1 00:00:00,090 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,720 continue to offer high quality educational resources for free. 5 00:00:10,720 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,280 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,280 --> 00:00:18,450 at ocw.mit.edu. 8 00:00:21,410 --> 00:00:23,760 DENNIS FREEMAN: Hello. 9 00:00:23,760 --> 00:00:29,160 Welcome to the first lecture in the last topic of this class. 10 00:00:29,160 --> 00:00:32,970 So we'll spend this lecture and the next lecture 11 00:00:32,970 --> 00:00:35,100 talking about modulation. 12 00:00:35,100 --> 00:00:38,880 Modulation, like sampling, is an excellent illustration 13 00:00:38,880 --> 00:00:41,490 of the power of thinking about signals in terms 14 00:00:41,490 --> 00:00:43,500 of their Fourier transforms. 15 00:00:43,500 --> 00:00:45,390 We'll see, just like we saw in sampling, 16 00:00:45,390 --> 00:00:47,940 that a problem that was potentially very 17 00:00:47,940 --> 00:00:51,120 complicated to understand, sampling, 18 00:00:51,120 --> 00:00:54,510 was very easy when you thought about it in the Fourier domain. 19 00:00:54,510 --> 00:00:56,010 And precisely the same thing happens 20 00:00:56,010 --> 00:00:58,200 with modulation, which is precisely the reason we're 21 00:00:58,200 --> 00:01:00,540 talking about it now. 22 00:01:00,540 --> 00:01:03,330 So I'll talk about modulation in the context of a communication 23 00:01:03,330 --> 00:01:05,040 system. 24 00:01:05,040 --> 00:01:06,360 That's just for convenience. 25 00:01:06,360 --> 00:01:09,030 In fact, modulation is used in lots of other places. 26 00:01:09,030 --> 00:01:11,890 In fact, next time, in the next lecture, 27 00:01:11,890 --> 00:01:14,250 I'll show you an application of modulation 28 00:01:14,250 --> 00:01:18,750 from my research group, where we used modulation 29 00:01:18,750 --> 00:01:22,950 to improve the resolution of an optical microscope-- 30 00:01:22,950 --> 00:01:27,210 nothing whatever to do with communications. 31 00:01:27,210 --> 00:01:29,010 But it's still-- the optical microscope-- 32 00:01:29,010 --> 00:01:32,190 the enhancement to resolution that we achieve 33 00:01:32,190 --> 00:01:34,470 is directly based on the principles 34 00:01:34,470 --> 00:01:36,970 that I'll start to describe today. 35 00:01:36,970 --> 00:01:39,840 So I'll talk about modulation today 36 00:01:39,840 --> 00:01:41,760 in the context of communications. 37 00:01:41,760 --> 00:01:44,400 Probably the most convenient, easiest 38 00:01:44,400 --> 00:01:47,310 to think about communication system to all of us, 39 00:01:47,310 --> 00:01:51,240 being humans, is speech. 40 00:01:51,240 --> 00:01:53,790 We use speech for communication all the time. 41 00:01:53,790 --> 00:01:57,570 It's very easy to think about how speech works 42 00:01:57,570 --> 00:01:58,710 as a communication medium. 43 00:01:58,710 --> 00:02:01,710 Somebody talks, somebody listens. 44 00:02:01,710 --> 00:02:03,540 One of the first ways of thinking about it 45 00:02:03,540 --> 00:02:08,430 as a technological feat was the telephone-- 46 00:02:08,430 --> 00:02:12,420 the idea being that you convert the sound that I'm emitting 47 00:02:12,420 --> 00:02:14,359 when I'm speaking by a microphone 48 00:02:14,359 --> 00:02:16,150 into an electrical representation that gets 49 00:02:16,150 --> 00:02:18,360 shot down a wire. 50 00:02:18,360 --> 00:02:22,800 Then at the other end, you take the electrical signal 51 00:02:22,800 --> 00:02:25,710 that's coming down the wire and turn it back into sound. 52 00:02:25,710 --> 00:02:28,980 That's the principle of a telephone-- 53 00:02:28,980 --> 00:02:31,440 works very well. 54 00:02:31,440 --> 00:02:32,880 Modulation comes up when we start 55 00:02:32,880 --> 00:02:37,560 to think about how would we generalize this notion 56 00:02:37,560 --> 00:02:39,930 for wireless communication. 57 00:02:39,930 --> 00:02:42,920 In particular, if we do cell phone communication, 58 00:02:42,920 --> 00:02:47,130 cell phones transmit the signal that is picked up 59 00:02:47,130 --> 00:02:49,690 from the microphone. 60 00:02:49,690 --> 00:02:53,410 That signal gets converted into electromagnetic signals. 61 00:02:53,410 --> 00:02:55,660 That's the basis by which cell phones communicate 62 00:02:55,660 --> 00:02:57,640 with the cell tower. 63 00:02:57,640 --> 00:02:59,560 The towers may communicate with other towers 64 00:02:59,560 --> 00:03:01,510 via lot of different kinds of technologies. 65 00:03:01,510 --> 00:03:03,310 I'm ignoring those for now. 66 00:03:03,310 --> 00:03:05,920 But the communication between the phone and the tower 67 00:03:05,920 --> 00:03:08,530 is via electromagnetic waves. 68 00:03:08,530 --> 00:03:10,570 And there's an interesting thing that 69 00:03:10,570 --> 00:03:14,500 happens when you try to recode the signal that would have been 70 00:03:14,500 --> 00:03:17,740 perfectly happy running down the copper wire, when 71 00:03:17,740 --> 00:03:21,630 you try to recode that signal into an electromagnetic wave. 72 00:03:21,630 --> 00:03:23,380 And that has to do with basic physics 73 00:03:23,380 --> 00:03:27,140 of electromagnetic waves, which I'm sure you all know. 74 00:03:27,140 --> 00:03:32,450 And so I'll just remind you of one simple idea. 75 00:03:32,450 --> 00:03:35,880 So for efficient transmission from 76 00:03:35,880 --> 00:03:40,600 an electrical representation to an electromagnetic wave, 77 00:03:40,600 --> 00:03:44,020 first off, that transduction is mediated by something 78 00:03:44,020 --> 00:03:45,490 we call an antenna. 79 00:03:45,490 --> 00:03:47,317 Antennas will take an electrical signal 80 00:03:47,317 --> 00:03:49,150 and convert it into an electromagnetic wave, 81 00:03:49,150 --> 00:03:51,250 and vice versa. 82 00:03:51,250 --> 00:03:54,670 But the efficiency with which an antenna works 83 00:03:54,670 --> 00:03:59,740 has to do, among other things, with its size. 84 00:03:59,740 --> 00:04:03,220 It's very difficult-- by which I mean it takes lots of power-- 85 00:04:07,720 --> 00:04:11,350 to transform a signal from an electrical representation 86 00:04:11,350 --> 00:04:15,910 to an E&M representation if the antenna is smaller-- 87 00:04:15,910 --> 00:04:21,579 is significantly smaller-- than the wavelength of interest. 88 00:04:21,579 --> 00:04:24,840 So it's not that you can't do it. 89 00:04:24,840 --> 00:04:27,290 It's that it takes lots of power. 90 00:04:27,290 --> 00:04:29,680 So if you don't want to burn a lot of power 91 00:04:29,680 --> 00:04:31,780 doing that transformation, then you 92 00:04:31,780 --> 00:04:33,580 need to make an antenna that's roughly 93 00:04:33,580 --> 00:04:38,710 the size of the wavelength that you're trying to transmit 94 00:04:38,710 --> 00:04:41,260 as an electromagnetic wave. 95 00:04:41,260 --> 00:04:44,140 So if we were thinking about this kind 96 00:04:44,140 --> 00:04:47,170 of a scheme for the communication of voice-- 97 00:04:47,170 --> 00:04:49,480 we've talked about voice many times-- 98 00:04:49,480 --> 00:04:51,190 Telephone-quality voice is usually 99 00:04:51,190 --> 00:04:54,280 defined to be frequencies between about 200 hertz 100 00:04:54,280 --> 00:04:55,990 and about 3 kilohertz. 101 00:04:55,990 --> 00:04:58,030 If I had a signal that was composed 102 00:04:58,030 --> 00:05:01,060 of those kinds of frequencies, how big 103 00:05:01,060 --> 00:05:05,260 should I make the antenna to get efficient coupling 104 00:05:05,260 --> 00:05:08,050 between the electrical representation 105 00:05:08,050 --> 00:05:10,750 in the cell phone and the electromagnetic wave 106 00:05:10,750 --> 00:05:12,550 that the cell phone wants to launch 107 00:05:12,550 --> 00:05:14,650 to get to the cell tower? 108 00:05:14,650 --> 00:05:16,040 So look at your neighbor. 109 00:05:19,270 --> 00:05:20,720 That involves turning your head. 110 00:05:20,720 --> 00:05:22,090 [LAUGHTER] 111 00:05:22,090 --> 00:05:25,490 Say hello and figure out how long should the antenna be. 112 00:05:30,979 --> 00:05:34,472 [INTERPOSING VOICES] 113 00:06:56,900 --> 00:07:00,180 DENNIS FREEMAN: Does everybody have an answer? 114 00:07:00,180 --> 00:07:02,360 Raise your hand if you don't have an answer. 115 00:07:02,360 --> 00:07:04,140 How do you like that for a switch of-- 116 00:07:04,140 --> 00:07:06,715 Raise your hand if you don't have an answer. 117 00:07:06,715 --> 00:07:07,590 You all have answers? 118 00:07:07,590 --> 00:07:10,260 OK, so how big should the antenna be-- 119 00:07:10,260 --> 00:07:11,590 1, 2, 3, 4, or 5? 120 00:07:17,187 --> 00:07:17,770 OK, very good. 121 00:07:17,770 --> 00:07:21,700 So the answer is really big. 122 00:07:21,700 --> 00:07:27,640 So you think about the relationship between wavelength 123 00:07:27,640 --> 00:07:31,540 and distance, so you can think about the relationship is 124 00:07:31,540 --> 00:07:32,860 given by speed. 125 00:07:32,860 --> 00:07:36,040 The speed of a wave is the wave length-- 126 00:07:36,040 --> 00:07:38,680 how long it goes per cycle-- 127 00:07:38,680 --> 00:07:42,580 times the number of cycles per second, f. 128 00:07:42,580 --> 00:07:44,470 So you can solve that expression to find out 129 00:07:44,470 --> 00:07:46,570 the wavelength in terms of the speed, which 130 00:07:46,570 --> 00:07:51,076 for an electromagnetic wave is the speed of light, 3 times 10 131 00:07:51,076 --> 00:07:53,110 to the eighth meters per second. 132 00:07:53,110 --> 00:07:57,580 The hardest one to launch is the lowest frequency. 133 00:07:57,580 --> 00:07:59,680 That'll take the biggest antenna. 134 00:07:59,680 --> 00:08:01,360 So the lowest frequency in telephony-- 135 00:08:01,360 --> 00:08:05,170 in telephone-quality speech-- is 200 hertz. 136 00:08:05,170 --> 00:08:09,940 So we get about 1,500 kilometers-- 137 00:08:09,940 --> 00:08:13,365 kind of big, kind of useless. 138 00:08:13,365 --> 00:08:14,740 If you were thinking about trying 139 00:08:14,740 --> 00:08:17,073 to make cellular communication and your antenna actually 140 00:08:17,073 --> 00:08:24,310 had to be 1,500 kilometers, that just isn't going to work. 141 00:08:24,310 --> 00:08:26,350 So what do we do? 142 00:08:26,350 --> 00:08:27,670 We obviously don't do this. 143 00:08:30,770 --> 00:08:33,580 So the answer is that you would need an antenna 144 00:08:33,580 --> 00:08:36,549 hundreds of miles in length. 145 00:08:36,549 --> 00:08:41,409 So what frequency should you be using 146 00:08:41,409 --> 00:08:44,440 if you wanted to build a phone that kind of fit in your hand-- 147 00:08:44,440 --> 00:08:46,810 10 centimeters or so? 148 00:08:46,810 --> 00:08:49,107 What would be the frequency-- 149 00:08:49,107 --> 00:08:51,190 what would be the interesting range of frequencies 150 00:08:51,190 --> 00:08:53,305 that you would want to use for such a device? 151 00:09:00,440 --> 00:09:02,870 And the answer is-- 152 00:09:02,870 --> 00:09:03,580 Of course. 153 00:09:03,580 --> 00:09:06,150 So the answer is, you go to the bottom one, 154 00:09:06,150 --> 00:09:07,690 and it's bigger than a gigahertz. 155 00:09:07,690 --> 00:09:10,090 And it's just running the same expression 156 00:09:10,090 --> 00:09:11,920 in the other direction. 157 00:09:11,920 --> 00:09:15,250 So you think about, the frequency that you would like 158 00:09:15,250 --> 00:09:18,430 would be the speed of light divided by the wavelength. 159 00:09:18,430 --> 00:09:21,220 If you wanted the wavelength to be 10 centimeters, 160 00:09:21,220 --> 00:09:23,860 then you would end up with a frequency 161 00:09:23,860 --> 00:09:25,480 on the order of gigahertz. 162 00:09:25,480 --> 00:09:27,400 And it shouldn't come as a surprise to you 163 00:09:27,400 --> 00:09:30,460 then that that's what we use in cellular communication. 164 00:09:30,460 --> 00:09:35,620 So a modern cell phone uses 2.1 gigahertz. 165 00:09:35,620 --> 00:09:40,570 So the point is that when we're thinking 166 00:09:40,570 --> 00:09:47,920 about how we would like to use the electromagnetic spectrum 167 00:09:47,920 --> 00:09:51,130 for a communications task, that spectrum is not 168 00:09:51,130 --> 00:09:54,190 necessarily well-matched to the communications 169 00:09:54,190 --> 00:09:55,960 problem of interest. 170 00:09:55,960 --> 00:09:57,970 You might think that that cellular example 171 00:09:57,970 --> 00:09:58,910 is an exception. 172 00:09:58,910 --> 00:10:00,670 In fact, that's the rule. 173 00:10:00,670 --> 00:10:03,520 If you try to have a signal of interest transmitted 174 00:10:03,520 --> 00:10:07,150 over a medium or stored on some sort of a medium, 175 00:10:07,150 --> 00:10:10,547 it is generally the case that there is a matching problem, 176 00:10:10,547 --> 00:10:12,130 that the characteristics of the medium 177 00:10:12,130 --> 00:10:14,260 don't match the characteristics of the message. 178 00:10:14,260 --> 00:10:16,960 And so part of communications engineering 179 00:10:16,960 --> 00:10:22,010 is trying to come up with a coding scheme that 180 00:10:22,010 --> 00:10:25,670 matches the characteristics of the message 181 00:10:25,670 --> 00:10:28,915 to the characteristics of the medium. 182 00:10:28,915 --> 00:10:30,290 And so what I'm going to do today 183 00:10:30,290 --> 00:10:34,700 is talk about some matching schemes based on modulation. 184 00:10:34,700 --> 00:10:39,860 So we just saw that if we wanted to do cellular communication 185 00:10:39,860 --> 00:10:44,210 of voice, voice might have a spectrum represented 186 00:10:44,210 --> 00:10:48,410 by this magnitude function, X, where 187 00:10:48,410 --> 00:10:53,910 the bandwidth is on the order of a few kilohertz. 188 00:10:53,910 --> 00:10:56,340 But we might want to transmit a signal that 189 00:10:56,340 --> 00:10:59,700 has the same information. 190 00:10:59,700 --> 00:11:01,340 However, we would like the frequencies 191 00:11:01,340 --> 00:11:05,460 to be up around 2 gigahertz. 192 00:11:05,460 --> 00:11:11,710 Which of these coding schemes, Y as a function of X, 193 00:11:11,710 --> 00:11:14,140 achieves this transformation? 194 00:11:14,140 --> 00:11:18,610 Take the stuff that you have in a low frequency range 195 00:11:18,610 --> 00:11:20,680 and shift it to a high frequency range. 196 00:11:20,680 --> 00:11:21,460 What would you do? 197 00:11:28,920 --> 00:11:33,570 The obvious answer is to stare with a blank face in it. 198 00:11:33,570 --> 00:11:36,704 It'll definitely come to you, right? 199 00:11:36,704 --> 00:11:38,370 You don't want to talk to somebody else. 200 00:11:38,370 --> 00:11:39,369 That would give it away. 201 00:11:42,154 --> 00:11:47,643 [INTERPOSING VOICES] 202 00:12:08,601 --> 00:12:11,810 DENNIS FREEMAN: So what's the relationship between X of t 203 00:12:11,810 --> 00:12:13,410 and Y of t? 204 00:12:13,410 --> 00:12:16,300 Is it relation 1, 2, 3, 4, or none of the above? 205 00:12:20,290 --> 00:12:24,950 OK, it's about 80% correct. 206 00:12:24,950 --> 00:12:26,960 So how do I think about it? 207 00:12:26,960 --> 00:12:29,050 So let's see, I want to figure out 208 00:12:29,050 --> 00:12:35,700 a relationship between the Fourier transform of Y and X. 209 00:12:35,700 --> 00:12:39,190 If I wanted to figure out the Fourier transform of Y, 210 00:12:39,190 --> 00:12:45,940 I would integrate Y of t e to the minus j omega t dt. 211 00:12:45,940 --> 00:12:47,850 That looks kind of right. 212 00:12:47,850 --> 00:12:50,059 And Y would be X of the t-- 213 00:12:50,059 --> 00:12:51,100 let's try the first one-- 214 00:12:51,100 --> 00:12:56,775 X of t e to the j omega c t e to the minus j omega t dt. 215 00:13:00,390 --> 00:13:06,580 Well, that's almost the Fourier transform of X. All I've done 216 00:13:06,580 --> 00:13:09,710 is shifted omega. 217 00:13:09,710 --> 00:13:18,970 So in fact, that's the same as X of j omega minus omega c. 218 00:13:18,970 --> 00:13:21,380 And that's what I want to do. 219 00:13:21,380 --> 00:13:26,230 So the idea is that, if you multiply 220 00:13:26,230 --> 00:13:30,010 by this complex exponential, the effect 221 00:13:30,010 --> 00:13:35,080 of that multiplication in time is to shift frequencies. 222 00:13:35,080 --> 00:13:37,120 Can somebody say that to me-- 223 00:13:37,120 --> 00:13:39,760 say the same transformation but in slightly different words? 224 00:13:43,958 --> 00:13:47,877 Guess that was kind of big. 225 00:13:47,877 --> 00:13:48,377 Yes. 226 00:13:48,377 --> 00:13:51,814 AUDIENCE: Couldn't you [INAUDIBLE] multiplication 227 00:13:51,814 --> 00:13:53,247 by the exponential in time. 228 00:13:53,247 --> 00:13:55,580 DENNIS FREEMAN: [INAUDIBLE] by the exponential in time-- 229 00:13:55,580 --> 00:13:56,300 AUDIENCE: Yeah equal to-- 230 00:13:56,300 --> 00:13:58,040 DENNIS FREEMAN: --should correspond to-- 231 00:13:58,040 --> 00:13:59,656 AUDIENCE: Which gives you a frequency [INAUDIBLE]. 232 00:13:59,656 --> 00:14:01,572 DENNIS FREEMAN: So generally, a multiplication 233 00:14:01,572 --> 00:14:02,865 in time corresponds to-- 234 00:14:02,865 --> 00:14:03,740 AUDIENCE: [INAUDIBLE] 235 00:14:03,740 --> 00:14:06,660 DENNIS FREEMAN: --convolution. 236 00:14:06,660 --> 00:14:10,410 So equivalently, instead of saying it shifted in frequency, 237 00:14:10,410 --> 00:14:12,610 you could say-- 238 00:14:12,610 --> 00:14:13,520 AUDIENCE: [INAUDIBLE] 239 00:14:13,520 --> 00:14:15,020 DENNIS FREEMAN: --it got convolved-- 240 00:14:15,020 --> 00:14:16,496 AUDIENCE: With delta. 241 00:14:16,496 --> 00:14:19,320 DENNIS FREEMAN: --with the delta function. 242 00:14:19,320 --> 00:14:22,410 So a different way of saying the same thing 243 00:14:22,410 --> 00:14:24,570 would be that you think about convolving 244 00:14:24,570 --> 00:14:29,400 with the delta function in frequency-- 245 00:14:29,400 --> 00:14:32,670 same thing. 246 00:14:32,670 --> 00:14:34,440 Now, we don't actually do this. 247 00:14:34,440 --> 00:14:36,023 When we're doing the cell phone thing, 248 00:14:36,023 --> 00:14:39,240 we don't actually multiply it by e to the j omega c t. 249 00:14:39,240 --> 00:14:39,920 Anyone know why? 250 00:14:43,550 --> 00:14:45,960 This is kind of the simplest way that you could imagine. 251 00:14:45,960 --> 00:14:47,830 I've taken frequency content centered near 0 252 00:14:47,830 --> 00:14:50,330 and turning it into frequency content centered near omega c. 253 00:14:50,330 --> 00:14:51,984 But that's not what we really do. 254 00:14:51,984 --> 00:14:53,150 Why don't we really do that? 255 00:14:53,150 --> 00:14:55,102 AUDIENCE: [INAUDIBLE] 256 00:14:55,102 --> 00:14:57,120 DENNIS FREEMAN: Exactly. 257 00:14:57,120 --> 00:15:02,130 We don't really do it, because the signals aren't real. 258 00:15:04,810 --> 00:15:06,640 So how do you know the signal is not real? 259 00:15:11,005 --> 00:15:15,855 AUDIENCE: Because magnitudes [INAUDIBLE]. 260 00:15:15,855 --> 00:15:18,600 DENNIS FREEMAN: If the signal had been real, 261 00:15:18,600 --> 00:15:20,225 the Fourier transform would have been-- 262 00:15:23,830 --> 00:15:25,750 If the signal had been real the Fourier 263 00:15:25,750 --> 00:15:28,370 transform would have been-- 264 00:15:28,370 --> 00:15:29,460 X of j omega. 265 00:15:34,407 --> 00:15:36,490 If the signal had been real, the Fourier transform 266 00:15:36,490 --> 00:15:37,300 would have been-- 267 00:15:40,228 --> 00:15:41,692 AUDIENCE: Symmetry. 268 00:15:41,692 --> 00:15:43,710 DENNIS FREEMAN: --some kind of symmetry. 269 00:15:43,710 --> 00:15:45,561 How do I see symmetry in that expression? 270 00:15:45,561 --> 00:15:46,394 AUDIENCE: Conjugate. 271 00:15:46,394 --> 00:15:46,791 AUDIENCE: Conjugate. 272 00:15:46,791 --> 00:15:48,320 DENNIS FREEMAN: Conjugate symmetry. 273 00:15:48,320 --> 00:15:50,675 How do I see conjugate symmetry in that expression? 274 00:15:55,910 --> 00:16:01,100 Well, this is cos omega t plus j sine omega t. 275 00:16:08,850 --> 00:16:12,810 So if this is a real signal, then it gives rise to something 276 00:16:12,810 --> 00:16:14,640 here that is symmetric. 277 00:16:14,640 --> 00:16:18,570 The cosine terms are all symmetric about the origin. 278 00:16:18,570 --> 00:16:21,120 And it has an imaginary part that 279 00:16:21,120 --> 00:16:23,397 is antisymmetric about the origin, 280 00:16:23,397 --> 00:16:25,230 because you add together a bunch of cosines. 281 00:16:25,230 --> 00:16:27,510 You can't get anything that's anything other than symmetric 282 00:16:27,510 --> 00:16:28,260 about the origin. 283 00:16:28,260 --> 00:16:29,490 You add together a bunch of sines. 284 00:16:29,490 --> 00:16:31,448 You can't get anything except something that is 285 00:16:31,448 --> 00:16:34,530 antisymmetric about the origin. 286 00:16:34,530 --> 00:16:37,510 So you know this thing has to be-- 287 00:16:37,510 --> 00:16:42,624 so a real signal would have had conjugate symmetry. 288 00:16:42,624 --> 00:16:44,290 The real part would have been symmetric. 289 00:16:44,290 --> 00:16:46,810 The imaginary part would be antisymmetric. 290 00:16:46,810 --> 00:16:50,946 And you can see that this is not conjugate symmetric. 291 00:16:50,946 --> 00:16:52,571 Everybody knows what I'm talking about? 292 00:16:52,571 --> 00:16:55,264 AUDIENCE: [INAUDIBLE] 293 00:16:55,264 --> 00:16:56,680 DENNIS FREEMAN: Conjugate symmetry 294 00:16:56,680 --> 00:16:59,680 would mean that the real part of the signal 295 00:16:59,680 --> 00:17:03,700 is symmetric about the origin, which means that if this 296 00:17:03,700 --> 00:17:06,069 is supposed to represent a real signal, 297 00:17:06,069 --> 00:17:09,490 then there should have been a reflection over here to make 298 00:17:09,490 --> 00:17:12,754 it symmetric about the origin. 299 00:17:12,754 --> 00:17:16,682 AUDIENCE: Just was it symmetric before? 300 00:17:16,682 --> 00:17:19,010 DENNIS FREEMAN: It was symmetric, 301 00:17:19,010 --> 00:17:21,930 but I shifted it by a complex number. 302 00:17:21,930 --> 00:17:27,290 I shifted it by cos omega c t plus j sine omega c t. 303 00:17:27,290 --> 00:17:33,018 So by that multiplication, I generated a complex number. 304 00:17:33,018 --> 00:17:35,880 AUDIENCE: [INAUDIBLE]. 305 00:17:35,880 --> 00:17:41,340 DENNIS FREEMAN: So this signal was complex valued and not 306 00:17:41,340 --> 00:17:43,836 conjugate symmetric. 307 00:17:43,836 --> 00:17:45,210 So the point is trying to get you 308 00:17:45,210 --> 00:17:47,130 to remember the kinds of things that we're supposed to know 309 00:17:47,130 --> 00:17:48,450 about Fourier transforms. 310 00:17:48,450 --> 00:17:54,240 So by shifting with a complex exponential, 311 00:17:54,240 --> 00:17:57,800 we wreck the realness of the original signal. 312 00:17:57,800 --> 00:18:00,900 The real original signal would have been conjugate symmetric 313 00:18:00,900 --> 00:18:02,880 in frequency. 314 00:18:02,880 --> 00:18:05,760 But the wrecking of it gave rise to a signal 315 00:18:05,760 --> 00:18:09,692 that was no longer conjugate symmetric in frequency. 316 00:18:12,350 --> 00:18:16,490 So we don't really modulate this way. 317 00:18:16,490 --> 00:18:18,470 But we do the obvious extension. 318 00:18:18,470 --> 00:18:21,720 What we would do is modulate with a cosine wave. 319 00:18:24,230 --> 00:18:26,570 So now, if instead of multiplying 320 00:18:26,570 --> 00:18:30,117 by a complex exponential, I multiply by the cosine omega c 321 00:18:30,117 --> 00:18:32,990 t, by Euler's expression, I can think about that 322 00:18:32,990 --> 00:18:37,100 as being the sum of two components-- 323 00:18:37,100 --> 00:18:40,520 one at plus omega c and one at minus omega c. 324 00:18:40,520 --> 00:18:42,950 And now, when I do the convolution, 325 00:18:42,950 --> 00:18:47,180 I get a signal that is conjugate symmetric. 326 00:18:47,180 --> 00:18:50,270 So when I convolve this with this one, 327 00:18:50,270 --> 00:18:52,530 this one gives me a copy of this here. 328 00:18:52,530 --> 00:18:54,230 And when I convolve this with this one, 329 00:18:54,230 --> 00:18:57,060 this one gives me a copy of this one down here. 330 00:18:57,060 --> 00:19:00,710 So now, the resulting signal, which I know by construction-- 331 00:19:00,710 --> 00:19:05,310 if this was real and this was real, then that's real-- 332 00:19:05,310 --> 00:19:08,426 I know by construction that this signal must have been real. 333 00:19:08,426 --> 00:19:10,050 But I can also see it in the transform, 334 00:19:10,050 --> 00:19:11,466 because I can see now that there's 335 00:19:11,466 --> 00:19:13,200 a symmetry that is consistent with it 336 00:19:13,200 --> 00:19:15,100 being conjugate symmetric. 337 00:19:15,100 --> 00:19:15,600 Yes. 338 00:19:15,600 --> 00:19:16,599 Somebody had a question? 339 00:19:20,800 --> 00:19:23,579 So that's what we mean by modulation. 340 00:19:23,579 --> 00:19:24,370 This is modulation. 341 00:19:24,370 --> 00:19:27,610 Modulation just is a fancy word that means multiply. 342 00:19:27,610 --> 00:19:31,300 So what we're going to do is multiply the signal 343 00:19:31,300 --> 00:19:32,860 by a carrier. 344 00:19:32,860 --> 00:19:36,490 The carrier is going to be a signal that carries 345 00:19:36,490 --> 00:19:38,530 the message through the medium. 346 00:19:38,530 --> 00:19:41,800 The carrier is chosen so that it goes 347 00:19:41,800 --> 00:19:44,500 efficiently through the medium. 348 00:19:44,500 --> 00:19:48,310 And then the carrier carries the message through the medium. 349 00:19:48,310 --> 00:19:51,190 So we think about this as modulation. 350 00:19:51,190 --> 00:19:54,010 And we want to be familiar with going back and forth 351 00:19:54,010 --> 00:19:55,760 between time and frequency. 352 00:19:55,760 --> 00:19:59,020 You can also think about the result of modulation in time. 353 00:19:59,020 --> 00:20:01,060 So if this were my message, which 354 00:20:01,060 --> 00:20:04,570 is intended to be represented as a low frequency, 355 00:20:04,570 --> 00:20:06,430 and this is my carrier, which is intended 356 00:20:06,430 --> 00:20:09,280 to be represented as a higher frequency, then 357 00:20:09,280 --> 00:20:12,550 when I modulate it, I get a modulated signal, 358 00:20:12,550 --> 00:20:14,000 by which I mean-- 359 00:20:14,000 --> 00:20:16,390 this is called amplitude modulation-- 360 00:20:16,390 --> 00:20:20,170 the amplitude of the carrier is modulated by the message. 361 00:20:20,170 --> 00:20:22,370 That's all we mean. 362 00:20:22,370 --> 00:20:26,350 So you can see that this transformation, which 363 00:20:26,350 --> 00:20:28,987 has the property of moving the information 364 00:20:28,987 --> 00:20:30,820 from a low frequency that's hard to transmit 365 00:20:30,820 --> 00:20:33,850 to a high frequency that is easy to transmit, 366 00:20:33,850 --> 00:20:40,240 that has the effect of doing a very particular pattern 367 00:20:40,240 --> 00:20:42,760 to the time-domain waveform. 368 00:20:45,091 --> 00:20:47,340 Now, it would be completely useless as a communication 369 00:20:47,340 --> 00:20:49,660 scheme if it weren't easy to invert. 370 00:20:52,770 --> 00:20:55,540 So imagine that I have this signal. 371 00:20:55,540 --> 00:20:58,950 And what I'd like to do is recover x. 372 00:20:58,950 --> 00:21:03,840 What should I do to recover x, the original message? 373 00:21:03,840 --> 00:21:06,180 So the idea is, I have an original message 374 00:21:06,180 --> 00:21:07,770 available in my cell phone. 375 00:21:07,770 --> 00:21:09,690 It gets modulated so that it can be launched 376 00:21:09,690 --> 00:21:11,070 into electromagnetic waves. 377 00:21:11,070 --> 00:21:14,670 The electromagnetic waves go to a receiver thousands of miles 378 00:21:14,670 --> 00:21:15,600 away. 379 00:21:15,600 --> 00:21:18,420 And now, the idea is to reconstruct my original signal 380 00:21:18,420 --> 00:21:19,320 x. 381 00:21:19,320 --> 00:21:20,880 What would I do-- 382 00:21:20,880 --> 00:21:24,705 what kind of a system would I use to recover x from y? 383 00:21:31,373 --> 00:21:32,677 AUDIENCE: [INAUDIBLE]. 384 00:21:32,677 --> 00:21:33,760 DENNIS FREEMAN: I'm sorry. 385 00:21:33,760 --> 00:21:35,191 AUDIENCE: Couldn't you divide out the cosine? 386 00:21:35,191 --> 00:21:37,470 DENNIS FREEMAN: You could divide out the cosine. 387 00:21:37,470 --> 00:21:45,330 So you could take x of t cosine omega c t times something-- 388 00:21:45,330 --> 00:21:48,600 what do I want to say-- a of t designed 389 00:21:48,600 --> 00:21:50,310 so that this times this is 1. 390 00:21:54,370 --> 00:21:56,270 That's kind of ugly. 391 00:21:56,270 --> 00:21:59,320 Anybody see anything ugly about that? 392 00:21:59,320 --> 00:21:59,820 Yeah. 393 00:21:59,820 --> 00:22:01,952 AUDIENCE: [INAUDIBLE] shift it back the other way? 394 00:22:01,952 --> 00:22:04,410 DENNIS FREEMAN: You could also shift it back the other way. 395 00:22:04,410 --> 00:22:04,951 That's right. 396 00:22:04,951 --> 00:22:07,027 So before we do that, why is this ugly? 397 00:22:07,027 --> 00:22:08,490 AUDIENCE: The zeros. 398 00:22:08,490 --> 00:22:10,770 DENNIS FREEMAN: The zeros. 399 00:22:10,770 --> 00:22:15,780 So if we wanted to take a signal that looks like a cosine wave 400 00:22:15,780 --> 00:22:20,970 and multiply it by some signal that generates 1, 401 00:22:20,970 --> 00:22:22,709 that's not too hard to do here. 402 00:22:22,709 --> 00:22:24,500 You would do that with something like this. 403 00:22:24,500 --> 00:22:26,250 But it becomes very hard to do here. 404 00:22:26,250 --> 00:22:31,320 So you would end up making some signal that does some awful-- 405 00:22:31,320 --> 00:22:33,390 So it would periodically be a mass. 406 00:22:36,290 --> 00:22:38,180 But you can do what you said. 407 00:22:38,180 --> 00:22:41,390 An alternative would be to multiply it 408 00:22:41,390 --> 00:22:45,530 by another cosine, which in the frequency domain 409 00:22:45,530 --> 00:22:46,760 is easy to think about. 410 00:22:46,760 --> 00:22:49,670 It would just shift it back-- 411 00:22:49,670 --> 00:22:52,460 so convolve with a pair of impulses 412 00:22:52,460 --> 00:22:56,690 to move something that was at DC out to some high frequency, 413 00:22:56,690 --> 00:22:59,180 convolve again to take the thing that was at high frequency 414 00:22:59,180 --> 00:23:01,340 and bring it back to DC. 415 00:23:01,340 --> 00:23:05,540 And you can think about that in either frequency or time. 416 00:23:05,540 --> 00:23:07,040 It's easy to think about it in time. 417 00:23:07,040 --> 00:23:08,748 If you think about it in time, here we've 418 00:23:08,748 --> 00:23:10,850 got the product of two cosines. 419 00:23:10,850 --> 00:23:12,890 But the product of two cosines is just 1/2 420 00:23:12,890 --> 00:23:16,640 plus half the cosine of double. 421 00:23:16,640 --> 00:23:18,500 Well, that's good. 422 00:23:18,500 --> 00:23:19,230 Why is that good? 423 00:23:19,230 --> 00:23:23,180 Well, if you multiply x of t by this, 424 00:23:23,180 --> 00:23:25,010 this is a super high frequency. 425 00:23:25,010 --> 00:23:30,390 If omega c was a high frequency, 2 omega c is even higher. 426 00:23:30,390 --> 00:23:36,410 So what you could do is remove x times 1/2 cos 2 omega c t 427 00:23:36,410 --> 00:23:39,890 with a low pass filter, since omega c 428 00:23:39,890 --> 00:23:41,630 is such a high frequency. 429 00:23:41,630 --> 00:23:44,480 And that would just leave you with half the message, which 430 00:23:44,480 --> 00:23:47,720 would be easy then to reconstruct, 431 00:23:47,720 --> 00:23:50,420 because what you would do is just put it through a low pass 432 00:23:50,420 --> 00:23:52,310 filter and then multiply by 2. 433 00:23:52,310 --> 00:23:54,920 You can similarly think about the same thing in frequency. 434 00:23:54,920 --> 00:23:58,460 If I took y and convolved it-- if I multiply 435 00:23:58,460 --> 00:24:03,020 in time by another cosine wave, that second cosine wave 436 00:24:03,020 --> 00:24:04,960 is a pair of impulses-- one at minus omega 437 00:24:04,960 --> 00:24:07,100 c and one at omega c. 438 00:24:07,100 --> 00:24:09,200 And now, when I convolve the y signal, 439 00:24:09,200 --> 00:24:13,870 this one shifts these two up, and this one 440 00:24:13,870 --> 00:24:15,760 shifts these two down. 441 00:24:15,760 --> 00:24:17,620 And two of them land on top of each other. 442 00:24:21,450 --> 00:24:26,770 But each of these was only of height 1/2. 443 00:24:26,770 --> 00:24:32,810 So by Euler's expression, cosine of something was 1/2 e 444 00:24:32,810 --> 00:24:34,720 to the whatever plus 1/2 e to the whatever. 445 00:24:34,720 --> 00:24:38,320 So I got 1/2's on each of those amplitudes. 446 00:24:38,320 --> 00:24:42,820 So the result is then that I have to multiply the low 447 00:24:42,820 --> 00:24:46,430 frequency part by a factor of two to undo the 1/2's. 448 00:24:51,340 --> 00:24:54,640 So this kind of a scheme is especially nice, 449 00:24:54,640 --> 00:24:58,510 because you can scramble together multiple messages 450 00:24:58,510 --> 00:25:02,820 and still get them separated at the destination. 451 00:25:02,820 --> 00:25:09,500 If you imagine having three similar transmitters that 452 00:25:09,500 --> 00:25:12,250 use their own omega c-- 453 00:25:12,250 --> 00:25:15,080 so the first one uses omega 1, omega 2, omega 3-- 454 00:25:17,700 --> 00:25:19,530 so if each one of the transmitters 455 00:25:19,530 --> 00:25:23,730 had their own frequency, and if the frequencies were far enough 456 00:25:23,730 --> 00:25:27,000 apart, and if the frequencies were all big compared 457 00:25:27,000 --> 00:25:31,470 to the message frequency, then you could combine them all 458 00:25:31,470 --> 00:25:35,440 and select out the one of interest 459 00:25:35,440 --> 00:25:42,290 by tuning the receiver, by choosing the demodulation 460 00:25:42,290 --> 00:25:43,020 frequency. 461 00:25:43,020 --> 00:25:46,190 So now, if the receiver chose omega c equals omega 1, 462 00:25:46,190 --> 00:25:49,000 you would decode message 1. 463 00:25:49,000 --> 00:25:54,110 If omega c were omega 2, you'd decode message 2. 464 00:25:54,110 --> 00:25:59,430 And that's because the medium works approximately linearly. 465 00:25:59,430 --> 00:26:01,940 So if you launch multiple waves into the air-- 466 00:26:05,360 --> 00:26:07,890 I don't want to get too much into electromagnetic theory-- 467 00:26:07,890 --> 00:26:10,620 so the presence of the antennas distort it from linearity. 468 00:26:10,620 --> 00:26:12,434 But once the antennas are all there, 469 00:26:12,434 --> 00:26:13,850 then it's perfectly linear system. 470 00:26:16,710 --> 00:26:21,180 And the thing that gets into the air as a result of a sum 471 00:26:21,180 --> 00:26:24,850 is the sum of the individual parts. 472 00:26:24,850 --> 00:26:27,540 So the idea then is illustrated here. 473 00:26:27,540 --> 00:26:31,080 If I had three different messages represented 474 00:26:31,080 --> 00:26:36,810 by different style houses, and each one of the messages was 475 00:26:36,810 --> 00:26:40,170 at a different frequency-- omega 1, omega 2, omega 3-- 476 00:26:40,170 --> 00:26:43,770 by tuning the omega c of the receiver, 477 00:26:43,770 --> 00:26:48,930 if I put omega c at omega 1, the convolution of this one 478 00:26:48,930 --> 00:26:51,810 would suck this one up to here. 479 00:26:51,810 --> 00:26:53,910 And by this one would lower that one down there. 480 00:26:53,910 --> 00:26:59,130 You get overlap of the lowest frequency pair. 481 00:26:59,130 --> 00:27:00,810 And so if you built a low pass filter 482 00:27:00,810 --> 00:27:06,450 of exactly the right width, you would decode message 1. 483 00:27:06,450 --> 00:27:11,800 Where, if you just changed the frequency of the demodulator-- 484 00:27:11,800 --> 00:27:15,760 if you make the demodulation frequency now be omega 2-- 485 00:27:15,760 --> 00:27:19,360 now, the effect of shifting that different amount 486 00:27:19,360 --> 00:27:23,260 means that the low pass filter recovers message 2, rather 487 00:27:23,260 --> 00:27:26,530 the message 1. 488 00:27:26,530 --> 00:27:29,760 So that's the idea. 489 00:27:29,760 --> 00:27:33,250 So that's the idea that we use in commercial AM radio. 490 00:27:33,250 --> 00:27:37,060 And that was, in fact, a revolutionary idea 491 00:27:37,060 --> 00:27:39,910 that enabled people to think about for the first time 492 00:27:39,910 --> 00:27:42,460 a communication system that did a lot of things 493 00:27:42,460 --> 00:27:45,220 that were very different from previous communication systems. 494 00:27:45,220 --> 00:27:49,000 In particular, it went at the speed of light. 495 00:27:49,000 --> 00:27:51,970 Even more importantly, or at least as important, 496 00:27:51,970 --> 00:27:56,040 is the fact that it was a broadcast system. 497 00:27:56,040 --> 00:28:00,040 So broadcast was an idea that was 498 00:28:00,040 --> 00:28:02,920 championed by David Sarnoff. 499 00:28:02,920 --> 00:28:06,170 Sarnoff was a visionary. 500 00:28:06,170 --> 00:28:10,670 He was the person who was very excited about the idea 501 00:28:10,670 --> 00:28:14,420 of broadcast, which is a little ironic, because he 502 00:28:14,420 --> 00:28:17,900 got his start with Marconi. 503 00:28:17,900 --> 00:28:20,370 Anybody ever hear of Marconi? 504 00:28:20,370 --> 00:28:22,760 Good, good, you're supposed to have heard of Marconi. 505 00:28:22,760 --> 00:28:24,860 So Sarnoff got his start-- 506 00:28:24,860 --> 00:28:26,930 this is Sarnoff; this is Marconi-- 507 00:28:26,930 --> 00:28:28,550 Sarnoff got his start with Marconi. 508 00:28:28,550 --> 00:28:33,170 Marconi made his mint with wireless telegraphy. 509 00:28:33,170 --> 00:28:35,870 Anybody ever hear of telegraphy? 510 00:28:35,870 --> 00:28:37,820 Of course not. 511 00:28:37,820 --> 00:28:41,480 So telegraphy, that's telegraph. 512 00:28:41,480 --> 00:28:42,440 Shake your heads yes. 513 00:28:42,440 --> 00:28:44,930 It's ancient, I realize. 514 00:28:44,930 --> 00:28:49,220 So Marconi made his fortune with wireless telegraphy. 515 00:28:49,220 --> 00:28:54,770 Telegraphy was what we call point to point. 516 00:28:54,770 --> 00:28:57,590 The idea in telegraphy was precisely the same 517 00:28:57,590 --> 00:29:01,450 as the idea of the US post office, 518 00:29:01,450 --> 00:29:03,070 except it was at the speed of light, 519 00:29:03,070 --> 00:29:05,140 or not quite the speed of light. 520 00:29:05,140 --> 00:29:10,330 So the idea in the post office is you take a sheet of paper 521 00:29:10,330 --> 00:29:13,540 and do something to it that makes it magically appear 522 00:29:13,540 --> 00:29:15,580 at somebody else's place. 523 00:29:15,580 --> 00:29:20,401 So point A communicated to point B. 524 00:29:20,401 --> 00:29:21,970 Telegraphy was precisely the same. 525 00:29:21,970 --> 00:29:25,150 You take your sheet of paper to the telegraph office. 526 00:29:25,150 --> 00:29:27,940 And somebody who's very skilled with their hands, 527 00:29:27,940 --> 00:29:31,510 or specifically with their finger, 528 00:29:31,510 --> 00:29:34,810 would do something that caused that piece of paper 529 00:29:34,810 --> 00:29:39,510 to be regenerated hundreds of miles away. 530 00:29:39,510 --> 00:29:42,110 Then that piece of paper got delivered. 531 00:29:42,110 --> 00:29:46,010 So message went from point A to point B. 532 00:29:46,010 --> 00:29:49,800 So telegraphy came long before Marconi. 533 00:29:49,800 --> 00:29:52,830 And it was a revolution in how you do communications. 534 00:29:52,830 --> 00:29:55,260 But it was point to point-- one person 535 00:29:55,260 --> 00:29:56,760 sent one message to one person. 536 00:29:59,980 --> 00:30:04,090 Sarnoff got his start in newspaper business. 537 00:30:04,090 --> 00:30:09,040 He was a Russian immigrant, impoverished, and had 538 00:30:09,040 --> 00:30:10,450 a newspaper route as a kid. 539 00:30:10,450 --> 00:30:12,430 And he was ambitious. 540 00:30:12,430 --> 00:30:15,440 Newspapers are broadcast. 541 00:30:15,440 --> 00:30:19,370 The idea in point to point and broadcast are very different. 542 00:30:19,370 --> 00:30:20,990 In broadcast, you're allowed to spend 543 00:30:20,990 --> 00:30:23,360 a fortune on the transmitter-- 544 00:30:23,360 --> 00:30:27,710 the printing press-- but not on the thing 545 00:30:27,710 --> 00:30:28,780 that the individuals get. 546 00:30:28,780 --> 00:30:31,450 The individuals get newsprint. 547 00:30:31,450 --> 00:30:34,730 So the paper and ink have to be cheap. 548 00:30:34,730 --> 00:30:37,580 The printing press doesn't. 549 00:30:37,580 --> 00:30:41,840 So that was Sarnoff's background. 550 00:30:41,840 --> 00:30:46,040 So he was interested as a kid in broadcast, in newspaper. 551 00:30:46,040 --> 00:30:50,000 But then he got his reputation working for Marconi in wireless 552 00:30:50,000 --> 00:30:51,110 telegraphy. 553 00:30:51,110 --> 00:30:54,830 Marconi, the inventor of radio, thought of a way of doing 554 00:30:54,830 --> 00:30:58,780 point-to-point telegraphy wirelessly via radio-- 555 00:30:58,780 --> 00:31:02,180 radio telegraphy. 556 00:31:02,180 --> 00:31:08,170 And he sold it to ships. 557 00:31:08,170 --> 00:31:11,980 And Sarnoff made his reputation, because he 558 00:31:11,980 --> 00:31:15,640 was the guy operating the radio telegraphy 559 00:31:15,640 --> 00:31:21,990 system at that Marconi company when the Titanic sank. 560 00:31:21,990 --> 00:31:24,540 Everybody know about the Titanic, right? 561 00:31:24,540 --> 00:31:25,440 Big ship sank. 562 00:31:33,255 --> 00:31:38,040 So Sarnoff was known as an amazing telegraphy operator. 563 00:31:38,040 --> 00:31:42,570 And he stayed at the station for 72 consecutive hours getting 564 00:31:42,570 --> 00:31:44,760 emergency messages from the Titanic, 565 00:31:44,760 --> 00:31:47,340 telling everybody everything that he could, 566 00:31:47,340 --> 00:31:50,040 trying to tell them the situation, whose family was 567 00:31:50,040 --> 00:31:52,380 in good shape, whose family was not in good shape. 568 00:31:52,380 --> 00:31:55,170 It had an enormous impact-- 569 00:31:55,170 --> 00:31:58,050 big enough that Congress made a law 570 00:31:58,050 --> 00:32:01,230 saying that every ship had to have wireless telegraphy. 571 00:32:01,230 --> 00:32:04,500 Now, that made Marconi extremely rich. 572 00:32:04,500 --> 00:32:08,410 And it indirectly made Sarnoff extremely rich too. 573 00:32:08,410 --> 00:32:13,020 Sarnoff then got very interested in extending 574 00:32:13,020 --> 00:32:15,330 the idea of radio, which then was 575 00:32:15,330 --> 00:32:19,000 point to point to broadcast. 576 00:32:19,000 --> 00:32:21,420 So the idea was to somehow-- 577 00:32:21,420 --> 00:32:24,690 he called it a radio music box. 578 00:32:24,690 --> 00:32:28,020 Somehow, it was supposed to be like a newspaper 579 00:32:28,020 --> 00:32:29,730 but at the speed of light. 580 00:32:29,730 --> 00:32:37,680 So the idea was to somehow make mass consumption of radio. 581 00:32:37,680 --> 00:32:43,560 At the time, radio was per ship, point to point. 582 00:32:43,560 --> 00:32:47,460 So you have the land-based station talking to ship A, 583 00:32:47,460 --> 00:32:48,990 or the land-based station talking 584 00:32:48,990 --> 00:32:51,480 to ship B, or ship A talking to ship B, 585 00:32:51,480 --> 00:32:53,100 but it was all point to point. 586 00:32:53,100 --> 00:32:57,000 Sarnoff's idea was, let's make a newspaper out of this. 587 00:32:57,000 --> 00:33:02,790 The key to doing that was making a cheap receiver. 588 00:33:02,790 --> 00:33:04,170 It's like newsprint. 589 00:33:04,170 --> 00:33:06,720 The printer can cost a mint. 590 00:33:06,720 --> 00:33:12,600 The transmitter for radio broadcast music 591 00:33:12,600 --> 00:33:15,299 is allowed to cost a mint, but the receivers 592 00:33:15,299 --> 00:33:16,590 are not allowed to cost a mint. 593 00:33:16,590 --> 00:33:18,600 That's like the newsprint. 594 00:33:18,600 --> 00:33:21,990 So the trick was to make an inexpensive receiver. 595 00:33:21,990 --> 00:33:25,500 The problem with making an expensive receiver is that 596 00:33:25,500 --> 00:33:29,670 the scheme that we just talked about, 597 00:33:29,670 --> 00:33:38,720 where you decode the signal by multiplying by cos omega c t-- 598 00:33:38,720 --> 00:33:41,910 omega c chosen to be the frequency that you want 599 00:33:41,910 --> 00:33:42,919 to listen to-- 600 00:33:42,919 --> 00:33:44,460 the problem with that-- that's called 601 00:33:44,460 --> 00:33:47,640 synchronous demodulation-- the problem with that 602 00:33:47,640 --> 00:33:51,880 is that you've got to be exactly synchronized. 603 00:33:51,880 --> 00:33:55,560 If you want to listen to the message at omega 2, 604 00:33:55,560 --> 00:34:01,050 omega c must equal omega 2, not omega 2 plus 3. 605 00:34:01,050 --> 00:34:04,560 So if you want to listen to a particular frequency, 606 00:34:04,560 --> 00:34:08,340 to a particular message, you had to have the frequency chosen 607 00:34:08,340 --> 00:34:10,830 to match the carrier of the message you 608 00:34:10,830 --> 00:34:13,110 wanted to listen to. 609 00:34:13,110 --> 00:34:15,239 Today, that's not so hard. 610 00:34:15,239 --> 00:34:17,130 The way we would make frequencies today 611 00:34:17,130 --> 00:34:18,810 is with crystal. 612 00:34:18,810 --> 00:34:23,190 Crystals are great because the frequencies are determined 613 00:34:23,190 --> 00:34:25,920 by the distances in crystal lattices, which 614 00:34:25,920 --> 00:34:30,179 are determined by nanomechanical processes very precisely. 615 00:34:30,179 --> 00:34:32,520 And in fact, we can make crystals with no problem 616 00:34:32,520 --> 00:34:35,760 with frequency resolutions of 10 to the minus seventh, 617 00:34:35,760 --> 00:34:37,230 even 10 to the minus eighth. 618 00:34:37,230 --> 00:34:42,060 So the errors are very small compared to the frequencies 619 00:34:42,060 --> 00:34:43,920 that we're trying to generate. 620 00:34:43,920 --> 00:34:47,010 Even that wouldn't be good enough. 621 00:34:47,010 --> 00:34:51,960 So back then-- so Sarnoff was working back in the 1800s-- 622 00:34:51,960 --> 00:34:53,850 back then, they couldn't possibly do 10 623 00:34:53,850 --> 00:34:55,147 to the minus eighth precision. 624 00:34:55,147 --> 00:34:56,730 They were doing something more like 10 625 00:34:56,730 --> 00:34:58,980 to the minus second precision. 626 00:34:58,980 --> 00:35:01,260 They didn't have a technology based on crystals. 627 00:35:05,890 --> 00:35:07,890 So it would have been impossible to match 628 00:35:07,890 --> 00:35:10,132 even to within a factor of 10 to the minus third. 629 00:35:10,132 --> 00:35:11,590 But even if they could have matched 630 00:35:11,590 --> 00:35:13,590 to within 10 to the minus eighth, which we could 631 00:35:13,590 --> 00:35:17,310 today, even that wouldn't work, because not only does 632 00:35:17,310 --> 00:35:21,620 the frequency have to match, the phase has to match. 633 00:35:21,620 --> 00:35:24,680 So if you're multiplying by cos omega c t 634 00:35:24,680 --> 00:35:26,660 and you want omega c to be omega 2, 635 00:35:26,660 --> 00:35:29,420 it better actually be the cosine of omega 2 636 00:35:29,420 --> 00:35:31,610 and not the sine of omega 2. 637 00:35:31,610 --> 00:35:34,310 Sine and cos differ by a phase. 638 00:35:34,310 --> 00:35:40,040 So the question is, what's the effect of phase when you're 639 00:35:40,040 --> 00:35:41,540 trying to demodulate a signal? 640 00:35:45,420 --> 00:35:47,610 So look at your neighbor. 641 00:35:47,610 --> 00:35:49,740 What would happen if you tried to demodulate 642 00:35:49,740 --> 00:35:54,870 by precisely the right frequency but you were slipped by phase? 643 00:36:05,270 --> 00:36:08,763 [INTERPOSING VOICES] 644 00:36:10,759 --> 00:36:17,745 AUDIENCE: [INAUDIBLE] over here and be much smaller at the end. 645 00:36:17,745 --> 00:36:21,737 [INTERPOSING VOICES] 646 00:36:24,232 --> 00:36:28,723 AUDIENCE: And then it's to the point [INAUDIBLE]. 647 00:36:28,723 --> 00:36:30,220 [INAUDIBLE] 648 00:36:30,220 --> 00:36:31,218 [INTERPOSING VOICES] 649 00:36:31,218 --> 00:36:40,200 AUDIENCE: And then if you [INAUDIBLE] 650 00:36:40,200 --> 00:36:45,190 [INTERPOSING VOICES] 651 00:37:14,910 --> 00:37:16,410 DENNIS FREEMAN: So what would happen 652 00:37:16,410 --> 00:37:20,340 if there is a shift between this carrier that 653 00:37:20,340 --> 00:37:24,600 was modulating the signal and the carrier that 654 00:37:24,600 --> 00:37:25,960 is demodulating the signal? 655 00:37:25,960 --> 00:37:27,460 What would happen if there's a phase 656 00:37:27,460 --> 00:37:29,730 shift of phi between those two? 657 00:37:29,730 --> 00:37:32,340 What's the effect of phi not being 0? 658 00:37:32,340 --> 00:37:34,440 Ideally, phi would be 0. 659 00:37:34,440 --> 00:37:35,445 Ideally, phi would be 0. 660 00:37:35,445 --> 00:37:36,945 What would happen if phi were not 0? 661 00:37:43,000 --> 00:37:46,654 You did all that talking, and you don't have-- 662 00:37:46,654 --> 00:37:47,154 Yes. 663 00:37:47,154 --> 00:37:52,130 AUDIENCE: So the low pass filter of the two will not work. 664 00:37:52,130 --> 00:37:53,630 It would be-- 665 00:37:53,630 --> 00:37:56,294 It will be lots more than what you want it 666 00:37:56,294 --> 00:37:58,084 to be, like 1/2 the original. 667 00:37:58,084 --> 00:37:59,750 DENNIS FREEMAN: So that's exactly right. 668 00:37:59,750 --> 00:38:02,500 So the effect of phi-- 669 00:38:02,500 --> 00:38:04,120 if you think about-- 670 00:38:04,120 --> 00:38:08,040 so now, you have cosine of some omega c t and cos omega c t 671 00:38:08,040 --> 00:38:09,000 plus phi. 672 00:38:09,000 --> 00:38:14,080 So you have cos A cos B. so that gives you 673 00:38:14,080 --> 00:38:16,577 the cos of the difference and the cos of the sum. 674 00:38:16,577 --> 00:38:18,160 The cos of the difference was supposed 675 00:38:18,160 --> 00:38:19,750 to be-- the difference was supposed to be 0. 676 00:38:19,750 --> 00:38:21,070 The cos of the difference would be 677 00:38:21,070 --> 00:38:22,486 the cos of zero, which would be 1, 678 00:38:22,486 --> 00:38:25,010 and that's where the 1/2 came from. 679 00:38:25,010 --> 00:38:29,170 But now, the difference isn't 0 anymore. 680 00:38:29,170 --> 00:38:32,440 So instead of getting 1/2 cos of 0, 681 00:38:32,440 --> 00:38:34,930 we get 1/2 cos of phi, which means 682 00:38:34,930 --> 00:38:40,270 that if phi is a constant, you just get the wrong amplitude. 683 00:38:40,270 --> 00:38:47,110 But if phi is a slowly varying signal, which it would be, 684 00:38:47,110 --> 00:38:50,882 even if you had a frequency reliability of one part in 10 685 00:38:50,882 --> 00:38:53,074 to the minus seventh-- 686 00:38:53,074 --> 00:38:55,240 one way of thinking about that would be that there's 687 00:38:55,240 --> 00:38:57,010 a slowly varying phase-- 688 00:38:57,010 --> 00:38:58,720 the effect of the slowly varying phase 689 00:38:58,720 --> 00:39:03,340 would be to make the amplitude vary with time. 690 00:39:03,340 --> 00:39:04,630 So we call that fading. 691 00:39:04,630 --> 00:39:07,600 And it would be a very distracting thing 692 00:39:07,600 --> 00:39:08,720 to have happen. 693 00:39:08,720 --> 00:39:11,710 So this kind of a technology would even 694 00:39:11,710 --> 00:39:15,190 be difficult today, when we can match frequencies very well. 695 00:39:15,190 --> 00:39:18,310 It was completely out of the question back in the 1800s, 696 00:39:18,310 --> 00:39:20,800 when they couldn't match frequencies that well. 697 00:39:20,800 --> 00:39:25,640 So the trick was to not only send the message 698 00:39:25,640 --> 00:39:27,140 but also send the carrier. 699 00:39:30,150 --> 00:39:34,020 So ideally, when we first talked about modulation, 700 00:39:34,020 --> 00:39:35,454 there is no C path. 701 00:39:35,454 --> 00:39:36,870 All you do is you take the signal, 702 00:39:36,870 --> 00:39:38,244 and you modulate it by a carrier, 703 00:39:38,244 --> 00:39:39,840 and you send that out on the antenna. 704 00:39:39,840 --> 00:39:46,320 Now, instead, add on a little bit of the carrier. 705 00:39:46,320 --> 00:39:51,010 That way what's in the air is the carrier and the message. 706 00:39:51,010 --> 00:39:53,160 So now when you receive it, somehow 707 00:39:53,160 --> 00:39:55,440 if you can receive the carrier, you 708 00:39:55,440 --> 00:39:57,720 can use the carrier to tell you information about not 709 00:39:57,720 --> 00:40:01,260 only the frequency but also the phase of the carrier. 710 00:40:01,260 --> 00:40:03,760 And you can use that then to demodulate the message. 711 00:40:03,760 --> 00:40:04,980 That's the idea. 712 00:40:04,980 --> 00:40:10,200 Notice that adding in a little bit C of the carrier 713 00:40:10,200 --> 00:40:16,560 is precisely the same as adding a constant to the message 714 00:40:16,560 --> 00:40:20,119 before you modulate-- mathematically identical. 715 00:40:20,119 --> 00:40:22,410 And that gives an easy way of thinking about the effect 716 00:40:22,410 --> 00:40:24,810 of this carrier. 717 00:40:24,810 --> 00:40:27,030 If you think about the message added to C 718 00:40:27,030 --> 00:40:35,650 and if C is big enough, you can make the message positive only. 719 00:40:35,650 --> 00:40:39,030 You remember in the previous illustration, 720 00:40:39,030 --> 00:40:41,730 every time the message went through 0, which might happen 721 00:40:41,730 --> 00:40:47,910 here, the modulating message went through 0 also, 722 00:40:47,910 --> 00:40:51,240 which means that sign changes were affected 723 00:40:51,240 --> 00:40:54,190 by a 180-degree phase shifts of the carrier, which 724 00:40:54,190 --> 00:40:55,950 was kind of a subtle thing. 725 00:40:55,950 --> 00:40:58,170 So now, the message appears entirely 726 00:40:58,170 --> 00:41:03,000 as the positive envelope of the carrier. 727 00:41:03,000 --> 00:41:05,280 Well, that's nice, because that makes it very easy 728 00:41:05,280 --> 00:41:11,080 to decode in a way that has no dependence whatever 729 00:41:11,080 --> 00:41:13,310 on the carrier frequency. 730 00:41:13,310 --> 00:41:16,390 If the carrier frequency is big enough-- 731 00:41:16,390 --> 00:41:19,300 if the frequency of the carrier is sufficiently larger 732 00:41:19,300 --> 00:41:21,490 than the maximum frequency of the message, 733 00:41:21,490 --> 00:41:26,380 there's a trivial way to decode such a system with a nonlinear 734 00:41:26,380 --> 00:41:27,830 circuit of this type. 735 00:41:27,830 --> 00:41:30,630 What's intended here is that you take the message that's 736 00:41:30,630 --> 00:41:36,190 received from the antenna, reconstruct y, which 737 00:41:36,190 --> 00:41:40,210 is intended to be the output message, such 738 00:41:40,210 --> 00:41:43,360 that if z, the signal on the antenna, 739 00:41:43,360 --> 00:41:49,730 exceeds the current value of the message, the diode comes on. 740 00:41:49,730 --> 00:41:53,200 And that makes the blue line, the decoded signal, 741 00:41:53,200 --> 00:41:55,420 rapidly go back up to the red line-- 742 00:41:55,420 --> 00:41:58,030 the thing that's coming off the antenna. 743 00:41:58,030 --> 00:42:01,930 But if the antenna signal shrinks below the blue line, 744 00:42:01,930 --> 00:42:05,250 let the blue line discharge, because there 745 00:42:05,250 --> 00:42:09,930 is an RC decay constant. 746 00:42:09,930 --> 00:42:13,190 So there's a fast attack through the diode, 747 00:42:13,190 --> 00:42:16,730 so that the blue quickly goes to the peak value of the red 748 00:42:16,730 --> 00:42:20,180 and slowly decays back toward 0. 749 00:42:20,180 --> 00:42:23,840 The result is that if the difference in frequency 750 00:42:23,840 --> 00:42:27,980 between the carrier frequency and the message frequencies 751 00:42:27,980 --> 00:42:30,470 is sufficiently large, you can effectively 752 00:42:30,470 --> 00:42:34,044 separate the blue from the red with a very simple circuit. 753 00:42:34,044 --> 00:42:35,335 And that's the way they do it-- 754 00:42:35,335 --> 00:42:39,320 or that's the way they did it in the early 1900s. 755 00:42:39,320 --> 00:42:41,310 But there's still a problem with that. 756 00:42:41,310 --> 00:42:43,760 The problem is that the messages-- 757 00:42:43,760 --> 00:42:47,420 audio of the type that I'm speaking, speech-- 758 00:42:47,420 --> 00:42:52,810 is characterized by an enormous peak-to-average ratio. 759 00:42:52,810 --> 00:42:56,290 The strongest pressures that are generated by speech 760 00:42:56,290 --> 00:43:00,030 are enormously more powerful than the average pressure 761 00:43:00,030 --> 00:43:01,360 that's generated in speech. 762 00:43:01,360 --> 00:43:04,150 You can see that in this diagram by these peaks. 763 00:43:04,150 --> 00:43:06,430 There are several things that generate peaks. 764 00:43:06,430 --> 00:43:12,550 Peaks are generated at about 60 or 70 hertz by my vocal chords. 765 00:43:12,550 --> 00:43:16,480 But they're also generated by my lips in plosives. 766 00:43:16,480 --> 00:43:19,810 When I do plosive, there's a sudden jump 767 00:43:19,810 --> 00:43:23,770 in the instantaneous frequency that's not there on average. 768 00:43:23,770 --> 00:43:26,590 And for normal speech, that ratio 769 00:43:26,590 --> 00:43:29,291 can be as high as 35 to 1. 770 00:43:29,291 --> 00:43:31,470 35 to 1, not a big deal. 771 00:43:31,470 --> 00:43:34,740 The problem is that power goes like the square of voltage. 772 00:43:34,740 --> 00:43:36,760 So 35 to 1 becomes 1,000 to 1. 773 00:43:36,760 --> 00:43:43,390 It takes 1,000 times more energy to code the peaks 774 00:43:43,390 --> 00:43:45,580 than it does to code the average. 775 00:43:45,580 --> 00:43:48,280 And the problem with that is that in this coding scheme 776 00:43:48,280 --> 00:43:51,550 that we talked about, you have to add 777 00:43:51,550 --> 00:43:55,720 a constant that is big enough so that the signal never 778 00:43:55,720 --> 00:43:57,680 goes negative. 779 00:43:57,680 --> 00:44:01,090 So the constant that you add has to go in proportion 780 00:44:01,090 --> 00:44:04,460 to the peak value. 781 00:44:04,460 --> 00:44:10,300 So you end up transmitting almost all of your power 782 00:44:10,300 --> 00:44:12,010 By the ratio of 1,000 to 1, that's 783 00:44:12,010 --> 00:44:13,780 the amount of power that gets used 784 00:44:13,780 --> 00:44:16,720 to transmit the carrier compared to the message. 785 00:44:16,720 --> 00:44:21,280 Well, that's a terrible scheme, if what we were trying to do 786 00:44:21,280 --> 00:44:23,230 is point to point. 787 00:44:23,230 --> 00:44:26,320 Imagine your cell phone, if you had 788 00:44:26,320 --> 00:44:32,080 to transmit enough power to, in the worst case, do the peaks, 789 00:44:32,080 --> 00:44:34,180 you would on average be transmitting power 790 00:44:34,180 --> 00:44:37,840 at 1,000 times the rate that you would necessarily have to. 791 00:44:37,840 --> 00:44:40,210 So that's OK for broadcast. 792 00:44:40,210 --> 00:44:43,900 So for example, WBZ broadcast radio, 793 00:44:43,900 --> 00:44:47,520 WBZ uses a 50-kilowatt transmitter. 794 00:44:47,520 --> 00:44:49,690 50 kilowatts is the amount of power 795 00:44:49,690 --> 00:44:53,830 that would otherwise be sufficient to generate 796 00:44:53,830 --> 00:44:57,950 500 100-watt light bulbs. 797 00:44:57,950 --> 00:44:59,200 That's a fair amount of power. 798 00:44:59,200 --> 00:45:02,980 Imagine the heat that comes off 500 100-watt light bulbs. 799 00:45:02,980 --> 00:45:05,470 That's how much power is being radiated 800 00:45:05,470 --> 00:45:09,460 by the antenna for WBZ. 801 00:45:09,460 --> 00:45:13,780 That power is not necessary to transmit the average message. 802 00:45:13,780 --> 00:45:18,370 It's necessary to transmit the peak message. 803 00:45:18,370 --> 00:45:20,350 You can imagine how long your cell phone 804 00:45:20,350 --> 00:45:26,210 battery would last if you were transmitting 50 kilowatts. 805 00:45:26,210 --> 00:45:27,820 That doesn't work. 806 00:45:27,820 --> 00:45:31,660 So that's how the broadcast radio 807 00:45:31,660 --> 00:45:33,850 takes advantage of broadcast. 808 00:45:33,850 --> 00:45:37,030 It makes no sense to use this coding scheme 809 00:45:37,030 --> 00:45:39,310 for a point-to-point system. 810 00:45:39,310 --> 00:45:41,740 It's fine if what you're trying to do 811 00:45:41,740 --> 00:45:44,470 is have one transmitter, WBZ, that 812 00:45:44,470 --> 00:45:46,540 services a million listeners. 813 00:45:46,540 --> 00:45:49,560 That's fine. 814 00:45:49,560 --> 00:45:51,490 The problem with this scheme for decoding 815 00:45:51,490 --> 00:45:55,000 is it still doesn't separate different channels. 816 00:45:55,000 --> 00:45:59,290 And the way to fix that was developed by Edwin Armstrong. 817 00:45:59,290 --> 00:46:01,420 So Sarnoff who was kind of the visionary. 818 00:46:01,420 --> 00:46:04,150 He had the idea for broadcast radio. 819 00:46:04,150 --> 00:46:08,980 He's the entrepreneurial type, who thought of how to do this. 820 00:46:08,980 --> 00:46:11,090 Armstrong was the technical genius. 821 00:46:11,090 --> 00:46:13,600 He knew how to do it. 822 00:46:13,600 --> 00:46:17,290 So Armstrong's idea here, which we call superheterodyne, 823 00:46:17,290 --> 00:46:21,230 was let's make the signal look like it always 824 00:46:21,230 --> 00:46:25,830 comes from omega i regardless of what channel it comes from. 825 00:46:25,830 --> 00:46:28,210 So omega i, the intermediate frequency, 826 00:46:28,210 --> 00:46:32,260 will always take whatever frequency you're interested in 827 00:46:32,260 --> 00:46:36,250 and turn it into omega i and will do that 828 00:46:36,250 --> 00:46:38,805 by just modulating. 829 00:46:38,805 --> 00:46:41,250 And the cleverness had to do with a lot 830 00:46:41,250 --> 00:46:42,720 of technical details. 831 00:46:42,720 --> 00:46:46,440 He worked out a scheme where this modulation was very easy. 832 00:46:46,440 --> 00:46:50,460 The cutoff-- the sidebands on the bandpass filters 833 00:46:50,460 --> 00:46:52,680 didn't have to be very steep, which 834 00:46:52,680 --> 00:46:55,680 made them easy to implement. 835 00:46:55,680 --> 00:46:59,370 The sharp band pass filters were all at that one 836 00:46:59,370 --> 00:47:01,170 intermediate frequency. 837 00:47:01,170 --> 00:47:05,510 So he had to generate one very sharp band pass filter, 838 00:47:05,510 --> 00:47:10,680 but that same sharp bandpass filter could then be used 839 00:47:10,680 --> 00:47:13,540 for all the different channels. 840 00:47:13,540 --> 00:47:18,300 So the idea then was use a coarse tunable filter 841 00:47:18,300 --> 00:47:22,320 to map the frequency of interest to omega i, 842 00:47:22,320 --> 00:47:25,500 put that through a very sharp filter, of which there 843 00:47:25,500 --> 00:47:29,280 is exactly one in each receiver, and then use 844 00:47:29,280 --> 00:47:43,530 this decoding scheme to demodulate the carrier, wi. 845 00:47:43,530 --> 00:47:45,210 That's how they did it. 846 00:47:45,210 --> 00:47:48,600 We would never do it that way. 847 00:47:48,600 --> 00:47:50,520 That's part of the theme of the course. 848 00:47:50,520 --> 00:47:52,380 We are interested in schemes that 849 00:47:52,380 --> 00:47:55,900 let us map continuous time to discrete time, 850 00:47:55,900 --> 00:47:57,000 that sort of thing. 851 00:47:57,000 --> 00:48:02,400 So one way we might do it is implement a radio digitally. 852 00:48:02,400 --> 00:48:06,030 So the idea would be, what if you took the antenna, 853 00:48:06,030 --> 00:48:11,190 put it through a sampler, turned the radio signal, 854 00:48:11,190 --> 00:48:13,590 which contains a gazillion number of bands-- 855 00:48:13,590 --> 00:48:17,610 for commercial radio, there's 100 channels in the frequency 856 00:48:17,610 --> 00:48:23,250 band 500 to 600 kilohertz-- 857 00:48:23,250 --> 00:48:26,340 but just take the whole signal off of the antenna, 858 00:48:26,340 --> 00:48:29,680 turn it into a bunch of bits, run that 859 00:48:29,680 --> 00:48:32,890 through some digital logic that does, 860 00:48:32,890 --> 00:48:36,300 by magic, picks out the one that you're interested in, 861 00:48:36,300 --> 00:48:39,430 generates a new stream of bits, yd, 862 00:48:39,430 --> 00:48:41,530 from which you can do bandlimited reconstruction. 863 00:48:41,530 --> 00:48:44,187 So this is the last two lectures-- 864 00:48:44,187 --> 00:48:45,520 do you do this, how you do this. 865 00:48:45,520 --> 00:48:48,680 Now, all we do is we put a particular algorithm in there. 866 00:48:48,680 --> 00:48:50,140 And we've got a radio. 867 00:48:50,140 --> 00:48:50,860 That's the idea. 868 00:48:55,050 --> 00:48:57,250 So the key to being able to do that 869 00:48:57,250 --> 00:49:01,300 is whether or not you can build that sampler. 870 00:49:01,300 --> 00:49:06,940 So what would be the required sampling time in order 871 00:49:06,940 --> 00:49:10,000 to make a digital radio? 872 00:49:10,000 --> 00:49:13,300 And since I'm running out of time, 873 00:49:13,300 --> 00:49:17,130 I'll just tell you that the important thing it's sampling-- 874 00:49:17,130 --> 00:49:22,900 it's what we did last time-- the answer is you need T, 875 00:49:22,900 --> 00:49:27,100 so that the sampling frequency is at least twice 876 00:49:27,100 --> 00:49:29,860 the maximum frequency of the thing 877 00:49:29,860 --> 00:49:33,160 that you're trying to code. 878 00:49:33,160 --> 00:49:39,800 So the biggest frequency here is 1,600 kilohertz. 879 00:49:39,800 --> 00:49:44,830 You need to sample that with omega sampling more than twice 880 00:49:44,830 --> 00:49:46,720 that frequency-- 881 00:49:46,720 --> 00:49:50,920 so bigger than 2 pi 1,600 kilohertz. 882 00:49:50,920 --> 00:49:54,490 And if you work that out, that leaves sampling time 883 00:49:54,490 --> 00:49:56,350 of about 1/3 of a microsecond. 884 00:49:56,350 --> 00:50:00,040 And the point is that that's easy to do these days. 885 00:50:00,040 --> 00:50:04,860 That's the kind of part that you get from DigiKey for $2. 886 00:50:04,860 --> 00:50:07,390 So that's easy. 887 00:50:07,390 --> 00:50:09,190 So the only thing that you need to do 888 00:50:09,190 --> 00:50:12,940 is worry about, well, then how much computation is there? 889 00:50:12,940 --> 00:50:15,769 And that also turns out to be easy. 890 00:50:15,769 --> 00:50:18,310 The principal thing you need to do is make a bandpass filter. 891 00:50:21,130 --> 00:50:25,210 The question is, how would you make a bandpass filter? 892 00:50:25,210 --> 00:50:28,330 And here are three possible systems 893 00:50:28,330 --> 00:50:30,850 for making a bandpass filter. 894 00:50:30,850 --> 00:50:37,450 Should I take my digitized antenna signal modulate 895 00:50:37,450 --> 00:50:43,320 low pass modulate, or modulate low pass modulate, modulate 896 00:50:43,320 --> 00:50:47,270 low pass modulate, cosine, sine, cosine, sine, 897 00:50:47,270 --> 00:50:52,700 or put it through a filter that looks just like that unit 898 00:50:52,700 --> 00:50:54,650 sample response, except that it's 899 00:50:54,650 --> 00:50:56,630 multiplied by cos omega c t n? 900 00:50:59,430 --> 00:51:03,050 Some number of those work. 901 00:51:03,050 --> 00:51:08,324 And I'll leave it for you to figure out which of those work. 902 00:51:08,324 --> 00:51:08,990 Good to see you. 903 00:51:08,990 --> 00:51:11,140 Have a good day.