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 00:00:21,029 --> 00:00:23,570 DENNIS FREEMAN: With that, what I want to do is think about-- 9 00:00:23,570 --> 00:00:25,370 finish up thinking about modulation. 10 00:00:25,370 --> 00:00:27,620 Last time we thought about modulation 11 00:00:27,620 --> 00:00:30,410 in a communications context. 12 00:00:30,410 --> 00:00:31,840 That's a very important context. 13 00:00:31,840 --> 00:00:37,310 It's a way of thinking about how we can use modulation to better 14 00:00:37,310 --> 00:00:41,000 match a signal to the medium. 15 00:00:41,000 --> 00:00:43,370 So we saw in particular that if we 16 00:00:43,370 --> 00:00:46,880 were trying to transmit human voice 17 00:00:46,880 --> 00:00:50,540 via electromagnetic waves, trying to simply 18 00:00:50,540 --> 00:00:54,770 launch an electrical representation of the voice 19 00:00:54,770 --> 00:00:57,957 into electromagnetic waves, just doesn't work very well. 20 00:00:57,957 --> 00:01:00,290 And that's because of the enormous frequency difference. 21 00:01:00,290 --> 00:01:02,164 We would rather have frequencies on the order 22 00:01:02,164 --> 00:01:07,130 of 2 gigahertz for efficient transmission 23 00:01:07,130 --> 00:01:12,500 through the atmosphere and human voices just are not 24 00:01:12,500 --> 00:01:14,840 centered on 2 gigahertz. 25 00:01:14,840 --> 00:01:17,930 We can use modulation to bridge that gap. 26 00:01:17,930 --> 00:01:21,770 Having done that, we get a number of other advantages too. 27 00:01:21,770 --> 00:01:24,770 So for example, we talked last time 28 00:01:24,770 --> 00:01:26,790 about the idea of broadcast radio, which 29 00:01:26,790 --> 00:01:28,970 was an enormous revolution. 30 00:01:28,970 --> 00:01:31,730 The idea of being able to instantly communicate 31 00:01:31,730 --> 00:01:36,020 lots of stuff like a newspaper, but instantaneously, 32 00:01:36,020 --> 00:01:40,040 not with a day or a week delay, was an enormous deal. 33 00:01:42,680 --> 00:01:44,620 Today what I want to think about-- 34 00:01:44,620 --> 00:01:46,730 I mean the way we motivated last time-- the way 35 00:01:46,730 --> 00:01:48,729 to do the modulation that we motivated last time 36 00:01:48,729 --> 00:01:51,530 was the idea of amplitude modulation. 37 00:01:51,530 --> 00:01:54,980 Amplitude modulation is a terribly non-linear process. 38 00:01:54,980 --> 00:01:56,750 We multiply two signals. 39 00:01:56,750 --> 00:02:00,620 That's pretty non-linear. 40 00:02:00,620 --> 00:02:03,350 And you could imagine that if we're 41 00:02:03,350 --> 00:02:06,460 willing to open up that can of worms 42 00:02:06,460 --> 00:02:09,650 so that we're willing to do a transformation that 43 00:02:09,650 --> 00:02:13,340 is non-linear, there's enormous numbers of them. 44 00:02:13,340 --> 00:02:16,010 So you don't have to just do amplitude modulation. 45 00:02:16,010 --> 00:02:17,420 You could also do-- 46 00:02:17,420 --> 00:02:18,890 here are three classics-- you can 47 00:02:18,890 --> 00:02:22,080 do phase modulation and frequency modulation as well. 48 00:02:22,080 --> 00:02:24,080 All of those systems work. 49 00:02:24,080 --> 00:02:25,970 Phase modulation, the idea is-- 50 00:02:25,970 --> 00:02:27,640 Well, first off, amplitude modulation. 51 00:02:27,640 --> 00:02:31,590 The idea was you modulate so-- this is the carrier. 52 00:02:31,590 --> 00:02:33,140 cos omega ct is the carrier. 53 00:02:33,140 --> 00:02:36,470 That's the thing that goes through the medium well. 54 00:02:36,470 --> 00:02:39,860 Now we want to somehow embed the message on the carrier. 55 00:02:39,860 --> 00:02:43,030 In AM, we did that by multiplying. 56 00:02:43,030 --> 00:02:46,600 We take the signal x of t and we multiply it. 57 00:02:46,600 --> 00:02:48,410 We take the signal of interest, x of t, 58 00:02:48,410 --> 00:02:52,100 and we multiply it by the carrier and transmit that. 59 00:02:52,100 --> 00:02:55,610 And we saw that, for reasons of decoding simplicity, 60 00:02:55,610 --> 00:02:59,060 it was convenient to add a constant term, which 61 00:02:59,060 --> 00:03:02,510 was essentially sending, not only the modulated carrier, 62 00:03:02,510 --> 00:03:06,380 but also the carrier alone, so that you could locally 63 00:03:06,380 --> 00:03:10,970 retrieve both the modulated message and the carrier. 64 00:03:10,970 --> 00:03:13,480 It simplified the modulation. 65 00:03:13,480 --> 00:03:15,500 But a different kind of alternative 66 00:03:15,500 --> 00:03:18,190 is, convey the message in the phase. 67 00:03:18,190 --> 00:03:18,690 Right? 68 00:03:18,690 --> 00:03:20,870 We all love phase, right? 69 00:03:20,870 --> 00:03:21,860 Nod your head yes. 70 00:03:21,860 --> 00:03:23,960 We all love phase. 71 00:03:23,960 --> 00:03:27,710 So what we could do instead is take the carrier 72 00:03:27,710 --> 00:03:32,400 and add a phase term in proportion to the message. 73 00:03:32,400 --> 00:03:38,370 That's a different way of coding the message on a carrier. 74 00:03:38,370 --> 00:03:42,130 Another way to do it is to code the frequency. 75 00:03:42,130 --> 00:03:46,290 So take the message, integrate it-- 76 00:03:46,290 --> 00:03:49,170 it is kind of a funny way to talk about FM. 77 00:03:49,170 --> 00:03:52,080 So you could use the message to modulate 78 00:03:52,080 --> 00:03:53,940 the frequency of the carrier, which 79 00:03:53,940 --> 00:03:55,770 would be very similar to using the message 80 00:03:55,770 --> 00:03:57,210 to modulate the phase. 81 00:03:57,210 --> 00:04:01,570 They, in fact, differ by an integral. 82 00:04:01,570 --> 00:04:03,970 And I'll talk about that more in a minute. 83 00:04:03,970 --> 00:04:08,730 So the idea then is even though the first thing they tried-- 84 00:04:08,730 --> 00:04:11,490 the first successful way of making broadcast radio 85 00:04:11,490 --> 00:04:14,010 --was via AM, we should think about the alternatives 86 00:04:14,010 --> 00:04:15,990 and try to figure out what are the advantages 87 00:04:15,990 --> 00:04:20,730 and disadvantages of different kinds of coding schemes. 88 00:04:20,730 --> 00:04:24,229 And also consistent with the goals of this course, 89 00:04:24,229 --> 00:04:25,770 in thinking about these alternatives, 90 00:04:25,770 --> 00:04:27,269 we're going to get a lot of practice 91 00:04:27,269 --> 00:04:29,220 in thinking about Fourier transforms. 92 00:04:29,220 --> 00:04:30,580 Which is good. 93 00:04:30,580 --> 00:04:34,770 So let's think about what would be the difference between AM 94 00:04:34,770 --> 00:04:38,280 modulation, where we convey the message by modulating 95 00:04:38,280 --> 00:04:41,190 the amplitude of the carrier versus FM, 96 00:04:41,190 --> 00:04:45,510 where we convey the message by modulating 97 00:04:45,510 --> 00:04:49,365 the instantaneous frequency of the carrier. 98 00:04:49,365 --> 00:04:51,490 And if we're just looking at time domain, which is, 99 00:04:51,490 --> 00:04:53,864 after all, probably the first place anybody should look-- 100 00:04:53,864 --> 00:04:57,230 if we just compare those two schemes in the time domain, 101 00:04:57,230 --> 00:05:00,100 there's some very big differences. 102 00:05:00,100 --> 00:05:04,060 You can see that when we amplitude modulate, 103 00:05:04,060 --> 00:05:08,850 the power that is used to transmit the message 104 00:05:08,850 --> 00:05:11,730 depends on the message. 105 00:05:11,730 --> 00:05:15,060 So there are places where there is not much power. 106 00:05:15,060 --> 00:05:17,820 There's not much energy in this signal. 107 00:05:17,820 --> 00:05:20,100 And there are places where there's lots. 108 00:05:20,100 --> 00:05:22,350 By contrast, down here you see something 109 00:05:22,350 --> 00:05:24,090 that has constant power. 110 00:05:24,090 --> 00:05:25,920 At least if you integrate over-- 111 00:05:25,920 --> 00:05:31,010 say you've got a 2 gigahertz carrier, 112 00:05:31,010 --> 00:05:34,670 then as long as you integrate over 10 or 20 cycles, which 113 00:05:34,670 --> 00:05:37,280 is a small fraction of a second, you're 114 00:05:37,280 --> 00:05:38,990 going to get the same answer, regardless 115 00:05:38,990 --> 00:05:42,860 of where in the message you do the integral. 116 00:05:42,860 --> 00:05:45,110 So there is obviously a power difference. 117 00:05:45,110 --> 00:05:49,460 There's no need to transmit the carrier in order 118 00:05:49,460 --> 00:05:52,040 to decode the FM signal, unless of course you're 119 00:05:52,040 --> 00:05:55,010 interested in understanding exactly where DC is. 120 00:05:55,010 --> 00:05:56,570 The interesting thing about audio 121 00:05:56,570 --> 00:05:59,900 is that we're very insensitive to DC. 122 00:05:59,900 --> 00:06:03,140 So that's not really very important for conveying 123 00:06:03,140 --> 00:06:05,720 speech sounds. 124 00:06:05,720 --> 00:06:08,510 But the original motivation for thinking 125 00:06:08,510 --> 00:06:13,830 about alternative ways of coding was in fact bandwidth. 126 00:06:13,830 --> 00:06:19,190 So the idea was there's a limited resource. 127 00:06:19,190 --> 00:06:21,440 Bandwidth is a limited resource. 128 00:06:21,440 --> 00:06:26,850 If person A is using 1 megahertz plus or minus 5 kilohertz, 129 00:06:26,850 --> 00:06:30,680 which is one of the broadcast AM frequency bands, 130 00:06:30,680 --> 00:06:32,435 then person B can't use that band. 131 00:06:35,120 --> 00:06:37,040 You can have only one transmitter 132 00:06:37,040 --> 00:06:39,620 for each of the radio bands. 133 00:06:39,620 --> 00:06:43,190 That makes bandwidth-- radio bandwidth --a resource. 134 00:06:43,190 --> 00:06:46,160 Now there's lots of it. 135 00:06:46,160 --> 00:06:49,880 There's lots of frequencies between a megahertz and 2 136 00:06:49,880 --> 00:06:51,650 gigahertz. 137 00:06:51,650 --> 00:06:55,640 On the other hand, there's lots of people who want it. 138 00:06:55,640 --> 00:06:57,287 So your local fire department thinks 139 00:06:57,287 --> 00:06:59,120 they ought to be able to talk to each other. 140 00:06:59,120 --> 00:07:00,579 Your local police department thinks 141 00:07:00,579 --> 00:07:02,411 they ought to be able to talk to each other. 142 00:07:02,411 --> 00:07:03,887 Your local ambulance service thinks 143 00:07:03,887 --> 00:07:05,720 they ought to be able to talk to each other. 144 00:07:05,720 --> 00:07:08,060 So there's lots of people with demands for it. 145 00:07:08,060 --> 00:07:10,100 So the idea of-- 146 00:07:10,100 --> 00:07:13,640 the original pursuit of FM was to try 147 00:07:13,640 --> 00:07:17,390 to think about a scheme that would use less bandwidth. 148 00:07:17,390 --> 00:07:20,990 If you're thinking about speech, you need about 3 kilohertz. 149 00:07:20,990 --> 00:07:25,220 3 kilohertz is considered telephone quality speech. 150 00:07:25,220 --> 00:07:27,530 It's not perfect, but it's good enough 151 00:07:27,530 --> 00:07:31,850 to get very good speech intelligibility across it. 152 00:07:31,850 --> 00:07:35,120 So I AM used plus or minus 5 kilohertz bandwidth. 153 00:07:35,120 --> 00:07:40,190 They allocated a band of 10 kilohertz for every station. 154 00:07:40,190 --> 00:07:46,530 The initial idea in FM to-- so take this expression for FM 155 00:07:46,530 --> 00:07:50,240 and think about an instantaneous frequency omega 156 00:07:50,240 --> 00:07:55,770 i, which is a frequency with a slightly different value 157 00:07:55,770 --> 00:07:57,210 from the carrier frequency. 158 00:07:57,210 --> 00:08:00,180 It's different because we're doing FM. 159 00:08:00,180 --> 00:08:03,570 And you can calculate the instantaneous frequency 160 00:08:03,570 --> 00:08:07,550 as the derivative of phase. 161 00:08:07,550 --> 00:08:12,180 So the derivative of phase gives you an omega c term out front. 162 00:08:12,180 --> 00:08:14,577 Omega ct is the carrier phase. 163 00:08:14,577 --> 00:08:16,160 And then there's the part of the phase 164 00:08:16,160 --> 00:08:17,960 that comes from the message. 165 00:08:17,960 --> 00:08:19,540 So the total instantaneous frequency 166 00:08:19,540 --> 00:08:23,120 is omega c plus the time derivative of phase. 167 00:08:23,120 --> 00:08:25,570 And since we're modulating frequency-- 168 00:08:25,570 --> 00:08:28,520 since we're modulating with the frequency, that turns out 169 00:08:28,520 --> 00:08:30,920 to be proportional to x. 170 00:08:30,920 --> 00:08:32,990 So the instantaneous frequency, like you 171 00:08:32,990 --> 00:08:35,659 would like for frequency modulation-- the instantaneous 172 00:08:35,659 --> 00:08:40,580 frequency --is a linear function of the message x. 173 00:08:40,580 --> 00:08:43,490 But it's proportional. 174 00:08:43,490 --> 00:08:49,130 So the reasoning was make k small. 175 00:08:49,130 --> 00:08:52,610 If the instantaneous deviations of frequency are small, 176 00:08:52,610 --> 00:08:56,830 say 10 to the minus 6th hertz-- 177 00:08:56,830 --> 00:08:58,150 really small. 178 00:08:58,150 --> 00:09:01,660 Well, if you could have 10 to the minus 6th hertz, 179 00:09:01,660 --> 00:09:08,670 you could pack 10 to the 6th stations in a hertz. 180 00:09:08,670 --> 00:09:10,050 Well that's pretty good. 181 00:09:10,050 --> 00:09:12,810 So instead of using 10 kilohertz to send one message, 182 00:09:12,810 --> 00:09:14,910 you could get 10 to the 6th messages 183 00:09:14,910 --> 00:09:16,690 in one hertz of bandwidth. 184 00:09:16,690 --> 00:09:18,180 That was the original motivation. 185 00:09:18,180 --> 00:09:24,300 That was an idea that was propagated at Bell Labs, who 186 00:09:24,300 --> 00:09:27,120 were one of the commercial entities who were seriously 187 00:09:27,120 --> 00:09:30,570 interested in trying to make money on broadcast radio. 188 00:09:30,570 --> 00:09:36,450 So the idea was maybe we could use FM to squeeze more signals 189 00:09:36,450 --> 00:09:40,430 in the available bandwidth. 190 00:09:40,430 --> 00:09:45,210 Well that turns out to be completely wrong. 191 00:09:45,210 --> 00:09:47,340 And that's why studying Fourier transforms 192 00:09:47,340 --> 00:09:50,370 is such a good example of how you can use Fourier transforms 193 00:09:50,370 --> 00:09:52,350 to figure things out. 194 00:09:52,350 --> 00:09:55,050 That argument is just completely wrong, 195 00:09:55,050 --> 00:09:58,260 as two lines of double-0 3 will show you. 196 00:09:58,260 --> 00:10:02,250 So here is our expression for the FM signal. 197 00:10:04,900 --> 00:10:07,120 You can see that it's complicated because it's 198 00:10:07,120 --> 00:10:09,240 the cosine of a sum. 199 00:10:09,240 --> 00:10:11,220 But we all know from trigonometry, 200 00:10:11,220 --> 00:10:17,630 the cos of a plus b is cos a cos b minus sine a sine b. 201 00:10:17,630 --> 00:10:23,300 So we can get an exact expression for this part-- 202 00:10:23,300 --> 00:10:25,660 we can expand this exactly as the cosine 203 00:10:25,660 --> 00:10:33,830 a times the cosine of b minus the sine of a times the sign 204 00:10:33,830 --> 00:10:35,912 of b. 205 00:10:35,912 --> 00:10:36,620 Well that's easy. 206 00:10:36,620 --> 00:10:38,750 Now what would happen to that expression 207 00:10:38,750 --> 00:10:41,490 if we made k very small? 208 00:10:41,490 --> 00:10:44,090 Well if we make k very small-- 209 00:10:44,090 --> 00:10:47,150 if k goes close to 0 --the cosine 210 00:10:47,150 --> 00:10:50,430 gets arbitrarily close to 1. 211 00:10:50,430 --> 00:10:54,060 Fine, that sounds OK. 212 00:10:54,060 --> 00:10:59,210 This sine of l times something does not go-- well the fallacy 213 00:10:59,210 --> 00:11:02,370 in the reasoning was that goes to 0. 214 00:11:02,370 --> 00:11:06,120 It does go to 0, but it actually goes to 0 slowly. 215 00:11:06,120 --> 00:11:09,400 It actually-- the limit approaches k times-- 216 00:11:09,400 --> 00:11:14,340 so the limit of the sine of theta, as theta gets small, 217 00:11:14,340 --> 00:11:14,940 is theta. 218 00:11:17,890 --> 00:11:21,300 So the idea wasn't quite right. 219 00:11:21,300 --> 00:11:23,610 It doesn't go arbitrarily close to 0. 220 00:11:23,610 --> 00:11:28,780 It gets arbitrarily close to the message. 221 00:11:28,780 --> 00:11:31,480 So that means that this expression, which 222 00:11:31,480 --> 00:11:38,380 looks horrible up here, is equivalent to this expression, 223 00:11:38,380 --> 00:11:41,280 which says that the signal that I'm transmitting 224 00:11:41,280 --> 00:11:43,320 is the carrier. 225 00:11:43,320 --> 00:11:54,310 Just like AM minus sine omega ct times the message. 226 00:11:54,310 --> 00:11:56,076 But that's just AM. 227 00:11:56,076 --> 00:11:57,610 I took the message times sine. 228 00:11:57,610 --> 00:11:58,290 Uh, it's sine. 229 00:11:58,290 --> 00:12:00,329 It's not cosine, who cares? 230 00:12:00,329 --> 00:12:02,120 That's a difference of 90 degrees of phase. 231 00:12:02,120 --> 00:12:03,500 Who cares? 232 00:12:03,500 --> 00:12:05,610 It's AM. 233 00:12:05,610 --> 00:12:09,730 The fallacy in the reasoning is that this k 234 00:12:09,730 --> 00:12:12,750 does not-- the sine of theta, when theta gets small, 235 00:12:12,750 --> 00:12:13,860 does not go to 0. 236 00:12:13,860 --> 00:12:15,970 It goes to theta. 237 00:12:15,970 --> 00:12:19,530 And so the limiting case for narrowband FM 238 00:12:19,530 --> 00:12:24,460 has precisely the same bandwidth as AM. 239 00:12:24,460 --> 00:12:27,060 Narrowband FM has the same bandwidth as AM, 240 00:12:27,060 --> 00:12:28,920 so therefore, this whole idea that you 241 00:12:28,920 --> 00:12:33,810 could use FM to squeeze more channels into a given bandwidth 242 00:12:33,810 --> 00:12:36,696 is just completely wrong. 243 00:12:36,696 --> 00:12:38,070 So it was the initial motivation. 244 00:12:38,070 --> 00:12:40,380 It's just wrong. 245 00:12:40,380 --> 00:12:46,432 In fact, what's good about FM is the other limit. 246 00:12:46,432 --> 00:12:47,640 Don't worry about narrowband. 247 00:12:47,640 --> 00:12:50,070 Worry about broadband. 248 00:12:50,070 --> 00:12:51,360 The value of FM-- 249 00:12:51,360 --> 00:12:53,040 and this was Armstrong again. 250 00:12:53,040 --> 00:12:56,580 The same guy who did the superheterodyne receiver, which 251 00:12:56,580 --> 00:12:59,520 made AM broadcast radio possible. 252 00:12:59,520 --> 00:13:02,850 Same guy went on to think about FM 253 00:13:02,850 --> 00:13:05,670 and he saw that the value of FM was, in fact, 254 00:13:05,670 --> 00:13:07,800 to use lots of bandwidth. 255 00:13:07,800 --> 00:13:09,084 Why would you do that? 256 00:13:09,084 --> 00:13:10,250 Well, we'll see in a minute. 257 00:13:10,250 --> 00:13:12,390 That the reason you want to use a lot of bandwidth 258 00:13:12,390 --> 00:13:17,070 is that you generate a robust signal that can be recovered, 259 00:13:17,070 --> 00:13:19,500 even when noise gets added to it. 260 00:13:19,500 --> 00:13:21,210 One of the big problems, especially 261 00:13:21,210 --> 00:13:24,411 with the early versions of AM broadcast radio, 262 00:13:24,411 --> 00:13:26,410 was that it had a lot of noise in the background 263 00:13:26,410 --> 00:13:28,620 we called static. 264 00:13:28,620 --> 00:13:31,650 Kind of like tsss, but a little bit more poppy 265 00:13:31,650 --> 00:13:34,230 and more irritating. 266 00:13:34,230 --> 00:13:37,620 And Armstrong figured out that there was a way 267 00:13:37,620 --> 00:13:42,390 to reduce that static by using more bandwidth to make 268 00:13:42,390 --> 00:13:43,750 a more robust signal. 269 00:13:43,750 --> 00:13:44,910 So that's the idea. 270 00:13:44,910 --> 00:13:48,930 And coincidentally, it gives us an excellent opportunity 271 00:13:48,930 --> 00:13:50,520 to practice our skills of figuring out 272 00:13:50,520 --> 00:13:52,110 Fourier transforms. 273 00:13:52,110 --> 00:13:56,170 So let's figure out the Fourier transform of an FM signal. 274 00:13:56,170 --> 00:13:56,670 Right? 275 00:13:56,670 --> 00:13:59,040 That'll be fun. 276 00:13:59,040 --> 00:14:01,260 Remember what the FM signal looked like? 277 00:14:01,260 --> 00:14:05,260 We saw back here, it's kind of horrendous looking. 278 00:14:05,260 --> 00:14:07,020 So the goal for the next five minutes 279 00:14:07,020 --> 00:14:08,728 is to take the Fourier transform of that. 280 00:14:12,340 --> 00:14:18,610 So let's think about phase coding. 281 00:14:18,610 --> 00:14:20,900 Phase and frequency coding are almost the same thing. 282 00:14:20,900 --> 00:14:24,640 All you do is, in one case you transmit a phase in proportion 283 00:14:24,640 --> 00:14:25,850 to x. 284 00:14:25,850 --> 00:14:27,670 That's phase modulation. 285 00:14:27,670 --> 00:14:29,170 In the case of frequency modulation, 286 00:14:29,170 --> 00:14:31,000 you transmit a phase proportionate 287 00:14:31,000 --> 00:14:35,730 to the integral of x, which is the same as modulating 288 00:14:35,730 --> 00:14:39,570 the instantaneous frequency by x. 289 00:14:39,570 --> 00:14:43,620 So let's think about phase modulating by sine omega mt. 290 00:14:43,620 --> 00:14:45,765 This is omega carrier and omega message. 291 00:14:48,550 --> 00:14:51,460 And I'm putting an m out front. 292 00:14:51,460 --> 00:14:53,260 The modulation depth. 293 00:14:53,260 --> 00:14:55,120 Just so that I can track what happens, 294 00:14:55,120 --> 00:14:59,470 as I turn up the amplitude of the signal. 295 00:14:59,470 --> 00:15:03,250 So let's think about, what does this signal look like? 296 00:15:03,250 --> 00:15:07,360 So we have cos a plus b. 297 00:15:07,360 --> 00:15:11,470 We expand that as cos a cos b minus sine a sine b. 298 00:15:11,470 --> 00:15:16,210 And let's start by looking at this first term, which 299 00:15:16,210 --> 00:15:20,980 is modulated something. 300 00:15:20,980 --> 00:15:22,550 The something is the hard part. 301 00:15:22,550 --> 00:15:25,900 So the something is the cosine of m sine omega mt. 302 00:15:25,900 --> 00:15:28,680 For the particular case, that the message is the sine omega 303 00:15:28,680 --> 00:15:30,580 mt. 304 00:15:30,580 --> 00:15:32,490 So let's think about that. 305 00:15:32,490 --> 00:15:34,460 So we'll start with the message. 306 00:15:34,460 --> 00:15:35,980 Sine omega mt. 307 00:15:35,980 --> 00:15:40,070 And we'll say that the modulation depth is 1. 308 00:15:40,070 --> 00:15:41,780 Now we have to take the cosine of that. 309 00:15:44,670 --> 00:15:48,100 So now we take the cosine of this signal. 310 00:15:48,100 --> 00:15:53,270 So this signal is going 0- 1- 0 minus 1- 0. 311 00:15:53,270 --> 00:15:58,930 So the cosine of that-- when the sine is 0, the cosine is 1. 312 00:15:58,930 --> 00:16:03,010 Then the sine is going toward 1. 313 00:16:03,010 --> 00:16:06,130 As the sine goes toward 1, the cosine starts going down. 314 00:16:06,130 --> 00:16:06,630 Right? 315 00:16:06,630 --> 00:16:10,870 The first part of the cosine waves, cosine t, as you 316 00:16:10,870 --> 00:16:12,040 increase from 0. 317 00:16:12,040 --> 00:16:13,180 It starts to decay. 318 00:16:13,180 --> 00:16:16,570 Same thing's happening here. 319 00:16:16,570 --> 00:16:22,290 But then by the time the sine gets up to 1, 320 00:16:22,290 --> 00:16:24,000 the sign starts going down. 321 00:16:24,000 --> 00:16:25,770 So the cosine starts going back up. 322 00:16:25,770 --> 00:16:27,478 Everybody see what I'm what I'm implying? 323 00:16:29,100 --> 00:16:30,960 So this waveform is a sine waveform. 324 00:16:30,960 --> 00:16:33,750 And this is the cosine of that waveform. 325 00:16:33,750 --> 00:16:35,130 That's what an FM signal is. 326 00:16:38,140 --> 00:16:41,560 As I increase the modulation depth, 327 00:16:41,560 --> 00:16:47,340 so make the modulation depth now 2, now the deviation from 1 328 00:16:47,340 --> 00:16:48,000 is bigger. 329 00:16:48,000 --> 00:16:48,510 Right? 330 00:16:48,510 --> 00:16:52,020 Before, the deviation for 1 was caused when 331 00:16:52,020 --> 00:16:53,920 the sine wave went up to 1. 332 00:16:53,920 --> 00:16:55,530 Now, the biggest deviation occurs when 333 00:16:55,530 --> 00:16:56,921 the sine wave gets up to 2. 334 00:16:56,921 --> 00:16:57,420 It's bigger. 335 00:17:00,110 --> 00:17:05,260 And then I may make it bigger and bigger. 336 00:17:05,260 --> 00:17:05,760 OK. 337 00:17:05,760 --> 00:17:07,290 By the time-- 338 00:17:07,290 --> 00:17:10,560 The big difference between 3 and 4-- 339 00:17:10,560 --> 00:17:12,794 3 and 4. 340 00:17:12,794 --> 00:17:14,460 Why is a big difference between 3 and 4? 341 00:17:17,010 --> 00:17:20,400 It's because there's a number between 3 and 4. 342 00:17:20,400 --> 00:17:22,680 Pi. 343 00:17:22,680 --> 00:17:25,609 By the time you get to pi, it rolls over. 344 00:17:28,250 --> 00:17:29,810 Everybody see that? 345 00:17:29,810 --> 00:17:37,010 I started out 1, 2, 3, just less than pi, 4, just bigger 346 00:17:37,010 --> 00:17:51,210 than pie, 5, 6, 7, 8, 9, 10, 20, 50. 347 00:17:51,210 --> 00:17:54,250 Horrendous, right? 348 00:17:54,250 --> 00:17:56,950 But I want you to find the Fourier transform of that. 349 00:17:56,950 --> 00:18:00,290 Actually, I want you to find the Fourier transform of that! 350 00:18:00,290 --> 00:18:04,630 But as a subtopic, I'll accept the Fourier transform of that. 351 00:18:04,630 --> 00:18:05,540 OK. 352 00:18:05,540 --> 00:18:06,680 Look at your neighbor. 353 00:18:06,680 --> 00:18:08,263 Tell me the Fourier transform of that. 354 00:19:59,810 --> 00:20:02,070 So what's the Fourier transform of the bottom thing? 355 00:20:05,240 --> 00:20:05,740 OK. 356 00:20:05,740 --> 00:20:06,900 Was a hard question. 357 00:20:06,900 --> 00:20:07,470 Yes! 358 00:20:07,470 --> 00:20:09,612 Yes. 359 00:20:09,612 --> 00:20:10,520 Oh nice! 360 00:20:10,520 --> 00:20:11,840 It was number 4, got it. 361 00:20:11,840 --> 00:20:18,870 [LAUGH] So what should I notice about this signal? 362 00:20:18,870 --> 00:20:21,080 Except what? 363 00:20:21,080 --> 00:20:23,570 As a sophisticated signal processor at this point, 364 00:20:23,570 --> 00:20:25,730 after all it's the penultimate lecture, 365 00:20:25,730 --> 00:20:27,550 you're already sophisticated. 366 00:20:27,550 --> 00:20:29,690 So as a sophisticated signal processor, 367 00:20:29,690 --> 00:20:32,180 what do you notice immediately about that waveform? 368 00:20:32,180 --> 00:20:33,440 It just cries out, I feel. 369 00:20:33,440 --> 00:20:35,580 It says? 370 00:20:35,580 --> 00:20:36,290 Periodic. 371 00:20:36,290 --> 00:20:38,400 Exactly. 372 00:20:38,400 --> 00:20:41,990 Even though it's horrendous, it's periodic. 373 00:20:41,990 --> 00:20:45,810 Why is that interesting? 374 00:20:45,810 --> 00:20:47,180 It has a series. 375 00:20:47,180 --> 00:20:49,260 Precisely. 376 00:20:49,260 --> 00:20:51,960 So all I need to do to figure out the transform, 377 00:20:51,960 --> 00:20:54,331 is figure out the series. 378 00:20:54,331 --> 00:20:54,830 OK. 379 00:20:54,830 --> 00:20:56,750 So let's do that. 380 00:20:56,750 --> 00:20:58,150 Now let's start. 381 00:20:58,150 --> 00:21:01,580 The same waveform on the top. m equals 1, just like before. 382 00:21:01,580 --> 00:21:05,210 Now I'm going to take the series of this periodic waveform. 383 00:21:08,650 --> 00:21:10,540 And I'm going to represent that down here. 384 00:21:10,540 --> 00:21:14,380 As you see, as I start to modulate this waveform, 385 00:21:14,380 --> 00:21:19,370 I'm getting two bumps, where I had one bump of this-- 386 00:21:19,370 --> 00:21:22,330 where I had one period of this. 387 00:21:22,330 --> 00:21:23,960 I'm getting two periods here. 388 00:21:27,690 --> 00:21:31,030 So that's the reason my first non-0 contribution 389 00:21:31,030 --> 00:21:32,280 is k equals 2. 390 00:21:35,190 --> 00:21:39,210 Also notice that if I turn to m the whole way down to 0, 391 00:21:39,210 --> 00:21:42,480 I would just get dc k equals 1. 392 00:21:42,480 --> 00:21:52,280 So for the small m's, I'm mostly getting dc and k equals 2. 393 00:21:52,280 --> 00:21:53,130 OK. 394 00:21:53,130 --> 00:21:56,580 Now as I turn up the amplitude, now I'm 395 00:21:56,580 --> 00:21:58,620 getting something that doesn't look quite as 396 00:21:58,620 --> 00:22:01,210 much like a sine wave. 397 00:22:01,210 --> 00:22:07,240 And that distortion is manifest as some k whose 4. 398 00:22:07,240 --> 00:22:08,500 Then I turn it up higher. 399 00:22:08,500 --> 00:22:12,460 Whoops, I hit the button a few times more than I intended to. 400 00:22:12,460 --> 00:22:15,640 If I go up to m equals 5, now it's wrapped around like this. 401 00:22:15,640 --> 00:22:21,580 And I can see I've got k equals 0, 2, 4, 6, and a little 8. 402 00:22:21,580 --> 00:22:28,430 Keep going up, up, up, up. 403 00:22:28,430 --> 00:22:33,980 And what you can see is that as I turn up the amplitude, 404 00:22:33,980 --> 00:22:37,780 I'm getting bigger and bigger k's. 405 00:22:37,780 --> 00:22:40,090 In fact, there's kind of a simple relationship. 406 00:22:40,090 --> 00:22:44,440 By the time I got up to m equals 50, I got about 50 k's. 407 00:22:44,440 --> 00:22:46,840 Roughly speaking. 408 00:22:46,840 --> 00:22:49,570 And that has something to do with the periodicity 409 00:22:49,570 --> 00:22:52,130 of the cosine being 6. 410 00:22:52,130 --> 00:22:52,630 2 pi. 411 00:22:55,490 --> 00:22:57,950 So the number of terms I'm getting 412 00:22:57,950 --> 00:23:02,840 is related to how big that signal is. 413 00:23:02,840 --> 00:23:05,270 I was able to represent this horrendous signal now 414 00:23:05,270 --> 00:23:07,520 by a series, but I asked you for-- what 415 00:23:07,520 --> 00:23:10,370 I'm really trying to do is find the transform of this. 416 00:23:10,370 --> 00:23:14,140 We just found the series of this, 417 00:23:14,140 --> 00:23:16,270 but I really want to find the transform of that. 418 00:23:16,270 --> 00:23:18,200 So I'd like to turn this into a transform. 419 00:23:18,200 --> 00:23:20,114 How do I turn a series in a transform? 420 00:23:26,750 --> 00:23:29,750 Turn each k into an impulse. 421 00:23:29,750 --> 00:23:32,030 So now I get a train of impulses. 422 00:23:32,030 --> 00:23:34,580 Each one of them has weights proportional 423 00:23:34,580 --> 00:23:37,010 to the length of the lines. 424 00:23:37,010 --> 00:23:40,010 Then each impulse gets located in frequency, instead of 425 00:23:40,010 --> 00:23:44,240 in k space, each impulse gets located in frequency 426 00:23:44,240 --> 00:23:46,760 at a multiple of 2 pi over omega m. 427 00:23:50,580 --> 00:23:53,420 So when I do that, I get the transform 428 00:23:53,420 --> 00:23:59,550 of this mass, which looks like this centered here, 429 00:23:59,550 --> 00:24:03,090 but then I modulate by the cosine. 430 00:24:03,090 --> 00:24:06,340 So modulation gives me two copies. 431 00:24:06,340 --> 00:24:09,010 I get the whole spectrum for the inside here 432 00:24:09,010 --> 00:24:10,240 and a duplicate over there. 433 00:24:10,240 --> 00:24:11,550 This time centered on omega c. 434 00:24:16,950 --> 00:24:18,620 That's all perfectly clear. 435 00:24:18,620 --> 00:24:20,600 Right? 436 00:24:20,600 --> 00:24:23,260 So then I just did this term. 437 00:24:23,260 --> 00:24:26,900 Now I have to worry about that term, but it's the same thing. 438 00:24:26,900 --> 00:24:29,450 Except now my periodicity is a little different. 439 00:24:29,450 --> 00:24:31,790 Now I'm taking the sine of the sine 440 00:24:31,790 --> 00:24:33,950 rather than the cos of the sine. 441 00:24:33,950 --> 00:24:38,210 So now as I crank up the waveforms, 442 00:24:38,210 --> 00:24:40,160 the picture looks slightly different. 443 00:24:40,160 --> 00:24:44,270 The principle harmonics are now odd 444 00:24:44,270 --> 00:24:47,020 because of the odd symmetry of the sine wave. 445 00:24:47,020 --> 00:24:48,700 I had evens when I used cosine. 446 00:24:48,700 --> 00:24:51,280 I have odds when I use sine. 447 00:24:51,280 --> 00:24:56,340 So now I'm filling in a bunch of odd harmonics, 448 00:24:56,340 --> 00:24:59,850 but the general pattern looks very similar. 449 00:24:59,850 --> 00:25:03,480 So that now the result looks very similar to the result 450 00:25:03,480 --> 00:25:05,070 from before. 451 00:25:05,070 --> 00:25:08,380 And the sum is the sum. 452 00:25:08,380 --> 00:25:15,150 Here's the Fourier transform of this waveform y of t. 453 00:25:15,150 --> 00:25:19,460 The point is it's huge bandwidth. 454 00:25:19,460 --> 00:25:22,940 The advantage of FM is not in conserving bandwidth, 455 00:25:22,940 --> 00:25:25,850 it's actually using bandwidth. 456 00:25:25,850 --> 00:25:29,480 So we're using-- this is the K, big case. 457 00:25:29,480 --> 00:25:32,490 So we're doing wideband FM. 458 00:25:32,490 --> 00:25:35,070 We're shoving frequency components 459 00:25:35,070 --> 00:25:39,630 all over the spectrum, but the advantage of that 460 00:25:39,630 --> 00:25:42,450 is that I get a signal that's very robust. 461 00:25:42,450 --> 00:25:46,490 Imagine if I were to add a small level of noise to this signal. 462 00:25:49,320 --> 00:25:52,032 You could still recover the signal. 463 00:25:52,032 --> 00:25:53,490 The only thing that's coded here is 464 00:25:53,490 --> 00:25:56,220 one sine wave that has one period 465 00:25:56,220 --> 00:25:57,480 across that whole length. 466 00:25:57,480 --> 00:26:00,630 You could recover that in the absence of, not only 467 00:26:00,630 --> 00:26:03,450 a little noise, but an enormous amount of noise. 468 00:26:03,450 --> 00:26:05,117 And that was what FM was good for. 469 00:26:05,117 --> 00:26:06,700 And that's what Armstrong figured out. 470 00:26:06,700 --> 00:26:07,783 And that's why we have FM. 471 00:26:07,783 --> 00:26:11,880 And that's why television, even HDTV, 472 00:26:11,880 --> 00:26:14,400 uses FM coding of the audio. 473 00:26:14,400 --> 00:26:15,840 Because it's resilient to noise. 474 00:26:18,550 --> 00:26:21,940 So that was kind of a motivation for thinking about modulation, 475 00:26:21,940 --> 00:26:23,980 in terms of communications. 476 00:26:23,980 --> 00:26:27,370 I don't want you to go away thinking about modulation 477 00:26:27,370 --> 00:26:31,540 as only useful for communications, 478 00:26:31,540 --> 00:26:34,420 so I want to close with a kind of unconventional 479 00:26:34,420 --> 00:26:37,300 use of modulation. 480 00:26:37,300 --> 00:26:39,190 I may have mentioned that I like microscopy. 481 00:26:41,740 --> 00:26:45,580 I'm going to show you how you can use modulation 482 00:26:45,580 --> 00:26:48,500 to improve microscopy. 483 00:26:48,500 --> 00:26:50,530 So this was an idea of Michael Mermelstein. 484 00:26:50,530 --> 00:26:54,250 Michael was a PhD student in my lab. 485 00:26:54,250 --> 00:26:55,930 He thought of it. 486 00:26:55,930 --> 00:26:58,450 Stan and Jay developed it. 487 00:26:58,450 --> 00:27:01,480 They all three got PhD's on this topic. 488 00:27:01,480 --> 00:27:05,980 Berthold is a professor in CS and five of us 489 00:27:05,980 --> 00:27:08,440 worked on this project. 490 00:27:08,440 --> 00:27:13,090 So the idea is improving microscopy with 6.003. 491 00:27:13,090 --> 00:27:16,840 With modulation in particular. 492 00:27:16,840 --> 00:27:17,830 You've all seen this. 493 00:27:17,830 --> 00:27:22,950 This is the double-0 3 model of a microscope. 494 00:27:22,950 --> 00:27:26,750 Double-0 3 microscope convolved with blurring function. 495 00:27:26,750 --> 00:27:28,070 Done. 496 00:27:28,070 --> 00:27:29,870 That's what a microscope is. 497 00:27:29,870 --> 00:27:32,570 That's the double-0 3 model of a microscope. 498 00:27:32,570 --> 00:27:34,400 Microscope is a low pass filter. 499 00:27:34,400 --> 00:27:37,677 Because all of the different spatial frequencies 500 00:27:37,677 --> 00:27:39,260 that are available in the target can't 501 00:27:39,260 --> 00:27:41,551 get through all the lenses for very fundamental reasons 502 00:27:41,551 --> 00:27:44,660 in physics, the high frequencies don't make it. 503 00:27:44,660 --> 00:27:46,100 The low frequencies do. 504 00:27:46,100 --> 00:27:48,830 The result is a blurred image. 505 00:27:48,830 --> 00:27:50,750 So I'm representing this as the target. 506 00:27:50,750 --> 00:27:52,700 It passes through the microscope, which 507 00:27:52,700 --> 00:27:54,770 is represented by a low pass filter 508 00:27:54,770 --> 00:27:56,510 and it comes out a blurry picture. 509 00:27:59,560 --> 00:28:05,770 Michael's idea was instead of illuminating the target 510 00:28:05,770 --> 00:28:07,360 with uniform light-- 511 00:28:07,360 --> 00:28:10,240 if you tear apart a microscope, a lot of the guts 512 00:28:10,240 --> 00:28:12,970 are intended to make a nice uniform light that 513 00:28:12,970 --> 00:28:15,010 goes across the target. 514 00:28:15,010 --> 00:28:18,000 Nice uniform illumination. 515 00:28:18,000 --> 00:28:20,090 Michael's idea was, let's not do that. 516 00:28:20,090 --> 00:28:23,970 Let's project stripes on it. 517 00:28:23,970 --> 00:28:25,800 That's a little bizarre. 518 00:28:25,800 --> 00:28:30,990 So instead of having a nice, blurry picture of a target, 519 00:28:30,990 --> 00:28:35,830 Michael wanted me to generate blurry pictures 520 00:28:35,830 --> 00:28:40,007 of stripey targets. 521 00:28:40,007 --> 00:28:42,340 But you're a sophisticated double-0 3 people, as well as 522 00:28:42,340 --> 00:28:44,644 Michael. 523 00:28:44,644 --> 00:28:45,810 Why does he want to do that? 524 00:28:51,270 --> 00:28:52,400 Different frequency. 525 00:28:55,080 --> 00:28:57,030 Phase modulated microscopy. 526 00:28:57,030 --> 00:28:58,830 That's why he wants to do this. 527 00:28:58,830 --> 00:29:01,200 We're going to modulate microscopy. 528 00:29:01,200 --> 00:29:02,930 OK, well what's that? 529 00:29:02,930 --> 00:29:08,520 OK, so Stan Hong was one of our TA's in double-0 3. 530 00:29:08,520 --> 00:29:11,700 And he wanted to explain the way this works 531 00:29:11,700 --> 00:29:14,374 using purely double-0 3 terms. 532 00:29:14,374 --> 00:29:16,290 He also knew that I would be receptive to that 533 00:29:16,290 --> 00:29:18,720 and that Bertold Horn, who teaches this course all 534 00:29:18,720 --> 00:29:21,490 the time, would be receptive to that kind of an argument. 535 00:29:21,490 --> 00:29:25,840 So Stan made a picture to illustrate that. 536 00:29:25,840 --> 00:29:26,930 So here Stan's picture. 537 00:29:32,210 --> 00:29:33,606 What do you see? 538 00:29:37,390 --> 00:29:38,460 Nice picture, right? 539 00:29:43,094 --> 00:29:43,760 What do you see? 540 00:29:47,000 --> 00:29:48,210 Great. 541 00:29:48,210 --> 00:29:49,440 Excellent. 542 00:29:49,440 --> 00:29:52,240 That's exactly right. 543 00:29:52,240 --> 00:29:54,240 Some of you close to the front, what do you see? 544 00:29:58,050 --> 00:30:00,620 Stripes. 545 00:30:00,620 --> 00:30:02,690 So what you see is stripes. 546 00:30:02,690 --> 00:30:04,836 And if I were sitting where you're sitting, 547 00:30:04,836 --> 00:30:05,960 I wouldn't see the stripes. 548 00:30:05,960 --> 00:30:06,470 Why is that? 549 00:30:11,180 --> 00:30:15,500 Because my eyes, like any optical system, blurs. 550 00:30:15,500 --> 00:30:19,430 Me being old, they blur more than you being young. 551 00:30:19,430 --> 00:30:21,960 You can see stripes better than I can. 552 00:30:21,960 --> 00:30:24,830 So if I were to sit there or if I read it-- 553 00:30:24,830 --> 00:30:27,350 or if you were to sit-in the back row it 554 00:30:27,350 --> 00:30:29,810 would be hard to see the stripes would you still see blur. 555 00:30:29,810 --> 00:30:32,540 You'd still see gray. 556 00:30:32,540 --> 00:30:33,485 So what do I do? 557 00:30:36,660 --> 00:30:38,996 This is double-0 3. 558 00:30:38,996 --> 00:30:40,620 We just did phase modulated microscopy, 559 00:30:40,620 --> 00:30:43,504 but what should I do next? 560 00:30:43,504 --> 00:30:44,545 I have a stripey picture. 561 00:30:54,110 --> 00:30:57,360 How did that picture get stripey? 562 00:30:57,360 --> 00:30:58,830 It got modulated. 563 00:30:58,830 --> 00:31:00,790 What should I do next? 564 00:31:00,790 --> 00:31:01,505 Demodulate it. 565 00:31:01,505 --> 00:31:02,380 So how do we do that? 566 00:31:08,380 --> 00:31:10,000 Do the same process when-- 567 00:31:10,000 --> 00:31:13,220 and that process was? 568 00:31:13,220 --> 00:31:15,480 So I put stripes-- 569 00:31:15,480 --> 00:31:17,270 I illuminated the picture with stripes 570 00:31:17,270 --> 00:31:19,400 in order to make the original. 571 00:31:19,400 --> 00:31:26,350 Now what I should do is illuminate the result 572 00:31:26,350 --> 00:31:27,000 with stripes. 573 00:31:31,260 --> 00:31:33,820 If I multiply the picture by stripes 574 00:31:33,820 --> 00:31:37,780 I'm phase modulating it. 575 00:31:37,780 --> 00:31:41,050 Multiply by cos omega whatever, right? 576 00:31:41,050 --> 00:31:43,090 So now I'll put the clicker down. 577 00:31:43,090 --> 00:31:44,170 No cheating, right? 578 00:31:44,170 --> 00:31:46,600 Nothing up my sleeves. 579 00:31:46,600 --> 00:31:49,030 And now I'm going to project that stripey pattern. 580 00:31:55,580 --> 00:31:56,402 Cute, huh? 581 00:32:00,560 --> 00:32:01,430 It's pretty amazing. 582 00:32:04,320 --> 00:32:06,310 So who is that? 583 00:32:06,310 --> 00:32:06,990 Fourier. 584 00:32:06,990 --> 00:32:07,489 Good. 585 00:32:11,160 --> 00:32:15,870 That's the principle behind Michael's microscope. 586 00:32:15,870 --> 00:32:19,320 And by the way, I'm not cheating. 587 00:32:19,320 --> 00:32:20,355 So if I roll the phase-- 588 00:32:26,210 --> 00:32:28,750 It works just like radio. 589 00:32:33,900 --> 00:32:35,370 Now how's it work? 590 00:32:35,370 --> 00:32:37,035 And how well-- so that was a demo. 591 00:32:40,680 --> 00:32:42,120 Stan's committee loved it. 592 00:32:47,580 --> 00:32:48,880 Here's the idea. 593 00:32:48,880 --> 00:32:52,260 So the poster was phase modulated Fourier. 594 00:32:52,260 --> 00:32:56,490 Fourier is a function of x and y. 595 00:32:56,490 --> 00:32:58,735 And you modulate the phase of carrier. 596 00:32:58,735 --> 00:33:01,530 The carrier now is the y displacement 597 00:33:01,530 --> 00:33:05,220 because we were putting this kind of line on it. 598 00:33:05,220 --> 00:33:06,720 So we modulated carrier. 599 00:33:06,720 --> 00:33:12,330 Cos omega cy was the carrier by the phase of Fourier. 600 00:33:12,330 --> 00:33:15,560 So we bumped the lines up and down. 601 00:33:15,560 --> 00:33:17,280 We being, Stan. 602 00:33:17,280 --> 00:33:20,280 Stan bumped the lines up and down in proportion 603 00:33:20,280 --> 00:33:23,450 to the brightness of Fourier. 604 00:33:23,450 --> 00:33:30,110 And that put Fourier's content centered on omega c. 605 00:33:30,110 --> 00:33:34,890 It was hard to see from the audience because your eyes-- 606 00:33:34,890 --> 00:33:36,590 the projector projected omega c. 607 00:33:36,590 --> 00:33:38,840 It's hard for you to see from the audience 608 00:33:38,840 --> 00:33:46,910 because your visible frequencies in space don't go that high, 609 00:33:46,910 --> 00:33:50,510 but you beat it with the stripey pattern, 610 00:33:50,510 --> 00:33:54,140 that modulates in space just the way it modulates in time. 611 00:33:54,140 --> 00:33:55,940 And it takes a copy of Fourier, which 612 00:33:55,940 --> 00:33:58,040 had been modulated up to omega c and bumps it 613 00:33:58,040 --> 00:33:59,807 back down to the visible. 614 00:33:59,807 --> 00:34:00,890 That's why you can see it. 615 00:34:05,340 --> 00:34:07,090 Well, we're going to see that in a moment. 616 00:34:07,090 --> 00:34:07,840 Good question. 617 00:34:10,900 --> 00:34:16,980 The idea is precisely the same as the superheterodyne radio. 618 00:34:16,980 --> 00:34:18,510 If this is the complicated picture 619 00:34:18,510 --> 00:34:20,010 that we would like to look at-- this 620 00:34:20,010 --> 00:34:21,780 is the Fourier transform or the complicated picture 621 00:34:21,780 --> 00:34:23,100 that we would like to look at. 622 00:34:23,100 --> 00:34:26,969 If we demodulate with this stripe at omega c, 623 00:34:26,969 --> 00:34:32,460 we're able to bounce down this pe house 624 00:34:32,460 --> 00:34:35,880 and we're able to see that part of the picture, 625 00:34:35,880 --> 00:34:38,460 even though those frequencies are too high for them 626 00:34:38,460 --> 00:34:40,156 to go through the microscope. 627 00:34:40,156 --> 00:34:41,989 There's a limited number of frequencies that 628 00:34:41,989 --> 00:34:43,500 will go through the microscope. 629 00:34:43,500 --> 00:34:44,960 Just like there's a limited number of frequencies 630 00:34:44,960 --> 00:34:46,440 that'll go through your eye. 631 00:34:46,440 --> 00:34:49,080 And we can take a band of frequencies, that don't make it 632 00:34:49,080 --> 00:34:52,830 through the microscope, beat it with this stripey pattern 633 00:34:52,830 --> 00:34:57,150 into a frequency range that does go through the microscope. 634 00:34:57,150 --> 00:35:01,170 Then we can change the stripey pattern. 635 00:35:01,170 --> 00:35:03,900 If we make stripey pattern with slightly higher frequency, 636 00:35:03,900 --> 00:35:08,770 we get a different part of the invisible spectrum. 637 00:35:08,770 --> 00:35:10,720 And we just keep repeating. 638 00:35:10,720 --> 00:35:13,420 There's a bit of an issue that it's a 2D transform. 639 00:35:13,420 --> 00:35:14,860 We have to worry about frequencies 640 00:35:14,860 --> 00:35:18,700 in x and frequencies in y, but we are all experts 641 00:35:18,700 --> 00:35:19,720 at this sort of thing. 642 00:35:19,720 --> 00:35:22,930 The difference between stripes this way and stripes this way 643 00:35:22,930 --> 00:35:26,080 is Fourier's this way and Fourier's that way. 644 00:35:26,080 --> 00:35:27,670 You rotate the stripe. 645 00:35:27,670 --> 00:35:29,685 It just rotates the Fourier transform. 646 00:35:33,400 --> 00:35:38,050 We can think about, in space we can modulate like that 647 00:35:38,050 --> 00:35:41,630 or we can modulate like that. 648 00:35:41,630 --> 00:35:44,420 And in fact, what we would like to do 649 00:35:44,420 --> 00:35:48,980 is modulate by a whole bunch, so that we could 650 00:35:48,980 --> 00:35:50,480 take all these little regions-- 651 00:35:50,480 --> 00:35:53,829 so say the circle corresponds to the radius 652 00:35:53,829 --> 00:35:56,120 of frequencies that get through the optical microscope. 653 00:35:56,120 --> 00:35:58,036 What we'd like to do is put that little radius 654 00:35:58,036 --> 00:36:02,460 at every possible place, one of the time, 655 00:36:02,460 --> 00:36:07,485 to bring down those frequencies in the target, one at a time. 656 00:36:10,320 --> 00:36:11,580 So the idea-- 657 00:36:11,580 --> 00:36:13,380 Then the problem becomes, how do you 658 00:36:13,380 --> 00:36:16,340 make so many stripey patterns? 659 00:36:16,340 --> 00:36:20,600 And you have to generate those stripey patterns at very 660 00:36:20,600 --> 00:36:22,280 high spatial frequencies. 661 00:36:22,280 --> 00:36:25,010 Very small distances. 662 00:36:25,010 --> 00:36:28,520 If you're going to beat a microscope-- 663 00:36:28,520 --> 00:36:30,320 Optical microscope resolutions are 664 00:36:30,320 --> 00:36:32,900 on the order of 500 nanometers. 665 00:36:32,900 --> 00:36:34,710 Wavelength of light. 666 00:36:34,710 --> 00:36:37,640 So you're going to have to make these patterns small compared 667 00:36:37,640 --> 00:36:40,170 to the wavelength of light. 668 00:36:40,170 --> 00:36:43,010 So Michael's idea was interference. 669 00:36:43,010 --> 00:36:47,350 You take two coherent laser beams, point them 670 00:36:47,350 --> 00:36:50,790 toward each other but at a slight angle, 671 00:36:50,790 --> 00:36:55,050 and they will interfere and make a stripey pattern. 672 00:36:55,050 --> 00:36:58,230 Then you turn on a different pair of beams 673 00:36:58,230 --> 00:37:01,770 and you get a different stripey pattern. 674 00:37:01,770 --> 00:37:02,598 Different. 675 00:37:02,598 --> 00:37:03,426 Different. 676 00:37:03,426 --> 00:37:04,050 Different. 677 00:37:04,050 --> 00:37:06,720 I'm showing on the left, the spatial, and on the right, 678 00:37:06,720 --> 00:37:08,900 the Fourier transform. 679 00:37:08,900 --> 00:37:12,770 Fourier transform for the stripey pattern. 680 00:37:12,770 --> 00:37:16,850 All light-- all pictures have some dc because there's 681 00:37:16,850 --> 00:37:19,630 no negative photons. 682 00:37:19,630 --> 00:37:21,100 So that's this. 683 00:37:21,100 --> 00:37:23,980 And these are coding-- 684 00:37:23,980 --> 00:37:28,210 the angle is coding the orientation 685 00:37:28,210 --> 00:37:30,090 and the distance is coding the pitch. 686 00:37:33,090 --> 00:37:34,230 Different stripey pattern. 687 00:37:34,230 --> 00:37:34,665 Different. 688 00:37:34,665 --> 00:37:35,165 Different. 689 00:37:35,165 --> 00:37:36,800 Different. 690 00:37:36,800 --> 00:37:38,524 There's a bunch of them. 691 00:37:48,490 --> 00:37:52,090 With 15 beams, you get 15k 2. 692 00:37:52,090 --> 00:37:54,540 Order 15 squared. 693 00:37:54,540 --> 00:37:56,950 That's the idea. 694 00:37:56,950 --> 00:37:58,270 And that was Michael. 695 00:37:58,270 --> 00:37:59,410 This is Stan. 696 00:37:59,410 --> 00:38:04,060 Stan figured out a way to build an apparatus 697 00:38:04,060 --> 00:38:09,070 to take the beam from a laser, break it into 15 parts, 698 00:38:09,070 --> 00:38:11,729 steer it with a bunch of mirrors, 699 00:38:11,729 --> 00:38:13,270 and point them all toward the center. 700 00:38:13,270 --> 00:38:17,190 So the target-- so the lasers coming in over here. 701 00:38:17,190 --> 00:38:20,330 There's these pick-off mirrors steering things around. 702 00:38:20,330 --> 00:38:23,790 This was some complicated optimization that Stan did. 703 00:38:23,790 --> 00:38:27,060 Here is the region of interest, which 704 00:38:27,060 --> 00:38:29,970 is this big on his microscope. 705 00:38:29,970 --> 00:38:33,870 And about 2 centimeters down, the beams 706 00:38:33,870 --> 00:38:36,090 converge on a specimen. 707 00:38:36,090 --> 00:38:38,307 Which is right there. 708 00:38:38,307 --> 00:38:40,640 So if we zoom in and you can see it a little bit better, 709 00:38:40,640 --> 00:38:43,830 there is a conventional microscope objective 710 00:38:43,830 --> 00:38:46,140 with 15 beams fired toward it. 711 00:38:46,140 --> 00:38:49,770 Now the idea is you put the specimen between the beams 712 00:38:49,770 --> 00:38:51,200 and the objective. 713 00:38:51,200 --> 00:38:55,650 And then you view the stripey illuminated target 714 00:38:55,650 --> 00:38:58,770 with the microscope objective. 715 00:38:58,770 --> 00:39:02,010 Here is a picture taken by Jay, where 716 00:39:02,010 --> 00:39:05,070 he took a bunch of small beads. 717 00:39:05,070 --> 00:39:09,690 It's easy to get plastic beads of very uniform dimension. 718 00:39:09,690 --> 00:39:13,340 So these are about 1 micron beads 719 00:39:13,340 --> 00:39:15,240 and about a gazillion of them. 720 00:39:15,240 --> 00:39:18,090 So he just made a solution of beads, 721 00:39:18,090 --> 00:39:20,730 put them on a glass slide, evaporated it 722 00:39:20,730 --> 00:39:25,620 so that he would have a random constellation of plastic beads 723 00:39:25,620 --> 00:39:27,930 all one micron in diameter. 724 00:39:27,930 --> 00:39:29,730 And here's one picture of it. 725 00:39:29,730 --> 00:39:31,740 And you can see the pixels. 726 00:39:31,740 --> 00:39:34,740 Each one of these squares is one pixel in the camera. 727 00:39:34,740 --> 00:39:37,309 I'm zooming in a lot. 728 00:39:37,309 --> 00:39:38,850 Keep in mind that these guys are only 729 00:39:38,850 --> 00:39:42,070 about a micron in dimension. 730 00:39:42,070 --> 00:39:45,490 But then if you change the stripey pattern, 731 00:39:45,490 --> 00:39:48,570 you get a slightly different picture. 732 00:39:48,570 --> 00:39:51,170 And if you change this stripey pattern again. 733 00:39:51,170 --> 00:39:52,620 And again. 734 00:39:52,620 --> 00:39:53,120 And again. 735 00:39:53,120 --> 00:39:53,550 And again. 736 00:39:53,550 --> 00:39:54,050 And again. 737 00:39:54,050 --> 00:39:54,700 And again. 738 00:39:54,700 --> 00:39:59,510 So he recorded then, 300 and some odd pictures 739 00:39:59,510 --> 00:40:01,920 of the same thing with different stripey patterns. 740 00:40:04,750 --> 00:40:07,690 Then he did some signal processing 741 00:40:07,690 --> 00:40:11,030 and turned that sequence of pictures into that picture. 742 00:40:14,380 --> 00:40:16,510 So you can see the resolution is up a bit. 743 00:40:16,510 --> 00:40:20,560 The resolution is very sub-pixel. 744 00:40:20,560 --> 00:40:23,020 You can see many resolution elements 745 00:40:23,020 --> 00:40:27,050 inside one pixel from the original picture. 746 00:40:27,050 --> 00:40:29,800 In fact, if you compare the original 747 00:40:29,800 --> 00:40:32,260 to the reconstruction-- 748 00:40:32,260 --> 00:40:37,840 so this is one picture taken with uniform illumination. 749 00:40:37,840 --> 00:40:42,580 This is the result of calculating some 300 pictures 750 00:40:42,580 --> 00:40:45,010 with structured illumination, with different structure 751 00:40:45,010 --> 00:40:46,630 in each picture. 752 00:40:46,630 --> 00:40:49,190 And you can see a lot better resolution here. 753 00:40:49,190 --> 00:40:50,800 In fact, you can see here there's 754 00:40:50,800 --> 00:40:53,320 something that looks like it might be a bead, 755 00:40:53,320 --> 00:40:57,800 but over here you can see more clearly it's really two beads. 756 00:40:57,800 --> 00:41:01,430 And if you take a solitary bead and plot the brightness 757 00:41:01,430 --> 00:41:03,260 through a line, you get this. 758 00:41:03,260 --> 00:41:05,630 But if you plot the brightness through a reconstruction, 759 00:41:05,630 --> 00:41:07,475 you get something much narrower. 760 00:41:10,200 --> 00:41:16,590 Stan then developed a method for scanning this around. 761 00:41:16,590 --> 00:41:19,590 He also developed a method for measuring the point spread 762 00:41:19,590 --> 00:41:21,037 function directly. 763 00:41:21,037 --> 00:41:22,620 Here I want to show you a measurement, 764 00:41:22,620 --> 00:41:25,590 same sort of apparatus, with a 200 nanometer bead. 765 00:41:25,590 --> 00:41:27,752 A much smaller bead. 766 00:41:27,752 --> 00:41:29,460 Keep in mind that the apparent brightness 767 00:41:29,460 --> 00:41:31,920 goes as the cube of diameter. 768 00:41:31,920 --> 00:41:34,230 So changing the diameter by a factor of five, 769 00:41:34,230 --> 00:41:39,520 changed the apparent brightness by 5 cubed. 770 00:41:39,520 --> 00:41:45,320 So here is a picture taken with Stan's microscope. 771 00:41:45,320 --> 00:41:49,450 And if we take a row of pixels and plot the brightness, 772 00:41:49,450 --> 00:41:52,300 this is the reconstructed image by using 773 00:41:52,300 --> 00:41:55,330 standing wave illumination, which is what we called it. 774 00:41:55,330 --> 00:41:58,900 And it's got an apparent diameter of 290. 775 00:41:58,900 --> 00:42:02,320 The 290 is bigger than the 200 because of the blurring 776 00:42:02,320 --> 00:42:04,600 of the microscope. 777 00:42:04,600 --> 00:42:09,250 The prediction, based on the angle of the laser beams-- 778 00:42:09,250 --> 00:42:10,720 the resolution of this microscope 779 00:42:10,720 --> 00:42:12,130 depends on the angle of the laser 780 00:42:12,130 --> 00:42:14,620 beams because the angle of the laser beams 781 00:42:14,620 --> 00:42:16,780 determines the pitch of the fringe. 782 00:42:16,780 --> 00:42:21,010 That the pitch of the stripey pattern. 783 00:42:21,010 --> 00:42:25,690 So the prediction, based on the angle of the beams 784 00:42:25,690 --> 00:42:30,870 from the apparatus, was that it should have been 250. 785 00:42:30,870 --> 00:42:32,700 And the fairest thing to compare that to 786 00:42:32,700 --> 00:42:35,430 is, what would it have been if we hadn't used any wave 787 00:42:35,430 --> 00:42:36,990 illumination? 788 00:42:36,990 --> 00:42:38,490 If we hadn't used any illumination, 789 00:42:38,490 --> 00:42:42,490 it would have been 1500 nanometers. 790 00:42:42,490 --> 00:42:44,400 So you can see, there's an enormous increase 791 00:42:44,400 --> 00:42:48,310 in the resolution by using this phase modulated microscopy. 792 00:42:48,310 --> 00:42:51,240 So the point is that you can use modulation for a lot 793 00:42:51,240 --> 00:42:53,121 other things than communications. 794 00:42:53,121 --> 00:42:54,870 Because the application and communications 795 00:42:54,870 --> 00:42:56,880 is terribly important, but here was 796 00:42:56,880 --> 00:42:59,280 a completely different application of modulation 797 00:42:59,280 --> 00:43:00,693 to improve microscopy. 798 00:43:03,870 --> 00:43:04,830 Thank you. 799 00:43:04,830 --> 00:43:06,060 See you next time. 800 00:43:06,060 --> 00:43:09,980 Fill out the subject evaluation.