WEBVTT 00:00:00.000 --> 00:00:02.555 The following content is provided under a Creative 00:00:02.555 --> 00:00:03.650 Commons license. 00:00:03.650 --> 00:00:06.600 Your support will help MIT OpenCourseWare continue to 00:00:06.600 --> 00:00:10.030 offer high quality educational resources for free. 00:00:10.030 --> 00:00:12.810 To make a donation or to view additional materials from 00:00:12.810 --> 00:00:16.560 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:16.560 --> 00:00:17.810 ocw.mit.edu. 00:00:21.080 --> 00:00:27.350 PROFESSOR: I want to pretty much finish up today talking 00:00:27.350 --> 00:00:31.420 about modulation. 00:00:31.420 --> 00:00:34.250 Part of the reason we don't spend much time on this is, I 00:00:34.250 --> 00:00:39.360 think most of you have seen some kind of undergraduate 00:00:39.360 --> 00:00:42.260 class in communication or signal 00:00:42.260 --> 00:00:45.480 processing or something. 00:00:45.480 --> 00:00:50.070 Where you do a lot of this because a lot of this is just 00:00:50.070 --> 00:00:54.170 nice exercises and -- well, nice or not nice, depending on 00:00:54.170 --> 00:00:58.130 the way you about it -- in multiplying waveforms by 00:00:58.130 --> 00:01:02.350 cosines and by sines and by complex exponentials. 00:01:02.350 --> 00:01:06.770 And you keep doing this, and doing this, and doing this and 00:01:06.770 --> 00:01:09.330 you get these long expressions. 00:01:09.330 --> 00:01:13.180 And and it all means something. 00:01:13.180 --> 00:01:16.000 So, most of you have seen a good deal of this. 00:01:16.000 --> 00:01:20.230 You probably haven't seen the Nyquist criterion before. 00:01:20.230 --> 00:01:23.360 I'm sure you haven't seen the Nyquist criterion done 00:01:23.360 --> 00:01:28.680 carefully, because I've never seen it done carefully before. 00:01:28.680 --> 00:01:32.990 I don't think it is done carefully anywhere else. 00:01:32.990 --> 00:01:34.470 And I'm sure you don't care about this. 00:01:34.470 --> 00:01:37.820 But, at some point, if you deal with this stuff, you will 00:01:37.820 --> 00:01:38.570 care about it. 00:01:38.570 --> 00:01:41.070 Because at some point you will need it. 00:01:41.070 --> 00:01:46.220 But anyway, we were looking at pulse amplitude modulation, 00:01:46.220 --> 00:01:48.050 you remember. 00:01:48.050 --> 00:01:51.600 We were looking at the modulated waveform. 00:01:51.600 --> 00:01:56.980 And this is all done down at baseband, now, remember. 00:01:56.980 --> 00:02:02.430 And the interesting problems down at baseband are, first, 00:02:02.430 --> 00:02:05.280 how do you choose a signal set. 00:02:05.280 --> 00:02:09.530 Namely, how do you choose those quantities u sub k, out 00:02:09.530 --> 00:02:12.240 of some set of possible values. 00:02:12.240 --> 00:02:15.670 So they have an appropriate distance between them. 00:02:15.670 --> 00:02:19.400 And, next, how do you choose this waveform p of t, which is 00:02:19.400 --> 00:02:23.500 called the pulse, that you're using. 00:02:23.500 --> 00:02:26.410 And then, after you've chosen those two things, there's 00:02:26.410 --> 00:02:28.710 nothing more to be done. 00:02:28.710 --> 00:02:33.130 You simply form the modulated waveform as u of t equals the 00:02:33.130 --> 00:02:36.400 sum of the signals which are coming in regularly at 00:02:36.400 --> 00:02:37.900 intervals of t. 00:02:37.900 --> 00:02:42.300 You multiply each one of them by this waveform, p of t. 00:02:42.300 --> 00:02:47.150 Or, alternatively, you think of the waveform as being a sum 00:02:47.150 --> 00:02:48.850 of impulses. 00:02:48.850 --> 00:02:51.270 Each impulse weighted by u sub k. 00:02:51.270 --> 00:02:54.520 And you think of passing this string of impulses through a 00:02:54.520 --> 00:02:58.190 filter with impulse response p of t. 00:02:58.190 --> 00:03:00.890 Which is usually the way that it's implemented, except of 00:03:00.890 --> 00:03:03.900 course you're not using ideal impulses. 00:03:03.900 --> 00:03:08.550 You're using sharp pulses which have a flat spectrum 00:03:08.550 --> 00:03:11.950 over the bandwidth of the filter p of t. 00:03:11.950 --> 00:03:15.340 But anyway, somehow or other you implement this. 00:03:15.340 --> 00:03:18.790 And that is this fundamental step we've been talking about 00:03:18.790 --> 00:03:21.920 for quite a while now, of how do you turn 00:03:21.920 --> 00:03:24.020 sequences into waveforms. 00:03:24.020 --> 00:03:27.640 How do you turn waveforms into sequences, it's a fundamental 00:03:27.640 --> 00:03:28.860 piece of source coding. 00:03:28.860 --> 00:03:31.890 It's a fundamental piece of channel coding. 00:03:31.890 --> 00:03:35.350 Here we're doing the channel coding part of it. 00:03:35.350 --> 00:03:38.050 Then we did something kind of flaky. 00:03:38.050 --> 00:03:42.040 We said that when we received this waveform, what we were 00:03:42.040 --> 00:03:46.290 going to do is to first pass it through another filter, and 00:03:46.290 --> 00:03:48.180 then sample it. 00:03:48.180 --> 00:03:51.950 If you've done the homework yet, you will probably have 00:03:51.950 --> 00:03:55.850 looked at what happens when you take an arbitrary linear 00:03:55.850 --> 00:04:02.750 operation on this received signal to try to retrieve what 00:04:02.750 --> 00:04:05.090 these signals u sub k are. 00:04:05.090 --> 00:04:09.700 And you will have found that this filtering and sampling is 00:04:09.700 --> 00:04:16.170 in fact not a general linear operation, but it's the only 00:04:16.170 --> 00:04:20.060 general linear operation that you're interested in as far as 00:04:20.060 --> 00:04:23.550 retrieving these signals from that waveform. 00:04:23.550 --> 00:04:28.270 So, in fact, this is, in fact, more general than it looks. 00:04:28.270 --> 00:04:32.910 There a confusing thing here. 00:04:32.910 --> 00:04:42.900 If you receive u of t at the receiver, and your question 00:04:42.900 --> 00:04:50.400 is, how do we get this sequence of samples u sub k 00:04:50.400 --> 00:05:01.740 out of it, and suppose that this pulse p of t or the -- 00:05:01.740 --> 00:05:05.310 I mean, suppose for example that the pulse is a narrow 00:05:05.310 --> 00:05:10.870 bandwidth pulse, and there's just no way you can perform 00:05:10.870 --> 00:05:14.680 linear operations and get these signals out from it. 00:05:14.680 --> 00:05:18.000 Is it possible to do nonlinear operations and get the signals 00:05:18.000 --> 00:05:20.310 out from it? 00:05:20.310 --> 00:05:22.270 And you ought to think about this question a little bit 00:05:22.270 --> 00:05:25.020 because it's kind of an interesting one. 00:05:28.760 --> 00:05:34.180 If I don't tell you what the signals u sub k are, if I ask 00:05:34.180 --> 00:05:39.260 you to build something which extracts these samples u sub 00:05:39.260 --> 00:05:44.600 k, without any idea of what signal set they're taken from, 00:05:44.600 --> 00:05:46.080 then there's nothing you can do better 00:05:46.080 --> 00:05:48.890 than a linear operation. 00:05:48.890 --> 00:05:53.810 And, in fact, if this pulse p of t has a bandwidth that's 00:05:53.810 --> 00:05:57.340 too narrow, you're just stuck. 00:05:57.340 --> 00:06:01.440 If I tell you that these u sub k are drawn from a particular 00:06:01.440 --> 00:06:05.620 signal set -- for example, suppose they're binary -- then 00:06:05.620 --> 00:06:09.950 you have an enormous extra amount of information. 00:06:09.950 --> 00:06:14.220 And you can, in fact, given this waveform, even though p 00:06:14.220 --> 00:06:17.230 of t is a very, very narrow band, you can still in 00:06:17.230 --> 00:06:20.600 principle get these binary signals out. 00:06:20.600 --> 00:06:23.070 So what's the game we're playing here? 00:06:23.070 --> 00:06:27.040 I mean, we're doing kind of a phony thing. 00:06:27.040 --> 00:06:31.810 We're restricting ourselves to only linear operations. 00:06:31.810 --> 00:06:35.600 We are restricting ourselves to retrieving these signals 00:06:35.600 --> 00:06:40.380 without knowing anything about what the signal set is. 00:06:40.380 --> 00:06:43.230 Which is not really the problem that we're interested 00:06:43.230 --> 00:06:44.620 in looking at. 00:06:44.620 --> 00:06:47.270 So what are we really doing? 00:06:47.270 --> 00:06:49.690 What we're really doing is trying to look at this 00:06:49.690 --> 00:06:53.240 question of modulation before we look at 00:06:53.240 --> 00:06:54.490 the question of noise. 00:06:58.350 --> 00:07:01.500 And after we start looking at noise, the thing that we're 00:07:01.500 --> 00:07:04.900 going to find is that this received waveform is a 00:07:04.900 --> 00:07:08.650 received waveform plus a lot of noise on top of it. 00:07:08.650 --> 00:07:11.790 And if there's a lot of noise on top of it, these non-linear 00:07:11.790 --> 00:07:14.295 operations you can think of are not going 00:07:14.295 --> 00:07:17.720 to work very well. 00:07:17.720 --> 00:07:23.400 And I guess the problem is, you just have to take that on 00:07:23.400 --> 00:07:24.640 faith right now. 00:07:24.640 --> 00:07:28.240 And after we look at random processes, and after we look 00:07:28.240 --> 00:07:31.550 at how to deal with the noise waveforms, you will in fact 00:07:31.550 --> 00:07:35.180 see that these operations we're talking about here are 00:07:35.180 --> 00:07:37.770 exactly the things we want to do. 00:07:37.770 --> 00:07:39.650 Now, I don't know whether this is the right 00:07:39.650 --> 00:07:41.730 way to do it or not. 00:07:41.730 --> 00:07:44.670 Dealing with all this phony stuff that you see in 00:07:44.670 --> 00:07:48.650 elementary courses before dealing with the real stuff. 00:07:48.650 --> 00:07:51.370 I think it probably is, because for most of you this 00:07:51.370 --> 00:07:54.440 is sort of familiar and we'll come back later 00:07:54.440 --> 00:07:56.430 and make it all right. 00:07:56.430 --> 00:08:00.210 But who knows? 00:08:00.210 --> 00:08:01.460 Anyway. 00:08:04.090 --> 00:08:10.370 what we want to do then is to find some composite filter, g, 00:08:10.370 --> 00:08:19.450 which is what happens when you take the filter p, or the 00:08:19.450 --> 00:08:24.685 pulse p, pass it through a filter q of t, what you get is 00:08:24.685 --> 00:08:26.960 the convolution of p and q. 00:08:26.960 --> 00:08:31.390 And, therefore, what comes out after this filtering is done 00:08:31.390 --> 00:08:35.710 is a received waveform which is just a sum over k of u sub 00:08:35.710 --> 00:08:39.650 k times g of t minus k t. 00:08:39.650 --> 00:08:43.000 In other words, these two filters are not doing anything 00:08:43.000 --> 00:08:44.070 extra for you. 00:08:44.070 --> 00:08:46.890 All they are is a way of putting part of the filter at 00:08:46.890 --> 00:08:50.050 the transmitter, part of the filter at the receiver. 00:08:50.050 --> 00:08:53.090 When you look at noise you'll find out that there is some 00:08:53.090 --> 00:08:56.120 real difference between what's done at the transmitter and 00:08:56.120 --> 00:08:59.210 what's done at the receiver, because the noise comes in 00:08:59.210 --> 00:09:01.670 between the transmitter and the receiver. 00:09:01.670 --> 00:09:04.640 But for now, it doesn't make any difference so the only 00:09:04.640 --> 00:09:07.200 thing we're interested in is the properties of this 00:09:07.200 --> 00:09:10.370 composite waveform, g of t. 00:09:10.370 --> 00:09:15.160 And what we find is that if we receive r of t, which is this 00:09:15.160 --> 00:09:19.910 waveform sum of u k g of t minus k t, and if we want to 00:09:19.910 --> 00:09:24.580 retrieve the coefficient u sub k, it becomes duck soup to do 00:09:24.580 --> 00:09:29.830 so if in fact this waveform is like a sampling waveform. 00:09:29.830 --> 00:09:37.540 In other words, if g of t is equal to 1 at t equals 0, and 00:09:37.540 --> 00:09:41.900 it's equal to 0 at each other sample point, then all we have 00:09:41.900 --> 00:09:46.170 to do is take this waveform, simply sample it each capital 00:09:46.170 --> 00:09:48.930 T seconds, and we get these coefficients out 00:09:48.930 --> 00:09:50.840 automatically. 00:09:50.840 --> 00:09:55.280 Now, I should warn you at this point that in the notes the 00:09:55.280 --> 00:10:02.030 scaling business is not done quite right. 00:10:02.030 --> 00:10:09.110 Sometimes we talk about g of t as a filter whose shifts are 00:10:09.110 --> 00:10:10.930 orthonormal to each other. 00:10:10.930 --> 00:10:14.470 And sometimes as a filter whose shifts are orthogonal to 00:10:14.470 --> 00:10:15.720 each other. 00:10:15.720 --> 00:10:19.950 I advise you not to worry about that, because you have 00:10:19.950 --> 00:10:22.190 to make changes about five times in the notes 00:10:22.190 --> 00:10:24.160 to make it all right. 00:10:24.160 --> 00:10:26.600 And I will put up a new version of the notes on the 00:10:26.600 --> 00:10:28.590 web which in fact does this right. 00:10:28.590 --> 00:10:30.700 It's not important. 00:10:30.700 --> 00:10:35.310 It's just this old question of, do you use a sinc function 00:10:35.310 --> 00:10:37.960 when you're dealing with strictly band-limited 00:10:37.960 --> 00:10:42.490 functions, or do you multiply the sinc function by 1 over 00:10:42.490 --> 00:10:45.880 the square root of t to make it orthonormal. 00:10:45.880 --> 00:10:50.840 Or do you multiply it by 1 over t to make it -- 00:10:50.840 --> 00:10:53.570 I mean, you can scale it in a number of different ways. 00:10:53.570 --> 00:10:56.120 And, fundamentally, it doesn't make any difference. 00:10:56.120 --> 00:10:58.290 It's just that if you want to get the right answer you have 00:10:58.290 --> 00:11:01.130 to scale it the right way. 00:11:01.130 --> 00:11:04.880 And it's not quite right in the notes. 00:11:04.880 --> 00:11:09.780 So I'll change it around and send the thing out to you. 00:11:09.780 --> 00:11:12.830 Then it says that T-spaced samples of r then reproduce u 00:11:12.830 --> 00:11:15.470 sub k without intersymbol interference. 00:11:15.470 --> 00:11:20.490 The Nyquist criterion is different from this business 00:11:20.490 --> 00:11:23.220 of the pulse being ideal Nyquist. 00:11:23.220 --> 00:11:26.610 Ideal Nyquist is talking about the time domain. 00:11:26.610 --> 00:11:29.680 It simply says the trivial thing you, want a pulse which 00:11:29.680 --> 00:11:33.390 has 0's at every sample point except for the sample point 00:11:33.390 --> 00:11:34.790 you're interested in. 00:11:34.790 --> 00:11:38.700 The Nyquist criterion translates that ideal Nyquist 00:11:38.700 --> 00:11:43.550 property, in time, to a property in frequency. 00:11:43.550 --> 00:11:46.950 And it says that the frequency, that the Fourier 00:11:46.950 --> 00:11:50.830 transform of g of t has to satisfy this 00:11:50.830 --> 00:11:52.680 relationship here. 00:11:52.680 --> 00:11:56.950 And there's this added condition on g of f that it 00:11:56.950 --> 00:12:00.820 has to go to 0 fast enough as f goes to infinity. 00:12:00.820 --> 00:12:04.230 But we won't worry about that today. 00:12:04.230 --> 00:12:11.570 So the condition is this: the picture that I showed you last 00:12:11.570 --> 00:12:13.610 time is this. 00:12:16.110 --> 00:12:23.200 We defined the nominal band, the Nyquist band, as the base 00:12:23.200 --> 00:12:26.800 bandwidth w equals 1 over 2t. 00:12:26.800 --> 00:12:31.030 That's the bandwidth that a sinc pulse would have if you 00:12:31.030 --> 00:12:32.790 were using a sinc pulse. 00:12:32.790 --> 00:12:35.910 The actual baseband limit, b, should be close to w. 00:12:39.290 --> 00:12:42.230 And most of the work that people do trying to build 00:12:42.230 --> 00:12:45.240 filters and things like this, since what we're trying to do 00:12:45.240 --> 00:12:48.360 is find a waveform that we're trying to transmit. 00:12:48.360 --> 00:12:51.250 And we're stuck with the FCC and all these other things 00:12:51.250 --> 00:12:56.280 that say, you better keep this band-limited. 00:12:56.280 --> 00:12:59.690 What we're going to do is to make the actual band of the 00:12:59.690 --> 00:13:07.420 waveform close to the nominal bandwidth. 00:13:07.420 --> 00:13:10.515 So we're going to assume that it's less than twice the 00:13:10.515 --> 00:13:12.130 nominal bandwidth. 00:13:12.130 --> 00:13:14.130 In other words, you can have a little bit of slop, but you 00:13:14.130 --> 00:13:15.700 can't have too much. 00:13:15.700 --> 00:13:18.750 When you try to design a filter that way, and you 00:13:18.750 --> 00:13:23.160 satisfy the Nyquist criterion, which is talking about all of 00:13:23.160 --> 00:13:26.190 these bands all the way out to infinity, and the fact they 00:13:26.190 --> 00:13:30.310 have to add up in a certain way, if you only have this 00:13:30.310 --> 00:13:34.270 band here and part of the next band -- 00:13:34.270 --> 00:13:39.430 if this function has to go to 0 before you get out to 2w, 00:13:39.430 --> 00:13:42.530 then the only thing you have to worry about is what does 00:13:42.530 --> 00:13:44.370 this waveform look like here. 00:13:44.370 --> 00:13:47.740 You take this and you pick it up. 00:13:47.740 --> 00:13:50.200 And you put it over here, and you add it up. 00:13:50.200 --> 00:13:54.340 You take this, you put it over here and you add it up. 00:13:54.340 --> 00:14:00.460 And if the pulse, p of t is real, then what you have over 00:14:00.460 --> 00:14:05.030 here is the complex conjugate of what you have here. 00:14:05.030 --> 00:14:08.220 And if you make this real in frequency also -- in other 00:14:08.220 --> 00:14:11.470 words, you have symmetry in time and symmetry in 00:14:11.470 --> 00:14:16.190 frequency, then this band edge symmetry requirement is that 00:14:16.190 --> 00:14:20.110 when you add this to this, you get something which is an 00:14:20.110 --> 00:14:25.260 ideal rectangular pulse. 00:14:25.260 --> 00:14:30.430 Now, if you think of taking this waveform here -- now you 00:14:30.430 --> 00:14:34.040 only have to worry about the positive frequency part of it. 00:14:34.040 --> 00:14:39.140 And you take that point right there, which is halfway 00:14:39.140 --> 00:14:40.700 between 0 and t. 00:14:40.700 --> 00:14:42.800 In other words, this is at t over 2. 00:14:45.770 --> 00:14:49.200 And it's also at frequency w. 00:14:49.200 --> 00:14:56.080 And you rotate this thing around by 180 degrees, then 00:14:56.080 --> 00:14:58.050 this comes up there. 00:14:58.050 --> 00:15:00.650 And it fills in this little slot up here. 00:15:00.650 --> 00:15:02.500 That's what the band edge symmetry means. 00:15:02.500 --> 00:15:02.800 Yes? 00:15:02.800 --> 00:15:04.050 AUDIENCE: [UNINTELLIGIBLE] 00:15:15.500 --> 00:15:18.080 PROFESSOR: Ah, yes. 00:15:18.080 --> 00:15:21.740 We could, if we wanted to, put various 00:15:21.740 --> 00:15:25.240 notches in this filter. 00:15:25.240 --> 00:15:30.540 But we've defined the bandwidth, b, as the largest 00:15:30.540 --> 00:15:38.240 frequency, f, such that g hat of f is 0 beyond b. 00:15:38.240 --> 00:15:43.590 In other words, everywhere beyond b, g hat of 00:15:43.590 --> 00:15:45.360 f is equal to 0. 00:15:45.360 --> 00:15:54.560 Now, if g hat f cuts down to 0, say, back here, there's no 00:15:54.560 --> 00:15:57.060 way you can meet the Nyquist criterion. 00:15:57.060 --> 00:15:59.750 Because there's no way you can build this thing up with all 00:15:59.750 --> 00:16:03.260 these out-of-band components so that you get something 00:16:03.260 --> 00:16:05.930 which is flat all the way out to w. 00:16:08.530 --> 00:16:11.740 So you simply can't have a filter which is band-limited 00:16:11.740 --> 00:16:15.370 to a frequency less than w. 00:16:15.370 --> 00:16:19.860 What you need is to use these out-of-band frequencies as a 00:16:19.860 --> 00:16:25.030 way to help you construct this ideal rectangular pulse. 00:16:25.030 --> 00:16:26.800 Through aliasing. 00:16:26.800 --> 00:16:29.230 In other words, the point here is when we're doing 00:16:29.230 --> 00:16:33.400 transmission of data, we know what the data is. 00:16:33.400 --> 00:16:37.020 We know what the filter is, and we can use 00:16:37.020 --> 00:16:39.590 aliasing to our advantage. 00:16:39.590 --> 00:16:42.920 When we were talking about data compression, aliasing 00:16:42.920 --> 00:16:44.910 just hurt us. 00:16:44.910 --> 00:16:48.240 Because we were trying to represent this waveform that 00:16:48.240 --> 00:16:50.420 we didn't have any control over. 00:16:50.420 --> 00:16:56.200 And the out-of-band parts added into the baseband parts 00:16:56.200 --> 00:16:57.040 and they clobbered us. 00:16:57.040 --> 00:17:00.270 Because we couldn't get the whole thing back again. 00:17:00.270 --> 00:17:04.670 Here, we're doing it the other way. 00:17:04.670 --> 00:17:07.420 In, other words we're starting out with a sequence. 00:17:07.420 --> 00:17:10.160 We're going to a waveform, and then we're trying to get the 00:17:10.160 --> 00:17:13.880 sequence back from the waveform. 00:17:13.880 --> 00:17:17.370 So it's really the opposite kind of problem. 00:17:17.370 --> 00:17:21.230 And here, the whole game, namely, the thing that Nyquist 00:17:21.230 --> 00:17:26.050 spotted, back in 1928, was you could use these out-of-band 00:17:26.050 --> 00:17:29.440 frequencies to, in fact, help you to get rid of this 00:17:29.440 --> 00:17:31.900 intersymbol interference. 00:17:31.900 --> 00:17:35.000 Because all you need to do is make these things add up to 00:17:35.000 --> 00:17:38.590 this so that you have something rectangular. 00:17:38.590 --> 00:17:40.120 And then when you do the samples, you 00:17:40.120 --> 00:17:41.370 have to write samples. 00:17:46.580 --> 00:17:50.600 Now, the problem that a filter designer comes to when saying 00:17:50.600 --> 00:17:56.680 this is to say, OK, how do I design a frequency response 00:17:56.680 --> 00:18:03.070 which has the property that it's going to go to 0 00:18:03.070 --> 00:18:05.800 quickly beyond w. 00:18:05.800 --> 00:18:09.940 Because the FCC, when we translate this up to passband, 00:18:09.940 --> 00:18:13.010 is going to tell us we can't have much energy outside of 00:18:13.010 --> 00:18:15.000 minus w to plus w. 00:18:15.000 --> 00:18:17.880 And if we can't design a good filter, it means we have to 00:18:17.880 --> 00:18:19.300 make w smaller. 00:18:19.300 --> 00:18:22.730 So we can keep ourselves within this given bandwidth. 00:18:22.730 --> 00:18:25.760 And we don't want to do that because that keeps us from 00:18:25.760 --> 00:18:27.670 transmitting much data. 00:18:27.670 --> 00:18:31.930 And then we can't sell our product, so suddenly we have 00:18:31.930 --> 00:18:37.350 to design something which uses all of this bandwidth that we 00:18:37.350 --> 00:18:39.630 have available. 00:18:39.630 --> 00:18:44.480 So what we want to do, then, is to design something where b 00:18:44.480 --> 00:18:49.350 is just a little bit more than w, but where also we get from 00:18:49.350 --> 00:18:52.650 t down to 0 very quickly. 00:18:52.650 --> 00:18:57.290 We could just use the square pulse to start with. 00:18:57.290 --> 00:18:59.510 And what's the trouble with that? 00:18:59.510 --> 00:19:03.900 This rectangular pulse, its inverse Fourier transform is 00:19:03.900 --> 00:19:05.740 the sinc pulse. 00:19:05.740 --> 00:19:09.350 The sinc pulse, because a discontinuity in the Fourier 00:19:09.350 --> 00:19:15.510 transform, it can't go to 0 any faster than is 1 over t. 00:19:15.510 --> 00:19:18.120 And suddenly it goes to 0 as 1 over t goes to 00:19:18.120 --> 00:19:19.970 0 very, very slowly. 00:19:19.970 --> 00:19:26.270 In other words, you have enormous problems over time. 00:19:26.270 --> 00:19:28.130 And you have enormous delay. 00:19:28.130 --> 00:19:31.150 And since you have so many pulses adding up together, 00:19:31.150 --> 00:19:34.410 everything has to be done extraordinarily carefully. 00:19:34.410 --> 00:19:40.280 So what you want is a pulse which remains equal to t over 00:19:40.280 --> 00:19:42.320 a y bandwidth here. 00:19:42.320 --> 00:19:45.170 Which gets down to 0 very, very fast. 00:19:45.170 --> 00:19:47.880 So the problem is, how do you design something which gets 00:19:47.880 --> 00:19:50.310 from here down to here very, very 00:19:50.310 --> 00:19:53.420 quickly and very smoothly. 00:19:53.420 --> 00:19:57.020 You want it to go smoothly because if you have any 00:19:57.020 --> 00:20:01.030 discontinuities in g of f, you're back to the problem 00:20:01.030 --> 00:20:05.950 where g of t goes to 0 as 1 over t. 00:20:05.950 --> 00:20:09.150 If you have a slope discontinuity, g of t is going 00:20:09.150 --> 00:20:12.440 to go to 0 as 1 over t squared. 00:20:12.440 --> 00:20:15.030 If you have a second derivative discontinuity, it's 00:20:15.030 --> 00:20:18.580 going to go to 0 as 1 over t cubed. 00:20:18.580 --> 00:20:22.790 Now, 1 over t cubed is not bad, and filter designers sort 00:20:22.790 --> 00:20:24.430 of live with that. 00:20:24.430 --> 00:20:28.220 So they design these filters which are raised cosine 00:20:28.220 --> 00:20:35.110 filters, which over the band here -- someday I'll get a pen 00:20:35.110 --> 00:20:38.660 that works -- are flat. 00:20:38.660 --> 00:20:44.730 Over this band here, from here to here, this is a squared 00:20:44.730 --> 00:20:47.680 cosine, analytically. 00:20:47.680 --> 00:20:51.410 And a squared cosine is just the same as a cosine which you 00:20:51.410 --> 00:20:56.100 take and displace up so it's centered right there. 00:20:56.100 --> 00:21:03.280 Well, excuse me, it's the same as a sine 00:21:03.280 --> 00:21:05.010 which is centered there. 00:21:05.010 --> 00:21:07.470 Which is what you get when you square a cosine 00:21:07.470 --> 00:21:12.060 pulse as 1/2 plus. 00:21:12.060 --> 00:21:13.540 Anyway, it looks like this. 00:21:16.360 --> 00:21:19.430 So our problem is, how do you design a filter which gets 00:21:19.430 --> 00:21:24.930 from here to there quickly but where the inverse transform 00:21:24.930 --> 00:21:28.310 also goes to 0 relatively quickly? 00:21:28.310 --> 00:21:33.720 Now, if you want to do this and you also face the fact 00:21:33.720 --> 00:21:38.670 that in a Nyquist criterion, any part of g hat of f which 00:21:38.670 --> 00:21:42.660 is imaginary, the Nyquist criterion says that what do 00:21:42.660 --> 00:21:45.420 you have to do in-band and what you have to do out of 00:21:45.420 --> 00:21:48.100 band have to add up there also. 00:21:48.100 --> 00:21:51.460 That doesn't help you at all in getting from this real 00:21:51.460 --> 00:21:54.770 number t here down to 0. 00:21:54.770 --> 00:21:59.170 So anything you do as a complex part of g hat of f is 00:21:59.170 --> 00:22:00.770 just wasted. 00:22:00.770 --> 00:22:04.040 I mean, your problem is getting from t to 0 with a 00:22:04.040 --> 00:22:05.800 smooth waveform. 00:22:05.800 --> 00:22:10.330 You would like to make g hat of f strictly real. 00:22:10.330 --> 00:22:12.020 You would like to make it symmetric. 00:22:12.020 --> 00:22:15.100 Why would you like to make it symmetric? 00:22:15.100 --> 00:22:20.230 Because this thing down here and this thing up here are 00:22:20.230 --> 00:22:22.030 really part of the same problem. 00:22:22.030 --> 00:22:25.820 If you find a good way to make a function go to 0 quickly up 00:22:25.820 --> 00:22:29.980 here, you might as well use the same thing over here. 00:22:29.980 --> 00:22:33.460 So you might as well wind up with a function which is 00:22:33.460 --> 00:22:36.540 symmetric and real. 00:22:36.540 --> 00:22:39.860 That leads us into the next thing we want to look at. 00:22:39.860 --> 00:22:42.350 That's a slightly flaky argument. 00:22:42.350 --> 00:22:47.230 We're going to find a better argument as we go. 00:22:47.230 --> 00:22:50.840 What we've said is, the real part of g hat of f has to 00:22:50.840 --> 00:22:54.970 satisfy this band edged symmetry condition. 00:22:54.970 --> 00:22:59.860 Choosing the imaginary part unequal to 0 simply increases 00:22:59.860 --> 00:23:02.320 the energy outside of the Nyquist band. 00:23:02.320 --> 00:23:09.290 You don't get any effect on reducing delay out of that. 00:23:09.290 --> 00:23:12.910 Thus, we're going to restrict g of f to be real. 00:23:12.910 --> 00:23:16.600 And we're going to also restrict it to be symmetric. 00:23:16.600 --> 00:23:18.630 Although that's less important. 00:23:18.630 --> 00:23:22.040 Now, when we start to look at noise, we're going to find out 00:23:22.040 --> 00:23:22.840 something else. 00:23:22.840 --> 00:23:25.360 We're going to find out that we want to make the magnitude 00:23:25.360 --> 00:23:30.790 of p of f equal to the magnitude of g of f. 00:23:30.790 --> 00:23:34.100 Now, magnitude doesn't make any difference. 00:23:34.100 --> 00:23:37.560 So we want the frequency characteristic of p of f to be 00:23:37.560 --> 00:23:41.770 the same as the frequency characteristic of q of f. 00:23:41.770 --> 00:23:45.270 In other words, there's no point descending a p of f 00:23:45.270 --> 00:23:49.570 which is a perfect sinc function and then using a very 00:23:49.570 --> 00:23:51.620 sloppy q of f. 00:23:51.620 --> 00:23:53.440 Because that's kind of silly. 00:23:53.440 --> 00:23:57.100 There's no point to using it very sloppy p of f and then 00:23:57.100 --> 00:24:01.040 using a very sharp q of f, because somehow when you start 00:24:01.040 --> 00:24:05.530 looking at noise, you're going to lose everything. 00:24:05.530 --> 00:24:10.350 Because the noise gets added to this pulse that you're 00:24:10.350 --> 00:24:11.440 transmitting. 00:24:11.440 --> 00:24:15.350 So, what we're going to find when we look at this later, is 00:24:15.350 --> 00:24:17.310 we really want to choose the magnitudes 00:24:17.310 --> 00:24:20.970 of these to be equal. 00:24:20.970 --> 00:24:26.180 Since g hat of f is equal to this product, and since we've 00:24:26.180 --> 00:24:29.920 already decided we want to make this real, what this 00:24:29.920 --> 00:24:36.110 means is that q hat of f is going to be equal to p complex 00:24:36.110 --> 00:24:37.950 conjugate of f. 00:24:37.950 --> 00:24:41.590 What that means is the filter q of t should be equal to the 00:24:41.590 --> 00:24:44.860 complex conjugate of p of minus t. 00:24:44.860 --> 00:24:50.410 You take a p of t, and you turn it around like this. 00:24:50.410 --> 00:24:54.400 If p of t is real, this is called a matched filter. 00:24:54.400 --> 00:24:57.000 And it's a filter which sort of collects everything which 00:24:57.000 --> 00:25:01.560 is in p of t, and all brings it up to 1p, which is what we 00:25:01.560 --> 00:25:03.300 would like to do here. 00:25:03.300 --> 00:25:08.110 So, anyway, when we do this it means that g of t is going to 00:25:08.110 --> 00:25:10.850 be this convolution of p of t. 00:25:10.850 --> 00:25:14.200 And q of t, which we can now write as the integral of p of 00:25:14.200 --> 00:25:19.920 tau, times p complex conjugate of t minus tau, p tau. 00:25:19.920 --> 00:25:24.520 And what we're interested in is, is this going to be ideal 00:25:24.520 --> 00:25:25.272 Nyquist or not. 00:25:25.272 --> 00:25:26.990 And what does that mean if it is ideal Nyquist? 00:25:35.660 --> 00:25:40.680 If g of t is ideal Nyquist, it means that the samples of g of 00:25:40.680 --> 00:25:45.890 t, times k times this signaling interval, t, have to 00:25:45.890 --> 00:25:49.360 have the property that these samples are equal to 1 for k 00:25:49.360 --> 00:25:53.900 equals 0, and 0 for k unequal to 0. 00:25:53.900 --> 00:25:55.760 What does that mean? 00:25:55.760 --> 00:25:59.510 If you look at this, that kind of looks like these 00:25:59.510 --> 00:26:02.590 orthogonality conditions that we've been dealing with, 00:26:02.590 --> 00:26:04.210 doesn't it? 00:26:04.210 --> 00:26:07.460 So that what it says is that this set of functions, p 00:26:07.460 --> 00:26:09.220 of t minus k t. 00:26:09.220 --> 00:26:14.630 In other words, the pulse p of t and all of it shifts by t, 00:26:14.630 --> 00:26:17.310 2t, 3t, and everything else. 00:26:17.310 --> 00:26:21.970 This set of pulses all have to be orthogonal to each other. 00:26:21.970 --> 00:26:24.890 And the thing which is a little screwed up in the notes 00:26:24.890 --> 00:26:27.750 is whether these are orthogonal or 00:26:27.750 --> 00:26:29.750 orthonormal, or what. 00:26:29.750 --> 00:26:31.320 And you need to make a few changes to 00:26:31.320 --> 00:26:34.100 make all of that right. 00:26:34.100 --> 00:26:36.720 These functions are all real L2 functions. 00:26:40.250 --> 00:26:43.030 But we're going to allow the possibility of complex 00:26:43.030 --> 00:26:44.190 functions for later. 00:26:44.190 --> 00:26:46.440 In other words, if we're transmitting a baseband 00:26:46.440 --> 00:26:50.120 waveform on a channel, how do you transmit 00:26:50.120 --> 00:26:51.370 an imaginary waveform? 00:26:53.800 --> 00:26:57.910 Well, I've never seen anything in electromagnetics, or in 00:26:57.910 --> 00:27:00.590 optics, or anything else, that lets me transmit 00:27:00.590 --> 00:27:02.940 an imaginary waveform. 00:27:02.940 --> 00:27:05.110 These are not physical. 00:27:05.110 --> 00:27:07.320 We will often think of baseband 00:27:07.320 --> 00:27:09.880 waveforms that are imaginary. 00:27:09.880 --> 00:27:12.090 Real and imaginary complex. 00:27:12.090 --> 00:27:18.700 And when we translate them up to baseband, we'll find 00:27:18.700 --> 00:27:19.700 something real. 00:27:19.700 --> 00:27:21.720 But the actual waveforms that get 00:27:21.720 --> 00:27:25.240 transmitted are always real. 00:27:25.240 --> 00:27:27.430 There's no way you can avoid that. 00:27:27.430 --> 00:27:29.140 That's real life. 00:27:29.140 --> 00:27:30.550 Real life is real. 00:27:30.550 --> 00:27:31.990 That's why they call it real, I guess. 00:27:31.990 --> 00:27:34.300 That's why they call it real life. 00:27:34.300 --> 00:27:36.520 I don't know. 00:27:36.520 --> 00:27:41.720 I mean it's more real than something imaginary, isn't it? 00:27:41.720 --> 00:27:45.430 So, anyway, what gets transmitted is real. 00:27:45.430 --> 00:27:50.750 But we'll allow p of t to be complex just for when we start 00:27:50.750 --> 00:27:56.030 dealing with something called QAM, which is our next topic. 00:27:56.030 --> 00:27:59.860 So in vector terms, the integral of u of tau times q 00:27:59.860 --> 00:28:03.900 of k t minus tau is the projection of u 00:28:03.900 --> 00:28:08.100 of t onto this waveform. 00:28:08.100 --> 00:28:11.360 And partly for that reason, q of t it's called the matched 00:28:11.360 --> 00:28:12.540 filter to p of t. 00:28:12.540 --> 00:28:17.060 In other words, you use this waveform here as a way of 00:28:17.060 --> 00:28:21.950 selecting out the parts of this waveform u of tau, u of 00:28:21.950 --> 00:28:24.680 t, that we're interested in. 00:28:24.680 --> 00:28:29.900 So that any way you look at it, we're going to use a pulse 00:28:29.900 --> 00:28:36.400 waveform, p of t, which has this property that its shifts 00:28:36.400 --> 00:28:38.640 are all orthogonal to each other. 00:28:38.640 --> 00:28:42.285 When we start studying noise, you will be very thankful that 00:28:42.285 --> 00:28:44.040 we did this. 00:28:44.040 --> 00:28:46.780 Because when you use pulses that are orthogonal to each 00:28:46.780 --> 00:28:49.410 other, you can break up the noise into 00:28:49.410 --> 00:28:51.390 an orthogonal expansion. 00:28:51.390 --> 00:28:54.190 And what goes on at one place is completely independent of 00:28:54.190 --> 00:28:56.360 what goes on at every other place. 00:28:56.360 --> 00:28:59.190 And we'll find out about this as we go. 00:28:59.190 --> 00:29:03.450 But, anyway, we have the nice property now, that anytime we 00:29:03.450 --> 00:29:09.130 find a function g of t, that satisfies the Nyquist 00:29:09.130 --> 00:29:14.770 criterion And any time we choose p of t and g of t so 00:29:14.770 --> 00:29:19.100 that their Fourier transforms have the same magnitude, then 00:29:19.100 --> 00:29:23.330 presto, we have freely gotten a set 00:29:23.330 --> 00:29:26.200 of orthonormal functions. 00:29:26.200 --> 00:29:29.130 Which just comes out in the wash. 00:29:29.130 --> 00:29:34.210 Before we worked very hard to get these truncated sinusoid 00:29:34.210 --> 00:29:39.310 expansions and sinc weighted sinusoid expansions, and all 00:29:39.310 --> 00:29:41.900 of this stuff to generate different 00:29:41.900 --> 00:29:45.610 orthonormal sets of waveforms. 00:29:45.610 --> 00:29:48.600 Suddenly, we just have an orthonormal set of waveforms 00:29:48.600 --> 00:29:52.040 and a very large set of orthonormal waveforms popping 00:29:52.040 --> 00:29:54.840 up and staring us in the face here. 00:29:54.840 --> 00:29:57.610 And in fact these are the waveforms we're going to use 00:29:57.610 --> 00:30:00.230 for communication, so they're nice things. 00:30:00.230 --> 00:30:03.390 Nobody uses sinc functions for communication. 00:30:03.390 --> 00:30:05.570 Nobody uses rectangular functions. 00:30:05.570 --> 00:30:08.720 You can't use either one of them because rectangular 00:30:08.720 --> 00:30:12.680 functions have lousy frequency characteristics. 00:30:12.680 --> 00:30:15.760 Sinc functions have lousy time characteristics. 00:30:15.760 --> 00:30:20.330 These functions p of t are sort of nice compromises. 00:30:20.330 --> 00:30:21.760 And they're orthonormal, again. 00:30:26.110 --> 00:30:28.120 Let's go on to the rest of modulation. 00:30:30.810 --> 00:30:34.710 We've been talking about baseband modulation. 00:30:34.710 --> 00:30:37.490 And when we're thinking about PAM, pulse amplitude 00:30:37.490 --> 00:30:42.000 modulation, we are thinking in terms of this sequence of 00:30:42.000 --> 00:30:44.160 symbols coming in. 00:30:44.160 --> 00:30:46.590 The symbols being turned into signals. 00:30:46.590 --> 00:30:49.910 The signals been turned into waveforms. 00:30:49.910 --> 00:30:53.990 And what comes out here, then, is some baseband waveform. 00:30:53.990 --> 00:30:56.420 That's what the Nyquist criterion is designed for. 00:30:56.420 --> 00:31:01.240 How do you make a baseband waveform which is very sharply 00:31:01.240 --> 00:31:04.020 cut off in frequency? 00:31:04.020 --> 00:31:07.180 Usually what we want to transmit is something at 00:31:07.180 --> 00:31:10.310 passband, so we somehow want to take this baseband 00:31:10.310 --> 00:31:14.020 waveform, convert it up to passband. 00:31:14.020 --> 00:31:16.480 We're then going to transmit it on a channel. 00:31:16.480 --> 00:31:23.280 I mean, why do we have to turn it into passband anyway? 00:31:23.280 --> 00:31:25.860 Well, if you did everything at baseband, you wouldn't have 00:31:25.860 --> 00:31:28.330 more than one channel available. 00:31:28.330 --> 00:31:30.120 I mean, wireless, you know you have all 00:31:30.120 --> 00:31:31.880 these different channels. 00:31:31.880 --> 00:31:34.990 The way it's done today, you use something called CDMA, 00:31:34.990 --> 00:31:38.770 where you're not breaking into narrow channels. 00:31:38.770 --> 00:31:41.740 But you should understand how to break it into narrow 00:31:41.740 --> 00:31:45.930 channels before understanding how to look at it as 00:31:45.930 --> 00:31:47.920 co-division multiple access. 00:31:47.920 --> 00:31:50.600 If you're using optics, you want to send things in 00:31:50.600 --> 00:31:53.160 different frequency bands. 00:31:53.160 --> 00:31:57.060 Whether it's optics or electromagnetics simply 00:31:57.060 --> 00:32:00.020 determines the frequency band you're looking at anyway. 00:32:00.020 --> 00:32:03.540 You can't propagate things in every frequency band. 00:32:03.540 --> 00:32:06.910 Things don't propagate very well at baseband. 00:32:06.910 --> 00:32:10.130 So for all of these reasons, we want to convert these 00:32:10.130 --> 00:32:13.430 baseband waveforms to passband. 00:32:13.430 --> 00:32:15.900 Why don't we generate them originally at passband? 00:32:20.070 --> 00:32:23.800 Because things are changing too fast there. 00:32:23.800 --> 00:32:26.990 I mean, you want to do digital signal processing to massage 00:32:26.990 --> 00:32:29.110 these signals, to do all the filtering. 00:32:29.110 --> 00:32:31.720 To do most of the other things you want to do. 00:32:31.720 --> 00:32:34.590 And you can do those very easily at baseband. 00:32:34.590 --> 00:32:36.920 And it's hard to do them at passband. 00:32:36.920 --> 00:32:41.330 So the generic way things are done is to first 00:32:41.330 --> 00:32:42.970 take a signal sequence. 00:32:42.970 --> 00:32:45.410 Convert it into a baseband waveform. 00:32:45.410 --> 00:32:48.620 Take the baseband waveform, convert it up to passband. 00:32:48.620 --> 00:32:51.040 And the passband is appropriate to whatever the 00:32:51.040 --> 00:32:52.100 channel is. 00:32:52.100 --> 00:32:52.910 You send it. 00:32:52.910 --> 00:32:57.320 You take it down from passband back to baseband, and then you 00:32:57.320 --> 00:33:00.570 filter and sample and get the waveform back again. 00:33:00.570 --> 00:33:01.890 You don't have to do this. 00:33:01.890 --> 00:33:06.320 We could generate the waveform directly at passband. 00:33:06.320 --> 00:33:08.980 There's a lot of research going on now trying to do 00:33:08.980 --> 00:33:13.720 this, which is trying to make things a little bit simpler. 00:33:13.720 --> 00:33:15.870 Well, it's not really trying to make things simpler. 00:33:15.870 --> 00:33:18.770 It's really trying to trying to pull a fast one 00:33:18.770 --> 00:33:21.340 on the FCC, I think. 00:33:21.340 --> 00:33:23.930 But, anyway, this is being done. 00:33:23.930 --> 00:33:27.610 And it doesn't go through this two-step process on the way. 00:33:27.610 --> 00:33:31.670 So it saves a little extra work. 00:33:34.220 --> 00:33:39.500 So what we're going to do with our PAM waveform, them, we're 00:33:39.500 --> 00:33:43.530 going to take u of t, which is the PAM waveform. 00:33:43.530 --> 00:33:44.860 And I'm sure you've all seen this. 00:33:44.860 --> 00:33:46.720 I hope you've all seen it somewhere or other. 00:33:46.720 --> 00:33:49.370 Because everybody likes to talk about this. 00:33:49.370 --> 00:33:53.640 Because you don't have to know anything to talk about this. 00:33:53.640 --> 00:33:55.200 So you take u of t. 00:33:55.200 --> 00:33:59.400 We multiply it by e to the 2 pi i f c t. 00:33:59.400 --> 00:34:02.550 In other words, you multiply it by a sine wave. 00:34:02.550 --> 00:34:05.080 A complex sine wave. 00:34:05.080 --> 00:34:08.110 When you do this, this thing is complex. 00:34:08.110 --> 00:34:09.910 You can't transmit it. 00:34:09.910 --> 00:34:11.290 So what do we do about that? 00:34:11.290 --> 00:34:12.660 Well, this is real. 00:34:12.660 --> 00:34:14.280 This is complex. 00:34:14.280 --> 00:34:18.140 If we add the complex conjugate of this, this plus 00:34:18.140 --> 00:34:20.980 its complex conjugate is real again. 00:34:20.980 --> 00:34:26.270 So we transmit this times this complex sinusoid, plus this 00:34:26.270 --> 00:34:30.250 other complex sinusoid, which is the complex 00:34:30.250 --> 00:34:31.650 conjugate of this. 00:34:31.650 --> 00:34:37.010 And you get 2 u of t times the cosine of 2 pi f c t. 00:34:37.010 --> 00:34:39.110 Which is just what you would do if you were implementing 00:34:39.110 --> 00:34:39.770 this anyway. 00:34:39.770 --> 00:34:44.180 You take the waveform u of t, you multiply it by cosine wave 00:34:44.180 --> 00:34:46.010 at the carrier frequency. 00:34:46.010 --> 00:34:49.040 And bingo, up it goes to carrier frequency. 00:34:49.040 --> 00:34:50.960 This is real. 00:34:50.960 --> 00:34:52.690 This was real. 00:34:52.690 --> 00:34:55.030 And everybody's happy. 00:34:55.030 --> 00:34:59.760 And in frequency, what this looks like, since all we're 00:34:59.760 --> 00:35:03.840 doing here is just, by this shift formula that we have for 00:35:03.840 --> 00:35:10.070 Fourier transforms, the multiplying a time waveform by 00:35:10.070 --> 00:35:13.390 an exponential -- 00:35:13.390 --> 00:35:20.180 by a complex sinusoid, is simply is the same as shifting 00:35:20.180 --> 00:35:21.760 the frequency response. 00:35:21.760 --> 00:35:25.170 So the Fourier transform of this is u hat of 00:35:25.170 --> 00:35:26.850 that minus f c. 00:35:26.850 --> 00:35:30.440 The Fourier transform of u f t times this is u hat of 00:35:30.440 --> 00:35:32.160 f the plus f c. 00:35:32.160 --> 00:35:35.200 And you start out with this waveform, whatever 00:35:35.200 --> 00:35:38.720 that shape is here. 00:35:38.720 --> 00:35:42.000 This, I tried to draw to satisfy the Nyquist criteria, 00:35:42.000 --> 00:35:42.780 which it does it. 00:35:42.780 --> 00:35:46.230 Satisfies that band edge symmetry condition. 00:35:46.230 --> 00:35:48.060 So this gets shifted up. 00:35:48.060 --> 00:35:50.200 It also gets shifted down. 00:35:50.200 --> 00:35:57.400 And the transmitted waveform then exists in the band which 00:35:57.400 --> 00:36:03.130 I'll call b sub u. b sub u is the bandwidth of u of t. 00:36:03.130 --> 00:36:05.440 Namely, it's this baseband bandwidth that we've been 00:36:05.440 --> 00:36:06.810 talking about. 00:36:06.810 --> 00:36:11.740 But, unfortunately, when we do this thing shifted up and this 00:36:11.740 --> 00:36:15.820 thing shifted down, the overall bandwidth here is now 00:36:15.820 --> 00:36:18.870 twice as much as it was before. 00:36:18.870 --> 00:36:21.960 Now, every communication engineer in the world, I 00:36:21.960 --> 00:36:25.540 think, measures bandwidth in the same way. 00:36:25.540 --> 00:36:28.220 When you talk about bandwidth, you're always talking about 00:36:28.220 --> 00:36:30.430 positive bandwidth. 00:36:30.430 --> 00:36:35.570 Because, back a long time ago, communication engineers didn't 00:36:35.570 --> 00:36:37.350 know about complex sinusoids. 00:36:37.350 --> 00:36:42.000 So everything was done in terms of cosines and sines. 00:36:42.000 --> 00:36:44.770 Which was very good, because back then communication 00:36:44.770 --> 00:36:47.620 engineers didn't have much else to do. 00:36:47.620 --> 00:36:51.270 So they had to learn to write everything twice. 00:36:51.270 --> 00:36:54.410 And now, since we have so many other things to worry about, 00:36:54.410 --> 00:36:56.890 we want to use complex sinusoids and only write 00:36:56.890 --> 00:36:58.120 things once. 00:36:58.120 --> 00:37:01.610 Well, in fact we have to write it twice here, but we don't 00:37:01.610 --> 00:37:04.720 write it twice very often. 00:37:04.720 --> 00:37:08.720 But, anyway, when this thing, which exists for minus b u up 00:37:08.720 --> 00:37:12.820 to plus b u gets translated up in frequency, we have 00:37:12.820 --> 00:37:18.060 something which exists from f c minus b u to f c plus b u. 00:37:18.060 --> 00:37:21.310 And this negative band is down here. 00:37:21.310 --> 00:37:25.470 Now, we're going to assume everywhere, usually without 00:37:25.470 --> 00:37:30.840 talking about it, that when we modulate this up in frequency, 00:37:30.840 --> 00:37:35.840 that the bandwidth here, this b u here, is less than the 00:37:35.840 --> 00:37:37.580 carrier frequency. 00:37:37.580 --> 00:37:40.330 In other words, when we translate it up in frequency, 00:37:40.330 --> 00:37:44.300 this and this do not intersect with each other. 00:37:44.300 --> 00:37:46.910 If this and this intersected with each other, it would be 00:37:46.910 --> 00:37:49.220 something very much like aliasing. 00:37:49.220 --> 00:37:52.210 You just couldn't to sort out from this, plus 00:37:52.210 --> 00:37:55.540 this, what this is. 00:37:55.540 --> 00:37:59.120 Here, if I drew it on paper at least, if you tell me what 00:37:59.120 --> 00:38:04.100 this is, I can figure out what that is. 00:38:04.100 --> 00:38:08.520 Namely, demodulating it independent of how we design 00:38:08.520 --> 00:38:12.470 the demodulator is, in some sense trivial, too. 00:38:12.470 --> 00:38:14.720 You just take this and you bring it back 00:38:14.720 --> 00:38:17.520 down to passband again. 00:38:17.520 --> 00:38:21.200 Well, anyway, since communication engineers define 00:38:21.200 --> 00:38:25.410 bandwidth in terms of positive frequencies, the bandwidth of 00:38:25.410 --> 00:38:29.590 this baseband waveform is b sub u. 00:38:29.590 --> 00:38:33.220 The bandwidth of this waveform is 2 b sub u. 00:38:35.860 --> 00:38:38.420 You can't get away from that. 00:38:38.420 --> 00:38:42.200 You have doubled the bandwidth, and you wind up 00:38:42.200 --> 00:38:44.870 with this plus this. 00:38:44.870 --> 00:38:47.060 And this looks kind of strange. 00:38:47.060 --> 00:38:49.590 So let's try to sort it out. 00:38:54.300 --> 00:38:59.240 The baseband waveform is limited to b. 00:38:59.240 --> 00:39:01.620 If it's shifted up to passband, the passband 00:39:01.620 --> 00:39:05.270 waveform becomes limited to 2 b. 00:39:05.270 --> 00:39:09.320 Might as well put these little u's in here. 00:39:09.320 --> 00:39:13.530 Because putting in little u's here is a way of getting 00:39:13.530 --> 00:39:17.570 around the problem of talking about baseband 00:39:17.570 --> 00:39:19.150 waveforms for a while. 00:39:19.150 --> 00:39:21.800 And then talking about passband waveforms. 00:39:21.800 --> 00:39:24.810 And one of them is always twice the other one. 00:39:29.010 --> 00:39:37.420 If you filter out this lower band here, now, what's the 00:39:37.420 --> 00:39:41.420 lower sideband here? 00:39:41.420 --> 00:39:45.010 Who thinks the lower sideband is this? 00:39:45.010 --> 00:39:49.550 Who thinks the lower sideband is this little thing here? 00:39:49.550 --> 00:39:51.230 Well, you all should think that because 00:39:51.230 --> 00:39:52.600 that's what it is. 00:39:52.600 --> 00:39:55.350 So when people talk about sidebands, what they're 00:39:55.350 --> 00:39:59.880 referring to, it's not, this is one sideband and this is 00:39:59.880 --> 00:40:00.890 another sideband. 00:40:00.890 --> 00:40:04.890 What they're referring to is this is one sideband and this 00:40:04.890 --> 00:40:07.910 is one sideband. 00:40:07.910 --> 00:40:10.710 This is stuff you all know, I'm sure. 00:40:10.710 --> 00:40:12.180 But haven't thought about for a while. 00:40:15.510 --> 00:40:21.410 If you filter out this lower sideband, then this resulting 00:40:21.410 --> 00:40:29.470 waveform, which now runs only in this upper sideband here, 00:40:29.470 --> 00:40:32.270 and since it has to be real it has this accompanying lower 00:40:32.270 --> 00:40:36.160 sideband down here going with it, but you then have the 00:40:36.160 --> 00:40:41.350 frequency band b sub u like you had before. 00:40:41.350 --> 00:40:45.170 So, in principle, you can design a communication system 00:40:45.170 --> 00:40:49.010 by translating things up in frequency by the carrier, and 00:40:49.010 --> 00:40:52.440 then chopping off that lower sideband. 00:40:52.440 --> 00:40:55.450 And then you haven't gained anything in frequency. 00:40:55.450 --> 00:40:59.390 And everything is essentially the same as it was before. 00:40:59.390 --> 00:41:02.160 Now, this used to be a very popular thing to do with 00:41:02.160 --> 00:41:03.920 analog communication. 00:41:03.920 --> 00:41:07.440 Partly because communication engineers felt the only thing 00:41:07.440 --> 00:41:10.690 they had to study back then was, how do you change things 00:41:10.690 --> 00:41:13.020 in frequency and how do you build filters. 00:41:13.020 --> 00:41:14.300 So they're very good at this. 00:41:14.300 --> 00:41:15.720 They love to do this. 00:41:15.720 --> 00:41:17.300 And this was their preferred way of 00:41:17.300 --> 00:41:18.380 dealing with the problem. 00:41:18.380 --> 00:41:22.000 They just got rid of that sideband, sent this positive 00:41:22.000 --> 00:41:24.720 sideband, and then somehow they would get it 00:41:24.720 --> 00:41:27.100 back down to here. 00:41:27.100 --> 00:41:30.150 Single sideband is hardly ever used for digital 00:41:30.150 --> 00:41:31.510 communication. 00:41:31.510 --> 00:41:35.300 It's not the usual way of doing things. 00:41:35.300 --> 00:41:39.320 Partly because these filters become very tricky when you're 00:41:39.320 --> 00:41:42.820 trying to send data at a high speed. 00:41:42.820 --> 00:41:46.840 All sorts of noise in here when you try to do this and 00:41:46.840 --> 00:41:49.510 you don't do it quite right. 00:41:49.510 --> 00:41:54.320 Affects you enormously, and people have just seen over the 00:41:54.320 --> 00:41:58.410 years that those systems don't work as well as the systems 00:41:58.410 --> 00:42:01.940 which do something else that we'll talk about later. 00:42:01.940 --> 00:42:04.900 Namely, QAM, which is what we want to talk about. 00:42:08.040 --> 00:42:10.220 If you don't do this filtering, you call this 00:42:10.220 --> 00:42:14.840 system a double sideband pulse amplitude modulation system. 00:42:14.840 --> 00:42:17.520 Which is what happens when you use pulse amplitude 00:42:17.520 --> 00:42:18.830 modulation. 00:42:18.830 --> 00:42:22.410 Namely, this baseband choosing a baseband pulse, which is the 00:42:22.410 --> 00:42:23.530 thing we're interested in. 00:42:23.530 --> 00:42:26.290 Because that's where the Nyquist criterion and all this 00:42:26.290 --> 00:42:28.160 neat stuff comes in. 00:42:28.160 --> 00:42:30.520 And then you translate it up in frequency. 00:42:30.520 --> 00:42:33.640 And you waste half the available frequency. 00:42:33.640 --> 00:42:37.390 If you don't care about frequency management, this is 00:42:37.390 --> 00:42:38.650 a fine thing to do. 00:42:38.650 --> 00:42:40.580 Nothing wrong with it. 00:42:40.580 --> 00:42:42.360 You just waste some frequency. 00:42:42.360 --> 00:42:44.490 It's the cheapest way to do things. 00:42:44.490 --> 00:42:46.270 And there are lots of cheap communication 00:42:46.270 --> 00:42:47.720 systems which do this. 00:42:50.380 --> 00:42:55.320 But if you're trying to send data, if you're concerned 00:42:55.320 --> 00:43:00.630 about the frequency efficiency of the system, then you're not 00:43:00.630 --> 00:43:01.880 going to do this. 00:43:05.230 --> 00:43:11.060 So what we're going to do is do something called quadrature 00:43:11.060 --> 00:43:12.310 amplitude modulation. 00:43:15.090 --> 00:43:19.190 QAM, which is what quadrature amplitude modulation stands 00:43:19.190 --> 00:43:23.890 for, solves the frequency waste problem of double 00:43:23.890 --> 00:43:28.650 sideband amplitude modulation by using a complex baseband 00:43:28.650 --> 00:43:31.430 waveform u of t. 00:43:31.430 --> 00:43:35.990 Before, what we were talking about is these signals which 00:43:35.990 --> 00:43:37.910 were one-dimensional signals. 00:43:37.910 --> 00:43:40.120 We would use these one-dimensional signals to 00:43:40.120 --> 00:43:44.260 modulate this waveform p of t. 00:43:44.260 --> 00:43:46.830 And we wound up with a real waveform. 00:43:46.830 --> 00:43:50.600 Now what we're going to do is use complex signals, which 00:43:50.600 --> 00:43:53.500 then have two dimensions. 00:43:53.500 --> 00:43:57.420 Use them to modulate the same sort of 00:43:57.420 --> 00:44:00.500 pulse, p of t, usually. 00:44:00.500 --> 00:44:03.740 And wind up with a complex baseband waveform. 00:44:03.740 --> 00:44:06.600 And then we're going to take that baseband waveform, 00:44:06.600 --> 00:44:08.630 translate it up in frequency. 00:44:08.630 --> 00:44:11.290 So when we do this, what do we get? 00:44:11.290 --> 00:44:15.430 We need a waveform to transmit which is real. 00:44:15.430 --> 00:44:20.460 So we're going to take u of t, which is complex. 00:44:20.460 --> 00:44:23.110 Translate, shift it up in frequency 00:44:23.110 --> 00:44:25.070 by the carrier frequency. 00:44:25.070 --> 00:44:30.230 So we get u of t times e to the 2 pi i f c t. 00:44:30.230 --> 00:44:33.870 To make it real, we have to add all this junk down at 00:44:33.870 --> 00:44:36.530 negative frequencies, which we'd just as soon not think 00:44:36.530 --> 00:44:39.210 about if we didn't have to. 00:44:39.210 --> 00:44:40.850 But they have to be there. 00:44:40.850 --> 00:44:46.160 So our total waveform is x sub t equal this sum of things. 00:44:46.160 --> 00:44:50.160 When you look at this and you take the real part of this, 00:44:50.160 --> 00:44:53.295 the real part of this, the imaginary part of this, and 00:44:53.295 --> 00:44:56.860 the imaginary part of this, as I'm sure most of you have seen 00:44:56.860 --> 00:45:02.430 before, x of t becomes 2 times the real part of u of t. 00:45:02.430 --> 00:45:04.990 Times this complex exponential. 00:45:09.180 --> 00:45:13.140 Which is equal to 2 times the real part of u of t times this 00:45:13.140 --> 00:45:18.200 cosine wave minus 2 times the imaginary part of u of t times 00:45:18.200 --> 00:45:19.350 a sine wave. 00:45:19.350 --> 00:45:26.400 Which says, you take this real part of this baseband waveform 00:45:26.400 --> 00:45:27.660 you've generated. 00:45:27.660 --> 00:45:30.170 You multiply it by cosine wave. 00:45:30.170 --> 00:45:32.290 You take the imaginary part, and you multiply 00:45:32.290 --> 00:45:33.670 it by a sine wave. 00:45:33.670 --> 00:45:36.640 For implementation, is one thing going to be real and the 00:45:36.640 --> 00:45:38.550 other thing imaginary? 00:45:38.550 --> 00:45:38.800 No. 00:45:38.800 --> 00:45:42.510 You can't make things that are imaginary, so you just deal 00:45:42.510 --> 00:45:44.600 with two real waveforms. 00:45:44.600 --> 00:45:47.180 And you call one of them the real part of u of t. 00:45:47.180 --> 00:45:49.615 You call the other one the imaginary part of u of t. 00:45:49.615 --> 00:45:52.180 And the imaginary part of u of t is in fact a 00:45:52.180 --> 00:45:53.780 real waveform again. 00:45:53.780 --> 00:45:57.470 So all this imaginary stuff is just in our imagination. 00:45:57.470 --> 00:46:00.030 And the actual waveforms look like this. 00:46:00.030 --> 00:46:03.730 You take one waveform which is generated. 00:46:03.730 --> 00:46:05.320 Multiply it by cosine. 00:46:05.320 --> 00:46:07.350 Take another waveform. 00:46:07.350 --> 00:46:10.940 Multiply it by sine. 00:46:10.940 --> 00:46:14.800 What about these factors of two here? 00:46:14.800 --> 00:46:19.530 The factors of two are things that drive everybody crazy. 00:46:19.530 --> 00:46:23.330 Everyone I talk to, I ask them how they manage to keep all 00:46:23.330 --> 00:46:24.430 this straight. 00:46:24.430 --> 00:46:27.180 And they all give me the same answer: they say they can't 00:46:27.180 --> 00:46:27.930 keep it straight. 00:46:27.930 --> 00:46:30.680 It's just too hard to keep it straight, and after they're 00:46:30.680 --> 00:46:36.390 all done, they try to figure out what the answer should be 00:46:36.390 --> 00:46:39.310 by looking at energy or something else. 00:46:39.310 --> 00:46:44.310 Or by just fudging things, which is what most people do. 00:46:44.310 --> 00:46:48.680 And part of the trouble is, you can do this two ways. 00:46:48.680 --> 00:46:50.790 You can do it three ways, in fact. 00:46:50.790 --> 00:46:56.730 You can either I want to view x of t as being some real 00:46:56.730 --> 00:47:01.910 function times the cosine wave and leave out that 2. 00:47:01.910 --> 00:47:06.330 And some other function, imaginary part of u t times 00:47:06.330 --> 00:47:08.570 the sine, and leave out the 2 there. 00:47:08.570 --> 00:47:09.820 And many people do that. 00:47:12.790 --> 00:47:14.310 And would that be better? 00:47:14.310 --> 00:47:17.470 But when you put the 2 in with the cosines and the sines, you 00:47:17.470 --> 00:47:21.580 have to put a 1/2 in here and a 1/2 in here. 00:47:21.580 --> 00:47:24.650 Most people, when they think about these things for a long 00:47:24.650 --> 00:47:28.360 time, they find it's far more convenient to be able to think 00:47:28.360 --> 00:47:35.610 of this positive frequency part of x of t as just u of t 00:47:35.610 --> 00:47:37.100 translated up in frequency. 00:47:37.100 --> 00:47:40.050 In other words, they like this diagram here. 00:47:44.960 --> 00:47:46.540 Which says you take this. 00:47:46.540 --> 00:47:48.260 You translate it up. 00:47:48.260 --> 00:47:51.180 And after you translate it up, you create something else down 00:47:51.180 --> 00:47:53.540 here, to make the whole thing real. 00:47:53.540 --> 00:47:57.410 But what we think of is this going up to this all the time. 00:47:57.410 --> 00:47:58.830 So that's one way of doing it. 00:47:58.830 --> 00:48:01.720 The other way of doing it is thinking in terms of sines and 00:48:01.720 --> 00:48:05.370 cosines, removing that 2 here. 00:48:05.370 --> 00:48:10.190 And who can imagine what the third way of doing it is? 00:48:10.190 --> 00:48:11.680 Just split the difference. 00:48:11.680 --> 00:48:15.030 Which means you put a square root of 2 in. 00:48:15.030 --> 00:48:17.610 And, in fact, that makes a whole lot of sense. 00:48:17.610 --> 00:48:22.160 Because then when you take the waveform u of t, translate it 00:48:22.160 --> 00:48:23.110 up in frequency. 00:48:23.110 --> 00:48:26.710 Make it real, you have the same energy in the baseband 00:48:26.710 --> 00:48:29.960 waveform as you have in the passband waveform. 00:48:29.960 --> 00:48:30.980 I'm not going to show that. 00:48:30.980 --> 00:48:35.060 You can just figure it out relatively easily. 00:48:35.060 --> 00:48:41.040 I mean, you know that the power in a cosine wave is 1/2, 00:48:41.040 --> 00:48:43.250 the power in a sine wave is 1/2. 00:48:43.250 --> 00:48:48.660 So when you're multiplying things by 1/2 in here -- well, 00:48:48.660 --> 00:48:51.210 this has a power of 1/2. 00:48:51.210 --> 00:48:54.410 This has a power of 1/2. 00:48:54.410 --> 00:48:57.840 And when you start looking at power here, you find out that 00:48:57.840 --> 00:49:01.730 that has to be a square root of 2 rather than 2. 00:49:01.730 --> 00:49:03.450 So there are three ways of doing it. 00:49:03.450 --> 00:49:06.710 People do it any one of three different ways. 00:49:06.710 --> 00:49:09.240 It doesn't make any difference, because any paper 00:49:09.240 --> 00:49:12.900 you read will start out doing it one way and then, as they 00:49:12.900 --> 00:49:15.740 go through various equations, they will start doing it a 00:49:15.740 --> 00:49:16.780 different way. 00:49:16.780 --> 00:49:18.820 And these factors of 2 multiply and 00:49:18.820 --> 00:49:20.450 multiply and multiply. 00:49:20.450 --> 00:49:23.510 And in big complicated papers, sometimes I've found that 00:49:23.510 --> 00:49:27.370 these add up to a factor of 8 or 16 or something else. 00:49:27.370 --> 00:49:29.910 By the time people are all done. 00:49:29.910 --> 00:49:33.510 And we will explain later why, in fact, you don't really care 00:49:33.510 --> 00:49:36.140 about that very much. 00:49:36.140 --> 00:49:37.620 But you can't just totally ignore 00:49:37.620 --> 00:49:40.510 those factors, so anyway. 00:49:40.510 --> 00:49:43.270 This is the way we will do it. 00:49:43.270 --> 00:49:47.230 We will try to be consistent about this, and 00:49:47.230 --> 00:49:48.480 usually we will be. 00:49:55.130 --> 00:49:58.690 The way we want to think about this conceptually is that 00:49:58.690 --> 00:50:02.450 quadrature amplitude modulation is going to take 00:50:02.450 --> 00:50:06.830 this complex waveform u of t. 00:50:06.830 --> 00:50:11.450 It's going to shift it up in frequency to f sub c, and then 00:50:11.450 --> 00:50:13.650 we're going to add the complex conjugate. 00:50:13.650 --> 00:50:16.590 Add it to form the real x sub t. 00:50:16.590 --> 00:50:17.780 In other words, we're going to think of it 00:50:17.780 --> 00:50:20.040 as a two-stage operation. 00:50:20.040 --> 00:50:23.230 First you take waveform, you translate it up. 00:50:23.230 --> 00:50:27.510 Then you take the real part or something, or add the negative 00:50:27.510 --> 00:50:29.400 frequency part. 00:50:29.400 --> 00:50:33.160 And we're going to think both of this double operation of 1 00:50:33.160 --> 00:50:35.310 going from u of t to the positive 00:50:35.310 --> 00:50:36.900 frequency part of things. 00:50:36.900 --> 00:50:39.530 And then, of looking at the real waveform that 00:50:39.530 --> 00:50:42.740 corresponds to that. 00:50:42.740 --> 00:50:46.380 What we're going to be doing here in terms of all of this 00:50:46.380 --> 00:50:50.360 is, we're going to start out with binary data. 00:50:50.360 --> 00:50:54.190 From the binary data, we're going to go to symbols. 00:50:54.190 --> 00:50:56.400 And we're going to go to symbols by taking a number of 00:50:56.400 --> 00:50:58.600 binary data -- 00:50:58.600 --> 00:51:00.810 a number of binary digits. 00:51:00.810 --> 00:51:03.760 Framing then into b tuples. 00:51:03.760 --> 00:51:06.670 Each b tuple will correspond to a set 00:51:06.670 --> 00:51:08.330 of 2 to the b symbols. 00:51:08.330 --> 00:51:12.340 We're going to map these symbols into complex signals. 00:51:12.340 --> 00:51:15.750 We're going to map the complex signals into a baseband 00:51:15.750 --> 00:51:17.080 waveform u of t. 00:51:17.080 --> 00:51:20.270 We're going to map the baseband waveform u of t into 00:51:20.270 --> 00:51:23.940 this positive frequency waveform u of t times this 00:51:23.940 --> 00:51:25.540 complex sinusoid. 00:51:25.540 --> 00:51:28.580 And finally we're going to add on the negative frequency part 00:51:28.580 --> 00:51:30.400 to wind up with x of t. 00:51:30.400 --> 00:51:32.550 What do you think we do at the receiver? 00:51:32.550 --> 00:51:35.330 As always, we do just the opposite. 00:51:35.330 --> 00:51:38.470 Namely, one of the reasons for wanting to think about this 00:51:38.470 --> 00:51:42.000 this way, is we want to use this layering idea. 00:51:42.000 --> 00:51:44.700 And the layering idea says, you start out with the 00:51:44.700 --> 00:51:46.850 received waveform x of t. 00:51:46.850 --> 00:51:49.780 And later on we'll have to add the noise to it. 00:51:49.780 --> 00:51:53.050 You go from there to the positive frequency part. 00:51:53.050 --> 00:51:55.410 You go from the positive frequency part. 00:51:55.410 --> 00:51:57.420 You shift it down to u of t. 00:51:57.420 --> 00:52:00.120 How we got from here to there, I'll explain in a minute. 00:52:00.120 --> 00:52:03.100 You go from here down to baseband again. 00:52:03.100 --> 00:52:06.530 You go from the baseband to the complex signals, which 00:52:06.530 --> 00:52:09.760 we're going to do simply by filtering and sampling. 00:52:09.760 --> 00:52:13.170 We go from the complex signals to the symbols, which is in 00:52:13.170 --> 00:52:14.830 fact a trivial operation. 00:52:14.830 --> 00:52:16.820 It's just a look-up operation. 00:52:16.820 --> 00:52:20.640 And then from there we un-segment things into binary 00:52:20.640 --> 00:52:21.370 digits again. 00:52:21.370 --> 00:52:23.030 So that's the whole system. 00:52:23.030 --> 00:52:25.700 And it has all these different pieces to it. 00:52:25.700 --> 00:52:29.000 I couldn't draw it as our favorite kind of diagram, 00:52:29.000 --> 00:52:30.670 because it has too many blocks in it. 00:52:30.670 --> 00:52:33.730 So that has to do. 00:52:36.320 --> 00:52:38.700 What we're going to do now is look at each of these pieces 00:52:38.700 --> 00:52:40.670 one at a time. 00:52:40.670 --> 00:52:45.440 And the first part is the complex QAM signal set. 00:52:45.440 --> 00:52:49.210 And, just for some notation here, so we'll be on the same 00:52:49.210 --> 00:52:57.170 page, but we use r to talk about the bits per second at 00:52:57.170 --> 00:53:02.830 which data is coming into this whole system. 00:53:02.830 --> 00:53:05.400 That's the figure you're interested in, when 00:53:05.400 --> 00:53:06.670 everything is done. 00:53:06.670 --> 00:53:10.020 How many bits per second can you transmit? 00:53:10.020 --> 00:53:13.180 We're going to segment this into b bits at a time. 00:53:13.180 --> 00:53:16.350 So we're going to have a symbol set with 2 to the b 00:53:16.350 --> 00:53:18.850 elements in it. 00:53:18.850 --> 00:53:22.920 We're going to map these m symbols, which are binary b 00:53:22.920 --> 00:53:27.820 tuples, into elements from the signal set. 00:53:27.820 --> 00:53:31.570 The signal rate, then, is r sub s, which is r over b. 00:53:31.570 --> 00:53:35.270 This is the number of signals per second that we're sending. 00:53:35.270 --> 00:53:41.940 In other words, t, this signal interval that we've always 00:53:41.940 --> 00:53:44.390 been using in everything we've been doing, is 00:53:44.390 --> 00:53:46.740 one over r sub s. 00:53:46.740 --> 00:53:49.710 So t is the signal interval. 00:53:49.710 --> 00:53:53.090 Every t seconds, you've got to send something. 00:53:53.090 --> 00:53:56.170 If you didn't send something every t seconds, the way this 00:53:56.170 --> 00:54:01.350 stuff coming in from the source would start piling up 00:54:01.350 --> 00:54:05.450 and your buffers would overflow and it wouldn't work. 00:54:05.450 --> 00:54:08.420 The signals u sub k are complex numbers. 00:54:08.420 --> 00:54:10.300 Or real 2-tuples. 00:54:10.300 --> 00:54:14.610 So we can, when we're trying to decide what signal set 00:54:14.610 --> 00:54:18.890 we're using, we can just draw our signals on a plane. 00:54:18.890 --> 00:54:23.030 The signal set is a constellation, then, of m 00:54:23.030 --> 00:54:26.040 complex numbers or real 2-tuples. 00:54:26.040 --> 00:54:29.120 So the problem of choosing the signal set is, how do you 00:54:29.120 --> 00:54:32.870 choose m points on a complex plane. 00:54:32.870 --> 00:54:36.190 What problem is that similar to? 00:54:36.190 --> 00:54:41.060 It's similar to the quantization problem where we 00:54:41.060 --> 00:54:44.590 were trying to choose m representation points. 00:54:44.590 --> 00:54:47.240 And it's very close to that problem. 00:54:47.240 --> 00:54:49.490 It's a very similar problems. 00:54:49.490 --> 00:54:52.180 Has a few small differences, but not many. 00:54:56.370 --> 00:54:59.920 But, before getting into that, we want to talk about a 00:54:59.920 --> 00:55:03.420 standard QAM signal set. 00:55:03.420 --> 00:55:08.620 In a minute I'll explain why people do that. 00:55:08.620 --> 00:55:14.040 And, as you might imagine, a standard QAM is just a square 00:55:14.040 --> 00:55:16.270 array of points. 00:55:16.270 --> 00:55:20.340 It's the simplest thing to do, and sometimes the simplest 00:55:20.340 --> 00:55:21.700 thing is the best. 00:55:21.700 --> 00:55:25.770 So it's determined by some distance, d, that you want to 00:55:25.770 --> 00:55:28.770 have between neighboring points. 00:55:28.770 --> 00:55:31.800 And given that distance, d, you just create a 00:55:31.800 --> 00:55:33.370 square array here. 00:55:33.370 --> 00:55:37.220 The square array means that m has to have an 00:55:37.220 --> 00:55:38.690 integer square root. 00:55:38.690 --> 00:55:41.740 This is drawn for m equals 16. 00:55:41.740 --> 00:55:45.460 If you look at this, you see that the real part of this 00:55:45.460 --> 00:55:48.350 it's the standard PAM set. 00:55:48.350 --> 00:55:53.040 The imaginary part is a standard PAM set, which says 00:55:53.040 --> 00:55:56.340 you can deal with the real part and the imaginary part 00:55:56.340 --> 00:55:57.190 separately. 00:55:57.190 --> 00:55:59.970 You take half the bits coming in and you choose your real 00:55:59.970 --> 00:56:01.180 part signal. 00:56:01.180 --> 00:56:03.090 Take the other half of the bits coming in. 00:56:03.090 --> 00:56:05.560 You form your imaginary part signal, and 00:56:05.560 --> 00:56:07.310 bingo, you're all done. 00:56:07.310 --> 00:56:11.670 The energy per 2D signal, we can find the energy for 2D 00:56:11.670 --> 00:56:14.540 signal by looking at it this way. 00:56:14.540 --> 00:56:18.610 It's two PAM systems running in parallel to each other. 00:56:18.610 --> 00:56:24.580 For the PAM system, the energy in one of these dimensions is 00:56:24.580 --> 00:56:28.150 then d squared times the square root of n squared, 00:56:28.150 --> 00:56:31.210 minus 1 divided by 12. 00:56:31.210 --> 00:56:35.120 But now we want to look at the energy which we have in both 00:56:35.120 --> 00:56:37.410 the real part and the imaginary part. 00:56:37.410 --> 00:56:39.300 So we need this extra factor. 00:56:39.300 --> 00:56:41.710 Well, we need to add together two of these things. 00:56:41.710 --> 00:56:46.660 So we wind up with d squared times m minus 1 divided by 6. 00:56:46.660 --> 00:56:47.910 Big deal. 00:56:54.060 --> 00:56:57.460 Choosing a good signal set is similar to choosing a 2D set 00:56:57.460 --> 00:57:01.180 of representation points in quantization. 00:57:01.180 --> 00:57:04.610 If you like to optimize things, you see this problem 00:57:04.610 --> 00:57:08.530 and you say, gee, at least there's something I can put my 00:57:08.530 --> 00:57:11.410 teeth into here. 00:57:11.410 --> 00:57:15.820 What's the best way to choose a signal set? 00:57:15.820 --> 00:57:18.010 And we found that for quantization that wasn't a 00:57:18.010 --> 00:57:21.270 terribly nice problem, although at least we had 00:57:21.270 --> 00:57:26.340 things like algorithms to try to choose reasonable sets. 00:57:26.340 --> 00:57:29.140 And we then looked at entropy quantization and things like 00:57:29.140 --> 00:57:32.120 this, and it was a certain amount of fun. 00:57:32.120 --> 00:57:34.560 Here, this problem it's just ugly. 00:57:34.560 --> 00:57:38.540 There's no other way to express it. 00:57:38.540 --> 00:57:40.080 I had to be convinced of this. 00:57:40.080 --> 00:57:43.440 I once spent an inordinate amount of time trying to find 00:57:43.440 --> 00:57:46.910 the best signal set with eight points in it, in two 00:57:46.910 --> 00:57:47.990 dimensions. 00:57:47.990 --> 00:57:52.820 How do you put eight single points in two dimensions in 00:57:52.820 --> 00:57:56.710 such a way that every point is distance at least d from every 00:57:56.710 --> 00:57:58.980 other point, and you minimize the energy 00:57:58.980 --> 00:58:02.100 of the set of points? 00:58:02.100 --> 00:58:04.520 The answer is just absolutely ugly. 00:58:04.520 --> 00:58:05.810 It has no symmetry. 00:58:05.810 --> 00:58:07.400 Nothing nice about it. 00:58:07.400 --> 00:58:10.000 You do the same thing for 16 points, and it's 00:58:10.000 --> 00:58:11.180 just an ugly problem. 00:58:11.180 --> 00:58:13.660 You do the same thing for any number of points. 00:58:13.660 --> 00:58:14.680 Except for four points. 00:58:14.680 --> 00:58:16.260 For four points, it's easy. 00:58:16.260 --> 00:58:19.730 For four points, you use standard QAM and it's the best 00:58:19.730 --> 00:58:21.220 thing to do. 00:58:21.220 --> 00:58:23.460 And that problem is easy. 00:58:23.460 --> 00:58:25.680 But you know, that's not much fun. 00:58:25.680 --> 00:58:28.260 Because you say, bleugh. 00:58:28.260 --> 00:58:32.650 So, partly for that reason, people use 00:58:32.650 --> 00:58:36.020 standard signal sets. 00:58:36.020 --> 00:58:39.370 Partly because you don't seem to be able to gain much by 00:58:39.370 --> 00:58:42.860 doing anything else. 00:58:42.860 --> 00:58:47.060 So that's about all we can say about standard -- oh, with 00:58:47.060 --> 00:58:49.810 eight signals, you can't use a standard signal set. 00:58:49.810 --> 00:58:52.510 That was one reason we had to worry about it. 00:58:52.510 --> 00:58:56.480 Back a long time ago, we were trying to design a 7200 bit 00:58:56.480 --> 00:58:58.440 per second modem. 00:58:58.440 --> 00:59:03.210 Back in the days when people did 2400 bits per second. 00:59:03.210 --> 00:59:06.250 And we managed to do 4800 bits per second by 00:59:06.250 --> 00:59:09.120 using QAM, big deal. 00:59:09.120 --> 00:59:12.320 And then we said, well, we can pile in an extra bit by using 00:59:12.320 --> 00:59:17.350 three bits per two dimensions instead of two bits. 00:59:17.350 --> 00:59:20.110 And spent this enormous amount of time trying to find a 00:59:20.110 --> 00:59:21.490 sensible signal set. 00:59:21.490 --> 00:59:23.580 I don't even want to tell you what it was, because it wasn't 00:59:23.580 --> 00:59:25.810 interesting at all. 00:59:25.810 --> 00:59:30.970 So, enough for signal sets. 00:59:30.970 --> 00:59:33.860 The next thing is, how do you turn the signals 00:59:33.860 --> 00:59:36.260 into complex waveforms? 00:59:36.260 --> 00:59:39.420 Namely, how do you go from the signals in two dimensions, 00:59:39.420 --> 00:59:45.100 complex signals into a baseband waveform u of t? 00:59:45.100 --> 00:59:48.890 Well, fortunately, Nyquist's theory is exactly the same 00:59:48.890 --> 00:59:52.380 here as it was when we were dealing with PAM. 00:59:52.380 --> 00:59:55.370 Everything we said before works here. 00:59:55.370 --> 00:59:58.270 The only difference is that you don't have to choose the 00:59:58.270 --> 01:00:00.820 pulse p of t to be real. 01:00:00.820 --> 01:00:03.240 But if you look back at what we did, we didn't assume that 01:00:03.240 --> 01:00:05.810 p of t was real before, anyway. 01:00:05.810 --> 01:00:09.120 We just said, you might as well choose it to be real. 01:00:09.120 --> 01:00:13.460 But you don't have to choose it to be real. 01:00:13.460 --> 01:00:19.680 Bandedge symmetry requires that g of t be real. 01:00:19.680 --> 01:00:21.230 Anybody know why that is? 01:00:24.300 --> 01:00:28.910 When you choose g of t to be real, the negative frequency 01:00:28.910 --> 01:00:31.420 part is the complex conjugate of the 01:00:31.420 --> 01:00:33.530 positive frequency part. 01:00:33.530 --> 01:00:38.570 Which is why, when we took this out-of-band stuff at 01:00:38.570 --> 01:00:42.070 negative frequencies, piled it into the positive frequencies, 01:00:42.070 --> 01:00:45.530 we got the same thing as if we simply rotate it around on the 01:00:45.530 --> 01:00:46.930 positive frequency. 01:00:46.930 --> 01:00:50.660 So that bandedge symmetry condition really requires that 01:00:50.660 --> 01:00:52.970 g of t be real. 01:00:52.970 --> 01:00:56.330 The orthogonality of t a t minus k t, this set of 01:00:56.330 --> 01:01:00.560 waveforms, requires g of t to be real. 01:01:00.560 --> 01:01:03.060 Neither of these things require p of t to be real. 01:01:03.060 --> 01:01:06.640 You can choose p of t to have any old phase characteristic 01:01:06.640 --> 01:01:10.590 you want to, but if we're choosing p of t -- if we're 01:01:10.590 --> 01:01:15.760 choosing p hat of f magnitude to be the square root of a 01:01:15.760 --> 01:01:21.440 Nyquist waveform, then you can choose this phase to be 01:01:21.440 --> 01:01:24.080 anything you want to make it. 01:01:24.080 --> 01:01:29.970 But you're just restricted in, aside from the phase, you're 01:01:29.970 --> 01:01:33.310 somewhat restricted in what p of t can be. 01:01:33.310 --> 01:01:33.620 OK. 01:01:33.620 --> 01:01:35.960 So we're going to make the nominal passband, Nyquist 01:01:35.960 --> 01:01:37.640 band, with 1 over t. 01:01:37.640 --> 01:01:40.360 Before we made the passband -- 01:01:40.360 --> 01:01:44.590 before we made the baseband bandwidth 1 over 2t. 01:01:44.590 --> 01:01:49.580 When we go up the passband we double the bandwidth so the 01:01:49.580 --> 01:01:52.210 Nyquist bandwidth is now 1 over t. 01:01:52.210 --> 01:01:54.800 The passband bandwidth is 1 over t. 01:01:54.800 --> 01:01:57.270 That's the only thing that's changed. 01:01:57.270 --> 01:02:00.150 Usually people design these filters, which they design at 01:02:00.150 --> 01:02:05.340 baseband, to go 5-10% over the Nyquist band. 01:02:05.340 --> 01:02:10.880 In other words, these filters are very, very sharp, usually. 01:02:10.880 --> 01:02:13.390 I mean, once you design a filter, it 01:02:13.390 --> 01:02:14.450 doesn't cost anything. 01:02:14.450 --> 01:02:17.480 You put it on a chip and that's the end of it. 01:02:17.480 --> 01:02:19.650 And the cost of it is zero. 01:02:19.650 --> 01:02:22.850 So it's just the cost to design it, so you might as 01:02:22.850 --> 01:02:24.100 well make it small. 01:02:28.360 --> 01:02:31.630 So finally, we want to go to base, 01:02:31.630 --> 01:02:33.540 from baseband to passband. 01:02:33.540 --> 01:02:36.240 We talked about this a little bit. 01:02:36.240 --> 01:02:40.990 In terms of these frequencies, the baseband frequency b sub 01:02:40.990 --> 01:02:43.420 u, we want to assume that that's less 01:02:43.420 --> 01:02:44.780 than the carrier frequency. 01:02:44.780 --> 01:02:49.540 This is this condition that we needed to make sure that the 01:02:49.540 --> 01:02:56.050 positive frequency part -- ah, here it is. 01:02:56.050 --> 01:02:58.550 That's the condition that makes sure that this is 01:02:58.550 --> 01:03:01.440 separated from that, and doesn't cause intersymbol 01:03:01.440 --> 01:03:04.740 interference between the two. 01:03:04.740 --> 01:03:08.110 So, everything we do, we'll make this assumption. 01:03:08.110 --> 01:03:13.540 I mean, a part of this is, if you're going to modulate with 01:03:13.540 --> 01:03:16.640 such a small carrier frequency, you might as well 01:03:16.640 --> 01:03:17.630 not do it at all. 01:03:17.630 --> 01:03:19.220 You might as well just generate the 01:03:19.220 --> 01:03:21.180 waveform you want directly. 01:03:21.180 --> 01:03:25.610 Because you don't gain that much by doing it at baseband. 01:03:25.610 --> 01:03:30.150 Because you don't really have a baseband in that case. 01:03:30.150 --> 01:03:34.480 So, u of t times e to the 2 pi i f c t is strictly in the 01:03:34.480 --> 01:03:36.880 positive frequency band, then. 01:03:36.880 --> 01:03:39.850 And these two bands don't overlap. 01:03:39.850 --> 01:03:42.110 As I said before, we're going to view this as 01:03:42.110 --> 01:03:43.670 two different steps. 01:03:43.670 --> 01:03:46.490 The first step is, I'm going to take this complex 01:03:46.490 --> 01:03:48.490 waveform, u of t. 01:03:48.490 --> 01:03:51.410 Multiply it by a complex sinusoid, which shifts me up 01:03:51.410 --> 01:03:52.720 in frequency. 01:03:52.720 --> 01:03:56.650 Just going to call that u passband of t. 01:03:56.650 --> 01:04:00.250 This is this passband signal that I want to think about. 01:04:00.250 --> 01:04:02.480 The thing that's up in positive frequencies. 01:04:02.480 --> 01:04:05.210 I'm going to ignore the thing at negative frequencies. 01:04:05.210 --> 01:04:11.910 And then I'm going to form the actual waveform x sub t, as 01:04:11.910 --> 01:04:14.640 this plus its conjugate. 01:04:26.000 --> 01:04:29.640 If you think a little bit now, you can see that since these 01:04:29.640 --> 01:04:33.770 two bands are separated, if you think in terms of complex 01:04:33.770 --> 01:04:38.720 waveforms, how do you retrieve this band up here by a 01:04:38.720 --> 01:04:42.750 waveform which has both bands in it? 01:04:42.750 --> 01:04:46.270 Well, you filter out what's at negative frequencies, OK? 01:04:48.800 --> 01:04:55.160 So we want to design a filter which filters out the negative 01:04:55.160 --> 01:04:58.660 frequencies and x of t, and only leaves the positive 01:04:58.660 --> 01:04:59.160 frequencies. 01:04:59.160 --> 01:05:03.210 In other words, you want a filter whose frequency 01:05:03.210 --> 01:05:08.470 response is just 1 for all positive frequencies, 0 for 01:05:08.470 --> 01:05:11.070 all negative frequencies. 01:05:11.070 --> 01:05:14.610 And that filter is called a Hilbert filter. 01:05:14.610 --> 01:05:19.020 Have any of you ever heard of a Hilbert filter before? 01:05:19.020 --> 01:05:21.460 I don't know of anybody that's ever built one. 01:05:21.460 --> 01:05:24.520 And we'll see why they don't built them in a while. 01:05:24.520 --> 01:05:26.420 But it's a nice idea. 01:05:26.420 --> 01:05:28.750 I mean, if you try to build one you'll find that it's 01:05:28.750 --> 01:05:35.820 harder to build -- 01:05:35.820 --> 01:05:39.150 you'll find that you have to implement four real filters in 01:05:39.150 --> 01:05:40.920 order to implement this filter. 01:05:40.920 --> 01:05:43.160 So we'll find out that's not the thing to do. 01:05:43.160 --> 01:05:46.400 But it's nice conceptually because it lets us study 01:05:46.400 --> 01:05:51.260 things like energy, power, and linearity, and 01:05:51.260 --> 01:05:52.510 all of these things. 01:05:56.140 --> 01:06:01.490 So the transmitter then becomes this thing you start 01:06:01.490 --> 01:06:04.990 out with a complex waveform of baseband. 01:06:04.990 --> 01:06:07.860 You shift it up in frequency. 01:06:07.860 --> 01:06:09.690 This gives you this high frequency 01:06:09.690 --> 01:06:12.960 waveform, u sub p of t. 01:06:12.960 --> 01:06:16.830 You then take 2 times the real part of that to 01:06:16.830 --> 01:06:18.450 find the real waveform. 01:06:18.450 --> 01:06:20.220 We won't worry about how to implement 01:06:20.220 --> 01:06:23.050 this, you just do it. 01:06:23.050 --> 01:06:26.440 This passband waveform, you then pass it through this 01:06:26.440 --> 01:06:29.190 Hilbert filter, which just chops off the negative 01:06:29.190 --> 01:06:31.000 frequency part of it. 01:06:31.000 --> 01:06:34.120 Gives you a complex waveform again. 01:06:34.120 --> 01:06:39.050 You multiply by e to the minus 2 pi i f c t, which takes this 01:06:39.050 --> 01:06:41.260 positive frequency waveform, shifts it 01:06:41.260 --> 01:06:43.610 back down to baseband. 01:06:43.610 --> 01:06:46.490 So this is a nice convenient way of thinking about, how do 01:06:46.490 --> 01:06:50.390 you go from baseband up to passband and passband down to 01:06:50.390 --> 01:06:52.150 baseband again. 01:06:52.150 --> 01:06:52.820 Now. 01:06:52.820 --> 01:06:58.050 If you want to view these vectors here as vectors, want 01:06:58.050 --> 01:07:03.980 to view u of t as a vector in L2, there's an important thing 01:07:03.980 --> 01:07:05.660 here going on. 01:07:05.660 --> 01:07:08.900 We'll have to talk about it a good deal later on. 01:07:08.900 --> 01:07:11.410 This is a complex waveform. 01:07:11.410 --> 01:07:14.970 You want to deal with it as a vector in complex L2. 01:07:14.970 --> 01:07:19.580 In complex L2, when we're dealing with vectors, we have 01:07:19.580 --> 01:07:23.390 scalars, which are complex numbers. 01:07:23.390 --> 01:07:28.150 When we start dealing with real parts of these things, we 01:07:28.150 --> 01:07:33.150 want to view the real parts as being elements of real L2. 01:07:33.150 --> 01:07:36.370 Where the scalars are real numbers. 01:07:36.370 --> 01:07:41.050 And what this says is that real L2 is not a subspace of 01:07:41.050 --> 01:07:43.480 complex L2. 01:07:43.480 --> 01:07:47.020 It's not a subspace because the scalars are different. 01:07:47.020 --> 01:07:50.350 This might sound like mathematical nitpicking. 01:07:50.350 --> 01:07:52.280 But, put it in the back of your mind. 01:07:52.280 --> 01:07:54.650 Because at some point it's going to come 01:07:54.650 --> 01:07:56.830 up and clobber you. 01:07:56.830 --> 01:07:59.810 And at that point, you will want to think that in fact 01:07:59.810 --> 01:08:04.950 real L2 is not a subspace of complex L2. 01:08:04.950 --> 01:08:08.390 When we start thinking about orthonormal expansions for u 01:08:08.390 --> 01:08:12.950 of t and orthonormal expansions for x of t, in fact 01:08:12.950 --> 01:08:15.600 you have to be quite careful about this. 01:08:15.600 --> 01:08:18.820 Because you take an orthonormal expansion here, 01:08:18.820 --> 01:08:21.840 translate it up into frequency. 01:08:21.840 --> 01:08:27.640 And you wind up with a bunch of complex waveforms. 01:08:27.640 --> 01:08:29.910 And they aren't real waveforms. 01:08:29.910 --> 01:08:32.430 And funny things start happening. 01:08:32.430 --> 01:08:34.090 So we'll deal with all of that later. 01:08:34.090 --> 01:08:37.100 This is just to warn you that we have to be 01:08:37.100 --> 01:08:39.550 careful about that. 01:08:39.550 --> 01:08:41.970 This is not the way people implement these things. 01:08:41.970 --> 01:08:46.410 Because these Hilbert filters are in fact for real filters. 01:08:49.020 --> 01:08:55.620 So the implementation is what you've seen. 01:08:55.620 --> 01:08:57.350 I mean, the implementation is old. 01:08:57.350 --> 01:09:00.940 And it's the way you want to build these things. 01:09:00.940 --> 01:09:05.290 You start out with two real baseband waveforms. 01:09:05.290 --> 01:09:07.940 One which we call the real part of u of t. 01:09:07.940 --> 01:09:11.130 One which we call the imaginary part of u of t. 01:09:11.130 --> 01:09:14.000 In this single diagram, one of them is the stuff 01:09:14.000 --> 01:09:15.270 that goes this way. 01:09:15.270 --> 01:09:17.880 And the other one is the stuff that goes this way. 01:09:17.880 --> 01:09:21.940 And if, in fact, you're using a standard QAM signal set, the 01:09:21.940 --> 01:09:25.490 two are completely independent of each other. 01:09:25.490 --> 01:09:31.310 So the real part of u of t is just the sum of these shifted 01:09:31.310 --> 01:09:34.160 pulses times the real parts. 01:09:34.160 --> 01:09:36.600 This is the sum of the shifted pulses 01:09:36.600 --> 01:09:39.590 times the complex parts. 01:09:39.590 --> 01:09:43.050 The defined u sub k prime is a real part, and u sub k double 01:09:43.050 --> 01:09:45.890 prime is the imaginary part. 01:09:45.890 --> 01:09:53.320 In the notes, this and this are called a sub k and a sub k 01:09:53.320 --> 01:09:55.170 double prime. 01:09:55.170 --> 01:09:57.600 Which doesn't correspond to anything else. 01:09:57.600 --> 01:10:00.430 So, this is the correct way of doing it. 01:10:00.430 --> 01:10:03.490 But, anyway, when you get all done, x sub t is 2 times the 01:10:03.490 --> 01:10:09.630 cosine of this low pass modulated PAM waveform. 01:10:09.630 --> 01:10:13.530 Minus 2 times the sine of this low pass 01:10:13.530 --> 01:10:16.240 PAM modulated waveform. 01:10:16.240 --> 01:10:19.910 So, QAM, when you look at it this way, is simply two 01:10:19.910 --> 01:10:23.160 different PAM systems. 01:10:23.160 --> 01:10:26.410 One of them modulated on a cosine carrier, one of them 01:10:26.410 --> 01:10:28.410 modulated on a sine carrier. 01:10:36.220 --> 01:10:40.220 And the picture of that is this. 01:10:40.220 --> 01:10:41.470 Aargh. 01:10:46.970 --> 01:10:50.940 Can't keep my notation straight. 01:10:50.940 --> 01:10:54.880 I'm sure it doesn't bother most of you that much, but it 01:10:54.880 --> 01:10:57.330 bothers me. 01:10:57.330 --> 01:10:59.410 All of those a's should be u's. 01:10:59.410 --> 01:11:02.460 They were a's last year, but they don't make 01:11:02.460 --> 01:11:05.290 any sense as a's. 01:11:05.290 --> 01:11:12.640 So the thing we're going to do now is we start out with the 01:11:12.640 --> 01:11:14.190 sequence of signals. 01:11:14.190 --> 01:11:15.550 The real part of the signals and the 01:11:15.550 --> 01:11:17.390 imaginary part of the signals. 01:11:17.390 --> 01:11:21.530 This is why it's called double side band quadrature carrier, 01:11:21.530 --> 01:11:23.050 because in fact we're doing two 01:11:23.050 --> 01:11:26.760 different things in parallel. 01:11:26.760 --> 01:11:31.010 We generate this as a pulse waveform. 01:11:31.010 --> 01:11:32.930 We filter it by p of t. 01:11:32.930 --> 01:11:36.850 We're thinking of p of t as a real waveform now. 01:11:36.850 --> 01:11:39.840 If you want p of t to be complex you have to modify 01:11:39.840 --> 01:11:40.950 this all a little bit. 01:11:40.950 --> 01:11:46.720 But there's no real reason to make p of t complex anyway. 01:11:46.720 --> 01:11:50.360 So when you get out of here, what you have is just this low 01:11:50.360 --> 01:11:53.980 pass real waveform, real PAM waveform. 01:11:53.980 --> 01:11:57.480 Here's another low pass real PAM waveform. 01:11:57.480 --> 01:12:01.540 You module this up by multiplying by cosine of 2 pi 01:12:01.540 --> 01:12:05.700 f c t, in fact, by 2 cosine of 2 pi f ct . 01:12:05.700 --> 01:12:09.330 You modulate this up by multiplying by minus sine. 01:12:09.330 --> 01:12:12.290 And you get the actual waveform that 01:12:12.290 --> 01:12:13.880 you're going to transmit. 01:12:13.880 --> 01:12:15.640 How do you demodulate this? 01:12:15.640 --> 01:12:20.130 Well, again, I'm sure you've seen it in one of the 01:12:20.130 --> 01:12:22.310 undergraduate courses you've taken. 01:12:22.310 --> 01:12:25.840 Because if you take this waveform, which is the sum of 01:12:25.840 --> 01:12:30.120 this and this, and you multiply this by cosine of 2 01:12:30.120 --> 01:12:32.330 pi f c t, what's going to happen? 01:12:34.960 --> 01:12:39.070 Taking this waveform and multiplying it by cosine is 01:12:39.070 --> 01:12:41.770 going to take this cosine waveform. 01:12:41.770 --> 01:12:45.560 Half of it goes up in frequency by f sub c. 01:12:45.560 --> 01:12:50.440 The other half goes down in frequency by f sub c of t. 01:12:50.440 --> 01:12:56.060 When you multiply by sine, the same thing happens. 01:12:56.060 --> 01:13:01.640 And all of the stuff at this double frequency term all gets 01:13:01.640 --> 01:13:02.940 filtered out. 01:13:02.940 --> 01:13:04.730 I mean, you have enough filtering to 01:13:04.730 --> 01:13:06.380 just wash that away. 01:13:06.380 --> 01:13:10.260 And you wind up, just with this one waveform which is the 01:13:10.260 --> 01:13:11.520 result of this. 01:13:11.520 --> 01:13:14.920 Another waveform which is the result of this. 01:13:14.920 --> 01:13:17.720 You can show that the two don't interfere at all, and 01:13:17.720 --> 01:13:19.980 you just have to do the multiplication 01:13:19.980 --> 01:13:20.810 to find this out. 01:13:20.810 --> 01:13:23.940 It looks a little bit like black magic when you look at 01:13:23.940 --> 01:13:24.730 it like this. 01:13:24.730 --> 01:13:28.400 Because when you're multiplying by a cosine wave, 01:13:28.400 --> 01:13:31.550 I mean it's easy to see what cosine squared does here. 01:13:31.550 --> 01:13:33.010 But it's a little harder to see what 01:13:33.010 --> 01:13:35.050 happens to all of this. 01:13:35.050 --> 01:13:39.480 And when we look at it the other way, which was this 01:13:39.480 --> 01:13:44.290 Hilbert filter kind of thing, when you look at in terms of 01:13:44.290 --> 01:13:48.750 the Hilbert filter it's quite clear that you can filter out 01:13:48.750 --> 01:13:51.550 the lower sideband and then you can just go back down to 01:13:51.550 --> 01:13:52.480 baseband again. 01:13:52.480 --> 01:13:55.550 So it's very clear that the whole thing works. 01:13:55.550 --> 01:13:58.630 Except you wouldn't implement it this way. 01:13:58.630 --> 01:14:02.300 Here you have to be more careful to see that works. 01:14:02.300 --> 01:14:05.080 But in fact you would. 01:14:09.160 --> 01:14:12.630 Well, after you get all done, then you get these baseband 01:14:12.630 --> 01:14:15.250 PAM waveforms back again. 01:14:15.250 --> 01:14:17.650 You sample them after filtering. 01:14:17.650 --> 01:14:20.920 And you're all done. 01:14:20.920 --> 01:14:26.630 With that, we are almost done with what we want to do with 01:14:26.630 --> 01:14:31.720 modulating up to passband and down to baseband. 01:14:31.720 --> 01:14:34.240 We'll spend a little bit of time reviewing a couple of 01:14:34.240 --> 01:14:37.550 minor points on this next time. 01:14:37.550 --> 01:14:40.660 Like, I guess, the main thing we have to talk about is how 01:14:40.660 --> 01:14:45.090 do you do frequency recovery, which is kind of a neat thing. 01:14:45.090 --> 01:14:48.000 And then we'll go on to talking about random processes 01:14:48.000 --> 01:14:50.420 and how you deal with noise. 01:14:50.420 --> 01:14:51.890 So. 01:14:51.890 --> 01:14:57.040 If you want to read ahead, we will probably have the notes 01:14:57.040 --> 01:15:01.820 on random processes on the web sometime tomorrow 01:15:01.820 --> 01:15:03.370 afternoon or Sunday. 01:15:03.370 --> 01:15:04.620 Thanks.