WEBVTT 00:00:01.620 --> 00:00:04.730 PROFESSOR: Today we finish up chapter 13. 00:00:04.730 --> 00:00:07.700 I think we are finished, but I'll take questions. 00:00:07.700 --> 00:00:12.520 We get into chapter 14, which starts to get into coding for 00:00:12.520 --> 00:00:15.440 bandwidth-limited channels. 00:00:15.440 --> 00:00:18.910 Here we're coding directly in terms of Euclidean space 00:00:18.910 --> 00:00:21.200 rather than in terms of Hamming space. 00:00:21.200 --> 00:00:24.150 And there's this wonderful quote from Neil Sloane. 00:00:24.150 --> 00:00:27.110 "Euclidean space coding is to Hamming space coding as 00:00:27.110 --> 00:00:31.660 classical music is to rock'n'roll." Meaning that, in 00:00:31.660 --> 00:00:34.740 the Euclidean space, things are continuous, whereas we 00:00:34.740 --> 00:00:36.950 were discrete back in Hamming space. 00:00:36.950 --> 00:00:42.110 Nonetheless, there are strong connections between the two, 00:00:42.110 --> 00:00:46.420 which in two lectures I'll basically only have a chance 00:00:46.420 --> 00:00:48.830 to hint at. 00:00:48.830 --> 00:00:52.860 You were supposed to hand in problem set nine last Friday, 00:00:52.860 --> 00:00:56.090 or today if you haven't already. 00:00:56.090 --> 00:00:59.220 Today, we'll hand out the solutions for nine, and then 00:00:59.220 --> 00:01:02.060 chapter 14 and its problems, which are problem set 10. 00:01:02.060 --> 00:01:04.610 These are not due. 00:01:04.610 --> 00:01:08.060 As always, I recommend you do them. 00:01:08.060 --> 00:01:11.650 The solutions will be handed out Wednesday. 00:01:11.650 --> 00:01:16.000 However, on the exam, the exam will not cover chapter 14. 00:01:16.000 --> 00:01:21.040 So everything we do this week is gravy. 00:01:21.040 --> 00:01:25.180 The exam is a week from tomorrow in the morning. 00:01:25.180 --> 00:01:28.960 Ashish will administer it because I won't be here. 00:01:28.960 --> 00:01:32.620 For this exam, it's closed book, except you're allowed to 00:01:32.620 --> 00:01:36.120 make five pages of notes, as you did three 00:01:36.120 --> 00:01:38.010 pages on the midterm. 00:01:38.010 --> 00:01:42.660 And I do recommend you bring a calculator, because for one of 00:01:42.660 --> 00:01:46.490 the problems the calculator will be helpful. 00:01:46.490 --> 00:01:49.900 And I was told that erasable memory should be cleared on 00:01:49.900 --> 00:01:55.410 the calculator, so point of honor if you do that. 00:01:55.410 --> 00:02:02.710 Any questions about the exam or chapter 13? 00:02:02.710 --> 00:02:03.100 Yes? 00:02:03.100 --> 00:02:06.010 AUDIENCE: Are you including the midterm [UNINTELLIGIBLE]? 00:02:06.010 --> 00:02:08.750 PROFESSOR: Yeah, oh, everything in the course up to 00:02:08.750 --> 00:02:11.730 chapter 13. 00:02:11.730 --> 00:02:14.660 The rule is everything that we covered in class. 00:02:14.660 --> 00:02:16.330 If we didn't cover it in class, then you're not 00:02:16.330 --> 00:02:19.520 responsible for it. 00:02:19.520 --> 00:02:20.770 Other questions? 00:02:23.140 --> 00:02:27.020 Any last questions on chapter 13? 00:02:27.020 --> 00:02:33.590 This is kind of the finale for binary codes. 00:02:33.590 --> 00:02:37.990 We found that, at least on the binary erasure channel, we had 00:02:37.990 --> 00:02:42.460 techniques that would allow us to design LDPC codes that 00:02:42.460 --> 00:02:46.060 would get us as close to capacity as we liked, or at 00:02:46.060 --> 00:02:48.930 least I made that assertion, directed you to where you 00:02:48.930 --> 00:02:50.600 could find that done in the literature. 00:02:50.600 --> 00:02:54.840 And I further asserted that, at least for binary symmetric 00:02:54.840 --> 00:02:57.820 input channels, like the binary additive white Gaussian 00:02:57.820 --> 00:03:03.320 noise channels, you could do much the same thing using 00:03:03.320 --> 00:03:07.350 techniques like density evolution or EXIT charts. 00:03:07.350 --> 00:03:10.280 Again, the issue is to design code, such as this iterative 00:03:10.280 --> 00:03:14.850 decoding, after many, many, many iterations, will, with 00:03:14.850 --> 00:03:19.180 probability one, converge to the correct solution (or with 00:03:19.180 --> 00:03:22.090 probably one minus epsilon). 00:03:22.090 --> 00:03:28.920 So that's the great conclusion to the story for binary codes, 00:03:28.920 --> 00:03:31.800 channels in which we want to have binary inputs. 00:03:31.800 --> 00:03:36.290 And way back in the beginning, we said that that was 00:03:36.290 --> 00:03:40.740 perfectly sufficient, really, for the general additive white 00:03:40.740 --> 00:03:42.630 Gaussian channel. 00:03:42.630 --> 00:03:46.560 When we were talking about binary code rates less than 00:03:46.560 --> 00:03:51.250 1/2, or nominal spectral efficiencies less than one bit 00:03:51.250 --> 00:03:55.690 per second per Hertz, which are equivalent statements. 00:03:55.690 --> 00:03:59.070 So what happens if we want to go over channels where we want 00:03:59.070 --> 00:04:03.000 to send at higher spectral efficiencies? 00:04:03.000 --> 00:04:07.630 We want to send four bits per second per Hertz, as we 00:04:07.630 --> 00:04:11.230 certainly do over, say, telephone line channels, which 00:04:11.230 --> 00:04:15.190 are 3,000, or 4,000-Hertz channels, over which we want 00:04:15.190 --> 00:04:18.610 to send tens of thousands of bits per second. 00:04:18.610 --> 00:04:22.780 Or many radio channels will support high numbers of bits 00:04:22.780 --> 00:04:25.560 per second per Hertz, and so forth. 00:04:25.560 --> 00:04:30.440 Well, obviously, you can't do that with binary codes. 00:04:30.440 --> 00:04:38.330 So, today and Wednesday in this chapter, we very quickly, 00:04:38.330 --> 00:04:42.390 of course, go through the series of techniques that have 00:04:42.390 --> 00:04:44.590 been developed to go over bandwidth-limited channels 00:04:44.590 --> 00:04:46.650 such as the bandwidth-limited additive white 00:04:46.650 --> 00:04:48.430 Gaussian noise channel. 00:04:48.430 --> 00:04:51.032 Clearly, you're going to need a non-binary constellation, 00:04:51.032 --> 00:04:53.820 and somehow code on that constellation. 00:04:57.330 --> 00:05:01.140 But what you'll see is that both the techniques and much 00:05:01.140 --> 00:05:05.880 of the development are very reminiscent 00:05:05.880 --> 00:05:07.660 of the binary case. 00:05:07.660 --> 00:05:11.290 Historically, the binary case was always done first, and 00:05:11.290 --> 00:05:15.820 then the generalizations to non-binary followed. 00:05:15.820 --> 00:05:17.450 And so we'll see -- 00:05:17.450 --> 00:05:20.630 you can start off with very simple, hand-designed 00:05:20.630 --> 00:05:24.160 constellations, as we were doing back in the early 00:05:24.160 --> 00:05:26.270 chapters, four and five. 00:05:26.270 --> 00:05:29.780 We played around with little m equals eight constellations 00:05:29.780 --> 00:05:32.790 and tried to optimize them from the point of view of, 00:05:32.790 --> 00:05:37.070 say, minimizing power for a fixed minimum distance between 00:05:37.070 --> 00:05:41.740 points in a certain number of dimensions in Euclidean space. 00:05:41.740 --> 00:05:44.000 Now, we want to go up to considerably longer 00:05:44.000 --> 00:05:45.310 constellations. 00:05:45.310 --> 00:05:50.080 So that was kind of at the level of Hamming codes and 00:05:50.080 --> 00:05:54.760 single parity-check codes and very elementary binary coding 00:05:54.760 --> 00:05:57.590 techniques like that. 00:05:57.590 --> 00:06:00.620 So we want more complicated codes, let's say moderate 00:06:00.620 --> 00:06:02.200 complexity codes. 00:06:02.200 --> 00:06:08.290 We will find things that are analogous to block codes, 00:06:08.290 --> 00:06:11.650 linear block codes, namely lattices. 00:06:11.650 --> 00:06:14.940 These are structures in Euclidean space that have the 00:06:14.940 --> 00:06:19.340 group property, as linear block codes do. 00:06:19.340 --> 00:06:22.960 And then we'll find trellis codes, which are the Euclidean 00:06:22.960 --> 00:06:26.450 space analog of convolutional codes. 00:06:26.450 --> 00:06:31.130 And basically, our goal in the next two lectures is to get to 00:06:31.130 --> 00:06:34.890 the level of a union-bound estimate analysis of the 00:06:34.890 --> 00:06:39.620 performance of these kinds of coding techniques, which, like 00:06:39.620 --> 00:06:44.460 their binary analogs, sort of sit in a moderate complexity, 00:06:44.460 --> 00:06:48.300 pretty good performance, get up to 6 dB coding gain, can't 00:06:48.300 --> 00:06:50.540 get to capacity. 00:06:50.540 --> 00:06:54.560 So that's as far as we'll be able to go in chapter 14. 00:06:54.560 --> 00:06:59.500 In the last page or two of chapter 14, I briefly mention 00:06:59.500 --> 00:07:01.130 higher performance techniques. 00:07:01.130 --> 00:07:04.160 And it's generally accepted that you can get as close to 00:07:04.160 --> 00:07:06.910 capacity on these bandwidth-limited channels, 00:07:06.910 --> 00:07:09.430 additive white Gaussian noise channels as you can on the 00:07:09.430 --> 00:07:13.030 binary channels, in some cases just by adapting binary 00:07:13.030 --> 00:07:17.190 techniques in a kind of force-fit way, and others that 00:07:17.190 --> 00:07:19.840 are somewhat more principled, I might say. 00:07:19.840 --> 00:07:25.290 But in any case, by adapting the binary techniques, you can 00:07:25.290 --> 00:07:29.330 get turbo codes, low-density parity-check codes, capacity 00:07:29.330 --> 00:07:32.540 approaching codes of various types for the 00:07:32.540 --> 00:07:34.670 bandwidth-limited channel. 00:07:34.670 --> 00:07:39.090 But I just hint how that can be done in one page, and I 00:07:39.090 --> 00:07:42.610 can't include it in the course. 00:07:42.610 --> 00:07:45.870 So that's where we're going. 00:07:45.870 --> 00:07:55.510 So basically, we're trying to design a constellation set of 00:07:55.510 --> 00:07:56.760 signal points. 00:07:59.940 --> 00:08:05.000 Constellation c or signal set or alphabet, 00:08:05.000 --> 00:08:07.950 script c, if you like. 00:08:07.950 --> 00:08:10.320 And we're doing the kind of things we were doing back in 00:08:10.320 --> 00:08:11.680 chapter four and five. 00:08:14.600 --> 00:08:18.510 What are some of the things that the parameters we have-- 00:08:18.510 --> 00:08:21.990 we're going to do this in n-dimensional space. 00:08:21.990 --> 00:08:26.020 We're going to be concerned with the size of the 00:08:26.020 --> 00:08:29.920 constellation, the log of the size, the number of log 2 of 00:08:29.920 --> 00:08:34.169 the size, the number of bits we can send. 00:08:34.169 --> 00:08:38.010 We call this m at one point. 00:08:38.010 --> 00:08:42.010 That will determine the number of bits we can send per n 00:08:42.010 --> 00:08:44.510 dimensions, or we can normalize it per two 00:08:44.510 --> 00:08:45.210 dimensions. 00:08:45.210 --> 00:08:47.260 Typically, I said, in bandwidth-limited regime, 00:08:47.260 --> 00:08:50.500 we'll normalize everything per two dimensions. 00:08:50.500 --> 00:08:54.370 We're going to have an average power of the constellation, 00:08:54.370 --> 00:08:56.840 which I'm just going to write like that. 00:08:56.840 --> 00:09:01.040 We're going to have a minimum squared distance between 00:09:01.040 --> 00:09:04.690 points in the constellation, which I'm going 00:09:04.690 --> 00:09:06.580 to write like that. 00:09:06.580 --> 00:09:10.190 And the idea is to optimize the trade-off 00:09:10.190 --> 00:09:12.210 between this and this. 00:09:12.210 --> 00:09:15.970 Clearly, if I take a given constellation and I scale it 00:09:15.970 --> 00:09:18.960 up, I'm going to get better minimum squared distance 00:09:18.960 --> 00:09:21.480 without the cost of increasing power. 00:09:21.480 --> 00:09:25.380 So I either fix the minimum squared distance between 00:09:25.380 --> 00:09:31.380 points and try to minimize the power, the average energy over 00:09:31.380 --> 00:09:39.470 the centroids of a bunch of spheres, or vice versa. 00:09:39.470 --> 00:09:44.370 So when we do this, this is kind of the game, we get into 00:09:44.370 --> 00:09:46.880 a classical problem called sphere-packing. 00:09:52.230 --> 00:09:57.250 If we say, hold the minimum squared distance constant, 00:09:57.250 --> 00:10:02.660 that means we want to have a certain minimum distance. 00:10:02.660 --> 00:10:03.910 Here I'll make it d_min over 2. 00:10:06.180 --> 00:10:08.980 That means that there's going to be a sphere of radius d_min 00:10:08.980 --> 00:10:14.170 over 2 around each point in this space, which must not be 00:10:14.170 --> 00:10:17.580 violated by the sphere of any other point. 00:10:17.580 --> 00:10:21.190 So basically, we're trying to pack spheres as closely 00:10:21.190 --> 00:10:23.200 together as we can. 00:10:23.200 --> 00:10:25.070 And we're maybe given the dimension. 00:10:25.070 --> 00:10:27.400 Here I have two dimensions to play with. 00:10:27.400 --> 00:10:30.520 We may be given the number, m. 00:10:30.520 --> 00:10:33.760 In general now, we're going to be thinking of m 00:10:33.760 --> 00:10:35.090 becoming very large. 00:10:35.090 --> 00:10:38.290 We want to go to arbitrarily large number of bits per 00:10:38.290 --> 00:10:39.890 second per Hertz. 00:10:39.890 --> 00:10:43.080 So think of schemes that will continue to work as we add 00:10:43.080 --> 00:10:46.390 more and more points. 00:10:46.390 --> 00:10:51.670 And if I do that in two dimensions, we get 00:10:51.670 --> 00:10:52.750 penny-packing. 00:10:52.750 --> 00:10:56.340 Take a bunch of seven pennies out of your pocket, and try to 00:10:56.340 --> 00:10:58.565 squeeze them together as tightly as you can. 00:10:58.565 --> 00:11:01.110 That's effectively what we're trying to do. 00:11:01.110 --> 00:11:05.800 They maintain their fixed radius, and you get something 00:11:05.800 --> 00:11:08.930 that quickly starts to look like what this is called, a 00:11:08.930 --> 00:11:13.910 hexagonal sphere-packing, because the centers line up on 00:11:13.910 --> 00:11:15.270 a hexagonal grid. 00:11:18.870 --> 00:11:21.910 And in fact, I forget exactly what we did. 00:11:21.910 --> 00:11:33.110 We did some of these back in the early chapters. 00:11:33.110 --> 00:11:37.860 And let me draw a 16-point constellation, which I 00:11:37.860 --> 00:11:41.440 remember we did draw. 00:11:41.440 --> 00:11:46.470 Here's a constellation consisting of 16 pennies. 00:11:46.470 --> 00:11:49.100 And subject to my drawing ability, it's packed as 00:11:49.100 --> 00:11:54.120 closely as possible for a fixed minimum distance between 00:11:54.120 --> 00:11:55.510 the points. 00:11:55.510 --> 00:12:01.470 And as far as anyone knows, this constellation, consisting 00:12:01.470 --> 00:12:06.350 a set of points on a hexagonal grid, is the most dense 00:12:06.350 --> 00:12:10.930 packing of 16 points in two dimensions, as measured by the 00:12:10.930 --> 00:12:13.860 average power of the centers. 00:12:13.860 --> 00:12:17.800 And it's not very symmetrical, but you can see it's based on 00:12:17.800 --> 00:12:21.380 a symmetrical packing. 00:12:21.380 --> 00:12:26.540 This hexagonal, this is basically a subset of 16 00:12:26.540 --> 00:12:30.690 points from the hexagonal lattice. 00:12:30.690 --> 00:12:34.210 If we extend these points in all directions-- 00:12:34.210 --> 00:12:36.020 again, I haven't done it very well-- 00:12:36.020 --> 00:12:42.040 but the hexagonal lattice is simply the 00:12:42.040 --> 00:12:43.440 underlying lattice here. 00:12:43.440 --> 00:12:50.185 Let me just draw a set of points that look like this. 00:12:53.786 --> 00:12:55.036 I'm going to fail again. 00:12:58.660 --> 00:13:00.560 Sorry, I'm better at rock'n'roll than I am at 00:13:00.560 --> 00:13:03.470 classical music. 00:13:03.470 --> 00:13:07.970 Anyway, this is the lattice, and it's called hexagonal 00:13:07.970 --> 00:13:10.960 because, if we draw the decision regions for this 00:13:10.960 --> 00:13:16.610 lattice, if we consider this as a set of points in a 00:13:16.610 --> 00:13:20.150 constellation and draw the decision regions, it looks 00:13:20.150 --> 00:13:28.080 like that, this little hexagon, defined by the six 00:13:28.080 --> 00:13:28.940 nearest neighbors. 00:13:28.940 --> 00:13:32.630 They define six sides of a hexagon. 00:13:32.630 --> 00:13:38.120 And so each lattice point has a little region of space that 00:13:38.120 --> 00:13:40.735 belongs to it, you can think of as its decision region. 00:13:44.330 --> 00:13:47.940 Now, what makes this a lattice? 00:13:47.940 --> 00:13:49.340 What's the definition of a lattice? 00:13:52.200 --> 00:13:53.560 We'll say an n-dimensional lattice. 00:13:57.010 --> 00:14:00.520 Very simple and elegant definition for a lattice. 00:14:00.520 --> 00:14:03.700 It's a discrete subset. 00:14:03.700 --> 00:14:11.570 That means a set of discrete points of Rn, sitting in 00:14:11.570 --> 00:14:15.925 Euclidean n-space, that has the group property. 00:14:29.130 --> 00:14:31.850 And you now all remember what that means. 00:14:31.850 --> 00:14:39.110 It means the group property, under ordinary addition of 00:14:39.110 --> 00:14:45.330 vectors in Euclidean n-space, addition or 00:14:45.330 --> 00:14:48.670 subtraction, of course. 00:14:48.670 --> 00:14:49.830 What are some of the implications of 00:14:49.830 --> 00:14:51.060 having a group property? 00:14:51.060 --> 00:14:54.880 That means that the sum of any two vectors is another vector 00:14:54.880 --> 00:14:57.470 in the lattice. 00:14:57.470 --> 00:14:59.470 Does the lattice have to include the 00:14:59.470 --> 00:15:00.910 0, the origin point? 00:15:06.670 --> 00:15:07.980 Anybody? 00:15:07.980 --> 00:15:08.530 Oh, good. 00:15:08.530 --> 00:15:09.820 I'm going to ask this on the final. 00:15:14.980 --> 00:15:15.270 Come on. 00:15:15.270 --> 00:15:16.260 Does it or doesn't it? 00:15:16.260 --> 00:15:17.800 AUDIENCE: It does. 00:15:17.800 --> 00:15:18.840 PROFESSOR: It does. 00:15:18.840 --> 00:15:19.580 Why? 00:15:19.580 --> 00:15:20.830 AUDIENCE: [INAUDIBLE]. 00:15:25.900 --> 00:15:29.050 PROFESSOR: OK, that's one good answer. 00:15:29.050 --> 00:15:32.050 It has to be closed under addition and subtraction. 00:15:32.050 --> 00:15:35.766 Subtraction of the point from itself is going to give you 0. 00:15:35.766 --> 00:15:38.770 A more mathematically trained person would probably, say, 00:15:38.770 --> 00:15:41.560 well, it obviously has to have an identity. 00:15:41.560 --> 00:15:43.180 Every group has an identity element. 00:15:45.890 --> 00:15:49.330 And that means, if lambda is in the lattice, does minus 00:15:49.330 --> 00:15:52.200 lambda have to be in the lattice? 00:15:52.200 --> 00:15:54.520 Yeah, every element has to have an additive 00:15:54.520 --> 00:15:58.850 inverse, and so forth. 00:15:58.850 --> 00:16:06.480 Well, so if I have a lattice like this, that means that one 00:16:06.480 --> 00:16:10.150 of these points has to be the origin, say this one. 00:16:15.740 --> 00:16:18.210 Now, of course, you can have a translate of a lattice, and 00:16:18.210 --> 00:16:21.060 geometrically it has all the same properties as the lattice 00:16:21.060 --> 00:16:25.180 itself, at any fixed t to all the lattice points. 00:16:25.180 --> 00:16:27.730 In other words, make the origin somewhere else. 00:16:27.730 --> 00:16:29.520 But that's not itself a group. 00:16:29.520 --> 00:16:30.800 That's the co-set of a group. 00:16:30.800 --> 00:16:33.680 It would translate as a co-set operation. 00:16:33.680 --> 00:16:42.100 So lambda must be a group, and lambda plus t, any lambda 00:16:42.100 --> 00:16:49.320 translate is a co-set of lambda. 00:16:49.320 --> 00:16:51.880 So most of the things you say about lambda are also true for 00:16:51.880 --> 00:16:55.330 lambda plus t, but in order to fix a group, we need the 00:16:55.330 --> 00:16:58.400 origin to be part of the points set. 00:16:58.400 --> 00:17:00.340 OK, if it's a group, what does that mean? 00:17:00.340 --> 00:17:03.420 That means, if we take any particular point, say this 00:17:03.420 --> 00:17:14.159 one, regard it as a vector, then call this lambda_1 -- 00:17:14.159 --> 00:17:18.000 or let me it g1, more suggestively. 00:17:18.000 --> 00:17:23.310 Does g1 plus g1 have to be in the group? 00:17:23.310 --> 00:17:24.119 Yes. 00:17:24.119 --> 00:17:26.760 So that forces this point to be in there, 00:17:26.760 --> 00:17:31.310 2g1, 3g1, so forth. 00:17:31.310 --> 00:17:37.860 All of these have to be in there, as well as minus g1, 00:17:37.860 --> 00:17:41.450 minus 2g1, and so forth. 00:17:41.450 --> 00:17:44.930 So basically, once we found a generator, all integers times 00:17:44.930 --> 00:17:47.650 that generator have to be in the lattice. 00:17:47.650 --> 00:17:52.170 So that means there's a cross-section along the axis 00:17:52.170 --> 00:17:56.460 defined by g1, such that these points are simply the 00:17:56.460 --> 00:18:02.050 integers, scaled by whatever the norm of g1 is. 00:18:06.730 --> 00:18:11.770 And of course, that holds for any of these neighbors here. 00:18:11.770 --> 00:18:14.190 You want to look for nearest neighbors of 0's. 00:18:14.190 --> 00:18:21.790 Take g2 and then, of course, all of its integer multiples 00:18:21.790 --> 00:18:22.940 have to be in the lattice. 00:18:22.940 --> 00:18:33.800 So we've got a set here that, in some sense, spans n-space. 00:18:33.800 --> 00:18:36.290 We have now two vectors, g1, g2, which are 00:18:36.290 --> 00:18:37.570 not linearly dependent. 00:18:37.570 --> 00:18:39.220 They're sitting in 2-space. 00:18:39.220 --> 00:18:43.080 So if we take all real linear combinations of these two 00:18:43.080 --> 00:18:46.490 vectors, we're going to get all of 2-space. 00:18:46.490 --> 00:18:49.460 If we take all integer linear combinations of these two 00:18:49.460 --> 00:18:53.640 vectors, they all must be in the lattice, right? 00:18:53.640 --> 00:18:55.860 By the group property? 00:18:55.860 --> 00:18:58.420 We've already proved that all the integer multiples of 00:18:58.420 --> 00:19:01.000 either of them is in the lattice, so any thing plus or 00:19:01.000 --> 00:19:03.010 minus those two. 00:19:03.010 --> 00:19:11.750 So certainly, the lattice includes the set of all 00:19:11.750 --> 00:19:19.410 integer multiples of a_i times gi, where i equals 1 to 2, 00:19:19.410 --> 00:19:25.410 where a1, a2 is in the two-dimensional integer, 00:19:25.410 --> 00:19:26.660 lattice Z-squared. 00:19:32.590 --> 00:19:33.840 Could there be any other points? 00:19:38.720 --> 00:19:41.500 There actually could, if you made a mistake and picked your 00:19:41.500 --> 00:19:45.810 initial vector to be 2g1, because then you wouldn't have 00:19:45.810 --> 00:19:47.290 taken account of g1. 00:19:47.290 --> 00:19:51.150 So you've got to take your two initial vectors to be two of 00:19:51.150 --> 00:19:53.380 the minimum norm points in the lattice. 00:19:53.380 --> 00:19:58.480 But if you do that, then you can easily prove that this is 00:19:58.480 --> 00:19:59.680 all of the lattice points. 00:19:59.680 --> 00:20:07.545 So in fact, a two-dimensional lattice is equal to a set of 00:20:07.545 --> 00:20:09.660 all integer linear combinations of two 00:20:09.660 --> 00:20:12.630 generators, which we can write as a little two-by-two 00:20:12.630 --> 00:20:18.270 generator matrix, G. Or very briefly, we could write lambda 00:20:18.270 --> 00:20:22.260 as G times Z-squared, and we can view it as just some 00:20:22.260 --> 00:20:26.040 linear transformation of Z-squared. 00:20:26.040 --> 00:20:30.660 So in this hexagonal lattice, we can view it as starting out 00:20:30.660 --> 00:20:32.820 with a square grid. 00:20:32.820 --> 00:20:37.090 Of course, Z-squared itself is a lattice. 00:20:37.090 --> 00:20:39.840 Z-squared is a group. 00:20:39.840 --> 00:20:43.320 This is just the square lattice in two dimensions. 00:20:43.320 --> 00:20:48.350 We have Z_n as a lattice in n dimensions, in general. 00:20:48.350 --> 00:20:53.300 So I take Z-squared, Z_n, more generally, and I distort it. 00:20:53.300 --> 00:20:59.620 I just run it through some g, which is a linear 00:20:59.620 --> 00:21:06.130 transformation, and slants some of the generator points. 00:21:06.130 --> 00:21:08.270 And that's a general way of getting a lattice. 00:21:08.270 --> 00:21:10.330 That's not the way we usually visualize 00:21:10.330 --> 00:21:11.400 the hexagonal lattice. 00:21:11.400 --> 00:21:14.660 We usually visualize it in such a way as to recognize 00:21:14.660 --> 00:21:18.000 that it has six-fold symmetry. 00:21:18.000 --> 00:21:21.220 But that's certainly a way to get it. 00:21:24.740 --> 00:21:28.426 And this has one interesting consequence. 00:21:28.426 --> 00:21:31.530 One of the measures we want for a lattice is the volume 00:21:31.530 --> 00:21:34.500 per lattice point. 00:21:34.500 --> 00:21:41.210 In here, the volume, you can divide this up into volumes 00:21:41.210 --> 00:21:43.770 that are all of equal size, and we call that 00:21:43.770 --> 00:21:45.180 the volume of A2. 00:21:45.180 --> 00:21:46.280 This lattice -- 00:21:46.280 --> 00:21:50.450 hexagonal lattice, is called A2, and it's the volume of 00:21:50.450 --> 00:21:53.200 that hexagon. 00:21:53.200 --> 00:21:57.410 1 over the volume of that hexagon is the density of 00:21:57.410 --> 00:21:59.600 points in 2-space, if you like. 00:21:59.600 --> 00:22:03.490 So density is 1/volume. 00:22:03.490 --> 00:22:06.930 I mean, think of every point as having a unique volume, and 00:22:06.930 --> 00:22:12.270 if we basically consider the cells around each lattice 00:22:12.270 --> 00:22:17.070 point, they fill up 2-space, or the perfect tessellation 00:22:17.070 --> 00:22:19.920 tiling of 2-space. 00:22:19.920 --> 00:22:22.360 These hexagons will. 00:22:22.360 --> 00:22:25.310 One easy consequence of the group property is that all the 00:22:25.310 --> 00:22:29.190 decision regions have to be congruent, in fact, just 00:22:29.190 --> 00:22:32.170 translates of one another by lattice points. 00:22:32.170 --> 00:22:35.920 The decision region around this point is the same as the 00:22:35.920 --> 00:22:39.510 decision region about 0, translated by a lattice point, 00:22:39.510 --> 00:22:41.360 namely the center of that decision region. 00:22:41.360 --> 00:22:45.950 I won't prove that but it's true and easy to prove. 00:22:45.950 --> 00:22:48.435 So that's one way of finding what's the volume. 00:22:48.435 --> 00:22:50.080 The volume will mean the volume of 00:22:50.080 --> 00:22:51.590 2-space per lattice point. 00:22:51.590 --> 00:22:53.700 But what's another way to do that? 00:22:53.700 --> 00:22:57.550 Another way is just to take this little parallelotope, 00:22:57.550 --> 00:23:03.410 whose volume is easier to calculate. 00:23:03.410 --> 00:23:07.810 And we can see that we can equally well tile 2-space with 00:23:07.810 --> 00:23:11.550 these little parallelograms, just anchor -- 00:23:11.550 --> 00:23:14.280 in this case, this is the one that's anchored in the lower 00:23:14.280 --> 00:23:16.830 left corner by the origin. 00:23:16.830 --> 00:23:19.120 Then we take another one that's anchored by this point. 00:23:19.120 --> 00:23:21.830 We take another one that's anchored by this point, 00:23:21.830 --> 00:23:23.760 another one that's anchored by this point. 00:23:23.760 --> 00:23:28.590 And it's clear that's another tiling of 2-space by regions, 00:23:28.590 --> 00:23:31.880 one for each lattice point that completely fills 2-space 00:23:31.880 --> 00:23:36.570 with overlaps only on the boundary, which don't count. 00:23:36.570 --> 00:23:40.140 So the volume of A2 is also the volume of that, what's 00:23:40.140 --> 00:23:42.780 called a fundamental parallelotope. 00:23:42.780 --> 00:23:46.690 And what's the volume of that fundamental parallelotope? 00:23:46.690 --> 00:23:52.290 Well, one way of calculating it is to take 00:23:52.290 --> 00:23:55.420 the volume of Z-squared. 00:23:55.420 --> 00:23:59.840 We can view this as the distortion of Z-squared, which 00:23:59.840 --> 00:24:01.090 looks like this. 00:24:03.690 --> 00:24:07.490 So if this is Z-squared, what's the fundamental volume 00:24:07.490 --> 00:24:09.320 of Z-squared? 00:24:09.320 --> 00:24:11.640 The volume of Z-squared per lattice point. 00:24:11.640 --> 00:24:19.150 This is (0,1,-1,1, so forth,1,1). 00:24:19.150 --> 00:24:21.520 What's the volume of Z-squared per-- 00:24:21.520 --> 00:24:22.720 AUDIENCE: 1 00:24:22.720 --> 00:24:23.370 PROFESSOR: --1. 00:24:23.370 --> 00:24:25.176 Clearly. 00:24:25.176 --> 00:24:26.850 And we have this decision region. 00:24:26.850 --> 00:24:30.300 This is little square side 1, or this parallelotope is also 00:24:30.300 --> 00:24:31.930 little square side 1. 00:24:31.930 --> 00:24:33.190 It's volume is 1. 00:24:36.220 --> 00:24:40.070 General geometric principle, if we take this and distort it 00:24:40.070 --> 00:24:47.570 by G, then what's the volume of, say, this parallelotope 00:24:47.570 --> 00:24:48.610 going to be? 00:24:48.610 --> 00:24:54.080 This parallelotope goes by G into this one, and its volume 00:24:54.080 --> 00:24:59.010 is the Jacobian, which is the determinant of G. 00:24:59.010 --> 00:25:04.950 So the volume of a lattice is the determinant of any 00:25:04.950 --> 00:25:06.730 generator matrix for that lattice. 00:25:06.730 --> 00:25:13.330 Just a nice little side fact, but it's an example of the 00:25:13.330 --> 00:25:15.890 kind of things you get in Euclidean space that you don't 00:25:15.890 --> 00:25:18.980 get in Hamming space. 00:25:18.980 --> 00:25:23.110 Of course, there are various generator matrices that you 00:25:23.110 --> 00:25:24.900 could use to generate this lattice. 00:25:24.900 --> 00:25:29.340 But whichever one you choose, this is going to be true. 00:25:29.340 --> 00:25:32.200 So this is an invariant among all the generating matrices. 00:25:34.880 --> 00:25:37.504 So that's a lattice. 00:25:37.504 --> 00:25:41.490 I haven't mentioned what's the most important thing for us 00:25:41.490 --> 00:25:42.065 about a lattice. 00:25:42.065 --> 00:25:45.610 A lattice is a group. 00:25:45.610 --> 00:25:57.340 What's the fundamental property of a group, which I 00:25:57.340 --> 00:26:00.580 guess was embodied in the alternate set of 00:26:00.580 --> 00:26:03.620 axioms for the group? 00:26:03.620 --> 00:26:09.480 If I take the group plus any element of the 00:26:09.480 --> 00:26:13.300 group, what do I get? 00:26:13.300 --> 00:26:14.510 AUDIENCE: [INAUDIBLE]. 00:26:14.510 --> 00:26:15.760 PROFESSOR: The group again. 00:26:19.000 --> 00:26:20.870 Let's interpret that geometrically. 00:26:20.870 --> 00:26:22.690 One of the nice things about this subject is you're 00:26:22.690 --> 00:26:24.010 constantly going back and forth 00:26:24.010 --> 00:26:27.100 between algebra and geometry. 00:26:27.100 --> 00:26:28.600 Geometrically, what does that mean? 00:26:28.600 --> 00:26:32.100 That means, if we take this lattice, say the origin is 00:26:32.100 --> 00:26:39.960 here, and we shift it, say, so that the origin is up here, 00:26:39.960 --> 00:26:41.350 then nothing changes. 00:26:41.350 --> 00:26:45.280 Everything is invariant to this shift. 00:26:45.280 --> 00:26:48.260 We can do this for any lattice point. 00:26:48.260 --> 00:26:50.580 That shows you that all the decision 00:26:50.580 --> 00:26:52.990 regions have to be congruent. 00:26:52.990 --> 00:27:04.000 It shows you that the number of nearest neighbors, the 00:27:04.000 --> 00:27:09.840 minimum distance from any point to its nearest neighbors 00:27:09.840 --> 00:27:12.470 is always the same for any lattice point. 00:27:12.470 --> 00:27:14.240 Basically -- 00:27:14.240 --> 00:27:18.030 it's a lattice is geometrically uniform. 00:27:18.030 --> 00:27:21.720 That means you can stand on any of the points and the 00:27:21.720 --> 00:27:24.630 world around you looks the same, as if you stood on any 00:27:24.630 --> 00:27:26.310 of the other points. 00:27:26.310 --> 00:27:30.160 Think of these as stars in the universe, and somebody 00:27:30.160 --> 00:27:33.770 blindfolds you and drops you on one of these points. 00:27:33.770 --> 00:27:35.510 Can you tell which one of those points you've been 00:27:35.510 --> 00:27:40.330 dropped on if nobody's written anything on these points? 00:27:40.330 --> 00:27:41.270 No, you can't tell. 00:27:41.270 --> 00:27:44.350 The world looks the same from whichever one you drop on. 00:27:44.350 --> 00:27:47.870 So that means the minimum square distance from any point 00:27:47.870 --> 00:27:49.150 is the same as every other. 00:27:49.150 --> 00:27:53.040 So the number of nearest neighbors at minimum square 00:27:53.040 --> 00:27:54.640 distance is the same. 00:27:54.640 --> 00:27:57.390 It means the total distance profile, starting from any 00:27:57.390 --> 00:27:58.950 point, is the same. 00:27:58.950 --> 00:28:02.230 It means if we were to take one of these points and send 00:28:02.230 --> 00:28:06.390 it over a Gaussian channel from a very large set, and we 00:28:06.390 --> 00:28:09.250 take one of the points from the interior and send it over 00:28:09.250 --> 00:28:12.570 an additive white Gaussian noise channel, the probability 00:28:12.570 --> 00:28:15.030 of error is going to be the same, regardless of which 00:28:15.030 --> 00:28:17.610 point we send it from, because the probability of error is 00:28:17.610 --> 00:28:22.050 the probability of falling outside your decision region. 00:28:22.050 --> 00:28:25.260 And since the decision region always has the same shape, 00:28:25.260 --> 00:28:28.410 it's invariant to which point you send. 00:28:28.410 --> 00:28:35.580 So this geometrical uniformity property, which that's the 00:28:35.580 --> 00:28:38.200 general name for it and it has all these specific 00:28:38.200 --> 00:28:45.780 consequences that I just listed, is the principal 00:28:45.780 --> 00:28:49.140 geometrical property of a lattice. 00:28:49.140 --> 00:28:51.650 This just follows from its group property. 00:28:54.170 --> 00:28:57.680 You don't have to be a lattice to have this property. 00:28:57.680 --> 00:29:01.350 For instance, the translate of a lattice is not a group, 00:29:01.350 --> 00:29:04.700 technically, but it obviously has the same geometrical 00:29:04.700 --> 00:29:06.360 uniformity property. 00:29:06.360 --> 00:29:09.850 And there are more elaborate examples of things that look 00:29:09.850 --> 00:29:11.730 less like lattices that are nonetheless 00:29:11.730 --> 00:29:12.740 geometrically uniform. 00:29:12.740 --> 00:29:14.350 On the other hand, lattices definitely are 00:29:14.350 --> 00:29:16.130 geometrically uniform. 00:29:16.130 --> 00:29:18.640 So this seems to be a good thing to do, because what's 00:29:18.640 --> 00:29:19.290 our objective? 00:29:19.290 --> 00:29:22.900 We're trying to have d_min-squared be the same for 00:29:22.900 --> 00:29:24.150 every point in our constellation. 00:29:26.990 --> 00:29:30.420 We want to tack these pennies as close together as we can. 00:29:30.420 --> 00:29:32.860 That means we want to hold d_min-squared constant for 00:29:32.860 --> 00:29:36.080 every point that we're going to use. 00:29:36.080 --> 00:29:39.650 And so lattices have that property. 00:29:39.650 --> 00:29:43.430 So that seems like a good way to do sphere-packing. 00:29:43.430 --> 00:29:48.800 And in fact, sphere-packing, in n-dimensions, is a very old 00:29:48.800 --> 00:29:50.050 mathematical subject. 00:29:53.130 --> 00:29:55.650 How many spheres can you pack around a three 00:29:55.650 --> 00:29:57.835 sphere, is it 12 or 13? 00:29:57.835 --> 00:30:01.400 That was debated in the 18th and 19th centuries, and I 00:30:01.400 --> 00:30:04.910 think it's only finally just been resolved a few years ago. 00:30:04.910 --> 00:30:09.780 This turns out to be not an easy problem. 00:30:09.780 --> 00:30:12.920 The densest sphere-packing in four dimensions was found in 00:30:12.920 --> 00:30:15.230 the mid-19th century. 00:30:15.230 --> 00:30:17.060 In eight dimensions, I think around the 00:30:17.060 --> 00:30:19.080 turn of the 20th century. 00:30:19.080 --> 00:30:22.620 In 16 dimensions, it's the Barnes-Wall lattice, which 00:30:22.620 --> 00:30:25.635 we'll see a little later, which was found in '54, about 00:30:25.635 --> 00:30:29.620 the same time as Reed-Muller codes, or maybe it was '59, 00:30:29.620 --> 00:30:33.150 maybe a little later than Reed-Muller codes. 00:30:33.150 --> 00:30:37.600 The Leech lattice was certainly found a little 00:30:37.600 --> 00:30:40.980 earlier than that, but around the same time as the Golay 00:30:40.980 --> 00:30:43.320 code, to which it's a very close cousin. 00:30:43.320 --> 00:30:43.920 And so forth. 00:30:43.920 --> 00:30:47.220 So you might think that these questions had all been 00:30:47.220 --> 00:30:50.010 settled, but in fact they're still open for most specific 00:30:50.010 --> 00:30:53.300 dimensions, n. 00:30:53.300 --> 00:30:56.680 Conway and Sloane have the authoritative book on this, 00:30:56.680 --> 00:30:59.690 which basically lists what we know about the densest 00:30:59.690 --> 00:31:02.600 sphere-packings in all dimensions. 00:31:02.600 --> 00:31:06.930 So here's our idea. 00:31:06.930 --> 00:31:10.520 Our idea is we're going to go consult the mathematicians and 00:31:10.520 --> 00:31:13.160 find out what the densest sphere-packings that they 00:31:13.160 --> 00:31:15.280 found are in various dimensions. 00:31:15.280 --> 00:31:17.910 In two dimensions, it's the hexagonal lattice. 00:31:17.910 --> 00:31:21.300 In one dimension, it's the integer, and basically the 00:31:21.300 --> 00:31:24.900 integers are the only packing with regularity properties in 00:31:24.900 --> 00:31:25.690 one dimension. 00:31:25.690 --> 00:31:29.380 And two dimensions, you could consider Z-squared, or you can 00:31:29.380 --> 00:31:31.260 consider A2. 00:31:31.260 --> 00:31:33.720 These are the two obvious candidates, the square and the 00:31:33.720 --> 00:31:35.560 hexagonal packings. 00:31:35.560 --> 00:31:38.840 In three dimensions, you get face-centered cubic packing. 00:31:38.840 --> 00:31:41.110 And what's the BCC? 00:31:41.110 --> 00:31:43.410 The base-centered cubic packing, I guess. 00:31:43.410 --> 00:31:44.930 AUDIENCE: Body-centered. 00:31:44.930 --> 00:31:45.490 PROFESSOR: Say again. 00:31:45.490 --> 00:31:46.700 AUDIENCE: Body center. 00:31:46.700 --> 00:31:48.250 PROFESSOR: Body-centered, thank you. 00:31:48.250 --> 00:31:50.510 I'm sure you're right. 00:31:50.510 --> 00:31:54.570 So anyway, there's some things that are known about this from 00:31:54.570 --> 00:31:59.250 classical mathematics and current mathematics. 00:31:59.250 --> 00:32:02.950 So our idea is we're going to try to find the densest 00:32:02.950 --> 00:32:08.930 possible lattice in n dimensions. 00:32:08.930 --> 00:32:12.970 But for digital communications, for coding, we 00:32:12.970 --> 00:32:16.480 only want a finite number of points from that 00:32:16.480 --> 00:32:18.970 constellation. 00:32:18.970 --> 00:32:22.240 So what we're going to do is we're going to create our 00:32:22.240 --> 00:32:26.560 constellation based on a lattice and 00:32:26.560 --> 00:32:30.130 on a region of n-space. 00:32:30.130 --> 00:32:31.860 And the idea is simple. 00:32:31.860 --> 00:32:35.060 We're just going to intersect the lattice. 00:32:35.060 --> 00:32:40.710 Or maybe let me allow a translate of a lattice with 00:32:40.710 --> 00:32:43.510 the region. 00:32:43.510 --> 00:32:48.580 And make this a finite, compact region. 00:32:48.580 --> 00:32:52.750 And this will give us a certain finite subset of the 00:32:52.750 --> 00:32:54.000 points in the lattice. 00:32:56.620 --> 00:32:58.980 Now, when I draw a constellation like this one, 00:32:58.980 --> 00:33:03.620 of a size 16, that's what I've done. 00:33:03.620 --> 00:33:05.540 I've basically taken A2. 00:33:05.540 --> 00:33:07.740 Or it's actually a translate of A2. 00:33:07.740 --> 00:33:10.800 I think the origin is in here somewhere. 00:33:10.800 --> 00:33:14.870 And I draw a sphere around the origin. 00:33:14.870 --> 00:33:17.330 And I take the 16 points that happen to 00:33:17.330 --> 00:33:18.560 fall within that sphere. 00:33:18.560 --> 00:33:21.470 And I call that my constellation. 00:33:21.470 --> 00:33:23.430 So that's the idea in general. 00:33:23.430 --> 00:33:27.050 Think of the region, r, as a cookie cutter. 00:33:27.050 --> 00:33:30.610 We're going to take the cookie cutter and chop out a finite 00:33:30.610 --> 00:33:35.900 number of points from this nice, regular, infinite grid. 00:33:35.900 --> 00:33:38.930 And that will give us a constellation. 00:33:38.930 --> 00:33:42.060 So especially for large constellations, we'll be able 00:33:42.060 --> 00:33:45.950 to make some approximations that will allow us to analyze 00:33:45.950 --> 00:33:48.660 that kind of constellation very easily. 00:33:48.660 --> 00:33:52.060 And actually, the best constellations that we know 00:33:52.060 --> 00:33:53.310 are of this type. 00:33:55.610 --> 00:33:58.810 We've taken this baseline, m-PAM. 00:33:58.810 --> 00:34:00.340 Well, what's m-PAM? 00:34:00.340 --> 00:34:08.790 We take the lattice plus t, let's say to be Z plus 1/2. 00:34:12.199 --> 00:34:16.980 So that means we don't take the origin, but we take the 00:34:16.980 --> 00:34:22.199 points 1/2 minus 1/2, 3/2. 00:34:22.199 --> 00:34:26.120 They're equally spaced points, but we've shifted them so as 00:34:26.120 --> 00:34:28.790 to be symmetric about the origin. 00:34:28.790 --> 00:34:32.460 5/2 minus minus, so forth. 00:34:32.460 --> 00:34:37.230 So this is Z plus 1/2 is our lattice. 00:34:37.230 --> 00:34:41.219 And then if we want 4-PAM, we take a cookie cutter 00:34:41.219 --> 00:34:44.070 consisting of this region. 00:34:44.070 --> 00:34:48.900 So this is the region going from minus 2 to 2. 00:34:48.900 --> 00:34:51.219 It's not unreasonable to expect that, if we take a 00:34:51.219 --> 00:34:56.670 region of length 4, we're going to get 4 points, if, 00:34:56.670 --> 00:34:59.940 again, it's symmetric around the corner, because each of 00:34:59.940 --> 00:35:03.800 these points takes up about one unit of space over here. 00:35:03.800 --> 00:35:08.650 So if we made it from minus 4 to plus 4, we'd have a region 00:35:08.650 --> 00:35:10.950 of length 8, and we'd get 8 points. 00:35:10.950 --> 00:35:13.510 And so forth. 00:35:13.510 --> 00:35:20.000 So that's how we do m-PAM, and then scale it if we want to. 00:35:20.000 --> 00:35:21.610 But this is the basic idea. 00:35:21.610 --> 00:35:28.040 Similarly, for m by m QAM, like everything else, it's 00:35:28.040 --> 00:35:30.420 just doing the same thing independently in two 00:35:30.420 --> 00:35:31.890 dimensions. 00:35:31.890 --> 00:35:39.780 Here we take our lattice to be Z plus 1/2 squared, which is 00:35:39.780 --> 00:35:47.120 just the offset Z-squared, offset so as to be four-way, 00:35:47.120 --> 00:35:50.200 to be roughly symmetrical to four-way 00:35:50.200 --> 00:35:53.620 rotations, 90-degree rotations. 00:35:53.620 --> 00:35:56.190 So here is Z-squared plus (1/2, 1/2). 00:35:59.690 --> 00:36:05.930 And then, again, here to get 16-QAM, for instance, we might 00:36:05.930 --> 00:36:12.860 take the region to be what? 00:36:12.860 --> 00:36:20.280 (-2,2)-squared which, again, has area 16. 00:36:20.280 --> 00:36:23.370 You might think of this as just taking the little square 00:36:23.370 --> 00:36:24.890 around each of these. 00:36:24.890 --> 00:36:27.746 Each of these squares has area 1. 00:36:27.746 --> 00:36:31.710 So when we take the union of all those 00:36:31.710 --> 00:36:33.160 squares, we get the region. 00:36:33.160 --> 00:36:35.480 That's another way of characterizing this region. 00:36:35.480 --> 00:36:39.530 It's just the union of all the decision regions of the points 00:36:39.530 --> 00:36:40.210 that we want. 00:36:40.210 --> 00:36:43.020 If you had started from the points, you could certainly 00:36:43.020 --> 00:36:45.200 invent a region that is the union of 00:36:45.200 --> 00:36:46.430 those decision regions. 00:36:46.430 --> 00:36:52.640 It would have an area equal to the number of points times the 00:36:52.640 --> 00:36:57.290 volume of 2-space per each point. 00:36:57.290 --> 00:36:59.380 That's the kind of approximation 00:36:59.380 --> 00:37:00.800 we're going to use. 00:37:00.800 --> 00:37:05.270 The bounding region is clearly going to have something like 00:37:05.270 --> 00:37:08.240 the number of points in the constellation, times the 00:37:08.240 --> 00:37:10.590 fundamental volume as its volume. 00:37:16.720 --> 00:37:18.610 So this is our idea. 00:37:18.610 --> 00:37:22.660 This is what we're going to call a lattice constellation. 00:37:22.660 --> 00:37:25.940 And we're going to investigate the properties of lattice 00:37:25.940 --> 00:37:29.860 constellations, not the most general thing we could think 00:37:29.860 --> 00:37:33.270 of but good enough. 00:37:33.270 --> 00:37:34.955 Seem to have the right sort of property. 00:37:38.490 --> 00:37:41.250 And we're going to take it at least as far as the 00:37:41.250 --> 00:37:45.090 union-bound estimate, so we can get a good estimate of 00:37:45.090 --> 00:37:48.010 probability of error if we take one of these 00:37:48.010 --> 00:37:51.660 constellations, like m-by-m QAM, and use it to transmit 00:37:51.660 --> 00:37:56.690 over a Gaussian noise channel. 00:37:56.690 --> 00:38:03.760 And furthermore, for large constellations, meaning as we 00:38:03.760 --> 00:38:06.790 get well up into the bandwidth-limited regions of 00:38:06.790 --> 00:38:10.730 large nominal spectral densities, we're going to be 00:38:10.730 --> 00:38:18.880 able to separate the analysis into lattice properties and 00:38:18.880 --> 00:38:20.130 region properties. 00:38:23.955 --> 00:38:28.600 And we'll basically be able to just add these up to get the 00:38:28.600 --> 00:38:37.980 overall analysis, by making some judicious approximations 00:38:37.980 --> 00:38:41.170 that become exact in the large constellation limit. 00:38:44.180 --> 00:38:50.970 So let's now restrict ourselves to lattices, or 00:38:50.970 --> 00:38:56.840 translates of lattices, and start to analyze them. 00:38:56.840 --> 00:39:07.630 What are their key parameters for our application, which is 00:39:07.630 --> 00:39:09.770 digital communications? 00:39:09.770 --> 00:39:13.090 Well, one is the minimum squared distance between 00:39:13.090 --> 00:39:15.050 lattice points. 00:39:15.050 --> 00:39:19.060 We've seen that doesn't depend on the particular point. 00:39:19.060 --> 00:39:20.940 So this is the minimum squared distance 00:39:20.940 --> 00:39:24.130 from any lattice point. 00:39:24.130 --> 00:39:25.680 The number of nearest neighbors. 00:39:31.320 --> 00:39:38.745 The volume per lattice point. 00:39:42.130 --> 00:39:45.140 And is there anything else? 00:39:50.020 --> 00:39:51.790 It seems to me there's a fourth one. 00:40:02.360 --> 00:40:04.010 No. 00:40:04.010 --> 00:40:05.610 I guess that's all we need to know. 00:40:11.370 --> 00:40:15.290 So I want to combine these into a key figure 00:40:15.290 --> 00:40:17.600 of merit for coding. 00:40:17.600 --> 00:40:29.510 Again, we did this back in chapter four for a complete 00:40:29.510 --> 00:40:30.320 constellation. 00:40:30.320 --> 00:40:34.340 We found the figure of merit by holding the average power 00:40:34.340 --> 00:40:38.620 fixed for a fixed number of points, m, on a fixed 00:40:38.620 --> 00:40:42.780 dimension, n, and maximizing the minimum square distance, 00:40:42.780 --> 00:40:46.160 or, conversely, holding the minimum squared distance fixed 00:40:46.160 --> 00:40:48.270 and minimizing the average power. 00:40:48.270 --> 00:40:52.440 So we get some figure of merit that was kind of normalized, 00:40:52.440 --> 00:40:55.050 and we could just optimize that one thing. 00:40:55.050 --> 00:41:01.900 The key figure of merit was called by 19th-century 00:41:01.900 --> 00:41:08.300 mathematician, Hermite's parameter, but we are going to 00:41:08.300 --> 00:41:13.070 call it the nominal coding gain because that's what it's 00:41:13.070 --> 00:41:14.320 going to turn out to be. 00:41:17.570 --> 00:41:20.455 And it's just defined as follows. 00:41:20.455 --> 00:41:25.360 The nominal coding gain of a lattice is just this minimum 00:41:25.360 --> 00:41:28.610 squared distance, so it goes up as the 00:41:28.610 --> 00:41:30.470 minimum squared distance. 00:41:30.470 --> 00:41:32.840 But then we want to normalize this. 00:41:32.840 --> 00:41:35.960 And the obvious thing to normalize it by, we want 00:41:35.960 --> 00:41:39.070 something that's going to scale as a 00:41:39.070 --> 00:41:41.275 two-dimensional quantity. 00:41:41.275 --> 00:41:44.970 If the lattice was scaled by alpha, then this would go up 00:41:44.970 --> 00:41:46.550 as alpha squared. 00:41:46.550 --> 00:41:49.670 So we want something that'll scale in the same way. 00:41:49.670 --> 00:41:53.800 And what we choose to normalize it by is the volume 00:41:53.800 --> 00:41:59.530 of lambda per two dimensions, if you like. 00:41:59.530 --> 00:42:02.405 So you can see immediately, this is invariant to scaling, 00:42:02.405 --> 00:42:06.740 that the nominal coding gain of lambda is the same as the 00:42:06.740 --> 00:42:10.610 nominal coding gain of alpha lambda. 00:42:10.610 --> 00:42:13.890 Is alpha lambda a lattice, by the way? 00:42:13.890 --> 00:42:17.360 If lambda's a group, and we have any scale factor alpha 00:42:17.360 --> 00:42:22.550 greater than 0, then alpha times lambda is also a group. 00:42:22.550 --> 00:42:24.350 Everything's just multiplied by alpha. 00:42:24.350 --> 00:42:26.990 So alpha lambda is a lattice. 00:42:26.990 --> 00:42:29.620 And this is actually one of the homework problems. 00:42:29.620 --> 00:42:33.010 If you scale everything by alpha, d_min-squared goes up 00:42:33.010 --> 00:42:34.530 by alpha squared. 00:42:34.530 --> 00:42:36.970 The volume goes up by alpha to the n, for an 00:42:36.970 --> 00:42:38.690 n-dimensional lattice. 00:42:38.690 --> 00:42:41.120 But when we take 2 over n, we're going to get back to 00:42:41.120 --> 00:42:42.360 alpha squared. 00:42:42.360 --> 00:42:48.890 So alpha c of lambda is alpha c of alpha lambda. 00:42:48.890 --> 00:42:51.570 So this is what you want. 00:42:51.570 --> 00:42:55.270 You want a kind of scale-free version of a lattice, because, 00:42:55.270 --> 00:42:57.370 obviously, when we use this, we're going to feel free to 00:42:57.370 --> 00:43:00.170 amplify it or compress it any way we want, 00:43:00.170 --> 00:43:02.880 so we do with QAM. 00:43:02.880 --> 00:43:07.230 All right, so this is normalized in that way. 00:43:07.230 --> 00:43:09.590 We prove in the homework it's also 00:43:09.590 --> 00:43:10.830 invariant to other things. 00:43:10.830 --> 00:43:15.510 If you took, say, an orthogonal transformation of 00:43:15.510 --> 00:43:20.650 this lattice, a rotation, reflection, multiply it by a 00:43:20.650 --> 00:43:28.540 matrix that doesn't change the scale, think of it 00:43:28.540 --> 00:43:32.270 geometrically as an orthogonal transformation, is that going 00:43:32.270 --> 00:43:33.520 to be a lattice? 00:43:35.560 --> 00:43:37.020 It is going to be a lattice. 00:43:37.020 --> 00:43:41.583 Everything's just going to have some h out here, times G 00:43:41.583 --> 00:43:44.080 times Z-squared, and that's still of 00:43:44.080 --> 00:43:46.830 the form of a lattice. 00:43:46.830 --> 00:43:49.680 Or call it U for orthogonal, unitary. 00:43:52.650 --> 00:43:56.090 So we multiply it by some orthogonal matrix, U. It's 00:43:56.090 --> 00:43:57.980 still a lattice. 00:43:57.980 --> 00:44:01.080 If we look in here, orthogonal transformations don't change 00:44:01.080 --> 00:44:02.330 squared distances. 00:44:05.270 --> 00:44:09.260 We're not going to change this -- it doesn't matter. 00:44:09.260 --> 00:44:13.300 Actually, I might. 00:44:16.430 --> 00:44:17.100 No. 00:44:17.100 --> 00:44:19.090 It's a rigid rotation, I'm sorry. 00:44:19.090 --> 00:44:20.740 Everything here stays the same. 00:44:20.740 --> 00:44:23.390 Hexagonal lattice still is hexagonal lattice, because 00:44:23.390 --> 00:44:28.330 it's a rigid rotation or reflection geometrically. 00:44:28.330 --> 00:44:32.750 And the volume clearly stays the same, so none of these 00:44:32.750 --> 00:44:33.670 things changes. 00:44:33.670 --> 00:44:37.990 And orthogonal transformation isn't going to change this 00:44:37.990 --> 00:44:39.400 quantity either. 00:44:39.400 --> 00:44:43.680 So we can rotate this or do whatever. 00:44:43.680 --> 00:44:46.990 And then, finally, there's an even more interesting one, 00:44:46.990 --> 00:44:52.860 which is, if we take the lattice, lambda to the m, this 00:44:52.860 --> 00:44:55.110 is simply the Cartesian product. 00:44:55.110 --> 00:45:02.490 It's the set of all lambda_1, lambda_2, lambda_m, such that 00:45:02.490 --> 00:45:06.580 each of these lambda_k is in Lambda. 00:45:06.580 --> 00:45:08.220 So it's the set of all n-tuples 00:45:08.220 --> 00:45:10.460 of elements of Lambda. 00:45:10.460 --> 00:45:16.200 This clearly lives in dimension mn, so it's a much 00:45:16.200 --> 00:45:18.770 higher-dimensional lattice. 00:45:18.770 --> 00:45:22.745 The simplest case is Z to the m - it lives in m dimensions, 00:45:22.745 --> 00:45:27.190 whereas Z is down in one dimension. 00:45:27.190 --> 00:45:29.230 So it lives in dimension mn. 00:45:29.230 --> 00:45:31.870 It's a way of constructing high-dimensional lattices from 00:45:31.870 --> 00:45:34.560 low-dimensional lattices. 00:45:34.560 --> 00:45:37.990 It's something that, in communications, we consider to 00:45:37.990 --> 00:45:41.180 be almost a non-event. 00:45:41.180 --> 00:45:46.940 Sending a sequence of elements of lambda is the same as 00:45:46.940 --> 00:45:48.360 sending a sequence of elements of lambda_m. 00:45:51.080 --> 00:45:55.340 In fact, if we just send a sequence of things from here, 00:45:55.340 --> 00:45:58.360 and a sequence of things from lambda, there's no difference 00:45:58.360 --> 00:46:00.010 from a receiver's point of view. 00:46:00.010 --> 00:46:04.630 So we regard these two lattices as equivalent, from a 00:46:04.630 --> 00:46:08.190 communications point of view. 00:46:08.190 --> 00:46:10.670 Well, what happens to d_min-squared? 00:46:10.670 --> 00:46:15.140 d_min-squared doesn't change, because we can always-- 00:46:15.140 --> 00:46:19.335 d_min-squared is, by the way, the weight of the smallest-- 00:46:19.335 --> 00:46:20.585 should have said that. 00:46:24.310 --> 00:46:27.110 You can always do things relative to 0. 00:46:27.110 --> 00:46:33.500 d_min-squared, therefore, is the squared norm or lowest 00:46:33.500 --> 00:46:37.230 squared energy of any non-zero lattice point, same as we did 00:46:37.230 --> 00:46:40.430 with Hamming codes, just by the group property. 00:46:40.430 --> 00:46:44.930 K_min is the number of such lowest 00:46:44.930 --> 00:46:48.340 Euclidean weight points. 00:46:48.340 --> 00:46:51.320 OK, here, what's the lowest weight point? 00:46:51.320 --> 00:46:54.110 It would look something like this, where this is one of the 00:46:54.110 --> 00:46:55.865 lowest weight points in lambda. 00:46:55.865 --> 00:47:00.010 So d_min-squared doesn't change. 00:47:00.010 --> 00:47:04.150 This changes, but we're not considering that. 00:47:04.150 --> 00:47:05.970 The volume, what is the volume? 00:47:05.970 --> 00:47:10.050 It turns out that it's not hard to show that V(lambda to 00:47:10.050 --> 00:47:13.700 the m) is V(lambda) to the m. 00:47:13.700 --> 00:47:19.150 And therefore, you can show that this is figure of merit 00:47:19.150 --> 00:47:21.740 doesn't change, even if you take Cartesian products. 00:47:25.490 --> 00:47:31.650 So for instance, I chose that the nominal coding gain of to 00:47:31.650 --> 00:47:38.020 the m is equal to the nominal coding gain of Z. Which is 00:47:38.020 --> 00:47:38.600 what, by the way? 00:47:38.600 --> 00:47:40.595 I should have done this as the first example. 00:47:43.490 --> 00:47:47.230 What's the minimum squared distance of Z? 00:47:47.230 --> 00:47:48.250 1. 00:47:48.250 --> 00:47:50.530 What's the volume of Z? 00:47:50.530 --> 00:47:51.670 1. 00:47:51.670 --> 00:47:54.870 So this is 1, or 0 dB. 00:47:54.870 --> 00:48:00.586 We're going to measure nominal coding gain in dB, which, 00:48:00.586 --> 00:48:04.410 well, we're getting us into communications. 00:48:04.410 --> 00:48:07.950 What's the nominal coding gain of the hexagonal lattice? 00:48:10.640 --> 00:48:14.420 By asking this question, I'm basically asking, is the 00:48:14.420 --> 00:48:20.150 hexagonal lattice denser than Z-squared in two dimensions 00:48:20.150 --> 00:48:21.920 than the square lattice in two dimensions? 00:48:21.920 --> 00:48:25.872 And if so, quantitatively, how much denser is it? 00:48:25.872 --> 00:48:27.800 AUDIENCE: [INAUDIBLE]. 00:48:27.800 --> 00:48:32.790 PROFESSOR: OK, well, there's a root 3 in it, but that isn't 00:48:32.790 --> 00:48:34.490 exactly the answer, I don't think. 00:48:38.780 --> 00:48:42.110 There are various ways that you can do it, but one of the 00:48:42.110 --> 00:48:46.910 ways is to take, as a generator matrix, (1, 0; 1/2, 00:48:46.910 --> 00:48:50.310 root 3 over 2). 00:48:50.310 --> 00:48:53.480 You like that as a generator matrix for the-- 00:48:53.480 --> 00:48:57.390 That's taking this and this as the two generators, and 00:48:57.390 --> 00:49:03.220 recognizing this as 1/2, and this as root 3 over 2. 00:49:03.220 --> 00:49:07.220 That makes the length of this equal to 1/4 plus 3/4, equals 00:49:07.220 --> 00:49:09.310 1 square root, and it's 1. 00:49:09.310 --> 00:49:12.550 So these are my two generators of length 1 in a hexagonal 00:49:12.550 --> 00:49:18.690 lattice of minimum squared distance 1. 00:49:18.690 --> 00:49:24.346 So based on that, I can say that the volume of A2 is what? 00:49:24.346 --> 00:49:25.596 AUDIENCE: [INAUDIBLE]. 00:49:27.820 --> 00:49:29.000 PROFESSOR: Root 3 over 2. 00:49:29.000 --> 00:49:33.490 Not hard to take the determinant of that, right? 00:49:33.490 --> 00:49:40.310 And what's the minimum squared distance in this scaling? 00:49:40.310 --> 00:49:43.480 What's the minimum non-zero weight -- 00:49:43.480 --> 00:49:46.610 squared norm, it's 1. 00:49:46.610 --> 00:49:53.190 So this is equal to 1 over root 3 over 2, which is equal 00:49:53.190 --> 00:49:56.060 to 2 over root 3. 00:49:56.060 --> 00:49:57.500 Which is equal to what? 00:49:57.500 --> 00:49:59.420 This is 3 dB. 00:49:59.420 --> 00:50:02.450 This is 4.8 dB, roughly. 00:50:02.450 --> 00:50:05.450 Square root is 2.4 dB. 00:50:05.450 --> 00:50:07.420 So this is about 0.6 dB. 00:50:12.850 --> 00:50:18.040 That's where it pays to be facile with dB's. 00:50:18.040 --> 00:50:21.720 This area, by the way, if you tried to compute the area of 00:50:21.720 --> 00:50:26.578 this little hexagon here, what would it be? 00:50:26.578 --> 00:50:29.230 If the area of this parallelotope is the square 00:50:29.230 --> 00:50:34.770 root of 3/2, what's the area of this hexagon? 00:50:34.770 --> 00:50:35.840 The same, right? 00:50:35.840 --> 00:50:37.920 They're both space filling when 00:50:37.920 --> 00:50:42.810 centered on lattice points. 00:50:42.810 --> 00:50:46.290 So they have to have the same area. 00:50:46.290 --> 00:50:48.880 So that's an easy way of computing the area of a 00:50:48.880 --> 00:50:54.350 hexagon, or of this particular hexagon anyway. 00:50:54.350 --> 00:50:57.020 But you would get the same result in either way. 00:50:57.020 --> 00:51:03.630 So comparing this with Z-squared, I now have a 00:51:03.630 --> 00:51:06.160 quantitative measure of how much 00:51:06.160 --> 00:51:10.740 denser A2 is than Z-squared. 00:51:10.740 --> 00:51:13.870 Put in one way, if I fix the minimum squared distance of 00:51:13.870 --> 00:51:16.440 each of them to be 1, if I'm doing penny-packing and I say 00:51:16.440 --> 00:51:20.160 I want pennies of a certain radius to be packed into 00:51:20.160 --> 00:51:26.530 two-dimensional space, then this is 1 over the volume, 00:51:26.530 --> 00:51:28.450 which is basically the density. 00:51:28.450 --> 00:51:33.140 The density of A2 is 2 over root 3, times [UNINTELLIGIBLE] 00:51:33.140 --> 00:51:35.540 as the density of points in z squared. 00:51:35.540 --> 00:51:40.480 I get more points per unit area with the same distance 00:51:40.480 --> 00:51:44.210 between them using A2 than I can with Z-squared. 00:51:44.210 --> 00:51:49.510 So that's the significance of this parameter. 00:51:49.510 --> 00:51:52.460 So it's an obvious thing for Hermite to have come up with. 00:51:52.460 --> 00:51:55.380 It's going to turn out to be equally fundamental for us. 00:52:00.920 --> 00:52:04.710 So that may be all we want to do on lattices 00:52:04.710 --> 00:52:07.754 for the time being. 00:52:07.754 --> 00:52:09.530 Yeah, seems to be. 00:52:14.150 --> 00:52:18.760 So we're talking about lattice constellations. 00:52:18.760 --> 00:52:21.110 The other thing we have to develop is the 00:52:21.110 --> 00:52:22.360 properties of regions. 00:52:25.490 --> 00:52:32.050 So let me talk about regions, R -- 00:52:32.050 --> 00:52:36.460 I'm thinking of a compact region in an 00:52:36.460 --> 00:52:38.397 n-dimensional space. 00:52:38.397 --> 00:52:42.160 So again, let me fix the dimension to n. 00:52:42.160 --> 00:52:44.690 And we're going to have a certain volume. 00:52:44.690 --> 00:52:49.570 That's obviously an important characteristic of a region. 00:52:49.570 --> 00:52:52.520 And I'm going to assume that this is finite. 00:52:52.520 --> 00:52:57.650 And what's the other thing that we basically want? 00:52:57.650 --> 00:52:59.920 Well, when I pick a constellation like this, by 00:52:59.920 --> 00:53:11.020 intersecting a lattice with a region, eventually I'm going 00:53:11.020 --> 00:53:14.695 to let the points in this constellation be equiprobable, 00:53:14.695 --> 00:53:18.170 as I always do in digital communications. 00:53:18.170 --> 00:53:19.190 16 points. 00:53:19.190 --> 00:53:22.570 Say they all have probably 1/16, the uniform probability 00:53:22.570 --> 00:53:23.970 over the discrete points. 00:53:29.750 --> 00:53:30.260 One of the approximations I am going to make -- so what's the 00:53:30.260 --> 00:53:33.430 average power of that constellation? 00:53:33.430 --> 00:53:41.270 We could figure it out by taking 1/16 times the energy. 00:53:41.270 --> 00:53:43.250 I should say the average energy, I'm sorry. 00:53:43.250 --> 00:53:44.300 We're not talking power here. 00:53:44.300 --> 00:53:47.450 We're talking energy, but I read it as P. What's the 00:53:47.450 --> 00:53:48.270 average energy? 00:53:48.270 --> 00:53:52.980 It's 1/16 times the L2 norm of each of these 00:53:52.980 --> 00:53:55.045 points, if you like that. 00:53:55.045 --> 00:53:59.940 It's simply the Euclidean squared norm. 00:54:02.760 --> 00:54:05.210 And we get some average. 00:54:05.210 --> 00:54:08.090 But here's the key approximation principle. 00:54:08.090 --> 00:54:11.240 I'm going to say the average power of this constellation, 00:54:11.240 --> 00:54:14.280 especially as the constellations get large, is 00:54:14.280 --> 00:54:16.860 simply going to be approximated as the average 00:54:16.860 --> 00:54:20.890 energy of a uniform distribution over this 00:54:20.890 --> 00:54:23.090 bounding region. 00:54:23.090 --> 00:54:27.670 Suppose I simply put a uniform continuous distribution over 00:54:27.670 --> 00:54:33.080 this circle, this sphere, two sphere that I've used to be my 00:54:33.080 --> 00:54:34.650 cookie cutter. 00:54:34.650 --> 00:54:36.950 I'm going to say that's going to be a good approximation to 00:54:36.950 --> 00:54:41.080 the average energy of the 16 discrete points, which are, by 00:54:41.080 --> 00:54:44.560 construction, they're spread uniformly over this region. 00:54:44.560 --> 00:54:49.600 The approximation is, I'm kind of approximating a unit 00:54:49.600 --> 00:54:52.770 impulse of weight there by distributed 00:54:52.770 --> 00:54:55.170 weight across its-- 00:54:55.170 --> 00:54:58.620 well, actually across its whole little region. 00:54:58.620 --> 00:55:02.280 If we have a hexagon around each one, uniform distribution 00:55:02.280 --> 00:55:04.206 over the hexagon. 00:55:04.206 --> 00:55:08.640 And if that's a reasonable approximation, then it's 00:55:08.640 --> 00:55:11.010 reasonable to say that the average power of the whole 00:55:11.010 --> 00:55:14.840 constellation is just the average power of a uniform 00:55:14.840 --> 00:55:16.175 distribution over this region. 00:55:19.100 --> 00:55:23.460 It's good exercise to try it for some moderate size cases. 00:55:23.460 --> 00:55:26.730 And you'll find it is a very good approximation. 00:55:26.730 --> 00:55:29.520 In any case, it's the one we're going to use. 00:55:29.520 --> 00:55:39.260 So the other key parameter is going to be P(R), which is, in 00:55:39.260 --> 00:55:53.665 words, the average energy of uniform distribution over R. 00:55:53.665 --> 00:56:02.500 Sorry, script R. Now, when we write this out, this actually 00:56:02.500 --> 00:56:05.730 gets a little formidable. 00:56:05.730 --> 00:56:09.160 So I think it's better to remember what we're actually 00:56:09.160 --> 00:56:11.920 trying to compute. 00:56:11.920 --> 00:56:14.700 Oh, and one last thing, we're going to 00:56:14.700 --> 00:56:16.200 normalize it per dimension. 00:56:22.480 --> 00:56:23.570 So what do I mean? 00:56:23.570 --> 00:56:28.630 What's a uniform distribution over R? 00:56:33.670 --> 00:56:44.030 This is P(x equals 1) over the volume of R for 00:56:44.030 --> 00:56:49.920 x in R and 0 otherwise. 00:56:56.330 --> 00:56:58.590 Uniform within, 0 outside. 00:56:58.590 --> 00:56:59.540 And what does it have to equal? 00:56:59.540 --> 00:57:03.040 It has to be equal to 1 over V(R), because its integral has 00:57:03.040 --> 00:57:07.260 to be equal to 1 if it's a probability distribution. 00:57:07.260 --> 00:57:17.790 So to compute P(R), we basically take the integral 00:57:17.790 --> 00:57:27.030 from over R, of this probability distribution, 1 00:57:27.030 --> 00:57:30.680 over V(R), times what? 00:57:30.680 --> 00:57:32.640 The energy per dimension. 00:57:32.640 --> 00:57:43.070 The total energy of x is x-squared, the L2 norm of x. 00:57:43.070 --> 00:57:44.910 Normalize per dimension -- we're going to put an 00:57:44.910 --> 00:57:47.080 n over there -- 00:57:47.080 --> 00:57:48.330 dx. 00:57:50.620 --> 00:57:54.470 So that's an equation for what it is I wanted to compute. 00:58:00.380 --> 00:58:02.770 I think it's much harder to remember the terms in this 00:58:02.770 --> 00:58:09.270 than to remember this verbal description of what we want. 00:58:09.270 --> 00:58:17.220 But you may be more literate than I. Now, let's define -- 00:58:17.220 --> 00:58:22.270 we want to normalize quantity comparable to Hermite or 00:58:22.270 --> 00:58:24.550 coding gain over here. 00:58:24.550 --> 00:58:27.960 The normalized quantity we're going to use -- 00:58:27.960 --> 00:58:35.170 we want to take P(R) -- and what should we normalize it by 00:58:35.170 --> 00:58:39.220 to make it scale-invariant, let's say again? 00:58:39.220 --> 00:58:43.350 Again, if I take the region, I scale the region by alpha and 00:58:43.350 --> 00:58:45.590 get alpha_R. 00:58:45.590 --> 00:58:48.550 What's going to happen to the average energy of a uniform 00:58:48.550 --> 00:58:49.850 distribution over R? 00:58:53.070 --> 00:58:54.450 AUDIENCE: [INAUDIBLE]. 00:58:54.450 --> 00:58:57.760 PROFESSOR: It's going to go up as alpha squared. 00:58:57.760 --> 00:59:02.430 Energy scarce, goes as the square of the scale factor. 00:59:02.430 --> 00:59:09.850 So again, I want to normalize it by V(R) over 2 to the n, 00:59:09.850 --> 00:59:12.390 because this is also something that will go up as alpha 00:59:12.390 --> 00:59:17.130 squared if I scale R by alpha. 00:59:20.960 --> 00:59:31.830 So this is what's known as the dimension-less or normalized-- 00:59:31.830 --> 00:59:33.380 it has both names-- 00:59:38.060 --> 00:59:39.310 second moment. 00:59:44.190 --> 00:59:50.970 Another name for this is the second moment per dimension of 00:59:50.970 --> 00:59:54.660 a uniform distribution over R. This is a normalized or 00:59:54.660 --> 00:59:58.620 dimension-less second moment. 00:59:58.620 --> 01:00:02.200 I think I call it normalized in the notes, but it's always 01:00:02.200 --> 01:00:05.100 called G in the literature. 01:00:05.100 --> 01:00:09.420 And let me just introduce here-- 01:00:09.420 --> 01:00:14.810 well, what's the normalized second moment of an interval? 01:00:14.810 --> 01:00:19.590 Say, a symmetric interval around the origin. 01:00:19.590 --> 01:00:22.510 G to the minus 1 to the 1, which is like the interval 01:00:22.510 --> 01:00:26.330 that we used to pull out the m-PAM signal structure. 01:00:26.330 --> 01:00:34.670 So let's take n to be equal to 1, R just to be the interval 01:00:34.670 --> 01:00:37.680 from -1 to +1. 01:00:37.680 --> 01:00:42.206 And V(R) equals what? 01:00:42.206 --> 01:00:44.380 It's the length of R, which is 2. 01:00:49.640 --> 01:00:50.970 What's P(R)? 01:00:54.860 --> 01:01:02.250 The integral from -1 to 1, x squared, dx, which is 2/3. 01:01:07.490 --> 01:01:09.716 I think so. 01:01:09.716 --> 01:01:10.966 AUDIENCE: [INAUDIBLE]. 01:01:14.150 --> 01:01:15.540 PROFESSOR: Right, 1/2. 01:01:15.540 --> 01:01:16.790 Thank you. 01:01:18.660 --> 01:01:19.910 That's 1/3. 01:01:24.290 --> 01:01:28.290 All right, so what is G(R) going to be, or G going to be? 01:01:28.290 --> 01:01:33.410 This is going to be P(R) over V(R)-squared. 01:01:36.070 --> 01:01:40.840 And so this is going to be 1/3 over 4, or 1/12. 01:01:45.050 --> 01:01:49.480 OK, that's not as nice as this over here, but 01:01:49.480 --> 01:01:51.940 it is what it is. 01:01:51.940 --> 01:01:56.000 There's a number that shows up in quantization, for instance, 01:01:56.000 --> 01:01:58.890 uniform scalar quantizer, you'll find a 1/12 in there. 01:01:58.890 --> 01:01:59.590 Why? 01:01:59.590 --> 01:02:00.840 It's because of this equation. 01:02:07.901 --> 01:02:11.830 If we ask again about the invariances of this-- 01:02:11.830 --> 01:02:15.000 this is the last homework problem for this week-- 01:02:15.000 --> 01:02:18.390 it's invariant not only to scaling, but, again, it's 01:02:18.390 --> 01:02:20.810 invariant to orthogonal transformations, because 01:02:20.810 --> 01:02:23.360 orthogonal transformations won't affect 01:02:23.360 --> 01:02:24.420 either of these things. 01:02:24.420 --> 01:02:28.210 A rigid rotation, about the origin, won't 01:02:28.210 --> 01:02:32.330 affect V(R) or P(R). 01:02:32.330 --> 01:02:35.110 And furthermore, it's invariant to Cartesian 01:02:35.110 --> 01:02:37.045 products, again. 01:02:37.045 --> 01:02:40.920 If we just have a series of regions, a region made up 01:02:40.920 --> 01:02:47.370 basically as the Cartesian product, like an n-cube is a 01:02:47.370 --> 01:02:48.540 Cartesian product. 01:02:48.540 --> 01:02:52.350 We just pick each of the dimensions independently from 01:02:52.350 --> 01:02:57.360 some set R. Then that's not going to affect the normalized 01:02:57.360 --> 01:02:58.450 second moment either. 01:02:58.450 --> 01:03:01.600 Exercise for the student. 01:03:01.600 --> 01:03:11.110 So from that result, I can say that G([-1 1]) to the m. 01:03:11.110 --> 01:03:12.360 Which is what? 01:03:14.640 --> 01:03:17.275 This is an m-cube of side 2. 01:03:20.570 --> 01:03:28.050 If I say each of the coordinates, x1, x2, x3, up to 01:03:28.050 --> 01:03:35.690 xm, has got to be xk n times [-1 1], then what I'm going to 01:03:35.690 --> 01:03:39.360 get, the region defined by that is an m-cubed, centered 01:03:39.360 --> 01:03:41.150 on the origin of side 2. 01:03:44.260 --> 01:03:47.585 So the normalized second moment of that 01:03:47.585 --> 01:03:48.835 is going to be 1/12. 01:03:52.650 --> 01:03:55.770 This is where the shortcut begins to help you. 01:03:59.340 --> 01:04:02.560 I guess I'm trying to put too much on one board. 01:04:02.560 --> 01:04:05.280 But the last thing I want to put up on here 01:04:05.280 --> 01:04:07.320 is the shaping gain. 01:04:07.320 --> 01:04:17.760 The shape gain of a region is defined as the shaping gain of 01:04:17.760 --> 01:04:23.120 the region is equal to 1/12. 01:04:23.120 --> 01:04:27.460 So we're kind of taking as a baseline, the m-cube, in any 01:04:27.460 --> 01:04:31.610 number of dimensions n, over G(the region). 01:04:36.190 --> 01:04:41.460 In other words, how much less is the dimension-less second 01:04:41.460 --> 01:04:44.830 moment, the normalized second moment, than what you would 01:04:44.830 --> 01:04:47.450 get for an m-cube? 01:04:47.450 --> 01:04:56.650 In every dimension except for one, a sphere is going to be a 01:04:56.650 --> 01:05:02.360 better region, from a second moment point of 01:05:02.360 --> 01:05:05.330 view, than an m-cube. 01:05:05.330 --> 01:05:07.460 An m-cube has corners. 01:05:07.460 --> 01:05:11.540 Really, it's pretty obvious that, if you want to minimize 01:05:11.540 --> 01:05:14.310 the normalized second moment in any number of dimensions, 01:05:14.310 --> 01:05:19.270 you would choose an m-sphere, because if you have any 01:05:19.270 --> 01:05:22.440 wrinkle that comes out of an m-sphere, anything that's not 01:05:22.440 --> 01:05:24.810 an m-sphere, you should try to squash it back in it. 01:05:24.810 --> 01:05:26.150 And that'll make it an m-sphere. 01:05:26.150 --> 01:05:29.430 You'll take higher energy and make it lower energy for the 01:05:29.430 --> 01:05:35.100 same little unit of Play-Doh or mass stuff. 01:05:35.100 --> 01:05:41.060 So a sphere will give you the best normalized second moment 01:05:41.060 --> 01:05:44.160 in any number of dimensions n, so you're always going to get 01:05:44.160 --> 01:05:46.830 a G(R) that's less than 1/12. 01:05:46.830 --> 01:05:48.220 That's good. 01:05:48.220 --> 01:05:52.000 And this measures the amount of gain. 01:05:52.000 --> 01:05:55.026 Again, you can measure this in dB. 01:05:55.026 --> 01:05:57.630 That's going to translate directly to dB. 01:06:03.080 --> 01:06:06.190 I suppose I could do an example for the sphere in two 01:06:06.190 --> 01:06:07.150 dimensions, but I won't. 01:06:07.150 --> 01:06:08.400 How are we doing on time? 01:06:20.080 --> 01:06:24.980 Let me see if I can do the union-bound estimate. 01:06:28.310 --> 01:06:33.870 That would be a trick, but I think it's doable, because 01:06:33.870 --> 01:06:35.120 this is where we're going. 01:06:38.160 --> 01:06:49.750 So the union-bound estimate, which is something we did way 01:06:49.750 --> 01:06:58.650 back in chapter five or something, is basically, given 01:06:58.650 --> 01:07:01.740 a constellation, which we're going to have a lattice 01:07:01.740 --> 01:07:04.890 constellation based on some lattice 01:07:04.890 --> 01:07:07.765 lambda and some region. 01:07:14.200 --> 01:07:19.410 So this is like our m-by-m QAM, or it might be this guy, 01:07:19.410 --> 01:07:24.930 which is better than 16-QAM, in some sense. 01:07:24.930 --> 01:07:26.640 In two senses, actually. 01:07:29.440 --> 01:07:31.670 The union-bound estimate, what is it? 01:07:34.310 --> 01:07:37.605 The probability of error is approximately equal to-- 01:07:40.280 --> 01:07:41.910 let me do it per block. 01:07:41.910 --> 01:07:46.120 So I'll have the number of nearest neighbors per block of 01:07:46.120 --> 01:07:54.650 our constellation times Q to the square root of 01:07:54.650 --> 01:07:57.890 d_min-squared of our constellation, 01:07:57.890 --> 01:07:58.960 over what is it? 01:07:58.960 --> 01:08:00.210 4 sigma squared? 01:08:05.930 --> 01:08:07.180 4 sigma squared. 01:08:15.600 --> 01:08:19.859 And if you remember how we got this, we took the union-bound, 01:08:19.859 --> 01:08:24.460 we took all the pairwise error probabilities, and from that, 01:08:24.460 --> 01:08:31.830 we get contributions from kd at every kd contributions at 01:08:31.830 --> 01:08:33.380 distance d. 01:08:33.380 --> 01:08:35.340 We added them all up in the union-bound. 01:08:35.340 --> 01:08:39.779 We said, well, for moderate complexity and performance, at 01:08:39.779 --> 01:08:41.300 least this is going to be dominated 01:08:41.300 --> 01:08:42.920 by the minimum distance. 01:08:42.920 --> 01:08:46.109 So this is the minimum distance terms. 01:08:46.109 --> 01:08:49.310 This is the number of minimum distance terms. 01:08:49.310 --> 01:08:50.739 And this is error probability -- 01:08:50.739 --> 01:08:53.220 the pairwise error probability we get for each minimum 01:08:53.220 --> 01:08:54.470 distance term. 01:09:00.300 --> 01:09:07.200 Now, to compute this, I'm going to use three 01:09:07.200 --> 01:09:12.390 approximations, but two important ones, this 01:09:12.390 --> 01:09:15.370 continuous approximation that I've already referred to. 01:09:18.670 --> 01:09:25.140 First, I'm going to say that the average power of my 01:09:25.140 --> 01:09:33.470 constellation is approximately equal to just the second 01:09:33.470 --> 01:09:46.609 moment of R. The average energy per dimension of R. 01:09:46.609 --> 01:09:51.080 And then I'm going to say, furthermore, how many points 01:09:51.080 --> 01:09:52.279 are there in the constellation? 01:09:52.279 --> 01:09:54.700 This will affect what rate I can go at. 01:09:57.300 --> 01:10:01.350 The average number of points in the constellation -- 01:10:01.350 --> 01:10:03.110 also mention this-- 01:10:03.110 --> 01:10:08.470 is going to be approximately equal to simply the volume of 01:10:08.470 --> 01:10:14.800 my bounding region, V(R), over the volume taken up by each 01:10:14.800 --> 01:10:16.590 point in my lattice. 01:10:20.330 --> 01:10:23.590 That makes sense, doesn't it? 01:10:23.590 --> 01:10:28.580 What I want to do here to get 16 points is I take a sphere 01:10:28.580 --> 01:10:32.590 whose area is roughly 16 times the area of one of these 01:10:32.590 --> 01:10:35.910 little hexagons or one of these little parallelograms. 01:10:35.910 --> 01:10:39.840 And I'll get about 16 points, because that's 01:10:39.840 --> 01:10:43.200 the density of it. 01:10:43.200 --> 01:10:48.400 And there's a final little approximation, which is that 01:10:48.400 --> 01:10:50.710 I'm going to assume that we're somewhere in the interior of 01:10:50.710 --> 01:10:55.450 the lattice, so that the average number of nearest 01:10:55.450 --> 01:10:59.760 neighbors in the constellation is simply equal to the number 01:10:59.760 --> 01:11:01.125 of nearest neighbors in the lattice. 01:11:01.125 --> 01:11:04.990 So that's a reasonable approximation, and that's a 01:11:04.990 --> 01:11:06.310 very obvious one. 01:11:06.310 --> 01:11:09.960 If I take any constellation based on the hexagonal 01:11:09.960 --> 01:11:12.720 lattice, for a large enough constellation, the average 01:11:12.720 --> 01:11:15.490 number of nearest neighbors is going to be 6. 01:11:15.490 --> 01:11:18.110 If I take it on the square lattice, the average number of 01:11:18.110 --> 01:11:19.990 nearest neighbors is going to be 4. 01:11:19.990 --> 01:11:25.380 But we know we're not terribly interested in that either. 01:11:28.730 --> 01:11:33.490 Let me do a little manipulation here. 01:11:33.490 --> 01:11:45.000 I'm going to write this as Q of d_min-squared of c, which 01:11:45.000 --> 01:11:49.600 clearly I'm going to also assume that d_min-squared of c 01:11:49.600 --> 01:11:53.460 is equal to d_min-squared of the lattice. 01:11:53.460 --> 01:11:55.940 So I'll make that d_min-squared of the lattice. 01:11:58.810 --> 01:12:07.670 And I want to normalize that by v of lambda, to the 2/n, 01:12:07.670 --> 01:12:12.620 whatever dimension n I'm in, to get something that we're 01:12:12.620 --> 01:12:13.450 familiar with. 01:12:13.450 --> 01:12:17.010 So I need a V(n) over 2 to the n. 01:12:17.010 --> 01:12:23.180 And anticipating a little, I'm going to put a V(R) 01:12:23.180 --> 01:12:25.864 over 2 to the n. 01:12:25.864 --> 01:12:32.035 And then over here, I'm going to put V(R) over-- 01:12:32.035 --> 01:12:33.450 Is this going to come out right?-- 01:12:33.450 --> 01:12:38.980 over 2 to the n, over-- 01:12:38.980 --> 01:12:40.230 this seems upside-down. 01:13:00.100 --> 01:13:01.570 Ah, no, it isn't upside-down. 01:13:05.960 --> 01:13:10.560 Because I'm using this expression here, I 01:13:10.560 --> 01:13:11.810 do want 1 over G(R). 01:13:16.340 --> 01:13:20.450 So this is 1 over G(R). 01:13:20.450 --> 01:13:29.170 If I multiply this by 1/12, now I get 1 over G(R), so I 01:13:29.170 --> 01:13:34.440 need to put a 1/12 here. 01:13:34.440 --> 01:13:42.255 And then I have a P(R) over 4 sigma squared. 01:13:44.810 --> 01:13:46.810 So does everyone agree I can write it that way? 01:13:50.260 --> 01:13:52.280 This'll be good. 01:13:52.280 --> 01:13:53.570 Go out with a bang. 01:13:56.200 --> 01:13:59.220 So what are all these things? 01:13:59.220 --> 01:14:03.995 This is the coding gain of lambda. 01:14:06.860 --> 01:14:13.090 This, according to this, is the size of the constellation 01:14:13.090 --> 01:14:15.960 to the 2 over n, approximately. 01:14:15.960 --> 01:14:23.900 The rate of the constellation is log 2 times the size of 01:14:23.900 --> 01:14:32.020 lambda, R. So the rate is equal to just 01:14:32.020 --> 01:14:33.750 log 2 of this quantity. 01:14:36.450 --> 01:14:42.680 So I can say 2 to the R. This is 2 to the R equals this. 01:14:42.680 --> 01:14:54.780 So this would be 2 to the 2 over n, 2R over n. 01:14:54.780 --> 01:14:56.880 Is that right? 01:14:56.880 --> 01:15:00.280 Again, I'm unsure if that's what I want. 01:15:00.280 --> 01:15:06.680 This is the shaping gain of the region, these three things 01:15:06.680 --> 01:15:12.790 together, times 3, times-- what's 01:15:12.790 --> 01:15:14.440 P(R) over sigma squared? 01:15:14.440 --> 01:15:15.960 That's the SNR. 01:15:15.960 --> 01:15:19.130 That's the average power per dimension. 01:15:19.130 --> 01:15:21.130 And this is the average noise power per dimension. 01:15:24.230 --> 01:15:26.740 I've used up these in that. 01:15:26.740 --> 01:15:28.530 So I've got a 1 over 1/12. 01:15:28.530 --> 01:15:29.890 That's 12 times that. 01:15:29.890 --> 01:15:32.270 So this is 3 times SNR. 01:15:35.340 --> 01:15:39.200 And let's see. 01:15:39.200 --> 01:15:45.657 This is 2R over n is just 2 to the-- 01:15:45.657 --> 01:15:47.525 AUDIENCE: Isn't that just 2R? 01:15:47.525 --> 01:15:49.753 Because [UNINTELLIGIBLE] normalized by the dimension. 01:15:49.753 --> 01:15:51.003 PROFESSOR: Yeah. 01:15:58.320 --> 01:16:05.830 And furthermore, it's minus, because it's upside-down. 01:16:05.830 --> 01:16:11.480 So it's 2 to the - 2R. 01:16:11.480 --> 01:16:15.360 Actually, what I want is the spectral efficiency. 01:16:15.360 --> 01:16:21.330 And the spectral efficiency is the rate per two dimensions. 01:16:21.330 --> 01:16:23.435 So this is just 2 to the minus rho. 01:16:30.060 --> 01:16:34.030 So with a little help for my friends, what's SNR 01:16:34.030 --> 01:16:35.280 over 2 to the rho? 01:16:39.320 --> 01:16:39.780 AUDIENCE: [INAUDIBLE]. 01:16:39.780 --> 01:16:42.210 PROFESSOR: It's approximately SNR norm. 01:16:42.210 --> 01:16:44.960 Certainly true as row becomes large. 01:16:44.960 --> 01:16:48.320 It's actually SNR over 2 to the rho minus 1. 01:16:48.320 --> 01:16:51.020 So this is SNR norm. 01:16:53.890 --> 01:16:58.830 OK, so with a little hand waving and a few high SNR 01:16:58.830 --> 01:17:09.960 approximations, we get that, in summary, for the UBE, the 01:17:09.960 --> 01:17:17.930 probability of error is approximately K_min(lambda) 01:17:17.930 --> 01:17:23.600 times Q to the square root of the coding gain of the 01:17:23.600 --> 01:17:29.320 lattice, times the shaping gain of the 01:17:29.320 --> 01:17:36.010 region, times 3SNR norm. 01:17:40.860 --> 01:17:42.230 And that's a wonderful approximation. 01:17:45.590 --> 01:17:48.380 Let me just call out some of its properties to you, and 01:17:48.380 --> 01:17:52.550 then we'll talk about them next time. 01:17:52.550 --> 01:17:55.450 But first of all, there's nothing here-- 01:17:55.450 --> 01:17:59.790 all these terms involve either lambda or R. So we have 01:17:59.790 --> 01:18:05.190 completely separated, in this analysis, the effects of the 01:18:05.190 --> 01:18:07.180 region from the effects of the lattice. 01:18:07.180 --> 01:18:12.240 We can choose the lattice and the region independently. 01:18:12.240 --> 01:18:18.810 And they independently affect how good a performance we get. 01:18:18.810 --> 01:18:26.260 For the baseline, which was either m-PAM or m-by-m QAM, we 01:18:26.260 --> 01:18:33.680 have, for the baseline PAM or QAM, we've set it up so that 01:18:33.680 --> 01:18:39.280 both the coding gain of the lattice and the shaping gain 01:18:39.280 --> 01:18:41.330 of the region are equal to 1. 01:18:41.330 --> 01:18:45.340 So for the baseline, this gives us what we already 01:18:45.340 --> 01:18:49.250 developed back in the early stages. 01:18:49.250 --> 01:18:51.130 I won't worry about this -- 01:18:51.130 --> 01:18:54.200 Q-hat, but it does give you the right coefficient out 01:18:54.200 --> 01:18:58.280 here, too, depending on whether you normalize it per 01:18:58.280 --> 01:18:59.960 two dimensions or what. 01:18:59.960 --> 01:19:04.850 It just gives you Q to the square root of 3SNR norm, 01:19:04.850 --> 01:19:07.590 which I hope you remember from chapter five. 01:19:07.590 --> 01:19:15.920 So this is our baseline curve for m-PAM or m-by-m QAM. 01:19:15.920 --> 01:19:22.100 We just plotted versus SNR norm, the probability of error 01:19:22.100 --> 01:19:25.860 per two dimensions, which is what I recommended. 01:19:25.860 --> 01:19:28.410 And we got some curve. 01:19:28.410 --> 01:19:29.880 So this is the baseline. 01:19:33.910 --> 01:19:37.390 And now we've got everything separated out in a nice way, 01:19:37.390 --> 01:19:40.350 so that if somebody gives us a lattice with a certain coding 01:19:40.350 --> 01:19:45.520 gain and a region with a certain shaping gain, those 01:19:45.520 --> 01:19:47.970 both just add up as gains. 01:19:47.970 --> 01:19:51.550 If that's all we had, forget the coefficient, we would just 01:19:51.550 --> 01:19:55.570 move to the right here, and it would give us effective 01:19:55.570 --> 01:19:59.800 reductions and the required SNR norm by the coding gain of 01:19:59.800 --> 01:20:03.240 the lattice and the shaping gain of the region. 01:20:03.240 --> 01:20:07.340 And we'd get the same curve moved out over here. 01:20:07.340 --> 01:20:11.690 Again, as in the binary case, typically we're going to get a 01:20:11.690 --> 01:20:15.580 larger number of near neighbors, leading to a larger 01:20:15.580 --> 01:20:17.070 coefficient out here. 01:20:17.070 --> 01:20:19.320 We want to normalize this per two dimensions. 01:20:19.320 --> 01:20:23.080 Per two dimensions, it's only 4 for the baseline. 01:20:23.080 --> 01:20:28.220 But in general, we're going to get a KS of lambda, the number 01:20:28.220 --> 01:20:30.810 of nearest neighbors per two dimensions, which is going to 01:20:30.810 --> 01:20:32.690 raise this a little. 01:20:32.690 --> 01:20:35.140 And that will lower the effective 01:20:35.140 --> 01:20:37.650 coding gain down here. 01:20:37.650 --> 01:20:42.350 But we've basically, by these large constellation, high SNR 01:20:42.350 --> 01:20:46.750 approximations, we've really reduced the analysis of 01:20:46.750 --> 01:20:49.890 lattice constellations to a very simple task, just to 01:20:49.890 --> 01:20:54.420 analyze what's the coding gain, what's the shaping gain, 01:20:54.420 --> 01:20:56.990 number of nearest neighbors, plot the performance curve, 01:20:56.990 --> 01:21:00.020 end of story. 01:21:00.020 --> 01:21:01.430 So we'll start there next time.