WEBVTT 00:00:00.000 --> 00:00:02.542 ANNOUNCER: The following content is provided under a Creative 00:00:02.542 --> 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. 00:00:30.410 --> 00:00:33.460 PROFESSOR: There's quite a bit I want to go through here. 00:00:33.460 --> 00:00:36.650 so we're going to talk today about modulation, 00:00:36.650 --> 00:00:39.950 which you've already gotten some notion of, 00:00:39.950 --> 00:00:42.920 and that's basically the task of matching a transmitted signal 00:00:42.920 --> 00:00:44.810 to the physical medium. 00:00:44.810 --> 00:00:48.710 And then we'll talk about demodulation, as well. 00:00:48.710 --> 00:00:49.210 Whoops. 00:00:52.820 --> 00:00:56.210 So just to remind you of how we got into this story, 00:00:56.210 --> 00:01:00.650 we started off talking about bits 00:01:00.650 --> 00:01:04.315 that we had to get across to a receiver from a source. 00:01:04.315 --> 00:01:05.940 And we've been spending quite some time 00:01:05.940 --> 00:01:09.440 now focusing on this piece of the system, which 00:01:09.440 --> 00:01:12.020 is taking the bits, converting them 00:01:12.020 --> 00:01:14.570 to actual physical samples of a voltage, 00:01:14.570 --> 00:01:16.580 for instance, and then trying to get them 00:01:16.580 --> 00:01:19.700 over some physical medium. 00:01:19.700 --> 00:01:22.530 And then, at the other end, converting back to bits, 00:01:22.530 --> 00:01:23.030 all right? 00:01:23.030 --> 00:01:25.955 So this is really a key piece of the system. 00:01:25.955 --> 00:01:27.830 If you can't get it over the physical medium, 00:01:27.830 --> 00:01:29.862 then you don't have anything. 00:01:29.862 --> 00:01:31.820 So we've been spending quite some time on that. 00:01:31.820 --> 00:01:34.430 We've talked about models for signals and models 00:01:34.430 --> 00:01:37.490 for systems, LTI models for systems, 00:01:37.490 --> 00:01:38.760 all in the time domain. 00:01:38.760 --> 00:01:40.790 And then we came to the frequency domain, which 00:01:40.790 --> 00:01:42.980 we said will make things a lot simpler, 00:01:42.980 --> 00:01:45.730 and actually is the way that people think about transmission 00:01:45.730 --> 00:01:50.000 on physical media, typically. 00:01:50.000 --> 00:01:53.420 OK, so the actual math. 00:01:53.420 --> 00:01:56.540 Well, we've seen-- this is just review. 00:01:56.540 --> 00:01:59.180 We've seen that, most recently, that you can actually 00:01:59.180 --> 00:02:03.650 represent any signal as a weighted combination 00:02:03.650 --> 00:02:06.440 of exponentials, so this was the transform domain 00:02:06.440 --> 00:02:08.000 representation. 00:02:08.000 --> 00:02:12.380 And the weights here were given by the discrete-time Fourier 00:02:12.380 --> 00:02:13.010 transform. 00:02:13.010 --> 00:02:15.620 So you give me a signal, I can find for you 00:02:15.620 --> 00:02:18.770 the discrete-time Fourier transform, which then tells me 00:02:18.770 --> 00:02:21.830 how to assemble complex exponentials to get 00:02:21.830 --> 00:02:23.660 the signal of interest, all right? 00:02:23.660 --> 00:02:27.110 And so this Fourier representation of the signal 00:02:27.110 --> 00:02:32.250 is really the frequency domain thinking. 00:02:32.250 --> 00:02:35.350 And then we saw how to apply that actually to a system. 00:02:35.350 --> 00:02:37.620 We have an LTI system, therefore characterized 00:02:37.620 --> 00:02:39.630 by a frequency response. 00:02:39.630 --> 00:02:41.550 We first introduced the frequency response 00:02:41.550 --> 00:02:44.190 as a way of thinking about what happens to cosines. 00:02:44.190 --> 00:02:46.920 Put a cosine in, and you get a cosine 00:02:46.920 --> 00:02:49.110 out that's scaled by the magnitude of the frequency 00:02:49.110 --> 00:02:50.790 response and with the phase shifted 00:02:50.790 --> 00:02:53.340 by the angle of the frequency response. 00:02:53.340 --> 00:02:57.030 And then we went from cosines, or in parallel we 00:02:57.030 --> 00:03:00.780 talked about exponential inputs, so inputs of this type. 00:03:00.780 --> 00:03:03.120 And now we have more generally a signal 00:03:03.120 --> 00:03:06.000 that's represented as a weighted combination of exponentials 00:03:06.000 --> 00:03:07.350 of that type. 00:03:07.350 --> 00:03:10.410 Out comes the same weighted combination of exponentials, 00:03:10.410 --> 00:03:12.750 except each one is scaled by the frequency 00:03:12.750 --> 00:03:14.940 response as appropriate. 00:03:14.940 --> 00:03:16.440 And then comparing that with what 00:03:16.440 --> 00:03:21.000 we expect as a spectral representation for the output. 00:03:21.000 --> 00:03:22.680 We get this key relationship, which 00:03:22.680 --> 00:03:26.550 is relating the input and the output of a system, 00:03:26.550 --> 00:03:30.340 an LTI system, that's governed by a frequency response. 00:03:30.340 --> 00:03:32.580 So now we're starting to think in terms 00:03:32.580 --> 00:03:35.520 of the spectral content of the input, all right, which 00:03:35.520 --> 00:03:37.770 is the frequency domain description of the signal that 00:03:37.770 --> 00:03:39.640 goes into the system. 00:03:39.640 --> 00:03:41.400 Then the frequency response of the system 00:03:41.400 --> 00:03:45.150 shaping that spectral content to give you the spectral content 00:03:45.150 --> 00:03:46.470 of what comes out, all right? 00:03:46.470 --> 00:03:48.150 So this is the language and the picture 00:03:48.150 --> 00:03:52.620 that we have, and it's all as simple as multiplication 00:03:52.620 --> 00:03:54.990 once you've figured out what the spectral content is 00:03:54.990 --> 00:03:57.240 of the signal of interest, once you have the frequency 00:03:57.240 --> 00:03:58.570 response of the system. 00:03:58.570 --> 00:04:01.440 So we've got to know how to do those pieces. 00:04:01.440 --> 00:04:03.780 And then we talked most specifically 00:04:03.780 --> 00:04:07.590 about a physical medium that's close to what you're 00:04:07.590 --> 00:04:11.560 doing in the lab, which is the medium of, well, 00:04:11.560 --> 00:04:14.120 an acoustic channel driven by a loudspeaker, 00:04:14.120 --> 00:04:15.690 and at the other end are a microphone 00:04:15.690 --> 00:04:17.290 to pick up the signal. 00:04:17.290 --> 00:04:19.740 And I showed you these typical characteristics 00:04:19.740 --> 00:04:23.370 of loud speakers, the kinds that you'll find listed everywhere, 00:04:23.370 --> 00:04:25.590 three different speakers. 00:04:25.590 --> 00:04:27.090 I mentioned last time that, when you 00:04:27.090 --> 00:04:30.370 look at frequency specs for speakers, 00:04:30.370 --> 00:04:33.930 people will typically only show you the magnitude specification 00:04:33.930 --> 00:04:37.350 because, for audio applications, the phase distortions are 00:04:37.350 --> 00:04:38.370 a little less important. 00:04:38.370 --> 00:04:41.280 They tend to not be picked up by the ear. 00:04:41.280 --> 00:04:44.820 All of these-- let's see they have passbands from-- 00:04:44.820 --> 00:04:47.190 this pointer's a little weak here, but-- 00:04:47.190 --> 00:04:51.460 passbands from around 100 Hertz to, let's say, 10 kilohertz. 00:04:51.460 --> 00:04:56.000 So in that region, they pass signals more or less uniformly 00:04:56.000 --> 00:04:59.040 in at least the magnitude characteristic, and then 00:04:59.040 --> 00:05:01.110 near the edges they taper off. 00:05:01.110 --> 00:05:06.780 And some speakers will have bigger passbands, 00:05:06.780 --> 00:05:10.090 and will taper off closer to DC. 00:05:10.090 --> 00:05:13.880 Other speakers actually will not pass frequencies 00:05:13.880 --> 00:05:17.453 till you get till about, oh, what is that, 120 Hertz or so 00:05:17.453 --> 00:05:18.870 on this characteristic, but you've 00:05:18.870 --> 00:05:22.500 got to get way up before you get anything through that speaker. 00:05:22.500 --> 00:05:24.990 But nominally, we can think of speakers, 00:05:24.990 --> 00:05:27.160 since they're aimed at audio applications-- 00:05:27.160 --> 00:05:27.660 what? 00:05:27.660 --> 00:05:31.560 The ear here is something on the order of, let's say, 100 Hertz 00:05:31.560 --> 00:05:34.930 to 10 kilohertz. 00:05:34.930 --> 00:05:39.630 OK, but the phase characteristic is important, too. 00:05:39.630 --> 00:05:41.220 It's not important, maybe, when you're 00:05:41.220 --> 00:05:43.560 talking about sending audio on a speaker, 00:05:43.560 --> 00:05:45.960 but in Audiocom in the lab, you're actually 00:05:45.960 --> 00:05:47.540 sending pulses across it. 00:05:47.540 --> 00:05:49.797 You're communicating something other than audio. 00:05:49.797 --> 00:05:51.630 You're actually trying to get a signal whose 00:05:51.630 --> 00:05:52.890 particular shape matters. 00:05:52.890 --> 00:05:54.720 It's not how you hear it, but what it looks 00:05:54.720 --> 00:05:57.390 like before you sample it, OK? 00:05:57.390 --> 00:06:00.210 So in settings like that, the phase characteristic 00:06:00.210 --> 00:06:01.610 is important, as well. 00:06:01.610 --> 00:06:04.440 Now, you haven't explicitly probed 00:06:04.440 --> 00:06:07.320 the frequency characteristic of the speaker you're using. 00:06:07.320 --> 00:06:09.300 You could do that, but instead you've 00:06:09.300 --> 00:06:11.977 been looking at things like step responses in the time domain 00:06:11.977 --> 00:06:13.560 and constructing eye diagrams, but you 00:06:13.560 --> 00:06:15.360 could look in the frequency domain 00:06:15.360 --> 00:06:18.090 and characterize your particular channel for your laptop sitting 00:06:18.090 --> 00:06:19.590 in a particular place. 00:06:19.590 --> 00:06:24.390 You could look to see what the magnitude and phase are like. 00:06:24.390 --> 00:06:28.320 OK, so I want to go through this exercise of looking 00:06:28.320 --> 00:06:29.850 at the spectral content of a signal 00:06:29.850 --> 00:06:32.130 you want to get across this audio channel, 00:06:32.130 --> 00:06:34.650 and then looking at how the audio channel shapes it, 00:06:34.650 --> 00:06:36.460 and then what you pick up at the other end. 00:06:36.460 --> 00:06:40.330 So just to give you a feel for how one thinks through this. 00:06:40.330 --> 00:06:44.100 So the input in the typical application 00:06:44.100 --> 00:06:47.310 you have for Audiocom you have-- 00:06:47.310 --> 00:06:50.010 let's see, if you wanted to signal just a 1 and then all 00:06:50.010 --> 00:06:51.780 0's. 00:06:51.780 --> 00:06:55.850 You would have 256 samples at height 1, 00:06:55.850 --> 00:06:59.430 and then everything from then on 0, OK? 00:06:59.430 --> 00:07:05.370 So what I'd like to do is think through how this pulse gets 00:07:05.370 --> 00:07:10.290 across the medium, but thinking it through in the frequency 00:07:10.290 --> 00:07:11.293 domain, all right? 00:07:11.293 --> 00:07:12.960 So the first thing we have to figure out 00:07:12.960 --> 00:07:16.443 is, what's the spectral content of this pulse? 00:07:16.443 --> 00:07:18.360 By the way, if we understand it for one pulse, 00:07:18.360 --> 00:07:20.040 then we can understand it-- then we know it 00:07:20.040 --> 00:07:21.748 for a sequence of pulses because if we're 00:07:21.748 --> 00:07:24.620 modeling the system as time-invariant, once we figure 00:07:24.620 --> 00:07:26.370 out what one pulse does, we can figure out 00:07:26.370 --> 00:07:28.080 what a later pulse will do. 00:07:28.080 --> 00:07:30.570 It's just the same response delayed in time, OK? 00:07:30.570 --> 00:07:32.100 So the key to it is understanding 00:07:32.100 --> 00:07:35.010 what happens with one pulse. 00:07:35.010 --> 00:07:36.630 So the spectral content of the signal 00:07:36.630 --> 00:07:39.180 is what we're interested in. 00:07:39.180 --> 00:07:42.030 And my question is, do you have any guesses 00:07:42.030 --> 00:07:45.480 as to what the spectral content might be, just roughly, 00:07:45.480 --> 00:07:47.040 qualitatively? 00:07:47.040 --> 00:07:52.220 Where do you think the energy of the signal is concentrated? 00:07:52.220 --> 00:07:54.830 What frequency ranges? 00:07:54.830 --> 00:07:55.400 Any thoughts? 00:07:59.080 --> 00:08:02.060 I'll need a hand up and a loud voice so I can figure out 00:08:02.060 --> 00:08:02.560 what's-- 00:08:06.240 --> 00:08:08.636 at least one? 00:08:08.636 --> 00:08:09.564 Yeah? 00:08:09.564 --> 00:08:10.030 AUDIENCE: Low frequency. 00:08:10.030 --> 00:08:11.655 PROFESSOR: Low frequency is a good idea 00:08:11.655 --> 00:08:14.660 because, for most of this signal, 00:08:14.660 --> 00:08:16.660 you've got essentially nothing happening, right? 00:08:16.660 --> 00:08:17.285 It's just flat. 00:08:17.285 --> 00:08:21.790 So you expect high spectral content at DC. 00:08:21.790 --> 00:08:25.540 But there is this sharp transition, 00:08:25.540 --> 00:08:27.730 so you might expect high frequencies associated 00:08:27.730 --> 00:08:28.250 with that. 00:08:28.250 --> 00:08:31.600 So do you think it might be low frequencies and then 00:08:31.600 --> 00:08:34.870 high frequencies, not much in between, or any thoughts? 00:08:38.350 --> 00:08:41.250 OK, well, let's look at what it actually is. 00:08:41.250 --> 00:08:43.140 Let's work it out. 00:08:43.140 --> 00:08:54.530 So we're talking about a signal that's at height 1. 00:08:54.530 --> 00:08:56.030 For-- let's do the general case. 00:08:56.030 --> 00:08:59.050 So let's say it's for n samples, so from 0 00:08:59.050 --> 00:09:04.430 to n minus 1 its height 1, and then it's 0 outside of that, 00:09:04.430 --> 00:09:04.930 OK? 00:09:04.930 --> 00:09:06.570 So suppose this is x of n. 00:09:09.250 --> 00:09:12.940 How do we determine the spectral content? 00:09:12.940 --> 00:09:14.470 Well, we've got to write the-- 00:09:14.470 --> 00:09:18.070 we've got to compute the DTFT, right? 00:09:18.070 --> 00:09:19.550 So what's the DTFT? 00:09:19.550 --> 00:09:28.510 It's a summation x m e to the minus j omega m over all m. 00:09:28.510 --> 00:09:31.360 But in this case, it simplifies, right? 00:09:31.360 --> 00:09:35.000 Because there are only a few non-zero values of the signal. 00:09:35.000 --> 00:09:36.770 So this is going to be-- 00:09:36.770 --> 00:09:37.270 let's see. 00:09:37.270 --> 00:09:40.720 It's going to be x0 e to the minus j omega 0 plus x1 00:09:40.720 --> 00:09:43.170 e the minus j omega 1, and so on. 00:09:43.170 --> 00:09:47.260 That's going to be 1 plus e to the minus j omega 00:09:47.260 --> 00:09:51.610 plus e to the minus j2 omega plus 00:09:51.610 --> 00:09:56.708 e to the minus j N minus 1 omega. 00:09:56.708 --> 00:09:57.500 So that's the DTFT. 00:10:00.030 --> 00:10:02.170 But till you work with that and get it in a form 00:10:02.170 --> 00:10:03.970 that you can make sense of, you still 00:10:03.970 --> 00:10:07.150 don't have a feel for where the frequency content is, right? 00:10:07.150 --> 00:10:09.400 You've got to-- the best way to get 00:10:09.400 --> 00:10:12.910 at that is to think of what the magnitude of this will be. 00:10:12.910 --> 00:10:14.590 And even then, it's not obvious how 00:10:14.590 --> 00:10:17.230 to think about the magnitude of a sum of complex numbers 00:10:17.230 --> 00:10:22.580 like this, so you've got to play with it a little more. 00:10:22.580 --> 00:10:24.718 OK, well, this is a geometric series, right? 00:10:24.718 --> 00:10:26.510 Each term is obtained from the previous one 00:10:26.510 --> 00:10:28.990 by multiplying by to the minus jam omega. 00:10:28.990 --> 00:10:32.420 And so if you've got the sum of a finite number 00:10:32.420 --> 00:10:34.980 of geometric series of this type, what do we have? 00:10:34.980 --> 00:10:42.790 We have that as the sum, right? 00:10:42.790 --> 00:10:43.990 You agree? 00:10:43.990 --> 00:10:48.350 So this was the factor by which we multiply each term. 00:10:48.350 --> 00:10:48.850 Sorry. 00:10:54.050 --> 00:10:56.100 And we've got N such terms, so you're summing 00:10:56.100 --> 00:10:57.560 N terms of a geometric series. 00:11:00.720 --> 00:11:05.340 Well, we might be getting closer here to extracting a magnitude, 00:11:05.340 --> 00:11:08.850 but you really want to do a little bit more massaging here. 00:11:08.850 --> 00:11:09.540 Let's see. 00:11:09.540 --> 00:11:15.900 If I make this e to the j omega N over 2, 00:11:15.900 --> 00:11:21.750 then here's e to the j omega N over 2, 00:11:21.750 --> 00:11:27.810 minus e to the minus j omega N over 2. 00:11:27.810 --> 00:11:32.642 And this is a trick we've done-- 00:11:32.642 --> 00:11:33.975 we've played a few times before. 00:11:48.100 --> 00:11:48.600 Right? 00:11:48.600 --> 00:11:51.290 I've just rearranged things. 00:11:51.290 --> 00:11:51.790 Let's see. 00:11:51.790 --> 00:11:53.320 How have I helped myself here? 00:11:53.320 --> 00:11:54.550 Have I helped myself at all? 00:12:00.298 --> 00:12:03.930 So what is-- what does that simplify to? 00:12:07.240 --> 00:12:10.090 Well, the factor in front I can write as some phase term e 00:12:10.090 --> 00:12:17.370 to the minus j omega N minus 1 over 2. 00:12:17.370 --> 00:12:18.370 And what's this? 00:12:21.070 --> 00:12:21.943 Anybody? 00:12:27.750 --> 00:12:28.260 Numerator? 00:12:28.260 --> 00:12:30.010 Does the numerator remind you of anything? 00:12:32.950 --> 00:12:33.450 Sine? 00:12:37.870 --> 00:12:39.810 Sine omega N over 2? 00:12:42.946 --> 00:12:51.280 And the denominator, sine omega over 2, right? 00:12:51.280 --> 00:12:54.520 So now it starts to look a little bit more manageable. 00:12:54.520 --> 00:12:56.680 If I wanted to get the magnitude of this, 00:12:56.680 --> 00:12:58.420 well, the magnitude of this is going 00:12:58.420 --> 00:13:04.690 to be the magnitude of this piece times the magnitude 00:13:04.690 --> 00:13:05.750 of that piece. 00:13:05.750 --> 00:13:08.110 What's the magnitude of the first term here? 00:13:08.110 --> 00:13:08.800 Just 1, right? 00:13:08.800 --> 00:13:12.650 It's e to the j something, so its magnitude is 1. 00:13:12.650 --> 00:13:16.690 So here's the magnitude of the DTFT, 00:13:16.690 --> 00:13:19.160 so that's the spectral characteristic, 00:13:19.160 --> 00:13:21.406 and that's something that we can plot. 00:13:21.406 --> 00:13:22.380 AUDIENCE: Question. 00:13:22.380 --> 00:13:24.140 PROFESSOR: OK. 00:13:24.140 --> 00:13:24.850 Sorry. 00:13:24.850 --> 00:13:25.686 Question, hi. 00:13:25.686 --> 00:13:28.552 AUDIENCE: [INAUDIBLE] 00:13:28.552 --> 00:13:29.260 PROFESSOR: Sorry. 00:13:29.260 --> 00:13:29.890 Say that again? 00:13:29.890 --> 00:13:34.330 AUDIENCE: [INAUDIBLE] 00:13:34.330 --> 00:13:37.680 PROFESSOR: Did I make a mistake somewhere here, or? 00:13:37.680 --> 00:13:38.440 Oh, this thing? 00:13:38.440 --> 00:13:39.910 This term here? 00:13:39.910 --> 00:13:42.550 I was trying to combine numerator and denominator here. 00:13:42.550 --> 00:13:45.623 AUDIENCE: [INAUDIBLE] 00:13:45.623 --> 00:13:46.540 PROFESSOR: Which part? 00:13:46.540 --> 00:13:47.200 Sorry. 00:13:47.200 --> 00:13:48.700 I have to stand where you are to see 00:13:48.700 --> 00:13:51.648 if I made a mistake because it's hard to see close up. 00:13:51.648 --> 00:13:56.151 AUDIENCE: Over the-- like, when we go from the 1 minus e 00:13:56.151 --> 00:13:58.870 to the minus 2 again to the other one, 00:13:58.870 --> 00:14:00.870 why are we dividing by two? 00:14:04.090 --> 00:14:06.410 PROFESSOR: Oh, what are we-- oh. 00:14:06.410 --> 00:14:07.480 Why are we dividing by 2? 00:14:07.480 --> 00:14:10.270 Because when you multiply it out, that's what it takes. 00:14:10.270 --> 00:14:14.980 I'm trying to group things to get something interpretable. 00:14:14.980 --> 00:14:17.260 I don't know what this is, but I know 00:14:17.260 --> 00:14:19.390 what the numerator over sine looks 00:14:19.390 --> 00:14:22.960 like, so I'm trying to make this a little bit more 00:14:22.960 --> 00:14:24.610 equally distributed, right? 00:14:24.610 --> 00:14:27.920 So if I pull out that factor, what's left? 00:14:27.920 --> 00:14:31.180 Looks like part of a sine. 00:14:31.180 --> 00:14:31.680 OK? 00:14:31.680 --> 00:14:34.140 So I'm just-- so it's this time this gives me 00:14:34.140 --> 00:14:35.790 the numerator here. 00:14:35.790 --> 00:14:38.710 And this times this gives me the denominator here. 00:14:38.710 --> 00:14:40.080 So it's just rearranging terms. 00:14:40.080 --> 00:14:43.510 We've used this trick before. 00:14:43.510 --> 00:14:45.427 Any trick that works twice is a method, OK? 00:14:45.427 --> 00:14:46.760 So we really have a method here. 00:14:46.760 --> 00:14:49.660 It's not just a trick. 00:14:49.660 --> 00:14:54.160 If it works three times, you can make a religion of it. 00:14:54.160 --> 00:14:59.920 OK, so that's the derivation we have here. 00:15:02.550 --> 00:15:06.720 What's the height of this at the origin? 00:15:06.720 --> 00:15:08.010 Let's just focus on that term. 00:15:17.280 --> 00:15:19.800 OK, so this is the magnitude we're talking about. 00:15:19.800 --> 00:15:22.920 What's the height at the origin, at omega equals 0? 00:15:22.920 --> 00:15:24.870 Well, now you can use L'Hopital's rule, right? 00:15:24.870 --> 00:15:27.630 Because omega is something that varies continuously. 00:15:27.630 --> 00:15:29.520 So for small values of the argument, you're 00:15:29.520 --> 00:15:33.810 really looking at something of height N. And then, 00:15:33.810 --> 00:15:37.410 when is the first time that this goes to 0? 00:15:37.410 --> 00:15:39.330 Well, for small values of frequency, 00:15:39.330 --> 00:15:43.680 the numerator is not changing sign, 00:15:43.680 --> 00:15:46.740 and this first goes to 0 when you 00:15:46.740 --> 00:15:52.003 get to omega equals 2 pi over N, to private capital N. 00:15:52.003 --> 00:15:53.670 So actually, instead of saying all that, 00:15:53.670 --> 00:15:55.045 I should just draw you a picture. 00:15:58.340 --> 00:16:00.160 There's a picture of one particular case. 00:16:00.160 --> 00:16:03.055 So this is a case where, actually, the pulse 00:16:03.055 --> 00:16:04.190 didn't start at 0. 00:16:04.190 --> 00:16:07.750 It was symmetrically located around 0, OK? 00:16:07.750 --> 00:16:13.550 It was a pulse of length 11 symmetrically located around 0. 00:16:17.570 --> 00:16:19.460 And because it was symmetrically located, 00:16:19.460 --> 00:16:22.640 this phase factor went away, and all you're left with 00:16:22.640 --> 00:16:27.310 is the sine omega N over 2 divided by sine omega over 2. 00:16:27.310 --> 00:16:31.770 So you're looking at the actual DTFT of a pulse of that type, 00:16:31.770 --> 00:16:32.270 OK? 00:16:32.270 --> 00:16:36.960 So this started at minus 5 and went to plus 5, 00:16:36.960 --> 00:16:41.600 and was 11 samples long and was 0 everywhere outside of that. 00:16:45.810 --> 00:16:47.240 So that's what this function looks 00:16:47.240 --> 00:16:51.753 like, the sine omega N over 2 divided by sine omega over 2. 00:16:51.753 --> 00:16:53.920 Does it remind you of a function you've seen before? 00:16:57.565 --> 00:16:58.065 Sinc? 00:16:58.065 --> 00:16:59.910 A sinc function? 00:16:59.910 --> 00:17:01.410 It's very close to a sinc. 00:17:01.410 --> 00:17:03.859 The sinc, though, had just frequency in the denominator. 00:17:03.859 --> 00:17:07.020 It didn't have sine of something in the denominator. 00:17:07.020 --> 00:17:10.380 And the reason this appears is, remember 00:17:10.380 --> 00:17:14.910 that transforms and frequency responses 00:17:14.910 --> 00:17:17.710 have to be periodic with period 2 pi. 00:17:17.710 --> 00:17:19.829 So it certainly wouldn't be possible 00:17:19.829 --> 00:17:21.960 for the transform of a signal to be a sinc 00:17:21.960 --> 00:17:24.172 because there's no periodicity in the sinc. 00:17:24.172 --> 00:17:25.630 But when you work it out carefully, 00:17:25.630 --> 00:17:28.170 you find that it's something close to a sinc, 00:17:28.170 --> 00:17:30.330 but one that has exactly the right periodicity, 00:17:30.330 --> 00:17:33.030 so this thing will repeat periodically 00:17:33.030 --> 00:17:35.470 with period 2 pi, exactly the way it's supposed to. 00:17:35.470 --> 00:17:38.240 So it's sort of sinc-like. 00:17:38.240 --> 00:17:41.460 For small values of omega, the sine 00:17:41.460 --> 00:17:43.590 is essentially just omega over 2, 00:17:43.590 --> 00:17:45.450 and it is essentially a sinc. 00:17:45.450 --> 00:17:47.550 But when you get to larger values of omega, 00:17:47.550 --> 00:17:50.215 this thing starts to play a role. 00:17:50.215 --> 00:17:51.090 AUDIENCE: [INAUDIBLE] 00:17:51.090 --> 00:17:51.990 PROFESSOR: Yeah? 00:17:51.990 --> 00:17:52.692 Sorry? 00:17:52.692 --> 00:17:54.780 AUDIENCE: Does the magnitude mean anything? 00:17:54.780 --> 00:17:57.197 PROFESSOR: I haven't-- I'm not plotting the magnitude now. 00:17:57.197 --> 00:17:59.290 I was plotting the actual DTFT for this case. 00:17:59.290 --> 00:18:03.150 So in this symmetric case, the actual DTFT 00:18:03.150 --> 00:18:10.570 is the sine N omega over 2 over sine omega over 2. 00:18:10.570 --> 00:18:13.050 So I was plotting the actual DTFT. 00:18:13.050 --> 00:18:18.930 And the magnitude of the DTFT I get just by taking 00:18:18.930 --> 00:18:21.173 absolute value, right? 00:18:21.173 --> 00:18:24.213 AUDIENCE: [INAUDIBLE] 00:18:24.213 --> 00:18:26.630 PROFESSOR: Well, I did this for a case of a pulse starting 00:18:26.630 --> 00:18:28.850 at time 0, OK? 00:18:28.850 --> 00:18:31.430 So this factor came purely from where I 00:18:31.430 --> 00:18:33.510 located this on the time axis. 00:18:33.510 --> 00:18:35.960 So different positions of this pulse on the time axis 00:18:35.960 --> 00:18:39.590 will modify this factor but won't touch this factor, OK? 00:18:39.590 --> 00:18:42.690 Shifts in time correspond to multiplication by e 00:18:42.690 --> 00:18:44.360 to the minus j omega something, right? 00:18:44.360 --> 00:18:46.310 You've seen that in recitation. 00:18:46.310 --> 00:18:48.680 So whenever you shift the pulse in time, 00:18:48.680 --> 00:18:52.010 what it does to the transform is it leaves this part intact 00:18:52.010 --> 00:18:53.450 and affects this factor, but that 00:18:53.450 --> 00:18:56.840 doesn't change the magnitude of the transform, OK? 00:18:56.840 --> 00:18:58.740 So in the case of a symmetrical pulse, 00:18:58.740 --> 00:19:00.950 symmetrical about the origin, actually 00:19:00.950 --> 00:19:03.950 that phase factor goes away, and the actual transform 00:19:03.950 --> 00:19:06.210 is just that piece without the e to the anything. 00:19:06.210 --> 00:19:06.710 Sorry. 00:19:06.710 --> 00:19:11.000 I jumped over a few steps in describing that. 00:19:11.000 --> 00:19:12.140 Just to go back a second. 00:19:15.210 --> 00:19:17.150 This is not a sinc function, but it's actually 00:19:17.150 --> 00:19:20.420 referred to as a periodic sinc. 00:19:20.420 --> 00:19:21.800 It's got a fancier name, also. 00:19:21.800 --> 00:19:25.160 It crops up all over the place. 00:19:25.160 --> 00:19:27.240 Height N at the origin and the first zero 00:19:27.240 --> 00:19:33.610 crossing at 2 pi over cap N. So as you make the pulse wider 00:19:33.610 --> 00:19:37.480 in time, you make it narrower in frequency, right? 00:19:37.480 --> 00:19:42.610 As N becomes larger, you make this wider in time. 00:19:42.610 --> 00:19:45.970 The main lobe of this frequency distribution 00:19:45.970 --> 00:19:47.770 gets more concentrated, and frequency gets 00:19:47.770 --> 00:19:49.540 closer to being a DC signal. 00:19:49.540 --> 00:19:50.350 Makes sense, right? 00:19:50.350 --> 00:19:52.600 The longer that this stays constant, 00:19:52.600 --> 00:19:55.480 the more the signal looks like just DC and the more 00:19:55.480 --> 00:20:00.130 the frequency is concentrated at the origin. 00:20:00.130 --> 00:20:03.760 But what you can see here is there's actually 00:20:03.760 --> 00:20:06.100 a full spread of frequencies. 00:20:06.100 --> 00:20:09.970 It's not that there's just low frequency for the flat parts 00:20:09.970 --> 00:20:11.710 and high frequency for the vertical edge 00:20:11.710 --> 00:20:12.668 and nothing in between. 00:20:12.668 --> 00:20:16.840 There's actually a full spread of frequency components 00:20:16.840 --> 00:20:21.610 that it takes to make up that step. 00:20:21.610 --> 00:20:24.030 OK. 00:20:24.030 --> 00:20:26.410 If you had a pulse that wasn't centered-- 00:20:26.410 --> 00:20:28.420 this is just to show you. 00:20:28.420 --> 00:20:31.470 Here is a pulse-- actually, this is not centered. 00:20:31.470 --> 00:20:34.260 It's only 10 long, but the magnitude here you're 00:20:34.260 --> 00:20:37.740 only seeing half the frequency scale, so 0 to pi 00:20:37.740 --> 00:20:39.630 essentially, except this is in terms 00:20:39.630 --> 00:20:43.830 of f, which is omega over 2 pi. 00:20:43.830 --> 00:20:46.020 You get the same kind of magnitude characteristic, 00:20:46.020 --> 00:20:48.060 but now because you've shifted it off-center, 00:20:48.060 --> 00:20:50.700 you've got a linear-phase characteristic, 00:20:50.700 --> 00:20:53.880 and what you're seeing here is a linear-phase characteristic, 00:20:53.880 --> 00:20:56.970 except every time you have a flip in sign, 00:20:56.970 --> 00:20:59.220 you jump the phase by 180 degrees, right? 00:20:59.220 --> 00:21:01.440 When you change the sign of something 00:21:01.440 --> 00:21:04.440 from a plus to a minus, that's like adding 180 degrees 00:21:04.440 --> 00:21:06.980 to the phase or subtracting 180 degrees to the phase. 00:21:06.980 --> 00:21:10.120 So you can spend time on all of this and make sense of it. 00:21:10.120 --> 00:21:15.540 But the basic idea is that you get the sinc-like distribution 00:21:15.540 --> 00:21:17.520 and frequency. 00:21:17.520 --> 00:21:20.010 OK, so let's get back to the particular pulse that's 00:21:20.010 --> 00:21:25.650 of interest to us, which is that pulse. 00:21:25.650 --> 00:21:29.230 So it's the same kind of thing, except N is 256. 00:21:29.230 --> 00:21:31.110 And what I've plotted for you here 00:21:31.110 --> 00:21:35.070 is the magnitude of the DTFT. 00:21:35.070 --> 00:21:37.400 It has the sinc-like shape. 00:21:37.400 --> 00:21:40.470 I haven't actually plotted it as a continuous function of omega. 00:21:40.470 --> 00:21:42.058 Instead, I've used the FFT. 00:21:42.058 --> 00:21:42.600 You remember? 00:21:42.600 --> 00:21:46.770 We talked about the Fast Fourier Transform. 00:21:46.770 --> 00:21:50.330 So what the fast Fourier transform is going to do 00:21:50.330 --> 00:21:57.120 is, if the actual magnitude DTFT was some continuous thing 00:21:57.120 --> 00:21:59.760 like this, the fast Fourier transform 00:21:59.760 --> 00:22:05.900 is going to give me samples of it, as many as I want. 00:22:05.900 --> 00:22:07.900 But the more samples I ask for, the more 00:22:07.900 --> 00:22:10.420 work I have to do, of course, OK? 00:22:10.420 --> 00:22:14.598 I asked for 48,000 samples of the DTFT 00:22:14.598 --> 00:22:16.390 so that I could get a nice big spread here. 00:22:18.970 --> 00:22:22.690 If your samples came from sampling at 48 kilohertz, 00:22:22.690 --> 00:22:29.530 for instance, then the rightmost end 00:22:29.530 --> 00:22:33.250 that corresponds to pi in terms of actual frequency 00:22:33.250 --> 00:22:35.740 would correspond to the sampling frequency divided by 2, 00:22:35.740 --> 00:22:38.410 so that's 24 kilohertz sitting there. 00:22:38.410 --> 00:22:42.580 So I actually have 24,000 points for 24,000 Hertz, 00:22:42.580 --> 00:22:45.640 so I've got one point at every Hertz position, 00:22:45.640 --> 00:22:48.232 but I could pick anything else. 00:22:48.232 --> 00:22:49.690 The other thing I wanted to mention 00:22:49.690 --> 00:22:52.270 was that the reason I could do this 00:22:52.270 --> 00:22:54.430 is because I'm using the FFT. 00:22:54.430 --> 00:22:57.790 Because I told you if I just did sort 00:22:57.790 --> 00:23:01.430 of simple-minded implementation of the formula, 00:23:01.430 --> 00:23:03.610 I would take order p squared computations, 00:23:03.610 --> 00:23:07.240 where p is the length of the signal that I'm looking at. 00:23:07.240 --> 00:23:11.375 If I use the FFT, I go from p squared to-- 00:23:14.045 --> 00:23:19.270 I go down to p log to the base 2 of p, all right? 00:23:19.270 --> 00:23:22.080 Well, the number of points I have here is 48,000, 00:23:22.080 --> 00:23:29.530 so going from p to log 2 p is going from 48,000 to 16, which 00:23:29.530 --> 00:23:31.670 is a factor of 3,000. 00:23:31.670 --> 00:23:33.670 So the difference is I'm sitting at the terminal 00:23:33.670 --> 00:23:37.540 and I hit the Return key to get the FFT, 00:23:37.540 --> 00:23:40.180 and maybe at 0.1 seconds later, I 00:23:40.180 --> 00:23:42.700 get the answer, versus if I didn't 00:23:42.700 --> 00:23:44.395 use the FFT I'd wait five minutes, 00:23:44.395 --> 00:23:46.270 and then I wouldn't be trying to put together 00:23:46.270 --> 00:23:48.970 these figures for you. 00:23:48.970 --> 00:23:52.860 OK, so the FFT really makes a real practical difference, 00:23:52.860 --> 00:23:58.690 and it really revolutionized how numerical computations 00:23:58.690 --> 00:23:59.830 were done. 00:23:59.830 --> 00:24:03.735 OK, so here you now see the full spectral distribution 00:24:03.735 --> 00:24:05.110 if you're willing to let your eye 00:24:05.110 --> 00:24:07.135 interpolate between these samples that I've got. 00:24:07.135 --> 00:24:10.240 You see the full spectral distribution 00:24:10.240 --> 00:24:12.220 of that rectangular pulse, OK? 00:24:12.220 --> 00:24:14.890 So a short pulse in time. 00:24:14.890 --> 00:24:17.530 It's got certainly high DC content, 00:24:17.530 --> 00:24:21.040 but it's got tremendous frequency distribution, all 00:24:21.040 --> 00:24:22.420 the way out to high frequencies. 00:24:22.420 --> 00:24:23.753 In fact, all the way to the end. 00:24:23.753 --> 00:24:26.560 You're still seeing frequency content. 00:24:26.560 --> 00:24:28.070 It's visible to the eye. 00:24:28.070 --> 00:24:30.040 So all the way out to 24,000 Hertz, 00:24:30.040 --> 00:24:33.010 and you could keep going all right? 00:24:33.010 --> 00:24:34.810 This is not a sinc. 00:24:34.810 --> 00:24:37.600 It's a sinc-like function because if you extended it, 00:24:37.600 --> 00:24:39.190 it would go back up again. 00:24:39.190 --> 00:24:42.520 It's got that period, 2 pi, exactly as it should. 00:24:42.520 --> 00:24:44.260 But for all intents and purposes, 00:24:44.260 --> 00:24:49.450 for small values of frequency, the sine omega over 2 here-- 00:24:49.450 --> 00:24:50.890 this is essentially omega over 2, 00:24:50.890 --> 00:24:53.665 and this is a sinc-like function, OK? 00:24:53.665 --> 00:24:55.665 Now, this is different from what you saw before. 00:24:55.665 --> 00:25:02.000 Before you had a constant segment and frequency. 00:25:02.000 --> 00:25:03.750 At least, I think it's different from what 00:25:03.750 --> 00:25:05.125 you've seen before, unless you've 00:25:05.125 --> 00:25:06.360 done examples in recitation. 00:25:06.360 --> 00:25:10.060 But before what we had was, for instance, 00:25:10.060 --> 00:25:14.720 trying to get a frequency response that 00:25:14.720 --> 00:25:19.430 was like this, right? 00:25:19.430 --> 00:25:23.150 We ended up with a unit sample response that 00:25:23.150 --> 00:25:24.980 was a sinc function in time. 00:25:28.103 --> 00:25:29.270 This is going the other way. 00:25:29.270 --> 00:25:32.610 This is a rectangular function in time, 00:25:32.610 --> 00:25:35.270 giving rise to a sinc-like distribution in frequency. 00:25:39.734 --> 00:25:43.210 All right. 00:25:43.210 --> 00:25:47.740 Let's zoom in a little bit just to see what we have here. 00:25:47.740 --> 00:25:50.310 And this is exactly what we expect to be seeing, 00:25:50.310 --> 00:25:51.660 the sinc-like distribution. 00:25:51.660 --> 00:25:54.030 The height there is 256, as it should be. 00:25:54.030 --> 00:25:56.910 That's the N. And then this should be at 2 pi 00:25:56.910 --> 00:25:59.910 over capital N. If you translate that to what it means 00:25:59.910 --> 00:26:05.370 in actual frequency, this first null there is at 187.5 Hertz. 00:26:05.370 --> 00:26:09.140 That's 48 kilohertz divided by N, so that's-- 00:26:17.410 --> 00:26:20.200 OK, so that's that number. 00:26:20.200 --> 00:26:22.200 So that gives you some idea of how this thing is 00:26:22.200 --> 00:26:23.550 spread in frequency. 00:26:26.590 --> 00:26:31.840 OK, so if this pulse is applied directly to the loudspeaker-- 00:26:31.840 --> 00:26:35.450 well, here, the loudspeaker passband 00:26:35.450 --> 00:26:38.890 goes from about 100 Hertz to, let's say, 10,000 Hertz. 00:26:38.890 --> 00:26:42.460 You see there's huge amounts of the energy that's 00:26:42.460 --> 00:26:46.180 not in the passband of the loudspeaker. 00:26:46.180 --> 00:26:46.680 All right? 00:26:46.680 --> 00:26:49.470 So this is not going to do very well if you just directly apply 00:26:49.470 --> 00:26:51.990 that pulse to the loudspeaker. 00:26:51.990 --> 00:26:56.073 You have to match the frequency content of the input 00:26:56.073 --> 00:26:57.990 to the frequency response of the system you're 00:26:57.990 --> 00:27:01.540 trying to get this over. 00:27:01.540 --> 00:27:02.040 OK. 00:27:04.990 --> 00:27:06.950 Now, just as an experiment before we 00:27:06.950 --> 00:27:09.890 get back to sending this across a loudspeaker, 00:27:09.890 --> 00:27:12.560 let's take a look at what happens 00:27:12.560 --> 00:27:17.392 if we send that rectangular pulse over a lowpass channel. 00:27:17.392 --> 00:27:19.100 So let's save the bandpass channel, which 00:27:19.100 --> 00:27:21.610 is a little more involved, for later, 00:27:21.610 --> 00:27:26.720 and just look at what happens if we 00:27:26.720 --> 00:27:30.120 send this pulse over a lowpass channel. 00:27:30.120 --> 00:27:33.710 So I'm thinking about a channel whose frequency 00:27:33.710 --> 00:27:36.450 characteristic-- 00:27:36.450 --> 00:27:39.320 this is h of omega-- 00:27:39.320 --> 00:27:41.120 it passes low frequencies, and then it 00:27:41.120 --> 00:27:43.321 truncates higher frequencies. 00:27:47.400 --> 00:27:47.900 OK? 00:27:47.900 --> 00:27:51.200 And what I'm going to do is send in an x of N, 00:27:51.200 --> 00:27:54.440 which is this pulse in time. 00:27:54.440 --> 00:28:00.410 256 at height 1, and then everything 0. 00:28:04.900 --> 00:28:07.430 All right, so if I'm thinking of it in the frequency domain, 00:28:07.430 --> 00:28:10.097 then I take the spectral content of the signal, which we've just 00:28:10.097 --> 00:28:13.990 worked out, and multiplied by the frequency response 00:28:13.990 --> 00:28:17.470 characteristic, which just basically selects out 00:28:17.470 --> 00:28:19.600 the frequency content that's in the passband 00:28:19.600 --> 00:28:21.550 and rejects everything else. 00:28:21.550 --> 00:28:24.460 And then that gives me the spectral content of the output, 00:28:24.460 --> 00:28:27.820 and I can translate that back to what 00:28:27.820 --> 00:28:29.060 happens in the time domain. 00:28:29.060 --> 00:28:33.160 So here is-- well, actually, let's take a zoomed-in version. 00:28:33.160 --> 00:28:39.040 I've taken a lowpass filter, where this cutoff corresponds 00:28:39.040 --> 00:28:40.960 to actually a cutoff at 400 Hertz 00:28:40.960 --> 00:28:44.740 if you're thinking in terms of the underlying waveform. 00:28:44.740 --> 00:28:48.850 So what we've done is take the rectangular pulse, 00:28:48.850 --> 00:28:50.230 put it through a lowpass filter-- 00:28:50.230 --> 00:28:53.020 and I'm assuming an ideal lowpass filter-- 00:28:53.020 --> 00:28:56.860 passes everything in this frequency band and nothing 00:28:56.860 --> 00:28:59.680 outside of it, OK? 00:28:59.680 --> 00:29:01.780 So here again, we see we're selecting out 00:29:01.780 --> 00:29:03.640 part of the spectral structure of the input, 00:29:03.640 --> 00:29:06.010 but there's huge amounts of the energy 00:29:06.010 --> 00:29:08.410 of the input that are being left out 00:29:08.410 --> 00:29:11.450 of the output of the filter. 00:29:11.450 --> 00:29:15.400 Now, in the time domain, look at what this corresponds to. 00:29:15.400 --> 00:29:18.790 It's an approximation to this pulse, 00:29:18.790 --> 00:29:21.132 but it's one in which all the high-frequency content has 00:29:21.132 --> 00:29:23.590 disappeared because we've only let the low-frequency pieces 00:29:23.590 --> 00:29:24.250 through. 00:29:24.250 --> 00:29:28.450 So what you have is a very rounded kind of pulse. 00:29:28.450 --> 00:29:34.450 It spreads out well over the 256 mark, so here's the 0 to 256, 00:29:34.450 --> 00:29:38.200 but this thing actually spills over into adjacent bit slots. 00:29:38.200 --> 00:29:40.850 We've taken out the high-frequency components, 00:29:40.850 --> 00:29:43.370 so it can't make any sharp turns anymore. 00:29:43.370 --> 00:29:46.900 It's got this lower-frequency wiggling. 00:29:46.900 --> 00:29:51.040 As you can imagine, if I made this even smaller, 00:29:51.040 --> 00:29:54.520 my wiggling would become even more leisurely here, 00:29:54.520 --> 00:29:57.767 and I would spill further into the adjacent bit slots, OK? 00:29:57.767 --> 00:29:59.350 So this is what lowpass filtering will 00:29:59.350 --> 00:30:03.810 do to that rectangular pulse. 00:30:03.810 --> 00:30:05.880 Any questions on this piece? 00:30:08.810 --> 00:30:09.310 OK. 00:30:13.672 --> 00:30:15.630 Now, we've actually seen examples of this type. 00:30:15.630 --> 00:30:18.140 I flashed these up last time. 00:30:18.140 --> 00:30:21.230 It's the same idea, except it was not just a single pulse. 00:30:21.230 --> 00:30:23.838 It was a succession of pulses like this. 00:30:23.838 --> 00:30:25.130 So you could do the same thing. 00:30:25.130 --> 00:30:27.500 Have a succession of pulses like this. 00:30:27.500 --> 00:30:31.520 Take its DTFT to assess the spectral content. 00:30:31.520 --> 00:30:34.520 I'm showing you not actually the DTFT but something proportional 00:30:34.520 --> 00:30:35.180 to it here. 00:30:35.180 --> 00:30:38.690 This is-- so ignore the labels there. 00:30:38.690 --> 00:30:40.550 Think of this as essentially the DTFT. 00:30:40.550 --> 00:30:43.770 To within the scale factor it is. 00:30:43.770 --> 00:30:47.120 So you can see that I have spectral content 00:30:47.120 --> 00:30:50.210 all the way out to the edges. 00:30:50.210 --> 00:30:52.310 But now send it through a lowpass channel 00:30:52.310 --> 00:30:55.040 which zeros out all the spectral content 00:30:55.040 --> 00:30:57.415 outside of some central region, and what 00:30:57.415 --> 00:30:58.790 you have coming out the other end 00:30:58.790 --> 00:31:01.560 is something that can't take the sharp turns anymore, 00:31:01.560 --> 00:31:03.410 so it's much more rounded. 00:31:03.410 --> 00:31:05.330 And as you narrow it down still further, 00:31:05.330 --> 00:31:08.870 you get even more rounding, and you get a spilling over. 00:31:08.870 --> 00:31:12.170 You see this sharply confined rectangular pulse now spills 00:31:12.170 --> 00:31:15.500 over into the adjacent slots. 00:31:15.500 --> 00:31:17.600 And you can go still further on that. 00:31:17.600 --> 00:31:19.880 The top plot here is the same as the bottom plot 00:31:19.880 --> 00:31:21.560 in the previous one. 00:31:21.560 --> 00:31:23.270 But this is just showing the sequence. 00:31:23.270 --> 00:31:27.350 So as you narrow it down further and further and further, 00:31:27.350 --> 00:31:29.990 what comes out the other end gets 00:31:29.990 --> 00:31:34.400 much less distinctive in its features, OK? 00:31:34.400 --> 00:31:37.370 It can only-- this only has very low-frequency content, 00:31:37.370 --> 00:31:39.380 so it can't do any sharp turns. 00:31:39.380 --> 00:31:42.010 And this is an eye diagram corresponding to-- 00:31:42.010 --> 00:31:45.140 in each of these received signals-- just 00:31:45.140 --> 00:31:48.113 to show you how detection gets difficult 00:31:48.113 --> 00:31:50.030 if you've got a lowpass channel and you've got 00:31:50.030 --> 00:31:54.050 this pulse that's not well-defined. 00:31:54.050 --> 00:31:56.240 Now, how might you actually-- 00:31:56.240 --> 00:31:58.520 how might you get a better-defined pulse 00:31:58.520 --> 00:31:59.580 for a given channel? 00:31:59.580 --> 00:32:03.020 So if I gave you this channel-- 00:32:03.020 --> 00:32:06.685 we sent in this pulse of length 256 samples, 00:32:06.685 --> 00:32:08.810 and we got something that we didn't like because it 00:32:08.810 --> 00:32:12.057 spilled over into other slots. 00:32:12.057 --> 00:32:13.640 The reason it spilled into other slots 00:32:13.640 --> 00:32:17.480 is we were cutting out too much of its spectral content. 00:32:17.480 --> 00:32:22.540 What could you do to this pulse to get more of its energy 00:32:22.540 --> 00:32:23.890 across that bandpass channel? 00:32:27.243 --> 00:32:28.180 Yeah? 00:32:28.180 --> 00:32:28.680 Sorry? 00:32:28.680 --> 00:32:30.120 AUDIENCE: Set a longer pulse. 00:32:30.120 --> 00:32:31.698 PROFESSOR: Set a longer pulse. 00:32:31.698 --> 00:32:32.490 How does that work? 00:32:32.490 --> 00:32:34.440 Now, you see, if you make N longer, 00:32:34.440 --> 00:32:37.680 you make the pulse longer, you shrink this correspondingly 00:32:37.680 --> 00:32:39.270 in the frequency domain, right? 00:32:39.270 --> 00:32:47.120 We said that on these sinc-type characteristics, 00:32:47.120 --> 00:32:49.890 the height was N. This first null was 00:32:49.890 --> 00:32:53.550 2 pi over N. Make the pulse longer, 00:32:53.550 --> 00:32:56.430 you pull the main lobe in tighter, and more of this 00:32:56.430 --> 00:32:58.780 is going to go through, OK? 00:32:58.780 --> 00:33:00.630 And you can see that clearly. 00:33:00.630 --> 00:33:03.160 That's something you can explore in your experiment. 00:33:03.160 --> 00:33:06.630 So if you wanted to get more clearly defined 00:33:06.630 --> 00:33:11.703 output for a given lowpass channel, 00:33:11.703 --> 00:33:13.870 you might want to increase the length of your pulse. 00:33:13.870 --> 00:33:16.245 Of course, that's going to slow down your signaling rate, 00:33:16.245 --> 00:33:18.090 so there's a trade-off involved, right? 00:33:22.740 --> 00:33:24.680 Now, we've actually-- this is just 00:33:24.680 --> 00:33:29.190 to sort of step back and point out that what we're seeing here 00:33:29.190 --> 00:33:35.510 are some properties that are inherent to Fourier transforms, 00:33:35.510 --> 00:33:36.010 OK? 00:33:36.010 --> 00:33:38.310 So it's typically the case that if you've 00:33:38.310 --> 00:33:42.155 got a signal that's wide in time, it's narrow in frequency. 00:33:42.155 --> 00:33:43.530 And if you make it wider in time, 00:33:43.530 --> 00:33:45.670 it gets narrower in frequency. 00:33:45.670 --> 00:33:48.630 In fact, as I mentioned up there, the uncertainty 00:33:48.630 --> 00:33:52.150 principle in physics really comes from this result. 00:33:52.150 --> 00:33:59.130 It says that the spread in time times the spread in frequency 00:33:59.130 --> 00:34:00.330 has some lower bound, OK? 00:34:00.330 --> 00:34:04.270 So this is some number that's strictly positive. 00:34:04.270 --> 00:34:07.260 So you can't make a signal arbitrarily concentrated 00:34:07.260 --> 00:34:10.020 in time and concentrated in frequency. 00:34:10.020 --> 00:34:11.550 If you make one small, the other one 00:34:11.550 --> 00:34:13.438 will have to grow correspondingly. 00:34:13.438 --> 00:34:15.480 So actually, the uncertainty principle in physics 00:34:15.480 --> 00:34:18.040 is precisely a theorem in Fourier transforms 00:34:18.040 --> 00:34:20.350 if you study it. 00:34:20.350 --> 00:34:25.260 Here is another such complementarity or duality. 00:34:25.260 --> 00:34:28.500 The smoother you make a signal in time, the more sharp 00:34:28.500 --> 00:34:30.590 it is in frequency. 00:34:30.590 --> 00:34:31.090 Let's see. 00:34:31.090 --> 00:34:33.840 We saw that here, for instance, right? 00:34:33.840 --> 00:34:36.840 We had a signal in time, namely the unit sample 00:34:36.840 --> 00:34:39.150 response of the ideal filter. 00:34:39.150 --> 00:34:42.050 This was-- this didn't have any sharp edges to it. 00:34:42.050 --> 00:34:44.820 It was a sinc function. 00:34:44.820 --> 00:34:47.219 You got something that's smooth in time, 00:34:47.219 --> 00:34:51.313 it ends up having sharp edges in frequency, OK? 00:34:51.313 --> 00:34:52.980 And the more smooth you make it in time, 00:34:52.980 --> 00:34:56.313 the sharper it gets in frequency. 00:34:56.313 --> 00:34:57.480 And this we've already seen. 00:34:57.480 --> 00:35:00.030 This is the kind of trade-off we're talking about. 00:35:00.030 --> 00:35:03.280 So these are characteristics to be on the lookout for. 00:35:03.280 --> 00:35:07.310 In fact, I just did a little experiment here. 00:35:07.310 --> 00:35:11.090 What if we decided not to try and send a rectangular pulse 00:35:11.090 --> 00:35:14.000 over, but we smoothed out the pulse a little bit 00:35:14.000 --> 00:35:16.200 to get rid of that sharp edge? 00:35:16.200 --> 00:35:20.630 So that's another way that you can try and get 00:35:20.630 --> 00:35:25.735 a pulse over a lowpass channel like this. 00:35:25.735 --> 00:35:27.110 So you see, what I'm trying to do 00:35:27.110 --> 00:35:30.740 is not have the sharp discontinuities, the 0 to 1 00:35:30.740 --> 00:35:32.000 and the 1 to 0. 00:35:32.000 --> 00:35:35.460 I want a more rounded behavior in the time domain 00:35:35.460 --> 00:35:39.950 so I get a sharper concentration in the frequency domain, 00:35:39.950 --> 00:35:41.850 and you can actually see how that works. 00:35:41.850 --> 00:35:54.180 So what I've actually done here is, instead of a signal that-- 00:35:54.180 --> 00:35:55.770 well, if you'll permit me, I don't 00:35:55.770 --> 00:35:59.010 want to draw these stem plots, the lollipop figures, 00:35:59.010 --> 00:36:03.000 because they get painful to draw, but just assume 00:36:03.000 --> 00:36:05.460 that this is 256 such things. 00:36:05.460 --> 00:36:08.220 That's the rectangular pulse we were trying to send before. 00:36:08.220 --> 00:36:10.320 What I've done now is instead say, 00:36:10.320 --> 00:36:11.880 well, let me around the edges, so I'm 00:36:11.880 --> 00:36:17.160 going to have a half cycle of a cosine for that edge, OK? 00:36:17.160 --> 00:36:18.870 So that's what-- these are the samples 00:36:18.870 --> 00:36:20.760 I'm going to use at this edge, and I'm 00:36:20.760 --> 00:36:23.640 going to have a half-cycle of a cosine at this edge. 00:36:23.640 --> 00:36:26.820 This is actually something that's used quite a bit 00:36:26.820 --> 00:36:28.898 in applications. 00:36:28.898 --> 00:36:30.190 All right, so what have I done? 00:36:30.190 --> 00:36:33.690 I've removed the sharp edge and gotten 00:36:33.690 --> 00:36:35.860 a much-rounded transition. 00:36:35.860 --> 00:36:39.330 In fact, if you're thinking of continuous functions, 00:36:39.330 --> 00:36:41.610 the original had a discontinuity, 00:36:41.610 --> 00:36:43.920 whereas here I've got to take two derivatives before I 00:36:43.920 --> 00:36:47.430 encounter a discontinuity because the function itself 00:36:47.430 --> 00:36:49.943 and its slope at these ends are well-matched, 00:36:49.943 --> 00:36:51.360 so I've got to differentiate twice 00:36:51.360 --> 00:36:52.630 before I get a discontinuity. 00:36:52.630 --> 00:36:56.160 So actually, it has quite some smoothness to it. 00:36:56.160 --> 00:36:58.980 Smooth in time means more tightly concentrated 00:36:58.980 --> 00:37:00.000 in frequency. 00:37:00.000 --> 00:37:05.050 So all I'm doing on the next slide 00:37:05.050 --> 00:37:08.440 is showing you, so your eye can compare, 00:37:08.440 --> 00:37:12.160 what the spectral content is of the original pulse 00:37:12.160 --> 00:37:13.438 and of this rounded pulse. 00:37:13.438 --> 00:37:14.980 And for the rounded pulse, I actually 00:37:14.980 --> 00:37:17.810 just flipped it over so that you can compare more easily. 00:37:17.810 --> 00:37:21.250 So this is the negative of the DTFT magnitude 00:37:21.250 --> 00:37:23.140 of the shaped pulse. 00:37:23.140 --> 00:37:25.690 And you can sort of see right away here 00:37:25.690 --> 00:37:29.000 that the frequency content has essentially settled out. 00:37:29.000 --> 00:37:32.230 It's almost all contained in this smaller region, 00:37:32.230 --> 00:37:34.590 whereas the rectangular pulse had 00:37:34.590 --> 00:37:37.160 a frequency content that went way off to high frequencies, 00:37:37.160 --> 00:37:37.660 right? 00:37:37.660 --> 00:37:42.250 Went off to 24 kilohertz, for instance, in our example. 00:37:42.250 --> 00:37:45.065 So the frequency content of the shaped pulse 00:37:45.065 --> 00:37:46.690 is much more tightly contained, and you 00:37:46.690 --> 00:37:48.398 have a much better chance of getting that 00:37:48.398 --> 00:37:50.480 across a lowpass channel. 00:37:50.480 --> 00:37:53.470 So this is just a little bit of shaping 00:37:53.470 --> 00:37:55.570 that can make a big difference in terms 00:37:55.570 --> 00:37:59.480 of adapting the signal you're trying to send to the channel. 00:37:59.480 --> 00:38:02.020 So if I look in the time domain, sending these two 00:38:02.020 --> 00:38:05.740 pulses over the same lowpass channel, 00:38:05.740 --> 00:38:08.780 you can do a visual comparison of what comes out. 00:38:08.780 --> 00:38:13.300 So here's the original 256 rectangular pulse coming out 00:38:13.300 --> 00:38:14.950 the other end. 00:38:14.950 --> 00:38:17.770 Here is my shaped pulse coming out the other end. 00:38:17.770 --> 00:38:20.980 And you can see the shaped pulse is much more tightly confined 00:38:20.980 --> 00:38:23.600 in the bit slot that I assigned to it, 00:38:23.600 --> 00:38:25.420 so it's much more tightly confined 00:38:25.420 --> 00:38:30.040 around the 256-sample width, OK? 00:38:30.040 --> 00:38:32.697 So this is another thing that is done, 00:38:32.697 --> 00:38:34.780 and it's done by thinking in the frequency domain. 00:38:34.780 --> 00:38:36.360 People designed these pulses thinking 00:38:36.360 --> 00:38:37.360 in the frequency domain. 00:38:37.360 --> 00:38:41.250 They're not doing convolution. 00:38:41.250 --> 00:38:44.160 OK, that was all lowpass, but we want 00:38:44.160 --> 00:38:49.075 to look at bandpass, so just a couple of quick examples there. 00:38:49.075 --> 00:38:50.325 So we're back to the speakers. 00:38:53.280 --> 00:38:56.460 And we're taking our rectangular pulse 00:38:56.460 --> 00:38:58.590 and applying it to the speaker. 00:38:58.590 --> 00:39:02.630 So here's what the spectrum looks like after ideal bandpass 00:39:02.630 --> 00:39:03.130 filtering. 00:39:03.130 --> 00:39:04.590 So I'm not actually filtering with the speaker 00:39:04.590 --> 00:39:05.550 characteristic. 00:39:05.550 --> 00:39:07.860 I'm assuming an ideal bandpass that 00:39:07.860 --> 00:39:10.650 extends from 100 Hertz to 10,000 Hertz 00:39:10.650 --> 00:39:13.450 and zeros out everything outside that range. 00:39:13.450 --> 00:39:16.950 So I send in the spectral characteristic 00:39:16.950 --> 00:39:18.890 of my rectangular pulse. 00:39:18.890 --> 00:39:20.880 I shape it with the bandpass. 00:39:20.880 --> 00:39:22.680 So what comes out is something that has 00:39:22.680 --> 00:39:23.982 this spectral characteristic. 00:39:23.982 --> 00:39:26.190 And you can see that the frequency content is sharply 00:39:26.190 --> 00:39:32.400 limited 10 kilohertz above 0, and then there 00:39:32.400 --> 00:39:35.010 is actually a central region that's entirely missing. 00:39:35.010 --> 00:39:39.820 So remember, this had to go originally up to 256, 00:39:39.820 --> 00:39:42.780 but because we've chopped out the center portion, 00:39:42.780 --> 00:39:46.410 we're only going out to 150-something out there. 00:39:46.410 --> 00:39:47.910 So actually, if I zoom in, you can 00:39:47.910 --> 00:39:50.500 see that a lot more closely. 00:39:50.500 --> 00:39:52.920 So this is a zoomed-in version of what 00:39:52.920 --> 00:39:58.340 comes out from a loudspeaker, from a bandpass filter, 00:39:58.340 --> 00:40:00.830 if I send in a rectangular pulse. 00:40:00.830 --> 00:40:03.512 The very low frequencies are entirely missing, 00:40:03.512 --> 00:40:05.220 and we saw in the previous characteristic 00:40:05.220 --> 00:40:08.260 that the very high frequencies are missing, as well. 00:40:08.260 --> 00:40:12.750 So what's the shape of the pulse that you get out? 00:40:12.750 --> 00:40:15.630 Not very good. 00:40:15.630 --> 00:40:17.370 Because the low frequencies are missing, 00:40:17.370 --> 00:40:19.620 this thing tends to sag in the middle. 00:40:19.620 --> 00:40:23.160 It can't hold up DC. 00:40:23.160 --> 00:40:25.680 And it can't make the very sharp transitions, either, 00:40:25.680 --> 00:40:27.550 so there is a more leisurely transition. 00:40:27.550 --> 00:40:30.060 But again, this can't stay at that level. 00:40:30.060 --> 00:40:32.040 It actually asymptotes. 00:40:32.040 --> 00:40:37.530 So the actual pulse occupies-- before it settles, occupies way 00:40:37.530 --> 00:40:41.760 over the 256 bits that I've allotted to it, OK? 00:40:41.760 --> 00:40:44.310 So taking that rectangular pulse and directly putting it 00:40:44.310 --> 00:40:48.090 on the speaker is going to give you something not pretty 00:40:48.090 --> 00:40:52.270 at the other end and something that you cannot signal with. 00:40:52.270 --> 00:40:58.140 So the question is what to do about that, OK? 00:40:58.140 --> 00:41:02.552 And the answer to that is this thing that we call modulation. 00:41:02.552 --> 00:41:04.260 We've already seen it in different forms. 00:41:08.198 --> 00:41:10.240 Let's think about it now in the frequency domain. 00:41:14.190 --> 00:41:18.130 OK, so here's what we're going to do. 00:41:18.130 --> 00:41:19.400 Want the big stick of chalk. 00:41:19.400 --> 00:41:19.900 OK. 00:41:27.430 --> 00:41:29.717 We're going to shape the spectral characteristics 00:41:29.717 --> 00:41:30.300 of the signal. 00:41:30.300 --> 00:41:34.600 We started off with some time-domain signal links, x N, 00:41:34.600 --> 00:41:38.920 corresponding DTFT x omega. 00:41:38.920 --> 00:41:42.010 It wasn't well-matched to our channel characteristics, 00:41:42.010 --> 00:41:46.600 and so what we're going to do is multiply 00:41:46.600 --> 00:41:51.260 by some carrier frequency. 00:41:51.260 --> 00:41:55.150 And this is simple amplitude modulation. 00:41:55.150 --> 00:41:57.790 We're just referring to it as modulation here. 00:42:05.210 --> 00:42:06.740 We get some signal out here. 00:42:12.400 --> 00:42:15.720 And the question is, what is the frequency characteristic 00:42:15.720 --> 00:42:17.226 of that signal? 00:42:17.226 --> 00:42:18.990 OK, we've already seen what the frequency 00:42:18.990 --> 00:42:21.355 characteristic of the input is. 00:42:21.355 --> 00:42:23.480 What's the frequency characteristic of that signal? 00:42:34.240 --> 00:42:40.260 So just to give you an example, we had our x of omega 00:42:40.260 --> 00:42:43.800 looking something like the sinc shape. 00:42:48.220 --> 00:42:49.570 Remember height N there? 00:42:56.630 --> 00:42:59.180 And the question is, what's the spectrum of this? 00:42:59.180 --> 00:43:00.980 And there's a computation up there 00:43:00.980 --> 00:43:03.560 that I don't want to go through, but it shows you 00:43:03.560 --> 00:43:05.510 that the answer is actually quite simple. 00:43:05.510 --> 00:43:08.120 So if I call this-- 00:43:08.120 --> 00:43:11.567 let me call this t of N because it's 00:43:11.567 --> 00:43:13.025 the signal we're going to transmit. 00:43:17.280 --> 00:43:21.510 Here's what the spectrum of the transmitted signal looks like. 00:43:26.610 --> 00:43:27.870 There is omega. 00:43:27.870 --> 00:43:29.430 There's pi. 00:43:29.430 --> 00:43:30.560 Here's omega c. 00:43:35.690 --> 00:43:37.800 OK, so the prescription is simple. 00:43:37.800 --> 00:43:42.430 Take the spectral characteristic and replicate it at omega c 00:43:42.430 --> 00:43:46.640 and replicate it at minus omega c, and scale by 1/2. 00:43:46.640 --> 00:43:52.310 So what you're going to get is this characteristic here, 00:43:52.310 --> 00:43:57.680 this characteristic here, and the height will be N over 2. 00:43:57.680 --> 00:43:58.580 It's that simple. 00:43:58.580 --> 00:44:00.347 So you can go through the math. 00:44:00.347 --> 00:44:02.180 When you're done with the math, what it says 00:44:02.180 --> 00:44:04.310 is that the spectral characteristic 00:44:04.310 --> 00:44:09.260 of this modulated signal is the spectral characteristic 00:44:09.260 --> 00:44:13.550 of the envelope of the baseband signal, or the envelope, 00:44:13.550 --> 00:44:17.642 but translated to the position of the carrier. 00:44:17.642 --> 00:44:19.100 So you can begin to see how this is 00:44:19.100 --> 00:44:21.980 going to help us shift a signal to get it 00:44:21.980 --> 00:44:23.323 across a bandpass channel. 00:44:23.323 --> 00:44:24.740 We started off with something that 00:44:24.740 --> 00:44:28.370 was not well-suited to the loudspeaker we now have a way 00:44:28.370 --> 00:44:30.740 to shift its energy to get it right 00:44:30.740 --> 00:44:33.170 in the passband of the speaker by picking the carrier 00:44:33.170 --> 00:44:34.830 frequency appropriately. 00:44:34.830 --> 00:44:35.330 All right? 00:44:39.990 --> 00:44:44.370 I think I'm going to skip some of this. 00:44:44.370 --> 00:44:47.060 But let's look at what this does. 00:44:47.060 --> 00:44:48.560 It's really this picture, but I just 00:44:48.560 --> 00:44:51.520 want to show you how it works with actual waveforms. 00:44:51.520 --> 00:44:57.650 So here is the rectangular pulse times the cosine. 00:44:57.650 --> 00:44:59.270 I've picked-- what did I pick? 00:44:59.270 --> 00:45:01.120 1 kilohertz as my carrier? 00:45:01.120 --> 00:45:02.210 Yeah. 00:45:02.210 --> 00:45:04.280 1,000 Hertz was the carrier. 00:45:04.280 --> 00:45:07.400 1,000 Hertz sits comfortably in the passband of a speaker, 00:45:07.400 --> 00:45:10.780 so it's a reasonable choice. 00:45:10.780 --> 00:45:13.480 By the way, in typical AM, the carrier frequencies 00:45:13.480 --> 00:45:15.458 are much higher than-- or the ratio 00:45:15.458 --> 00:45:17.500 of the carrier frequency to the rate of variation 00:45:17.500 --> 00:45:20.387 of the envelope is much higher than what 00:45:20.387 --> 00:45:21.470 we have in these examples. 00:45:21.470 --> 00:45:25.060 So the audio channel is actually very challenging. 00:45:25.060 --> 00:45:28.060 OK, so this is my modulated signal. 00:45:28.060 --> 00:45:31.130 The question is, what does its spectrum look like? 00:45:31.130 --> 00:45:34.660 So I run it through the FFT, and indeed I 00:45:34.660 --> 00:45:38.470 get the replication here. 00:45:38.470 --> 00:45:39.970 Let's zoom in a little bit so we can 00:45:39.970 --> 00:45:41.178 see it a little more closely. 00:45:47.030 --> 00:45:49.390 So what I have is those two sinc-like spectral 00:45:49.390 --> 00:45:53.440 characteristics, but translated to sit centered at 1,000 00:45:53.440 --> 00:45:55.240 Hertz and minus 1,000 Hertz. 00:45:55.240 --> 00:45:58.270 Remember that this is 0 out here, OK? 00:45:58.270 --> 00:45:59.770 And the height, well, it's now it's 00:45:59.770 --> 00:46:02.980 128, which is half the 256 that I had before. 00:46:14.000 --> 00:46:16.720 So in terms of positioning this within where 00:46:16.720 --> 00:46:18.680 the loudspeaker will transmit it, 00:46:18.680 --> 00:46:20.935 we were taking the lower cutoff of the loudspeaker 00:46:20.935 --> 00:46:22.900 as being around 100 Hertz. 00:46:22.900 --> 00:46:24.920 This is where 100 Hertz sits. 00:46:24.920 --> 00:46:26.350 So you can see that a huge amount 00:46:26.350 --> 00:46:28.730 of the energy of the pulse is getting through. 00:46:28.730 --> 00:46:29.980 It's at the wrong frequencies. 00:46:29.980 --> 00:46:32.300 We'll have to deal with getting it back. 00:46:32.300 --> 00:46:36.300 But at least getting the energy across is working here, OK? 00:46:36.300 --> 00:46:38.927 And the upper cutoff of the speaker is way off over here. 00:46:38.927 --> 00:46:40.510 So this is 10,000, but I'm showing you 00:46:40.510 --> 00:46:45.520 a zoomed-in version, so the upper frequency is way off. 00:46:45.520 --> 00:46:47.650 In fact, that also brings up the idea 00:46:47.650 --> 00:46:50.830 that you could actually do this trick multiple times. 00:46:50.830 --> 00:46:53.470 You could actually pick another carrier frequency somewhat 00:46:53.470 --> 00:46:56.517 higher than this with some other modulating signal on it 00:46:56.517 --> 00:46:57.850 and tuck that in there, as well. 00:46:57.850 --> 00:47:00.790 So you could simultaneously transmit messages 00:47:00.790 --> 00:47:03.190 on multiple carriers through that same speaker, 00:47:03.190 --> 00:47:05.210 and you'll be exploring that, as well. 00:47:05.210 --> 00:47:05.710 OK. 00:47:08.500 --> 00:47:10.960 So what does the-- what's the time-domain signal that 00:47:10.960 --> 00:47:13.490 corresponds to this looks like? 00:47:13.490 --> 00:47:17.650 So when you get it, you impress this modulated signal 00:47:17.650 --> 00:47:20.830 on the loudspeaker, on this bandpass filter. 00:47:20.830 --> 00:47:23.760 What's the output of the bandpass filter? 00:47:23.760 --> 00:47:28.390 You can see it's almost exactly what you put in, OK? 00:47:28.390 --> 00:47:30.300 There's a little bit of distortion 00:47:30.300 --> 00:47:34.290 at the different places, but it's basically exactly 00:47:34.290 --> 00:47:36.000 the pulse that you put in. 00:47:36.000 --> 00:47:38.340 So almost all the energy has gone through, 00:47:38.340 --> 00:47:41.280 and you don't have significant distortion 00:47:41.280 --> 00:47:43.200 because you've squarely placed the energy 00:47:43.200 --> 00:47:44.580 in the passband of the filter. 00:47:44.580 --> 00:47:47.880 OK, now how do we recover? 00:47:47.880 --> 00:47:52.470 How do we get back the original baseband signal? 00:47:52.470 --> 00:47:56.970 Well, it turns out it's very easy. 00:47:56.970 --> 00:47:59.800 Let me actually do it in pictures, 00:47:59.800 --> 00:48:03.030 and then we'll look at the math. 00:48:03.030 --> 00:48:05.940 This is what's coming in to our receiver, 00:48:05.940 --> 00:48:10.140 and now we've got to process this to get back a signal that 00:48:10.140 --> 00:48:12.850 has this spectrum. 00:48:12.850 --> 00:48:15.010 We've learned a trick, which is modulation. 00:48:15.010 --> 00:48:17.680 Multiply it by a cosine, and you'll take the spectrum 00:48:17.680 --> 00:48:22.257 and replicate it at omega c and at minus omega c. 00:48:22.257 --> 00:48:23.840 I'm going to use the same trick again. 00:48:23.840 --> 00:48:27.130 I'm going to take the received signal multiplied 00:48:27.130 --> 00:48:30.010 by a cosine of the same carrier frequency, 00:48:30.010 --> 00:48:32.300 and what's that going to do? 00:48:32.300 --> 00:48:37.500 Well, my scale is getting bigger here each time. 00:48:41.250 --> 00:48:45.180 So here's my omega c. 00:48:45.180 --> 00:48:47.640 Here's my minus omega c. 00:48:47.640 --> 00:48:50.520 What's coming in is this signal. 00:48:50.520 --> 00:48:52.890 I'm going to multiply it by cosine omega c, so what 00:48:52.890 --> 00:48:54.510 does that do in the spectrum? 00:48:54.510 --> 00:48:58.770 It takes this spectrum, replicates it at omega c. 00:48:58.770 --> 00:48:59.890 So what does that do? 00:48:59.890 --> 00:49:05.330 Well, it puts a piece here, and it puts a piece here, 00:49:05.330 --> 00:49:08.580 at 2 omega c, right? 00:49:08.580 --> 00:49:11.612 Because I've taken this spectrum-- 00:49:11.612 --> 00:49:13.320 you've got to imagine the change of scale 00:49:13.320 --> 00:49:14.760 so I can draw all of this. 00:49:14.760 --> 00:49:16.320 I've taken this spectrum and I've 00:49:16.320 --> 00:49:19.800 placed it centered on omega c. 00:49:19.800 --> 00:49:21.270 And this is now-- 00:49:21.270 --> 00:49:22.845 this is the N over 2 here. 00:49:22.845 --> 00:49:24.720 Oh, but now it's going to be N over 4, right? 00:49:24.720 --> 00:49:28.140 Because I divide by 2. 00:49:28.140 --> 00:49:30.840 And then I take the same spectrum 00:49:30.840 --> 00:49:33.630 and I center it on minus omega c. 00:49:33.630 --> 00:49:37.440 OK, so I've got the N over 4 piece 00:49:37.440 --> 00:49:41.250 here, but I'm going to have the other-- 00:49:41.250 --> 00:49:43.080 oh, sorry. 00:49:43.080 --> 00:49:45.610 I drew it in the wrong place. 00:49:45.610 --> 00:49:47.520 I'm going to center it at omega c. 00:49:47.520 --> 00:49:52.590 So this is going to end up at minus 2 omega c. 00:49:52.590 --> 00:49:54.280 And the replication here. 00:49:54.280 --> 00:49:58.380 So there's going to be a second one sitting here at the origin. 00:49:58.380 --> 00:50:01.190 So the net effect at the origin is 00:50:01.190 --> 00:50:10.460 that I get the original spectrum, but scaled by a half, 00:50:10.460 --> 00:50:14.360 and then I get vestiges of this, if you like, 00:50:14.360 --> 00:50:18.188 centered at twice the carrier frequency, OK? 00:50:18.188 --> 00:50:19.980 And that's actually what the algebra shows. 00:50:19.980 --> 00:50:22.290 The algebra is very simple. 00:50:22.290 --> 00:50:24.700 You're receiving this. 00:50:24.700 --> 00:50:26.720 Multiply it again by a cosine. 00:50:26.720 --> 00:50:29.870 So take the received signal and multiply it again by a cosine. 00:50:29.870 --> 00:50:33.290 Well, if you substitute for what the received signal is, 00:50:33.290 --> 00:50:36.380 you get x n times cosine squared. 00:50:36.380 --> 00:50:39.110 Cosine squared, if you use a standard identity, 00:50:39.110 --> 00:50:40.595 splits into these two terms. 00:50:43.140 --> 00:50:43.640 Let's see. 00:50:43.640 --> 00:50:45.610 Do I have that right? 00:50:45.610 --> 00:50:46.250 Yeah. 00:50:46.250 --> 00:50:52.100 So here's the 0.5 x n sitting here at the origin, 00:50:52.100 --> 00:50:57.210 and here's another term, which is x n times the cosine. 00:50:57.210 --> 00:50:59.420 So this is like a modulated signal, 00:50:59.420 --> 00:51:03.030 but modulated by twice the carrier frequency. 00:51:03.030 --> 00:51:05.060 So what does this translate to? 00:51:05.060 --> 00:51:10.730 Well, it's actually 0.5 times x n times cosine 2 omega c. 00:51:10.730 --> 00:51:12.890 What does that do in the spectral domain? 00:51:12.890 --> 00:51:16.190 It's going to take half of that and place it at minus 2 00:51:16.190 --> 00:51:18.410 omega c and plus 2 omega c, so it's completely 00:51:18.410 --> 00:51:20.310 consistent with this picture. 00:51:20.310 --> 00:51:22.310 So what is it that we have to do now to extract 00:51:22.310 --> 00:51:23.268 the signal of interest? 00:51:26.100 --> 00:51:28.920 Just a lowpass filtering here, OK? 00:51:28.920 --> 00:51:32.850 So if you can select out this piece with a lowpass filter, 00:51:32.850 --> 00:51:34.800 you've recovered the signal of interest. 00:51:34.800 --> 00:51:36.570 You can adjust the scale factor, too, 00:51:36.570 --> 00:51:39.470 so you can have a lowpass filter with a gain of 2 00:51:39.470 --> 00:51:41.320 and you've recovered your original signal. 00:51:41.320 --> 00:51:43.410 All right, we'll build on Monday. 00:51:43.410 --> 00:51:46.380 And that'll be the last lecture on this material. 00:51:46.380 --> 00:51:49.150 Relative to the calendar, we're just sliding forward, 00:51:49.150 --> 00:51:51.210 so we'll wrap up this stuff on Monday 00:51:51.210 --> 00:51:54.830 and then continue with packets.