WEBVTT 00:00:00.090 --> 00:00:02.490 The following content is provided under a Creative 00:00:02.490 --> 00:00:04.030 Commons license. 00:00:04.030 --> 00:00:06.330 Your support will help MIT OpenCourseWare 00:00:06.330 --> 00:00:10.720 continue to offer high quality educational resources for free. 00:00:10.720 --> 00:00:13.320 To make a donation or view additional materials 00:00:13.320 --> 00:00:17.280 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.280 --> 00:00:18.450 at ocw.mit.edu. 00:00:21.410 --> 00:00:23.760 DENNIS FREEMAN: Hello. 00:00:23.760 --> 00:00:29.160 Welcome to the first lecture in the last topic of this class. 00:00:29.160 --> 00:00:32.970 So we'll spend this lecture and the next lecture 00:00:32.970 --> 00:00:35.100 talking about modulation. 00:00:35.100 --> 00:00:38.880 Modulation, like sampling, is an excellent illustration 00:00:38.880 --> 00:00:41.490 of the power of thinking about signals in terms 00:00:41.490 --> 00:00:43.500 of their Fourier transforms. 00:00:43.500 --> 00:00:45.390 We'll see, just like we saw in sampling, 00:00:45.390 --> 00:00:47.940 that a problem that was potentially very 00:00:47.940 --> 00:00:51.120 complicated to understand, sampling, 00:00:51.120 --> 00:00:54.510 was very easy when you thought about it in the Fourier domain. 00:00:54.510 --> 00:00:56.010 And precisely the same thing happens 00:00:56.010 --> 00:00:58.200 with modulation, which is precisely the reason we're 00:00:58.200 --> 00:01:00.540 talking about it now. 00:01:00.540 --> 00:01:03.330 So I'll talk about modulation in the context of a communication 00:01:03.330 --> 00:01:05.040 system. 00:01:05.040 --> 00:01:06.360 That's just for convenience. 00:01:06.360 --> 00:01:09.030 In fact, modulation is used in lots of other places. 00:01:09.030 --> 00:01:11.890 In fact, next time, in the next lecture, 00:01:11.890 --> 00:01:14.250 I'll show you an application of modulation 00:01:14.250 --> 00:01:18.750 from my research group, where we used modulation 00:01:18.750 --> 00:01:22.950 to improve the resolution of an optical microscope-- 00:01:22.950 --> 00:01:27.210 nothing whatever to do with communications. 00:01:27.210 --> 00:01:29.010 But it's still-- the optical microscope-- 00:01:29.010 --> 00:01:32.190 the enhancement to resolution that we achieve 00:01:32.190 --> 00:01:34.470 is directly based on the principles 00:01:34.470 --> 00:01:36.970 that I'll start to describe today. 00:01:36.970 --> 00:01:39.840 So I'll talk about modulation today 00:01:39.840 --> 00:01:41.760 in the context of communications. 00:01:41.760 --> 00:01:44.400 Probably the most convenient, easiest 00:01:44.400 --> 00:01:47.310 to think about communication system to all of us, 00:01:47.310 --> 00:01:51.240 being humans, is speech. 00:01:51.240 --> 00:01:53.790 We use speech for communication all the time. 00:01:53.790 --> 00:01:57.570 It's very easy to think about how speech works 00:01:57.570 --> 00:01:58.710 as a communication medium. 00:01:58.710 --> 00:02:01.710 Somebody talks, somebody listens. 00:02:01.710 --> 00:02:03.540 One of the first ways of thinking about it 00:02:03.540 --> 00:02:08.430 as a technological feat was the telephone-- 00:02:08.430 --> 00:02:12.420 the idea being that you convert the sound that I'm emitting 00:02:12.420 --> 00:02:14.359 when I'm speaking by a microphone 00:02:14.359 --> 00:02:16.150 into an electrical representation that gets 00:02:16.150 --> 00:02:18.360 shot down a wire. 00:02:18.360 --> 00:02:22.800 Then at the other end, you take the electrical signal 00:02:22.800 --> 00:02:25.710 that's coming down the wire and turn it back into sound. 00:02:25.710 --> 00:02:28.980 That's the principle of a telephone-- 00:02:28.980 --> 00:02:31.440 works very well. 00:02:31.440 --> 00:02:32.880 Modulation comes up when we start 00:02:32.880 --> 00:02:37.560 to think about how would we generalize this notion 00:02:37.560 --> 00:02:39.930 for wireless communication. 00:02:39.930 --> 00:02:42.920 In particular, if we do cell phone communication, 00:02:42.920 --> 00:02:47.130 cell phones transmit the signal that is picked up 00:02:47.130 --> 00:02:49.690 from the microphone. 00:02:49.690 --> 00:02:53.410 That signal gets converted into electromagnetic signals. 00:02:53.410 --> 00:02:55.660 That's the basis by which cell phones communicate 00:02:55.660 --> 00:02:57.640 with the cell tower. 00:02:57.640 --> 00:02:59.560 The towers may communicate with other towers 00:02:59.560 --> 00:03:01.510 via lot of different kinds of technologies. 00:03:01.510 --> 00:03:03.310 I'm ignoring those for now. 00:03:03.310 --> 00:03:05.920 But the communication between the phone and the tower 00:03:05.920 --> 00:03:08.530 is via electromagnetic waves. 00:03:08.530 --> 00:03:10.570 And there's an interesting thing that 00:03:10.570 --> 00:03:14.500 happens when you try to recode the signal that would have been 00:03:14.500 --> 00:03:17.740 perfectly happy running down the copper wire, when 00:03:17.740 --> 00:03:21.630 you try to recode that signal into an electromagnetic wave. 00:03:21.630 --> 00:03:23.380 And that has to do with basic physics 00:03:23.380 --> 00:03:27.140 of electromagnetic waves, which I'm sure you all know. 00:03:27.140 --> 00:03:32.450 And so I'll just remind you of one simple idea. 00:03:32.450 --> 00:03:35.880 So for efficient transmission from 00:03:35.880 --> 00:03:40.600 an electrical representation to an electromagnetic wave, 00:03:40.600 --> 00:03:44.020 first off, that transduction is mediated by something 00:03:44.020 --> 00:03:45.490 we call an antenna. 00:03:45.490 --> 00:03:47.317 Antennas will take an electrical signal 00:03:47.317 --> 00:03:49.150 and convert it into an electromagnetic wave, 00:03:49.150 --> 00:03:51.250 and vice versa. 00:03:51.250 --> 00:03:54.670 But the efficiency with which an antenna works 00:03:54.670 --> 00:03:59.740 has to do, among other things, with its size. 00:03:59.740 --> 00:04:03.220 It's very difficult-- by which I mean it takes lots of power-- 00:04:07.720 --> 00:04:11.350 to transform a signal from an electrical representation 00:04:11.350 --> 00:04:15.910 to an E&M representation if the antenna is smaller-- 00:04:15.910 --> 00:04:21.579 is significantly smaller-- than the wavelength of interest. 00:04:21.579 --> 00:04:24.840 So it's not that you can't do it. 00:04:24.840 --> 00:04:27.290 It's that it takes lots of power. 00:04:27.290 --> 00:04:29.680 So if you don't want to burn a lot of power 00:04:29.680 --> 00:04:31.780 doing that transformation, then you 00:04:31.780 --> 00:04:33.580 need to make an antenna that's roughly 00:04:33.580 --> 00:04:38.710 the size of the wavelength that you're trying to transmit 00:04:38.710 --> 00:04:41.260 as an electromagnetic wave. 00:04:41.260 --> 00:04:44.140 So if we were thinking about this kind 00:04:44.140 --> 00:04:47.170 of a scheme for the communication of voice-- 00:04:47.170 --> 00:04:49.480 we've talked about voice many times-- 00:04:49.480 --> 00:04:51.190 Telephone-quality voice is usually 00:04:51.190 --> 00:04:54.280 defined to be frequencies between about 200 hertz 00:04:54.280 --> 00:04:55.990 and about 3 kilohertz. 00:04:55.990 --> 00:04:58.030 If I had a signal that was composed 00:04:58.030 --> 00:05:01.060 of those kinds of frequencies, how big 00:05:01.060 --> 00:05:05.260 should I make the antenna to get efficient coupling 00:05:05.260 --> 00:05:08.050 between the electrical representation 00:05:08.050 --> 00:05:10.750 in the cell phone and the electromagnetic wave 00:05:10.750 --> 00:05:12.550 that the cell phone wants to launch 00:05:12.550 --> 00:05:14.650 to get to the cell tower? 00:05:14.650 --> 00:05:16.040 So look at your neighbor. 00:05:19.270 --> 00:05:20.720 That involves turning your head. 00:05:20.720 --> 00:05:22.090 [LAUGHTER] 00:05:22.090 --> 00:05:25.490 Say hello and figure out how long should the antenna be. 00:05:30.979 --> 00:05:34.472 [INTERPOSING VOICES] 00:06:56.900 --> 00:07:00.180 DENNIS FREEMAN: Does everybody have an answer? 00:07:00.180 --> 00:07:02.360 Raise your hand if you don't have an answer. 00:07:02.360 --> 00:07:04.140 How do you like that for a switch of-- 00:07:04.140 --> 00:07:06.715 Raise your hand if you don't have an answer. 00:07:06.715 --> 00:07:07.590 You all have answers? 00:07:07.590 --> 00:07:10.260 OK, so how big should the antenna be-- 00:07:10.260 --> 00:07:11.590 1, 2, 3, 4, or 5? 00:07:17.187 --> 00:07:17.770 OK, very good. 00:07:17.770 --> 00:07:21.700 So the answer is really big. 00:07:21.700 --> 00:07:27.640 So you think about the relationship between wavelength 00:07:27.640 --> 00:07:31.540 and distance, so you can think about the relationship is 00:07:31.540 --> 00:07:32.860 given by speed. 00:07:32.860 --> 00:07:36.040 The speed of a wave is the wave length-- 00:07:36.040 --> 00:07:38.680 how long it goes per cycle-- 00:07:38.680 --> 00:07:42.580 times the number of cycles per second, f. 00:07:42.580 --> 00:07:44.470 So you can solve that expression to find out 00:07:44.470 --> 00:07:46.570 the wavelength in terms of the speed, which 00:07:46.570 --> 00:07:51.076 for an electromagnetic wave is the speed of light, 3 times 10 00:07:51.076 --> 00:07:53.110 to the eighth meters per second. 00:07:53.110 --> 00:07:57.580 The hardest one to launch is the lowest frequency. 00:07:57.580 --> 00:07:59.680 That'll take the biggest antenna. 00:07:59.680 --> 00:08:01.360 So the lowest frequency in telephony-- 00:08:01.360 --> 00:08:05.170 in telephone-quality speech-- is 200 hertz. 00:08:05.170 --> 00:08:09.940 So we get about 1,500 kilometers-- 00:08:09.940 --> 00:08:13.365 kind of big, kind of useless. 00:08:13.365 --> 00:08:14.740 If you were thinking about trying 00:08:14.740 --> 00:08:17.073 to make cellular communication and your antenna actually 00:08:17.073 --> 00:08:24.310 had to be 1,500 kilometers, that just isn't going to work. 00:08:24.310 --> 00:08:26.350 So what do we do? 00:08:26.350 --> 00:08:27.670 We obviously don't do this. 00:08:30.770 --> 00:08:33.580 So the answer is that you would need an antenna 00:08:33.580 --> 00:08:36.549 hundreds of miles in length. 00:08:36.549 --> 00:08:41.409 So what frequency should you be using 00:08:41.409 --> 00:08:44.440 if you wanted to build a phone that kind of fit in your hand-- 00:08:44.440 --> 00:08:46.810 10 centimeters or so? 00:08:46.810 --> 00:08:49.107 What would be the frequency-- 00:08:49.107 --> 00:08:51.190 what would be the interesting range of frequencies 00:08:51.190 --> 00:08:53.305 that you would want to use for such a device? 00:09:00.440 --> 00:09:02.870 And the answer is-- 00:09:02.870 --> 00:09:03.580 Of course. 00:09:03.580 --> 00:09:06.150 So the answer is, you go to the bottom one, 00:09:06.150 --> 00:09:07.690 and it's bigger than a gigahertz. 00:09:07.690 --> 00:09:10.090 And it's just running the same expression 00:09:10.090 --> 00:09:11.920 in the other direction. 00:09:11.920 --> 00:09:15.250 So you think about, the frequency that you would like 00:09:15.250 --> 00:09:18.430 would be the speed of light divided by the wavelength. 00:09:18.430 --> 00:09:21.220 If you wanted the wavelength to be 10 centimeters, 00:09:21.220 --> 00:09:23.860 then you would end up with a frequency 00:09:23.860 --> 00:09:25.480 on the order of gigahertz. 00:09:25.480 --> 00:09:27.400 And it shouldn't come as a surprise to you 00:09:27.400 --> 00:09:30.460 then that that's what we use in cellular communication. 00:09:30.460 --> 00:09:35.620 So a modern cell phone uses 2.1 gigahertz. 00:09:35.620 --> 00:09:40.570 So the point is that when we're thinking 00:09:40.570 --> 00:09:47.920 about how we would like to use the electromagnetic spectrum 00:09:47.920 --> 00:09:51.130 for a communications task, that spectrum is not 00:09:51.130 --> 00:09:54.190 necessarily well-matched to the communications 00:09:54.190 --> 00:09:55.960 problem of interest. 00:09:55.960 --> 00:09:57.970 You might think that that cellular example 00:09:57.970 --> 00:09:58.910 is an exception. 00:09:58.910 --> 00:10:00.670 In fact, that's the rule. 00:10:00.670 --> 00:10:03.520 If you try to have a signal of interest transmitted 00:10:03.520 --> 00:10:07.150 over a medium or stored on some sort of a medium, 00:10:07.150 --> 00:10:10.547 it is generally the case that there is a matching problem, 00:10:10.547 --> 00:10:12.130 that the characteristics of the medium 00:10:12.130 --> 00:10:14.260 don't match the characteristics of the message. 00:10:14.260 --> 00:10:16.960 And so part of communications engineering 00:10:16.960 --> 00:10:22.010 is trying to come up with a coding scheme that 00:10:22.010 --> 00:10:25.670 matches the characteristics of the message 00:10:25.670 --> 00:10:28.915 to the characteristics of the medium. 00:10:28.915 --> 00:10:30.290 And so what I'm going to do today 00:10:30.290 --> 00:10:34.700 is talk about some matching schemes based on modulation. 00:10:34.700 --> 00:10:39.860 So we just saw that if we wanted to do cellular communication 00:10:39.860 --> 00:10:44.210 of voice, voice might have a spectrum represented 00:10:44.210 --> 00:10:48.410 by this magnitude function, X, where 00:10:48.410 --> 00:10:53.910 the bandwidth is on the order of a few kilohertz. 00:10:53.910 --> 00:10:56.340 But we might want to transmit a signal that 00:10:56.340 --> 00:10:59.700 has the same information. 00:10:59.700 --> 00:11:01.340 However, we would like the frequencies 00:11:01.340 --> 00:11:05.460 to be up around 2 gigahertz. 00:11:05.460 --> 00:11:11.710 Which of these coding schemes, Y as a function of X, 00:11:11.710 --> 00:11:14.140 achieves this transformation? 00:11:14.140 --> 00:11:18.610 Take the stuff that you have in a low frequency range 00:11:18.610 --> 00:11:20.680 and shift it to a high frequency range. 00:11:20.680 --> 00:11:21.460 What would you do? 00:11:28.920 --> 00:11:33.570 The obvious answer is to stare with a blank face in it. 00:11:33.570 --> 00:11:36.704 It'll definitely come to you, right? 00:11:36.704 --> 00:11:38.370 You don't want to talk to somebody else. 00:11:38.370 --> 00:11:39.369 That would give it away. 00:11:42.154 --> 00:11:47.643 [INTERPOSING VOICES] 00:12:08.601 --> 00:12:11.810 DENNIS FREEMAN: So what's the relationship between X of t 00:12:11.810 --> 00:12:13.410 and Y of t? 00:12:13.410 --> 00:12:16.300 Is it relation 1, 2, 3, 4, or none of the above? 00:12:20.290 --> 00:12:24.950 OK, it's about 80% correct. 00:12:24.950 --> 00:12:26.960 So how do I think about it? 00:12:26.960 --> 00:12:29.050 So let's see, I want to figure out 00:12:29.050 --> 00:12:35.700 a relationship between the Fourier transform of Y and X. 00:12:35.700 --> 00:12:39.190 If I wanted to figure out the Fourier transform of Y, 00:12:39.190 --> 00:12:45.940 I would integrate Y of t e to the minus j omega t dt. 00:12:45.940 --> 00:12:47.850 That looks kind of right. 00:12:47.850 --> 00:12:50.059 And Y would be X of the t-- 00:12:50.059 --> 00:12:51.100 let's try the first one-- 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. 00:13:00.390 --> 00:13:06.580 Well, that's almost the Fourier transform of X. All I've done 00:13:06.580 --> 00:13:09.710 is shifted omega. 00:13:09.710 --> 00:13:18.970 So in fact, that's the same as X of j omega minus omega c. 00:13:18.970 --> 00:13:21.380 And that's what I want to do. 00:13:21.380 --> 00:13:26.230 So the idea is that, if you multiply 00:13:26.230 --> 00:13:30.010 by this complex exponential, the effect 00:13:30.010 --> 00:13:35.080 of that multiplication in time is to shift frequencies. 00:13:35.080 --> 00:13:37.120 Can somebody say that to me-- 00:13:37.120 --> 00:13:39.760 say the same transformation but in slightly different words? 00:13:43.958 --> 00:13:47.877 Guess that was kind of big. 00:13:47.877 --> 00:13:48.377 Yes. 00:13:48.377 --> 00:13:51.814 AUDIENCE: Couldn't you [INAUDIBLE] multiplication 00:13:51.814 --> 00:13:53.247 by the exponential in time. 00:13:53.247 --> 00:13:55.580 DENNIS FREEMAN: [INAUDIBLE] by the exponential in time-- 00:13:55.580 --> 00:13:56.300 AUDIENCE: Yeah equal to-- 00:13:56.300 --> 00:13:58.040 DENNIS FREEMAN: --should correspond to-- 00:13:58.040 --> 00:13:59.656 AUDIENCE: Which gives you a frequency [INAUDIBLE]. 00:13:59.656 --> 00:14:01.572 DENNIS FREEMAN: So generally, a multiplication 00:14:01.572 --> 00:14:02.865 in time corresponds to-- 00:14:02.865 --> 00:14:03.740 AUDIENCE: [INAUDIBLE] 00:14:03.740 --> 00:14:06.660 DENNIS FREEMAN: --convolution. 00:14:06.660 --> 00:14:10.410 So equivalently, instead of saying it shifted in frequency, 00:14:10.410 --> 00:14:12.610 you could say-- 00:14:12.610 --> 00:14:13.520 AUDIENCE: [INAUDIBLE] 00:14:13.520 --> 00:14:15.020 DENNIS FREEMAN: --it got convolved-- 00:14:15.020 --> 00:14:16.496 AUDIENCE: With delta. 00:14:16.496 --> 00:14:19.320 DENNIS FREEMAN: --with the delta function. 00:14:19.320 --> 00:14:22.410 So a different way of saying the same thing 00:14:22.410 --> 00:14:24.570 would be that you think about convolving 00:14:24.570 --> 00:14:29.400 with the delta function in frequency-- 00:14:29.400 --> 00:14:32.670 same thing. 00:14:32.670 --> 00:14:34.440 Now, we don't actually do this. 00:14:34.440 --> 00:14:36.023 When we're doing the cell phone thing, 00:14:36.023 --> 00:14:39.240 we don't actually multiply it by e to the j omega c t. 00:14:39.240 --> 00:14:39.920 Anyone know why? 00:14:43.550 --> 00:14:45.960 This is kind of the simplest way that you could imagine. 00:14:45.960 --> 00:14:47.830 I've taken frequency content centered near 0 00:14:47.830 --> 00:14:50.330 and turning it into frequency content centered near omega c. 00:14:50.330 --> 00:14:51.984 But that's not what we really do. 00:14:51.984 --> 00:14:53.150 Why don't we really do that? 00:14:53.150 --> 00:14:55.102 AUDIENCE: [INAUDIBLE] 00:14:55.102 --> 00:14:57.120 DENNIS FREEMAN: Exactly. 00:14:57.120 --> 00:15:02.130 We don't really do it, because the signals aren't real. 00:15:04.810 --> 00:15:06.640 So how do you know the signal is not real? 00:15:11.005 --> 00:15:15.855 AUDIENCE: Because magnitudes [INAUDIBLE]. 00:15:15.855 --> 00:15:18.600 DENNIS FREEMAN: If the signal had been real, 00:15:18.600 --> 00:15:20.225 the Fourier transform would have been-- 00:15:23.830 --> 00:15:25.750 If the signal had been real the Fourier 00:15:25.750 --> 00:15:28.370 transform would have been-- 00:15:28.370 --> 00:15:29.460 X of j omega. 00:15:34.407 --> 00:15:36.490 If the signal had been real, the Fourier transform 00:15:36.490 --> 00:15:37.300 would have been-- 00:15:40.228 --> 00:15:41.692 AUDIENCE: Symmetry. 00:15:41.692 --> 00:15:43.710 DENNIS FREEMAN: --some kind of symmetry. 00:15:43.710 --> 00:15:45.561 How do I see symmetry in that expression? 00:15:45.561 --> 00:15:46.394 AUDIENCE: Conjugate. 00:15:46.394 --> 00:15:46.791 AUDIENCE: Conjugate. 00:15:46.791 --> 00:15:48.320 DENNIS FREEMAN: Conjugate symmetry. 00:15:48.320 --> 00:15:50.675 How do I see conjugate symmetry in that expression? 00:15:55.910 --> 00:16:01.100 Well, this is cos omega t plus j sine omega t. 00:16:08.850 --> 00:16:12.810 So if this is a real signal, then it gives rise to something 00:16:12.810 --> 00:16:14.640 here that is symmetric. 00:16:14.640 --> 00:16:18.570 The cosine terms are all symmetric about the origin. 00:16:18.570 --> 00:16:21.120 And it has an imaginary part that 00:16:21.120 --> 00:16:23.397 is antisymmetric about the origin, 00:16:23.397 --> 00:16:25.230 because you add together a bunch of cosines. 00:16:25.230 --> 00:16:27.510 You can't get anything that's anything other than symmetric 00:16:27.510 --> 00:16:28.260 about the origin. 00:16:28.260 --> 00:16:29.490 You add together a bunch of sines. 00:16:29.490 --> 00:16:31.448 You can't get anything except something that is 00:16:31.448 --> 00:16:34.530 antisymmetric about the origin. 00:16:34.530 --> 00:16:37.510 So you know this thing has to be-- 00:16:37.510 --> 00:16:42.624 so a real signal would have had conjugate symmetry. 00:16:42.624 --> 00:16:44.290 The real part would have been symmetric. 00:16:44.290 --> 00:16:46.810 The imaginary part would be antisymmetric. 00:16:46.810 --> 00:16:50.946 And you can see that this is not conjugate symmetric. 00:16:50.946 --> 00:16:52.571 Everybody knows what I'm talking about? 00:16:52.571 --> 00:16:55.264 AUDIENCE: [INAUDIBLE] 00:16:55.264 --> 00:16:56.680 DENNIS FREEMAN: Conjugate symmetry 00:16:56.680 --> 00:16:59.680 would mean that the real part of the signal 00:16:59.680 --> 00:17:03.700 is symmetric about the origin, which means that if this 00:17:03.700 --> 00:17:06.069 is supposed to represent a real signal, 00:17:06.069 --> 00:17:09.490 then there should have been a reflection over here to make 00:17:09.490 --> 00:17:12.754 it symmetric about the origin. 00:17:12.754 --> 00:17:16.682 AUDIENCE: Just was it symmetric before? 00:17:16.682 --> 00:17:19.010 DENNIS FREEMAN: It was symmetric, 00:17:19.010 --> 00:17:21.930 but I shifted it by a complex number. 00:17:21.930 --> 00:17:27.290 I shifted it by cos omega c t plus j sine omega c t. 00:17:27.290 --> 00:17:33.018 So by that multiplication, I generated a complex number. 00:17:33.018 --> 00:17:35.880 AUDIENCE: [INAUDIBLE]. 00:17:35.880 --> 00:17:41.340 DENNIS FREEMAN: So this signal was complex valued and not 00:17:41.340 --> 00:17:43.836 conjugate symmetric. 00:17:43.836 --> 00:17:45.210 So the point is trying to get you 00:17:45.210 --> 00:17:47.130 to remember the kinds of things that we're supposed to know 00:17:47.130 --> 00:17:48.450 about Fourier transforms. 00:17:48.450 --> 00:17:54.240 So by shifting with a complex exponential, 00:17:54.240 --> 00:17:57.800 we wreck the realness of the original signal. 00:17:57.800 --> 00:18:00.900 The real original signal would have been conjugate symmetric 00:18:00.900 --> 00:18:02.880 in frequency. 00:18:02.880 --> 00:18:05.760 But the wrecking of it gave rise to a signal 00:18:05.760 --> 00:18:09.692 that was no longer conjugate symmetric in frequency. 00:18:12.350 --> 00:18:16.490 So we don't really modulate this way. 00:18:16.490 --> 00:18:18.470 But we do the obvious extension. 00:18:18.470 --> 00:18:21.720 What we would do is modulate with a cosine wave. 00:18:24.230 --> 00:18:26.570 So now, if instead of multiplying 00:18:26.570 --> 00:18:30.117 by a complex exponential, I multiply by the cosine omega c 00:18:30.117 --> 00:18:32.990 t, by Euler's expression, I can think about that 00:18:32.990 --> 00:18:37.100 as being the sum of two components-- 00:18:37.100 --> 00:18:40.520 one at plus omega c and one at minus omega c. 00:18:40.520 --> 00:18:42.950 And now, when I do the convolution, 00:18:42.950 --> 00:18:47.180 I get a signal that is conjugate symmetric. 00:18:47.180 --> 00:18:50.270 So when I convolve this with this one, 00:18:50.270 --> 00:18:52.530 this one gives me a copy of this here. 00:18:52.530 --> 00:18:54.230 And when I convolve this with this one, 00:18:54.230 --> 00:18:57.060 this one gives me a copy of this one down here. 00:18:57.060 --> 00:19:00.710 So now, the resulting signal, which I know by construction-- 00:19:00.710 --> 00:19:05.310 if this was real and this was real, then that's real-- 00:19:05.310 --> 00:19:08.426 I know by construction that this signal must have been real. 00:19:08.426 --> 00:19:10.050 But I can also see it in the transform, 00:19:10.050 --> 00:19:11.466 because I can see now that there's 00:19:11.466 --> 00:19:13.200 a symmetry that is consistent with it 00:19:13.200 --> 00:19:15.100 being conjugate symmetric. 00:19:15.100 --> 00:19:15.600 Yes. 00:19:15.600 --> 00:19:16.599 Somebody had a question? 00:19:20.800 --> 00:19:23.579 So that's what we mean by modulation. 00:19:23.579 --> 00:19:24.370 This is modulation. 00:19:24.370 --> 00:19:27.610 Modulation just is a fancy word that means multiply. 00:19:27.610 --> 00:19:31.300 So what we're going to do is multiply the signal 00:19:31.300 --> 00:19:32.860 by a carrier. 00:19:32.860 --> 00:19:36.490 The carrier is going to be a signal that carries 00:19:36.490 --> 00:19:38.530 the message through the medium. 00:19:38.530 --> 00:19:41.800 The carrier is chosen so that it goes 00:19:41.800 --> 00:19:44.500 efficiently through the medium. 00:19:44.500 --> 00:19:48.310 And then the carrier carries the message through the medium. 00:19:48.310 --> 00:19:51.190 So we think about this as modulation. 00:19:51.190 --> 00:19:54.010 And we want to be familiar with going back and forth 00:19:54.010 --> 00:19:55.760 between time and frequency. 00:19:55.760 --> 00:19:59.020 You can also think about the result of modulation in time. 00:19:59.020 --> 00:20:01.060 So if this were my message, which 00:20:01.060 --> 00:20:04.570 is intended to be represented as a low frequency, 00:20:04.570 --> 00:20:06.430 and this is my carrier, which is intended 00:20:06.430 --> 00:20:09.280 to be represented as a higher frequency, then 00:20:09.280 --> 00:20:12.550 when I modulate it, I get a modulated signal, 00:20:12.550 --> 00:20:14.000 by which I mean-- 00:20:14.000 --> 00:20:16.390 this is called amplitude modulation-- 00:20:16.390 --> 00:20:20.170 the amplitude of the carrier is modulated by the message. 00:20:20.170 --> 00:20:22.370 That's all we mean. 00:20:22.370 --> 00:20:26.350 So you can see that this transformation, which 00:20:26.350 --> 00:20:28.987 has the property of moving the information 00:20:28.987 --> 00:20:30.820 from a low frequency that's hard to transmit 00:20:30.820 --> 00:20:33.850 to a high frequency that is easy to transmit, 00:20:33.850 --> 00:20:40.240 that has the effect of doing a very particular pattern 00:20:40.240 --> 00:20:42.760 to the time-domain waveform. 00:20:45.091 --> 00:20:47.340 Now, it would be completely useless as a communication 00:20:47.340 --> 00:20:49.660 scheme if it weren't easy to invert. 00:20:52.770 --> 00:20:55.540 So imagine that I have this signal. 00:20:55.540 --> 00:20:58.950 And what I'd like to do is recover x. 00:20:58.950 --> 00:21:03.840 What should I do to recover x, the original message? 00:21:03.840 --> 00:21:06.180 So the idea is, I have an original message 00:21:06.180 --> 00:21:07.770 available in my cell phone. 00:21:07.770 --> 00:21:09.690 It gets modulated so that it can be launched 00:21:09.690 --> 00:21:11.070 into electromagnetic waves. 00:21:11.070 --> 00:21:14.670 The electromagnetic waves go to a receiver thousands of miles 00:21:14.670 --> 00:21:15.600 away. 00:21:15.600 --> 00:21:18.420 And now, the idea is to reconstruct my original signal 00:21:18.420 --> 00:21:19.320 x. 00:21:19.320 --> 00:21:20.880 What would I do-- 00:21:20.880 --> 00:21:24.705 what kind of a system would I use to recover x from y? 00:21:31.373 --> 00:21:32.677 AUDIENCE: [INAUDIBLE]. 00:21:32.677 --> 00:21:33.760 DENNIS FREEMAN: I'm sorry. 00:21:33.760 --> 00:21:35.191 AUDIENCE: Couldn't you divide out the cosine? 00:21:35.191 --> 00:21:37.470 DENNIS FREEMAN: You could divide out the cosine. 00:21:37.470 --> 00:21:45.330 So you could take x of t cosine omega c t times something-- 00:21:45.330 --> 00:21:48.600 what do I want to say-- a of t designed 00:21:48.600 --> 00:21:50.310 so that this times this is 1. 00:21:54.370 --> 00:21:56.270 That's kind of ugly. 00:21:56.270 --> 00:21:59.320 Anybody see anything ugly about that? 00:21:59.320 --> 00:21:59.820 Yeah. 00:21:59.820 --> 00:22:01.952 AUDIENCE: [INAUDIBLE] shift it back the other way? 00:22:01.952 --> 00:22:04.410 DENNIS FREEMAN: You could also shift it back the other way. 00:22:04.410 --> 00:22:04.951 That's right. 00:22:04.951 --> 00:22:07.027 So before we do that, why is this ugly? 00:22:07.027 --> 00:22:08.490 AUDIENCE: The zeros. 00:22:08.490 --> 00:22:10.770 DENNIS FREEMAN: The zeros. 00:22:10.770 --> 00:22:15.780 So if we wanted to take a signal that looks like a cosine wave 00:22:15.780 --> 00:22:20.970 and multiply it by some signal that generates 1, 00:22:20.970 --> 00:22:22.709 that's not too hard to do here. 00:22:22.709 --> 00:22:24.500 You would do that with something like this. 00:22:24.500 --> 00:22:26.250 But it becomes very hard to do here. 00:22:26.250 --> 00:22:31.320 So you would end up making some signal that does some awful-- 00:22:31.320 --> 00:22:33.390 So it would periodically be a mass. 00:22:36.290 --> 00:22:38.180 But you can do what you said. 00:22:38.180 --> 00:22:41.390 An alternative would be to multiply it 00:22:41.390 --> 00:22:45.530 by another cosine, which in the frequency domain 00:22:45.530 --> 00:22:46.760 is easy to think about. 00:22:46.760 --> 00:22:49.670 It would just shift it back-- 00:22:49.670 --> 00:22:52.460 so convolve with a pair of impulses 00:22:52.460 --> 00:22:56.690 to move something that was at DC out to some high frequency, 00:22:56.690 --> 00:22:59.180 convolve again to take the thing that was at high frequency 00:22:59.180 --> 00:23:01.340 and bring it back to DC. 00:23:01.340 --> 00:23:05.540 And you can think about that in either frequency or time. 00:23:05.540 --> 00:23:07.040 It's easy to think about it in time. 00:23:07.040 --> 00:23:08.748 If you think about it in time, here we've 00:23:08.748 --> 00:23:10.850 got the product of two cosines. 00:23:10.850 --> 00:23:12.890 But the product of two cosines is just 1/2 00:23:12.890 --> 00:23:16.640 plus half the cosine of double. 00:23:16.640 --> 00:23:18.500 Well, that's good. 00:23:18.500 --> 00:23:19.230 Why is that good? 00:23:19.230 --> 00:23:23.180 Well, if you multiply x of t by this, 00:23:23.180 --> 00:23:25.010 this is a super high frequency. 00:23:25.010 --> 00:23:30.390 If omega c was a high frequency, 2 omega c is even higher. 00:23:30.390 --> 00:23:36.410 So what you could do is remove x times 1/2 cos 2 omega c t 00:23:36.410 --> 00:23:39.890 with a low pass filter, since omega c 00:23:39.890 --> 00:23:41.630 is such a high frequency. 00:23:41.630 --> 00:23:44.480 And that would just leave you with half the message, which 00:23:44.480 --> 00:23:47.720 would be easy then to reconstruct, 00:23:47.720 --> 00:23:50.420 because what you would do is just put it through a low pass 00:23:50.420 --> 00:23:52.310 filter and then multiply by 2. 00:23:52.310 --> 00:23:54.920 You can similarly think about the same thing in frequency. 00:23:54.920 --> 00:23:58.460 If I took y and convolved it-- if I multiply 00:23:58.460 --> 00:24:03.020 in time by another cosine wave, that second cosine wave 00:24:03.020 --> 00:24:04.960 is a pair of impulses-- one at minus omega 00:24:04.960 --> 00:24:07.100 c and one at omega c. 00:24:07.100 --> 00:24:09.200 And now, when I convolve the y signal, 00:24:09.200 --> 00:24:13.870 this one shifts these two up, and this one 00:24:13.870 --> 00:24:15.760 shifts these two down. 00:24:15.760 --> 00:24:17.620 And two of them land on top of each other. 00:24:21.450 --> 00:24:26.770 But each of these was only of height 1/2. 00:24:26.770 --> 00:24:32.810 So by Euler's expression, cosine of something was 1/2 e 00:24:32.810 --> 00:24:34.720 to the whatever plus 1/2 e to the whatever. 00:24:34.720 --> 00:24:38.320 So I got 1/2's on each of those amplitudes. 00:24:38.320 --> 00:24:42.820 So the result is then that I have to multiply the low 00:24:42.820 --> 00:24:46.430 frequency part by a factor of two to undo the 1/2's. 00:24:51.340 --> 00:24:54.640 So this kind of a scheme is especially nice, 00:24:54.640 --> 00:24:58.510 because you can scramble together multiple messages 00:24:58.510 --> 00:25:02.820 and still get them separated at the destination. 00:25:02.820 --> 00:25:09.500 If you imagine having three similar transmitters that 00:25:09.500 --> 00:25:12.250 use their own omega c-- 00:25:12.250 --> 00:25:15.080 so the first one uses omega 1, omega 2, omega 3-- 00:25:17.700 --> 00:25:19.530 so if each one of the transmitters 00:25:19.530 --> 00:25:23.730 had their own frequency, and if the frequencies were far enough 00:25:23.730 --> 00:25:27.000 apart, and if the frequencies were all big compared 00:25:27.000 --> 00:25:31.470 to the message frequency, then you could combine them all 00:25:31.470 --> 00:25:35.440 and select out the one of interest 00:25:35.440 --> 00:25:42.290 by tuning the receiver, by choosing the demodulation 00:25:42.290 --> 00:25:43.020 frequency. 00:25:43.020 --> 00:25:46.190 So now, if the receiver chose omega c equals omega 1, 00:25:46.190 --> 00:25:49.000 you would decode message 1. 00:25:49.000 --> 00:25:54.110 If omega c were omega 2, you'd decode message 2. 00:25:54.110 --> 00:25:59.430 And that's because the medium works approximately linearly. 00:25:59.430 --> 00:26:01.940 So if you launch multiple waves into the air-- 00:26:05.360 --> 00:26:07.890 I don't want to get too much into electromagnetic theory-- 00:26:07.890 --> 00:26:10.620 so the presence of the antennas distort it from linearity. 00:26:10.620 --> 00:26:12.434 But once the antennas are all there, 00:26:12.434 --> 00:26:13.850 then it's perfectly linear system. 00:26:16.710 --> 00:26:21.180 And the thing that gets into the air as a result of a sum 00:26:21.180 --> 00:26:24.850 is the sum of the individual parts. 00:26:24.850 --> 00:26:27.540 So the idea then is illustrated here. 00:26:27.540 --> 00:26:31.080 If I had three different messages represented 00:26:31.080 --> 00:26:36.810 by different style houses, and each one of the messages was 00:26:36.810 --> 00:26:40.170 at a different frequency-- omega 1, omega 2, omega 3-- 00:26:40.170 --> 00:26:43.770 by tuning the omega c of the receiver, 00:26:43.770 --> 00:26:48.930 if I put omega c at omega 1, the convolution of this one 00:26:48.930 --> 00:26:51.810 would suck this one up to here. 00:26:51.810 --> 00:26:53.910 And by this one would lower that one down there. 00:26:53.910 --> 00:26:59.130 You get overlap of the lowest frequency pair. 00:26:59.130 --> 00:27:00.810 And so if you built a low pass filter 00:27:00.810 --> 00:27:06.450 of exactly the right width, you would decode message 1. 00:27:06.450 --> 00:27:11.800 Where, if you just changed the frequency of the demodulator-- 00:27:11.800 --> 00:27:15.760 if you make the demodulation frequency now be omega 2-- 00:27:15.760 --> 00:27:19.360 now, the effect of shifting that different amount 00:27:19.360 --> 00:27:23.260 means that the low pass filter recovers message 2, rather 00:27:23.260 --> 00:27:26.530 the message 1. 00:27:26.530 --> 00:27:29.760 So that's the idea. 00:27:29.760 --> 00:27:33.250 So that's the idea that we use in commercial AM radio. 00:27:33.250 --> 00:27:37.060 And that was, in fact, a revolutionary idea 00:27:37.060 --> 00:27:39.910 that enabled people to think about for the first time 00:27:39.910 --> 00:27:42.460 a communication system that did a lot of things 00:27:42.460 --> 00:27:45.220 that were very different from previous communication systems. 00:27:45.220 --> 00:27:49.000 In particular, it went at the speed of light. 00:27:49.000 --> 00:27:51.970 Even more importantly, or at least as important, 00:27:51.970 --> 00:27:56.040 is the fact that it was a broadcast system. 00:27:56.040 --> 00:28:00.040 So broadcast was an idea that was 00:28:00.040 --> 00:28:02.920 championed by David Sarnoff. 00:28:02.920 --> 00:28:06.170 Sarnoff was a visionary. 00:28:06.170 --> 00:28:10.670 He was the person who was very excited about the idea 00:28:10.670 --> 00:28:14.420 of broadcast, which is a little ironic, because he 00:28:14.420 --> 00:28:17.900 got his start with Marconi. 00:28:17.900 --> 00:28:20.370 Anybody ever hear of Marconi? 00:28:20.370 --> 00:28:22.760 Good, good, you're supposed to have heard of Marconi. 00:28:22.760 --> 00:28:24.860 So Sarnoff got his start-- 00:28:24.860 --> 00:28:26.930 this is Sarnoff; this is Marconi-- 00:28:26.930 --> 00:28:28.550 Sarnoff got his start with Marconi. 00:28:28.550 --> 00:28:33.170 Marconi made his mint with wireless telegraphy. 00:28:33.170 --> 00:28:35.870 Anybody ever hear of telegraphy? 00:28:35.870 --> 00:28:37.820 Of course not. 00:28:37.820 --> 00:28:41.480 So telegraphy, that's telegraph. 00:28:41.480 --> 00:28:42.440 Shake your heads yes. 00:28:42.440 --> 00:28:44.930 It's ancient, I realize. 00:28:44.930 --> 00:28:49.220 So Marconi made his fortune with wireless telegraphy. 00:28:49.220 --> 00:28:54.770 Telegraphy was what we call point to point. 00:28:54.770 --> 00:28:57.590 The idea in telegraphy was precisely the same 00:28:57.590 --> 00:29:01.450 as the idea of the US post office, 00:29:01.450 --> 00:29:03.070 except it was at the speed of light, 00:29:03.070 --> 00:29:05.140 or not quite the speed of light. 00:29:05.140 --> 00:29:10.330 So the idea in the post office is you take a sheet of paper 00:29:10.330 --> 00:29:13.540 and do something to it that makes it magically appear 00:29:13.540 --> 00:29:15.580 at somebody else's place. 00:29:15.580 --> 00:29:20.401 So point A communicated to point B. 00:29:20.401 --> 00:29:21.970 Telegraphy was precisely the same. 00:29:21.970 --> 00:29:25.150 You take your sheet of paper to the telegraph office. 00:29:25.150 --> 00:29:27.940 And somebody who's very skilled with their hands, 00:29:27.940 --> 00:29:31.510 or specifically with their finger, 00:29:31.510 --> 00:29:34.810 would do something that caused that piece of paper 00:29:34.810 --> 00:29:39.510 to be regenerated hundreds of miles away. 00:29:39.510 --> 00:29:42.110 Then that piece of paper got delivered. 00:29:42.110 --> 00:29:46.010 So message went from point A to point B. 00:29:46.010 --> 00:29:49.800 So telegraphy came long before Marconi. 00:29:49.800 --> 00:29:52.830 And it was a revolution in how you do communications. 00:29:52.830 --> 00:29:55.260 But it was point to point-- one person 00:29:55.260 --> 00:29:56.760 sent one message to one person. 00:29:59.980 --> 00:30:04.090 Sarnoff got his start in newspaper business. 00:30:04.090 --> 00:30:09.040 He was a Russian immigrant, impoverished, and had 00:30:09.040 --> 00:30:10.450 a newspaper route as a kid. 00:30:10.450 --> 00:30:12.430 And he was ambitious. 00:30:12.430 --> 00:30:15.440 Newspapers are broadcast. 00:30:15.440 --> 00:30:19.370 The idea in point to point and broadcast are very different. 00:30:19.370 --> 00:30:20.990 In broadcast, you're allowed to spend 00:30:20.990 --> 00:30:23.360 a fortune on the transmitter-- 00:30:23.360 --> 00:30:27.710 the printing press-- but not on the thing 00:30:27.710 --> 00:30:28.780 that the individuals get. 00:30:28.780 --> 00:30:31.450 The individuals get newsprint. 00:30:31.450 --> 00:30:34.730 So the paper and ink have to be cheap. 00:30:34.730 --> 00:30:37.580 The printing press doesn't. 00:30:37.580 --> 00:30:41.840 So that was Sarnoff's background. 00:30:41.840 --> 00:30:46.040 So he was interested as a kid in broadcast, in newspaper. 00:30:46.040 --> 00:30:50.000 But then he got his reputation working for Marconi in wireless 00:30:50.000 --> 00:30:51.110 telegraphy. 00:30:51.110 --> 00:30:54.830 Marconi, the inventor of radio, thought of a way of doing 00:30:54.830 --> 00:30:58.780 point-to-point telegraphy wirelessly via radio-- 00:30:58.780 --> 00:31:02.180 radio telegraphy. 00:31:02.180 --> 00:31:08.170 And he sold it to ships. 00:31:08.170 --> 00:31:11.980 And Sarnoff made his reputation, because he 00:31:11.980 --> 00:31:15.640 was the guy operating the radio telegraphy 00:31:15.640 --> 00:31:21.990 system at that Marconi company when the Titanic sank. 00:31:21.990 --> 00:31:24.540 Everybody know about the Titanic, right? 00:31:24.540 --> 00:31:25.440 Big ship sank. 00:31:33.255 --> 00:31:38.040 So Sarnoff was known as an amazing telegraphy operator. 00:31:38.040 --> 00:31:42.570 And he stayed at the station for 72 consecutive hours getting 00:31:42.570 --> 00:31:44.760 emergency messages from the Titanic, 00:31:44.760 --> 00:31:47.340 telling everybody everything that he could, 00:31:47.340 --> 00:31:50.040 trying to tell them the situation, whose family was 00:31:50.040 --> 00:31:52.380 in good shape, whose family was not in good shape. 00:31:52.380 --> 00:31:55.170 It had an enormous impact-- 00:31:55.170 --> 00:31:58.050 big enough that Congress made a law 00:31:58.050 --> 00:32:01.230 saying that every ship had to have wireless telegraphy. 00:32:01.230 --> 00:32:04.500 Now, that made Marconi extremely rich. 00:32:04.500 --> 00:32:08.410 And it indirectly made Sarnoff extremely rich too. 00:32:08.410 --> 00:32:13.020 Sarnoff then got very interested in extending 00:32:13.020 --> 00:32:15.330 the idea of radio, which then was 00:32:15.330 --> 00:32:19.000 point to point to broadcast. 00:32:19.000 --> 00:32:21.420 So the idea was to somehow-- 00:32:21.420 --> 00:32:24.690 he called it a radio music box. 00:32:24.690 --> 00:32:28.020 Somehow, it was supposed to be like a newspaper 00:32:28.020 --> 00:32:29.730 but at the speed of light. 00:32:29.730 --> 00:32:37.680 So the idea was to somehow make mass consumption of radio. 00:32:37.680 --> 00:32:43.560 At the time, radio was per ship, point to point. 00:32:43.560 --> 00:32:47.460 So you have the land-based station talking to ship A, 00:32:47.460 --> 00:32:48.990 or the land-based station talking 00:32:48.990 --> 00:32:51.480 to ship B, or ship A talking to ship B, 00:32:51.480 --> 00:32:53.100 but it was all point to point. 00:32:53.100 --> 00:32:57.000 Sarnoff's idea was, let's make a newspaper out of this. 00:32:57.000 --> 00:33:02.790 The key to doing that was making a cheap receiver. 00:33:02.790 --> 00:33:04.170 It's like newsprint. 00:33:04.170 --> 00:33:06.720 The printer can cost a mint. 00:33:06.720 --> 00:33:12.600 The transmitter for radio broadcast music 00:33:12.600 --> 00:33:15.299 is allowed to cost a mint, but the receivers 00:33:15.299 --> 00:33:16.590 are not allowed to cost a mint. 00:33:16.590 --> 00:33:18.600 That's like the newsprint. 00:33:18.600 --> 00:33:21.990 So the trick was to make an inexpensive receiver. 00:33:21.990 --> 00:33:25.500 The problem with making an expensive receiver is that 00:33:25.500 --> 00:33:29.670 the scheme that we just talked about, 00:33:29.670 --> 00:33:38.720 where you decode the signal by multiplying by cos omega c t-- 00:33:38.720 --> 00:33:41.910 omega c chosen to be the frequency that you want 00:33:41.910 --> 00:33:42.919 to listen to-- 00:33:42.919 --> 00:33:44.460 the problem with that-- that's called 00:33:44.460 --> 00:33:47.640 synchronous demodulation-- the problem with that 00:33:47.640 --> 00:33:51.880 is that you've got to be exactly synchronized. 00:33:51.880 --> 00:33:55.560 If you want to listen to the message at omega 2, 00:33:55.560 --> 00:34:01.050 omega c must equal omega 2, not omega 2 plus 3. 00:34:01.050 --> 00:34:04.560 So if you want to listen to a particular frequency, 00:34:04.560 --> 00:34:08.340 to a particular message, you had to have the frequency chosen 00:34:08.340 --> 00:34:10.830 to match the carrier of the message you 00:34:10.830 --> 00:34:13.110 wanted to listen to. 00:34:13.110 --> 00:34:15.239 Today, that's not so hard. 00:34:15.239 --> 00:34:17.130 The way we would make frequencies today 00:34:17.130 --> 00:34:18.810 is with crystal. 00:34:18.810 --> 00:34:23.190 Crystals are great because the frequencies are determined 00:34:23.190 --> 00:34:25.920 by the distances in crystal lattices, which 00:34:25.920 --> 00:34:30.179 are determined by nanomechanical processes very precisely. 00:34:30.179 --> 00:34:32.520 And in fact, we can make crystals with no problem 00:34:32.520 --> 00:34:35.760 with frequency resolutions of 10 to the minus seventh, 00:34:35.760 --> 00:34:37.230 even 10 to the minus eighth. 00:34:37.230 --> 00:34:42.060 So the errors are very small compared to the frequencies 00:34:42.060 --> 00:34:43.920 that we're trying to generate. 00:34:43.920 --> 00:34:47.010 Even that wouldn't be good enough. 00:34:47.010 --> 00:34:51.960 So back then-- so Sarnoff was working back in the 1800s-- 00:34:51.960 --> 00:34:53.850 back then, they couldn't possibly do 10 00:34:53.850 --> 00:34:55.147 to the minus eighth precision. 00:34:55.147 --> 00:34:56.730 They were doing something more like 10 00:34:56.730 --> 00:34:58.980 to the minus second precision. 00:34:58.980 --> 00:35:01.260 They didn't have a technology based on crystals. 00:35:05.890 --> 00:35:07.890 So it would have been impossible to match 00:35:07.890 --> 00:35:10.132 even to within a factor of 10 to the minus third. 00:35:10.132 --> 00:35:11.590 But even if they could have matched 00:35:11.590 --> 00:35:13.590 to within 10 to the minus eighth, which we could 00:35:13.590 --> 00:35:17.310 today, even that wouldn't work, because not only does 00:35:17.310 --> 00:35:21.620 the frequency have to match, the phase has to match. 00:35:21.620 --> 00:35:24.680 So if you're multiplying by cos omega c t 00:35:24.680 --> 00:35:26.660 and you want omega c to be omega 2, 00:35:26.660 --> 00:35:29.420 it better actually be the cosine of omega 2 00:35:29.420 --> 00:35:31.610 and not the sine of omega 2. 00:35:31.610 --> 00:35:34.310 Sine and cos differ by a phase. 00:35:34.310 --> 00:35:40.040 So the question is, what's the effect of phase when you're 00:35:40.040 --> 00:35:41.540 trying to demodulate a signal? 00:35:45.420 --> 00:35:47.610 So look at your neighbor. 00:35:47.610 --> 00:35:49.740 What would happen if you tried to demodulate 00:35:49.740 --> 00:35:54.870 by precisely the right frequency but you were slipped by phase? 00:36:05.270 --> 00:36:08.763 [INTERPOSING VOICES] 00:36:10.759 --> 00:36:17.745 AUDIENCE: [INAUDIBLE] over here and be much smaller at the end. 00:36:17.745 --> 00:36:21.737 [INTERPOSING VOICES] 00:36:24.232 --> 00:36:28.723 AUDIENCE: And then it's to the point [INAUDIBLE]. 00:36:28.723 --> 00:36:30.220 [INAUDIBLE] 00:36:30.220 --> 00:36:31.218 [INTERPOSING VOICES] 00:36:31.218 --> 00:36:40.200 AUDIENCE: And then if you [INAUDIBLE] 00:36:40.200 --> 00:36:45.190 [INTERPOSING VOICES] 00:37:14.910 --> 00:37:16.410 DENNIS FREEMAN: So what would happen 00:37:16.410 --> 00:37:20.340 if there is a shift between this carrier that 00:37:20.340 --> 00:37:24.600 was modulating the signal and the carrier that 00:37:24.600 --> 00:37:25.960 is demodulating the signal? 00:37:25.960 --> 00:37:27.460 What would happen if there's a phase 00:37:27.460 --> 00:37:29.730 shift of phi between those two? 00:37:29.730 --> 00:37:32.340 What's the effect of phi not being 0? 00:37:32.340 --> 00:37:34.440 Ideally, phi would be 0. 00:37:34.440 --> 00:37:35.445 Ideally, phi would be 0. 00:37:35.445 --> 00:37:36.945 What would happen if phi were not 0? 00:37:43.000 --> 00:37:46.654 You did all that talking, and you don't have-- 00:37:46.654 --> 00:37:47.154 Yes. 00:37:47.154 --> 00:37:52.130 AUDIENCE: So the low pass filter of the two will not work. 00:37:52.130 --> 00:37:53.630 It would be-- 00:37:53.630 --> 00:37:56.294 It will be lots more than what you want it 00:37:56.294 --> 00:37:58.084 to be, like 1/2 the original. 00:37:58.084 --> 00:37:59.750 DENNIS FREEMAN: So that's exactly right. 00:37:59.750 --> 00:38:02.500 So the effect of phi-- 00:38:02.500 --> 00:38:04.120 if you think about-- 00:38:04.120 --> 00:38:08.040 so now, you have cosine of some omega c t and cos omega c t 00:38:08.040 --> 00:38:09.000 plus phi. 00:38:09.000 --> 00:38:14.080 So you have cos A cos B. so that gives you 00:38:14.080 --> 00:38:16.577 the cos of the difference and the cos of the sum. 00:38:16.577 --> 00:38:18.160 The cos of the difference was supposed 00:38:18.160 --> 00:38:19.750 to be-- the difference was supposed to be 0. 00:38:19.750 --> 00:38:21.070 The cos of the difference would be 00:38:21.070 --> 00:38:22.486 the cos of zero, which would be 1, 00:38:22.486 --> 00:38:25.010 and that's where the 1/2 came from. 00:38:25.010 --> 00:38:29.170 But now, the difference isn't 0 anymore. 00:38:29.170 --> 00:38:32.440 So instead of getting 1/2 cos of 0, 00:38:32.440 --> 00:38:34.930 we get 1/2 cos of phi, which means 00:38:34.930 --> 00:38:40.270 that if phi is a constant, you just get the wrong amplitude. 00:38:40.270 --> 00:38:47.110 But if phi is a slowly varying signal, which it would be, 00:38:47.110 --> 00:38:50.882 even if you had a frequency reliability of one part in 10 00:38:50.882 --> 00:38:53.074 to the minus seventh-- 00:38:53.074 --> 00:38:55.240 one way of thinking about that would be that there's 00:38:55.240 --> 00:38:57.010 a slowly varying phase-- 00:38:57.010 --> 00:38:58.720 the effect of the slowly varying phase 00:38:58.720 --> 00:39:03.340 would be to make the amplitude vary with time. 00:39:03.340 --> 00:39:04.630 So we call that fading. 00:39:04.630 --> 00:39:07.600 And it would be a very distracting thing 00:39:07.600 --> 00:39:08.720 to have happen. 00:39:08.720 --> 00:39:11.710 So this kind of a technology would even 00:39:11.710 --> 00:39:15.190 be difficult today, when we can match frequencies very well. 00:39:15.190 --> 00:39:18.310 It was completely out of the question back in the 1800s, 00:39:18.310 --> 00:39:20.800 when they couldn't match frequencies that well. 00:39:20.800 --> 00:39:25.640 So the trick was to not only send the message 00:39:25.640 --> 00:39:27.140 but also send the carrier. 00:39:30.150 --> 00:39:34.020 So ideally, when we first talked about modulation, 00:39:34.020 --> 00:39:35.454 there is no C path. 00:39:35.454 --> 00:39:36.870 All you do is you take the signal, 00:39:36.870 --> 00:39:38.244 and you modulate it by a carrier, 00:39:38.244 --> 00:39:39.840 and you send that out on the antenna. 00:39:39.840 --> 00:39:46.320 Now, instead, add on a little bit of the carrier. 00:39:46.320 --> 00:39:51.010 That way what's in the air is the carrier and the message. 00:39:51.010 --> 00:39:53.160 So now when you receive it, somehow 00:39:53.160 --> 00:39:55.440 if you can receive the carrier, you 00:39:55.440 --> 00:39:57.720 can use the carrier to tell you information about not 00:39:57.720 --> 00:40:01.260 only the frequency but also the phase of the carrier. 00:40:01.260 --> 00:40:03.760 And you can use that then to demodulate the message. 00:40:03.760 --> 00:40:04.980 That's the idea. 00:40:04.980 --> 00:40:10.200 Notice that adding in a little bit C of the carrier 00:40:10.200 --> 00:40:16.560 is precisely the same as adding a constant to the message 00:40:16.560 --> 00:40:20.119 before you modulate-- mathematically identical. 00:40:20.119 --> 00:40:22.410 And that gives an easy way of thinking about the effect 00:40:22.410 --> 00:40:24.810 of this carrier. 00:40:24.810 --> 00:40:27.030 If you think about the message added to C 00:40:27.030 --> 00:40:35.650 and if C is big enough, you can make the message positive only. 00:40:35.650 --> 00:40:39.030 You remember in the previous illustration, 00:40:39.030 --> 00:40:41.730 every time the message went through 0, which might happen 00:40:41.730 --> 00:40:47.910 here, the modulating message went through 0 also, 00:40:47.910 --> 00:40:51.240 which means that sign changes were affected 00:40:51.240 --> 00:40:54.190 by a 180-degree phase shifts of the carrier, which 00:40:54.190 --> 00:40:55.950 was kind of a subtle thing. 00:40:55.950 --> 00:40:58.170 So now, the message appears entirely 00:40:58.170 --> 00:41:03.000 as the positive envelope of the carrier. 00:41:03.000 --> 00:41:05.280 Well, that's nice, because that makes it very easy 00:41:05.280 --> 00:41:11.080 to decode in a way that has no dependence whatever 00:41:11.080 --> 00:41:13.310 on the carrier frequency. 00:41:13.310 --> 00:41:16.390 If the carrier frequency is big enough-- 00:41:16.390 --> 00:41:19.300 if the frequency of the carrier is sufficiently larger 00:41:19.300 --> 00:41:21.490 than the maximum frequency of the message, 00:41:21.490 --> 00:41:26.380 there's a trivial way to decode such a system with a nonlinear 00:41:26.380 --> 00:41:27.830 circuit of this type. 00:41:27.830 --> 00:41:30.630 What's intended here is that you take the message that's 00:41:30.630 --> 00:41:36.190 received from the antenna, reconstruct y, which 00:41:36.190 --> 00:41:40.210 is intended to be the output message, such 00:41:40.210 --> 00:41:43.360 that if z, the signal on the antenna, 00:41:43.360 --> 00:41:49.730 exceeds the current value of the message, the diode comes on. 00:41:49.730 --> 00:41:53.200 And that makes the blue line, the decoded signal, 00:41:53.200 --> 00:41:55.420 rapidly go back up to the red line-- 00:41:55.420 --> 00:41:58.030 the thing that's coming off the antenna. 00:41:58.030 --> 00:42:01.930 But if the antenna signal shrinks below the blue line, 00:42:01.930 --> 00:42:05.250 let the blue line discharge, because there 00:42:05.250 --> 00:42:09.930 is an RC decay constant. 00:42:09.930 --> 00:42:13.190 So there's a fast attack through the diode, 00:42:13.190 --> 00:42:16.730 so that the blue quickly goes to the peak value of the red 00:42:16.730 --> 00:42:20.180 and slowly decays back toward 0. 00:42:20.180 --> 00:42:23.840 The result is that if the difference in frequency 00:42:23.840 --> 00:42:27.980 between the carrier frequency and the message frequencies 00:42:27.980 --> 00:42:30.470 is sufficiently large, you can effectively 00:42:30.470 --> 00:42:34.044 separate the blue from the red with a very simple circuit. 00:42:34.044 --> 00:42:35.335 And that's the way they do it-- 00:42:35.335 --> 00:42:39.320 or that's the way they did it in the early 1900s. 00:42:39.320 --> 00:42:41.310 But there's still a problem with that. 00:42:41.310 --> 00:42:43.760 The problem is that the messages-- 00:42:43.760 --> 00:42:47.420 audio of the type that I'm speaking, speech-- 00:42:47.420 --> 00:42:52.810 is characterized by an enormous peak-to-average ratio. 00:42:52.810 --> 00:42:56.290 The strongest pressures that are generated by speech 00:42:56.290 --> 00:43:00.030 are enormously more powerful than the average pressure 00:43:00.030 --> 00:43:01.360 that's generated in speech. 00:43:01.360 --> 00:43:04.150 You can see that in this diagram by these peaks. 00:43:04.150 --> 00:43:06.430 There are several things that generate peaks. 00:43:06.430 --> 00:43:12.550 Peaks are generated at about 60 or 70 hertz by my vocal chords. 00:43:12.550 --> 00:43:16.480 But they're also generated by my lips in plosives. 00:43:16.480 --> 00:43:19.810 When I do plosive, there's a sudden jump 00:43:19.810 --> 00:43:23.770 in the instantaneous frequency that's not there on average. 00:43:23.770 --> 00:43:26.590 And for normal speech, that ratio 00:43:26.590 --> 00:43:29.291 can be as high as 35 to 1. 00:43:29.291 --> 00:43:31.470 35 to 1, not a big deal. 00:43:31.470 --> 00:43:34.740 The problem is that power goes like the square of voltage. 00:43:34.740 --> 00:43:36.760 So 35 to 1 becomes 1,000 to 1. 00:43:36.760 --> 00:43:43.390 It takes 1,000 times more energy to code the peaks 00:43:43.390 --> 00:43:45.580 than it does to code the average. 00:43:45.580 --> 00:43:48.280 And the problem with that is that in this coding scheme 00:43:48.280 --> 00:43:51.550 that we talked about, you have to add 00:43:51.550 --> 00:43:55.720 a constant that is big enough so that the signal never 00:43:55.720 --> 00:43:57.680 goes negative. 00:43:57.680 --> 00:44:01.090 So the constant that you add has to go in proportion 00:44:01.090 --> 00:44:04.460 to the peak value. 00:44:04.460 --> 00:44:10.300 So you end up transmitting almost all of your power 00:44:10.300 --> 00:44:12.010 By the ratio of 1,000 to 1, that's 00:44:12.010 --> 00:44:13.780 the amount of power that gets used 00:44:13.780 --> 00:44:16.720 to transmit the carrier compared to the message. 00:44:16.720 --> 00:44:21.280 Well, that's a terrible scheme, if what we were trying to do 00:44:21.280 --> 00:44:23.230 is point to point. 00:44:23.230 --> 00:44:26.320 Imagine your cell phone, if you had 00:44:26.320 --> 00:44:32.080 to transmit enough power to, in the worst case, do the peaks, 00:44:32.080 --> 00:44:34.180 you would on average be transmitting power 00:44:34.180 --> 00:44:37.840 at 1,000 times the rate that you would necessarily have to. 00:44:37.840 --> 00:44:40.210 So that's OK for broadcast. 00:44:40.210 --> 00:44:43.900 So for example, WBZ broadcast radio, 00:44:43.900 --> 00:44:47.520 WBZ uses a 50-kilowatt transmitter. 00:44:47.520 --> 00:44:49.690 50 kilowatts is the amount of power 00:44:49.690 --> 00:44:53.830 that would otherwise be sufficient to generate 00:44:53.830 --> 00:44:57.950 500 100-watt light bulbs. 00:44:57.950 --> 00:44:59.200 That's a fair amount of power. 00:44:59.200 --> 00:45:02.980 Imagine the heat that comes off 500 100-watt light bulbs. 00:45:02.980 --> 00:45:05.470 That's how much power is being radiated 00:45:05.470 --> 00:45:09.460 by the antenna for WBZ. 00:45:09.460 --> 00:45:13.780 That power is not necessary to transmit the average message. 00:45:13.780 --> 00:45:18.370 It's necessary to transmit the peak message. 00:45:18.370 --> 00:45:20.350 You can imagine how long your cell phone 00:45:20.350 --> 00:45:26.210 battery would last if you were transmitting 50 kilowatts. 00:45:26.210 --> 00:45:27.820 That doesn't work. 00:45:27.820 --> 00:45:31.660 So that's how the broadcast radio 00:45:31.660 --> 00:45:33.850 takes advantage of broadcast. 00:45:33.850 --> 00:45:37.030 It makes no sense to use this coding scheme 00:45:37.030 --> 00:45:39.310 for a point-to-point system. 00:45:39.310 --> 00:45:41.740 It's fine if what you're trying to do 00:45:41.740 --> 00:45:44.470 is have one transmitter, WBZ, that 00:45:44.470 --> 00:45:46.540 services a million listeners. 00:45:46.540 --> 00:45:49.560 That's fine. 00:45:49.560 --> 00:45:51.490 The problem with this scheme for decoding 00:45:51.490 --> 00:45:55.000 is it still doesn't separate different channels. 00:45:55.000 --> 00:45:59.290 And the way to fix that was developed by Edwin Armstrong. 00:45:59.290 --> 00:46:01.420 So Sarnoff who was kind of the visionary. 00:46:01.420 --> 00:46:04.150 He had the idea for broadcast radio. 00:46:04.150 --> 00:46:08.980 He's the entrepreneurial type, who thought of how to do this. 00:46:08.980 --> 00:46:11.090 Armstrong was the technical genius. 00:46:11.090 --> 00:46:13.600 He knew how to do it. 00:46:13.600 --> 00:46:17.290 So Armstrong's idea here, which we call superheterodyne, 00:46:17.290 --> 00:46:21.230 was let's make the signal look like it always 00:46:21.230 --> 00:46:25.830 comes from omega i regardless of what channel it comes from. 00:46:25.830 --> 00:46:28.210 So omega i, the intermediate frequency, 00:46:28.210 --> 00:46:32.260 will always take whatever frequency you're interested in 00:46:32.260 --> 00:46:36.250 and turn it into omega i and will do that 00:46:36.250 --> 00:46:38.805 by just modulating. 00:46:38.805 --> 00:46:41.250 And the cleverness had to do with a lot 00:46:41.250 --> 00:46:42.720 of technical details. 00:46:42.720 --> 00:46:46.440 He worked out a scheme where this modulation was very easy. 00:46:46.440 --> 00:46:50.460 The cutoff-- the sidebands on the bandpass filters 00:46:50.460 --> 00:46:52.680 didn't have to be very steep, which 00:46:52.680 --> 00:46:55.680 made them easy to implement. 00:46:55.680 --> 00:46:59.370 The sharp band pass filters were all at that one 00:46:59.370 --> 00:47:01.170 intermediate frequency. 00:47:01.170 --> 00:47:05.510 So he had to generate one very sharp band pass filter, 00:47:05.510 --> 00:47:10.680 but that same sharp bandpass filter could then be used 00:47:10.680 --> 00:47:13.540 for all the different channels. 00:47:13.540 --> 00:47:18.300 So the idea then was use a coarse tunable filter 00:47:18.300 --> 00:47:22.320 to map the frequency of interest to omega i, 00:47:22.320 --> 00:47:25.500 put that through a very sharp filter, of which there 00:47:25.500 --> 00:47:29.280 is exactly one in each receiver, and then use 00:47:29.280 --> 00:47:43.530 this decoding scheme to demodulate the carrier, wi. 00:47:43.530 --> 00:47:45.210 That's how they did it. 00:47:45.210 --> 00:47:48.600 We would never do it that way. 00:47:48.600 --> 00:47:50.520 That's part of the theme of the course. 00:47:50.520 --> 00:47:52.380 We are interested in schemes that 00:47:52.380 --> 00:47:55.900 let us map continuous time to discrete time, 00:47:55.900 --> 00:47:57.000 that sort of thing. 00:47:57.000 --> 00:48:02.400 So one way we might do it is implement a radio digitally. 00:48:02.400 --> 00:48:06.030 So the idea would be, what if you took the antenna, 00:48:06.030 --> 00:48:11.190 put it through a sampler, turned the radio signal, 00:48:11.190 --> 00:48:13.590 which contains a gazillion number of bands-- 00:48:13.590 --> 00:48:17.610 for commercial radio, there's 100 channels in the frequency 00:48:17.610 --> 00:48:23.250 band 500 to 600 kilohertz-- 00:48:23.250 --> 00:48:26.340 but just take the whole signal off of the antenna, 00:48:26.340 --> 00:48:29.680 turn it into a bunch of bits, run that 00:48:29.680 --> 00:48:32.890 through some digital logic that does, 00:48:32.890 --> 00:48:36.300 by magic, picks out the one that you're interested in, 00:48:36.300 --> 00:48:39.430 generates a new stream of bits, yd, 00:48:39.430 --> 00:48:41.530 from which you can do bandlimited reconstruction. 00:48:41.530 --> 00:48:44.187 So this is the last two lectures-- 00:48:44.187 --> 00:48:45.520 do you do this, how you do this. 00:48:45.520 --> 00:48:48.680 Now, all we do is we put a particular algorithm in there. 00:48:48.680 --> 00:48:50.140 And we've got a radio. 00:48:50.140 --> 00:48:50.860 That's the idea. 00:48:55.050 --> 00:48:57.250 So the key to being able to do that 00:48:57.250 --> 00:49:01.300 is whether or not you can build that sampler. 00:49:01.300 --> 00:49:06.940 So what would be the required sampling time in order 00:49:06.940 --> 00:49:10.000 to make a digital radio? 00:49:10.000 --> 00:49:13.300 And since I'm running out of time, 00:49:13.300 --> 00:49:17.130 I'll just tell you that the important thing it's sampling-- 00:49:17.130 --> 00:49:22.900 it's what we did last time-- the answer is you need T, 00:49:22.900 --> 00:49:27.100 so that the sampling frequency is at least twice 00:49:27.100 --> 00:49:29.860 the maximum frequency of the thing 00:49:29.860 --> 00:49:33.160 that you're trying to code. 00:49:33.160 --> 00:49:39.800 So the biggest frequency here is 1,600 kilohertz. 00:49:39.800 --> 00:49:44.830 You need to sample that with omega sampling more than twice 00:49:44.830 --> 00:49:46.720 that frequency-- 00:49:46.720 --> 00:49:50.920 so bigger than 2 pi 1,600 kilohertz. 00:49:50.920 --> 00:49:54.490 And if you work that out, that leaves sampling time 00:49:54.490 --> 00:49:56.350 of about 1/3 of a microsecond. 00:49:56.350 --> 00:50:00.040 And the point is that that's easy to do these days. 00:50:00.040 --> 00:50:04.860 That's the kind of part that you get from DigiKey for $2. 00:50:04.860 --> 00:50:07.390 So that's easy. 00:50:07.390 --> 00:50:09.190 So the only thing that you need to do 00:50:09.190 --> 00:50:12.940 is worry about, well, then how much computation is there? 00:50:12.940 --> 00:50:15.769 And that also turns out to be easy. 00:50:15.769 --> 00:50:18.310 The principal thing you need to do is make a bandpass filter. 00:50:21.130 --> 00:50:25.210 The question is, how would you make a bandpass filter? 00:50:25.210 --> 00:50:28.330 And here are three possible systems 00:50:28.330 --> 00:50:30.850 for making a bandpass filter. 00:50:30.850 --> 00:50:37.450 Should I take my digitized antenna signal modulate 00:50:37.450 --> 00:50:43.320 low pass modulate, or modulate low pass modulate, modulate 00:50:43.320 --> 00:50:47.270 low pass modulate, cosine, sine, cosine, sine, 00:50:47.270 --> 00:50:52.700 or put it through a filter that looks just like that unit 00:50:52.700 --> 00:50:54.650 sample response, except that it's 00:50:54.650 --> 00:50:56.630 multiplied by cos omega c t n? 00:50:59.430 --> 00:51:03.050 Some number of those work. 00:51:03.050 --> 00:51:08.324 And I'll leave it for you to figure out which of those work. 00:51:08.324 --> 00:51:08.990 Good to see you. 00:51:08.990 --> 00:51:11.140 Have a good day.