WEBVTT 00:00:00.000 --> 00:00:02.490 The following content is provided under a Creative 00:00:02.490 --> 00:00:04.059 Commons license. 00:00:04.059 --> 00:00:06.360 Your support will help MIT OpenCourseWare 00:00:06.360 --> 00:00:10.720 continue to offer high quality educational resources for free. 00:00:10.720 --> 00:00:13.350 To make a donation or view additional materials 00:00:13.350 --> 00:00:17.290 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.290 --> 00:00:18.294 at ocw.mit.edu. 00:00:26.493 --> 00:00:27.410 HARI BALAKRISHNAN: Hi. 00:00:27.410 --> 00:00:28.670 My name is Hari Balakrishnan. 00:00:28.670 --> 00:00:30.380 I'm your replacement professor. 00:00:30.380 --> 00:00:32.240 You guys know about replacement referees? 00:00:32.240 --> 00:00:32.843 AUDIENCE: Yes. 00:00:32.843 --> 00:00:33.760 HARI BALAKRISHNAN: No. 00:00:33.760 --> 00:00:36.080 So I'm the other professor in the class. 00:00:36.080 --> 00:00:39.910 And I generally start with materials that's a little bit 00:00:39.910 --> 00:00:40.410 later. 00:00:40.410 --> 00:00:41.840 But what we thought we'd do today 00:00:41.840 --> 00:00:44.720 is to talk about this Audiocom library. 00:00:44.720 --> 00:00:46.220 So before I get started and show you 00:00:46.220 --> 00:00:49.550 how to use this and all the different ways in which things 00:00:49.550 --> 00:00:52.190 will break because it's something the real world, 00:00:52.190 --> 00:00:53.550 let me see a show of hands. 00:00:53.550 --> 00:00:56.720 How many people have tried to install or run or do something 00:00:56.720 --> 00:00:58.230 with it? 00:00:58.230 --> 00:01:00.450 OK, how many people don't have any idea 00:01:00.450 --> 00:01:02.470 of what I'm talking about? 00:01:02.470 --> 00:01:03.290 OK. 00:01:03.290 --> 00:01:05.770 And how many people have managed to get something to work? 00:01:05.770 --> 00:01:07.940 By "work," it's if you got the preamble to decode, 00:01:07.940 --> 00:01:09.880 it means something is working. 00:01:09.880 --> 00:01:13.110 All right how many people have got nothing? 00:01:13.110 --> 00:01:14.580 OK. 00:01:14.580 --> 00:01:16.050 Well, it'll all work. 00:01:16.050 --> 00:01:19.830 It's just a question of figuring out how to get it to work. 00:01:19.830 --> 00:01:21.185 OK, so here's the story. 00:01:25.640 --> 00:01:27.980 In order to do something practical with all the theory 00:01:27.980 --> 00:01:29.772 that you're learning, it's important to try 00:01:29.772 --> 00:01:31.160 to implement something. 00:01:31.160 --> 00:01:34.307 Now, the theory we're learning applies to a wide range 00:01:34.307 --> 00:01:35.390 of communication channels. 00:01:35.390 --> 00:01:38.210 It applies to radio. 00:01:38.210 --> 00:01:43.580 It applies to wired links-- you know, ethernets or cables. 00:01:43.580 --> 00:01:49.460 It applies to optical links like infrared or free-space 00:01:49.460 --> 00:01:50.990 optical links. 00:01:50.990 --> 00:01:53.420 And it also applies to audio. 00:01:53.420 --> 00:01:56.150 And we've decided to use audio as the vehicle 00:01:56.150 --> 00:01:59.730 to bring these ideas into the lab for two reasons. 00:01:59.730 --> 00:02:02.480 The first and most important reason 00:02:02.480 --> 00:02:05.380 is that it turns out it's the easiest piece of hardware. 00:02:05.380 --> 00:02:07.130 It's the most convenient piece of hardware 00:02:07.130 --> 00:02:09.181 that everybody has access to. 00:02:09.181 --> 00:02:10.639 All you actually need is a computer 00:02:10.639 --> 00:02:12.620 with a microphone and a speaker. 00:02:12.620 --> 00:02:15.470 And in fact, I would imagine that in the next couple 00:02:15.470 --> 00:02:17.550 of years, it'll run on this thing too. 00:02:17.550 --> 00:02:20.330 So you could write apps on this that will make this work. 00:02:20.330 --> 00:02:24.380 Already, we have stuff working on an Android and iPhone 00:02:24.380 --> 00:02:26.518 to do much of the stuff. 00:02:26.518 --> 00:02:28.310 So the first one is just-- it's everywhere. 00:02:28.310 --> 00:02:29.120 Everybody has it. 00:02:29.120 --> 00:02:31.660 You don't have to do anything special to get it. 00:02:31.660 --> 00:02:34.160 The second is that it turns out it actually illustrates many 00:02:34.160 --> 00:02:35.910 of the problems that we're talking about-- 00:02:35.910 --> 00:02:42.280 noise and this idea of how linear time-invariant systems-- 00:02:42.280 --> 00:02:44.330 an understanding of linear time-invariant systems 00:02:44.330 --> 00:02:49.520 applies to what we're trying to build to communication channels 00:02:49.520 --> 00:02:53.030 and how you use the ideas you've learned about LTI systems 00:02:53.030 --> 00:02:56.308 to improve the performance of whatever you implement. 00:02:59.060 --> 00:03:01.920 I'm going to restrict myself to this Audiocom system, 00:03:01.920 --> 00:03:04.490 but these ideas apply across the board. 00:03:04.490 --> 00:03:07.988 So let me tell you how this system does its job. 00:03:07.988 --> 00:03:10.030 And this goes back to what Professor Verghese was 00:03:10.030 --> 00:03:11.920 telling you before. 00:03:11.920 --> 00:03:13.900 Ultimately, all of the information 00:03:13.900 --> 00:03:17.720 is modulated on top of a carrier. 00:03:17.720 --> 00:03:21.939 And a carrier, as you know, are sinusoid. 00:03:29.763 --> 00:03:36.310 So time goes this way, and this is the amplitude, or which 00:03:36.310 --> 00:03:38.303 we also call the voltage. 00:03:38.303 --> 00:03:39.720 So it's a time-invariant waveform. 00:03:43.990 --> 00:03:46.440 What your receiver is capable-- so 00:03:46.440 --> 00:03:48.490 the transmitter is capable of generating 00:03:48.490 --> 00:03:51.410 these kinds of waveforms at different frequencies. 00:03:51.410 --> 00:03:59.860 So this might be, for example, 1 kilohertz or 1,000 hertz. 00:03:59.860 --> 00:04:01.240 We can go up to-- 00:04:01.240 --> 00:04:04.420 I've got mine to work up to 20 kilohertz, which is fortunate 00:04:04.420 --> 00:04:06.820 because I can't hear it. 00:04:06.820 --> 00:04:10.795 And there's a wide range of frequencies this could work in. 00:04:10.795 --> 00:04:12.700 In the lab, I found that things tend 00:04:12.700 --> 00:04:15.360 to work between 1 kilohertz and only about 3 or 4 kilohertz. 00:04:15.360 --> 00:04:17.829 So that's the range we're talking about. 00:04:17.829 --> 00:04:22.350 So you pick a carrier waveform for your transmission. 00:04:22.350 --> 00:04:24.730 What the receiver's capable of doing is-- 00:04:24.730 --> 00:04:25.910 it's a digital receiver. 00:04:25.910 --> 00:04:29.140 So what it's capable of doing is receiving data, 00:04:29.140 --> 00:04:32.020 receiving signals from the sound card at a certain number 00:04:32.020 --> 00:04:34.090 of samples per second. 00:04:34.090 --> 00:04:37.540 We're going to call that the sampling rate. 00:04:37.540 --> 00:04:39.660 We've used this term before. 00:04:39.660 --> 00:04:46.060 The sampling rate can be anything. 00:04:46.060 --> 00:04:47.890 By default in the system, we've picked 00:04:47.890 --> 00:04:52.570 the highest possible sampling rate of 48 kilohertz. 00:04:52.570 --> 00:04:54.670 In some cases, you might have to yank it down. 00:04:54.670 --> 00:04:56.980 You might have to go as low as 8 kilohertz. 00:04:56.980 --> 00:04:58.840 But 48 kilohertz seems to work in the lab. 00:04:58.840 --> 00:05:02.493 It works on a bunch of machines that I've looked at. 00:05:02.493 --> 00:05:03.410 So we have two things. 00:05:03.410 --> 00:05:06.720 We have the 1 kilohertz, which is the carrier waveform. 00:05:06.720 --> 00:05:09.020 We'll use the notation FC for that. 00:05:09.020 --> 00:05:11.980 And we can receive samples at 48 kilohertz. 00:05:11.980 --> 00:05:13.480 And that's how the transmitter also, 00:05:13.480 --> 00:05:16.270 when it wants to send stuff on the air, 00:05:16.270 --> 00:05:17.400 it picks a sampling rate. 00:05:17.400 --> 00:05:18.733 It's going to pick 48 kilohertz. 00:05:18.733 --> 00:05:22.210 The receiver and sender need to agree on the sampling rate. 00:05:22.210 --> 00:05:24.052 What happens then is very straightforward. 00:05:24.052 --> 00:05:25.510 If you have a carrier waveform that 00:05:25.510 --> 00:05:27.730 looks like this, what 1 kilohertz means 00:05:27.730 --> 00:05:30.940 is that we do the cycle 1,000 times a second, which 00:05:30.940 --> 00:05:36.520 means that one period of the cycle is 1/1000th of a second, 00:05:36.520 --> 00:05:37.773 or 1 millisecond. 00:05:42.980 --> 00:05:47.430 Now, if I sample this at 48,000, or 48 kilohertz, 00:05:47.430 --> 00:05:51.190 or 48,000 times a second, in one of these periods, 00:05:51.190 --> 00:05:53.330 how many samples do I get? 00:05:53.330 --> 00:05:54.865 AUDIENCE: [INAUDIBLE] 00:05:54.865 --> 00:05:57.490 HARI BALAKRISHNAN: This is where you have to tell me something. 00:06:00.920 --> 00:06:01.713 Sorry? 00:06:01.713 --> 00:06:02.640 AUDIENCE: 48. 00:06:02.640 --> 00:06:03.880 HARI BALAKRISHNAN: 48. 00:06:03.880 --> 00:06:04.380 Great. 00:06:04.380 --> 00:06:06.320 So I sample at 48,000 times a second. 00:06:06.320 --> 00:06:09.150 Each of these things is 1/1000th of a second, so I get 48. 00:06:09.150 --> 00:06:12.450 What that means is that we pick a sample here. 00:06:12.450 --> 00:06:13.440 We pick a sample here. 00:06:13.440 --> 00:06:15.360 We pick a sample here. 00:06:15.360 --> 00:06:20.640 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and we keep going. 00:06:20.640 --> 00:06:22.560 12, 12, et cetera. 00:06:22.560 --> 00:06:25.290 So we get 48, OK? 00:06:25.290 --> 00:06:29.980 So if I were to send a pure sinusoid at the receiver-- 00:06:29.980 --> 00:06:32.730 and the way the sinusoid is transmitted 00:06:32.730 --> 00:06:36.570 is by providing these samples, each sample 00:06:36.570 --> 00:06:39.270 being equal to this voltage value picked from this curve. 00:06:39.270 --> 00:06:40.890 And I send each of those samples, 00:06:40.890 --> 00:06:42.810 and each of those in our implementation 00:06:42.810 --> 00:06:45.030 is a floating-point number, OK? 00:06:45.030 --> 00:06:46.740 Like a 32-bit floating-point number. 00:06:46.740 --> 00:06:48.000 I pick a number here, 0. 00:06:48.000 --> 00:06:49.542 And then I pick something here, which 00:06:49.542 --> 00:06:52.445 is the value of the sinusoid at this point in time, 00:06:52.445 --> 00:06:54.570 and this, and this, and this, and all the way down. 00:06:54.570 --> 00:06:56.190 And I send those samples. 00:06:56.190 --> 00:06:58.740 The receiver is going to be listening at 48 kilohertz 00:06:58.740 --> 00:07:01.320 and picking up whatever it's getting on the audio channel. 00:07:01.320 --> 00:07:04.040 And that's what it's going to be assuming. 00:07:04.040 --> 00:07:05.790 That's what it's going to assume was sent. 00:07:05.790 --> 00:07:07.207 Of course, in the real world, if I 00:07:07.207 --> 00:07:08.850 send at 1 volt, what might be received 00:07:08.850 --> 00:07:11.220 is 0.01 volts, or 0.2 volts, or whatever, 00:07:11.220 --> 00:07:16.950 because signals may, in fact, lose amplitude 00:07:16.950 --> 00:07:19.150 as you transmit them over the air. 00:07:19.150 --> 00:07:20.880 So that's the overall context. 00:07:20.880 --> 00:07:23.625 I'm going to start by showing you a few things here. 00:07:27.430 --> 00:07:31.690 The first thing I'll show is this grapher program 00:07:31.690 --> 00:07:33.290 that actually is part of the package. 00:07:33.290 --> 00:07:36.330 But you need some GTK toolkits for this to work. 00:07:36.330 --> 00:07:40.990 So you need this for the lab, but this is just a useful test. 00:07:40.990 --> 00:07:43.010 If I transmit a waveform, this program 00:07:43.010 --> 00:07:46.580 assumes that I'm transmitting data at 1 kilohertz. 00:07:46.580 --> 00:07:50.390 If I send data at 1 kilohertz-- and this is a pure sinusoid. 00:07:50.390 --> 00:07:53.000 Let me do that I'll explain its parameters in a bit. 00:08:01.347 --> 00:08:02.329 [TONE PLAYS] 00:08:02.329 --> 00:08:08.050 You can see that what's going on here is if I'm silent, 00:08:08.050 --> 00:08:10.660 the points get demodulated at plus 1 00:08:10.660 --> 00:08:12.490 and minus 1 that correspond to the top 00:08:12.490 --> 00:08:14.510 and the bottom of the waveform. 00:08:14.510 --> 00:08:17.510 [CLAPPING] 00:08:17.510 --> 00:08:21.042 And if I make a noise, the effect of noise get visible. 00:08:21.042 --> 00:08:23.500 And in fact, you can see that sometimes the guy on the left 00:08:23.500 --> 00:08:26.315 goes to the right, and vice versa, OK? 00:08:26.315 --> 00:08:27.940 So that's actually a very visual effect 00:08:27.940 --> 00:08:29.607 of what noise does to your transmission. 00:08:29.607 --> 00:08:31.120 I'm going to show this again. 00:08:31.120 --> 00:08:33.700 I'm going to do this again so you can see what's going on. 00:08:33.700 --> 00:08:35.330 As the amount of noise increases, 00:08:35.330 --> 00:08:36.789 you can see that-- you can imagine that what's 00:08:36.789 --> 00:08:38.372 going on here in the signal-- and this 00:08:38.372 --> 00:08:40.120 is called bipolar signaling, where 00:08:40.120 --> 00:08:45.520 you might send bit 0 as a particular kind of sinusoid, 00:08:45.520 --> 00:08:48.220 and a bit 1 at the other way, as the other kind. 00:08:48.220 --> 00:08:50.617 So the idea would be that the 0's get mapped to the left. 00:08:50.617 --> 00:08:51.950 The 1's get mapped to the right. 00:08:51.950 --> 00:08:53.200 And as we increase the amount of noise, 00:08:53.200 --> 00:08:55.180 you can start to see that they cross over, 00:08:55.180 --> 00:08:58.263 and we'd be making errors in our estimation. 00:09:03.939 --> 00:09:04.890 [TONE PLAYS] 00:09:04.890 --> 00:09:06.200 So things are working fine now. 00:09:06.200 --> 00:09:10.280 But as I [CLAPS] make noise, or as you start talking, 00:09:10.280 --> 00:09:11.922 and you create additional noise, we're 00:09:11.922 --> 00:09:13.880 going to start to see the effects of the noise. 00:09:13.880 --> 00:09:16.250 And this is annoying, so I'm going to turn it off. 00:09:16.250 --> 00:09:16.760 OK? 00:09:16.760 --> 00:09:19.460 So that's the first order way in which you determine 00:09:19.460 --> 00:09:21.020 that things are working. 00:09:21.020 --> 00:09:23.455 You don't need to use the graphing program to do that. 00:09:23.455 --> 00:09:25.205 And this is actually annoying to the eyes, 00:09:25.205 --> 00:09:27.980 so let me turn that off too. 00:09:27.980 --> 00:09:30.740 So you run, and the documentation we've given you 00:09:30.740 --> 00:09:34.300 tells you how you go about step by step trying to debug this. 00:09:34.300 --> 00:09:36.730 So let me tell you what these options mean. 00:09:36.730 --> 00:09:38.762 It's all written up, so you don't 00:09:38.762 --> 00:09:39.970 have to take notes right now. 00:09:39.970 --> 00:09:42.490 Everything is there. 00:09:42.490 --> 00:09:45.850 In that command line, the dash little 00:09:45.850 --> 00:09:48.940 s 256 refers to the number of samples per bit. 00:09:48.940 --> 00:09:51.050 I'll get to that point in a second. 00:09:51.050 --> 00:09:54.550 The dash capital S 1 means that we send-- 00:09:54.550 --> 00:09:56.230 capital S is the kind of source. 00:09:56.230 --> 00:09:59.050 You know, it's something about the nature of the source. 00:09:59.050 --> 00:10:01.450 Capital S 1 means that my entire information 00:10:01.450 --> 00:10:03.940 is sent as 1's, which means that I'm sending 00:10:03.940 --> 00:10:05.320 a pure sinusoidal carrier. 00:10:05.320 --> 00:10:08.636 All I'm doing is sending the sinusoidal carrier. 00:10:08.636 --> 00:10:10.930 Dash g just means, show me some graphs. 00:10:10.930 --> 00:10:14.530 And n refers to the number of bits I wish to send, OK? 00:10:14.530 --> 00:10:16.500 So these are all documented, and you 00:10:16.500 --> 00:10:18.050 don't have to worry about it. 00:10:18.050 --> 00:10:21.430 So what we're going to do now is actually to show you 00:10:21.430 --> 00:10:26.450 what happens when we transmit some data. 00:10:26.450 --> 00:10:28.450 This is actually the very first task in the lab. 00:10:28.450 --> 00:10:31.148 You transmit some data, and you plot what 00:10:31.148 --> 00:10:32.440 the noise histogram looks like. 00:10:38.200 --> 00:10:39.640 [TONE PLAYS] 00:10:41.512 --> 00:10:42.970 So what we did was we sent a couple 00:10:42.970 --> 00:10:48.200 hundred bits at some appropriate number of samples per bit. 00:10:48.200 --> 00:10:50.200 You know, the way you tell if stuff's working is 00:10:50.200 --> 00:10:51.350 that line there. 00:10:51.350 --> 00:10:54.790 If it received the preamble-- and there's this long preamble. 00:10:54.790 --> 00:10:56.590 I'll explain what that word means. 00:10:56.590 --> 00:10:59.980 But if you receive the preamble, it means that stuff's working. 00:10:59.980 --> 00:11:02.697 If you didn't get the preamble, it means that-- 00:11:02.697 --> 00:11:04.780 it doesn't mean that things are completely broken. 00:11:04.780 --> 00:11:07.660 It just means you got to do a little bit more work. 00:11:07.660 --> 00:11:09.760 We also plot out the signal-to-noise ratio 00:11:09.760 --> 00:11:10.830 of the transmission. 00:11:10.830 --> 00:11:11.830 And that shows up there. 00:11:11.830 --> 00:11:16.780 If that signal-to-noise ratio is something like 20 dB or 15 dB, 00:11:16.780 --> 00:11:18.280 things are fine. 00:11:18.280 --> 00:11:21.340 This system doesn't really work below 15 or 10 decibels. 00:11:21.340 --> 00:11:23.470 It means that the signal-to-noise ratio 00:11:23.470 --> 00:11:25.000 is too low, which means you either 00:11:25.000 --> 00:11:26.625 have to yank up the volume, or you have 00:11:26.625 --> 00:11:28.030 to go to a quieter location. 00:11:28.030 --> 00:11:30.070 Or-- and I'll explain this-- you have 00:11:30.070 --> 00:11:32.890 to change some of the parameters in the program. 00:11:32.890 --> 00:11:36.050 Now, what does this graph look like? 00:11:36.050 --> 00:11:38.990 We'll ignore this for a second. 00:11:38.990 --> 00:11:42.260 For this number of samples, that was the normal distribution. 00:11:42.260 --> 00:11:45.817 At 0.44, you could look at it, and it sort of looks Gaussian. 00:11:45.817 --> 00:11:46.900 It's not quite a Gaussian. 00:11:46.900 --> 00:11:48.970 I'll explain why in a little bit. 00:11:48.970 --> 00:11:51.730 What this picture show is this is what the noise-- 00:11:51.730 --> 00:11:54.085 this is what the received samples look like. 00:11:54.085 --> 00:11:58.850 And it has a mean value that's in the center. 00:11:58.850 --> 00:12:02.740 And then it's kind of got a shape on both sides. 00:12:02.740 --> 00:12:06.610 This picture in the middle are the samples post-demodulation. 00:12:06.610 --> 00:12:08.860 It's actually want you received at the receiver 00:12:08.860 --> 00:12:11.710 after we ran the demodulation step. 00:12:11.710 --> 00:12:13.300 And the stuff on top-- 00:12:13.300 --> 00:12:15.820 the blue refers to what was transmitted, which in this case 00:12:15.820 --> 00:12:17.350 was a pure carrier. 00:12:17.350 --> 00:12:19.460 The green shows what was received, 00:12:19.460 --> 00:12:22.210 which is some noisy version of the carrier, 00:12:22.210 --> 00:12:24.820 because we didn't send information other than just 00:12:24.820 --> 00:12:27.340 the carrier. 00:12:27.340 --> 00:12:29.030 This is effectively the first lab task. 00:12:29.030 --> 00:12:31.238 We have to do nothing more than run this a few times. 00:12:31.238 --> 00:12:34.870 And most of the first task in the lab, the first two 00:12:34.870 --> 00:12:38.140 tasks in the lab is just making sure that the stuff works. 00:12:40.940 --> 00:12:43.310 Now, this entire system-- 00:12:43.310 --> 00:12:45.300 you know, everything works if this thing 00:12:45.300 --> 00:12:46.610 called the preamble is decoded. 00:12:46.610 --> 00:12:47.652 So what is this preamble? 00:12:47.652 --> 00:12:49.315 And why do we need it? 00:12:49.315 --> 00:12:50.690 Well, the problem is that, as you 00:12:50.690 --> 00:12:55.790 saw in this graphing receiver when I showed this to you, 00:12:55.790 --> 00:12:58.500 the audio hardware is going to be-- 00:12:58.500 --> 00:13:00.278 what it's doing is it's always listening. 00:13:00.278 --> 00:13:01.820 When you run the program, it's always 00:13:01.820 --> 00:13:03.440 listening on the channel. 00:13:03.440 --> 00:13:05.492 So it's getting data. 00:13:05.492 --> 00:13:06.950 Even when I'm not sending anything, 00:13:06.950 --> 00:13:08.760 it's getting something. 00:13:08.760 --> 00:13:11.240 There's always something on the audio channel. 00:13:11.240 --> 00:13:13.490 So the question is, how does the receiver 00:13:13.490 --> 00:13:15.220 know that what it's receiving is part 00:13:15.220 --> 00:13:17.690 of a legitimate transmission? 00:13:17.690 --> 00:13:19.320 All communication systems need to solve 00:13:19.320 --> 00:13:21.320 this problem of synchronizing between the sender 00:13:21.320 --> 00:13:23.840 and receiver, and that's done using the preamble. 00:13:23.840 --> 00:13:28.010 The preamble is nothing more than a well-known bit sequence, 00:13:28.010 --> 00:13:30.080 sequence of bits, that the sender and receiver 00:13:30.080 --> 00:13:31.372 both agree on. 00:13:31.372 --> 00:13:33.830 So in this case, our preamble is that long sequence there-- 00:13:33.830 --> 00:13:37.000 1 0 1 1 0 1 1 1, et cetera. 00:13:37.000 --> 00:13:39.200 And there's some guidelines and rules of thumb 00:13:39.200 --> 00:13:41.390 that go into what makes a preamble good 00:13:41.390 --> 00:13:42.950 and what makes it not so good. 00:13:42.950 --> 00:13:44.658 We're not going to worry about that here. 00:13:44.658 --> 00:13:46.430 I'm just telling you that's the preamble. 00:13:46.430 --> 00:13:48.440 So as long as the receiver is successfully 00:13:48.440 --> 00:13:50.533 able to decode the preamble, it means 00:13:50.533 --> 00:13:52.700 that it knows that there's a legitimate transmission 00:13:52.700 --> 00:13:55.590 over the air, and it can start listening to it. 00:13:55.590 --> 00:13:57.710 So if you don't get the preamble, 00:13:57.710 --> 00:14:01.300 all bets are off as to what the heck's going on on the channel. 00:14:01.300 --> 00:14:04.670 And sometimes, you may not get the preamble. 00:14:04.670 --> 00:14:07.630 Now, one of the things you would have to do, 00:14:07.630 --> 00:14:14.210 you would want to do, is when don't get a preamble, usually, 00:14:14.210 --> 00:14:16.910 it's a sign that either some samples are 00:14:16.910 --> 00:14:20.180 being lost because the audio hardware isn't 00:14:20.180 --> 00:14:21.350 able to cope with it. 00:14:21.350 --> 00:14:25.130 And the documentation describes how you deal with that problem. 00:14:25.130 --> 00:14:27.290 Or it means that the sampling rate is too high, 00:14:27.290 --> 00:14:30.078 and maybe something is running on your computer that's-- 00:14:30.078 --> 00:14:30.620 I don't know. 00:14:30.620 --> 00:14:32.578 You have some video going on in the background, 00:14:32.578 --> 00:14:35.885 and it's not keeping up to read at 48,000 hertz. 00:14:35.885 --> 00:14:37.760 Or it's a sign that the signal-to-noise ratio 00:14:37.760 --> 00:14:38.523 is too low. 00:14:38.523 --> 00:14:40.190 Maybe you're in a very crowded location, 00:14:40.190 --> 00:14:42.177 or maybe the volume is too low. 00:14:42.177 --> 00:14:43.760 So those are usually the ways in which 00:14:43.760 --> 00:14:48.690 you go about fixing this problem with the preamble. 00:14:48.690 --> 00:14:50.960 Now I want to show you one more noise graph. 00:14:50.960 --> 00:14:54.550 What I'd like you to do now is actually make-- 00:14:54.550 --> 00:14:57.972 we're going to try to do this with some music on the side. 00:14:57.972 --> 00:14:59.930 What I want to show you is that as you increase 00:14:59.930 --> 00:15:01.670 the amount of noise, the noise is 00:15:01.670 --> 00:15:05.500 captured in the variance of the Gaussian distribution. 00:15:05.500 --> 00:15:07.870 The more the noise, the bigger the variance 00:15:07.870 --> 00:15:09.090 in the distribution. 00:15:09.090 --> 00:15:12.370 So I'm going to try to do this by sending 1,000 bits. 00:15:12.370 --> 00:15:16.520 So this will take a little bit of time. 00:15:16.520 --> 00:15:18.318 So let's go send it without-- 00:15:18.318 --> 00:15:21.734 [TONE PLAYS] 00:15:24.627 --> 00:15:26.710 So we'll send this without a huge amount of noise. 00:15:26.710 --> 00:15:29.705 I mean, there's some ambient noise in the room. 00:15:29.705 --> 00:15:31.330 And this will take a little bit of time 00:15:31.330 --> 00:15:36.285 to demodulate and get working. 00:15:36.285 --> 00:15:37.980 There's a couple of errors. 00:15:37.980 --> 00:15:40.840 The bit error rate is 0.002 bits, so we had an error here. 00:15:43.698 --> 00:15:44.990 And we saw something like this. 00:15:44.990 --> 00:15:46.480 It isn't quite a Gaussian in this case. 00:15:46.480 --> 00:15:47.772 But we saw something like that. 00:15:51.290 --> 00:15:58.100 Now, what I'd like you to do is kind of start-- 00:15:58.100 --> 00:16:00.500 when I say yes, just start making some noise. 00:16:00.500 --> 00:16:02.840 Just clap, or just talk, or whatever. 00:16:02.840 --> 00:16:04.620 Just turn on your phones. 00:16:04.620 --> 00:16:05.170 Do something. 00:16:05.170 --> 00:16:07.162 [CHUCKLING] 00:16:08.158 --> 00:16:09.121 [TONE PLAYS] 00:16:09.121 --> 00:16:09.621 Yeah. 00:16:09.621 --> 00:16:10.579 [CHEERING AND APPLAUSE] 00:16:10.579 --> 00:16:12.447 All right, let's see what-- 00:16:16.690 --> 00:16:19.320 All right, we're done. 00:16:19.320 --> 00:16:19.820 All right. 00:16:19.820 --> 00:16:21.218 [LAUGHTER] 00:16:21.218 --> 00:16:22.150 AUDIENCE: I had to. 00:16:22.150 --> 00:16:23.988 [CHUCKLING] 00:16:23.988 --> 00:16:26.030 HARI BALAKRISHNAN: I was actually going to play-- 00:16:26.030 --> 00:16:27.980 I found some nice YouTube clips of a music 00:16:27.980 --> 00:16:30.520 group, one of the MIT a cappella music groups. 00:16:30.520 --> 00:16:33.090 OK, we really got toasted here because you guys started 00:16:33.090 --> 00:16:35.530 talking, and we couldn't recover the preamble. 00:16:35.530 --> 00:16:36.952 But we've probably got-- 00:16:36.952 --> 00:16:38.185 AUDIENCE: [LAUGHTER] 00:16:38.185 --> 00:16:40.810 HARI BALAKRISHNAN: That was the noise distribution that we saw. 00:16:40.810 --> 00:16:43.143 And you can see that it's actually kind of a [INAUDIBLE] 00:16:43.143 --> 00:16:44.040 distribution. 00:16:44.040 --> 00:16:45.810 I'll show you what this picture is. 00:16:45.810 --> 00:16:48.210 This is a picture that essentially 00:16:48.210 --> 00:16:53.400 will become your best friend, or maybe your worst enemy. 00:16:53.400 --> 00:16:55.200 This is called an eye diagram. 00:16:55.200 --> 00:16:57.247 And it looks completely messed up. 00:16:57.247 --> 00:16:58.830 So I'm going to show you what that is. 00:16:58.830 --> 00:17:01.380 And we're going to talk about it. 00:17:01.380 --> 00:17:02.710 So let me explain to you-- 00:17:02.710 --> 00:17:03.990 but this noise was too high. 00:17:03.990 --> 00:17:05.790 And the point here is that that's shown in this picture 00:17:05.790 --> 00:17:08.490 here where we ended up with enough of a variation that we 00:17:08.490 --> 00:17:10.230 couldn't distinguish between 0's and 1's. 00:17:10.230 --> 00:17:12.020 We sent something which had a little bit 00:17:12.020 --> 00:17:13.020 of 0's in the beginning. 00:17:13.020 --> 00:17:15.270 And then the entire 1 distribution 00:17:15.270 --> 00:17:18.430 was spread between 0.1 volts to 0.9 volts. 00:17:18.430 --> 00:17:19.950 So the variance is extremely high. 00:17:22.710 --> 00:17:24.822 Now let's do one thing. 00:17:24.822 --> 00:17:26.530 I'm going to change this to just transmit 00:17:26.530 --> 00:17:29.260 random pieces of information here. 00:17:29.260 --> 00:17:31.390 When I don't give anything, it means that the data 00:17:31.390 --> 00:17:34.360 that's being sent is just a random sequence of 0's and 1's. 00:17:34.360 --> 00:17:35.950 Let me change this to 200 bits. 00:17:39.429 --> 00:17:40.920 [TONE PLAYS] 00:17:45.890 --> 00:17:47.930 All right, so we got those bits through. 00:17:47.930 --> 00:17:52.372 And that's a beautiful eye diagram. 00:17:52.372 --> 00:17:53.580 I'll explain what this means. 00:17:53.580 --> 00:17:56.030 But the point is that when you see the separation, 00:17:56.030 --> 00:17:57.680 and you see a point in the middle 00:17:57.680 --> 00:18:02.277 here with a big gap on this eye diagram graph. 00:18:02.277 --> 00:18:03.860 And then you see a separation between, 00:18:03.860 --> 00:18:05.420 these were the 0's, and those were 00:18:05.420 --> 00:18:07.520 the 1's, which means you could threshold somewhere 00:18:07.520 --> 00:18:10.550 in the middle and separate out which bits were 0's and which 00:18:10.550 --> 00:18:11.540 bits were 1. 00:18:11.540 --> 00:18:14.210 It means you're cooking things a bit, OK? 00:18:14.210 --> 00:18:19.050 So you can see that there's a distribution here 00:18:19.050 --> 00:18:21.440 of what the 1's looked like in the empirical data. 00:18:21.440 --> 00:18:24.380 There's a distribution here of what the 0's look like. 00:18:24.380 --> 00:18:26.660 This is not going to be a Gaussian at all. 00:18:26.660 --> 00:18:30.327 I'll explain why that is in a moment, OK? 00:18:30.327 --> 00:18:31.910 So I have to do two more things today, 00:18:31.910 --> 00:18:34.010 and then I'll turn it over to Professor Verghese. 00:18:34.010 --> 00:18:35.593 The first one I want to explain to you 00:18:35.593 --> 00:18:40.400 is what this eye diagram is and why it's kind of useful. 00:18:40.400 --> 00:18:42.860 And why, as you add more noise into the system, 00:18:42.860 --> 00:18:46.215 the combination of something called intersymbol interference 00:18:46.215 --> 00:18:49.580 when noise gets in the way of decoding the bits. 00:18:52.780 --> 00:18:55.750 Now, on this channel, there are two things 00:18:55.750 --> 00:18:57.770 that distort the quality of communication. 00:18:57.770 --> 00:19:00.442 The first is noise, and you kind of saw the effect of that. 00:19:00.442 --> 00:19:01.900 The second is this thing that we've 00:19:01.900 --> 00:19:06.740 been studying by modeling it as a linear time-invariant system. 00:19:06.740 --> 00:19:11.560 The idea is that when you have a sequence of 0's on the input, 00:19:11.560 --> 00:19:15.880 and then you go into a sequence of 1 samples, 00:19:15.880 --> 00:19:19.930 that sudden sharp input transition does not immediately 00:19:19.930 --> 00:19:21.610 get captured at the receiver. 00:19:21.610 --> 00:19:24.640 It takes time for the 0 to settle into a 1 and a 1 00:19:24.640 --> 00:19:25.510 to settle into a 0. 00:19:25.510 --> 00:19:28.870 And you can see that in this picture here. 00:19:28.870 --> 00:19:30.460 Now let's focus in in this picture 00:19:30.460 --> 00:19:36.380 here and look carefully at this place here. 00:19:41.540 --> 00:19:42.040 Oops. 00:19:44.640 --> 00:19:46.200 All right, let's do this again. 00:19:46.200 --> 00:19:48.550 I learned about this, like, 15 minutes ago. 00:19:48.550 --> 00:19:49.650 So bear with me. 00:20:02.320 --> 00:20:04.360 All right, so they were 0's at the bottom, 00:20:04.360 --> 00:20:06.370 and then we bumped up to 1 on the input. 00:20:06.370 --> 00:20:09.640 The input bumped from 0 to a 1. 00:20:09.640 --> 00:20:11.230 This is after we demodulate it. 00:20:11.230 --> 00:20:13.570 So if things work perfectly, you would 00:20:13.570 --> 00:20:15.670 see at the output the 0 immediately goes to a 1. 00:20:15.670 --> 00:20:17.050 But what do you actually see? 00:20:17.050 --> 00:20:18.850 You see that it takes a while for the 1 00:20:18.850 --> 00:20:20.830 to settle down, right? 00:20:20.830 --> 00:20:22.900 It goes from 0 to 1 sharply on the input, 00:20:22.900 --> 00:20:24.670 but it takes a while to go from 0 to 1. 00:20:24.670 --> 00:20:26.440 And then I go up like that. 00:20:26.440 --> 00:20:28.000 And then I want to go from 1 to 0. 00:20:28.000 --> 00:20:30.227 And you can see that as it comes down to 0, 00:20:30.227 --> 00:20:31.560 it takes a while to settle down. 00:20:35.160 --> 00:20:37.912 Do you guys all understand why that happens? 00:20:37.912 --> 00:20:40.250 Like, I don't mean the physics of why it happens. 00:20:40.250 --> 00:20:44.380 But what I mean is how that idea relates 00:20:44.380 --> 00:20:47.500 to this idea of the unit step response and the unit sample 00:20:47.500 --> 00:20:48.140 response? 00:20:48.140 --> 00:20:50.140 This is nothing more than the unit step response 00:20:50.140 --> 00:20:51.460 of this channel, right? 00:20:51.460 --> 00:20:54.370 I have a 0, and I've assumed I'm on 0 for a while. 00:20:54.370 --> 00:20:58.220 I bump up to a 1, and it takes a while for it to go from 0 to 1. 00:20:58.220 --> 00:21:00.670 And similarly, it takes a while for it to go from a 1 back 00:21:00.670 --> 00:21:02.560 to a 0. 00:21:02.560 --> 00:21:09.590 Now, suppose I end up switching 0, 1, 1, 0, 0, 1, 0, 1, 00:21:09.590 --> 00:21:11.440 0, 1 very, very quickly. 00:21:11.440 --> 00:21:14.240 And I don't give enough time for the whole thing to settle down. 00:21:14.240 --> 00:21:20.830 In other words, as I go from a 0 to a 1, 00:21:20.830 --> 00:21:23.762 before it settles down into 1, if the next bit is a 0, 00:21:23.762 --> 00:21:24.970 and then I start coming down. 00:21:24.970 --> 00:21:28.270 And before the next bit settles down to 0, I get to a 1. 00:21:28.270 --> 00:21:30.040 And before it comes back to 0-- 00:21:30.040 --> 00:21:31.370 I keep doing that. 00:21:31.370 --> 00:21:34.480 What I'm going to end up with is this combination 00:21:34.480 --> 00:21:38.140 of 0's and 1's that are sort of random start 00:21:38.140 --> 00:21:40.900 confusing the receiver because we're not giving enough time 00:21:40.900 --> 00:21:42.770 for the 0 to settle down. 00:21:42.770 --> 00:21:44.620 And similarly, when we go from 0 to 1, 00:21:44.620 --> 00:21:48.910 we're not giving enough time for the 1 to settle down. 00:21:48.910 --> 00:21:51.010 That's what this eye diagram was referring 00:21:51.010 --> 00:21:53.070 to where we found in this confused case-- 00:21:53.070 --> 00:21:55.570 and then when you have noise, some of the 0's get moved to 1 00:21:55.570 --> 00:21:57.580 anyway, and 1's get moved to 0. 00:21:57.580 --> 00:22:01.180 And we end up with this very crowded picture. 00:22:01.180 --> 00:22:02.860 The way you tackle this problem is it 00:22:02.860 --> 00:22:04.750 all has to do with the number of samples 00:22:04.750 --> 00:22:07.690 that you decide to use within 1 bit. 00:22:07.690 --> 00:22:09.820 So coming back to this picture, if I 00:22:09.820 --> 00:22:11.990 do this at 48,000 samples per second, 00:22:11.990 --> 00:22:17.530 in any one of these carriers, I have 48 samples that I use. 00:22:17.530 --> 00:22:23.140 But now I get to decide how many samples corresponds 00:22:23.140 --> 00:22:25.390 to 1 bit of information? 00:22:25.390 --> 00:22:28.060 When I have a bit that I want to transmit-- let's say a 1-- 00:22:31.860 --> 00:22:37.065 how many samples? 00:22:39.960 --> 00:22:43.410 What that effectively means is to transmit a 1, 00:22:43.410 --> 00:22:48.540 how many of these periods of this waveform 00:22:48.540 --> 00:22:51.090 do I want to use to transmit a 1? 00:22:51.090 --> 00:22:57.970 So for example, if I decide I want to send a 1 00:22:57.970 --> 00:23:00.730 as three carriers of the waveform, 00:23:00.730 --> 00:23:04.750 then in any one of these carriers, I have 48 samples. 00:23:04.750 --> 00:23:07.720 Therefore, I represent a 1 as 48 times 3, 00:23:07.720 --> 00:23:13.762 which is 144 samples per bit. 00:23:17.210 --> 00:23:20.560 Now, to transmit a 0, I could do a bunch of different things. 00:23:20.560 --> 00:23:24.520 I could decide to keep the channel silent. 00:23:24.520 --> 00:23:26.270 If I do that, it's called on-off keying. 00:23:26.270 --> 00:23:27.740 And that's what we're using here. 00:23:27.740 --> 00:23:30.910 So we're sending 1 as-- 00:23:30.910 --> 00:23:33.730 if we decide to use 144 samples per bit, 00:23:33.730 --> 00:23:37.660 then it means we represent a 1 as 124 samples, which 00:23:37.660 --> 00:23:40.110 corresponds to three of these. 00:23:40.110 --> 00:23:44.017 And then we represent 0 as nothing. 00:23:44.017 --> 00:23:45.100 So we don't send anything. 00:23:47.680 --> 00:23:49.840 If we do that, this is called on-off keying 00:23:49.840 --> 00:23:52.930 because we send on for 1, where we send a sinusoid. 00:23:52.930 --> 00:23:56.700 And for sending us a 0 bit, we send nothing. 00:23:56.700 --> 00:24:01.600 Now, if the number of samples I select per bit is too small, 00:24:01.600 --> 00:24:04.330 you end up with this effect that I don't give enough time 00:24:04.330 --> 00:24:06.240 for the 0 to settle down to-- 00:24:06.240 --> 00:24:09.130 when I make the transition from a 0 to a 1, 00:24:09.130 --> 00:24:14.350 if I pick too small samples per bit, before we settle into a 1, 00:24:14.350 --> 00:24:16.163 the next bit may show up as a 0. 00:24:16.163 --> 00:24:17.830 And then before we settle down into a 0, 00:24:17.830 --> 00:24:19.750 the next bit shows up perhaps as a 1. 00:24:19.750 --> 00:24:22.660 And we end up commingling the 0's and the 1's. 00:24:22.660 --> 00:24:24.880 And we're not able at the receiver 00:24:24.880 --> 00:24:28.510 to tell the difference between these different-volted samples 00:24:28.510 --> 00:24:29.530 after we demodulate. 00:24:29.530 --> 00:24:31.910 So we don't know what happened. 00:24:31.910 --> 00:24:34.540 So if the eye diagram is a way to capture 00:24:34.540 --> 00:24:36.950 that, what the eye diagram does is-- 00:24:36.950 --> 00:24:38.620 it's a kind of a clever hack. 00:24:38.620 --> 00:24:41.140 What it does is-- 00:24:41.140 --> 00:24:42.370 thought I had it somewhere. 00:24:42.370 --> 00:24:43.202 There it is. 00:24:46.028 --> 00:24:47.570 The way you generate an eye diagram-- 00:24:47.570 --> 00:24:49.987 you don't have to worry about writing the software for it. 00:24:49.987 --> 00:24:52.450 But you will look at a lot of these kinds of pictures 00:24:52.450 --> 00:24:55.000 in lab 5. 00:24:55.000 --> 00:24:57.630 You transmit a random sequence of bits. 00:24:57.630 --> 00:24:59.770 And then you look at the samples that 00:24:59.770 --> 00:25:02.550 were received at the receiver after demodulation was done. 00:25:07.100 --> 00:25:10.205 The output of the demodulation is a set of voltage values. 00:25:14.020 --> 00:25:18.910 What you do is you look at three bit periods. 00:25:18.910 --> 00:25:20.740 In other words, you look at a sequence 00:25:20.740 --> 00:25:23.250 of time, a number of samples that corresponds to three bit 00:25:23.250 --> 00:25:23.750 periods. 00:25:23.750 --> 00:25:27.370 So for example, if I pick 144 samples per bit, what I do 00:25:27.370 --> 00:25:32.155 is I take 144 and multiply that by 3 bit periods. 00:25:34.720 --> 00:25:37.850 And I look at that many samples. 00:25:37.850 --> 00:25:42.510 So for every three bit periods, I take all the samples, 00:25:42.510 --> 00:25:44.590 and I plot them, OK? 00:25:44.590 --> 00:25:47.660 So a particular sequence of bits-- in this case, 00:25:47.660 --> 00:25:50.080 it might be a 0 and a 1 and a 0. 00:25:50.080 --> 00:25:54.330 This is a 0 followed by a 1 followed by a 0. 00:25:54.330 --> 00:25:56.860 A different sequence of bits could be a 1 followed 00:25:56.860 --> 00:25:58.392 by 1 followed by 1. 00:25:58.392 --> 00:26:00.100 A different sequence of bits could be a 0 00:26:00.100 --> 00:26:01.840 followed by 0 followed by 0. 00:26:01.840 --> 00:26:03.730 So there are eight combinations of sequences 00:26:03.730 --> 00:26:05.020 of three bits each. 00:26:05.020 --> 00:26:05.915 So there are eight-- 00:26:05.915 --> 00:26:08.800 you have this clean eye diagram like this. 00:26:08.800 --> 00:26:12.640 You're going to see a variety of different lines corresponding 00:26:12.640 --> 00:26:16.410 to all the places where different 3-bit sequences 00:26:16.410 --> 00:26:17.900 appeared in your input. 00:26:17.900 --> 00:26:20.410 So for any given 3-bit sequence of the input, 00:26:20.410 --> 00:26:23.860 there's a 3-bit sample sequence at the output that corresponds 00:26:23.860 --> 00:26:28.120 to a number of samples equal to the samples per bit multiplied 00:26:28.120 --> 00:26:29.420 by 3. 00:26:29.420 --> 00:26:33.010 And each of those generates one of those trajectories 00:26:33.010 --> 00:26:36.430 through this picture. 00:26:36.430 --> 00:26:38.110 If you generate that picture, and you 00:26:38.110 --> 00:26:40.090 find that there's a very clean gap 00:26:40.090 --> 00:26:43.120 between all possible combinations of these bit 00:26:43.120 --> 00:26:46.420 sequences, as is in this case here, 00:26:46.420 --> 00:26:50.380 then it means that you're very likely to be able to decode. 00:26:50.380 --> 00:26:52.660 Because what you can do is, essentially, the receiver 00:26:52.660 --> 00:26:56.230 can decide that when 1's happen, it 00:26:56.230 --> 00:27:00.250 corresponds to something over here at 0.35 or 0.4 volts. 00:27:00.250 --> 00:27:02.620 When a 0 happens, it may not be a 0, 00:27:02.620 --> 00:27:04.840 but it may correspond to something like a 0.1 volt 00:27:04.840 --> 00:27:07.060 here, which means I can pick the middle point 00:27:07.060 --> 00:27:09.250 and slice it at that middle point 00:27:09.250 --> 00:27:13.192 to determine whether the received bits were 0 or 1. 00:27:13.192 --> 00:27:15.400 Now, if we were to run this experiment again and make 00:27:15.400 --> 00:27:17.350 a lot of noise-- 00:27:17.350 --> 00:27:19.630 let's try that-- what would happen 00:27:19.630 --> 00:27:22.870 is we may not be able to decode at all. 00:27:22.870 --> 00:27:25.900 So I'm going to request you guys to make a little bit of noise 00:27:25.900 --> 00:27:28.000 after I start. 00:27:28.000 --> 00:27:29.630 And then we'll see how it goes. 00:27:29.630 --> 00:27:30.661 All right, start. 00:27:30.661 --> 00:27:31.623 [WHISTLING] 00:27:31.623 --> 00:27:32.585 [TONE PLAYS] 00:27:32.585 --> 00:27:35.070 [APPLAUSE] 00:27:35.070 --> 00:27:37.570 AUDIENCE: [SCREAMS] 00:27:37.615 --> 00:27:38.990 HARI BALAKRISHNAN: You know what? 00:27:38.990 --> 00:27:40.040 It wasn't loud enough. 00:27:40.040 --> 00:27:41.350 It worked great. 00:27:41.350 --> 00:27:44.920 But let's look at what the eye diagram looks like. 00:27:44.920 --> 00:27:47.267 Well, it's a little worse than the other one, 00:27:47.267 --> 00:27:49.100 but I think the energy is down in this room. 00:27:49.100 --> 00:27:51.970 So let's do it one more time. 00:27:51.970 --> 00:27:54.880 I want to be in a position where nothing decodes. 00:27:54.880 --> 00:27:55.780 [TONE PLAYS] 00:27:55.780 --> 00:27:59.210 [SCREAMING AND APPLAUSE] 00:28:02.130 --> 00:28:05.160 Well, 15% better is, in fact, my preamble decoded, 00:28:05.160 --> 00:28:05.970 which is amazing. 00:28:05.970 --> 00:28:08.167 [INTERPOSING VOICES] 00:28:08.167 --> 00:28:10.250 HARI BALAKRISHNAN: And that's what it looked like. 00:28:10.250 --> 00:28:15.570 OK, this is an eye diagram even a mother wouldn't like. 00:28:15.570 --> 00:28:17.367 All right, so I'm going to stop here. 00:28:17.367 --> 00:28:19.200 Before I turn it over to Professor Verghese, 00:28:19.200 --> 00:28:20.623 are there any questions? 00:28:20.623 --> 00:28:21.290 Anything at all. 00:28:21.290 --> 00:28:22.748 I know some of you have had issues. 00:28:22.748 --> 00:28:25.570 And I met one or two of you this morning to fix your computers. 00:28:25.570 --> 00:28:26.970 And I'm happy to do that. 00:28:26.970 --> 00:28:28.795 I'll be in the lab from 4 o'clock. 00:28:28.795 --> 00:28:30.420 We'll get it working on your computers. 00:28:30.420 --> 00:28:31.962 If it doesn't work, then you're going 00:28:31.962 --> 00:28:34.080 to have to use the lab machines. 00:28:34.080 --> 00:28:35.920 Let's take some questions or comments. 00:28:35.920 --> 00:28:38.070 Any people have any questions about this stuff? 00:28:38.070 --> 00:28:39.780 Do people get an idea of what's going on 00:28:39.780 --> 00:28:41.070 and what do you need to do? 00:28:41.070 --> 00:28:44.346 Once you get this working, the labs sort of-- 00:28:44.346 --> 00:28:45.390 they'll write themselves. 00:28:45.390 --> 00:28:48.615 You just have to do a little bit of work. 00:28:48.615 --> 00:28:51.420 Questions? 00:28:51.420 --> 00:28:53.370 This can go up to higher frequencies too. 00:28:53.370 --> 00:28:54.967 So if the sound's annoying your ears, 00:28:54.967 --> 00:28:56.300 you can actually get it to work. 00:28:56.300 --> 00:28:58.820 I know one of you guys is trying to get this to work 00:28:58.820 --> 00:29:03.170 with ultrasonic reception. 00:29:03.170 --> 00:29:06.530 That's challenging, but we can probably help you on. 00:29:06.530 --> 00:29:07.138 Yeah. 00:29:07.138 --> 00:29:09.130 AUDIENCE: What do we get for the uniform noise 00:29:09.130 --> 00:29:12.145 since [INAUDIBLE]? 00:29:12.145 --> 00:29:13.520 HARI BALAKRISHNAN: Uniform noise? 00:29:13.520 --> 00:29:14.145 AUDIENCE: Yeah. 00:29:14.145 --> 00:29:17.795 So if we just have another tone on the other side, 00:29:17.795 --> 00:29:21.610 how do we get to [INAUDIBLE]? 00:29:21.610 --> 00:29:22.990 HARI BALAKRISHNAN: Yeah, you mean 00:29:22.990 --> 00:29:25.115 if you were to make another transmission at exactly 00:29:25.115 --> 00:29:26.110 the same frequency? 00:29:26.110 --> 00:29:28.930 The beauty of this is that there's different demodulation 00:29:28.930 --> 00:29:29.853 schemes. 00:29:29.853 --> 00:29:32.020 Right now, we're using something called the envelope 00:29:32.020 --> 00:29:35.290 demodulation, which Professor Verghese talked about before. 00:29:35.290 --> 00:29:37.780 All that's doing is it's taking the absolute value 00:29:37.780 --> 00:29:41.140 of every received sample and then running a simple averaging 00:29:41.140 --> 00:29:41.910 filter. 00:29:41.910 --> 00:29:43.520 And that's what you write in the lab. 00:29:43.520 --> 00:29:45.760 It's a very, very simple demodulation. 00:29:45.760 --> 00:29:48.200 We'll study something called quadrature demodulation 00:29:48.200 --> 00:29:51.310 in probably next lecture or the one 00:29:51.310 --> 00:29:54.370 after that, which means that'll have the property that, if you 00:29:54.370 --> 00:29:57.340 transmit at a certain frequency, and somebody else interferes 00:29:57.340 --> 00:29:59.680 and transmits at a different carrier frequency, 00:29:59.680 --> 00:30:02.710 you're still going to be able to recover your transmission. 00:30:02.710 --> 00:30:07.353 But if the other transmitter has significant signal strength 00:30:07.353 --> 00:30:09.520 in the same frequencies that you're transmitting in, 00:30:09.520 --> 00:30:12.170 then he's going to start to look like noise to you, 00:30:12.170 --> 00:30:13.460 and that's going to not work. 00:30:13.460 --> 00:30:15.418 So what happened here is when you guys were all 00:30:15.418 --> 00:30:19.150 whistling and were clapping and so on, 00:30:19.150 --> 00:30:21.580 there were signals generated across all frequencies, 00:30:21.580 --> 00:30:23.890 including the frequency at which I was transmitting. 00:30:23.890 --> 00:30:27.130 And that's what caused the signal to have noise. 00:30:27.130 --> 00:30:29.680 It's not like you were all transmitting at 1,000 hertz. 00:30:29.680 --> 00:30:32.530 It just so happens that that combination 00:30:32.530 --> 00:30:35.840 of noise you were making had a component at 1,000 hertz. 00:30:35.840 --> 00:30:37.141 Does that answer your question? 00:30:37.141 --> 00:30:38.323 AUDIENCE: Yeah. 00:30:38.323 --> 00:30:40.740 HARI BALAKRISHNAN: Any other questions, comments, remarks? 00:30:45.610 --> 00:30:46.580 OK. 00:30:46.580 --> 00:30:47.955 GEORGE VERGHESE: Yeah, we've been 00:30:47.955 --> 00:30:50.170 talking about modeling the baseband channel. 00:30:50.170 --> 00:30:52.870 And we've said we'll focus on LTI channels. 00:30:52.870 --> 00:30:55.480 So I just wanted to take advantage of this being 00:30:55.480 --> 00:30:58.840 up here to have you think about what this tells you 00:30:58.840 --> 00:31:01.330 about how close to linear this channel is, 00:31:01.330 --> 00:31:03.370 and how close to time-invariant it is. 00:31:03.370 --> 00:31:04.120 What do you think? 00:31:04.120 --> 00:31:06.830 What we're seeing here is, for instance, 00:31:06.830 --> 00:31:09.730 a step response to something that's at 0 00:31:09.730 --> 00:31:12.250 and then goes up to 1 and stays at 1. 00:31:12.250 --> 00:31:16.070 We're seeing a superposition of many such 0-to-1 transitions, 00:31:16.070 --> 00:31:21.630 some 1-to-0 transitions, later 1-to-0 transitions, and so on. 00:31:21.630 --> 00:31:23.380 So we're really looking at a superposition 00:31:23.380 --> 00:31:27.250 of step responses staggered in time and going from 0 to or 1 00:31:27.250 --> 00:31:28.870 to 0. 00:31:28.870 --> 00:31:32.450 Do you think time invariance is maybe a good assumption 00:31:32.450 --> 00:31:35.252 for this channel? 00:31:35.252 --> 00:31:35.960 Plausible, right? 00:31:35.960 --> 00:31:39.470 Because the stuff that we get here looks a lot like the stuff 00:31:39.470 --> 00:31:41.180 we're getting here. 00:31:41.180 --> 00:31:45.200 The deviations might be noise, but the more structured parts 00:31:45.200 --> 00:31:48.900 of the waveform here match the structured part of the waveform 00:31:48.900 --> 00:31:49.400 here. 00:31:49.400 --> 00:31:53.090 And what you're really seeing is the loudspeaker here 00:31:53.090 --> 00:31:54.920 reverberating through the room. 00:31:54.920 --> 00:31:57.620 You're all staying fixed, so the echoes 00:31:57.620 --> 00:31:58.910 are from fixed locations. 00:31:58.910 --> 00:32:00.470 The walls are fixed. 00:32:00.470 --> 00:32:04.340 And so what we're seeing is the step response of the room, 00:32:04.340 --> 00:32:05.330 in effect. 00:32:05.330 --> 00:32:07.610 If you hit the room a little bit later, 00:32:07.610 --> 00:32:10.550 well, you get the same response, but a little later. 00:32:10.550 --> 00:32:12.250 Does this look like it's very linear? 00:32:12.250 --> 00:32:14.880 Would linearity be a good assumption for this channel? 00:32:17.760 --> 00:32:19.740 I mean this is very partial information, 00:32:19.740 --> 00:32:22.930 but does it give you enough to judge? 00:32:22.930 --> 00:32:25.825 So why do you think linearity might be good here? 00:32:25.825 --> 00:32:27.700 Were you saying, yes, it's a good assumption? 00:32:27.700 --> 00:32:29.510 AUDIENCE: Yeah, because time is a pattern. 00:32:29.510 --> 00:32:30.760 GEORGE VERGHESE: Because what? 00:32:30.760 --> 00:32:34.696 AUDIENCE: Like, if you have a signal go five seconds 00:32:34.696 --> 00:32:37.877 from a different signal, it'll still appear in the center. 00:32:37.877 --> 00:32:40.210 GEORGE VERGHESE: But that's the time-invariance argument 00:32:40.210 --> 00:32:41.127 that you're giving me. 00:32:41.127 --> 00:32:42.680 What's the linearity argument? 00:32:42.680 --> 00:32:43.294 Yeah. 00:32:43.294 --> 00:32:45.270 AUDIENCE: When you scale the intact input, 00:32:45.270 --> 00:32:46.878 the output gets scaled accordingly. 00:32:46.878 --> 00:32:48.670 GEORGE VERGHESE: OK, so when you scale it-- 00:32:48.670 --> 00:32:50.173 so we're not really seeing too much. 00:32:50.173 --> 00:32:51.340 Well, we are seeing scaling. 00:32:51.340 --> 00:32:55.080 What kind of scaling are you seeing here? 00:32:55.080 --> 00:32:57.330 We have 0-to-1 transitions, but we also 00:32:57.330 --> 00:33:00.150 have 1-to-0 kinds of transitions, right? 00:33:00.150 --> 00:33:02.910 What would you want to see for a linear channel for how 00:33:02.910 --> 00:33:05.310 the 1-to-0 transition behaves relative 00:33:05.310 --> 00:33:07.410 to the 0-to-1 transition? 00:33:07.410 --> 00:33:10.264 What would you expect on the linear channel? 00:33:10.264 --> 00:33:10.764 Yeah. 00:33:10.764 --> 00:33:12.165 AUDIENCE: [INAUDIBLE] 00:33:12.165 --> 00:33:13.790 GEORGE VERGHESE: So from superposition, 00:33:13.790 --> 00:33:17.020 you would really expect to see the same shape on the downslope 00:33:17.020 --> 00:33:18.880 that you see on the upslope, right? 00:33:18.880 --> 00:33:21.070 So when you look at the upward transition here, 00:33:21.070 --> 00:33:22.140 you see a certain shape. 00:33:22.140 --> 00:33:26.120 Well, it's not quite matched on the downward slope here. 00:33:26.120 --> 00:33:28.900 And the reason is that the particular simple demodulation 00:33:28.900 --> 00:33:31.630 that we're using here for this on-off scheme 00:33:31.630 --> 00:33:35.652 actually makes that not look very linear for signals 00:33:35.652 --> 00:33:37.360 that are for channels that have this kind 00:33:37.360 --> 00:33:38.592 of an overshoot to them. 00:33:38.592 --> 00:33:40.300 But the other scheme that Hari mentioned, 00:33:40.300 --> 00:33:43.570 the quadrature demodulation, will actually do a lot better. 00:33:43.570 --> 00:33:44.780 We, in fact, saw that, right? 00:33:44.780 --> 00:33:52.990 We saw that there was modulation that essentially pulled out 00:33:52.990 --> 00:33:56.050 the absolute value of what you were sending. 00:33:56.050 --> 00:33:59.380 And then there was another modulation scheme that actually 00:33:59.380 --> 00:34:01.350 pulled out the signal itself. 00:34:01.350 --> 00:34:03.820 And so if you're doing things like taking absolute values 00:34:03.820 --> 00:34:05.570 somewhere in the middle there, then you're 00:34:05.570 --> 00:34:09.150 going to start losing the ability to model it as linear. 00:34:09.150 --> 00:34:12.250 OK, so linear time-invariant models 00:34:12.250 --> 00:34:14.260 are not necessarily good for all channels. 00:34:14.260 --> 00:34:17.150 It's something that you've got to look for. 00:34:17.150 --> 00:34:20.132 There are good reasons to try and structure a channel 00:34:20.132 --> 00:34:22.090 so that it's close to linear and time-invariant 00:34:22.090 --> 00:34:26.679 because then you can do a lot of analysis and design for it. 00:34:26.679 --> 00:34:28.330 OK, so I want to continue talking 00:34:28.330 --> 00:34:33.520 about our models for LTI channels. 00:34:33.520 --> 00:34:35.800 And still in the time domain, next time, 00:34:35.800 --> 00:34:38.510 we'll start to look at this in the frequency domain. 00:34:38.510 --> 00:34:42.400 Hari mentioned frequency several times here. 00:34:42.400 --> 00:34:45.880 So we're talking about an LTI channel, Linear 00:34:45.880 --> 00:34:47.440 and Time-Invariant. 00:34:47.440 --> 00:34:51.560 We talked about characterizing it by its unit sample response. 00:34:51.560 --> 00:34:52.600 And so let's see. 00:34:52.600 --> 00:34:58.760 Unit sample response means put in a unit sample function. 00:34:58.760 --> 00:35:00.280 You get out an output that you're 00:35:00.280 --> 00:35:03.040 going to call the unit sample response. 00:35:03.040 --> 00:35:06.490 Put in an input x(n). 00:35:06.490 --> 00:35:08.950 That is a summation of such things. 00:35:08.950 --> 00:35:18.330 Let's say x(k) delta of n minus k summed over all k, right? 00:35:18.330 --> 00:35:20.170 That's the general input represented 00:35:20.170 --> 00:35:22.870 as a weighted combination of unit samples. 00:35:22.870 --> 00:35:25.730 Well, what comes out in that case? 00:35:25.730 --> 00:35:26.470 What is y of n? 00:35:29.480 --> 00:35:31.730 If we're talking about a linear time-invariant system, 00:35:31.730 --> 00:35:33.400 then it's the same weighted combination 00:35:33.400 --> 00:35:36.620 of the responses to these unit samples. 00:35:36.620 --> 00:35:43.885 So we're going to get summation x(k) h of n minus k, right? 00:35:43.885 --> 00:35:46.780 So this is the convolution expression 00:35:46.780 --> 00:35:48.440 that we talked about last time. 00:35:48.440 --> 00:35:50.470 So what I want to do in the rest of the lecture 00:35:50.470 --> 00:35:53.770 is give you some other ways to think about this. 00:35:56.752 --> 00:36:01.870 Our notation for this was y of n equals 00:36:01.870 --> 00:36:08.460 x convolved with h evaluated at time n, right? 00:36:14.010 --> 00:36:16.140 Another way to think about this-- let's see. 00:36:16.140 --> 00:36:19.100 Let me actually first do an example 00:36:19.100 --> 00:36:22.780 and then give you another way to think about this. 00:36:22.780 --> 00:36:26.370 Suppose I have a system whose effect 00:36:26.370 --> 00:36:32.350 is to multiply the input by A, some number A, 00:36:32.350 --> 00:36:38.350 and delay by some number D. 00:36:38.350 --> 00:36:41.358 And I tell you that this is LTI. 00:36:41.358 --> 00:36:43.400 Actually, I don't have to tell you that it's LTI. 00:36:43.400 --> 00:36:45.320 You can prove that it's LTI. 00:36:45.320 --> 00:36:48.260 If I tell you have a system whose only action on the input 00:36:48.260 --> 00:36:51.140 is to delay the input by capital D 00:36:51.140 --> 00:36:53.540 and scale the input by capital A, 00:36:53.540 --> 00:36:56.690 you can actually prove that it satisfies time-invariance 00:36:56.690 --> 00:37:00.020 and that you can superimpose, OK? 00:37:00.020 --> 00:37:01.397 So this is LTI. 00:37:01.397 --> 00:37:02.855 So what's the unit sample response? 00:37:06.967 --> 00:37:09.050 If I put in the unit sample function of the input, 00:37:09.050 --> 00:37:09.860 what's the output? 00:37:17.918 --> 00:37:18.870 AUDIENCE: A delta n. 00:37:18.870 --> 00:37:20.060 GEORGE VERGHESE: Anyone? 00:37:20.060 --> 00:37:21.440 AUDIENCE: A delta n minus D. 00:37:21.440 --> 00:37:28.630 GEORGE VERGHESE: Yeah, A delta of n minus d, right? 00:37:28.630 --> 00:37:32.930 And if I put some general input function 00:37:32.930 --> 00:37:36.028 in here, what's the output? 00:37:36.028 --> 00:37:37.480 AUDIENCE: [INAUDIBLE] 00:37:37.480 --> 00:37:40.077 GEORGE VERGHESE: Without telling you about convolution. 00:37:40.077 --> 00:37:41.410 Somebody-- I heard a voice here. 00:37:41.410 --> 00:37:42.036 Yeah. 00:37:42.036 --> 00:37:43.994 AUDIENCE: A of s times [INAUDIBLE] D. 00:37:43.994 --> 00:37:44.952 GEORGE VERGHESE: Right. 00:37:49.280 --> 00:37:50.780 We didn't have to do any convolution 00:37:50.780 --> 00:37:54.260 to figure this out, right, because I described the system 00:37:54.260 --> 00:37:56.210 to you in a simple way. 00:37:56.210 --> 00:37:58.440 You could tell me what it does to the output. 00:37:58.440 --> 00:38:00.350 All right, so I want to give you another way 00:38:00.350 --> 00:38:07.050 to think about a system where the unit sample 00:38:07.050 --> 00:38:12.590 response of h events. 00:38:12.590 --> 00:38:16.000 I'm talking about a system with the unit sample response 00:38:16.000 --> 00:38:16.900 h of n. 00:38:20.130 --> 00:38:21.220 So what does that mean? 00:38:21.220 --> 00:38:25.020 That means that at time 0, I evaluate 0. 00:38:25.020 --> 00:38:31.300 When I put out and put in a unit sample at time 1, 00:38:31.300 --> 00:38:32.490 I get some h1. 00:38:32.490 --> 00:38:37.910 At time 2, get some h2, and so on. 00:38:37.910 --> 00:38:42.320 This is a hn, OK? 00:38:42.320 --> 00:38:49.040 So if I give you a system and tell you that the unit sample 00:38:49.040 --> 00:38:53.420 response is this function, h, Here's 00:38:53.420 --> 00:38:55.640 a way to think of what it is. 00:38:55.640 --> 00:39:00.170 Inside there, here's what my system 00:39:00.170 --> 00:39:01.460 can be thought of as doing. 00:39:04.040 --> 00:39:06.140 Inside here, I've got many parallel paths. 00:39:06.140 --> 00:39:27.980 I've got a system that scales by h0, delays by 0, 00:39:27.980 --> 00:39:35.172 and in parallel with the system that scales by-- 00:39:35.172 --> 00:39:36.130 let me put it up here-- 00:39:39.190 --> 00:39:49.180 scales by h1, delay by 1, and so on. 00:39:54.330 --> 00:39:57.490 OK, so all of these in parallel-- 00:39:57.490 --> 00:39:58.790 each one is very simple. 00:39:58.790 --> 00:40:00.970 Each one is as simple as the example I showed you. 00:40:04.000 --> 00:40:07.060 OK, I've got a whole bunch of these parallel systems, 00:40:07.060 --> 00:40:09.430 each one as simple as this. 00:40:09.430 --> 00:40:13.120 If I put a unit sample in here, what comes out? 00:40:13.120 --> 00:40:15.370 It's going to be exactly that, right, because the unit 00:40:15.370 --> 00:40:19.050 sample would get scaled by 0, delayed by nothing, 00:40:19.050 --> 00:40:20.040 will come out there. 00:40:20.040 --> 00:40:24.220 Then unit sample will get through this path scaled by h1, 00:40:24.220 --> 00:40:26.020 delayed by 1, come out there. 00:40:26.020 --> 00:40:29.830 When you assemble all of these with this summer here, 00:40:29.830 --> 00:40:32.360 you're going to get exactly that response. 00:40:32.360 --> 00:40:34.930 So here's another way to think about what's sitting inside 00:40:34.930 --> 00:40:38.300 a system, an LTI system whose unit sample response 00:40:38.300 --> 00:40:42.130 is given to be that, OK? 00:40:42.130 --> 00:40:45.410 So if I put x and n, what is it that's going to come out? 00:40:52.580 --> 00:40:55.070 If I put x and n, what comes out? 00:40:55.070 --> 00:40:57.950 Well, through this path, I get x(n) scaled by 0 00:40:57.950 --> 00:40:59.690 and delayed by nothing. 00:40:59.690 --> 00:41:01.970 Through this path, I get x(n) scaled by h1 00:41:01.970 --> 00:41:04.280 and delayed by 1, and so on. 00:41:04.280 --> 00:41:23.000 So what comes out? y of n is equal to h(m) x of n minus m 00:41:23.000 --> 00:41:26.310 over all m, OK? 00:41:26.310 --> 00:41:28.050 So I take x(n). 00:41:28.050 --> 00:41:29.200 I shift it by nothing. 00:41:29.200 --> 00:41:30.390 I scale by h0. 00:41:30.390 --> 00:41:31.600 That's one of these terms. 00:41:31.600 --> 00:41:34.110 The term corresponding to m equals 0. 00:41:34.110 --> 00:41:37.800 I take x(n), scale it by h1, delay it by 1. 00:41:37.800 --> 00:41:40.360 That gives me the term with m equals 1. 00:41:40.360 --> 00:41:42.870 So here's another way to write it, OK? 00:41:42.870 --> 00:41:44.400 Last time, I said that actually you 00:41:44.400 --> 00:41:46.890 can write convolution in this form 00:41:46.890 --> 00:41:49.423 or with the operation reversed. 00:41:49.423 --> 00:41:51.840 So it actually doesn't matter which order you write things 00:41:51.840 --> 00:41:52.800 in. 00:41:52.800 --> 00:41:55.890 This would be something you might write as h 00:41:55.890 --> 00:42:00.150 convolved with x at time n. 00:42:00.150 --> 00:42:03.210 So two different ways of writing the convolution and two 00:42:03.210 --> 00:42:06.390 different ways of thinking about how the output gets 00:42:06.390 --> 00:42:09.410 represented that way. 00:42:09.410 --> 00:42:11.960 You can easily get from one representation to the other 00:42:11.960 --> 00:42:14.080 by just making a change of variables. 00:42:14.080 --> 00:42:17.380 Like, let n minus k equals m, and you 00:42:17.380 --> 00:42:19.130 get from this representation to the other. 00:42:19.130 --> 00:42:21.800 This is a more mechanistic way of thinking about why 00:42:21.800 --> 00:42:24.680 these two representations work. 00:42:24.680 --> 00:42:28.040 OK, so with that is a given, let me show you 00:42:28.040 --> 00:42:34.160 how to actually carry out these operations graphically. 00:42:34.160 --> 00:42:37.158 In either form, here is a simple graphical way 00:42:37.158 --> 00:42:38.450 to think about what's going on. 00:42:48.450 --> 00:42:50.770 So to determine y at time n, this 00:42:50.770 --> 00:42:53.410 is the operation that I have to carry out, all right? 00:42:53.410 --> 00:42:57.040 To define y at time n, the output at time n, 00:42:57.040 --> 00:42:59.950 I've got to find a way to implement this operation. 00:43:09.460 --> 00:43:12.420 So how are we going to think of this graphically? 00:43:12.420 --> 00:43:14.500 I want to sketch the signal x. 00:43:14.500 --> 00:43:16.800 I want to sketch the signal h and then 00:43:16.800 --> 00:43:21.480 do something with these two signals to construct this, OK? 00:43:21.480 --> 00:43:26.280 So when I plot x, and I plot h to implement 00:43:26.280 --> 00:43:31.680 this operation, what's the name on my time axis? 00:43:31.680 --> 00:43:36.790 Is it m that I'm going to stick here or k or what? 00:43:36.790 --> 00:43:37.970 I want to implement this. 00:43:37.970 --> 00:43:41.740 I want to draw these two time functions, the functions of k, 00:43:41.740 --> 00:43:42.500 right? 00:43:42.500 --> 00:43:46.610 And it's just the number that I'm specifying here. 00:43:46.610 --> 00:43:52.550 OK, so on the k-axis, I'm going to take x and plot it. 00:43:52.550 --> 00:43:53.840 So x is some time function. 00:44:01.430 --> 00:44:02.180 Here's my x. 00:44:07.830 --> 00:44:11.110 I won't label them all because that would get crowded. 00:44:11.110 --> 00:44:14.380 And just to keep things clean, let's change colors here. 00:44:16.970 --> 00:44:19.150 How am I going to plot h of n minus k? 00:44:19.150 --> 00:44:21.910 Well, let's start thinking about the n equals 0 case. 00:44:21.910 --> 00:44:26.275 So for n equals 0, I've got to plot h of minus k. 00:44:26.275 --> 00:44:30.310 So how does h of minus k relate to the unit sample response 00:44:30.310 --> 00:44:31.480 h of k or h of n? 00:44:36.350 --> 00:44:40.010 If I tell you that I have a system-- 00:44:40.010 --> 00:44:42.870 we had an example up here, didn't we? 00:44:42.870 --> 00:44:45.710 I lost it. 00:44:45.710 --> 00:44:55.920 If I tell you that I have a system with this unit sample 00:44:55.920 --> 00:44:56.700 response-- 00:44:59.830 --> 00:45:01.040 let's do something simple. 00:45:01.040 --> 00:45:09.330 Let's say that this is 1/2 to the n times u of n. 00:45:09.330 --> 00:45:13.325 So for positive time, it's kind of a decaying geometric series. 00:45:16.450 --> 00:45:19.860 And for negative time, it's 0, OK? 00:45:19.860 --> 00:45:23.690 So that's an example of a unit sample response of a system. 00:45:23.690 --> 00:45:27.040 So if that's h of n, what does h of minus k look like? 00:45:29.670 --> 00:45:30.385 Yeah, anyone? 00:45:30.385 --> 00:45:31.260 AUDIENCE: Reflection. 00:45:31.260 --> 00:45:31.540 GEORGE VERGHESE: Sorry? 00:45:31.540 --> 00:45:32.750 AUDIENCE: Reflection [INAUDIBLE].. 00:45:32.750 --> 00:45:34.458 GEORGE VERGHESE: It's the reflection, OK? 00:45:34.458 --> 00:45:38.660 So if I want for n equals 0 to plot just the h of minus k 00:45:38.660 --> 00:45:42.800 that I need here, it's that reversed and plotted. 00:45:47.590 --> 00:45:49.490 OK, so this is an h of minus k here. 00:45:53.330 --> 00:45:55.040 What about h of minus k? 00:45:55.040 --> 00:45:56.630 Sorry, what about h of n minus k? 00:45:56.630 --> 00:46:01.880 Suppose I had n equals 3 now. 00:46:01.880 --> 00:46:03.630 How do I get h of 3 minus k? 00:46:10.520 --> 00:46:15.900 So I want to get h of 3 minus k. 00:46:15.900 --> 00:46:18.800 So that corresponds to sliding this over. 00:46:18.800 --> 00:46:21.620 Do I slide it to the right by 3, to the left by 3? 00:46:25.850 --> 00:46:28.160 You can tell by looking at the argument here. 00:46:28.160 --> 00:46:32.030 Whatever used to happen at 0 has now got to happen k equals 3. 00:46:32.030 --> 00:46:34.350 So that means a rightward shift. 00:46:34.350 --> 00:46:40.490 OK, so you flip this over, and then you slide it by n steps, 00:46:40.490 --> 00:46:41.450 OK? 00:46:41.450 --> 00:46:45.860 So you take the h of k, flip it around to get h of minus k, 00:46:45.860 --> 00:46:48.343 slide it by n steps. 00:46:48.343 --> 00:46:50.510 So if n is positive, you're sliding it to the right. 00:46:50.510 --> 00:46:53.095 If n is negative, you're sliding it to the left. 00:46:53.095 --> 00:46:54.470 So now you're on a single figure. 00:46:54.470 --> 00:46:56.660 You've managed to plot these two. 00:46:56.660 --> 00:46:58.448 What's the remaining operation? 00:47:01.140 --> 00:47:03.600 What you've got to do is the point-by-point product 00:47:03.600 --> 00:47:06.360 of these two waveforms and sum over the entire time axis. 00:47:06.360 --> 00:47:08.910 It's like taking a dot product, right? 00:47:08.910 --> 00:47:12.420 So you're going to take this value of the red curve-- 00:47:12.420 --> 00:47:15.910 well, actually, let me slide it over. 00:47:15.910 --> 00:47:17.280 Let's do this case. 00:47:17.280 --> 00:47:19.770 This is the slid-over case, right? 00:47:22.950 --> 00:47:26.130 You're going to take every one of the purple values multiplied 00:47:26.130 --> 00:47:29.250 by the white and sum over the entire time axis. 00:47:29.250 --> 00:47:31.880 That's an implementation of this. 00:47:31.880 --> 00:47:35.745 So in recitation tomorrow, you'll get practice on this. 00:47:35.745 --> 00:47:38.090 But what you want to think of, sort 00:47:38.090 --> 00:47:44.120 of the mantra for graphical implementation of convolution 00:47:44.120 --> 00:47:48.680 is you've got to do the flip of one of the time functions, 00:47:48.680 --> 00:48:00.030 slide by n, and then the product, OK? 00:48:00.030 --> 00:48:03.053 So you slide it to get one particular value of n. 00:48:03.053 --> 00:48:05.220 If you want the next value of n, you slide it 1 over 00:48:05.220 --> 00:48:06.553 and go through this whole thing. 00:48:06.553 --> 00:48:09.380 So it's flipping, sliding by the right number of spots, 00:48:09.380 --> 00:48:10.380 doing the inner product. 00:48:10.380 --> 00:48:12.490 That gives you one value of the answer. 00:48:12.490 --> 00:48:13.600 And then you repeat. 00:48:13.600 --> 00:48:17.210 All right, you'll get more practice in recitation.