1 00:00:00,000 --> 00:00:02,203 FEMALE VOICE: The following content is provided under a 2 00:00:02,203 --> 00:00:03,640 creative commons license. 3 00:00:03,640 --> 00:00:06,730 Your support will help MIT OpenCourseWare continue to 4 00:00:06,730 --> 00:00:09,970 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 --> 00:00:19,360 ocw.mit.edu. 8 00:00:19,360 --> 00:00:24,350 PROFESSOR: We've just started talking about wireless 9 00:00:24,350 --> 00:00:26,290 communication. 10 00:00:26,290 --> 00:00:31,230 We spent a lot of time talking about how do you communicate, 11 00:00:31,230 --> 00:00:35,690 essentially on wire lines, where essentially the only 12 00:00:35,690 --> 00:00:38,510 problem is white Gaussian noise. 13 00:00:38,510 --> 00:00:42,230 Namely, you transmit a signal, noise gets added to it, you 14 00:00:42,230 --> 00:00:47,080 receive the sum of the transmitted signal plus noise 15 00:00:47,080 --> 00:00:50,160 and all of the going from base band to pass band. 16 00:00:50,160 --> 00:00:55,300 All of that stuff is all messy analytically, but essentially 17 00:00:55,300 --> 00:00:59,470 all that's happening is that you're going from -- 18 00:00:59,470 --> 00:01:04,470 you take a signal, you move up to pass band, you add noise, 19 00:01:04,470 --> 00:01:07,210 you move back to base band, in which case you get the 20 00:01:07,210 --> 00:01:10,610 original transmitted signal plus the noise 21 00:01:10,610 --> 00:01:13,000 moved back to -- 22 00:01:13,000 --> 00:01:14,610 move back -- 23 00:01:14,610 --> 00:01:17,770 moved to base band. 24 00:01:17,770 --> 00:01:19,700 So what we wound up with there was a 25 00:01:19,700 --> 00:01:22,150 relatively simple situation. 26 00:01:22,150 --> 00:01:25,480 Wireless communication is not so simple. 27 00:01:25,480 --> 00:01:30,030 As you all know, if you use a cell phone, and I'm sure most 28 00:01:30,030 --> 00:01:34,630 of you do, because of something called fading -- 29 00:01:34,630 --> 00:01:37,870 what we want to understand is where this fading comes from, 30 00:01:37,870 --> 00:01:42,990 how it arises and some of the things you might do about it. 31 00:01:42,990 --> 00:01:45,350 One of the things you should recognize right at the 32 00:01:45,350 --> 00:01:48,250 beginning -- and we won't spend much time 33 00:01:48,250 --> 00:01:50,380 talking about it -- 34 00:01:50,380 --> 00:01:56,490 is if you happen to be trying to use a cellular phone and 35 00:01:56,490 --> 00:01:59,830 there's a big wall, which is perfectly reflecting right in 36 00:01:59,830 --> 00:02:02,400 front of you and the base station you're trying to 37 00:02:02,400 --> 00:02:05,020 communicate with is on the other side of that wall, 38 00:02:05,020 --> 00:02:07,070 you're not going to get through. 39 00:02:07,070 --> 00:02:10,960 In other words, there are some situations where no matter how 40 00:02:10,960 --> 00:02:14,240 you design a cellular phone, you just can't get any 41 00:02:14,240 --> 00:02:15,790 communication. 42 00:02:15,790 --> 00:02:17,650 It's part of life. 43 00:02:17,650 --> 00:02:22,130 What we want to do, however, is to make sure that when you 44 00:02:22,130 --> 00:02:24,890 can get communication, you will get 45 00:02:24,890 --> 00:02:27,290 it as well as possible. 46 00:02:27,290 --> 00:02:31,330 You will prolong the period, but you're still communicating 47 00:02:31,330 --> 00:02:34,060 while the channel's getting worse and worse -- 48 00:02:34,060 --> 00:02:38,150 and part of the way to do that is to understand what it is in 49 00:02:38,150 --> 00:02:43,950 the physical mechanisms that's making the problem difficult. 50 00:02:43,950 --> 00:02:48,780 OK, so to start out with this, we'll start out kind of easy. 51 00:02:48,780 --> 00:02:54,620 We'll assume an input, which is just a cosine of 2 pi ft at 52 00:02:54,620 --> 00:02:57,100 some ficed antenna -- 53 00:02:57,100 --> 00:03:03,610 and this is radiating outwards and the electric field -- 54 00:03:06,170 --> 00:03:10,640 anywhere in free space, if you go a long distance away. 55 00:03:10,640 --> 00:03:13,740 If you go a very short distance from the antenna, if 56 00:03:13,740 --> 00:03:16,590 you study electromagnetism, you know that all sorts of 57 00:03:16,590 --> 00:03:18,520 crazy things are going on. 58 00:03:18,520 --> 00:03:21,830 When you get very far away, there's something called the 59 00:03:21,830 --> 00:03:26,330 far field and essentially what happens is that all of these 60 00:03:26,330 --> 00:03:31,000 disturbances close in just sort of disappear and very far 61 00:03:31,000 --> 00:03:34,830 away what's happening is that the field strength -- 62 00:03:34,830 --> 00:03:37,090 and this is true for the magnetic field also, 63 00:03:37,090 --> 00:03:39,710 which is just in -- 64 00:03:39,710 --> 00:03:41,990 if the electric field is this way, the magnetic 65 00:03:41,990 --> 00:03:43,390 field is that way. 66 00:03:43,390 --> 00:03:46,870 They're both propogating outwards and they're going 67 00:03:46,870 --> 00:03:51,730 down as 1 over r, just like when we deal with linear 68 00:03:51,730 --> 00:03:55,720 systems, we never talk about voltage and current anymore. 69 00:03:55,720 --> 00:03:59,270 This is why we use a square root of minus 1 70 00:03:59,270 --> 00:04:02,700 at this point -- 71 00:04:02,700 --> 00:04:05,120 because after you learn about voltage and current, you don't 72 00:04:05,120 --> 00:04:07,640 have to talk about them anymore. 73 00:04:07,640 --> 00:04:11,030 You just deal with one of them for the function that exists 74 00:04:11,030 --> 00:04:13,440 someplace and the other one just follows 75 00:04:13,440 --> 00:04:15,120 along from the impedance. 76 00:04:15,120 --> 00:04:16,950 Same thing happens for electric fields 77 00:04:16,950 --> 00:04:18,240 and magnetic fields. 78 00:04:18,240 --> 00:04:21,240 You don't have to bother about both of them, unless you're 79 00:04:21,240 --> 00:04:24,770 really trying to solve Maxwell's field equations, 80 00:04:24,770 --> 00:04:27,700 which is a very challenging endeavor. 81 00:04:27,700 --> 00:04:30,670 I admire any of you who can do this. 82 00:04:30,670 --> 00:04:33,900 I used to be able to do it many, many years ago and I've 83 00:04:33,900 --> 00:04:37,900 given up on it because I decided that's for younger 84 00:04:37,900 --> 00:04:39,150 people than me. 85 00:04:41,640 --> 00:04:45,840 This far field has a very simple kind of behavior, 86 00:04:45,840 --> 00:04:50,810 because what happens is it has to go down as 1 over r. 87 00:04:50,810 --> 00:04:53,450 How do you know it has to go down as 1 over r? 88 00:04:53,450 --> 00:05:00,090 Namely, the distance away from this radiating antenna. 89 00:05:00,090 --> 00:05:03,480 If you look at a sphere, which is very, very far away from 90 00:05:03,480 --> 00:05:08,220 the radiating antenna, all you find is this field that's 91 00:05:08,220 --> 00:05:11,750 radiating outwards and you look at how much energy is 92 00:05:11,750 --> 00:05:14,150 radiating outwards. 93 00:05:14,150 --> 00:05:18,470 The energy which is radiating outwards, there's no place to 94 00:05:18,470 --> 00:05:21,930 lose it because we're transmitting just an open 95 00:05:21,930 --> 00:05:25,490 space now -- or at least that's what we're imagining at 96 00:05:25,490 --> 00:05:26,080 this point. 97 00:05:26,080 --> 00:05:29,940 So we're transmitting out in open space. 98 00:05:29,940 --> 00:05:33,270 Energy doesn't get beaten up any place so it just keeps 99 00:05:33,270 --> 00:05:37,890 going out until it disappears out of the outer edges of the 100 00:05:37,890 --> 00:05:39,870 known universe. 101 00:05:39,870 --> 00:05:42,440 It travels out there with the speed of light. 102 00:05:42,440 --> 00:05:47,380 That's what Maxwell's laws say if you solve them. 103 00:05:47,380 --> 00:05:51,840 Since this sphere has an area which is proportional to r 104 00:05:51,840 --> 00:05:56,190 squared, the only thing that can be happening in this wave, 105 00:05:56,190 --> 00:05:59,710 which is propogating outward spherically, is it has to be 106 00:05:59,710 --> 00:06:02,920 going down as 1 over r, because that's the only way 107 00:06:02,920 --> 00:06:05,640 for the power to balance out. 108 00:06:05,640 --> 00:06:13,210 You can't be creating power out in this free space and you 109 00:06:13,210 --> 00:06:15,410 can't be losing power. 110 00:06:15,410 --> 00:06:19,280 Whatever you're sending is just radiating outwards, so we 111 00:06:19,280 --> 00:06:22,450 have to have this 1 over r dependence any time you're 112 00:06:22,450 --> 00:06:24,160 dealing with free space. 113 00:06:24,160 --> 00:06:26,410 The other thing that's happening is we have an 114 00:06:26,410 --> 00:06:27,860 antenna pattern. 115 00:06:27,860 --> 00:06:31,240 The antenna pattern is a function of two angles. 116 00:06:31,240 --> 00:06:34,670 We'll think of theta as being the angle around this way and 117 00:06:34,670 --> 00:06:37,230 psi as being the angle this way. 118 00:06:37,230 --> 00:06:39,310 If you like to think of angles in some 119 00:06:39,310 --> 00:06:42,380 other way, be my guest. 120 00:06:42,380 --> 00:06:45,520 The only thing is, when we're radiating outwards through 121 00:06:45,520 --> 00:06:49,330 this sphere, if the antenna is directional, it's going to be 122 00:06:49,330 --> 00:06:52,770 radiating more in some directions than it is in other 123 00:06:52,770 --> 00:06:57,330 directions and this simply takes that into account. 124 00:06:57,330 --> 00:07:00,120 I'm not going to pay any attention to this at all. 125 00:07:00,120 --> 00:07:02,530 It just exists and it's there. 126 00:07:02,530 --> 00:07:05,080 For people who design antennas, that's an important 127 00:07:05,080 --> 00:07:07,840 issue: How do you make antennas which are 128 00:07:07,840 --> 00:07:11,410 directional, which will shine their energy in one direction 129 00:07:11,410 --> 00:07:13,210 rather than another? 130 00:07:13,210 --> 00:07:14,930 We're not going to go into that at all. 131 00:07:14,930 --> 00:07:18,290 We're just representing the fact that it's there. 132 00:07:18,290 --> 00:07:23,740 This is just a factor which says, how much loss do you 133 00:07:23,740 --> 00:07:26,940 have in the antenna and how much of the power that you're 134 00:07:26,940 --> 00:07:30,450 radiating goes in each one of these directions and how does 135 00:07:30,450 --> 00:07:33,660 it depend on the frequency that you're transmitting at? 136 00:07:33,660 --> 00:07:36,480 Antennas are sometimes designed to be rather 137 00:07:36,480 --> 00:07:38,530 frequency dependent, so they're tuned 138 00:07:38,530 --> 00:07:40,120 to a certain frequency. 139 00:07:40,120 --> 00:07:42,700 They work very well at that frequency and on other 140 00:07:42,700 --> 00:07:46,170 frequencies, they just go to pot. 141 00:07:46,170 --> 00:07:50,090 The other part of this equation is that if I send a 142 00:07:50,090 --> 00:07:53,060 signal, as it radiates outward, it's going to be 143 00:07:53,060 --> 00:07:58,210 radiating outward at the speed of light. 144 00:07:58,210 --> 00:08:02,980 Therefore, whatever I receive is going to be delayed by the 145 00:08:02,980 --> 00:08:07,700 distance that I am away divided by the speed of light. 146 00:08:07,700 --> 00:08:10,980 So this equation is something you don't really have to know 147 00:08:10,980 --> 00:08:14,530 any electromagnetics to derive. 148 00:08:14,530 --> 00:08:17,010 If you want to find out what this term is, yes, you'd need 149 00:08:17,010 --> 00:08:18,620 some electromagnetics. 150 00:08:18,620 --> 00:08:21,390 All this is saying is the power has to be going down as 151 00:08:21,390 --> 00:08:25,880 1 over r and you have a propogation delay, which has 152 00:08:25,880 --> 00:08:30,430 to be going as r divided by the speed of light. 153 00:08:30,430 --> 00:08:36,600 Now, if we look at what happens when we put a 154 00:08:36,600 --> 00:08:41,670 receiving antenna out at some distance r away from the 155 00:08:41,670 --> 00:08:50,360 transmitting antenna and in this direction theta and psi, 156 00:08:50,360 --> 00:08:55,030 that receiving antenna is going to distort this 157 00:08:55,030 --> 00:08:59,970 electromagnetic wave locally around the receiving antenna, 158 00:08:59,970 --> 00:09:04,160 but it's not going to distort the whole thing. 159 00:09:04,160 --> 00:09:06,780 In other words, its power is radiating outwards. 160 00:09:06,780 --> 00:09:10,090 It doesn't know anything about this little receiving antenna 161 00:09:10,090 --> 00:09:15,060 until it gets close to it, then the electromagnetic wave 162 00:09:15,060 --> 00:09:17,110 gets distorted somewhat. 163 00:09:17,110 --> 00:09:19,090 The only thing that's happening because of this 164 00:09:19,090 --> 00:09:26,140 receiving antenna is that there's some added antenna 165 00:09:26,140 --> 00:09:29,290 pattern due to the receiving antenna -- namely, some added 166 00:09:29,290 --> 00:09:38,090 attenuation which multiplies by the attentuation in the 167 00:09:38,090 --> 00:09:39,260 source antenna. 168 00:09:39,260 --> 00:09:43,150 You have both the source antenna pattern, the receiving 169 00:09:43,150 --> 00:09:44,290 antenna pattern. 170 00:09:44,290 --> 00:09:45,590 We put the two together. 171 00:09:45,590 --> 00:09:49,100 We call that alpha and we don't bother about it anymore 172 00:09:49,100 --> 00:09:51,030 except recognizing that it might change 173 00:09:51,030 --> 00:09:52,420 with frequency also. 174 00:09:55,840 --> 00:09:59,980 Then we have this propagation delay between transmitting 175 00:09:59,980 --> 00:10:05,650 antenna and receiving antenna If you're transmitting in free 176 00:10:05,650 --> 00:10:11,150 space from one antenna to another antenna, this is what 177 00:10:11,150 --> 00:10:15,830 happens, with some arbitrary pattern here for the two 178 00:10:15,830 --> 00:10:19,340 antennas, which depends on whether they radiate 179 00:10:19,340 --> 00:10:21,810 spherically or whether they radiate in 180 00:10:21,810 --> 00:10:24,270 some sort of direction. 181 00:10:24,270 --> 00:10:25,760 That's the received wave form. 182 00:10:34,480 --> 00:10:41,060 That said, if you look at it, at the received field -- 183 00:10:44,690 --> 00:10:47,970 this is supposed to be valid for any -- 184 00:10:47,970 --> 00:10:49,220 oh, sorry. 185 00:10:53,560 --> 00:10:54,810 Yes, that would help. 186 00:10:59,300 --> 00:11:02,260 This equation is supposed to be valid for any frequency we 187 00:11:02,260 --> 00:11:05,400 want to transmit at, at any time, and if we're 188 00:11:05,400 --> 00:11:08,940 transmitting two signals, both together, the response is 189 00:11:08,940 --> 00:11:11,980 going to be the sum of the response to one signal and a 190 00:11:11,980 --> 00:11:13,400 response to the other signal. 191 00:11:13,400 --> 00:11:17,670 In other words, Maxwell's laws are linear and therefore the 192 00:11:17,670 --> 00:11:21,490 response that you get when you solve Maxwell's laws -- 193 00:11:21,490 --> 00:11:23,730 I've never solved Maxwell's laws for anything this 194 00:11:23,730 --> 00:11:27,670 complicated and probably none of you had either. 195 00:11:27,670 --> 00:11:34,770 You might have, but anyway, they are linear and therefore, 196 00:11:34,770 --> 00:11:39,300 in fact, what's going on is that this gives you a system 197 00:11:39,300 --> 00:11:46,640 function which says what the response is to any given input 198 00:11:46,640 --> 00:11:48,090 that you might want to transmit. 199 00:11:48,090 --> 00:11:50,260 It says the response -- 200 00:11:50,260 --> 00:11:54,860 at frequency f to a sinusoidal input, which is what we were 201 00:11:54,860 --> 00:11:55,800 assuming before. 202 00:11:55,800 --> 00:12:00,140 We were assuming that we transmitted cosine 2 pi ft. 203 00:12:00,140 --> 00:12:04,100 The system function is just this antenna pattern times e 204 00:12:04,100 --> 00:12:08,780 to the minus 2 pi i times f times the distance away 205 00:12:08,780 --> 00:12:11,840 divided by c divided by r. 206 00:12:11,840 --> 00:12:15,870 Namely, this takes into account the propogation delay. 207 00:12:15,870 --> 00:12:18,150 The only thing it doesn't take into account is 208 00:12:18,150 --> 00:12:21,190 what the input is. 209 00:12:21,190 --> 00:12:24,900 The received field is then the real part of the system 210 00:12:24,900 --> 00:12:28,950 function times what we already assume that we were going to 211 00:12:28,950 --> 00:12:30,550 be transmitting. 212 00:12:30,550 --> 00:12:34,410 Namely, the real part -- e to the 2 pi i ft. 213 00:12:34,410 --> 00:12:37,660 So at this point, what we're doing is taking into account 214 00:12:37,660 --> 00:12:41,720 the fact that the solution to Maxwell's equations is going 215 00:12:41,720 --> 00:12:46,190 to be linear and therefore we can just add up what happens 216 00:12:46,190 --> 00:12:50,160 for each frequency of input. 217 00:12:50,160 --> 00:12:55,160 What you notice from looking at that is that when we have a 218 00:12:55,160 --> 00:12:59,170 fixed transmitting antenna, a fixed receiving antenna and 219 00:12:59,170 --> 00:13:02,890 free space between them, we are right back to the problem 220 00:13:02,890 --> 00:13:04,350 that we started with. 221 00:13:04,350 --> 00:13:07,965 Namely, white Gaussian noise on a channel, because nothing 222 00:13:07,965 --> 00:13:09,430 is varying with time. 223 00:13:09,430 --> 00:13:11,360 There isn't any fading. 224 00:13:11,360 --> 00:13:13,380 Nothing interesting is going on. 225 00:13:13,380 --> 00:13:15,680 This is sort of like the case of microwave towers. 226 00:13:15,680 --> 00:13:19,820 Microwave towers are set up and they have nice directional 227 00:13:19,820 --> 00:13:25,280 antennas, nice horns which are blowing at each other. 228 00:13:25,280 --> 00:13:28,320 Nothing changes except every once in awhile is a rainstorm 229 00:13:28,320 --> 00:13:31,830 or something and the communication goes to pot. 230 00:13:31,830 --> 00:13:34,220 Usually, you're just transmitting as if it was 231 00:13:34,220 --> 00:13:38,650 white Gaussian noise and you usually view microwave towers 232 00:13:38,650 --> 00:13:42,710 sending microwave as being almost equivalent to wire line 233 00:13:42,710 --> 00:13:44,980 communication. 234 00:13:44,980 --> 00:13:46,650 So there's nothing very interesting there. 235 00:13:52,980 --> 00:13:59,700 Now if the receiving antenna starts to move -- 236 00:13:59,700 --> 00:14:03,140 at this point, we are transmitting from a fixed 237 00:14:03,140 --> 00:14:04,850 sending antenna. 238 00:14:04,850 --> 00:14:09,690 We have a receiving antenna, which for example, is in a car 239 00:14:09,690 --> 00:14:12,570 where somebody's running along, driving a car with 240 00:14:12,570 --> 00:14:15,280 their feet and talking on two cellular phones at 241 00:14:15,280 --> 00:14:17,020 the same time -- 242 00:14:17,020 --> 00:14:21,880 and the car's driving along at 100 miles an hour and 243 00:14:21,880 --> 00:14:24,340 something's going to happen soon, but it 244 00:14:24,340 --> 00:14:26,830 hasn't happened yet. 245 00:14:26,830 --> 00:14:30,530 At this point what we're interested in is not the 246 00:14:30,530 --> 00:14:34,200 response of some fixed place r, but what we're interested 247 00:14:34,200 --> 00:14:37,620 in is the electromagnetic field in the absence of 248 00:14:37,620 --> 00:14:41,930 receiver at this point, which is moving. 249 00:14:41,930 --> 00:14:48,640 We're interested in the electromagnetic field at a 250 00:14:48,640 --> 00:14:54,830 point r 0 plus v2, where v is the velocity of this vehicle. 251 00:14:54,830 --> 00:14:57,010 We'll assume for the time being that the vehicle is 252 00:14:57,010 --> 00:15:01,680 going directly away from the transmitting antenna. 253 00:15:01,680 --> 00:15:04,110 If it's going at some angle, it just changes these 254 00:15:04,110 --> 00:15:05,410 equations a little bit. 255 00:15:08,480 --> 00:15:12,660 And the electric field there, before we put the car in the 256 00:15:12,660 --> 00:15:16,605 car changes all of the field patterns, but just changes it 257 00:15:16,605 --> 00:15:18,020 in a local way again. 258 00:15:18,020 --> 00:15:20,680 Just like when we put the receiving end antenna in a 259 00:15:20,680 --> 00:15:24,230 fixed location, it changed all the local field equation, but 260 00:15:24,230 --> 00:15:27,150 it didn't change anything globally. 261 00:15:27,150 --> 00:15:30,500 Again, what we have when we put in the 262 00:15:30,500 --> 00:15:32,290 receiving antenna -- 263 00:15:32,290 --> 00:15:36,690 is now the electric field at a point r, which is 264 00:15:36,690 --> 00:15:37,950 varying with time. 265 00:15:37,950 --> 00:15:39,710 There's a time dependence with this now. 266 00:15:39,710 --> 00:15:42,730 1 over r 0 plus v.t. 267 00:15:42,730 --> 00:15:46,360 times the real part of this antenna pattern -- which we'll 268 00:15:46,360 --> 00:15:52,440 assume remains fixed -- times e to the 2 pi if times t minus 269 00:15:52,440 --> 00:15:55,270 r 0 plus vt over c. 270 00:15:55,270 --> 00:16:01,950 This is for a velocity away from the antenna. 271 00:16:01,950 --> 00:16:05,450 That's just what this same equation says, but we can now 272 00:16:05,450 --> 00:16:10,920 interpret this nicely if we take the vt over c and combine 273 00:16:10,920 --> 00:16:16,680 it with the ft here and then we get something which looks 274 00:16:16,680 --> 00:16:22,790 like this antenna pattern again -- 275 00:16:22,790 --> 00:16:27,900 e to the 2 pi i f times 1 minus v over c times t. 276 00:16:27,900 --> 00:16:35,580 This v over c here is just this term here. 277 00:16:35,580 --> 00:16:39,050 It's coming down to there. 278 00:16:39,050 --> 00:16:42,650 Nothing mysterious has happened here, but what you 279 00:16:42,650 --> 00:16:45,000 see is this well known phenomenon 280 00:16:45,000 --> 00:16:47,600 called Doppler shift. 281 00:16:47,600 --> 00:16:52,890 If you throw some screaming person over a cliff, what 282 00:16:52,890 --> 00:16:57,740 you'll hear coming back to you is a scream in velocity much 283 00:16:57,740 --> 00:17:01,980 smaller than the actual scream that the person is actually 284 00:17:01,980 --> 00:17:03,580 transmitting to you. 285 00:17:03,580 --> 00:17:04,700 You're all familiar with this. 286 00:17:04,700 --> 00:17:07,900 You're familiar with having planes fly overhead and when 287 00:17:07,900 --> 00:17:11,500 you hear them coming towards you, you hear a 288 00:17:11,500 --> 00:17:12,860 higher pitched sound. 289 00:17:12,860 --> 00:17:16,060 When it passes by you -- this is a nicer example than 290 00:17:16,060 --> 00:17:19,190 throwing somebody over a cliff, obviously -- 291 00:17:19,190 --> 00:17:23,360 and you then hear a lower frequency sound as the plane 292 00:17:23,360 --> 00:17:26,180 starts moving away from you. 293 00:17:26,180 --> 00:17:29,180 This Doppler shift is a well known phenomena as far as 294 00:17:29,180 --> 00:17:30,910 sound is concerned. 295 00:17:30,910 --> 00:17:36,470 The same phenomena exists with electromagnetic radiation. 296 00:17:36,470 --> 00:17:39,660 And there's nothing more to it than just this -- 297 00:17:39,660 --> 00:17:43,460 it's just that as you are transmitting from here to a 298 00:17:43,460 --> 00:17:48,480 point which is moving away, it keeps taking longer for the 299 00:17:48,480 --> 00:17:51,900 electromagnetic wave to get out to there than it takes to 300 00:17:51,900 --> 00:17:55,430 get here, so if you look at the wave fronts going along, 301 00:17:55,430 --> 00:17:59,560 the peak of the wave as it travels along, the peak of the 302 00:17:59,560 --> 00:18:02,790 wave takes a little longer to get out here than it took to 303 00:18:02,790 --> 00:18:07,540 get here, which means that from the viewpoint of the 304 00:18:07,540 --> 00:18:11,530 receiver, it looks like the receive frequency is much 305 00:18:11,530 --> 00:18:14,390 smaller than it was when it was actually being 306 00:18:14,390 --> 00:18:16,530 transmitted. 307 00:18:16,530 --> 00:18:20,320 We get this thing called the Doppler shift. 308 00:18:20,320 --> 00:18:23,340 Just to get some idea of the magnitude of this Doppler 309 00:18:23,340 --> 00:18:27,390 shift, what you're interested in is the speed of the vehicle 310 00:18:27,390 --> 00:18:29,060 divided by the speed of light. 311 00:18:29,060 --> 00:18:38,110 That's the relative change in the 312 00:18:38,110 --> 00:18:42,040 frequency that you observe. 313 00:18:42,040 --> 00:18:45,210 The situation here is quite different than it is in sound. 314 00:18:45,210 --> 00:18:47,470 Sound travels rather slowly. 315 00:18:47,470 --> 00:18:51,040 Light travels pretty fast and therefore, you need a really 316 00:18:51,040 --> 00:18:55,430 rapidly speeding vehicle to make this be any appreciable 317 00:18:55,430 --> 00:18:57,960 fraction of one. 318 00:18:57,960 --> 00:19:01,620 So it looks like this is a very small effect. 319 00:19:01,620 --> 00:19:07,930 The trouble is, what you are multiplying this effect by is 320 00:19:07,930 --> 00:19:10,620 the carrier frequency, which can be up in 321 00:19:10,620 --> 00:19:13,410 the gigahertz range. 322 00:19:13,410 --> 00:19:16,880 To look at it another way, we are looking at situations 323 00:19:16,880 --> 00:19:22,070 where the wavelength is small fractions of the meter. 324 00:19:22,070 --> 00:19:26,270 What this equation says is that any time this receiving 325 00:19:26,270 --> 00:19:31,820 vehicle moves by one wavelength, namely a small 326 00:19:31,820 --> 00:19:39,750 fraction of a meter, the crest of this wave goes from maximum 327 00:19:39,750 --> 00:19:43,520 down to minimum back up to maximum again. 328 00:19:43,520 --> 00:19:47,180 In other words, in a quarter wavelength, it will go from 329 00:19:47,180 --> 00:19:51,470 maximum down to zero. 330 00:19:51,470 --> 00:19:56,090 What you are observing at this carrier frequency is very, 331 00:19:56,090 --> 00:20:02,430 very different and it keeps changing rather rapidly. 332 00:20:02,430 --> 00:20:05,610 All these other terms are just junk, of course. 333 00:20:05,610 --> 00:20:09,070 This f times r 0 over c -- 334 00:20:09,070 --> 00:20:12,430 all that is is just a fixed phase difference, so we don't 335 00:20:12,430 --> 00:20:14,350 care about that. 336 00:20:14,350 --> 00:20:17,350 This is just some fixed term. 337 00:20:17,350 --> 00:20:21,260 This quantity here is changing with t also. 338 00:20:24,110 --> 00:20:26,630 If we're thinking of a distance away, it's several 339 00:20:26,630 --> 00:20:30,730 kilometers and we're thinking of the amount of time for this 340 00:20:30,730 --> 00:20:35,490 to become an appreciable fraction of one. 341 00:20:35,490 --> 00:20:38,990 For this to becoming an appreciable fraction of this, 342 00:20:38,990 --> 00:20:41,130 that's a pretty long time. 343 00:20:41,130 --> 00:20:45,170 The amount of time for this to change appreciably is seconds 344 00:20:45,170 --> 00:20:46,720 or minutes. 345 00:20:46,720 --> 00:20:51,000 The amount of time for this to go through a wavelength change 346 00:20:51,000 --> 00:20:52,250 is milliseconds. 347 00:20:54,390 --> 00:20:58,850 So despite the fact that you see this sitting in an 348 00:20:58,850 --> 00:21:03,160 important place down there, this is not important. 349 00:21:03,160 --> 00:21:07,260 Everything that goes on as far as fading is concerned is tied 350 00:21:07,260 --> 00:21:09,040 up with that term there. 351 00:21:09,040 --> 00:21:10,290 This is important. 352 00:21:19,200 --> 00:21:22,460 So we now have a system which is linear. 353 00:21:22,460 --> 00:21:25,320 We still have the linear field equation, but it's not time 354 00:21:25,320 --> 00:21:26,950 invariant anymore. 355 00:21:26,950 --> 00:21:28,490 It's changing with time. 356 00:21:28,490 --> 00:21:31,340 The response is changing with time. 357 00:21:31,340 --> 00:21:34,070 You send an exponential, what you received. 358 00:21:34,070 --> 00:21:36,590 If you have a linear time invariant system, when you 359 00:21:36,590 --> 00:21:39,570 send an exponential of frequency f, you receive an 360 00:21:39,570 --> 00:21:41,680 exponential with frequency f. 361 00:21:41,680 --> 00:21:44,890 The only thing that a linear time invariant system can do 362 00:21:44,890 --> 00:21:49,060 is change the phase of that signal and change the 363 00:21:49,060 --> 00:21:50,340 amplitude of it. 364 00:21:50,340 --> 00:21:53,270 Can't do anything more complicated than that. 365 00:21:53,270 --> 00:21:56,740 That's why we love to study it; because it's so simple. 366 00:21:56,740 --> 00:21:58,640 Now we have something more. 367 00:21:58,640 --> 00:22:01,820 We have a system that can also change the frequency of what's 368 00:22:01,820 --> 00:22:04,450 getting received. 369 00:22:04,450 --> 00:22:07,120 This small change down here -- 370 00:22:07,120 --> 00:22:09,960 and of course, if obstacles get in the way or something 371 00:22:09,960 --> 00:22:12,850 then there's this huge shadowing difference and all 372 00:22:12,850 --> 00:22:14,820 those important things, but you can't do 373 00:22:14,820 --> 00:22:16,670 anything about that. 374 00:22:16,670 --> 00:22:19,640 You can do something about this, which is why we're 375 00:22:19,640 --> 00:22:20,890 focusing on that. 376 00:22:24,150 --> 00:22:25,770 Let's go to the next example. 377 00:22:29,270 --> 00:22:32,740 Incidentally, that example is no problem at all for 378 00:22:32,740 --> 00:22:33,660 communication. 379 00:22:33,660 --> 00:22:35,790 I'll show you why in a little bit. 380 00:22:35,790 --> 00:22:39,220 You can get around that problem very, very easily and 381 00:22:39,220 --> 00:22:40,100 I'll show you why. 382 00:22:40,100 --> 00:22:43,140 This is a problem you can't get around so easily. 383 00:22:43,140 --> 00:22:46,520 Here we have a vehicle, which is travelling, say, at 60 384 00:22:46,520 --> 00:22:48,520 kilometers an hour. 385 00:22:48,520 --> 00:22:51,640 Person's talking on his two cell phones, has his eyes 386 00:22:51,640 --> 00:22:54,140 closed because something surprising is happening, 387 00:22:54,140 --> 00:22:56,840 there's this big reflecting wall right in front of him and 388 00:22:56,840 --> 00:23:01,160 he doesn't see it at all, so he's speeding into this wall. 389 00:23:01,160 --> 00:23:02,820 We're going to analyze this problem right 390 00:23:02,820 --> 00:23:05,650 before he hits the wall. 391 00:23:05,650 --> 00:23:07,170 We have two paths here. 392 00:23:07,170 --> 00:23:11,820 We have one path which is the path from here out to the 393 00:23:11,820 --> 00:23:16,940 vehicle, which has a length, r of t -- 394 00:23:16,940 --> 00:23:19,940 this is the distance away from the sending antenna to the 395 00:23:19,940 --> 00:23:21,400 receiving antenna. 396 00:23:21,400 --> 00:23:25,530 We have another path which has a length d, then it gets 397 00:23:25,530 --> 00:23:29,580 reflected and this distances is d minus r of t. 398 00:23:29,580 --> 00:23:31,000 The total length -- 399 00:23:31,000 --> 00:23:33,870 and you're adding up this length with this length -- is 400 00:23:33,870 --> 00:23:36,450 2d minus r of t. 401 00:23:36,450 --> 00:23:39,680 The reason is -- do I have an extra picture there? 402 00:23:39,680 --> 00:23:44,520 No, I didn't make my extra pretty picture. 403 00:23:44,520 --> 00:23:49,570 The reason is that one way to deal with electromagnetic 404 00:23:49,570 --> 00:23:55,870 radiation is when you see a wall, the thing that happens 405 00:23:55,870 --> 00:24:03,600 is that you get a reflection which is coming back this way. 406 00:24:03,600 --> 00:24:10,020 The reflection has a strength which is equal to the 407 00:24:10,020 --> 00:24:14,280 radiation that you would get if there weren't any wall -- 408 00:24:14,280 --> 00:24:16,980 uhhuh except, of course, that you've changed directions. 409 00:24:16,980 --> 00:24:21,100 In other words, this wall has generated a new plane wave 410 00:24:21,100 --> 00:24:26,210 which is going backwards, which is just enough to make 411 00:24:26,210 --> 00:24:30,070 the electric field strength on this wall equal to zero, 412 00:24:30,070 --> 00:24:33,190 because we're assuming a perfectly reflecting wall. 413 00:24:33,190 --> 00:24:37,280 You can satisfy Maxwell's equations by having this 414 00:24:37,280 --> 00:24:40,160 incoming electromagnetic wave. 415 00:24:40,160 --> 00:24:42,950 You would like to have an outgoing electromagnetic wave, 416 00:24:42,950 --> 00:24:45,430 but you can't do that because there's no way for the wave to 417 00:24:45,430 --> 00:24:46,670 get through it. 418 00:24:46,670 --> 00:24:49,960 The only way you can do it is to generate a new wave, which 419 00:24:49,960 --> 00:24:54,560 is moving backwards, which cancels out the incoming wave 420 00:24:54,560 --> 00:24:57,560 right at this point. 421 00:24:57,560 --> 00:25:10,160 We really have a path here of length 2d minus r of t and as 422 00:25:10,160 --> 00:25:15,060 a result of that, the electric wave has two components. 423 00:25:15,060 --> 00:25:18,750 One is the component we were dealing with before where 424 00:25:18,750 --> 00:25:23,080 there's this Doppler shift because this is moving away 425 00:25:23,080 --> 00:25:26,050 from the sending antenna. 426 00:25:26,050 --> 00:25:28,240 The other term -- 427 00:25:28,240 --> 00:25:32,280 in fact, we are moving closer to the wall and the distance 428 00:25:32,280 --> 00:25:35,460 in this path is getting shorter and shorter as doom 429 00:25:35,460 --> 00:25:37,820 approaches. 430 00:25:37,820 --> 00:25:40,630 Here we have a positive Doppler shift. 431 00:25:40,630 --> 00:25:43,500 Here we have a negative Doppler shift and we have 432 00:25:43,500 --> 00:25:46,060 these junk terms in both places. 433 00:25:46,060 --> 00:25:48,030 One is fr 0 over c. 434 00:25:48,030 --> 00:25:51,930 One is 2d minus rc over c. 435 00:25:51,930 --> 00:25:54,280 Here we have r0 plus vt. 436 00:25:54,280 --> 00:25:58,180 Here we have 2d minus r0 zero minus v.t.. 437 00:25:58,180 --> 00:26:04,620 As we said before, this term and this term are not changing 438 00:26:04,620 --> 00:26:07,110 very rapidly. 439 00:26:07,110 --> 00:26:10,740 It makes it a little easier to analyze this if we say, let's 440 00:26:10,740 --> 00:26:13,770 suppose that this is equal to this. 441 00:26:13,770 --> 00:26:17,700 In other words, we'd like to look at this right before the 442 00:26:17,700 --> 00:26:20,240 car strikes the wall. 443 00:26:20,240 --> 00:26:23,810 That also is where this approximation is best because 444 00:26:23,810 --> 00:26:27,340 for those of you who have studied electromagnetism, you 445 00:26:27,340 --> 00:26:30,830 know that if a plane wave impinges on a wall, funny 446 00:26:30,830 --> 00:26:33,900 things happen. 447 00:26:33,900 --> 00:26:38,390 If you look very far away from the wall, you will find this 448 00:26:38,390 --> 00:26:42,360 electromagnetic wave, which looks like a plane wave if the 449 00:26:42,360 --> 00:26:44,750 wall is distant from the source. 450 00:26:44,750 --> 00:26:48,270 So this electromagnetic wave coming in, there's this wall 451 00:26:48,270 --> 00:26:53,190 in here and what happens to the electromagnetic radiation 452 00:26:53,190 --> 00:26:56,080 is outside of the wall that's going to go out 453 00:26:56,080 --> 00:26:57,730 past the wall -- 454 00:26:57,730 --> 00:27:00,350 and because of Maxwell's equations, it just sort of 455 00:27:00,350 --> 00:27:03,320 gathers together beyond the wall and it 456 00:27:03,320 --> 00:27:04,640 sort of comes together. 457 00:27:04,640 --> 00:27:10,080 What you find is a disturbance, which is just 458 00:27:10,080 --> 00:27:11,900 around the wall. 459 00:27:11,900 --> 00:27:15,790 Far away from the wall, you get the same electromagnetic 460 00:27:15,790 --> 00:27:19,400 radiation that you had before and close to the wall you have 461 00:27:19,400 --> 00:27:21,240 this disturbance. 462 00:27:21,240 --> 00:27:25,690 If you look at the situation -- 463 00:27:31,170 --> 00:27:33,930 here it is. 464 00:27:33,930 --> 00:27:37,240 If you look at what happens here and the wall is not big 465 00:27:37,240 --> 00:27:41,980 enough, if the wall is very small, this reflection is 466 00:27:41,980 --> 00:27:45,520 really going to look like what happens when you have an 467 00:27:45,520 --> 00:27:49,960 electromagnetic wave hitting the wall and the wall then 468 00:27:49,960 --> 00:27:54,470 re-radiates an electromagnetic wave, which very far away, 469 00:27:54,470 --> 00:27:57,640 this wall just looks like a point source. 470 00:27:57,640 --> 00:28:03,040 What you have is instead of a 1 over r, 1 over 2r minus 1 471 00:28:03,040 --> 00:28:08,050 over 2d minus r attenuation, you have a 1 over d 472 00:28:08,050 --> 00:28:13,680 attenuation multiplied by a 1 over d minus r attenuation. 473 00:28:13,680 --> 00:28:15,540 If you didn't get all of that, fine. 474 00:28:15,540 --> 00:28:17,150 Doesn't make any difference. 475 00:28:17,150 --> 00:28:24,320 The point that I'm trying to make is that this analysis is 476 00:28:24,320 --> 00:28:27,430 really limited to the case where there the wall is rather 477 00:28:27,430 --> 00:28:31,380 small, where the wall is very large, because otherwise you 478 00:28:31,380 --> 00:28:36,340 won't have just this plane wave radiation effect. 479 00:28:39,680 --> 00:28:48,250 What we wind up with these two terms instead of one term. 480 00:28:48,250 --> 00:28:52,020 If I throw away all of the phase terms and I assume that 481 00:28:52,020 --> 00:28:54,410 the denominators are equal -- 482 00:28:54,410 --> 00:28:59,440 I'm going through this for some sort of reason -- 483 00:28:59,440 --> 00:29:03,660 I wind up with two sinusoids: e to the blah 484 00:29:03,660 --> 00:29:08,350 and e to blah plus. 485 00:29:08,350 --> 00:29:13,060 When I take the real part of the sum of two sinusoids, and 486 00:29:13,060 --> 00:29:18,190 I look in all of the high school books I can think of 487 00:29:18,190 --> 00:29:21,560 about elementary geometry and playing around with sine 488 00:29:21,560 --> 00:29:26,500 waves, what I find is that this collapses into 2 alpha 489 00:29:26,500 --> 00:29:29,840 times the sine of 2 pi ftv over c times the 490 00:29:29,840 --> 00:29:32,600 sine of 2 pi ft. 491 00:29:32,600 --> 00:29:37,720 In other words, it collapses into a sinusoidal term, which 492 00:29:37,720 --> 00:29:42,710 is the major part of this term, ft, and the major part 493 00:29:42,710 --> 00:29:43,490 of this term. 494 00:29:43,490 --> 00:29:46,960 In other words, I can cancel out the terms here that are 495 00:29:46,960 --> 00:29:49,100 the same as the terms here. 496 00:29:49,100 --> 00:29:51,930 When I cancel out those same terms, that's the 497 00:29:51,930 --> 00:29:54,790 term that comes out. 498 00:29:54,790 --> 00:29:57,280 When I look at the other terms, I get an e to the minus 499 00:29:57,280 --> 00:30:01,490 vc over t and an e to the plus vc over to t. 500 00:30:01,490 --> 00:30:05,280 When we look at e to the minus vc over to t and e to the plus 501 00:30:05,280 --> 00:30:09,110 vc over t, even I know how to deal with that. 502 00:30:09,110 --> 00:30:13,130 It looks like either a cosine term or sine term, depending 503 00:30:13,130 --> 00:30:16,420 on whether the sines in the same or 504 00:30:16,420 --> 00:30:17,750 the sines are different. 505 00:30:24,300 --> 00:30:29,635 What we have at that point is this sinusoid is really a 506 00:30:29,635 --> 00:30:33,650 sinusoid at the carrier that we're transmitting at. 507 00:30:33,650 --> 00:30:38,960 This term here is really something which expands and 508 00:30:38,960 --> 00:30:40,660 contracts slowly. 509 00:30:40,660 --> 00:30:46,070 So it's a beat, which says that if you're transmitting 510 00:30:46,070 --> 00:30:56,170 from this source at this receiver, what you're hearing 511 00:30:56,170 --> 00:31:02,070 is something which contracts, expands, then contracts, then 512 00:31:02,070 --> 00:31:03,410 expands again. 513 00:31:03,410 --> 00:31:06,890 There's nothing you can do about that problem either. 514 00:31:06,890 --> 00:31:11,440 There's just no energy there part of the time. 515 00:31:11,440 --> 00:31:18,440 This sine term here is running along, sort of changing from 516 00:31:18,440 --> 00:31:22,360 maximum to minimum at a few milliseconds time period. 517 00:31:22,360 --> 00:31:25,280 If you have this vehicle traveling at 60 518 00:31:25,280 --> 00:31:36,000 kilometers per hour -- 519 00:31:36,000 --> 00:31:38,330 you just work out the numbers there with the velocity of 520 00:31:38,330 --> 00:31:41,060 light and all of that stuff. 521 00:31:41,060 --> 00:31:43,700 You find that you really can't communicate over your 522 00:31:43,700 --> 00:31:47,690 cellphone in that situation, because of these peak 523 00:31:47,690 --> 00:31:49,200 frequencies. 524 00:31:49,200 --> 00:31:51,210 It's too fast. 525 00:31:54,500 --> 00:32:01,550 It's too slow to ignore and ride over it and it's too fast 526 00:32:01,550 --> 00:32:03,410 to be able to get all of your data 527 00:32:03,410 --> 00:32:05,950 transmitted before it happens. 528 00:32:05,950 --> 00:32:08,990 It sort of is a catastrophe. 529 00:32:08,990 --> 00:32:12,510 So that's what happens because of Doppler shift. 530 00:32:12,510 --> 00:32:15,710 You get a response which is periodically fading at the 531 00:32:15,710 --> 00:32:16,960 Doppler frequency. 532 00:32:16,960 --> 00:32:19,875 This is called multipath fading or fast fading. 533 00:32:19,875 --> 00:32:23,540 It's called fast fading because it happens so fast. 534 00:32:23,540 --> 00:32:27,240 It's called multipath fading because it happens because of 535 00:32:27,240 --> 00:32:32,030 multiple paths, which each have lengths which are 536 00:32:32,030 --> 00:32:35,980 changing relative to each other. 537 00:32:35,980 --> 00:32:44,590 Let's go back and look at the thing we had before, where we 538 00:32:44,590 --> 00:32:46,050 just had a moving antenna. 539 00:32:50,170 --> 00:32:53,230 Here you have a Doppler shift also. 540 00:32:53,230 --> 00:32:56,860 Why doesn't it bother you? 541 00:32:56,860 --> 00:33:01,060 Here you have this Doppler shift and you're transmitting, 542 00:33:01,060 --> 00:33:05,560 let's say, a gigahertz and what the receiver is getting 543 00:33:05,560 --> 00:33:07,640 is the gighertz minus -- 544 00:33:10,250 --> 00:33:14,260 perhaps a kilohertz or something. 545 00:33:14,260 --> 00:33:15,690 So why isn't that a problem? 546 00:33:18,340 --> 00:33:26,640 If I demodulate at the carrier frequency, I'm sort of in bad 547 00:33:26,640 --> 00:33:29,490 luck because I have a signal then which is 548 00:33:29,490 --> 00:33:30,890 changing very rapidly. 549 00:33:30,890 --> 00:33:36,130 I have something which looks like a time varying system. 550 00:33:36,130 --> 00:33:37,380 But what's going to happen? 551 00:33:39,720 --> 00:33:42,840 If I use the same kind of frequency recovery system that 552 00:33:42,840 --> 00:33:46,910 we talked about earlier, that frequency recovery system has 553 00:33:46,910 --> 00:33:51,400 all the time in the world to track that frequency which is 554 00:33:51,400 --> 00:33:54,670 one gigahertz minus a kilohertz. 555 00:33:54,670 --> 00:33:59,190 It can track it perfectly, which says so what happens is 556 00:33:59,190 --> 00:34:02,240 we start out with a signal. 557 00:34:02,240 --> 00:34:05,400 We move it up in frequency by one gigahertz. 558 00:34:05,400 --> 00:34:08,890 The Doppler shift moves it down by a kilohertz. 559 00:34:08,890 --> 00:34:13,260 We track that frequency, then we move it down again by a 560 00:34:13,260 --> 00:34:16,880 gigahertz minus a kilohertz and everything works fine and 561 00:34:16,880 --> 00:34:21,060 nobody even knows that there's any Doppler shift there. 562 00:34:21,060 --> 00:34:23,830 So the problem is not Doppler shift. 563 00:34:23,830 --> 00:34:26,700 The problem is multiple Doppler shifts which are at 564 00:34:26,700 --> 00:34:30,940 different speeds relative to each other. 565 00:34:30,940 --> 00:34:34,570 That's an important point and we will come back to it as we 566 00:34:34,570 --> 00:34:35,820 move along. 567 00:34:39,420 --> 00:34:43,130 If you put all of those phases that we neglected in the 568 00:34:43,130 --> 00:34:47,030 analysis that we just went through, this is the equation 569 00:34:47,030 --> 00:34:49,000 that arises. 570 00:34:49,000 --> 00:34:52,790 I write this down not because it's important, but because 571 00:34:52,790 --> 00:34:56,060 the notes -- this is in lecture 20 -- 572 00:34:56,060 --> 00:34:57,310 have an error. 573 00:35:00,760 --> 00:35:03,700 In the sine term, it fails to put an i in, which should be 574 00:35:03,700 --> 00:35:10,250 there and this is the correct term and that is off like 575 00:35:10,250 --> 00:35:12,360 ninety degrees. 576 00:35:12,360 --> 00:35:18,470 If you write this down, you will then see what's going on. 577 00:35:18,470 --> 00:35:22,920 Equation 7 in the notes has an e to the 2 pi f.t. 578 00:35:22,920 --> 00:35:23,730 minus f.d. 579 00:35:23,730 --> 00:35:30,090 over c, which is not the right thing. 580 00:35:30,090 --> 00:35:33,210 As we said, the fading is due to Doppler spread between 581 00:35:33,210 --> 00:35:35,370 different paths. 582 00:35:35,370 --> 00:35:39,070 The single Doppler shift does not bother us at all. 583 00:35:39,070 --> 00:35:43,500 If you have a vehicle which is traveling away from the 584 00:35:43,500 --> 00:35:48,390 sending antenna and you have some kind of reflector, which 585 00:35:48,390 --> 00:35:51,410 is not something you're running into, but which is a 586 00:35:51,410 --> 00:35:55,290 reflector above, a reflector below or something like that, 587 00:35:55,290 --> 00:35:58,300 the thing which is going to happen is that both of those 588 00:35:58,300 --> 00:36:01,940 paths then are going to have roughly the same Doppler shift 589 00:36:01,940 --> 00:36:04,850 in them and if they both have roughly the same Doppler 590 00:36:04,850 --> 00:36:10,230 shift, it's not going to be this kind of beat cancellation 591 00:36:10,230 --> 00:36:12,850 that we have here, which is something you really 592 00:36:12,850 --> 00:36:14,100 can't get rid of. 593 00:36:16,490 --> 00:36:19,550 The other thing that this points out again is that this 594 00:36:19,550 --> 00:36:25,770 variation is going to be in terms of minutes or seconds 595 00:36:25,770 --> 00:36:29,100 and anything you're doing to track the signal is going to 596 00:36:29,100 --> 00:36:32,570 be adequate for that until you get to the point where there 597 00:36:32,570 --> 00:36:35,410 just isn't enough energy anymore and then of course you 598 00:36:35,410 --> 00:36:37,130 have to move to a different base station 599 00:36:37,130 --> 00:36:38,380 or something else. 600 00:36:42,120 --> 00:36:48,100 Want go through one more example of electromagnetism 601 00:36:48,100 --> 00:36:50,720 because it's so surprising -- at least, it was surprising to 602 00:36:50,720 --> 00:36:53,630 me when I found this out. 603 00:36:53,630 --> 00:36:56,840 If you have a sending antenna -- think of this as a base 604 00:36:56,840 --> 00:37:00,100 station, which is high up at about maybe 605 00:37:00,100 --> 00:37:03,130 15 meters or something. 606 00:37:03,130 --> 00:37:08,750 It's sending to some receive antenna, which is at some 607 00:37:08,750 --> 00:37:10,510 height above the ground -- 608 00:37:10,510 --> 00:37:13,250 usually quite a bit smaller. 609 00:37:13,250 --> 00:37:18,020 Suppose there is some more or less partly reflecting plane, 610 00:37:18,020 --> 00:37:20,560 like a road. 611 00:37:20,560 --> 00:37:24,940 Here we have a vehicle which is travelling along a road and 612 00:37:24,940 --> 00:37:28,680 we have ascending antenna, which is also close to the 613 00:37:28,680 --> 00:37:33,030 road, which is sending the signal so we have two paths, 614 00:37:33,030 --> 00:37:36,200 one which is the direct path from sending antenna to 615 00:37:36,200 --> 00:37:37,710 receive antenna. 616 00:37:37,710 --> 00:37:42,130 The other is a reflecting path which goes down here and comes 617 00:37:42,130 --> 00:37:44,780 back up again. 618 00:37:44,780 --> 00:37:48,120 The rather surprising thing here, the thing that I 619 00:37:48,120 --> 00:37:50,660 couldn't believe when I saw it, because it contradicted 620 00:37:50,660 --> 00:37:56,340 all of my intuition, is that when r gets quite big, the 621 00:37:56,340 --> 00:37:59,860 difference between these two path lengths goes to 0. 622 00:38:03,670 --> 00:38:05,610 Is that surprising to anybody else? 623 00:38:09,220 --> 00:38:14,390 Anybody awake enough to be surprised? 624 00:38:14,390 --> 00:38:18,690 This difference in path length here really goes 625 00:38:18,690 --> 00:38:20,430 down as 1 over r. 626 00:38:20,430 --> 00:38:23,800 That's an easy geometric problem to solve. 627 00:38:23,800 --> 00:38:27,630 You just write down what this length is and the sum of these 628 00:38:27,630 --> 00:38:30,380 two lengths and you will find when you do it that the 629 00:38:30,380 --> 00:38:34,280 difference is proportional to 1 over r. 630 00:38:34,280 --> 00:38:38,370 What happens then is that as r gets big, these two path 631 00:38:38,370 --> 00:38:41,810 lengths get closer and closer together. 632 00:38:41,810 --> 00:38:44,540 As they get closer and closer together, eventually they're 633 00:38:44,540 --> 00:38:48,000 much closer than one wavelength to each other. 634 00:38:48,000 --> 00:38:51,290 When they're much closer than one wavelength to each other, 635 00:38:51,290 --> 00:38:58,140 the thing that happens is that we get a reflection in the 636 00:38:58,140 --> 00:38:59,990 electric field here, so. 637 00:38:59,990 --> 00:39:02,520 This field and this field are going to be 638 00:39:02,520 --> 00:39:04,640 canceling each other. 639 00:39:04,640 --> 00:39:07,540 They'll be canceling each other, except for this phase 640 00:39:07,540 --> 00:39:11,000 difference which is proportional to 1 over r, and 641 00:39:11,000 --> 00:39:14,820 the phase difference which is proportional to 1 over r is 642 00:39:14,820 --> 00:39:17,640 the only thing that gives us any power here at all. 643 00:39:17,640 --> 00:39:20,810 As we move further and further away, that phase difference 644 00:39:20,810 --> 00:39:25,500 goes down as 1 over r, which means what's happening is the 645 00:39:25,500 --> 00:39:28,920 overall electric field that you receive here, instead of 646 00:39:28,920 --> 00:39:32,620 going down as 1 over r, which it would in free space, is 647 00:39:32,620 --> 00:39:35,460 going down as 1 over r squared. 648 00:39:35,460 --> 00:39:41,970 Since it's going down as 1 over r squared, it means that 649 00:39:41,970 --> 00:39:46,600 the power that we're receiving is proportional to 1 over r 650 00:39:46,600 --> 00:39:50,210 4th instead of 1 over r squared. 651 00:39:50,210 --> 00:39:53,430 The analysis of this is not particularly important. 652 00:39:53,430 --> 00:39:57,550 Why it is that surfaces such as macadam reflects so well is 653 00:39:57,550 --> 00:40:00,370 not particularly important to us either. 654 00:40:00,370 --> 00:40:10,630 The point is that when you look at all the problems of 655 00:40:10,630 --> 00:40:17,460 electromagnetic radiation, in actual situations, you find 656 00:40:17,460 --> 00:40:21,000 some situations which behave like this, you find some 657 00:40:21,000 --> 00:40:26,920 situations which behave like this nice plane 658 00:40:26,920 --> 00:40:28,500 wave and free space. 659 00:40:28,500 --> 00:40:32,640 You find other things where in fact the radiation goes down 660 00:40:32,640 --> 00:40:38,010 as 1 over r 6th, rather than 1 over r squared, so that we 661 00:40:38,010 --> 00:40:44,790 wind up with each path has its own particular kind of 662 00:40:44,790 --> 00:40:48,870 attenuation in with path lengths and it's kind of hard 663 00:40:48,870 --> 00:40:52,640 to figure out what all of those are. 664 00:40:52,640 --> 00:40:55,890 Then you go on and say that things are even worse than 665 00:40:55,890 --> 00:40:59,450 this because sometimes you're communicating through a wall 666 00:40:59,450 --> 00:41:03,960 which is only partly absorbing and if you're transmitting 667 00:41:03,960 --> 00:41:08,720 through a wall which is partly absorbing, then the 668 00:41:08,720 --> 00:41:13,790 attenuation is exponential in the width of the wall. 669 00:41:13,790 --> 00:41:17,180 You have a lot of different paths, all of which are very 670 00:41:17,180 --> 00:41:21,760 messy electromagnetic radiation problems and all of 671 00:41:21,760 --> 00:41:27,730 which have attenuations which range from 1 over r squared to 672 00:41:27,730 --> 00:41:31,470 1 over r 6th, sometimes with an exponential thrown in for 673 00:41:31,470 --> 00:41:36,300 good measure, which says that if you really try to find the 674 00:41:36,300 --> 00:41:41,040 electromagnetic field at a wireless cell phone, you're in 675 00:41:41,040 --> 00:41:43,840 very deep trouble. 676 00:41:43,840 --> 00:41:47,060 It's certainly not something that your cellular phone is 677 00:41:47,060 --> 00:41:50,050 going to solve for you and it's certainly not something 678 00:41:50,050 --> 00:41:52,540 that you're going to have solved ahead of time and 679 00:41:52,540 --> 00:41:56,310 program into your cellular phone or into the base station 680 00:41:56,310 --> 00:41:58,200 or anything else. 681 00:41:58,200 --> 00:42:00,130 The question is, what do we do about this? 682 00:42:06,260 --> 00:42:08,960 One thing is, we're not going to study these electromagnetic 683 00:42:08,960 --> 00:42:13,010 phenomena any further because it's a losing game. 684 00:42:13,010 --> 00:42:15,280 If the thing that you're interested in is finding out 685 00:42:15,280 --> 00:42:19,170 where to place base stations, all of this kind of analysis 686 00:42:19,170 --> 00:42:22,250 is useful and you can go much further with it and you should 687 00:42:22,250 --> 00:42:24,060 go much further with it. 688 00:42:24,060 --> 00:42:28,000 If your interest is in, how do you build third or fourth or 689 00:42:28,000 --> 00:42:32,960 fifth or 20th generation wireless systems? 690 00:42:32,960 --> 00:42:36,400 All the electromagnetics that you study is only going to 691 00:42:36,400 --> 00:42:39,920 give you gross ideas of what kind of phenomena you 692 00:42:39,920 --> 00:42:41,280 have to deal with. 693 00:42:41,280 --> 00:42:44,000 We already have some idea of the kind of phenomena we have 694 00:42:44,000 --> 00:42:44,690 to deal with. 695 00:42:44,690 --> 00:42:48,940 We have to deal with paths which have different 696 00:42:48,940 --> 00:42:52,610 attenuations on them, which have different propogation 697 00:42:52,610 --> 00:42:56,350 delays on them, and all of these multiple paths are 698 00:42:56,350 --> 00:43:01,040 things that we have to somehow deal with without analyzing 699 00:43:01,040 --> 00:43:02,270 them in detail. 700 00:43:02,270 --> 00:43:04,010 The question is, how do we do this? 701 00:43:09,960 --> 00:43:15,140 Everything that we've done so far is called rate tracing. 702 00:43:15,140 --> 00:43:17,940 In fact, even with all the complexity that we're dealing 703 00:43:17,940 --> 00:43:22,320 with now, we have highly oversimplified it. 704 00:43:22,320 --> 00:43:25,610 Each of these paths that we have is going to give rise to 705 00:43:25,610 --> 00:43:27,290 an attenuation. 706 00:43:27,290 --> 00:43:32,360 We'll now call the attenuation beta of j of t and a 707 00:43:32,360 --> 00:43:36,280 propogation delay, which we'll call passive j of t and the 708 00:43:36,280 --> 00:43:40,690 propogation delay is just what we get by assuming that a 709 00:43:40,690 --> 00:43:46,020 plane wave is going from source to destination and we 710 00:43:46,020 --> 00:43:49,230 add up the distance from source to reflector to 711 00:43:49,230 --> 00:43:54,660 destination and that gives us this propogation delay. 712 00:43:54,660 --> 00:43:58,710 These are going to vary with time, but in our rate tracing 713 00:43:58,710 --> 00:44:00,600 approximation, we've assumed that they're 714 00:44:00,600 --> 00:44:02,460 independent of frequency. 715 00:44:02,460 --> 00:44:05,045 I originally assumed that the antenna pattern was a function 716 00:44:05,045 --> 00:44:06,160 of frequency. 717 00:44:06,160 --> 00:44:08,960 We didn't want to say anything about that. 718 00:44:08,960 --> 00:44:12,500 So we have a total of capital J paths. 719 00:44:12,500 --> 00:44:16,670 we put in an input of cosine 2 pi ft, what's going to come 720 00:44:16,670 --> 00:44:22,710 out is an electromagnetic radiation, which is a sum of 721 00:44:22,710 --> 00:44:27,920 these different attenuation factors times e to the 2 pi ft 722 00:44:27,920 --> 00:44:30,830 minus this propogation delay. 723 00:44:34,230 --> 00:44:36,510 Everything you can do with ray tracing is 724 00:44:36,510 --> 00:44:38,600 included in this formula. 725 00:44:38,600 --> 00:44:42,780 What you might be able to do in a wireless system is you 726 00:44:42,780 --> 00:44:45,720 might, by looking at the received wave form and knowing 727 00:44:45,720 --> 00:44:48,630 things about the transmitted wave form, you might be able 728 00:44:48,630 --> 00:44:52,190 to figure out what these attenuation factors are and 729 00:44:52,190 --> 00:44:56,290 what these propogation delay factors are. 730 00:44:56,290 --> 00:44:59,740 Just like when we tried to do frequency recovery, we can 731 00:44:59,740 --> 00:45:02,420 find out what the transmitted frequency was. 732 00:45:02,420 --> 00:45:06,460 We can play the same sorts of games here, but they're harder 733 00:45:06,460 --> 00:45:09,300 and we'll talk about that later. 734 00:45:09,300 --> 00:45:13,650 If you ever hear the term rake receiver, a rake receiver is a 735 00:45:13,650 --> 00:45:17,800 receiver that in fact measures all this stuff and responds to 736 00:45:17,800 --> 00:45:23,450 it and we'll talk about that probably next Monday. 737 00:45:23,450 --> 00:45:26,040 If we want to look at that the reflecting wall just as an 738 00:45:26,040 --> 00:45:31,500 example of what this formula means: beta 1 of t, namely for 739 00:45:31,500 --> 00:45:33,180 the direct path. 740 00:45:33,180 --> 00:45:36,840 We have an attenuation, which is the magnitude of the 741 00:45:36,840 --> 00:45:41,550 antenna patterns divided by r0 plus vt. 742 00:45:41,550 --> 00:45:46,150 For the attenuation on the return path from the wall, 743 00:45:46,150 --> 00:45:48,350 it's the same alpha. 744 00:45:48,350 --> 00:45:49,790 Why is it the same alpha? 745 00:45:49,790 --> 00:45:52,100 Because we assumed it was the same alpha to make things 746 00:45:52,100 --> 00:45:54,600 simple for ourselves -- divided by 2d 747 00:45:54,600 --> 00:45:58,330 minus r0 minus vt. 748 00:45:58,330 --> 00:46:03,280 If you look at these propogation delay terms, the 749 00:46:03,280 --> 00:46:09,900 propogation delay terms are r0 plus vt divided by c and this 750 00:46:09,900 --> 00:46:12,450 gives us the Doppler shift that we're interested in here. 751 00:46:12,450 --> 00:46:16,520 We're also going to have an extra term here, which is 752 00:46:16,520 --> 00:46:20,200 really caused by the phase change at the transmitting 753 00:46:20,200 --> 00:46:23,830 antenna and the phase change at the receiving antenna and a 754 00:46:23,830 --> 00:46:26,600 phase change at a reflector, if there's any there. 755 00:46:26,600 --> 00:46:30,980 We have the same sort of term at both of these places. 756 00:46:30,980 --> 00:46:34,580 The reason that I talk about that is if you look at the 757 00:46:34,580 --> 00:46:38,260 electromagnetic wave that you received for this reflecting 758 00:46:38,260 --> 00:46:42,690 wall problem that we've talked about a good deal, the second 759 00:46:42,690 --> 00:46:47,320 term is there with a negative sign rather than a plus sign. 760 00:46:47,320 --> 00:46:51,080 Here, everything is put in with plus signs. 761 00:46:51,080 --> 00:46:56,220 You can create negative signs by phase changes of pi. 762 00:46:56,220 --> 00:46:59,790 So the assumption is, we put in a phase change of pi as 763 00:46:59,790 --> 00:47:02,830 part of this term here. 764 00:47:02,830 --> 00:47:06,860 So all of these terms can be expressed in this general 765 00:47:06,860 --> 00:47:08,110 forum here. 766 00:47:13,280 --> 00:47:15,800 As we said before, you only have two choices with a 767 00:47:15,800 --> 00:47:17,660 cellular system. 768 00:47:17,660 --> 00:47:21,810 You cannot solve the electromagnetic field problem 769 00:47:21,810 --> 00:47:23,480 at the cell phone. 770 00:47:23,480 --> 00:47:26,250 The person using the cell phone is not going to do it. 771 00:47:26,250 --> 00:47:28,120 The cell phone is not going to do it. 772 00:47:28,120 --> 00:47:30,890 The base station is not going to do it and you're not going 773 00:47:30,890 --> 00:47:34,630 to store all those changes because these radiations 774 00:47:34,630 --> 00:47:39,160 change remarkably within a period of just small fractions 775 00:47:39,160 --> 00:47:41,070 of one meter. 776 00:47:41,070 --> 00:47:46,230 You have a coverage here, which in fact, is at least 777 00:47:46,230 --> 00:47:51,780 area coverage of one kilometer times one kilometer. 778 00:47:51,780 --> 00:47:55,180 Then the reflectors are going to be moving also, so you 779 00:47:55,180 --> 00:47:57,620 can't deal with them very easily either. 780 00:47:57,620 --> 00:47:59,890 It's a hopeless problem, to try to solve the 781 00:47:59,890 --> 00:48:05,520 electromagnetic problem and store it some place. 782 00:48:05,520 --> 00:48:08,920 Electromagnetism helps us to limit the range and likelihood 783 00:48:08,920 --> 00:48:13,320 of choices, but it doesn't help in actual detection. 784 00:48:13,320 --> 00:48:16,560 So we're now going to deal with the kind of thing we just 785 00:48:16,560 --> 00:48:21,680 talked about, which is this sort of general expression for 786 00:48:21,680 --> 00:48:26,680 electric field in terms of attenuation factors and phase 787 00:48:26,680 --> 00:48:31,880 changes, as opposed to anything which is 788 00:48:31,880 --> 00:48:33,150 wh much more detailed. 789 00:48:35,860 --> 00:48:41,130 We're going to define a channel system function as 790 00:48:41,130 --> 00:48:45,905 just this sum of these attenuation terms times phase 791 00:48:45,905 --> 00:48:47,230 change terms. 792 00:48:47,230 --> 00:48:51,000 The reason that we're doing this is that if we put in an 793 00:48:51,000 --> 00:48:59,530 input -- e to the 2 pi ift, then what we get is this 794 00:48:59,530 --> 00:49:04,510 system function here -- times x of t, which is e to the 2 pi 795 00:49:04,510 --> 00:49:07,860 ift, so we get this quantity here. 796 00:49:07,860 --> 00:49:10,640 Here's the minus 2 pi if tau -- 797 00:49:13,250 --> 00:49:18,380 that's that term coming down there and here is the e to the 798 00:49:18,380 --> 00:49:21,970 2 pi ift coming down here. 799 00:49:21,970 --> 00:49:30,210 So all of this term and this term are both included in this 800 00:49:30,210 --> 00:49:31,700 system response term. 801 00:49:34,730 --> 00:49:38,335 This is linear also, so we know what the response is to 802 00:49:38,335 --> 00:49:40,380 an exponential that we put in. 803 00:49:40,380 --> 00:49:42,490 I'm cheating you a little bit here by going 804 00:49:42,490 --> 00:49:48,540 from real to complex. 805 00:49:48,540 --> 00:49:53,890 And the notes do that a little more carefully, but we ought 806 00:49:53,890 --> 00:49:56,150 to be used to that now. 807 00:49:56,150 --> 00:50:01,800 If I put in an input, x hat of f e to the 2 pi ift df and 808 00:50:01,800 --> 00:50:02,800 integrate it. 809 00:50:02,800 --> 00:50:06,240 In other words, if I put in an arbitrary input, which I 810 00:50:06,240 --> 00:50:09,900 represent in terms of its Fourier transform, I now know 811 00:50:09,900 --> 00:50:12,980 what the response is to every x-hat of f. 812 00:50:12,980 --> 00:50:16,340 It's given by this response here. 813 00:50:16,340 --> 00:50:20,360 So I just integrate over that and I find that the response y 814 00:50:20,360 --> 00:50:26,680 of t to an arbitrary input now is the integral of x-hat of f, 815 00:50:26,680 --> 00:50:31,360 h-hat of ft, e to the 2 pi ift df -- 816 00:50:31,360 --> 00:50:35,590 namely, the same game that we always play. 817 00:50:35,590 --> 00:50:40,320 Namely, that's just the system analysis way of looking at 818 00:50:40,320 --> 00:50:44,950 arbitrary systems in terms of their Fourier transforms. 819 00:50:44,950 --> 00:50:51,370 So the output y of t is just this integral here. 820 00:50:51,370 --> 00:51:00,200 Important point: When you look at this, this says this is the 821 00:51:00,200 --> 00:51:03,700 same as any old linear time invariant system. 822 00:51:03,700 --> 00:51:07,060 This is not a linear time invariant system. 823 00:51:07,060 --> 00:51:11,650 If you try to take the Fourier transform of this to get y-hat 824 00:51:11,650 --> 00:51:18,880 of f, you're not going to get x-hat of f times h-hat of f. 825 00:51:18,880 --> 00:51:22,820 Namely, this is not equal to that. 826 00:51:22,820 --> 00:51:25,490 Why isn't it equal to it? 827 00:51:25,490 --> 00:51:29,550 First reason is, try to take the Fourier transform of this 828 00:51:29,550 --> 00:51:32,640 and see what you get over here and when you deal with the 829 00:51:32,640 --> 00:51:36,750 fact that there's a t in here, you will find that there's 830 00:51:36,750 --> 00:51:38,630 nothing you can do. 831 00:51:38,630 --> 00:51:41,200 You'll find you're stuck. 832 00:51:41,200 --> 00:51:45,570 So you can't derive this equation here. 833 00:51:45,570 --> 00:51:52,360 Next, argument is this quantity here is not a 834 00:51:52,360 --> 00:51:54,510 function of t. 835 00:51:54,510 --> 00:51:58,730 This quantity here is a function of t. 836 00:51:58,730 --> 00:52:02,880 You can't have a quality between something which is not 837 00:52:02,880 --> 00:52:06,530 a function of t and something else which is a function of t. 838 00:52:06,530 --> 00:52:08,930 It just can't happen. 839 00:52:08,930 --> 00:52:12,960 The final argument is, if you look at what's happening here, 840 00:52:12,960 --> 00:52:16,070 when you put in a single frequency -- when you put in x 841 00:52:16,070 --> 00:52:21,510 of t equals e to the 2 pi ift, what comes out? 842 00:52:21,510 --> 00:52:24,880 In terms of this reflecting wall example, the thing that 843 00:52:24,880 --> 00:52:29,040 came out was not one sinusoid, but two sinusoids. 844 00:52:29,040 --> 00:52:32,190 One sinusoid a little bit above the carrier, the other 845 00:52:32,190 --> 00:52:34,990 sinusoid a little bit below the carrier. 846 00:52:34,990 --> 00:52:38,950 In other words, because of Doppler shifts, when you put 847 00:52:38,950 --> 00:52:43,960 in a sinusoid, what comes out is not a sinusoid, but a 848 00:52:43,960 --> 00:52:45,500 modulated sinusoid. 849 00:52:45,500 --> 00:52:50,960 It's something spread out over a region of frequencies. 850 00:52:50,960 --> 00:52:54,590 So one of your favorite tools for dealing with linear time 851 00:52:54,590 --> 00:52:57,960 invariant systems is no longer adequate. 852 00:52:57,960 --> 00:52:59,990 Doesn't work. 853 00:52:59,990 --> 00:53:02,860 Just make a mark of that, because every time you see a 854 00:53:02,860 --> 00:53:06,610 problem like this, everytime I see it, the first thing I try 855 00:53:06,610 --> 00:53:11,190 to do is go through that most familiar and most favorite 856 00:53:11,190 --> 00:53:15,320 form of linear system analysis, which is the Fourier 857 00:53:15,320 --> 00:53:19,550 transform of convolution, is the same as multiplication in 858 00:53:19,550 --> 00:53:20,680 the frequency domain. 859 00:53:20,680 --> 00:53:22,560 You cannot do that anymore. 860 00:53:22,560 --> 00:53:26,680 However, convolution still work, so that's the next thing 861 00:53:26,680 --> 00:53:27,930 we want to look at. 862 00:53:31,710 --> 00:53:35,960 So the thing that we have is the output of the system is 863 00:53:35,960 --> 00:53:39,590 now going to be this integral over frequency -- 864 00:53:39,590 --> 00:53:43,660 x-hat of f times the frequency response function for the 865 00:53:43,660 --> 00:53:46,300 linear but time varying system, times 866 00:53:46,300 --> 00:53:47,550 e to the 2 pi ift. 867 00:53:49,960 --> 00:53:54,670 This is what we derived on the last page, on the last slide. 868 00:53:54,670 --> 00:54:00,730 It is the thing which just automatically happens here. 869 00:54:00,730 --> 00:54:04,530 If I let h of tau and t be the inverse Fourier transform of 870 00:54:04,530 --> 00:54:06,810 h-hat of ft -- 871 00:54:06,810 --> 00:54:12,020 and here what I'm doing is I'm regarding t as a parameter. 872 00:54:12,020 --> 00:54:15,970 So this is a function of tau now and this is a function of 873 00:54:15,970 --> 00:54:20,240 tau for a given t, I can take the Fourier transform of this. 874 00:54:20,240 --> 00:54:23,420 Take the Fourier transform of any old thing at all, so long 875 00:54:23,420 --> 00:54:25,380 as it's L2. 876 00:54:25,380 --> 00:54:28,150 I will sort of half-pretend it's L2. 877 00:54:28,150 --> 00:54:30,590 We'll worry about that later. 878 00:54:30,590 --> 00:54:35,260 So this is the Fourier transform of this. 879 00:54:35,260 --> 00:54:39,020 So then, the thing that happens is that y of t is 880 00:54:39,020 --> 00:54:44,280 going to be equal to this quantity here, except in place 881 00:54:44,280 --> 00:54:49,060 of the system function, h and f of t, for a particular value 882 00:54:49,060 --> 00:54:53,130 of t, I'm going to put in this inverse Fourier transform. 883 00:54:53,130 --> 00:54:56,200 So it'll be h of tau and t, e to the minus 2 884 00:54:56,200 --> 00:54:58,130 pi i of tau d tau. 885 00:55:01,360 --> 00:55:07,080 This integral here is just h-hat of f and t. 886 00:55:07,080 --> 00:55:14,210 If I take this quantity here and I move this term inside 887 00:55:14,210 --> 00:55:16,960 and I interchange orders of integration -- 888 00:55:16,960 --> 00:55:19,940 incidentally, when we're dealing with wireless, we're 889 00:55:19,940 --> 00:55:23,040 going to forget about all of the nice things that we know 890 00:55:23,040 --> 00:55:24,830 about L2 functions. 891 00:55:24,830 --> 00:55:27,890 There's just too much new stuff that's going on here to 892 00:55:27,890 --> 00:55:30,410 worry about that. 893 00:55:30,410 --> 00:55:35,240 So what you want to do is just take Fourier transforms like 894 00:55:35,240 --> 00:55:38,460 while, interchange orders of integration, interchange 895 00:55:38,460 --> 00:55:42,790 everything you want to and simply forgot about all the 896 00:55:42,790 --> 00:55:45,340 mathematical problems that might arise. 897 00:55:45,340 --> 00:55:48,280 After you understand this, at that point, go back and 898 00:55:48,280 --> 00:55:50,660 straighten out the mathematical issues. 899 00:55:50,660 --> 00:55:54,320 This in fact is the way we deal with any problem, or the 900 00:55:54,320 --> 00:55:56,850 way you should deal with any problem. 901 00:55:56,850 --> 00:55:59,870 You don't bring the mathematics in unless it's 902 00:55:59,870 --> 00:56:01,380 going to help you solve the problem. 903 00:56:01,380 --> 00:56:05,370 You don't bring it in to frustrate yourselves. 904 00:56:05,370 --> 00:56:10,060 So the thing we're going to do now is to interchange these 905 00:56:10,060 --> 00:56:11,220 orders of integration. 906 00:56:11,220 --> 00:56:16,360 We're going to integrate over tau on the outside and f on 907 00:56:16,360 --> 00:56:17,100 the inside. 908 00:56:17,100 --> 00:56:20,020 So we're going to bring the function of tau outside. 909 00:56:20,020 --> 00:56:22,190 This is an integral in tau. 910 00:56:22,190 --> 00:56:25,810 The function of f is going to go on the inside. we have e to 911 00:56:25,810 --> 00:56:28,620 the 2 pi ft -- 912 00:56:28,620 --> 00:56:30,630 that's this quantity here. 913 00:56:35,200 --> 00:56:39,170 We have this term there. 914 00:56:39,170 --> 00:56:43,130 When we look at this, we see something very nice because 915 00:56:43,130 --> 00:56:46,660 this is in fact just the Fourier transform of 916 00:56:46,660 --> 00:56:50,030 x of t minus tau. 917 00:56:50,030 --> 00:56:53,740 When we take that out, what we get is the integral of x of t 918 00:56:53,740 --> 00:56:58,570 minus tau times h of tau and t be tau. 919 00:56:58,570 --> 00:57:03,450 In other words, this is time varying convolution. 920 00:57:03,450 --> 00:57:07,970 Nice, simple equation, makes a lot of sense. 921 00:57:07,970 --> 00:57:15,430 It says you have this impulse response here. h of tau and t 922 00:57:15,430 --> 00:57:21,140 now can be interpreted as the response at time t to an 923 00:57:21,140 --> 00:57:24,250 impulse tau seconds earlier. 924 00:57:24,250 --> 00:57:27,410 If you have a system which is changing very, very very, 925 00:57:27,410 --> 00:57:32,340 slowly, then this is essentially just a function of 926 00:57:32,340 --> 00:57:38,330 tau and it's the usual impulse response that you're familiar 927 00:57:38,330 --> 00:57:42,430 with -- namely, this convolution equation gives you 928 00:57:42,430 --> 00:57:50,870 the response at time t to an impulse tau seconds early. 929 00:57:50,870 --> 00:57:54,600 Now we just have something which says this is a linear 930 00:57:54,600 --> 00:57:58,790 time varying filter and in all the cases, we're interested in 931 00:57:58,790 --> 00:58:02,480 this linear time varying filter, changes its impulse 932 00:58:02,480 --> 00:58:06,830 response very slowly as time changes. 933 00:58:06,830 --> 00:58:09,850 Relatively fast change with tau, relatively 934 00:58:09,850 --> 00:58:11,500 slow change with t. 935 00:58:14,250 --> 00:58:16,650 So this is very similar to linear time invariant 936 00:58:16,650 --> 00:58:17,820 convolution. 937 00:58:17,820 --> 00:58:21,490 Channel behaves like a slowly time varying filter and that's 938 00:58:21,490 --> 00:58:22,780 the bottom line of this. 939 00:58:28,020 --> 00:58:31,940 For these ray tracing models we were looking at, the system 940 00:58:31,940 --> 00:58:39,240 function is a sum of terms at the sum of attenuations times 941 00:58:39,240 --> 00:58:41,900 phase change terms. 942 00:58:41,900 --> 00:58:45,250 If we take the inverse Fourier transform of this -- 943 00:58:45,250 --> 00:58:50,170 remember, we're taking the inverse Fourier transform on f 944 00:58:50,170 --> 00:58:51,740 and putting in a tau. 945 00:58:51,740 --> 00:58:55,150 So we're taking this inverse Fourier transform for a 946 00:58:55,150 --> 00:58:57,920 particular t. 947 00:58:57,920 --> 00:58:59,990 We have a function of f and of t. 948 00:58:59,990 --> 00:59:04,390 We take the inverse Fourier transform with respect to the 949 00:59:04,390 --> 00:59:06,290 f and get a tau here. 950 00:59:06,290 --> 00:59:09,970 This then becomes this quantity here. 951 00:59:09,970 --> 00:59:11,960 How do I interpret that? 952 00:59:11,960 --> 00:59:15,330 If I look at a single term here, what is it? 953 00:59:15,330 --> 00:59:19,450 A single term here is just an attentuation factor, a 954 00:59:19,450 --> 00:59:22,420 constant times a sinusoid. 955 00:59:22,420 --> 00:59:25,800 At a particular value of t, this is just the 956 00:59:25,800 --> 00:59:27,710 constant here also. 957 00:59:27,710 --> 00:59:32,280 So for a particular t, all I have is a sinusoid. 958 00:59:32,280 --> 00:59:36,170 What's the inverse Fourier transform of a sinusoid? 959 00:59:36,170 --> 00:59:39,360 I told you all along that it doesn't exist, but for the 960 00:59:39,360 --> 00:59:43,190 time being we will assume that it's what you learned early, 961 00:59:43,190 --> 00:59:46,230 that the inverse Foruier transform of the sinusoid is 962 00:59:46,230 --> 00:59:49,050 an impulse. 963 00:59:49,050 --> 00:59:52,230 So here we are with our impulse there. 964 00:59:52,230 --> 00:59:59,570 This says that the response at time t to an impulse at tau is 965 00:59:59,570 --> 01:00:05,370 going to be zero unless tau is equal to one of these 966 01:00:05,370 --> 01:00:07,460 propogation delay terms. 967 01:00:07,460 --> 01:00:10,580 In other words, I have a system where I'm putting in an 968 01:00:10,580 --> 01:00:14,250 input and this input comes in. 969 01:00:14,250 --> 01:00:18,620 The response to the input is a number of different path 970 01:00:18,620 --> 01:00:22,980 delays and at each path delay, what I'm going to get out of 971 01:00:22,980 --> 01:00:26,470 the system is just a delayed and attenuated version 972 01:00:26,470 --> 01:00:28,530 of what I put in. 973 01:00:28,530 --> 01:00:33,650 Namely, the system isn't a function of frequency at all. 974 01:00:33,650 --> 01:00:36,130 That's what I get through using ray tracing. 975 01:00:36,130 --> 01:00:38,080 I mean, it's one of the consequences 976 01:00:38,080 --> 01:00:40,380 of using ray tracing. 977 01:00:40,380 --> 01:00:44,080 So what I wind up with is a system function which is a 978 01:00:44,080 --> 01:00:48,490 string of impulses and the output then, the convolution 979 01:00:48,490 --> 01:00:51,600 of this with that -- 980 01:00:51,600 --> 01:00:53,920 you're probably better at working with impulses than I 981 01:00:53,920 --> 01:00:58,240 am -- and it's y of t is just the sum of these attenuation 982 01:00:58,240 --> 01:01:04,970 terms times x of t at these various delays. 983 01:01:04,970 --> 01:01:09,030 So what we're doing is we're putting in an arbitrary input. 984 01:01:09,030 --> 01:01:13,030 What's coming out is that attenuated input coming out at 985 01:01:13,030 --> 01:01:15,150 various different times, due to these 986 01:01:15,150 --> 01:01:16,560 various different paths. 987 01:01:16,560 --> 01:01:19,540 I get various paths that are delaying the input by 988 01:01:19,540 --> 01:01:22,620 different amounts and out that delayed input 989 01:01:22,620 --> 01:01:25,360 comes at various times. 990 01:01:25,360 --> 01:01:29,280 This is a nice sanity check because if you think about it, 991 01:01:29,280 --> 01:01:32,620 that's exactly what ought to come out of here. 992 01:01:32,620 --> 01:01:35,250 On the other hand, you ought to wonder about this impulsive 993 01:01:35,250 --> 01:01:39,520 impulse response, because that clearly doesn't make any sense 994 01:01:39,520 --> 01:01:40,570 physically. 995 01:01:40,570 --> 01:01:43,620 So what's going on here? 996 01:01:43,620 --> 01:01:48,830 The thing that's going on is that when we started, we said, 997 01:01:48,830 --> 01:01:54,510 if we're putting in a narrow band input, we don't care 998 01:01:54,510 --> 01:01:58,730 about the frequency response because the frequency response 999 01:01:58,730 --> 01:02:02,810 on these different paths cannot change very quickly and 1000 01:02:02,810 --> 01:02:06,710 therefore, we're just going to have a fixed frequency 1001 01:02:06,710 --> 01:02:08,420 response term. 1002 01:02:08,420 --> 01:02:11,340 Then we've worked with that thing which is not a function 1003 01:02:11,340 --> 01:02:17,460 of frequency and then finally we get down here, where in 1004 01:02:17,460 --> 01:02:20,430 fact, what we're doing is looking at the output. 1005 01:02:20,430 --> 01:02:24,400 Due to a bunch of delayed input terms, if this input 1006 01:02:24,400 --> 01:02:25,800 term here -- 1007 01:02:25,800 --> 01:02:28,890 if x of t is in fact the narrow band term -- 1008 01:02:28,890 --> 01:02:30,910 I guess the way to see that as to look at -- 1009 01:02:33,830 --> 01:02:35,080 where do I look at it? 1010 01:02:38,200 --> 01:02:41,770 I want to look at this expression here. 1011 01:02:41,770 --> 01:02:44,040 If I have a narrow band or maybe -- 1012 01:02:48,950 --> 01:02:52,320 I guess this one is better here. 1013 01:02:52,320 --> 01:02:53,790 Let's look at this expression. 1014 01:02:57,320 --> 01:03:02,340 If my input is in fact narrow band, it's only going to be 1015 01:03:02,340 --> 01:03:06,250 non zero over a small range of frequencies. 1016 01:03:06,250 --> 01:03:09,830 If it's only non zero over a small range of frequencies, I 1017 01:03:09,830 --> 01:03:13,000 don't care what this is, except over that small range 1018 01:03:13,000 --> 01:03:14,340 of frequencies. 1019 01:03:14,340 --> 01:03:17,020 All of this gets filtered out. 1020 01:03:17,020 --> 01:03:20,380 In other words, this filters out this, opposite of the 1021 01:03:20,380 --> 01:03:21,400 usual case. 1022 01:03:21,400 --> 01:03:24,530 So I don't care about what this is at different frequency 1023 01:03:24,530 --> 01:03:28,400 ranges and therefore, we simply have the consequences 1024 01:03:28,400 --> 01:03:31,300 of this, which is something that you see in linear system 1025 01:03:31,300 --> 01:03:32,580 theory all the time. 1026 01:03:32,580 --> 01:03:35,540 We've sort of ruled out impulses and sine waves 1027 01:03:35,540 --> 01:03:39,010 because they don't carry information, but in terms of 1028 01:03:39,010 --> 01:03:41,840 looking at things as intermediate points and going 1029 01:03:41,840 --> 01:03:44,570 through filters and things like that, they're perfectly 1030 01:03:44,570 --> 01:03:45,630 fine 1031 01:03:45,630 --> 01:03:50,570 So here, for simplicity, we assume that these channel 1032 01:03:50,570 --> 01:03:54,800 filters really do not have any -- 1033 01:03:54,800 --> 01:03:58,260 don't respond to frequency changes, whereas in fact they 1034 01:03:58,260 --> 01:04:01,800 do, and all we're doing is modelling them in certain 1035 01:04:01,800 --> 01:04:05,970 frequency bands, which, when we get all done, is what 1036 01:04:05,970 --> 01:04:14,010 really gets rid of all our problems, due to this sort of 1037 01:04:14,010 --> 01:04:18,160 input because this input is smooth now and therefore, the 1038 01:04:18,160 --> 01:04:19,790 output is smooth also. 1039 01:04:26,530 --> 01:04:29,310 The next thing I want to spend a little bit of time on and 1040 01:04:29,310 --> 01:04:33,230 we'll come back to it next time is, how do you deal with 1041 01:04:33,230 --> 01:04:36,290 all of this at baseband? 1042 01:04:36,290 --> 01:04:41,830 I should warn you here that the notes don't do a terribly 1043 01:04:41,830 --> 01:04:44,090 good job of this. 1044 01:04:44,090 --> 01:04:48,580 They have all of the results that you need. 1045 01:04:48,580 --> 01:04:51,010 They don't seem to put them in a very nice, 1046 01:04:51,010 --> 01:04:52,260 well organized fashion. 1047 01:04:52,260 --> 01:04:54,130 I'm not sure there is a nice, well organized 1048 01:04:54,130 --> 01:04:57,090 fashion to put them in. 1049 01:04:57,090 --> 01:04:59,270 But anyway there's a lot of stuff going on when you try to 1050 01:04:59,270 --> 01:05:03,690 move this down from pass band down to baseband. 1051 01:05:03,690 --> 01:05:07,620 I will try to change the notes a little bit to make it clear, 1052 01:05:07,620 --> 01:05:10,690 but I'm not sure that I can. 1053 01:05:10,690 --> 01:05:14,190 The kind of system that we're looking at now is our usual 1054 01:05:14,190 --> 01:05:18,310 QAM type system, which can be generalized somewhat. 1055 01:05:18,310 --> 01:05:20,750 We have a binary input coming in. 1056 01:05:20,750 --> 01:05:22,850 We have a baseband encoder. 1057 01:05:22,850 --> 01:05:27,080 That baseband encoder is creating baseband signals, 1058 01:05:27,080 --> 01:05:31,360 which are being added together to give us a baseband complex 1059 01:05:31,360 --> 01:05:33,230 input to the channel. 1060 01:05:33,230 --> 01:05:36,890 This is being frequency modulated up to some function, 1061 01:05:36,890 --> 01:05:41,390 x of t, which is just the real part of ut times e to the 2 pi 1062 01:05:41,390 --> 01:05:44,280 if of ct as usual. 1063 01:05:44,280 --> 01:05:51,740 So we now have a real part of this signal here, modulated up 1064 01:05:51,740 --> 01:05:53,630 by the carrier frequency. 1065 01:05:53,630 --> 01:05:56,870 This is going through what we'll now regard as a time 1066 01:05:56,870 --> 01:05:59,860 varying channel filter. 1067 01:05:59,860 --> 01:06:02,740 We talked a little bit about channel filters before when we 1068 01:06:02,740 --> 01:06:07,900 were talking about Nyquist theory, because we said in 1069 01:06:07,900 --> 01:06:11,920 general, you want to take your input, you want to pass it 1070 01:06:11,920 --> 01:06:14,160 through a pulse p of t. 1071 01:06:14,160 --> 01:06:18,370 That goes through another filter, which is some h of t 1072 01:06:18,370 --> 01:06:21,140 and that goes into another filter, which is at the 1073 01:06:21,140 --> 01:06:25,390 receiver, which is q of t and you want the product of t and 1074 01:06:25,390 --> 01:06:28,290 you want the convolution of all of those to satisfy the 1075 01:06:28,290 --> 01:06:30,110 Nyquist criteria. 1076 01:06:30,110 --> 01:06:33,970 So we're back with that in spades because now this is 1077 01:06:33,970 --> 01:06:37,180 varying with t also. 1078 01:06:37,180 --> 01:06:38,600 So this goes through this channel 1079 01:06:38,600 --> 01:06:43,010 filter, up at pass band. 1080 01:06:43,010 --> 01:06:46,480 The channel filter up at pass band, we've seen that one of 1081 01:06:46,480 --> 01:06:49,350 the things that it can be viewed as doing is putting 1082 01:06:49,350 --> 01:06:53,200 Doppler shift into this input. 1083 01:06:53,200 --> 01:06:58,600 So what comes in at some frequency f is now going to be 1084 01:06:58,600 --> 01:07:02,710 coming through here at some slightly different frequency. 1085 01:07:02,710 --> 01:07:05,730 We then add white noise to it. 1086 01:07:05,730 --> 01:07:09,790 We then get y of t out, which has now been shifted around 1087 01:07:09,790 --> 01:07:12,520 and smudged in frequency a little bit. 1088 01:07:12,520 --> 01:07:15,100 We go through a frequency demodulation. 1089 01:07:15,100 --> 01:07:18,390 We get down frequency to modulation by 1090 01:07:18,390 --> 01:07:20,450 this carrier frequency. 1091 01:07:20,450 --> 01:07:28,420 We get down to v of t, which is now a baseband complex 1092 01:07:28,420 --> 01:07:31,770 function again, which is supposed to be the same as 1093 01:07:31,770 --> 01:07:35,370 this except for the white Gaussian noise and except for 1094 01:07:35,370 --> 01:07:39,200 the fact we've gone through this filtering operation here. 1095 01:07:39,200 --> 01:07:43,780 Then we want to do base band detection at this point. 1096 01:07:43,780 --> 01:07:46,680 What we would like to do and what will make life a little 1097 01:07:46,680 --> 01:07:51,190 easier for ourselves because we all got sick of this 1098 01:07:51,190 --> 01:07:54,510 business of looking at filters at baseband and also looking 1099 01:07:54,510 --> 01:07:58,470 at filters at pass band, we would like to be able to take 1100 01:07:58,470 --> 01:08:02,680 some baseband equivalent of this filter here. 1101 01:08:02,680 --> 01:08:04,780 So what we're going to do is look at the baseband 1102 01:08:04,780 --> 01:08:06,080 equivalent. 1103 01:08:09,050 --> 01:08:12,930 The system function at baseband corresponding to this 1104 01:08:12,930 --> 01:08:17,760 system at carrier frequency will just be this response 1105 01:08:17,760 --> 01:08:20,370 moved down by s of t. 1106 01:08:20,370 --> 01:08:23,630 In other words, when you take what comes out of here, you 1107 01:08:23,630 --> 01:08:30,270 multiply it, and you shift it down in frequency by s sub c, 1108 01:08:30,270 --> 01:08:34,615 what happens is that this gets shifted down in frequency by s 1109 01:08:34,615 --> 01:08:39,770 sub c and this channel filter gets shifted down in frequency 1110 01:08:39,770 --> 01:08:43,620 by f of c and therefore, what happens is the effect of 1111 01:08:43,620 --> 01:08:48,320 passing y of t through a base band filter -- 1112 01:08:48,320 --> 01:08:54,586 h-hat of f plus fc and t or 0 for f, less than or 1113 01:08:54,586 --> 01:08:56,420 equal to minus fc. 1114 01:08:56,420 --> 01:09:01,030 So far, this is all kind of straightforward and not too 1115 01:09:01,030 --> 01:09:02,280 mysterious. 1116 01:09:09,810 --> 01:09:13,290 So you wind up then in -- 1117 01:09:13,290 --> 01:09:17,010 and this is pure analogy to what we did before -- 1118 01:09:17,010 --> 01:09:23,210 the output is then going to be the integral of this Fourier 1119 01:09:23,210 --> 01:09:26,560 transform of the input. 1120 01:09:26,560 --> 01:09:31,433 The system function for the baseband filter times e to the 1121 01:09:31,433 --> 01:09:33,530 2 pi ift df. 1122 01:09:33,530 --> 01:09:37,280 Same equation as we had before, but before we did it 1123 01:09:37,280 --> 01:09:43,350 at pass band and now we're doing it at baseband . 1124 01:09:43,350 --> 01:09:49,290 For the ray tracing model that we looked at, this function 1125 01:09:49,290 --> 01:09:55,420 here down at baseband is going to be the same as it was at 1126 01:09:55,420 --> 01:10:01,280 pass band, except in place of the 2 pi i tau jft, we now 1127 01:10:01,280 --> 01:10:07,840 have f plus the carrier frequency times tau jft. 1128 01:10:07,840 --> 01:10:12,020 So when we take the inverse Fourier transform of this, 1129 01:10:12,020 --> 01:10:14,240 we're given parameter t. 1130 01:10:14,240 --> 01:10:18,320 What we're going to wind up with is this quantity here. 1131 01:10:18,320 --> 01:10:22,380 This is the same as we had before, with the difference 1132 01:10:22,380 --> 01:10:26,820 that now we're stuck with this carrier frequency times 1133 01:10:26,820 --> 01:10:31,450 propogation delay, which is occurring in time t. 1134 01:10:31,450 --> 01:10:36,520 We still have the same delta functions here and the output 1135 01:10:36,520 --> 01:10:39,210 v of t is the same time variant 1136 01:10:39,210 --> 01:10:43,570 convolution as we had before. 1137 01:10:43,570 --> 01:10:46,680 I'm doing this much too fast for you to follow 1138 01:10:46,680 --> 01:10:48,530 this in real time. 1139 01:10:48,530 --> 01:10:51,160 What I'm saying is, this is exactly the same thing as we 1140 01:10:51,160 --> 01:10:56,630 did before and the only thing new that happens is now this 1141 01:10:56,630 --> 01:11:02,110 carrier frequency term is coming in here on this term. 1142 01:11:02,110 --> 01:11:06,330 If we're using the ray tracing model then, v of t is equal to 1143 01:11:06,330 --> 01:11:08,860 this quantity here. 1144 01:11:08,860 --> 01:11:13,530 The reason I write this down is that this shows you quite 1145 01:11:13,530 --> 01:11:18,970 simply what's going on in terms of the Doppler shift, 1146 01:11:18,970 --> 01:11:23,480 because these terms here, these delay terms are changing 1147 01:11:23,480 --> 01:11:27,470 with Doppler shift and now we're going to see exactly 1148 01:11:27,470 --> 01:11:28,720 what they do to us. 1149 01:11:36,010 --> 01:11:38,250 So we're going to represent the propogation 1150 01:11:38,250 --> 01:11:40,070 delay on each path. 1151 01:11:40,070 --> 01:11:43,840 That's tau j of t, which is tau j prime, which is the 1152 01:11:43,840 --> 01:11:49,960 propogation delay of time zero minus the Doppler shift times 1153 01:11:49,960 --> 01:11:52,460 t divided by f. 1154 01:11:52,460 --> 01:11:56,720 This is just Doppler shift, which is a frequency term, and 1155 01:11:56,720 --> 01:11:59,770 it's increasing linearly. 1156 01:11:59,770 --> 01:12:04,030 The propogation delay is increasing linearly with t. 1157 01:12:04,030 --> 01:12:14,960 The Doppler shift is a shift in frequency and, as I think 1158 01:12:14,960 --> 01:12:19,030 we saw before, you need a 1 over f to compensate for this 1159 01:12:19,030 --> 01:12:19,770 Doppler shift. 1160 01:12:19,770 --> 01:12:23,010 Namely, this is in terms of hertz. 1161 01:12:23,010 --> 01:12:24,710 This is in terms of hertz. 1162 01:12:24,710 --> 01:12:26,300 This is in terms of time. 1163 01:12:26,300 --> 01:12:28,280 This is in terms of time. 1164 01:12:28,280 --> 01:12:31,020 So you need this frequency here to be dimensionally right 1165 01:12:31,020 --> 01:12:32,790 at any rate. 1166 01:12:32,790 --> 01:12:36,580 If we're using narrow band communication, then v of t is 1167 01:12:36,580 --> 01:12:40,290 just going to be this difference here, which is just 1168 01:12:40,290 --> 01:12:46,180 the last equation with the Doppler shift put into it. 1169 01:12:46,180 --> 01:12:52,600 So all of this says that when we're now looking at things at 1170 01:12:52,600 --> 01:12:57,060 base band instead of a pass band, what has happened is 1171 01:12:57,060 --> 01:13:00,750 that this term has been added. 1172 01:13:00,750 --> 01:13:05,280 Before, we just had these delay terms here and we had 1173 01:13:05,280 --> 01:13:07,330 these attenuation terms. 1174 01:13:07,330 --> 01:13:12,090 Now what's happening is because we are modulating up 1175 01:13:12,090 --> 01:13:16,885 with carrier frequency s sub c, we then get a Doppler shift 1176 01:13:16,885 --> 01:13:19,140 that moves us down a little bit. 1177 01:13:19,140 --> 01:13:21,730 We're then moving down by s sub c. 1178 01:13:21,730 --> 01:13:25,090 What we wind up with is something which is off by that 1179 01:13:25,090 --> 01:13:27,290 Doppler shift. 1180 01:13:27,290 --> 01:13:30,180 If we only had a single term, we wouldn't be 1181 01:13:30,180 --> 01:13:31,940 demodulating by s sub c. 1182 01:13:31,940 --> 01:13:35,180 We'd be demodulating by something which would get rid 1183 01:13:35,180 --> 01:13:36,780 of this term for us. 1184 01:13:36,780 --> 01:13:39,840 Then all we'd have is just some arbitrary phase term 1185 01:13:39,840 --> 01:13:41,300 here, because this isn't important. 1186 01:13:41,300 --> 01:13:43,350 This is just a phase term. 1187 01:13:43,350 --> 01:13:46,960 This is the term which is important. 1188 01:13:46,960 --> 01:13:49,276 The reason for going through all of this -- 1189 01:13:49,276 --> 01:13:52,780 and I'm going to come back to this next time -- 1190 01:13:52,780 --> 01:13:57,510 is that when you use frequency recovery at the receiver and 1191 01:13:57,510 --> 01:14:02,420 you always use frequency recovery at the receiver, 1192 01:14:02,420 --> 01:14:05,270 what's going to happen is you're not going to shift 1193 01:14:05,270 --> 01:14:07,500 down by s of c. 1194 01:14:07,500 --> 01:14:13,000 You're going to shift down s of c, plus some average value 1195 01:14:13,000 --> 01:14:14,940 of Doppler shift. 1196 01:14:14,940 --> 01:14:18,020 If they have a bunch of terms, each with different Doppler 1197 01:14:18,020 --> 01:14:23,330 shifts in them, when you try to measure how much Doppler -- 1198 01:14:23,330 --> 01:14:26,540 when you try to measure what the received carrier is, 1199 01:14:26,540 --> 01:14:28,950 you're going to be frustrated by all of these different 1200 01:14:28,950 --> 01:14:29,910 Doppler shifts. 1201 01:14:29,910 --> 01:14:32,760 You're going to come up with some average value in your 1202 01:14:32,760 --> 01:14:36,510 frequency circuit, which means that in place of this 1203 01:14:36,510 --> 01:14:40,890 quantity, you will get something which replaces each 1204 01:14:40,890 --> 01:14:45,820 of these D sub j's by not the actual Doppler shift, but by 1205 01:14:45,820 --> 01:14:48,300 how far this Doppler shift is away from the 1206 01:14:48,300 --> 01:14:50,680 main Doppler shift. 1207 01:14:50,680 --> 01:14:55,010 So these terms here, which are the things that make this 1208 01:14:55,010 --> 01:15:00,090 system function in here change with time, are in fact 1209 01:15:00,090 --> 01:15:04,390 changing according to how far away each of these Doppler 1210 01:15:04,390 --> 01:15:07,770 shifts are from the main Doppler shift, which means 1211 01:15:07,770 --> 01:15:11,400 that the amount of time this system function in here is 1212 01:15:11,400 --> 01:15:18,770 going to remain stable depends on how much the Doppler shifts 1213 01:15:18,770 --> 01:15:22,060 vary from each other on these different paths. 1214 01:15:22,060 --> 01:15:24,840 In other words, the important thing is the Doppler spread 1215 01:15:24,840 --> 01:15:27,580 between the biggest Doppler shifts and the smallest 1216 01:15:27,580 --> 01:15:29,200 Doppler shifts. 1217 01:15:29,200 --> 01:15:32,930 That Doppler spread is going to determine how long you've 1218 01:15:32,930 --> 01:15:36,700 got something that looks like a linear time invariant system 1219 01:15:36,700 --> 01:15:40,530 function which says that every once in awhile, if you're 1220 01:15:40,530 --> 01:15:44,390 trying to measure what's going on at the channel, at 1221 01:15:44,390 --> 01:15:47,050 intervals of time approximately one over two 1222 01:15:47,050 --> 01:15:49,760 times the Doppler spread, you're going to have to change 1223 01:15:49,760 --> 01:15:52,050 those measurements. 1224 01:15:52,050 --> 01:15:55,900 So whether you can make cellular telephony work or not 1225 01:15:55,900 --> 01:16:00,150 depends on whether you can make measurements at a speed 1226 01:16:00,150 --> 01:16:04,830 which is equal to one over two times the Doppler spread. 1227 01:16:04,830 --> 01:16:07,930 I'm going to do more about that next time. 1228 01:16:07,930 --> 01:16:10,890 I don't expect you to understand it now. 1229 01:16:10,890 --> 01:16:15,350 Maybe after you read about it and we talk about it more, 1230 01:16:15,350 --> 01:16:16,600 because in fact that's -- 1231 01:16:20,760 --> 01:16:24,980 if you look at this question of Doppler spread and what its 1232 01:16:24,980 --> 01:16:28,910 effect is on how long a system looks like it's time 1233 01:16:28,910 --> 01:16:32,630 invariant, this is one of the key parameters to 1234 01:16:32,630 --> 01:16:36,110 understanding how any kind of wireless 1235 01:16:36,110 --> 01:16:37,360 system is going to work.