 00:02:26,490 receiver, which is in a vehicle for example, which is 44 00:02:26,490 --> 00:02:31,680 moving away from or towards the transmitter. 45 00:02:31,680 --> 00:02:36,480 So these are the, or in a sense an easy way of keeping 46 00:02:36,480 --> 00:02:40,740 track of that physical situation. 47 00:02:40,740 --> 00:02:46,490 So the response at passband to a real waveform, x of t, it's 48 00:02:46,490 --> 00:02:52,600 going to be just beta sub t times x of t minus tau of t. 49 00:02:52,600 --> 00:02:55,020 In other words, here's the delay here. 50 00:02:55,020 --> 00:02:57,040 Here's the attenuation here and we just 51 00:02:57,040 --> 00:02:58,290 have a single path. 52 00:03:00,770 --> 00:03:03,390 Now if you start out with a baseband waveform, if you 53 00:03:03,390 --> 00:03:07,680 start out with a complex baseband waveform, say u of t, 54 00:03:07,680 --> 00:03:12,230 you're going to shift this up first to a positive frequency 55 00:03:12,230 --> 00:03:17,040 band, which we'll call u sub p of t, which is u of t e to the 56 00:03:17,040 --> 00:03:22,950 2 pi ifct and you're going to transmit it as x of t, which 57 00:03:22,950 --> 00:03:27,410 is equal to two times a real part of that. 58 00:03:27,410 --> 00:03:29,900 Half the time we've been dealing with going through 59 00:03:29,900 --> 00:03:34,340 this modulation from baseband to passband in terms of 60 00:03:34,340 --> 00:03:36,160 cosines and sines. 61 00:03:36,160 --> 00:03:39,210 Half the time we've been doing it in terms of complex 62 00:03:39,210 --> 00:03:40,980 exponentials. 63 00:03:40,980 --> 00:03:43,780 One of the problems that appears as soon as you get 64 00:03:43,780 --> 00:03:48,320 into wireless is that if you really like to do things in 65 00:03:48,320 --> 00:03:52,120 terms of cosines and sines, you really run into a lot of 66 00:03:52,120 --> 00:03:57,460 trouble here because with phases varying all over the 67 00:03:57,460 --> 00:04:02,890 place, it just becomes very, very difficult to track what's 68 00:04:02,890 --> 00:04:07,280 going on with all of the cosine terms and all of the 69 00:04:07,280 --> 00:04:08,520 sine terms. 70 00:04:08,520 --> 00:04:13,670 So what we sort of have to wind up with here is this, in 71 00:04:13,670 --> 00:04:17,010 a sense, simpler but more abstract viewpoint where we 72 00:04:17,010 --> 00:04:20,820 think of modulation as being a two step process, where we 73 00:04:20,820 --> 00:04:24,750 first take the take the low pass waveform, move it up in 74 00:04:24,750 --> 00:04:29,340 frequency and then we add on the negative frequency part as 75 00:04:29,340 --> 00:04:32,185 an afterthought and then at the receiver, what we're going 76 00:04:32,185 --> 00:04:35,930 to do is have a Hilbert filter, which you would never 77 00:04:35,930 --> 00:04:36,780 build, of course. 78 00:04:36,780 --> 00:04:39,500 You would always build it in terms of cosines and sines, 79 00:04:39,500 --> 00:04:42,670 but conceptually have a Hilbert filter which blocks 80 00:04:42,670 --> 00:04:45,670 off that negative frequency part and then you just shift 81 00:04:45,670 --> 00:04:48,020 things down in frequency again. 82 00:04:48,020 --> 00:04:52,320 So that's the way we're thinking about this here, so 83 00:04:52,320 --> 00:04:55,500 we're going to have u of t shifted up and then 84 00:04:55,500 --> 00:05:00,570 transmitted as 2 times the real part of up of t and 85 00:05:00,570 --> 00:05:02,610 therefore, it's going to get received -- 86 00:05:02,610 --> 00:05:06,880 the received waveform at the passband. 87 00:05:06,880 --> 00:05:10,810 We're still using the timing at the transmitter. 88 00:05:10,810 --> 00:05:14,460 That's the purpose of going through all this analysis, to 89 00:05:14,460 --> 00:05:17,520 look at what's going on at transmit time and what's going 90 00:05:17,520 --> 00:05:20,150 on at receive time. 91 00:05:20,150 --> 00:05:24,340 So in terms of transmit time, this received waveform at 92 00:05:24,340 --> 00:05:30,510 passband is just the attenuation term times x of t 93 00:05:30,510 --> 00:05:35,040 minus tau of t, which is two times the real part of theta 94 00:05:35,040 --> 00:05:37,990 of t times this positive frequency part. 95 00:05:37,990 --> 00:05:41,280 Now when you write it out in all of its glory, it's 2 times 96 00:05:41,280 --> 00:05:45,410 the real part of the attenuation times the input 97 00:05:45,410 --> 00:05:51,750 you started with, but now this has been delayed by tau of t 98 00:05:51,750 --> 00:05:57,320 and e to the 2 pi ifct, but this was the carrier that was 99 00:05:57,320 --> 00:06:01,030 put on at the transmitter, physically at the transmitter 100 00:06:01,030 --> 00:06:04,130 and now we're at the receiver, so that carrier has gotten 101 00:06:04,130 --> 00:06:07,980 delayed by a factor of tau sub t. 102 00:06:07,980 --> 00:06:10,890 The question is, how do you demodulate this? 103 00:06:10,890 --> 00:06:14,660 How do you come down again from transmitter to receiver? 104 00:06:14,660 --> 00:06:17,830 What we've thought all along and what makes a lot of sense 105 00:06:17,830 --> 00:06:21,150 until you start putting these delays in, which are really a 106 00:06:21,150 --> 00:06:24,150 pain in the neck, is that what we're going to do is just 107 00:06:24,150 --> 00:06:27,750 multiply this positive frequency waveform by e to the 108 00:06:27,750 --> 00:06:31,860 minus 2 pi i f sub ct, but that's not what we're going to 109 00:06:31,860 --> 00:06:35,440 do because the receiver sitting there and the receiver 110 00:06:35,440 --> 00:06:38,260 has to recover timing and it has to 111 00:06:38,260 --> 00:06:41,760 recover carrier frequency. 112 00:06:41,760 --> 00:06:44,770 So what's it going to do? 113 00:06:44,770 --> 00:06:48,700 It's going to figure out what the timing is and the timing 114 00:06:48,700 --> 00:06:53,480 that it wants is at time 0, it wants to be seeing what the 115 00:06:53,480 --> 00:06:56,730 transmitter transmitted at transmit time 0. 116 00:06:56,730 --> 00:07:00,250 In other words, if you look at what the transmitter was doing 117 00:07:00,250 --> 00:07:03,530 at time 0, it was sending, say, the first bit that was 118 00:07:03,530 --> 00:07:05,040 going to be sent. 119 00:07:05,040 --> 00:07:08,500 In receiving this, we want to have our timing so we're 120 00:07:08,500 --> 00:07:10,920 looking at that first bit sent. 121 00:07:10,920 --> 00:07:15,340 So our timing is going to be shifted from what it was then. 122 00:07:15,340 --> 00:07:17,140 So we're going to take this new timing. 123 00:07:17,140 --> 00:07:21,900 The receiver clock time is now going to be t prime, which is 124 00:07:21,900 --> 00:07:25,890 t minus this delay term. 125 00:07:25,890 --> 00:07:28,910 If you get confused between the minuses and the pluses 126 00:07:28,910 --> 00:07:33,470 here, don't worry about it. 127 00:07:33,470 --> 00:07:36,610 I get confused about them too and the only way I can 128 00:07:36,610 --> 00:07:40,410 straighten them out as to put down one or the other and then 129 00:07:40,410 --> 00:07:44,720 spend ten minutes looking at it and try to figure out -- 130 00:07:44,720 --> 00:07:46,500 after I write it down -- 131 00:07:46,500 --> 00:07:49,150 whether it should really be a plus or a minus and I think 132 00:07:49,150 --> 00:07:54,115 these sines are right, but I'm not going to try to argue why 133 00:07:54,115 --> 00:07:56,140 in realtime. 134 00:07:56,140 --> 00:08:00,820 But anyway, this received waveform now, as a function of 135 00:08:00,820 --> 00:08:06,440 received time, is going to be the received waveform and in 136 00:08:06,440 --> 00:08:10,150 terms of transmit time -- 137 00:08:10,150 --> 00:08:15,100 and in place of t now, I'm going to have t prime so that 138 00:08:15,100 --> 00:08:20,090 since you t prime is equal to t minus tau of t, this is why 139 00:08:20,090 --> 00:08:22,290 it's t prime plus tau of t. 140 00:08:22,290 --> 00:08:25,590 In other words, what I'm looking at now, at time 0 at 141 00:08:25,590 --> 00:08:28,560 the receiver is what was being transmitted a 142 00:08:28,560 --> 00:08:29,910 little while ago. 143 00:08:29,910 --> 00:08:34,590 So that's this term and that's going to be the real part of 144 00:08:34,590 --> 00:08:39,070 this, where I've taken account of all of these shift terms. 145 00:08:39,070 --> 00:08:43,260 Now if you look at this carefully, you see that u of t 146 00:08:43,260 --> 00:08:47,170 minus t of t is turned into tau of t prime. 147 00:08:47,170 --> 00:08:53,420 This term has turned into 2 pi ifct prime, so everything is 148 00:08:53,420 --> 00:08:54,700 fine there. 149 00:08:54,700 --> 00:08:58,920 What I should have done with this term is to compensate for 150 00:08:58,920 --> 00:09:04,010 the fact that I'm now looking at it in a different time and 151 00:09:04,010 --> 00:09:08,520 at this point, what I'm going to do is to say, this quantity 152 00:09:08,520 --> 00:09:11,310 is changing so slowly with time that I 153 00:09:11,310 --> 00:09:13,090 don't care about that. 154 00:09:13,090 --> 00:09:17,920 If you really try to adjust this to make it right, it's a 155 00:09:17,920 --> 00:09:20,000 terrible mess. 156 00:09:20,000 --> 00:09:23,430 So we're just not going to worry about it. 157 00:09:23,430 --> 00:09:26,280 So what we receive then is approximately equal to this 158 00:09:26,280 --> 00:09:29,900 where the approximation is due to the fact that I'm not 159 00:09:29,900 --> 00:09:34,070 evaluating this attentuation term at exactly the right 160 00:09:34,070 --> 00:09:37,170 time, because it's a pain in the neck to do so. 161 00:09:41,350 --> 00:09:45,550 All of this stuff with wireless, you simply have to 162 00:09:45,550 --> 00:09:48,150 make approximations all over the place. 163 00:09:48,150 --> 00:09:50,930 You have to make crazy modeling assumptions all over 164 00:09:50,930 --> 00:09:53,470 the place, because the physical medium is so 165 00:09:53,470 --> 00:09:57,740 complicated that you can't do much else and the whole 166 00:09:57,740 --> 00:10:02,180 question is trying to make the right approximations and get 167 00:10:02,180 --> 00:10:04,960 some sense of which thing's very fast and which thing's 168 00:10:04,960 --> 00:10:06,900 very slowly. 169 00:10:06,900 --> 00:10:08,620 So that's the equation that we had. 170 00:10:08,620 --> 00:10:13,800 What we receive now at passband, but at the received 171 00:10:13,800 --> 00:10:18,050 time scale is this quantity here. 172 00:10:18,050 --> 00:10:20,150 So what's the receiver going to do? 173 00:10:20,150 --> 00:10:24,640 The receiver, at its time is going to demodulate by 174 00:10:24,640 --> 00:10:28,890 multiplying by e to the minus 2 pi ifct prime. 175 00:10:28,890 --> 00:10:32,090 First, we're going to take away this 2 times the real 176 00:10:32,090 --> 00:10:35,960 sine by going through the Hilbert filter, which just 177 00:10:35,960 --> 00:10:37,210 removes this. 178 00:10:39,930 --> 00:10:42,810 At that point, we have a complex positive frequency 179 00:10:42,810 --> 00:10:46,390 waveform and then we're going to multiply that positive 180 00:10:46,390 --> 00:10:52,315 frequency waveform by e to the minus 2 pi ifct prime, where t 181 00:10:52,315 --> 00:10:55,860 prime is the only thing the receiver knows anything about. 182 00:10:55,860 --> 00:10:58,650 So after you shift the baseband with a recovered 183 00:10:58,650 --> 00:11:03,820 carrier in receiver time, what you get in receiver time is v 184 00:11:03,820 --> 00:11:07,980 of t prime as equal to beta of t prime; namely, the 185 00:11:07,980 --> 00:11:11,820 attenuation at t prime -- or this is the approximate part 186 00:11:11,820 --> 00:11:14,080 of it -- times what was actually set. 187 00:11:18,270 --> 00:11:23,300 So now, think of this as saying, suppose this delay 188 00:11:23,300 --> 00:11:26,210 term is actually changing with time. 189 00:11:26,210 --> 00:11:32,780 Namely, it's some tau 0 minus a velocity term, vt divided by 190 00:11:32,780 --> 00:11:35,120 the velocity of light. 191 00:11:35,120 --> 00:11:40,590 In other words, the time delay that you incur, where the 192 00:11:40,590 --> 00:11:44,480 change in the time delay that you incur, is due to the time 193 00:11:44,480 --> 00:11:47,620 that it takes light to travel from where the receiver was to 194 00:11:47,620 --> 00:11:50,220 where the receiver is now. 195 00:11:50,220 --> 00:11:55,650 We can also write that as tau 0 minus the Doppler shift 196 00:11:55,650 --> 00:12:01,120 times the time divided by the carrier frequency. 197 00:12:01,120 --> 00:12:04,030 Now here's another approximation, because what 198 00:12:04,030 --> 00:12:06,180 we're sending -- 199 00:12:06,180 --> 00:12:09,380 I mean, this quantity here is exact. 200 00:12:09,380 --> 00:12:13,180 When you try to convert from the velocity, which you're 201 00:12:13,180 --> 00:12:16,100 going to, for what it does in frequency, which is the 202 00:12:16,100 --> 00:12:20,090 Doppler shift, it's a function of the frequency. 203 00:12:20,090 --> 00:12:23,120 Here what we're assuming is that all the frequencies we're 204 00:12:23,120 --> 00:12:26,480 dealing with are so close to the carrier frequency that we 205 00:12:26,480 --> 00:12:28,500 can just ignore everything else. 206 00:12:28,500 --> 00:12:38,170 So we're just writing this as the Doppler shift divided by 207 00:12:38,170 --> 00:12:40,170 this carrier frequency. 208 00:12:40,170 --> 00:12:45,510 Now if you view the passband waveform, going into a 209 00:12:45,510 --> 00:12:50,220 baseband of modulation and you think of doing this in 210 00:12:50,220 --> 00:12:53,200 transmit time, what are you going to do? 211 00:12:53,200 --> 00:12:58,100 In terms of transmit time, you've got to get rid of not 212 00:12:58,100 --> 00:13:02,220 this term, but this thing here. 213 00:13:02,220 --> 00:13:06,070 So in terms of transmit time, you're going to be multiplying 214 00:13:06,070 --> 00:13:10,520 by e to the minus 2 pi i times carrier frequency minus the 215 00:13:10,520 --> 00:13:13,740 Doppler shift. 216 00:13:13,740 --> 00:13:16,460 So the thing that's happening is that time at the 217 00:13:16,460 --> 00:13:22,300 transmitter gets spread out a little bit at the receiver. 218 00:13:22,300 --> 00:13:26,480 In other words, as this receiving antenna is moving 219 00:13:26,480 --> 00:13:30,910 away from the transmitter, a period of time like this at 220 00:13:30,910 --> 00:13:35,210 the transmitter is a period of time like this at the receiver 221 00:13:35,210 --> 00:13:38,000 and therefore, a certain number of cycles of carrier at 222 00:13:38,000 --> 00:13:41,110 the transmitter looks like a different number of cycles of 223 00:13:41,110 --> 00:13:43,110 carrier at the receiver. 224 00:13:43,110 --> 00:13:46,450 Therefore, if you're dealing with transmit time, you're 225 00:13:46,450 --> 00:13:48,830 taking account of this Doppler shift. 226 00:13:48,830 --> 00:13:52,240 You're multiplying not by the carrier frequency, but by the 227 00:13:52,240 --> 00:13:54,660 carrier frequency minus the Doppler shift. 228 00:13:54,660 --> 00:14:00,020 If you do it in receiver time, you're just multiplying by the 229 00:14:00,020 --> 00:14:11,450 by this carrier frequency, because the trouble is, your 230 00:14:11,450 --> 00:14:14,230 clock is running a little slow. 231 00:14:14,230 --> 00:14:16,880 Because your clock is running a little slow, you think 232 00:14:16,880 --> 00:14:20,410 you're demodulating at the actual carrier frequency, 233 00:14:20,410 --> 00:14:22,720 whereas in terms of what the transmitter thinks, you're 234 00:14:22,720 --> 00:14:25,180 doing something else. 235 00:14:25,180 --> 00:14:28,440 If you all got confused when you studied relativity, this 236 00:14:28,440 --> 00:14:32,550 is exactly the same problem you got confused about there. 237 00:14:32,550 --> 00:14:36,120 There's nothing very mysterious about it or hard 238 00:14:36,120 --> 00:14:39,230 about it, except that to most of us, time is a very 239 00:14:39,230 --> 00:14:41,850 fundamental quantity and you don't like 240 00:14:41,850 --> 00:14:43,690 to monkey with that. 241 00:14:43,690 --> 00:14:46,780 If you can't count on time being what it's supposed to 242 00:14:46,780 --> 00:14:48,990 be, you're really in deep trouble. 243 00:14:48,990 --> 00:14:53,290 When you try to write equations that do that, it 244 00:14:53,290 --> 00:14:55,450 makes things very, very tough also. 245 00:14:55,450 --> 00:15:02,210 So anyway, the result after we look at this in both ways, is 246 00:15:02,210 --> 00:15:05,700 that you get the same answer each way, but the receiver 247 00:15:05,700 --> 00:15:10,340 sees the carrier frequency and sees this carrier frequency in 248 00:15:10,340 --> 00:15:13,390 transmitter time and it sees this 249 00:15:13,390 --> 00:15:16,480 frequency in received time. 250 00:15:16,480 --> 00:15:19,960 Of course, the receiver only sees received time because the 251 00:15:19,960 --> 00:15:23,260 receiver sitting there trying to measure from the waveform 252 00:15:23,260 --> 00:15:27,420 coming in what that carrier frequency is and what that 253 00:15:27,420 --> 00:15:28,670 time base is. 254 00:15:32,100 --> 00:15:34,190 So now it gets to be fun. 255 00:15:34,190 --> 00:15:37,580 We look at multiple paths. 256 00:15:37,580 --> 00:15:47,460 When you look at it in transmit time, what gets 257 00:15:47,460 --> 00:15:50,710 received at the receiver, according to the 258 00:15:50,710 --> 00:15:52,760 transmit clock -- 259 00:15:52,760 --> 00:15:54,990 so here we are at the transmitter and we're peering 260 00:15:54,990 --> 00:16:03,270 at what's going on at this distant receiver and you now 261 00:16:03,270 --> 00:16:05,770 have the thing that we've talked about in the notes 262 00:16:05,770 --> 00:16:08,390 where you have all these different paths. 263 00:16:08,390 --> 00:16:12,660 Each path is associated with a certain attenuation and each 264 00:16:12,660 --> 00:16:17,370 path is associated with a time delay. 265 00:16:17,370 --> 00:16:20,950 This can be written in terms of what has gotten modulated 266 00:16:20,950 --> 00:16:26,170 up from baseband as 2 times the real part of this sum -- 267 00:16:26,170 --> 00:16:29,870 of beta j of t times the positive frequency part of 268 00:16:29,870 --> 00:16:34,500 what got transmitted and then it now has this delay in it. 269 00:16:34,500 --> 00:16:37,420 This can be rewritten in the same way that we did before, 270 00:16:37,420 --> 00:16:40,580 but now we just have all of these paths in there instead 271 00:16:40,580 --> 00:16:41,830 of one path. 272 00:16:45,520 --> 00:16:48,490 If you look at this expression then, we have all of these 273 00:16:48,490 --> 00:16:50,530 propogation delay terms. 274 00:16:50,530 --> 00:16:54,660 We have this baseband input, which has been delayed in 275 00:16:54,660 --> 00:16:59,880 different ways for each path and you have have this 276 00:16:59,880 --> 00:17:04,260 exponential, which has been delayed also. 277 00:17:04,260 --> 00:17:08,310 So at this point, the receiver is in a real pickle because if 278 00:17:08,310 --> 00:17:10,630 the receiver is smart enough to see -- yes? 279 00:17:10,630 --> 00:17:17,680 AUDIENCE: [UNINTELLIGIBLE PHRASE] 280 00:17:17,680 --> 00:17:20,450 PROFESSOR: The receiver doesn't know anything. 281 00:17:20,450 --> 00:17:22,600 It's like you're driving in your car, you're talking on 282 00:17:22,600 --> 00:17:25,930 your cell phone and your cell phone -- 283 00:17:25,930 --> 00:17:28,960 I mean, the only way your cell phone knows that you're moving 284 00:17:28,960 --> 00:17:33,380 is that the cell phone is getting some waveform coming 285 00:17:33,380 --> 00:17:38,190 in and it has to tell from that waveform what's going on. 286 00:17:38,190 --> 00:17:41,400 So it's going to do things like the carrier tracking that 287 00:17:41,400 --> 00:17:45,510 we were talking about before and it's not going to have too 288 00:17:45,510 --> 00:17:48,560 much trouble because all of these changes are taking place 289 00:17:48,560 --> 00:17:52,190 relatively slowly, relative to this very high -- 290 00:17:55,080 --> 00:17:58,100 either one gigabits or two gigabits or four gigabits, or 291 00:17:58,100 --> 00:18:00,350 what have you, carrier frequency. 292 00:18:00,350 --> 00:18:03,890 So all these changes look like almost nothing, but at the 293 00:18:03,890 --> 00:18:06,830 same time, they're fairly important because they keep 294 00:18:06,830 --> 00:18:09,820 changing faces. 295 00:18:09,820 --> 00:18:15,950 But anyway, in terms of the transmitter clock, this is 296 00:18:15,950 --> 00:18:19,960 what you actually receive. 297 00:18:19,960 --> 00:18:24,220 So at this point, what the receiver has to do is it has 298 00:18:24,220 --> 00:18:28,190 to retrieve clock time and the clock time that it retrieves 299 00:18:28,190 --> 00:18:29,460 is going to be -- 300 00:18:38,260 --> 00:18:42,690 receiver clock time is -- 301 00:18:42,690 --> 00:18:47,170 is t minus tau 0 sub t -- 302 00:18:47,170 --> 00:18:52,830 so it's t prime equals t minus tau 0 sub t. 303 00:18:52,830 --> 00:18:56,330 That's the same thing it was before, except that this tau 304 00:18:56,330 --> 00:18:58,830 zero of t here doesn't really amount to 305 00:18:58,830 --> 00:19:00,830 anything physical anymore. 306 00:19:00,830 --> 00:19:04,140 It's just whatever the circuitry that we have trying 307 00:19:04,140 --> 00:19:07,870 to recover carrier and trying to recover timing -- 308 00:19:07,870 --> 00:19:11,220 it's just whatever that happens to come up with, so 309 00:19:11,220 --> 00:19:15,520 with this best sense of the timing it should use if it's 310 00:19:15,520 --> 00:19:17,870 going to try to detect what the signal is. 311 00:19:17,870 --> 00:19:21,630 So it's some arbitrary value and again, this 312 00:19:21,630 --> 00:19:23,600 is varying in time. 313 00:19:23,600 --> 00:19:28,060 So again what we have is now this expression becomes really 314 00:19:28,060 --> 00:19:36,150 messy, because instead of having a receiver time 315 00:19:36,150 --> 00:19:42,710 variation, which cancels out what the actual variation in 316 00:19:42,710 --> 00:19:47,110 path length is, you really have two terms that are 317 00:19:47,110 --> 00:19:51,320 different and you have the same thing in terms of this 318 00:19:51,320 --> 00:19:53,170 phase, which is changing and this is the 319 00:19:53,170 --> 00:19:55,740 more important term. 320 00:19:55,740 --> 00:19:58,020 But anyway, you have that. 321 00:19:58,020 --> 00:19:59,840 You can then demodulate. 322 00:19:59,840 --> 00:20:01,920 What do you do when you demodulate? 323 00:20:01,920 --> 00:20:06,840 You take this thing in receiver time and you multiply 324 00:20:06,840 --> 00:20:11,360 by e to the 2 pi i, e to the minus 2 pi i -- 325 00:20:11,360 --> 00:20:14,890 this carrier frequency, what you think it is, times this 326 00:20:14,890 --> 00:20:17,960 time reference, what you think that is -- 327 00:20:17,960 --> 00:20:22,320 So that term has disappeared and what we then wind up with 328 00:20:22,320 --> 00:20:26,190 is the carrier frequency times these two terms here. 329 00:20:26,190 --> 00:20:27,780 That's to the left there. 330 00:20:27,780 --> 00:20:32,810 These two terms turn out to have both Doppler shift terms 331 00:20:32,810 --> 00:20:37,190 in them and also time spread terms. 332 00:20:37,190 --> 00:20:42,600 So when we write that out in terms of Doppler shift, what 333 00:20:42,600 --> 00:20:49,420 we wind up with is the input, which is then delayed by some 334 00:20:49,420 --> 00:20:52,590 arbitrary amount of time, where this is our best 335 00:20:52,590 --> 00:20:56,010 estimate of the right time to use and this is the actual 336 00:20:56,010 --> 00:20:57,910 time on that path. 337 00:20:57,910 --> 00:21:01,080 The thing that's happening here, which you don't see in 338 00:21:01,080 --> 00:21:06,560 the notes unfortunately, in the notes, it pretty much 339 00:21:06,560 --> 00:21:10,440 looks at things as far as transmit time is concerned. 340 00:21:10,440 --> 00:21:14,070 Therefore, what you see is expressions for impulse 341 00:21:14,070 --> 00:21:18,710 response where the impulse response is at some time much 342 00:21:18,710 --> 00:21:20,570 later than time 0. 343 00:21:20,570 --> 00:21:21,620 You have a -- 344 00:21:21,620 --> 00:21:25,030 I mean, you put in an impulse of time 0, what you see is 345 00:21:25,030 --> 00:21:29,230 stuff dribbling out over some very tiny interval, but a good 346 00:21:29,230 --> 00:21:32,340 deal later than what got put in. 347 00:21:32,340 --> 00:21:36,510 What we want to do now is to shift that so that in fact, 348 00:21:36,510 --> 00:21:39,120 when we look at filtering and things like that, we have a 349 00:21:39,120 --> 00:21:43,190 filter which starts a little bit before 0 and ends a little 350 00:21:43,190 --> 00:21:48,540 bit after 0 and that's the whole purpose of this. 351 00:21:48,540 --> 00:21:52,310 If you don't think my purposes just to confuse you, it really 352 00:21:52,310 --> 00:21:57,440 isn't, although I realize this is very confusing, but you 353 00:21:57,440 --> 00:21:59,970 sort of have to go through with it to see why these time 354 00:21:59,970 --> 00:22:02,950 shifts appear in the places where they do. 355 00:22:02,950 --> 00:22:06,910 But anyway, up here in the phase, you wind up with the 356 00:22:06,910 --> 00:22:10,050 Doppler shift terms and down here, when you're worrying 357 00:22:10,050 --> 00:22:14,450 about the input waveform, you're worrying about these 358 00:22:14,450 --> 00:22:15,520 time shift terms. 359 00:22:15,520 --> 00:22:18,950 We'll see we'll see how they come in a little later. 360 00:22:24,210 --> 00:22:27,990 To make the expression a little bit easier, let me look 361 00:22:27,990 --> 00:22:33,310 at tau j prime of t as tau j minus tau 0 and let me look at 362 00:22:33,310 --> 00:22:37,480 the Doppler differential as the Doppler that actually 363 00:22:37,480 --> 00:22:41,040 occurs on path j minus what the receiver thinks the 364 00:22:41,040 --> 00:22:42,860 Doppler is. 365 00:22:42,860 --> 00:22:47,010 This is this receiver term that we've demodulated by and 366 00:22:47,010 --> 00:22:50,220 therefore, we've really gotten rid of this term. 367 00:22:50,220 --> 00:22:54,460 Then when we rewrite this received baseband waveform -- 368 00:22:54,460 --> 00:22:59,110 and I've now gotten rid of the t primes because everything 369 00:22:59,110 --> 00:23:02,980 we've done up until now, we've just automatically assumed 370 00:23:02,980 --> 00:23:06,060 that the receiver timing and the transmit timing, we didn't 371 00:23:06,060 --> 00:23:09,120 have to be careful about it and it wasn't until we got to 372 00:23:09,120 --> 00:23:11,370 wireless that we do have to be careful about it. 373 00:23:11,370 --> 00:23:15,420 So now we're throwing it out again and what we have is 374 00:23:15,420 --> 00:23:19,360 these attenuations, which are very slowly varied. 375 00:23:19,360 --> 00:23:22,990 These terms, which are now delayed by this difference 376 00:23:22,990 --> 00:23:27,650 between the actual delay and what the receiver is assuming 377 00:23:27,650 --> 00:23:28,750 that the light ought to be. 378 00:23:28,750 --> 00:23:32,600 So these are really just differential delays relative 379 00:23:32,600 --> 00:23:34,560 to each other. 380 00:23:34,560 --> 00:23:37,770 These are Doppler shifts relative to each other. 381 00:23:37,770 --> 00:23:40,940 So that's the whole receive baseband waveform then. 382 00:23:46,820 --> 00:23:50,300 If you look at this, it's not apparent where the carrier 383 00:23:50,300 --> 00:23:54,770 frequency is coming in and the carrier frequency is coming in 384 00:23:54,770 --> 00:23:59,030 because these Doppler shifts have the carrier 385 00:23:59,030 --> 00:24:00,140 frequency in them. 386 00:24:00,140 --> 00:24:04,250 Namely, the Doppler shift is the velocity times the carrier 387 00:24:04,250 --> 00:24:06,520 frequency divided by the speed of light. 388 00:24:09,110 --> 00:24:11,710 One of the things you have to be very careful with in the 389 00:24:11,710 --> 00:24:14,990 wireless business, is that everybody thinks it's a neat 390 00:24:14,990 --> 00:24:17,210 idea to be able to be able to go up to higher and higher 391 00:24:17,210 --> 00:24:19,570 frequencies. 392 00:24:19,570 --> 00:24:23,440 When you go from one gigahertz up to four gigahertz, one of 393 00:24:23,440 --> 00:24:25,540 the things that you have to deal with is that every 394 00:24:25,540 --> 00:24:29,020 Doppler shift you're dealing with is now four times higher 395 00:24:29,020 --> 00:24:30,270 than it was before. 396 00:24:33,880 --> 00:24:37,650 So there's one big disadvantage in operating at 397 00:24:37,650 --> 00:24:39,530 extremely high frequencies. 398 00:24:39,530 --> 00:24:44,080 It's not necessarily a bad thing, but it's there and if 399 00:24:44,080 --> 00:24:47,540 you have a bunch of different paths which are all acting in 400 00:24:47,540 --> 00:24:50,520 different ways, you suddenly wind up 401 00:24:50,520 --> 00:24:53,330 with some real problems. 402 00:24:53,330 --> 00:24:58,050 The received baseband waveform on path j is going to change 403 00:24:58,050 --> 00:25:00,270 completely over an interval -- 404 00:25:00,270 --> 00:25:08,380 1 over 4 times the magnitude of this 405 00:25:08,380 --> 00:25:10,680 differential Doppler shift. 406 00:25:10,680 --> 00:25:13,600 The Doppler shift might be positive or negative, but if 407 00:25:13,600 --> 00:25:17,050 you look at this expression up here, that's the amount of 408 00:25:17,050 --> 00:25:20,020 time that it takes for this quantity to 409 00:25:20,020 --> 00:25:23,120 change by pi over 2. 410 00:25:23,120 --> 00:25:27,940 When this changes by pi over 2, this exponential goes from 411 00:25:27,940 --> 00:25:32,580 its maximum one down to zero or it goes from zero to minus 412 00:25:32,580 --> 00:25:39,430 one or it goes from minus one to zero or from zero up to 413 00:25:39,430 --> 00:25:42,530 plus one, and that's a pretty major change. 414 00:25:47,910 --> 00:25:51,940 So the interval which is required to do that is 1 over 415 00:25:51,940 --> 00:25:53,600 4 times that Doppler shift. 416 00:25:56,370 --> 00:26:00,490 The Doppler spread now is defined as the difference 417 00:26:00,490 --> 00:26:04,490 between the maximum Doppler shift and the 418 00:26:04,490 --> 00:26:08,390 minimum Doppler shift. 419 00:26:08,390 --> 00:26:13,580 If we choose this Doppler shift D sub 0 that we had, 420 00:26:13,580 --> 00:26:16,280 because of what the receiver was doing, seeing all these 421 00:26:16,280 --> 00:26:18,740 different Doppler shifts and trying to come up with 422 00:26:18,740 --> 00:26:24,960 something in the middle, that's pretty close to what 423 00:26:24,960 --> 00:26:30,330 the receiver thinks it should be using as far as receive 424 00:26:30,330 --> 00:26:31,600 time is concerned. 425 00:26:31,600 --> 00:26:35,860 So it's the maximum plus the minimum divided by 2 and then 426 00:26:35,860 --> 00:26:40,060 each of these differential Doppler shifts is going to be 427 00:26:40,060 --> 00:27:00,430 somewhere between that overall Doppler spread minus Doppler 428 00:27:00,430 --> 00:27:03,970 spread over 2 and plus Doppler spread over 2. 429 00:27:03,970 --> 00:27:05,220 It goes from -- 430 00:27:09,000 --> 00:27:17,130 minus D over 2 up to plus D over 2 and this difference in 431 00:27:17,130 --> 00:27:25,130 here, is strangely enough, D. So what we've done is by 432 00:27:25,130 --> 00:27:27,770 adjusting our received timing -- 433 00:27:27,770 --> 00:27:31,520 and slowing it down or speeding it up if necessary, 434 00:27:31,520 --> 00:27:34,850 is we wind up with Doppler shifts that are sort of spread 435 00:27:34,850 --> 00:27:39,580 around from negative Doppler shifts to plus Doppler shifts. 436 00:27:39,580 --> 00:27:43,520 What that means is, if you look at this expression, it 437 00:27:43,520 --> 00:27:46,660 takes a period of time, about -- 438 00:27:49,570 --> 00:27:55,780 here we had 1 over 4 times the differential Doppler, which is 439 00:27:55,780 --> 00:28:02,000 now 1 over 2 times this overall Doppler spread. 440 00:28:02,000 --> 00:28:07,270 So what that says is, there's a coherence time on the 441 00:28:07,270 --> 00:28:12,150 channel of about 1 over 2 times the Doppler spread. 442 00:28:12,150 --> 00:28:16,090 That's important because that says the channel is going to 443 00:28:16,090 --> 00:28:21,570 look like the same thing for a period of time, which is about 444 00:28:21,570 --> 00:28:26,330 1 over 2 times this Doppler spread term. 445 00:28:26,330 --> 00:28:31,270 I think this will be D instead of D 0. 446 00:28:31,270 --> 00:28:34,050 So you find out what the Doppler spread is and the 447 00:28:34,050 --> 00:28:36,970 reason you want to find out what the Doppler spread is is 448 00:28:36,970 --> 00:28:41,920 that that tells you how long the channel remains stable. 449 00:28:41,920 --> 00:28:43,400 Why do you want to know how long the 450 00:28:43,400 --> 00:28:45,460 channel remains stable? 451 00:28:45,460 --> 00:28:47,590 If you're going to do detection or anything like 452 00:28:47,590 --> 00:28:50,170 that, if you're going to filter the received waveform 453 00:28:50,170 --> 00:28:54,040 to try to find out what the input waveform is, you would 454 00:28:54,040 --> 00:28:55,780 like to know what the channel is doing. 455 00:28:55,780 --> 00:28:59,060 You'd like to be able to measure the channel. 456 00:28:59,060 --> 00:29:02,450 If you're going to measure the channel, you really want to 457 00:29:02,450 --> 00:29:06,100 know how long it stays the same before you have to 458 00:29:06,100 --> 00:29:07,970 measure it again. 459 00:29:07,970 --> 00:29:10,300 If you look at all these cellular systems and you look 460 00:29:10,300 --> 00:29:14,360 at every other wireless system in the world, all of them do 461 00:29:14,360 --> 00:29:17,780 one of one of two things: either periodically they 462 00:29:17,780 --> 00:29:21,760 measure what the channel is so they can use it and the other 463 00:29:21,760 --> 00:29:28,040 thing they do is every once in awhile, they send a pilot tone 464 00:29:28,040 --> 00:29:32,760 and they use the pilot tone to try to figure out what kind of 465 00:29:32,760 --> 00:29:33,930 channel they have. 466 00:29:33,930 --> 00:29:37,510 This is the thing which tells you how often you have to send 467 00:29:37,510 --> 00:29:39,290 that pilots tone. 468 00:29:39,290 --> 00:29:42,700 If you know about estimation theory, it's also the thing 469 00:29:42,700 --> 00:29:45,210 that tells you whether you have any chance or not of 470 00:29:45,210 --> 00:29:47,930 estimating what the channel is, because it takes you a 471 00:29:47,930 --> 00:29:52,110 certain amount of time in the presence of noise to estimate 472 00:29:52,110 --> 00:29:54,410 something and if that something you're trying to 473 00:29:54,410 --> 00:29:58,520 estimate it's changing faster than you can estimate it, then 474 00:29:58,520 --> 00:30:00,030 you don't have a prayer of a chance. 475 00:30:02,930 --> 00:30:04,490 So yes, this is important. 476 00:30:04,490 --> 00:30:05,620 This is the prime coherence. 477 00:30:05,620 --> 00:30:08,860 It's one of the main parameters of a wireless 478 00:30:08,860 --> 00:30:11,330 channel you want to understand. 479 00:30:11,330 --> 00:30:13,900 It tells you how long the channel will remain 480 00:30:13,900 --> 00:30:16,400 essentially the same before it changes and 481 00:30:16,400 --> 00:30:18,390 becomes something different. 482 00:30:20,960 --> 00:30:24,420 Both of these quantities are very approximate. 483 00:30:24,420 --> 00:30:27,680 If anyone wants to tell you that it's twice what I say it 484 00:30:27,680 --> 00:30:31,060 is here, I certainly wouldn't quarrel with them. 485 00:30:31,060 --> 00:30:34,060 I mean, it's like talking about the bandwidth of a 486 00:30:34,060 --> 00:30:35,740 waveform them that you send. 487 00:30:35,740 --> 00:30:38,090 I mean, you look at the spectrum of what you send and 488 00:30:38,090 --> 00:30:40,900 it's going to be something which sort of tails off. 489 00:30:40,900 --> 00:30:43,400 If you want to call it the maximum out there where it's 490 00:30:43,400 --> 00:30:47,940 going to zero or the minimum, where it starts to go to zero 491 00:30:47,940 --> 00:30:50,590 or what have you, it doesn't make much difference. 492 00:30:50,590 --> 00:30:52,790 Here, it's a worse situation because you 493 00:30:52,790 --> 00:30:54,590 don't shape this waveform. 494 00:30:54,590 --> 00:30:58,210 It's just given to you and you find something which is going 495 00:30:58,210 --> 00:30:59,800 to be tailing off slowly. 496 00:30:59,800 --> 00:31:06,040 It's going to have little bits of nothingness way out there. 497 00:31:06,040 --> 00:31:09,890 All this is is just one very approximate number, which 498 00:31:09,890 --> 00:31:13,600 tells you what that Doppler spread is and therefore, which 499 00:31:13,600 --> 00:31:17,740 tells you how long the channel is going to remain stable and 500 00:31:17,740 --> 00:31:21,030 yes, in half of that time the channel starts to change and 501 00:31:21,030 --> 00:31:24,300 in twice that time, the channel might be still sort of 502 00:31:24,300 --> 00:31:25,450 like it was before. 503 00:31:25,450 --> 00:31:28,570 So this is only a very approximate way of looking at 504 00:31:28,570 --> 00:31:32,000 these things, but it gives you an idea about what you can do 505 00:31:32,000 --> 00:31:33,380 and what you can't do. 506 00:31:33,380 --> 00:31:36,170 It gives you an idea, for example, if you say, can I 507 00:31:36,170 --> 00:31:41,550 take this wireless system I've been using and use it at a 508 00:31:41,550 --> 00:31:44,360 frequency that's 10 times as high -- 509 00:31:44,360 --> 00:31:47,480 and you know when you use it at a frequency 10 times as 510 00:31:47,480 --> 00:31:51,910 high that the stability of the channel is going to be 10 511 00:31:51,910 --> 00:31:53,650 times as small. 512 00:31:53,650 --> 00:31:55,930 You know if you were having trouble measuring the channel 513 00:31:55,930 --> 00:31:58,930 before, you know you don't have a prayer of a chance when 514 00:31:58,930 --> 00:32:02,030 you go up to that higher frequency, so it so it tells 515 00:32:02,030 --> 00:32:03,280 you things like that. 516 00:32:05,660 --> 00:32:08,390 There are two other sort of related quantities, which are 517 00:32:08,390 --> 00:32:13,210 called multipath spread and coherence frequency. 518 00:32:13,210 --> 00:32:17,130 So we have Doppler spread on the one hand, which is a 519 00:32:17,130 --> 00:32:23,840 physical phenomena and that gives rise to coherence time, 520 00:32:23,840 --> 00:32:27,720 which says how long the channel will stay the same and 521 00:32:27,720 --> 00:32:29,110 in the opposite domain -- 522 00:32:29,110 --> 00:32:34,360 and these are almost dual quantities of each other, you 523 00:32:34,360 --> 00:32:38,220 have something called multipath spread, which -- 524 00:32:38,220 --> 00:32:40,800 and multipath spread is pretty much what it 525 00:32:40,800 --> 00:32:41,690 sounds like it is. 526 00:32:41,690 --> 00:32:44,290 You have a bunch of different paths. 527 00:32:44,290 --> 00:32:48,690 You send a very short duration waveform and things start 528 00:32:48,690 --> 00:32:53,070 trickling in on the shortest path and then you got a big 529 00:32:53,070 --> 00:32:58,650 bunch of stuff in all the paths which are sort of the 530 00:32:58,650 --> 00:33:03,480 same lanegth and then they sort of dribble away again. 531 00:33:03,480 --> 00:33:06,570 So the waveform that you see coming in is spread out from 532 00:33:06,570 --> 00:33:09,830 what you send because of these multiple paths and the 533 00:33:09,830 --> 00:33:13,570 multiple paths having different delays on them. 534 00:33:13,570 --> 00:33:19,560 Because you have all of these different delay paths, you're 535 00:33:19,560 --> 00:33:22,330 going to have an impulse response for this channel, 536 00:33:22,330 --> 00:33:25,580 which has a certain duration to it. 537 00:33:25,580 --> 00:33:29,210 When you take the Fourier transform of that, you find a 538 00:33:29,210 --> 00:33:34,480 certain duration for Fourier transform. 539 00:33:34,480 --> 00:33:37,240 As we're going to see in just a minute, that's the thing 540 00:33:37,240 --> 00:33:41,550 that tells you how stable this channel is in frequency. 541 00:33:41,550 --> 00:33:47,620 If you measure it at one frequency, how far do you have 542 00:33:47,620 --> 00:33:50,960 to go in frequency before you find something which is 543 00:33:50,960 --> 00:33:55,460 totally separate from what you measure down here? 544 00:33:55,460 --> 00:33:59,650 This is something the cuts both ways. 545 00:33:59,650 --> 00:34:03,520 Having a large frequency coherence and having a large 546 00:34:03,520 --> 00:34:07,460 time coherence can be good or it can be bad. 547 00:34:07,460 --> 00:34:10,000 If you have a very large time coherence and a very large 548 00:34:10,000 --> 00:34:12,910 frequency coherence, you can measure the channel once and 549 00:34:12,910 --> 00:34:16,290 you can use it for a long time and you can use it over a 550 00:34:16,290 --> 00:34:19,360 broadband width and everything looks very nice. 551 00:34:19,360 --> 00:34:22,290 If the channel happens to be faded over that long period of 552 00:34:22,290 --> 00:34:25,440 time and over that long interval of frequency, you're 553 00:34:25,440 --> 00:34:27,480 really in the soup. 554 00:34:27,480 --> 00:34:30,500 On the other hand, if you have a very small time coherence 555 00:34:30,500 --> 00:34:33,540 and a very small frequency coherence, if the channel 556 00:34:33,540 --> 00:34:36,030 doesn't look good there, you move to a different time 557 00:34:36,030 --> 00:34:39,710 interval or you moved to a different frequency interval 558 00:34:39,710 --> 00:34:42,440 and you can transmit again. 559 00:34:42,440 --> 00:34:46,940 So both of these quantities have good aspects to them and 560 00:34:46,940 --> 00:34:50,900 bad aspects to them, but if you want to evaluate what a 561 00:34:50,900 --> 00:34:55,550 physical wireless channel is and somebody lets you ask two 562 00:34:55,550 --> 00:35:00,510 questions, the two questions you ought to ask are, what is 563 00:35:00,510 --> 00:35:03,450 the time coherence and what is the frequency coherence? 564 00:35:03,450 --> 00:35:05,570 Those are the first things you want to find out. 565 00:35:05,570 --> 00:35:10,080 Everything else is sort of secondary. 566 00:35:10,080 --> 00:35:14,500 So trying to claim that the frequency coherence here 567 00:35:14,500 --> 00:35:18,160 almost has a matter duality, as 1 over 568 00:35:18,160 --> 00:35:20,440 twice the time spread. 569 00:35:20,440 --> 00:35:26,170 Let's take a slide to see why that is. 570 00:35:29,700 --> 00:35:38,160 If I take the impulse response at baseband of this channel, 571 00:35:38,160 --> 00:35:41,100 which I wrote down here -- 572 00:35:41,100 --> 00:35:48,310 here's the impulse response and you get that directly from 573 00:35:48,310 --> 00:35:52,130 the baseband response for giving input waveform. 574 00:35:52,130 --> 00:35:55,600 With all these different paths, what this impulse 575 00:35:55,600 --> 00:36:02,400 response is at a particular time t -- namely, this is the 576 00:36:02,400 --> 00:36:08,770 response at time t to an impulse tau seconds earlier. 577 00:36:08,770 --> 00:36:11,520 In other words, when I'm using this notation here, I'm really 578 00:36:11,520 --> 00:36:15,400 thinking of t as a receiver time and t minus tau as 579 00:36:15,400 --> 00:36:17,490 transmit time. 580 00:36:17,490 --> 00:36:19,530 But anyway, this is what it is. 581 00:36:19,530 --> 00:36:25,920 You take the Fourier transform of this for a particular t -- 582 00:36:25,920 --> 00:36:27,240 in other words, we talked about this a 583 00:36:27,240 --> 00:36:29,070 little bit last time. 584 00:36:29,070 --> 00:36:34,090 You have a function of two parameters here and at any 585 00:36:34,090 --> 00:36:38,420 given t at the receiver, what you see is a spread of 586 00:36:38,420 --> 00:36:41,650 different inputs coming in from the transmitter. 587 00:36:41,650 --> 00:36:45,540 What we're interested in now is at that particular time t 588 00:36:45,540 --> 00:36:49,870 at the receiver, what's the Fourier transform, when you 589 00:36:49,870 --> 00:36:53,440 take the Fourier transform on tau to see what kinds of 590 00:36:53,440 --> 00:36:56,100 frequencies are coming in? 591 00:36:56,100 --> 00:37:03,090 We take the Fourier transform of a unit delta function and 592 00:37:03,090 --> 00:37:04,990 you just get an exponential function. 593 00:37:04,990 --> 00:37:09,820 So the transform of that delta is just this e to the minus 2 594 00:37:09,820 --> 00:37:14,080 pi if times tau sub j prime of t. 595 00:37:14,080 --> 00:37:17,820 This is this differential delay that we're faced with 596 00:37:17,820 --> 00:37:20,860 and we still have the Doppler shift over here. 597 00:37:20,860 --> 00:37:24,100 So when we look at it this way, we have both the Doppler 598 00:37:24,100 --> 00:37:29,640 shift sitting there and we have this delay 599 00:37:29,640 --> 00:37:31,840 shift sitting up here. 600 00:37:31,840 --> 00:37:35,700 Now the question we ask is, how much does the frequency 601 00:37:35,700 --> 00:37:39,280 have to change before this quantity is going to change 602 00:37:39,280 --> 00:37:41,380 materially? 603 00:37:41,380 --> 00:37:44,330 The answer is the same as it was before. 604 00:37:44,330 --> 00:37:49,590 When this quantity here changes by pi over 2, you have 605 00:37:49,590 --> 00:37:52,330 a totally different result than you had before. 606 00:37:52,330 --> 00:37:55,970 So the question is, how much does a frequency have to 607 00:37:55,970 --> 00:38:05,650 change in order for f times tau j prime of t to the change 608 00:38:05,650 --> 00:38:10,260 by a factor of one-fourth. 609 00:38:10,260 --> 00:38:14,780 So the answer that we come up with is, this has to go from 610 00:38:14,780 --> 00:38:23,740 minus L over 2 to plus L over 2 and this is where we get the 611 00:38:23,740 --> 00:38:25,630 result that we were just talking about, that the 612 00:38:25,630 --> 00:38:34,290 frequency coherence is 1 over 2 times the multipath spread 613 00:38:34,290 --> 00:38:35,540 of the channel. 614 00:38:44,580 --> 00:38:49,180 The next thing we want to do in this long process -- 615 00:38:49,180 --> 00:38:53,860 it's a process that we've sort of been doing all along. 616 00:38:53,860 --> 00:39:00,440 The whole purpose of looking at waveforms and looking at 617 00:39:00,440 --> 00:39:04,220 digital communication is to be able to go from sequences to 618 00:39:04,220 --> 00:39:06,060 waveforms -- 619 00:39:06,060 --> 00:39:08,190 and what do you think the next thing we're going to do is? 620 00:39:08,190 --> 00:39:10,140 We're going to go from these waveforms -- 621 00:39:10,140 --> 00:39:13,245 we've now got them down to baseband waveforms and we want 622 00:39:13,245 --> 00:39:17,840 to go back to looking at sequences. 623 00:39:17,840 --> 00:39:23,650 So what we would like to do is to take this input waveform, 624 00:39:23,650 --> 00:39:26,170 view it in terms of the sampling theorem -- 625 00:39:26,170 --> 00:39:29,880 better thing to do would be to view it in terms of input 626 00:39:29,880 --> 00:39:34,400 pulses, but that's too complicated for now, so we'll 627 00:39:34,400 --> 00:39:38,950 just take the input waveform and view it as a sequence of 628 00:39:38,950 --> 00:39:41,390 data that we're sending -- the u sub k's 629 00:39:41,390 --> 00:39:45,560 times these sinc waveforms. 630 00:39:45,560 --> 00:39:47,070 Same thing we've been doing all along. 631 00:39:47,070 --> 00:39:51,190 We take a sequence of numbers, we put little sine x over 632 00:39:51,190 --> 00:39:54,160 x-hats around them and that gives us a 633 00:39:54,160 --> 00:39:57,600 band limited waveform. 634 00:39:57,600 --> 00:40:00,760 The baseband output is then -- 635 00:40:00,760 --> 00:40:02,210 you can look at the same way. 636 00:40:02,210 --> 00:40:04,380 The baseband output is going to be 637 00:40:04,380 --> 00:40:07,630 limited to the same frequency. 638 00:40:07,630 --> 00:40:11,880 Question here: You have these Doppler shifts. 639 00:40:11,880 --> 00:40:16,550 So when I send a frequency which is right up at the limit 640 00:40:16,550 --> 00:40:20,990 of w over 2, what I get back from that is going to be 641 00:40:20,990 --> 00:40:23,320 spread out a little bit from that. 642 00:40:23,320 --> 00:40:27,720 So if I send a waveform that's limited to w over 2, I'm going 643 00:40:27,720 --> 00:40:30,120 to get back a waveform which is a little 644 00:40:30,120 --> 00:40:31,750 bit bigger than that. 645 00:40:31,750 --> 00:40:35,230 What do I do about that? 646 00:40:35,230 --> 00:40:37,130 I worried about it in the notes a little bit and I'm 647 00:40:37,130 --> 00:40:41,960 probably going to take it out from the notes, because as a 648 00:40:41,960 --> 00:40:46,400 practical matter, the amount of Doppler shift that you get 649 00:40:46,400 --> 00:40:54,380 is such a negligible fraction of the kind of bandwidth you 650 00:40:54,380 --> 00:40:57,020 would be using for transmission that you just 651 00:40:57,020 --> 00:40:59,430 want to forget about that. 652 00:40:59,430 --> 00:41:02,670 So we are going to forget about it and what's coming in 653 00:41:02,670 --> 00:41:07,090 is again a band limited function with maybe a little 654 00:41:07,090 --> 00:41:09,930 bit of squishiness on it. 655 00:41:09,930 --> 00:41:14,870 We get this baseband output, which we can then view in 656 00:41:14,870 --> 00:41:16,840 terms of the sampling theorem. 657 00:41:16,840 --> 00:41:20,320 We can sample the output and what do we get 658 00:41:20,320 --> 00:41:22,020 when we're all done? 659 00:41:22,020 --> 00:41:28,390 Here's the nice thing that we can jump to: The output at 660 00:41:28,390 --> 00:41:32,920 this free time m is then the sum -- 661 00:41:32,920 --> 00:41:35,780 namely, it's the convolution, it's the discrete convolution 662 00:41:35,780 --> 00:41:42,210 of what went in at time m, at time m minus 1, m minus 2, m 663 00:41:42,210 --> 00:41:45,840 plus 1, m plus 2 -- 664 00:41:45,840 --> 00:41:51,320 and what we're doing is, we're designing this recovery time 665 00:41:51,320 --> 00:41:56,870 so the main peak is about a time 0, so k goes from minus 666 00:41:56,870 --> 00:42:00,890 something to plus something here and then we wind up with 667 00:42:00,890 --> 00:42:06,020 these channel terms, which are again, sampled at this point 668 00:42:06,020 --> 00:42:09,690 because we can sample them, because it's only these low 669 00:42:09,690 --> 00:42:13,850 frequency components of the channel response that make any 670 00:42:13,850 --> 00:42:15,000 difference. 671 00:42:15,000 --> 00:42:19,300 They're the only things that give us this output here and 672 00:42:19,300 --> 00:42:23,280 this output has already been filtered 673 00:42:23,280 --> 00:42:26,540 down to this low frequency. 674 00:42:26,540 --> 00:42:29,880 When I look what these components are, these are big 675 00:42:29,880 --> 00:42:31,080 messes again. 676 00:42:31,080 --> 00:42:37,370 They're sums over all of these different paths. 677 00:42:37,370 --> 00:42:45,050 So what we're winding up with is a function -- 678 00:42:45,050 --> 00:42:52,680 if I sketch g of tau and t as a function of tau, what I'm 679 00:42:52,680 --> 00:42:55,120 going to find is a bunch of different terms. 680 00:43:08,760 --> 00:43:13,420 Now I'm going to filter this so every one of these turns 681 00:43:13,420 --> 00:43:16,730 into a little sinc x over x and then I'm 682 00:43:16,730 --> 00:43:18,590 going to sample it. 683 00:43:18,590 --> 00:43:22,960 So what I wind up with then is this goes into something which 684 00:43:22,960 --> 00:43:26,130 looks like this. 685 00:43:26,130 --> 00:43:29,065 It's going to look like something which goes up and 686 00:43:29,065 --> 00:43:32,340 comes down again, and I'm just interested in what it's sample 687 00:43:32,340 --> 00:43:38,140 values are and I might wind up with five or six sample values 688 00:43:38,140 --> 00:43:39,840 and that expresses the whole thing. 689 00:43:44,100 --> 00:43:46,030 I mean, this is something you have to think about a little 690 00:43:46,030 --> 00:43:49,490 bit because it's something that we've done about 10 times 691 00:43:49,490 --> 00:43:54,120 already, but now when we do it here it looks very different 692 00:43:54,120 --> 00:43:55,780 from what we've done before -- 693 00:43:58,280 --> 00:44:00,970 except this is what it is analytically. 694 00:44:00,970 --> 00:44:11,570 This is what it is in pictures and that's where it comes out. 695 00:44:17,470 --> 00:44:20,830 The thing that's going on here then is we started to look at 696 00:44:20,830 --> 00:44:23,690 this physical modeling in terms of ray tracing and 697 00:44:23,690 --> 00:44:36,170 things like that and what we're doing at this point is 698 00:44:36,170 --> 00:44:40,180 we're modeling what's going on at the frequency bandwidth 699 00:44:40,180 --> 00:44:41,840 that we're using -- 700 00:44:41,840 --> 00:44:45,300 namely, not the carrier frequency anymore, but the 701 00:44:45,300 --> 00:44:47,550 baseband bandwidth. 702 00:44:47,550 --> 00:44:51,380 We're viewing this filter as having filtered these things 703 00:44:51,380 --> 00:44:56,650 out for us so we wind up with a baseband discrete filter 704 00:44:56,650 --> 00:45:00,420 that filters these impulses and aggregates them under 705 00:45:00,420 --> 00:45:02,270 discrete caps. 706 00:45:02,270 --> 00:45:07,780 So at this point, when we look at this filter for what the 707 00:45:07,780 --> 00:45:11,630 channel is doing, instead of seeing a bunch of impulses, we 708 00:45:11,630 --> 00:45:14,120 see a rather smooth waveform. 709 00:45:14,120 --> 00:45:18,110 If we look at what contributes to each tap, we see a large 710 00:45:18,110 --> 00:45:20,775 number of different paths, which all 711 00:45:20,775 --> 00:45:24,720 contribute to the same tap. 712 00:45:24,720 --> 00:45:28,710 If the multipath spread is not too large 713 00:45:28,710 --> 00:45:31,230 relative to 1 over w -- 714 00:45:31,230 --> 00:45:35,600 if if the multipath spread is small relative to 1 over w, 715 00:45:35,600 --> 00:45:40,510 then you can really represent this whole thing in one path. 716 00:45:40,510 --> 00:45:42,790 In other words, if the multipath spread is very, very 717 00:45:42,790 --> 00:45:47,500 small and the sampling time interval is big, everything 718 00:45:47,500 --> 00:45:51,980 just appears under one tap and that's called flat fading. 719 00:45:51,980 --> 00:45:56,490 When you have flat fading, the output that comes out is 720 00:45:56,490 --> 00:46:00,360 really just a faded version of the input and there isn't any 721 00:46:00,360 --> 00:46:01,710 time dispersion on it. 722 00:46:01,710 --> 00:46:02,960 It all just -- 723 00:46:06,730 --> 00:46:10,420 it's an undistorted version of what got sent, except it's 724 00:46:10,420 --> 00:46:16,640 very slowly fading and coming back up again and fading again 725 00:46:16,640 --> 00:46:17,950 and coming back up again. 726 00:46:21,600 --> 00:46:25,790 If you have a slightly wider bandwidth, then you need 727 00:46:25,790 --> 00:46:28,660 multiple taps. 728 00:46:28,660 --> 00:46:32,610 If you look at most of the of the systems that are being 729 00:46:32,610 --> 00:46:37,110 built today, the largest number of taps they ever used 730 00:46:37,110 --> 00:46:40,780 in this kind of implementation is three to five or something 731 00:46:40,780 --> 00:46:44,310 in that order, so it's not a huge number of taps that we're 732 00:46:44,310 --> 00:46:47,560 talking about, because the bandwidths are not huge. 733 00:46:53,990 --> 00:47:03,740 But anyway, statistically what we wind up with is that if the 734 00:47:03,740 --> 00:47:09,090 bandwidth that we're using is relatively narrow and it's not 735 00:47:09,090 --> 00:47:17,430 too many times what this term 1 over L is -- 736 00:47:17,430 --> 00:47:20,870 namely, this multipath spread, then we wind up with a small 737 00:47:20,870 --> 00:47:22,910 number of paths. 738 00:47:22,910 --> 00:47:26,440 We can sort of assume that we have a large number of paths 739 00:47:26,440 --> 00:47:30,130 coming in on each tap and then what we're doing is that each 740 00:47:30,130 --> 00:47:34,810 tap strength is going to be a sum of a bunch of terms which 741 00:47:34,810 --> 00:47:38,250 are essentially random. 742 00:47:38,250 --> 00:47:40,670 If you add up a bunch of complex terms that are 743 00:47:40,670 --> 00:47:42,480 essentially random, what do you get? 744 00:47:45,240 --> 00:47:48,850 If you add up a very humongous number of them, you get 745 00:47:48,850 --> 00:47:50,260 something which is Gaussian. 746 00:47:54,890 --> 00:47:58,460 If you were modeling these things statistically, you 747 00:47:58,460 --> 00:48:00,250 would like them to be Gaussian. 748 00:48:00,250 --> 00:48:01,870 So what do you do? 749 00:48:01,870 --> 00:48:04,610 You assume that they're Gaussian, which is what 750 00:48:04,610 --> 00:48:07,180 everyone does. 751 00:48:07,180 --> 00:48:13,240 As a slight excuse for that, what we're really interested 752 00:48:13,240 --> 00:48:17,900 in if you're designing cellular systems, you're 753 00:48:17,900 --> 00:48:21,720 trying to build cellular receivers which will deal with 754 00:48:21,720 --> 00:48:24,460 all of the different circumstances that they come 755 00:48:24,460 --> 00:48:28,850 up against and as you walk around or drive around using 756 00:48:28,850 --> 00:48:34,360 these things, if you look at the ensemble of different 757 00:48:34,360 --> 00:48:39,150 situations that these phones are faced with, over that 758 00:48:39,150 --> 00:48:43,180 ensemble of things, each of these taps is pretty much 759 00:48:43,180 --> 00:48:47,090 going to look like a Gaussian random variable. 760 00:48:47,090 --> 00:48:51,270 If you look at one sample path of one little cellular phone, 761 00:48:51,270 --> 00:48:54,430 then no, it won't look like that -- but as far as a design 762 00:48:54,430 --> 00:48:58,300 tool, it looks like something which you can model as 763 00:48:58,300 --> 00:49:04,040 Gaussian because you're going to view each of these taps in 764 00:49:04,040 --> 00:49:08,390 this filter, in this time varying filter, the sum of 765 00:49:08,390 --> 00:49:12,430 lots of unrelated components and therefore, we're going to 766 00:49:12,430 --> 00:49:18,040 take the density of these taps and a statistical model as 767 00:49:18,040 --> 00:49:19,410 being jointly Gaussian -- 768 00:49:22,320 --> 00:49:25,700 a proper Gaussian random variable for each of them. 769 00:49:25,700 --> 00:49:29,430 What happens if different taps is jointly Gaussian? 770 00:49:29,430 --> 00:49:35,030 So what we wind up with is a model for wireless channels 771 00:49:35,030 --> 00:49:41,820 then, where the channel itself is a Gaussian random process 772 00:49:41,820 --> 00:49:46,050 and as a Gaussian random process both in tau and in t, 773 00:49:46,050 --> 00:49:52,460 it's easier to look at if we look at it in terms of these 774 00:49:52,460 --> 00:49:53,490 discrete samples. 775 00:49:53,490 --> 00:49:58,770 Namely, we're going to look at the channel now, not in terms 776 00:49:58,770 --> 00:50:00,880 of a wavelength type of thing. 777 00:50:00,880 --> 00:50:04,920 We're just going to look at it as a bunch of taps, which the 778 00:50:04,920 --> 00:50:10,720 receiver has to add over to get to the received waveform. 779 00:50:10,720 --> 00:50:13,660 We're going to model each of those taps as being Gaussian. 780 00:50:13,660 --> 00:50:17,810 We're going to model in time t as being stationary. 781 00:50:17,810 --> 00:50:23,450 We're going to model them in time tau as coming up and 782 00:50:23,450 --> 00:50:27,180 going back down again and existing over a multipath 783 00:50:27,180 --> 00:50:30,260 spread that's about L in duration. 784 00:50:33,000 --> 00:50:39,430 So the phase is going to be uniform with this density. 785 00:50:39,430 --> 00:50:43,270 This is this two dimensional independent Gaussian density 786 00:50:43,270 --> 00:50:47,010 that we've looked at so many times and if you look at this, 787 00:50:47,010 --> 00:50:50,560 it has circular symmetry to it, which means if you look at 788 00:50:50,560 --> 00:50:54,940 it in an amplituding phase, the phase is random, uniform 789 00:50:54,940 --> 00:50:57,790 between zero and 2 pi. 790 00:50:57,790 --> 00:51:03,700 The energy in it is exponential and the magnitude 791 00:51:03,700 --> 00:51:07,050 is this, which comes from that -- 792 00:51:07,050 --> 00:51:11,310 which is what you call a Rayleigh density. 793 00:51:11,310 --> 00:51:14,840 So Rayleigh fading channels are simply channels which you 794 00:51:14,840 --> 00:51:18,820 have modeled as having taps, which are 795 00:51:18,820 --> 00:51:21,590 really random variables. 796 00:51:21,590 --> 00:51:28,030 Real and imaginary parts, each of them have the same 797 00:51:28,030 --> 00:51:29,200 distribution. 798 00:51:29,200 --> 00:51:32,320 I mean, you'd be very very surprised if you looked at 799 00:51:32,320 --> 00:51:37,640 these taps after the way we've derived them and you didn't 800 00:51:37,640 --> 00:51:40,260 find the same distribution on the real part as 801 00:51:40,260 --> 00:51:41,510 the imaginary part. 802 00:51:48,260 --> 00:51:51,900 If you look at the real part and imaginary part, they're 803 00:51:51,900 --> 00:51:56,100 simply coming as an arbitrary phase, which comes from some 804 00:51:56,100 --> 00:51:59,800 arbitrary demodulating frequency that you've used 805 00:51:59,800 --> 00:52:02,670 with a phase in there that doesn't have any physical 806 00:52:02,670 --> 00:52:04,540 connotation at all. 807 00:52:04,540 --> 00:52:07,650 I mean, what is real is what you choose to call real and 808 00:52:07,650 --> 00:52:11,490 what is imaginary is what you choose to call imaginary. 809 00:52:11,490 --> 00:52:15,650 If you made your time reference just a little bit, a 810 00:52:15,650 --> 00:52:18,870 smidgen away from where your time reference is, what is 811 00:52:18,870 --> 00:52:21,840 real would become imaginary and what is imaginary would 812 00:52:21,840 --> 00:52:25,250 become real again and you can't expect the channel to 813 00:52:25,250 --> 00:52:29,140 know what time reference you've chosen. 814 00:52:29,140 --> 00:52:32,350 I mean, this is sort of a big philosophical argument, but 815 00:52:32,350 --> 00:52:36,400 you certainly wouldn't expect these random variables to have 816 00:52:36,400 --> 00:52:38,900 a distribution which is anything other 817 00:52:38,900 --> 00:52:41,220 than uniform in phase. 818 00:52:41,220 --> 00:52:45,750 So that's the distribution we get and what we now have is we 819 00:52:45,750 --> 00:52:53,170 changed this awful physical situation into a very simple 820 00:52:53,170 --> 00:52:58,070 analytical situation where what we're doing is modeling 821 00:52:58,070 --> 00:53:03,350 this complex channel as just a bunch of taps in time, at some 822 00:53:03,350 --> 00:53:07,190 bandwidth that we're using and each of these taps is now 823 00:53:07,190 --> 00:53:14,820 Gaussian and the output is going to be the discrete sum 824 00:53:14,820 --> 00:53:18,710 as we take a discrete sequence of inputs, pass it through 825 00:53:18,710 --> 00:53:23,100 this discrete filter and then we add up these outputs and at 826 00:53:23,100 --> 00:53:25,840 that point, we have to worry about how do we detect things 827 00:53:25,840 --> 00:53:27,730 from what's coming out? 828 00:53:27,730 --> 00:53:31,040 So they say here -- this is a really flaky modeling 829 00:53:31,040 --> 00:53:36,370 assumption and it's only partly justified by looking at 830 00:53:36,370 --> 00:53:40,890 lots of different situations that a given cellular phone 831 00:53:40,890 --> 00:53:44,310 would be placed in. 832 00:53:44,310 --> 00:53:47,790 If we look at this whole ensemble, it makes a little 833 00:53:47,790 --> 00:53:56,380 bit of sense, but really what we're doing here is saying 834 00:53:56,380 --> 00:54:00,280 what we want is a vehicle to let us understand how to 835 00:54:00,280 --> 00:54:02,010 receive these waveforms. 836 00:54:02,010 --> 00:54:04,350 We know there is going to be fading. 837 00:54:04,350 --> 00:54:07,290 Since we know there is going to be fading, we have to find 838 00:54:07,290 --> 00:54:10,520 some kind of statistical model for it if we're going to find 839 00:54:10,520 --> 00:54:13,100 sensible things to do and this is the one 840 00:54:13,100 --> 00:54:14,350 we're going to pick. 841 00:54:16,800 --> 00:54:20,530 It's very common instead of using a Rayleigh fading model 842 00:54:20,530 --> 00:54:22,870 to use a Rician fading model. 843 00:54:22,870 --> 00:54:28,620 A Rician fading model makes sense because if you have a 844 00:54:28,620 --> 00:54:34,790 line of sight directly to a base station, the waveform you 845 00:54:34,790 --> 00:54:38,090 get from the base station is going to be kind of strong and 846 00:54:38,090 --> 00:54:41,790 all of these waveforms you get reflected from other obstacles 847 00:54:41,790 --> 00:54:45,320 and so forth are going to be rather weak. 848 00:54:45,320 --> 00:54:48,300 So in that case, you're going to have one strong received 849 00:54:48,300 --> 00:54:52,290 waveform, which is one big peak and everything else is 850 00:54:52,290 --> 00:54:53,840 going to be very weak. 851 00:54:56,910 --> 00:54:59,490 The thing that that says is the kind of statistics that 852 00:54:59,490 --> 00:55:04,020 you want are not a sum of a lot of little tiny things, but 853 00:55:04,020 --> 00:55:07,910 the sum of one big thing and lots of tiny things. 854 00:55:07,910 --> 00:55:09,910 That's not a good model either. 855 00:55:09,910 --> 00:55:13,340 What you would really like is a sum of a bunch of medium 856 00:55:13,340 --> 00:55:17,420 sized things most of the time, but about as far as anybody 857 00:55:17,420 --> 00:55:26,280 gets talking about statistical models for fading channels is 858 00:55:26,280 --> 00:55:30,090 talking about Rician model and talking about Rayleigh models 859 00:55:30,090 --> 00:55:32,820 and then worrying about what happens the different paths of 860 00:55:32,820 --> 00:55:34,970 these filters. 861 00:55:34,970 --> 00:55:38,590 The trouble with Rician random variables is they're very 862 00:55:38,590 --> 00:55:39,710 complicated. 863 00:55:39,710 --> 00:55:42,050 They have vessel functions in them. 864 00:55:42,050 --> 00:55:47,100 If you read the notes I passed out today, you find that when 865 00:55:47,100 --> 00:55:51,690 you try to detect things, when they've gone through this kind 866 00:55:51,690 --> 00:55:57,780 of Rician fading, everything is just a bloody mess. 867 00:55:57,780 --> 00:56:00,690 That's the way it is. 868 00:56:03,210 --> 00:56:07,850 Since we are talking now about a statistical model for these 869 00:56:07,850 --> 00:56:12,463 channels, we would like to have something called a tap 870 00:56:12,463 --> 00:56:19,040 gain correlation function, which is going to tell us how 871 00:56:19,040 --> 00:56:22,540 each of these taps vary in time. 872 00:56:25,650 --> 00:56:33,020 That's just the expected value of Gkm and G of k prime n. 873 00:56:33,020 --> 00:56:36,050 What we're going to assume, and what is really a very good 874 00:56:36,050 --> 00:56:40,220 assumption, is if you look at the things that add up as taps 875 00:56:40,220 --> 00:56:45,870 under a particular value of k and under some other value of 876 00:56:45,870 --> 00:56:50,040 k -- in other words, all of the paths that are coming in 877 00:56:50,040 --> 00:56:53,710 in one range of delays and all the paths that come in on 878 00:56:53,710 --> 00:56:56,450 another range of delays, those things are pretty much 879 00:56:56,450 --> 00:56:58,190 independent of each other. 880 00:56:58,190 --> 00:57:05,080 So almost always, when you start calculating tap gain 881 00:57:05,080 --> 00:57:07,840 correlation functions, you assume that this is 882 00:57:07,840 --> 00:57:10,370 independent when k is unequal to k prime. 883 00:57:10,370 --> 00:57:13,340 So really the only thing you're concerned about here is 884 00:57:13,340 --> 00:57:18,030 how these quantities vary in time. 885 00:57:18,030 --> 00:57:20,830 We've already talked about how they vary in time. 886 00:57:20,830 --> 00:57:26,860 In other words, that was why we talked about all of this 887 00:57:26,860 --> 00:57:30,520 business about time coherence and Doppler shifts. 888 00:57:30,520 --> 00:57:33,610 The thing which is causing these things to change in time 889 00:57:33,610 --> 00:57:36,000 is these Doppler shifts. 890 00:57:36,000 --> 00:57:39,040 We have already decided that the amount of time the channel 891 00:57:39,040 --> 00:57:46,040 remain stable is about 1 over 2 times this Doppler spread. 892 00:57:46,040 --> 00:57:50,390 Here we have an analytical, statistical way of doing the 893 00:57:50,390 --> 00:57:52,570 same thing. 894 00:57:52,570 --> 00:57:56,840 In other words, you can look at this correlation function 895 00:57:56,840 --> 00:58:00,710 here and you can see how long it takes before it drops 896 00:58:00,710 --> 00:58:02,800 essentially to zero. 897 00:58:02,800 --> 00:58:08,270 So you can define the time coherence in terms of this 898 00:58:08,270 --> 00:58:12,900 duration here just as well as the way we've already done. 899 00:58:12,900 --> 00:58:17,410 In other words, how big does n have to get before this thing 900 00:58:17,410 --> 00:58:19,660 gets small? 901 00:58:19,660 --> 00:58:22,080 What's the advantage of doing it this way instead of in 902 00:58:22,080 --> 00:58:25,060 terms of Doppler shifts? 903 00:58:25,060 --> 00:58:27,160 If you want to measure it, you don't have to go out and 904 00:58:27,160 --> 00:58:29,210 measure where these paths are. 905 00:58:29,210 --> 00:58:34,670 If you want to measure it, you just take a cellular phone and 906 00:58:34,670 --> 00:58:37,490 you measure where these things are coming in and how they 907 00:58:37,490 --> 00:58:41,380 change in time and that gives you a direct view of what the 908 00:58:41,380 --> 00:58:43,970 time coherence is. 909 00:58:43,970 --> 00:58:49,360 Why that W was there, there's a problem in the problem set 910 00:58:49,360 --> 00:58:51,720 that will give you some insight into why it's there, 911 00:58:51,720 --> 00:58:55,720 but otherwise it's just arbitrary. 912 00:58:55,720 --> 00:58:58,290 I want to spend the rest of the time talking about 913 00:58:58,290 --> 00:59:03,440 Rayleigh fading, because Rayleigh fading is something 914 00:59:03,440 --> 00:59:05,420 you can analyze easily. 915 00:59:10,320 --> 00:59:12,280 We're going to do something even simpler. 916 00:59:12,280 --> 00:59:15,110 We're going to do Rayleigh fading where you have 917 00:59:15,110 --> 00:59:17,380 a single tap model. 918 00:59:17,380 --> 00:59:21,040 In other words, where the bandwidth that you're using is 919 00:59:21,040 --> 00:59:28,090 small enough that all these different paths all fall in 920 00:59:28,090 --> 00:59:29,630 the same delay range. 921 00:59:29,630 --> 00:59:37,690 In other words, it's where the multipath spread, L, is small 922 00:59:37,690 --> 00:59:42,370 relative to 1 over W. That's what's called flat fading, so 923 00:59:42,370 --> 00:59:45,550 we're assuming flat fading and we're assuming that it changes 924 00:59:45,550 --> 00:59:48,240 in time according to a Rayleigh model. 925 00:59:50,980 --> 00:59:59,030 Suppose that we do something really simple, like trying the 926 00:59:59,030 --> 01:00:02,060 antipodal signalling that we've been using all along, 927 01:00:02,060 --> 01:00:06,130 which is really neat when you have Gaussian noise. 928 01:00:06,130 --> 01:00:14,390 It's just about the best kind of signaling that you can use. 929 01:00:14,390 --> 01:00:18,410 It doesn't work at all here and it doesn't work at all 930 01:00:18,410 --> 01:00:23,360 because since this channel is Rayleigh fading, you don't 931 01:00:23,360 --> 01:00:27,570 know what the phase is of the channel, so you send a bit and 932 01:00:27,570 --> 01:00:31,180 what comes in is something which, sure, it has an 933 01:00:31,180 --> 01:00:34,880 amplitude to it, but it has a completely random phase to it, 934 01:00:34,880 --> 01:00:36,530 so you can't tell the difference between 935 01:00:36,530 --> 01:00:38,450 one and minus one. 936 01:00:38,450 --> 01:00:42,350 The only way you can tell is to send first one signal and 937 01:00:42,350 --> 01:00:45,620 then another signal, but if you send one signal and 938 01:00:45,620 --> 01:00:49,910 another signal, you might as well choose a single pattern 939 01:00:49,910 --> 01:00:52,720 which lets you do this sensibly. 940 01:00:52,720 --> 01:00:55,840 So the particular thing that we'll look at is something 941 01:00:55,840 --> 01:00:58,880 called post position modulation. 942 01:00:58,880 --> 01:01:01,850 This is essentially the same as about ten other schemes and 943 01:01:01,850 --> 01:01:06,580 we'll talk about that a little more later, but what you're 944 01:01:06,580 --> 01:01:10,630 going to use is two degrees of freedom, which is two 945 01:01:10,630 --> 01:01:13,060 different time instants. 946 01:01:13,060 --> 01:01:15,930 In the first instant, you're going to send an a. 947 01:01:15,930 --> 01:01:19,680 In the second instant, you're going to send a zero where 948 01:01:19,680 --> 01:01:22,850 conversely, you're going to send a zero in the first 949 01:01:22,850 --> 01:01:26,750 degree of freedom and an a in the second degree of freedom. 950 01:01:26,750 --> 01:01:30,280 So you can do this either with two different frequencies, you 951 01:01:30,280 --> 01:01:32,570 can do it with two different time instants. 952 01:01:32,570 --> 01:01:34,960 You can do it in all sorts of different ways. 953 01:01:34,960 --> 01:01:39,660 Any time you have two degrees of freedom, you either use one 954 01:01:39,660 --> 01:01:42,940 degree of freedom or you use the other degree of freedom. 955 01:01:42,940 --> 01:01:45,870 It's not like the situation we've talked about before, 956 01:01:45,870 --> 01:01:49,320 because these degrees of freedom are now complex 957 01:01:49,320 --> 01:01:50,910 degrees of freedom. 958 01:01:50,910 --> 01:01:55,420 If I send the one, it's going to come in as some complex 959 01:01:55,420 --> 01:02:01,140 number, which is going to have uniform phase. 960 01:02:01,140 --> 01:02:09,240 So my hypotheses now are one hypothesis is a and zero and 961 01:02:09,240 --> 01:02:13,260 the other hypothesis is zero and a. 962 01:02:13,260 --> 01:02:18,550 What's going to happen with these hypotheses is that 963 01:02:18,550 --> 01:02:21,440 first, the channel is going to do its thing to them. 964 01:02:21,440 --> 01:02:25,430 In other words, if I send this, the channel is going to 965 01:02:25,430 --> 01:02:27,470 take that a. 966 01:02:27,470 --> 01:02:31,780 It's going to multiply it by a circularly symmetric Gaussian 967 01:02:31,780 --> 01:02:37,150 random variable, so it's going to come as some blob and in 968 01:02:37,150 --> 01:02:41,410 the other dimension, what I send is a zero, so what comes 969 01:02:41,410 --> 01:02:43,170 in there is nothing. 970 01:02:43,170 --> 01:02:47,890 I'm going to add noise to this, so when I send a zero, 971 01:02:47,890 --> 01:02:49,530 I'm going to get -- 972 01:02:49,530 --> 01:02:51,950 from the channel, I get a blob in this 973 01:02:51,950 --> 01:02:53,490 first degree of freedom. 974 01:02:53,490 --> 01:02:55,440 The noise adds another blob, which is 975 01:02:55,440 --> 01:02:58,110 also circularly symmetric. 976 01:02:58,110 --> 01:03:01,600 What comes in the other degree of freedom is zero on the 977 01:03:01,600 --> 01:03:04,120 channel and a blob of noise. 978 01:03:04,120 --> 01:03:08,660 So I'm going to base my detection on either a big blob 979 01:03:08,660 --> 01:03:12,310 or a little blob and on the other hypothesis, I got a 980 01:03:12,310 --> 01:03:15,610 little blob here and a big blob there. 981 01:03:15,610 --> 01:03:21,210 So the question is, you have to look at this pair of blobs 982 01:03:21,210 --> 01:03:23,380 and decide from it what you think was 983 01:03:23,380 --> 01:03:25,840 the most likely signals. 984 01:03:25,840 --> 01:03:34,000 Sounds hard, but it turns out that it's easy, because look, 985 01:03:34,000 --> 01:03:39,820 under the hypothesis H 0, what you're going to receive -- 986 01:03:39,820 --> 01:03:43,800 and these are a times a complex 987 01:03:43,800 --> 01:03:47,310 variable, G 0 plus Z 0. 988 01:03:47,310 --> 01:03:50,090 Both of these are Gaussian random variables, both zero 989 01:03:50,090 --> 01:03:55,630 mean, both circularly symmetric and V 1 is going to 990 01:03:55,630 --> 01:03:57,630 be just Z 1. 991 01:03:57,630 --> 01:04:01,340 So here I have a sum of two Gaussian random variables. 992 01:04:01,340 --> 01:04:04,840 Here I have one Gaussian random variable. 993 01:04:04,840 --> 01:04:17,190 V 0 is complex Gaussian and its variance is a squared plus 994 01:04:17,190 --> 01:04:21,550 N 0 times W, because that's what the noise is. 995 01:04:21,550 --> 01:04:27,320 So I look at the probability density of these two complex 996 01:04:27,320 --> 01:04:31,330 Gaussian random variables, sample value v 0 and v 1, 997 01:04:31,330 --> 01:04:33,350 conditional on H 0. 998 01:04:33,350 --> 01:04:35,700 What I have is some constant here -- 999 01:04:35,700 --> 01:04:37,370 I don't even care about it -- 1000 01:04:37,370 --> 01:04:41,470 times e need to the minus v 0 squared divided by a 1001 01:04:41,470 --> 01:04:45,070 squared plus wn 0. 1002 01:04:45,070 --> 01:04:49,700 This is the variance of V 0, because V 0 was a 1003 01:04:49,700 --> 01:04:51,820 squared times G 0. 1004 01:04:51,820 --> 01:04:54,000 That's the a squared there. 1005 01:04:54,000 --> 01:05:00,170 Variance is Z 0 WN 0 and for the other random variable -- 1006 01:05:00,170 --> 01:05:03,260 that's the little blob random variable -- 1007 01:05:03,260 --> 01:05:06,660 probability density of that is minus v 1 squared 1008 01:05:06,660 --> 01:05:11,510 divided by WN 0. 1009 01:05:11,510 --> 01:05:14,640 For the alternative hypothesis, I get the same 1010 01:05:14,640 --> 01:05:17,470 constant out front, which I haven't even bother to write 1011 01:05:17,470 --> 01:05:23,480 down, times e to the minus v 0 squared over WN 0. 1012 01:05:23,480 --> 01:05:26,430 Here are the other variables: The one where we have the big 1013 01:05:26,430 --> 01:05:32,060 blob, so that's the variance a squared plus WN 0. 1014 01:05:32,060 --> 01:05:36,980 I take the log likelihood ratio, which is to take the 1015 01:05:36,980 --> 01:05:42,400 logarithm of the ratio of these two hypotheses, of these 1016 01:05:42,400 --> 01:05:48,450 two likelihoods, and when I take this quantity and divide 1017 01:05:48,450 --> 01:05:52,890 it by this quantity, these are both up in the exponents, so 1018 01:05:52,890 --> 01:05:56,590 I'm just taking this and subtracting this. 1019 01:05:56,590 --> 01:06:00,450 When I subtract this quantity from this quantity and then 1020 01:06:00,450 --> 01:06:05,000 subtract this quantity from this quantity, what I get is 1021 01:06:05,000 --> 01:06:10,110 just this quantity here. 1022 01:06:10,110 --> 01:06:14,480 I mean, you can see it's simple because this difference 1023 01:06:14,480 --> 01:06:16,970 is going to be the same as this difference, except they 1024 01:06:16,970 --> 01:06:18,670 have a different sign. 1025 01:06:18,670 --> 01:06:22,060 So it's just algebra to find out what happens when you when 1026 01:06:22,060 --> 01:06:25,190 you take this minus this and then you take 1027 01:06:25,190 --> 01:06:27,980 minus this plus this. 1028 01:06:27,980 --> 01:06:30,660 This is what you get. 1029 01:06:30,660 --> 01:06:34,400 Then you look at that for a bit and you say, how do I find 1030 01:06:34,400 --> 01:06:37,580 the probability of error from this? 1031 01:06:37,580 --> 01:06:41,210 Before we find the probability of error, the first thing we 1032 01:06:41,210 --> 01:06:47,050 have to do is say, what's the maximum 1033 01:06:47,050 --> 01:06:50,500 likelihood rule to use here? 1034 01:06:50,500 --> 01:06:54,530 Maximum likelihood rule is you take the log likelihood ratio 1035 01:06:54,530 --> 01:06:57,730 and if it's positive, you choose v0. 1036 01:06:57,730 --> 01:07:01,770 If it's negative, you choose v1 and if it's 0 -- if 0 was 1037 01:07:01,770 --> 01:07:05,390 zero probability, it doesn't matter what you do. 1038 01:07:05,390 --> 01:07:09,140 So what we're interested in now is, what is the 1039 01:07:09,140 --> 01:07:13,800 probability that this quantity here is going to 1040 01:07:13,800 --> 01:07:17,460 be bigger than zero? 1041 01:07:17,460 --> 01:07:24,720 But you see, this isn't hard because when you look at the 1042 01:07:24,720 --> 01:07:29,430 difference between v 0 squared and v 1 squared, v 0 squared 1043 01:07:29,430 --> 01:07:32,430 is an exponential random variable. 1044 01:07:32,430 --> 01:07:37,490 It's just the energy of a complex Gaussian. 1045 01:07:37,490 --> 01:07:41,210 So we're taking one exponential random variable, 1046 01:07:41,210 --> 01:07:44,710 subtracting off another exponential random variable 1047 01:07:44,710 --> 01:07:49,790 and we're just looking at it over on the region where these 1048 01:07:49,790 --> 01:07:53,170 things are positive. 1049 01:07:53,170 --> 01:07:59,470 When you integrate that, this is the answer that you get. 1050 01:07:59,470 --> 01:08:00,875 I mean, it's not hard to integrate it. 1051 01:08:00,875 --> 01:08:02,125 I just don't -- 1052 01:08:04,800 --> 01:08:07,850 and if you look at the notes, the notes carry it out in all 1053 01:08:07,850 --> 01:08:12,900 of its gory detail, but it really is about two steps. 1054 01:08:12,900 --> 01:08:17,350 The thing we want to look at is this result here, because 1055 01:08:17,350 --> 01:08:21,370 it looks so different from all of the error probability 1056 01:08:21,370 --> 01:08:23,610 results that we've gotten before. 1057 01:08:23,610 --> 01:08:26,880 I mean, before, error probabilities go down 1058 01:08:26,880 --> 01:08:33,670 exponentially as a square of the signal amplitudes that 1059 01:08:33,670 --> 01:08:37,350 we're using and they go down exponentially with the energy 1060 01:08:37,350 --> 01:08:38,890 that we're using. 1061 01:08:38,890 --> 01:08:46,230 Here, this crazy thing is going down as 1 over 2 plus 1062 01:08:46,230 --> 01:08:48,270 energy to noise ratio. 1063 01:08:48,270 --> 01:09:02,390 This is 1 over 2 plus Ed over N 0. 1064 01:09:02,390 --> 01:09:04,880 So it really goes down slowly. 1065 01:09:04,880 --> 01:09:07,440 So what's going on? 1066 01:09:07,440 --> 01:09:10,500 I mean, why is it that with all of these Gaussian random 1067 01:09:10,500 --> 01:09:14,440 variables where things go down so fast, we're really in such 1068 01:09:14,440 --> 01:09:15,690 deep trouble here? 1069 01:09:18,810 --> 01:09:23,010 The trouble is, every time the fading is significant and the 1070 01:09:23,010 --> 01:09:27,340 fading is significant whenever -- 1071 01:09:27,340 --> 01:09:32,630 under hypothesis 0, if you don't have much 1072 01:09:32,630 --> 01:09:34,190 of a channel -- 1073 01:09:34,190 --> 01:09:37,110 namely, if both v 0 and v 1 are both very -- 1074 01:09:41,480 --> 01:09:45,940 if both the real part of G 0 and the imaginary part of G 0 1075 01:09:45,940 --> 01:09:47,750 are both very small -- 1076 01:09:47,750 --> 01:09:51,160 which happens with reasonable probability -- 1077 01:09:51,160 --> 01:09:59,280 then you don't have a chance of trying to decode correctly 1078 01:09:59,280 --> 01:10:02,740 because the two signals are both going to look the same, 1079 01:10:02,740 --> 01:10:06,660 because both of them are just Gaussian random variables. 1080 01:10:06,660 --> 01:10:12,220 So as a result of that, when you're dealing with Rayleigh 1081 01:10:12,220 --> 01:10:17,580 fading, you really have to find something else to do in 1082 01:10:17,580 --> 01:10:19,700 order to make this work. 1083 01:10:19,700 --> 01:10:22,000 I mean, if you have Rayleigh fading and you just try to 1084 01:10:22,000 --> 01:10:26,320 communicate in the way that we tried to communicate before -- 1085 01:10:26,320 --> 01:10:30,700 just sending individual bits and hoping for the best, 1086 01:10:30,700 --> 01:10:32,890 you're in very deep trouble. 1087 01:10:32,890 --> 01:10:36,660 So next time we'll start to talk about ways of dealing 1088 01:10:36,660 --> 01:10:41,170 with this using diversity, using channel measurement, 1089 01:10:41,170 --> 01:10:44,400 using all sorts of other things, but -- 1090 01:10:44,400 --> 01:10:46,190 I don't know what happened to the person who was supposed to 1091 01:10:46,190 --> 01:10:53,380 evaluate the class, but maybe they will come in next week. 1092 01:10:53,380 --> 01:10:54,900 I don't Know.