WEBVTT 00:00:00.000 --> 00:00:02.360 The following content is provided under a Create 00:00:02.360 --> 00:00:03.630 Commons license. 00:00:03.630 --> 00:00:06.600 Your support will help MIT OpenCourseWare continue to 00:00:06.600 --> 00:00:09.970 offer high quality education resources for free. 00:00:09.970 --> 00:00:12.870 To make a donation or to view additional materials from 00:00:12.870 --> 00:00:15.280 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:15.280 --> 00:00:16.530 ocw.mit.edu. 00:00:21.070 --> 00:00:23.380 PROFESSOR: Let's get started then. 00:00:23.380 --> 00:00:26.260 We went through Rayleigh fading very, very 00:00:26.260 --> 00:00:29.200 quickly last time. 00:00:29.200 --> 00:00:34.110 I want to spend a little more time on it today since it's 00:00:34.110 --> 00:00:40.950 one of the sort of classical models of wireless channels. 00:00:40.950 --> 00:00:43.400 And it's good to understand how it works. 00:00:43.400 --> 00:00:46.070 And it's good to also understand what all the 00:00:46.070 --> 00:00:51.260 assumptions that are made when one assumes Rayleigh fading, 00:00:51.260 --> 00:00:54.220 because they're really quite a few of them. 00:00:54.220 --> 00:00:58.560 OK, so what we're doing is we're assuming flat fading. 00:00:58.560 --> 00:01:00.960 In other words when we talk about flat fading, we're 00:01:00.960 --> 00:01:06.830 talking about fading where if you generate a discrete model 00:01:06.830 --> 00:01:10.500 for the channel, that discrete model is just going to have 00:01:10.500 --> 00:01:12.280 one path in it. 00:01:12.280 --> 00:01:16.460 In other words, the output is going to look like a faded 00:01:16.460 --> 00:01:18.290 version of the input. 00:01:18.290 --> 00:01:21.700 It'll be shifted in phase because of the unknown phase 00:01:21.700 --> 00:01:22.760 in the channel. 00:01:22.760 --> 00:01:26.770 It'll be attenuated by some random amount. 00:01:26.770 --> 00:01:29.590 But if you look at the waveform, it'll look like the 00:01:29.590 --> 00:01:32.860 waveform that you transmitted except for the noise. 00:01:36.260 --> 00:01:39.070 And that's what really is represented by this one tap 00:01:39.070 --> 00:01:41.260 model that we've been looking at. 00:01:41.260 --> 00:01:45.980 In general we've said that you can model a pretty arbitrary 00:01:45.980 --> 00:01:49.590 channel for purposes of somewhat narrow band 00:01:49.590 --> 00:01:56.070 communication by using a sequence of taps where usually 00:01:56.070 --> 00:01:59.510 for want of something better to do, we model those taps as 00:01:59.510 --> 00:02:03.800 being Gaussian random variables, complex Gaussian 00:02:03.800 --> 00:02:13.630 random variables with zero mean, and variables which are 00:02:13.630 --> 00:02:15.340 circularly symmetric. 00:02:15.340 --> 00:02:19.350 And we assume for not any very good reason, that the taps are 00:02:19.350 --> 00:02:21.100 independent of each other. 00:02:21.100 --> 00:02:23.370 I mean we have to make some assumptions or we can't start 00:02:23.370 --> 00:02:27.490 to make any progress on trying to analyze these channels. 00:02:27.490 --> 00:02:30.830 But we should realize that all of these assumptions are 00:02:30.830 --> 00:02:33.940 subject to a certain amount of question. 00:02:33.940 --> 00:02:34.300 OK. 00:02:34.300 --> 00:02:42.760 When we assume a single tap model, and these tap models 00:02:42.760 --> 00:02:46.630 are always given with the number of the tap given first, 00:02:46.630 --> 00:02:48.340 and the time given second. 00:02:48.340 --> 00:02:50.760 So what we're assuming here is the only tap is 00:02:50.760 --> 00:02:52.580 the tap at time 0. 00:02:52.580 --> 00:02:56.270 And it's at time 0 because we're assuming the receiver 00:02:56.270 --> 00:02:59.080 timing is locked at transmitter timing. 00:02:59.080 --> 00:03:01.160 And we're just going to get rid of the zero because 00:03:01.160 --> 00:03:03.740 there's only one tap, and call this G sub m. 00:03:03.740 --> 00:03:08.210 We're also going to pretty much assume that G sub m stays 00:03:08.210 --> 00:03:10.270 relatively constant for a relatively 00:03:10.270 --> 00:03:11.560 long amount of time. 00:03:14.440 --> 00:03:17.680 Except as far as this analysis of Rayleigh fading goes, we 00:03:17.680 --> 00:03:19.890 don't have to assume that. 00:03:19.890 --> 00:03:23.860 Because in fact, when we're assuming Rayleigh fading, the 00:03:23.860 --> 00:03:28.180 analysis that we're going to follow, the receiver doesn't 00:03:28.180 --> 00:03:30.960 know anything about the channel at all, except that 00:03:30.960 --> 00:03:32.980 it's a single tap model. 00:03:32.980 --> 00:03:35.760 And therefore what the receiver does is it goes 00:03:35.760 --> 00:03:39.700 through maximum likelihood detection assuming that that 00:03:39.700 --> 00:03:45.010 single tap is just a complex Gaussian random variable. 00:03:45.010 --> 00:03:48.030 OK when you have a complex Gaussian random variable as 00:03:48.030 --> 00:03:52.060 you've seen in the problem sets and we've noted a number 00:03:52.060 --> 00:03:57.720 of times, the energy in that complex Gaussian random 00:03:57.720 --> 00:04:00.630 variable is exponential. 00:04:00.630 --> 00:04:08.620 And the magnitude is just a square root of the magnitude 00:04:08.620 --> 00:04:10.310 squared, namely the energy. 00:04:10.310 --> 00:04:13.460 And that has a Rayleigh distribution which looks like 00:04:13.460 --> 00:04:20.750 this, namely the probability density of how much 00:04:20.750 --> 00:04:22.210 response you get. 00:04:22.210 --> 00:04:23.980 We'll base this law here. 00:04:23.980 --> 00:04:28.810 And the phase of course, is equally likely to be anything. 00:04:28.810 --> 00:04:31.990 Namely the phase is uniform and random. 00:04:31.990 --> 00:04:34.610 This density looks like this. 00:04:34.610 --> 00:04:37.910 I wanted to draw this so it would emphasize the fact that 00:04:37.910 --> 00:04:41.290 the magnitude is in fact, always nonnegative. 00:04:41.290 --> 00:04:43.830 But also to emphasize the fact there's a whole lot of 00:04:43.830 --> 00:04:46.670 probability down here where there's 00:04:46.670 --> 00:04:48.990 very, very little channel. 00:04:48.990 --> 00:04:52.680 And this is in fact what gives rise to the fact that if you 00:04:52.680 --> 00:04:58.150 try to communicate over Rayleigh fading, and you don't 00:04:58.150 --> 00:05:00.860 make any use of diversity-- and we'll talk later about 00:05:00.860 --> 00:05:02.290 diversity-- 00:05:02.290 --> 00:05:05.660 in fact you can't communicate very well at all. 00:05:05.660 --> 00:05:09.270 And that's because of this very bad region here where the 00:05:09.270 --> 00:05:11.230 channel is very badly faded. 00:05:11.230 --> 00:05:16.210 You send a bit on this channel which is very badly faded, and 00:05:16.210 --> 00:05:18.010 there's nothing much the receiver can do. 00:05:18.010 --> 00:05:22.670 And that's the thing we want to try to get a 00:05:22.670 --> 00:05:24.000 real feeling for. 00:05:24.000 --> 00:05:26.970 OK so the output of the channel when you put in an 00:05:26.970 --> 00:05:31.230 input U sub m, and we'll think of this as being a binary 00:05:31.230 --> 00:05:33.470 digit for this time being. 00:05:33.470 --> 00:05:37.830 So the output is going to be the input times this tap 00:05:37.830 --> 00:05:40.500 variable, which is this complex Gaussian random 00:05:40.500 --> 00:05:44.800 variable plus a noise random variable, which we're also 00:05:44.800 --> 00:05:49.330 assuming is complex Gaussian and circularly symmetric. 00:05:49.330 --> 00:05:53.020 OK so what we have, if you're going to make two hypotheses 00:05:53.020 --> 00:05:56.780 about two possible values of U sub m, look at what this 00:05:56.780 --> 00:05:58.790 random phase does here. 00:05:58.790 --> 00:06:06.660 No matter what U sub m you transmit in one epoch of time, 00:06:06.660 --> 00:06:10.130 the channel is going to rotate this around by a completely 00:06:10.130 --> 00:06:11.420 random phase. 00:06:11.420 --> 00:06:14.020 It's going to add a noise to it which has a completely 00:06:14.020 --> 00:06:15.330 random phase. 00:06:15.330 --> 00:06:17.240 And the output is going to come out. 00:06:17.240 --> 00:06:21.530 And the output has a completely random phase. 00:06:21.530 --> 00:06:25.610 Namely the phase of the output cannot possibly tell you 00:06:25.610 --> 00:06:28.280 anything about what input you're 00:06:28.280 --> 00:06:29.980 putting into the channel. 00:06:29.980 --> 00:06:34.530 OK so in other words, in this model that we're using, the 00:06:34.530 --> 00:06:37.990 phase is completely useless. 00:06:37.990 --> 00:06:45.270 And if we want to talk about anything connected to using 00:06:45.270 --> 00:06:49.630 likelihoods, the only thing we can use is the magnitude of 00:06:49.630 --> 00:06:50.730 the output. 00:06:50.730 --> 00:06:52.110 OK. 00:06:52.110 --> 00:06:54.930 Now, why don't we just analyze it in terms of the magnitude 00:06:54.930 --> 00:06:56.060 of the output? 00:06:56.060 --> 00:07:00.360 Well when you analyze these problems, Gaussian things are 00:07:00.360 --> 00:07:05.680 usually much easier to analyze than things like this. 00:07:05.680 --> 00:07:07.590 Not always, I mean we have to get used to 00:07:07.590 --> 00:07:09.080 analyzing all of them. 00:07:09.080 --> 00:07:12.400 But this particular problem of Rayleigh fading is really 00:07:12.400 --> 00:07:14.310 easier to analyze in terms of these 00:07:14.310 --> 00:07:16.220 Gaussian random variables. 00:07:16.220 --> 00:07:19.510 But it's easier to understand in terms of recognizing that 00:07:19.510 --> 00:07:23.320 the only thing you can make any use is these magnitudes. 00:07:28.190 --> 00:07:32.040 OK if we only use one complex degree of freedom in a signal, 00:07:32.040 --> 00:07:37.540 namely if we try to send some signal and we only use one 00:07:37.540 --> 00:07:41.730 input to the channel, then we only get one output. 00:07:41.730 --> 00:07:45.800 Namely we sent U sub 0, we get V sub 0. 00:07:45.800 --> 00:07:49.330 And we try to decide from V sub 0 what was sent. 00:07:49.330 --> 00:07:52.850 We're really in a very bad pickle at that point. 00:07:52.850 --> 00:07:56.050 Because the only thing that makes any sense, since we can 00:07:56.050 --> 00:08:01.090 only use the magnitudes, is to send a very small magnitude or 00:08:01.090 --> 00:08:03.530 a positive magnitude. 00:08:03.530 --> 00:08:05.550 Magnitudes are positive anyway. 00:08:05.550 --> 00:08:08.170 So if you're going to send binary signals, this is your 00:08:08.170 --> 00:08:11.520 only choice-- if it makes any sense-- 00:08:11.520 --> 00:08:13.760 if you make this larger than zero you're 00:08:13.760 --> 00:08:15.980 just wasting energy. 00:08:15.980 --> 00:08:18.060 So you only have this choice. 00:08:18.060 --> 00:08:20.700 And you can choose the amplitude a that you're using. 00:08:20.700 --> 00:08:23.020 But that's the only thing that you can do. 00:08:23.020 --> 00:08:28.340 OK, this is a very nasty thing to analyze for one thing. 00:08:28.340 --> 00:08:29.820 It gives you a very large error 00:08:29.820 --> 00:08:31.780 probability for another thing. 00:08:31.780 --> 00:08:34.330 Nobody uses it for another thing. 00:08:34.330 --> 00:08:38.080 And therefore almost all systems of trying to transmit 00:08:38.080 --> 00:08:42.160 in this kind of Rayleigh fading, always use at least 00:08:42.160 --> 00:08:44.520 two sample values. 00:08:44.520 --> 00:08:47.180 In other words, instead of just putting one complex 00:08:47.180 --> 00:08:49.970 degree of freedom into the channel, you're going to put 00:08:49.970 --> 00:08:52.930 two complex degrees of freedom into the channel. 00:08:52.930 --> 00:08:55.860 And the thing that we're going to analyze, because it's the 00:08:55.860 --> 00:08:58.700 easiest thing to do in this discrete time model we've 00:08:58.700 --> 00:09:03.930 developed, is to think of modeling hypothesis 0 as 00:09:03.930 --> 00:09:08.320 sending two symbols U sub 0 and U sub 1 will make U sub 0 00:09:08.320 --> 00:09:12.380 equal to a, and U sub 1 equal to 0. 00:09:12.380 --> 00:09:15.530 And the alternative case, if we're going to try to send 00:09:15.530 --> 00:09:18.300 input 1, this is binary transmission. 00:09:18.300 --> 00:09:21.660 You can talk about more than binary transmission, but 00:09:21.660 --> 00:09:24.400 binary is awful enough. 00:09:24.400 --> 00:09:27.680 You get U sub 0 and U sub 1 is equal to 0 and a. 00:09:27.680 --> 00:09:31.480 So what you're going to be doing here in this 00:09:31.480 --> 00:09:35.030 pulse-position modulation, is choosing one of these two 00:09:35.030 --> 00:09:38.040 different epochs to put the data in. 00:09:38.040 --> 00:09:41.000 So in one case, you put all your energy in the first one. 00:09:41.000 --> 00:09:44.690 In the other case, you put all your energy in the second one. 00:09:44.690 --> 00:09:47.340 Mathematically, this is completely equivalent to 00:09:47.340 --> 00:09:50.190 frequency-shift keying, that's completely equivalent to 00:09:50.190 --> 00:09:52.320 phase-shift keying. 00:09:52.320 --> 00:09:54.740 And if we had a little more time, we 00:09:54.740 --> 00:09:55.890 could talk about that. 00:09:55.890 --> 00:09:58.750 And I'll probably put an appendix in which talks about 00:09:58.750 --> 00:10:00.020 those two systems. 00:10:00.020 --> 00:10:02.310 But in fact, it's completely the same thing. 00:10:02.310 --> 00:10:04.810 It's just that you're using different complex degrees of 00:10:04.810 --> 00:10:06.910 freedom than we're using here. 00:10:06.910 --> 00:10:10.500 So we're really analyzing FSK and PSK. 00:10:10.500 --> 00:10:13.160 And that's where people usually come up with these 00:10:13.160 --> 00:10:17.240 analyses of Rayleigh fading. 00:10:17.240 --> 00:10:25.000 OK when we have input 0, what we receive then, is the 0 is 00:10:25.000 --> 00:10:29.470 going to be the input a, times the magnitude of the channel 00:10:29.470 --> 00:10:32.060 of time 0, plus a noise variable. 00:10:32.060 --> 00:10:35.290 The noise is complex Gaussian, remember. 00:10:35.290 --> 00:10:39.110 The second input is just going to be the noise variable. 00:10:39.110 --> 00:10:42.120 Alternatively, if we're sending the second symbol, 00:10:42.120 --> 00:10:44.740 which means we put our energy into the second degree of 00:10:44.740 --> 00:10:48.440 freedom, it means that what we're going to get is the 0 00:10:48.440 --> 00:10:50.430 was just going to be the noise. 00:10:50.430 --> 00:10:53.840 And the second output is going to be the 00:10:53.840 --> 00:10:56.020 signal plus the noise. 00:10:56.020 --> 00:11:00.740 And remember, both this variable and this variable are 00:11:00.740 --> 00:11:02.790 both complex Gaussian. 00:11:02.790 --> 00:11:04.390 The phase doesn't mean anything. 00:11:04.390 --> 00:11:10.080 So what we can use is simply the magnitude. 00:11:10.080 --> 00:11:16.250 OK, so when we have hypothesis equal to 0, what comes out is 00:11:16.250 --> 00:11:20.710 going to be, the 0 is going to be a complex 00:11:20.710 --> 00:11:22.120 Gaussian random variable. 00:11:22.120 --> 00:11:24.920 Let me introduce a new piece of notation now. 00:11:24.920 --> 00:11:28.870 Because it gets to be a real mess to constantly talk about 00:11:28.870 --> 00:11:32.640 a Gaussian complex random variable, and talk about it's 00:11:32.640 --> 00:11:36.310 real part and imaginary part as being independent Gaussian. 00:11:36.310 --> 00:11:41.230 So I'll just call this normal complex. 00:11:41.230 --> 00:11:44.550 And this first thing is the mean, which is a real and 00:11:44.550 --> 00:11:47.510 imaginary part, but it's zero in most of the 00:11:47.510 --> 00:11:48.980 things we deal with. 00:11:48.980 --> 00:11:53.850 And the second one is the mean square value of this random 00:11:53.850 --> 00:11:55.660 variable V sub zero. 00:11:55.660 --> 00:12:01.640 So this quantity here is now twice the variance of the real 00:12:01.640 --> 00:12:05.635 part of V sub 0, and twice the imaginary part of 00:12:05.635 --> 00:12:09.910 the variance of v0. 00:12:09.910 --> 00:12:14.170 We scaled the noise in a peculiar way here. 00:12:14.170 --> 00:12:19.040 And I apologize for all of the mess that 00:12:19.040 --> 00:12:21.510 occurs when we do this. 00:12:21.510 --> 00:12:26.100 Because sometimes we think of the noise as having variance N 00:12:26.100 --> 00:12:29.720 sub 0 over 2 in each real and imaginary degree of freedom. 00:12:29.720 --> 00:12:33.310 And therefore N sub 0 in a complex degree of freedom. 00:12:33.310 --> 00:12:37.560 And sometimes we think of it as having variance N sub 0 W 00:12:37.560 --> 00:12:39.930 Where does that difference come from? 00:12:39.930 --> 00:12:44.060 It's this infernal problem of the sampling theorem being so 00:12:44.060 --> 00:12:47.810 critical in most of the models that we talk about. 00:12:47.810 --> 00:12:51.920 OK because when you use the sampling theorem, the sinc x 00:12:51.920 --> 00:12:56.080 over x waveforms that we use are not orthonormal, they're 00:12:56.080 --> 00:12:57.700 orthogonal. 00:12:57.700 --> 00:13:03.390 And this factor of W appears exactly because of that. 00:13:03.390 --> 00:13:08.240 They appear because the magnitude of the signal is a, 00:13:08.240 --> 00:13:10.690 and the energy and the power in the 00:13:10.690 --> 00:13:13.020 signal is then a-squared. 00:13:13.020 --> 00:13:13.910 OK. 00:13:13.910 --> 00:13:17.050 In this case the power in the signal is not quite a-squared 00:13:17.050 --> 00:13:21.950 because we only send energy in one or the other of alternate 00:13:21.950 --> 00:13:23.150 degrees of freedom. 00:13:23.150 --> 00:13:31.620 So therefore, if we look at a time one second, we get W 00:13:31.620 --> 00:13:33.810 complex degrees of freedom to use. 00:13:33.810 --> 00:13:41.990 We only send energy in half of those so that the actual power 00:13:41.990 --> 00:13:47.290 that we're sending is a-squared divided by 2. 00:13:47.290 --> 00:13:47.730 OK. 00:13:47.730 --> 00:13:50.810 Because of that, when we normalize the noise the same 00:13:50.810 --> 00:13:55.090 way the signal is normalized, we get this 00:13:55.090 --> 00:13:57.600 variance W N sub 0. 00:13:57.600 --> 00:14:00.910 If you're confused by that, everyone is confused by it. 00:14:00.910 --> 00:14:04.220 Everyone I know, when they go through calculations like 00:14:04.220 --> 00:14:07.410 this, they always start out with some arbitrary fudge 00:14:07.410 --> 00:14:08.840 factor like this. 00:14:08.840 --> 00:14:12.120 And after they get all done, they think it through or more 00:14:12.120 --> 00:14:14.785 likely they look it up in a book to see what somebody else 00:14:14.785 --> 00:14:16.390 has gotten. 00:14:16.390 --> 00:14:19.100 And then they sweat about it a little bit, and they finally 00:14:19.100 --> 00:14:21.430 decide what it ought to be. 00:14:21.430 --> 00:14:23.190 And that's just the way it is. 00:14:23.190 --> 00:14:27.710 It's the problem of having both the sampling theorem and 00:14:27.710 --> 00:14:29.800 orthonormal waveforms sitting around. 00:14:29.800 --> 00:14:34.400 It's also the problem of multiplying the power by 2 as 00:14:34.400 --> 00:14:36.110 soon as we go to passband. 00:14:36.110 --> 00:14:40.370 Because both of those things together generate all of this 00:14:40.370 --> 00:14:41.090 difficulty. 00:14:41.090 --> 00:14:43.710 But anyway, this is the way it is. 00:14:43.710 --> 00:14:47.820 And the important thing for us is that what we can have now 00:14:47.820 --> 00:14:49.950 is under these two hypotheses. 00:14:49.950 --> 00:14:53.300 We just have two Gaussian random variables, complex 00:14:53.300 --> 00:14:55.030 Gaussian random variables. 00:14:55.030 --> 00:14:59.690 And in one case, the larger mean square value is in one. 00:14:59.690 --> 00:15:01.500 And in the other case the larger mean square 00:15:01.500 --> 00:15:02.910 value is in the other. 00:15:09.900 --> 00:15:10.380 OK. 00:15:10.380 --> 00:15:12.470 So just reviewing that. 00:15:12.470 --> 00:15:17.450 If H is equal to zero, V sub 0 and V sub 1 are these complex 00:15:17.450 --> 00:15:19.250 Gaussian random variables. 00:15:19.250 --> 00:15:23.590 If H is equal to one, then we have this set of Gaussian 00:15:23.590 --> 00:15:25.190 random variables. 00:15:25.190 --> 00:15:29.560 The probability density of V sub 0 and V sub 1, and now 00:15:29.560 --> 00:15:34.460 it's more convenient to use the real and imaginary parts 00:15:34.460 --> 00:15:36.230 for the Gaussian density. 00:15:36.230 --> 00:15:40.260 Anytime you're working problems of this type, try 00:15:40.260 --> 00:15:45.300 both densities using real and imaginary parts, and using 00:15:45.300 --> 00:15:49.300 magnitude in phase, and see which one is easier. 00:15:49.300 --> 00:15:52.400 Here it turns out that the easiest thing is just to use 00:15:52.400 --> 00:15:56.230 the ordinary conventional density over real and 00:15:56.230 --> 00:15:58.060 imaginary parts. 00:15:58.060 --> 00:16:01.450 And what we wind up with is this Gaussian density. 00:16:04.130 --> 00:16:10.190 On V sub 0 the density is V sub 0 squared divided by the 00:16:10.190 --> 00:16:13.440 variance a-squared plus W N sub 0. 00:16:13.440 --> 00:16:18.760 And on V sub 1, it's this Gaussian density V sub 1 00:16:18.760 --> 00:16:21.250 squared divided by W N sub 0. 00:16:21.250 --> 00:16:23.920 Just because here we have this variance. 00:16:23.920 --> 00:16:26.330 Here we have this variance. 00:16:26.330 --> 00:16:29.910 OK, on the alternative hypothesis when H is equal to 00:16:29.910 --> 00:16:33.880 one, you have the same thing but the denominators are 00:16:33.880 --> 00:16:35.010 switched around. 00:16:35.010 --> 00:16:38.250 When you take the likelihood ratio, you want to take the 00:16:38.250 --> 00:16:41.660 ratio of this, to the ratio of this. 00:16:41.660 --> 00:16:44.420 If you look at it and you take the logarithm of that, you're 00:16:44.420 --> 00:16:46.650 taking the ratio of this to this. 00:16:46.650 --> 00:16:49.140 Incidentally the coefficient here, you could write it out 00:16:49.140 --> 00:16:50.320 if you want to. 00:16:50.320 --> 00:16:53.630 It's 1 over the square root of blah, times 1 over the square 00:16:53.630 --> 00:16:54.700 root of blah. 00:16:54.700 --> 00:16:58.350 But if you recognize that the coefficient here has to be the 00:16:58.350 --> 00:17:00.550 same as the coefficient here, you don't have 00:17:00.550 --> 00:17:01.850 to worry about it. 00:17:01.850 --> 00:17:05.320 So when you take the log likelihood ratio, you get this 00:17:05.320 --> 00:17:06.700 divided by this. 00:17:06.700 --> 00:17:08.790 You have the same form in both cases. 00:17:08.790 --> 00:17:13.050 In one case, you have this term minus this term and this 00:17:13.050 --> 00:17:14.840 term minus this term. 00:17:14.840 --> 00:17:24.340 And the other case well, for V sub 0, you have this term 00:17:24.340 --> 00:17:25.480 minus this term. 00:17:25.480 --> 00:17:28.830 And for V sub 1 you have this term minus this term. 00:17:28.830 --> 00:17:31.440 Because of the symmetry between the two, this just 00:17:31.440 --> 00:17:34.230 comes out to V sub 0 squared minus V sub 1 00:17:34.230 --> 00:17:36.010 squared times a-squared. 00:17:36.010 --> 00:17:39.040 And when you do the algebra, the denominator is a-squared 00:17:39.040 --> 00:17:43.330 plus W N sub 0 times W N sub 0. 00:17:43.330 --> 00:17:43.850 OK. 00:17:43.850 --> 00:17:48.600 What do we do for making a maximum likelihood decision? 00:17:48.600 --> 00:17:53.150 Maximum likelihood is map when the threshold is equal to 1, 00:17:53.150 --> 00:17:55.530 which is when the logarithm of the correct 00:17:55.530 --> 00:17:57.510 threshold is equal 0. 00:17:57.510 --> 00:18:02.040 Which says that you take this quantity, and if it's 00:18:02.040 --> 00:18:07.550 nonnegative, you choose H equals zero. 00:18:07.550 --> 00:18:10.490 And if it's negative, you choose H equals one. 00:18:10.490 --> 00:18:15.200 Which says you compare V sub 0 squared and V sub 1 squared. 00:18:15.200 --> 00:18:18.700 And whichever one is larger, that's the one you choose. 00:18:18.700 --> 00:18:22.660 And if you go back and look at the problem, it's pretty 00:18:22.660 --> 00:18:25.650 obvious that that's what you want to do anyway. 00:18:25.650 --> 00:18:27.770 I mean you'd be very, very surprised when you're 00:18:27.770 --> 00:18:31.280 comparing two Gaussian random variables where one of them 00:18:31.280 --> 00:18:33.410 has a larger variance than the other. 00:18:33.410 --> 00:18:35.490 And on the other hypothesis, the absolon 00:18:35.490 --> 00:18:38.650 has the larger variance. 00:18:38.650 --> 00:18:41.950 If you came up with any rule other than to take the 00:18:41.950 --> 00:18:46.360 magnitude squares and to then compare those two magnitude 00:18:46.360 --> 00:18:49.410 squares, you would go back and look at the problem again 00:18:49.410 --> 00:18:52.810 realizing you must have done something wrong. 00:18:52.810 --> 00:18:57.960 But anyway when you deal with problems like this, I advise 00:18:57.960 --> 00:19:01.430 you to take log likelihood ratio anyway. 00:19:01.430 --> 00:19:04.800 Because every once in awhile you find something which comes 00:19:04.800 --> 00:19:07.430 out in a somewhat peculiar way. 00:19:07.430 --> 00:19:10.370 But anyway, here there's nothing peculiar. 00:19:10.370 --> 00:19:12.950 So what we have to do now is to find the 00:19:12.950 --> 00:19:14.680 probability of error. 00:19:14.680 --> 00:19:16.480 Now what's the probability of error? 00:19:24.990 --> 00:19:34.250 OK if we actually transmit zero, then V sub 0 squared is 00:19:34.250 --> 00:19:36.160 exponential. 00:19:36.160 --> 00:19:42.600 It's exponential with this mean. 00:19:42.600 --> 00:19:45.830 Namely this is the mean of V sub 0 squared. 00:19:45.830 --> 00:19:48.710 And V sub 1 is exponential with this mean. 00:19:48.710 --> 00:19:52.340 In other words, this is a big exponential. 00:19:52.340 --> 00:19:54.310 And this is a little exponential. 00:19:54.310 --> 00:19:57.370 The two of them have probability densities that 00:19:57.370 --> 00:19:58.620 look like this. 00:20:01.170 --> 00:20:02.610 This is not going to work. 00:20:07.550 --> 00:20:11.620 The big one has a probability density that looks like this. 00:20:14.710 --> 00:20:16.680 And the little one-- 00:20:16.680 --> 00:20:19.220 this is big-- 00:20:19.220 --> 00:20:21.490 and the little one has a probability density 00:20:21.490 --> 00:20:23.140 that looks like this. 00:20:23.140 --> 00:20:28.190 And what you want to do is to subtract a random variable 00:20:28.190 --> 00:20:30.250 with this density from a random 00:20:30.250 --> 00:20:33.140 variable with this density. 00:20:33.140 --> 00:20:37.630 So you're convolving two exponential densities with 00:20:37.630 --> 00:20:39.300 each other. 00:20:39.300 --> 00:20:42.860 And unfortunately, you're taking the differences of two. 00:20:42.860 --> 00:20:47.040 So you're convolving the negatives of this with this. 00:20:47.040 --> 00:20:48.660 And then you have to integrate the thing. 00:20:48.660 --> 00:20:53.290 And it's just something you have to do. 00:20:53.290 --> 00:21:01.080 And the answer is, the probability of error is then 2 00:21:01.080 --> 00:21:05.570 plus a-squared over W N sub 0 to the minus 1. 00:21:05.570 --> 00:21:08.250 OK that is really an awful result. 00:21:08.250 --> 00:21:12.090 Because that says that if you increase the energy that 00:21:12.090 --> 00:21:15.910 you're using, the probability of error goes down very, very, 00:21:15.910 --> 00:21:17.390 very slowly. 00:21:17.390 --> 00:21:20.320 And if you look at this picture you think about it a 00:21:20.320 --> 00:21:23.490 little bit, it should be clear that that's the only thing 00:21:23.490 --> 00:21:26.050 that can happen. 00:21:26.050 --> 00:21:26.420 OK. 00:21:26.420 --> 00:21:30.660 Because if you increase a-squared a little bit, it's 00:21:30.660 --> 00:21:32.970 not going to save you much here. 00:21:32.970 --> 00:21:35.670 Because when you have a bigger a-squared, it's just going to 00:21:35.670 --> 00:21:38.570 move down the value of g bar that's 00:21:38.570 --> 00:21:39.960 going to give you trouble. 00:21:39.960 --> 00:21:44.900 Namely when you double a, the value of magnitude of g that 00:21:44.900 --> 00:21:48.130 gives you trouble just goes down by a factor of two. 00:21:48.130 --> 00:21:52.850 When that goes down by a factor two, this bad part of 00:21:52.850 --> 00:21:57.330 the curve just goes down in a quadratic way. 00:21:57.330 --> 00:22:01.400 Well that's what this is telling us. 00:22:01.400 --> 00:22:03.730 OK. 00:22:03.730 --> 00:22:08.010 I mean the thing that we see is a quadratic and a. 00:22:08.010 --> 00:22:10.590 So we're sort of assured that we're doing the right thing. 00:22:10.590 --> 00:22:13.750 And we're sort of also assured that the reason why this 00:22:13.750 --> 00:22:17.370 result is so awful, is just that sometimes the fading is 00:22:17.370 --> 00:22:21.640 so bad there's nothing you can do about it. 00:22:21.640 --> 00:22:23.790 OK now the signal power as we said before is 00:22:23.790 --> 00:22:25.150 a-squared over 2. 00:22:25.150 --> 00:22:27.040 Since half the inputs are zero. 00:22:27.040 --> 00:22:29.710 So we can put twice as much energy into the 00:22:29.710 --> 00:22:31.440 ones that are non-zero. 00:22:31.440 --> 00:22:35.520 And therefore when you put this in terms of the average 00:22:35.520 --> 00:22:39.240 signal energy that you're sending, what we get is E sub 00:22:39.240 --> 00:22:41.070 b over N sub 0. 00:22:41.070 --> 00:22:42.020 OK. 00:22:42.020 --> 00:22:46.550 So that again says exactly the same thing that this does. 00:22:46.550 --> 00:22:51.060 It's worthwhile keeping both of these notions around, 00:22:51.060 --> 00:22:55.460 because we have done something kind of peculiar here. 00:22:55.460 --> 00:22:57.590 I should mention it for you. 00:22:57.590 --> 00:23:02.580 As soon as you're looking at a fading channel, the power that 00:23:02.580 --> 00:23:05.380 you're talking about becomes a little peculiar. 00:23:05.380 --> 00:23:07.030 Because remember when we were looking 00:23:07.030 --> 00:23:09.020 at white noise channels? 00:23:09.020 --> 00:23:12.600 What we were looking at is the power at the receiver, the 00:23:12.600 --> 00:23:16.650 signal power as received at the receiver. 00:23:16.650 --> 00:23:19.890 Now at this point, we still want to talk about E sub b. 00:23:19.890 --> 00:23:23.320 We still want to isolate this problem from the attenuation 00:23:23.320 --> 00:23:26.590 that occurs just because of distance and things like. 00:23:26.590 --> 00:23:31.680 Because of that, when we model g, this original model that we 00:23:31.680 --> 00:23:39.220 use here was a model in which the magnitude 00:23:39.220 --> 00:23:41.840 of g had mean one. 00:23:41.840 --> 00:23:44.500 And we made it have mean one so that the energy 00:23:44.500 --> 00:23:45.560 would come out right. 00:23:45.560 --> 00:23:48.470 Which is another reason why you get confused with these E 00:23:48.470 --> 00:23:50.940 sub b over N sub 0 terms. 00:23:50.940 --> 00:23:57.720 OK so anyway, that's the answer. 00:23:57.720 --> 00:24:03.760 And E sub b is in terms of the received energy using the 00:24:03.760 --> 00:24:06.360 average value of fading. 00:24:06.360 --> 00:24:09.800 OK we next want to look at non-coherent detection. 00:24:09.800 --> 00:24:12.270 Non-coherent detection is another thing that 00:24:12.270 --> 00:24:15.370 communication engineers use all the time, talk 00:24:15.370 --> 00:24:16.760 about all the time. 00:24:16.760 --> 00:24:20.670 And you have to understand what the difference is between 00:24:20.670 --> 00:24:24.060 coherent transmission and incoherent transmission. 00:24:24.060 --> 00:24:27.850 The general idea is that when you're doing incoherent 00:24:27.850 --> 00:24:30.590 detection, you're assuming that you don't know what the 00:24:30.590 --> 00:24:32.590 phase of the channel is. 00:24:32.590 --> 00:24:35.960 And somehow you want to do your detection without knowing 00:24:35.960 --> 00:24:37.520 that phase. 00:24:37.520 --> 00:24:39.870 The difference between Rayleigh fading on this kind 00:24:39.870 --> 00:24:43.990 of channel and incoherent detection, is that with 00:24:43.990 --> 00:24:48.140 incoherent detection the receiver is assumed to know 00:24:48.140 --> 00:24:52.200 what the magnitude of the channel is, but not the phase. 00:24:52.200 --> 00:24:55.640 It's harder to measure the phase of the channel than it 00:24:55.640 --> 00:24:57.060 is the measure of the magnitude. 00:24:57.060 --> 00:25:01.010 Because the phase changes very, very fast. 00:25:01.010 --> 00:25:03.810 If you look at these equations we have for what the response 00:25:03.810 --> 00:25:08.610 of the channel is, you see the phase changing many, many 00:25:08.610 --> 00:25:13.240 times during the time where the amplitude of the fading 00:25:13.240 --> 00:25:15.560 changes by just a little bit. 00:25:15.560 --> 00:25:20.100 So a very common assumption that people make when trying 00:25:20.100 --> 00:25:23.910 to do detection is that it's incoherent. 00:25:23.910 --> 00:25:27.250 Partly, people get used to analyzing incoherent 00:25:27.250 --> 00:25:29.480 communication. 00:25:29.480 --> 00:25:31.450 And I've seen this so many times. 00:25:31.450 --> 00:25:34.740 And they insist on building communication systems using 00:25:34.740 --> 00:25:36.360 incoherent detection. 00:25:36.360 --> 00:25:38.300 They will swear up and down there's no way you 00:25:38.300 --> 00:25:39.740 can measure the phase. 00:25:39.740 --> 00:25:42.840 And what they're really saying is that's the only kind of 00:25:42.840 --> 00:25:45.250 communication they understand. 00:25:45.250 --> 00:25:48.090 And because that's the only thing they understand, they 00:25:48.090 --> 00:25:51.260 become very, very upset if anyone suggests that you ought 00:25:51.260 --> 00:25:53.050 to try to measure the phase. 00:25:53.050 --> 00:25:56.830 But that's a tale for another day. 00:26:00.310 --> 00:26:04.050 OK so now we want to look at the case where we're assuming 00:26:04.050 --> 00:26:07.560 that we know the magnitude of the channel. 00:26:07.560 --> 00:26:11.570 It's just some quantity that we'll call g tilde. 00:26:11.570 --> 00:26:16.390 We're assuming that the same magnitude occurs both on U sub 00:26:16.390 --> 00:26:17.810 0 and U sub 1. 00:26:17.810 --> 00:26:20.430 We're going to use the same transmission system that we 00:26:20.430 --> 00:26:24.190 used before, namely pulse-position modulation. 00:26:24.190 --> 00:26:28.340 We'll either put our energy in U sub 0 or we'll put our 00:26:28.340 --> 00:26:29.750 energy in U sub 1. 00:26:29.750 --> 00:26:31.830 We'll try to detect what's going on. 00:26:31.830 --> 00:26:34.770 But we just give the detector this little extra amount of 00:26:34.770 --> 00:26:37.920 ability of knowing what the channel is. 00:26:37.920 --> 00:26:42.050 I'm going to talk more later about how you can use this 00:26:42.050 --> 00:26:45.190 knowledge of what the channel is, and how you can measure 00:26:45.190 --> 00:26:46.050 what the channel is. 00:26:46.050 --> 00:26:48.710 But for now we just assume that we know it. 00:26:48.710 --> 00:26:51.450 So the phase is random and independent 00:26:51.450 --> 00:26:53.050 of everything else. 00:26:53.050 --> 00:27:00.400 So under hypothesis H equals zero, we have the output of 00:27:00.400 --> 00:27:01.060 the channel. 00:27:01.060 --> 00:27:07.690 And times 0 is whatever input level we put in a, times what 00:27:07.690 --> 00:27:12.190 the channel does to us, times e to this random phase. 00:27:12.190 --> 00:27:13.760 And V sub 1 it's just-- 00:27:19.650 --> 00:27:23.320 plus Z sub 0. 00:27:23.320 --> 00:27:28.050 And in the other case we have V sub 1 equals Z sub 1. 00:27:28.050 --> 00:27:32.355 And under the other hypothesis V sub 1 is this input with a 00:27:32.355 --> 00:27:37.820 random phase but a known magnitude and again, a 00:27:37.820 --> 00:27:39.040 Gaussian random variable. 00:27:39.040 --> 00:27:42.250 Phases are independent of the hypothesis. 00:27:42.250 --> 00:27:44.000 The phases are independent of the 00:27:44.000 --> 00:27:45.660 magnitudes which are known. 00:27:45.660 --> 00:27:48.470 The phases are independent of everything and therefore, we 00:27:48.470 --> 00:27:51.130 just want to forget about them. 00:27:51.130 --> 00:27:54.350 So the question is, how do we make a maximum likelihood 00:27:54.350 --> 00:27:57.510 decision on this problem? 00:27:57.510 --> 00:27:59.080 Well you look at the problem. 00:27:59.080 --> 00:28:03.510 And for the same reason as before you say, it's obvious 00:28:03.510 --> 00:28:06.070 how to make a maximum likelihood decision just from 00:28:06.070 --> 00:28:09.450 all the symmetry that you have. 00:28:09.450 --> 00:28:12.770 If the magnitude of V sub 0 is bigger than the magnitude of V 00:28:12.770 --> 00:28:16.000 sub 1, V sub 0 corresponds to this little bit of extra 00:28:16.000 --> 00:28:17.960 energy that you have. 00:28:17.960 --> 00:28:22.010 So if V sub 0, the magnitude of V sub 0 is positive, is 00:28:22.010 --> 00:28:24.790 bigger than the magnitude of V sub 1, you want to 00:28:24.790 --> 00:28:26.960 choose H equals 0. 00:28:26.960 --> 00:28:30.740 And alternatively you'll want to choose H equals 1. 00:28:30.740 --> 00:28:33.250 It's obvious right? 00:28:33.250 --> 00:28:37.040 I've tried for years to find a way to prove that. 00:28:37.040 --> 00:28:40.280 And the only way I can prove it is by going into Bessel 00:28:40.280 --> 00:28:43.340 functions which is the way that everybody else proves it. 00:28:43.340 --> 00:28:46.040 And this seems like absolute foolishness to me. 00:28:46.040 --> 00:28:48.930 And if any of you can find a way to do this, I would be 00:28:48.930 --> 00:28:51.000 delighted to hear it. 00:28:51.000 --> 00:28:55.120 I will be in great admiration of you. 00:28:55.120 --> 00:28:57.020 Because I'm sure there has to be an easy way to 00:28:57.020 --> 00:28:58.430 look at this problem. 00:28:58.430 --> 00:29:00.860 And I just can't find it. 00:29:00.860 --> 00:29:03.440 OK anyway, we're not going to worry about all these Bessel 00:29:03.440 --> 00:29:08.670 functions, because that's just arithmetic in a sense. 00:29:08.670 --> 00:29:12.040 So we're just going to say well it can be proven using 00:29:12.040 --> 00:29:13.380 all of this machinery. 00:29:13.380 --> 00:29:16.610 So what we really want to find is what is the probability of 00:29:16.610 --> 00:29:19.420 error when we make that decision. 00:29:19.420 --> 00:29:23.180 And when we make that decision, namely what we're 00:29:23.180 --> 00:29:26.740 looking for is the probability of this magnitude then, is 00:29:26.740 --> 00:29:34.370 bigger than this magnitude when H equals one is the 00:29:34.370 --> 00:29:35.420 correct hypothesis. 00:29:35.420 --> 00:29:38.440 Because that's the probability of error then. 00:29:38.440 --> 00:29:40.130 So you have these two different terms. 00:29:40.130 --> 00:29:43.230 You just go through all of the junk that's in the appendix to 00:29:43.230 --> 00:29:46.080 the notes we passed out last time. 00:29:46.080 --> 00:29:48.080 If you want to go through that, I think it's great. 00:29:48.080 --> 00:29:50.510 It's an interesting analysis. 00:29:50.510 --> 00:29:52.510 Certainly not going to do it now. 00:29:52.510 --> 00:29:55.490 When you get done doing that you find out the probability 00:29:55.490 --> 00:30:00.540 of error is exactly one half times e to the minus a-squared 00:30:00.540 --> 00:30:04.250 times this known magnitude of the channel. 00:30:04.250 --> 00:30:08.670 I mean, a-squared and g tilde have to appear together here. 00:30:08.670 --> 00:30:11.860 OK because what's coming out of the channel, the magnitude 00:30:11.860 --> 00:30:15.370 of what's coming out of the channel without noise is just 00:30:15.370 --> 00:30:17.510 a times g tilde. 00:30:17.510 --> 00:30:19.810 They both come together everywhere. 00:30:19.810 --> 00:30:24.560 And therefore, they have to come together anytime you're 00:30:24.560 --> 00:30:27.860 talking about optimal detection, probability of 00:30:27.860 --> 00:30:29.700 error, or anything else. 00:30:29.700 --> 00:30:31.030 So these two appear together. 00:30:31.030 --> 00:30:34.350 We have the same noise term down here as we had before. 00:30:34.350 --> 00:30:37.650 Because again we're using a sampling theorem analysis and 00:30:37.650 --> 00:30:42.510 the noise in each of these random variables is W N sub 0. 00:30:42.510 --> 00:30:44.630 OK so that's a little surprising that that's what 00:30:44.630 --> 00:30:47.270 the noise is. 00:30:47.270 --> 00:30:52.130 If you knew the phase also, if the detector knew both the 00:30:52.130 --> 00:30:55.200 magnitude and the phase of the channel, it would be the 00:30:55.200 --> 00:30:57.520 conventional Gaussian problem that we've 00:30:57.520 --> 00:31:00.000 analyzed many times before. 00:31:00.000 --> 00:31:04.770 And the solution would be that probability of error is equal 00:31:04.770 --> 00:31:08.420 to Q of a-squared times g tilde squared 00:31:08.420 --> 00:31:11.750 divided by W N sub 0. 00:31:11.750 --> 00:31:13.890 Now if you remember the estimates we've come up with 00:31:13.890 --> 00:31:17.350 and the bounds we've come up with on the Q function, the 00:31:17.350 --> 00:31:21.940 simplest bound that we came up with was this. 00:31:21.940 --> 00:31:25.490 Namely you take this thing, you take one half of it, which 00:31:25.490 --> 00:31:28.070 is the Gaussian density with the coefficient. 00:31:28.070 --> 00:31:29.970 You multiply it by one half. 00:31:29.970 --> 00:31:33.740 So this is the simplest estimate we can get of this. 00:31:33.740 --> 00:31:37.280 On the other hand when this quantity is large, a much 00:31:37.280 --> 00:31:40.930 better estimate of this is to have that estimate which has a 00:31:40.930 --> 00:31:47.030 1 over the square root of pi times W N sub 0 over a-squared 00:31:47.030 --> 00:31:48.560 g tilde squared in it. 00:31:48.560 --> 00:31:50.600 So we have that term extra. 00:31:50.600 --> 00:31:53.520 Which says that when this is large, whenever we're 00:31:53.520 --> 00:31:57.430 communicating at all reasonably, this probability 00:31:57.430 --> 00:32:03.230 of error is much smaller than this probability of error. 00:32:03.230 --> 00:32:06.345 However you talk to any communication engineer, and 00:32:06.345 --> 00:32:09.090 they'll say when you have a good signal noise ratio, 00:32:09.090 --> 00:32:11.130 incoherent detection is virtually as 00:32:11.130 --> 00:32:14.300 good as coherent detection. 00:32:14.300 --> 00:32:15.550 And why did they say that? 00:32:18.640 --> 00:32:22.920 Well it's because the probability of error goes down 00:32:22.920 --> 00:32:24.860 so quickly with energy here. 00:32:24.860 --> 00:32:27.330 It's going down as a square of an exponent. 00:32:27.330 --> 00:32:31.590 Well it's going down as an exponent in the energy. 00:32:31.590 --> 00:32:37.160 The question you want to ask is how much extra energy do I 00:32:37.160 --> 00:32:40.120 have to use? 00:32:40.120 --> 00:32:43.070 If I'm using coherent detection, how much more 00:32:43.070 --> 00:32:47.520 energy does an incoherent detector need at the input in 00:32:47.520 --> 00:32:49.930 order to get the same results? 00:32:49.930 --> 00:32:52.300 And then you see the question is very different. 00:32:52.300 --> 00:32:55.580 Because if I increase this quantity just a little bit, 00:32:55.580 --> 00:33:00.230 this probability of error goes down like a bat. 00:33:00.230 --> 00:33:01.590 OK. 00:33:01.590 --> 00:33:05.100 So what happens then, when you compare these two terms is 00:33:05.100 --> 00:33:09.220 that as the signal to noise ratio gets larger and larger, 00:33:09.220 --> 00:33:12.100 the amount of extra energy you need to make incoherent 00:33:12.100 --> 00:33:16.470 detection work as well as coherent detection goes down 00:33:16.470 --> 00:33:17.720 with 1 over a-squared. 00:33:21.250 --> 00:33:26.280 Which says that these communication engineers who 00:33:26.280 --> 00:33:30.770 swear that they like incoherent detection in fact, 00:33:30.770 --> 00:33:33.500 have something on their side. 00:33:33.500 --> 00:33:35.380 because they don't have to assume so 00:33:35.380 --> 00:33:36.830 much about the channel. 00:33:36.830 --> 00:33:39.620 They have something which is more robust. 00:33:39.620 --> 00:33:43.560 And in fact what's turning out here, is that even though this 00:33:43.560 --> 00:33:47.140 error probability is a little bigger than this error 00:33:47.140 --> 00:33:50.640 probability, there's only a very negligible amount of 00:33:50.640 --> 00:33:54.270 extra dB required to make the two the same. 00:33:54.270 --> 00:33:57.320 So it only costs a little bit of extra energy to be able to 00:33:57.320 --> 00:34:01.710 use incoherent detection instead of coherent detection. 00:34:01.710 --> 00:34:04.640 OK so this is very strange now. 00:34:04.640 --> 00:34:07.810 We have a nice error probability which is almost as 00:34:07.810 --> 00:34:11.140 good as the Gaussian error probability 00:34:11.140 --> 00:34:13.890 using incoherent detection. 00:34:13.890 --> 00:34:18.110 This is assuming that the channel, that the receiver 00:34:18.110 --> 00:34:22.440 knows what g tilde is. 00:34:22.440 --> 00:34:25.450 But now we go back and think about this, and look at our 00:34:25.450 --> 00:34:29.340 detection rule, which is the optimal detection rule. 00:34:29.340 --> 00:34:32.950 And the optimal detection rule is no matter what g tilde 00:34:32.950 --> 00:34:36.700 happens to be, we compare the magnitude of V sub 0 with the 00:34:36.700 --> 00:34:39.640 magnitude of V sub 1. 00:34:39.640 --> 00:34:42.020 In other words, we have analyzed this assuming that we 00:34:42.020 --> 00:34:43.350 know what g tilde is. 00:34:43.350 --> 00:34:45.660 We know what the gain of the channel is. 00:34:45.660 --> 00:34:47.900 But the receiver doesn't pay any attention to it. 00:34:50.420 --> 00:34:56.590 OK so now we have this very peculiar situation where 00:34:56.590 --> 00:35:01.910 incoherent detection with a known value channel is almost 00:35:01.910 --> 00:35:06.150 as good as coherent detection is. 00:35:06.150 --> 00:35:10.180 But at the same time Rayleigh fading gives this awful error 00:35:10.180 --> 00:35:12.800 probability. 00:35:12.800 --> 00:35:16.580 So now you have the final part of the argument take this 00:35:16.580 --> 00:35:22.270 probability of error, multiply it by the probability density 00:35:22.270 --> 00:35:27.690 of g tilde squared, integrate it to find out what the 00:35:27.690 --> 00:35:31.090 average error probability is when we average over the 00:35:31.090 --> 00:35:32.900 channel fading. 00:35:32.900 --> 00:35:34.340 And guess what answer you get? 00:35:37.830 --> 00:35:39.690 Well you ought to be able to guess it if you've looked at 00:35:39.690 --> 00:35:41.050 the homework already. 00:35:41.050 --> 00:35:43.190 Because in the homework you actually go through this 00:35:43.190 --> 00:35:44.260 integration. 00:35:44.260 --> 00:35:47.170 And bingo you get the Rayleigh fading result. 00:35:47.170 --> 00:35:50.510 Which says that the problem with Rayleigh fading is not 00:35:50.510 --> 00:35:52.960 any lack of knowledge about the channel. 00:35:52.960 --> 00:35:57.580 Knowing what the channel is would not give you, even 00:35:57.580 --> 00:36:00.010 knowing what the phase is of the channel would not give you 00:36:00.010 --> 00:36:01.020 a lot of extra help. 00:36:01.020 --> 00:36:03.790 The only help in knowing what the phase is, is to get this 00:36:03.790 --> 00:36:05.680 result instead of this result. 00:36:05.680 --> 00:36:07.870 And even that won't help you much. 00:36:07.870 --> 00:36:13.070 The problem is anytime you're dealing with Rayleigh fading, 00:36:13.070 --> 00:36:17.170 the channel has faded so badly a large fraction of the time, 00:36:17.170 --> 00:36:19.110 that you can't get an acceptable 00:36:19.110 --> 00:36:22.030 probability of error. 00:36:22.030 --> 00:36:23.730 OK so now we have to stop and think. 00:36:23.730 --> 00:36:25.080 What do you do about this? 00:36:34.190 --> 00:36:37.910 Well you have two general kinds of techniques to use at 00:36:37.910 --> 00:36:39.270 this point. 00:36:39.270 --> 00:36:42.330 OK and one of them is to try to measure the 00:36:42.330 --> 00:36:45.490 channel at the receiver. 00:36:45.490 --> 00:36:48.320 You take the measurement of the channel at the receiver. 00:36:48.320 --> 00:36:50.530 You send it to the transmitter. 00:36:50.530 --> 00:36:54.810 And the transmitter then does something to compensate for 00:36:54.810 --> 00:36:56.510 the amount of fading. 00:36:56.510 --> 00:37:00.210 One thing that the transmitter can do is anytime the channel 00:37:00.210 --> 00:37:03.080 is badly faded, it increases the amount of 00:37:03.080 --> 00:37:05.620 power that it's sending. 00:37:05.620 --> 00:37:09.220 That's what typical voice systems do. 00:37:09.220 --> 00:37:11.700 And the other thing that you can do is change the rate at 00:37:11.700 --> 00:37:13.830 which you're transmitting. 00:37:13.830 --> 00:37:16.700 You can do all sorts of things with the transmitter if you 00:37:16.700 --> 00:37:18.180 know what the channel is. 00:37:18.180 --> 00:37:21.300 You can respond to it in various ways. 00:37:21.300 --> 00:37:24.310 And all these different communication systems have 00:37:24.310 --> 00:37:28.250 various ways of dealing with that. 00:37:28.250 --> 00:37:31.780 And we'll talk a little about that on Wednesday when we talk 00:37:31.780 --> 00:37:33.030 about CDMA. 00:37:35.450 --> 00:37:38.270 The other thing you can do about it is use something 00:37:38.270 --> 00:37:40.050 called diversity. 00:37:40.050 --> 00:37:46.270 And the idea of diversity is that instead of sending this 00:37:46.270 --> 00:37:49.580 one bit, trying to use as few degrees of freedom as 00:37:49.580 --> 00:37:54.610 possible, you try to send your bits using as many degrees of 00:37:54.610 --> 00:37:56.320 freedom as possible. 00:37:56.320 --> 00:37:59.560 If you can use a large number of degrees of freedom, and if 00:37:59.560 --> 00:38:02.280 the fading is independent on these different degrees of 00:38:02.280 --> 00:38:05.790 freedom, then in fact you gain something. 00:38:05.790 --> 00:38:09.490 Because instead of having one random variable which can 00:38:09.490 --> 00:38:14.350 totally cripple you, you have lots of random variables. 00:38:14.350 --> 00:38:16.670 And if any one of them is good, you get through. 00:38:16.670 --> 00:38:20.960 So you get a benefit out of diversity. 00:38:20.960 --> 00:38:23.510 OK so that's our next topic. 00:38:27.460 --> 00:38:29.860 Namely, how do you measure the channel? 00:38:29.860 --> 00:38:33.000 Because if you're going to use diversity it's a help to know 00:38:33.000 --> 00:38:34.010 the channel. 00:38:34.010 --> 00:38:37.080 If you're going to use coding, coding is just another way to 00:38:37.080 --> 00:38:39.290 get diversity. 00:38:39.290 --> 00:38:42.180 Again your coding will work better if you know what the 00:38:42.180 --> 00:38:43.950 channel is. 00:38:43.950 --> 00:38:48.590 So somehow we would like to be able to measure the channel 00:38:48.590 --> 00:38:51.480 and send it back to the transmitter if we want to 00:38:51.480 --> 00:38:55.830 alter the power, the rate of the transmitter, and to let 00:38:55.830 --> 00:39:01.780 the receiver use it if the receiver is going to. 00:39:01.780 --> 00:39:06.430 Well we have seen that when you use one bit on just a 00:39:06.430 --> 00:39:10.060 couple of degrees of freedom, knowing what the channel is 00:39:10.060 --> 00:39:11.780 does not do you much help. 00:39:11.780 --> 00:39:15.890 If you use coding or if you use one bit and spread it over 00:39:15.890 --> 00:39:18.920 a large number of degrees of freedom, then knowing what the 00:39:18.920 --> 00:39:21.740 channel is gives you a great deal. 00:39:21.740 --> 00:39:25.240 This is one of the basic confusions that everyone has 00:39:25.240 --> 00:39:27.170 when they deal with Rayleigh fading. 00:39:27.170 --> 00:39:29.400 Because when you look at a Rayleigh faded channel, the 00:39:29.400 --> 00:39:32.580 first thing you analyze is this incredibly small number 00:39:32.580 --> 00:39:34.160 of degrees of freedom. 00:39:34.160 --> 00:39:35.770 And you say wow, that's awful. 00:39:35.770 --> 00:39:39.880 There's no way to deal with that. 00:39:39.880 --> 00:39:42.500 And then you start looking for something. 00:39:42.500 --> 00:39:45.510 And you say, well diversity helps me. 00:39:45.510 --> 00:39:48.140 But in general, this is the general scheme of things that 00:39:48.140 --> 00:39:50.110 we're going to use. 00:39:50.110 --> 00:39:51.020 OK. 00:39:51.020 --> 00:39:54.600 So as we said channel measurement helps if diversity 00:39:54.600 --> 00:39:55.820 is available. 00:39:55.820 --> 00:39:58.940 Why does that help when diversity is available? 00:39:58.940 --> 00:40:02.790 OK, think of sending this one bit. 00:40:02.790 --> 00:40:06.000 You get one reception, and then 00:40:06.000 --> 00:40:07.970 you get another reception. 00:40:07.970 --> 00:40:11.200 And on this reception, you get one amount of fading. 00:40:11.200 --> 00:40:15.470 On this reception you get another amount of fading. 00:40:15.470 --> 00:40:19.350 If I don't know how much fading there is it doesn't 00:40:19.350 --> 00:40:21.010 help me an awful lot. 00:40:21.010 --> 00:40:22.260 It helps me some. 00:40:22.260 --> 00:40:25.300 But if I know that this channel is faded badly and 00:40:25.300 --> 00:40:28.320 this channel is not faded, then I'm going to use what 00:40:28.320 --> 00:40:32.310 comes out here instead of what comes out here. 00:40:32.310 --> 00:40:35.990 And then my detector is going to work much, much better. 00:40:35.990 --> 00:40:40.100 When you look at diversity results, always ask yourself a 00:40:40.100 --> 00:40:42.360 couple of questions. 00:40:42.360 --> 00:40:45.620 Is the detector using knowledge of what the strength 00:40:45.620 --> 00:40:50.420 of the channel is on these two diversity outputs? 00:40:50.420 --> 00:40:53.360 Is the transmitter using it's knowledge of what those 00:40:53.360 --> 00:40:54.430 channels are? 00:40:54.430 --> 00:40:57.640 You get very different results for diversity depending on the 00:40:57.640 --> 00:41:02.670 answers to both of those questions. 00:41:02.670 --> 00:41:08.660 OK, so if you have a multi-tap model for a channel-- 00:41:08.660 --> 00:41:13.000 OK remember the multi-tap models that we came up with. 00:41:13.000 --> 00:41:18.970 We we're looking at transmission using multipath. 00:41:18.970 --> 00:41:22.470 And we had multipath in different ranges of a delay. 00:41:22.470 --> 00:41:29.950 We came up with a model which gave us multiple taps for a 00:41:29.950 --> 00:41:32.440 discrete model of the channel. 00:41:32.440 --> 00:41:35.560 You get a large number of taps if you're using broadband 00:41:35.560 --> 00:41:36.430 communication. 00:41:36.430 --> 00:41:40.210 Because using broadband communication 1 over W becomes 00:41:40.210 --> 00:41:41.250 very small. 00:41:41.250 --> 00:41:45.200 And therefore these ranges of delay become very small. 00:41:45.200 --> 00:41:48.030 And if you're using very narrow band communication, 00:41:48.030 --> 00:41:50.180 that's when you have the flat fading. 00:41:50.180 --> 00:41:54.080 Namely flat fading is not flat fading, it's fading which is 00:41:54.080 --> 00:41:56.740 flat over the bandwidth that you're using. 00:41:56.740 --> 00:41:59.370 So if you use a broader bandwidth and you have 00:41:59.370 --> 00:42:02.580 multiple taps, then these taps are going to be independent of 00:42:02.580 --> 00:42:03.580 each other. 00:42:03.580 --> 00:42:06.170 And you automatically have diversity. 00:42:06.170 --> 00:42:08.520 So the question is how do you use that. 00:42:08.520 --> 00:42:10.300 Well if you're going to use it, you better be able to 00:42:10.300 --> 00:42:12.270 measure it. 00:42:12.270 --> 00:42:16.310 OK so now we're going to try to figure out how to do that 00:42:16.310 --> 00:42:17.560 measurement. 00:42:20.170 --> 00:42:24.600 And the first thing to do is to assume the simplest 00:42:24.600 --> 00:42:26.180 possible thing. 00:42:26.180 --> 00:42:29.610 I mean, suppose you know how many taps the channel has. 00:42:29.610 --> 00:42:32.850 Suppose it has k sub 0 channel taps. 00:42:32.850 --> 00:42:35.400 So the channel looks like this, G sub 0, G sub 00:42:35.400 --> 00:42:38.120 1, and G sub 2. 00:42:38.120 --> 00:42:42.170 You're transmitting a sequence of inputs. 00:42:42.170 --> 00:42:45.960 OK remember all of this stuff came from trying to model a 00:42:45.960 --> 00:42:49.430 channel in terms of discrete inputs, where you're sending 00:42:49.430 --> 00:42:52.810 one input each one over W seconds. 00:42:52.810 --> 00:42:55.270 So you put in a sequence of inputs. 00:42:55.270 --> 00:42:58.510 You have these three different channel taps here. 00:42:58.510 --> 00:43:05.920 And what comes out when you put in a single bit here or a 00:43:05.920 --> 00:43:08.120 single symbol. 00:43:08.120 --> 00:43:11.110 You get something out when this tap right away. 00:43:11.110 --> 00:43:15.430 You get something out here one time unit later. 00:43:15.430 --> 00:43:19.720 You get something out here, one epoch still later. 00:43:19.720 --> 00:43:22.870 So all of these outputs get added up. 00:43:22.870 --> 00:43:32.290 And therefore the output here at time m, is the input at 00:43:32.290 --> 00:43:39.160 time m times this tap, plus the input at time m minus 1 00:43:39.160 --> 00:43:42.800 times this tap, plus the input at time m minus 00:43:42.800 --> 00:43:44.740 2 times this tap. 00:43:44.740 --> 00:43:48.370 Because it takes these inputs that long to go through there. 00:43:48.370 --> 00:43:52.360 All this is is just digital convolution, OK. 00:43:52.360 --> 00:43:54.370 I'm just drawing it out in the figure so you see 00:43:54.370 --> 00:43:55.090 what's going on. 00:43:55.090 --> 00:43:57.990 Because otherwise you tend to think everything happens at 00:43:57.990 --> 00:43:59.560 one instant of time. 00:43:59.560 --> 00:44:03.080 Then we're adding this white Gaussian noise. 00:44:03.080 --> 00:44:05.230 When we're talking about digital systems, white 00:44:05.230 --> 00:44:08.600 Gaussian noise just means that each of these random variables 00:44:08.600 --> 00:44:11.050 are independent at each other random variable. 00:44:11.050 --> 00:44:12.760 They all have the same variance. 00:44:12.760 --> 00:44:14.900 And the real parts and imaginary parts 00:44:14.900 --> 00:44:15.960 have the same variance. 00:44:15.960 --> 00:44:17.310 And they're independent of each other. 00:44:17.310 --> 00:44:20.660 Namely these are all normal random variables. 00:44:20.660 --> 00:44:27.270 Since we're sending a, or minus a, or something with 00:44:27.270 --> 00:44:30.960 magnitude a, we want to divide by a out here, if we want to 00:44:30.960 --> 00:44:34.940 figure out anything about what these taps are. 00:44:34.940 --> 00:44:39.550 OK so suppose that what we send now is a bunch of zeros, 00:44:39.550 --> 00:44:43.240 followed by a single input, followed by a bunch of zeros. 00:44:43.240 --> 00:44:44.730 What comes out? 00:44:44.730 --> 00:44:48.320 Well the thing that comes out is at the point that this big 00:44:48.320 --> 00:44:57.630 input comes in, we get a G sub 0 out at the time 00:44:57.630 --> 00:44:58.960 that you put in a. 00:44:58.960 --> 00:45:01.430 I mean we're leaving out propagation delay here. 00:45:01.430 --> 00:45:04.900 We got a times G sub 1, the next epoch. 00:45:04.900 --> 00:45:08.370 Then we get a times G sub 2, the next epoch. 00:45:08.370 --> 00:45:11.770 And by that time the input is completely out of the filter. 00:45:11.770 --> 00:45:14.270 And we get zeros after that. 00:45:14.270 --> 00:45:19.410 So if you put in a bunch of zeros and then a single a, you 00:45:19.410 --> 00:45:21.550 got a nice measurement of the channel. 00:45:21.550 --> 00:45:25.050 There's Gaussian noise added to each of these inputs. 00:45:25.050 --> 00:45:28.710 But in fact you do get a reading of 00:45:28.710 --> 00:45:29.960 each channel output. 00:45:29.960 --> 00:45:34.980 When you divide these by the a here, then you get something 00:45:34.980 --> 00:45:42.410 which is a measurement of the appropriate tap G plus 00:45:42.410 --> 00:45:44.690 Gaussian noise on it. 00:45:44.690 --> 00:45:47.930 OK now you try to make an estimation from this. 00:45:47.930 --> 00:45:50.040 And the trouble is we don't want to say much about 00:45:50.040 --> 00:45:52.020 estimation theory. 00:45:52.020 --> 00:45:55.430 But in fact the notes gives you a very brief introduction 00:45:55.430 --> 00:45:57.690 into estimation. 00:45:57.690 --> 00:46:00.080 There are two well-known kinds of estimation. 00:46:00.080 --> 00:46:03.550 One of them is maximum likelihood estimation. 00:46:03.550 --> 00:46:07.530 And the other one is minimum mean square error estimation. 00:46:07.530 --> 00:46:11.510 Maximum likelihood estimation is in fact exactly the same as 00:46:11.510 --> 00:46:13.630 maximum likelihood detection. 00:46:13.630 --> 00:46:16.000 Namely you look at the likelihoods which is the 00:46:16.000 --> 00:46:21.910 probabilities of the outputs given the inputs. 00:46:21.910 --> 00:46:23.630 And what's the input in this problem? 00:46:27.430 --> 00:46:30.140 The input is these channel variables. 00:46:30.140 --> 00:46:33.840 Because that's the thing we're trying to measure in this 00:46:33.840 --> 00:46:34.860 measurement problem. 00:46:34.860 --> 00:46:37.310 We assume that the probing signal is known. 00:46:37.310 --> 00:46:40.035 It's just a bunch of zeros, followed by a, followed by a 00:46:40.035 --> 00:46:41.100 bunch of zeros. 00:46:41.100 --> 00:46:42.220 So we know that. 00:46:42.220 --> 00:46:45.070 We're trying to estimate these things. 00:46:45.070 --> 00:46:48.020 So these are the variables that we're trying to estimate. 00:46:48.020 --> 00:46:50.880 So we try to find the probability density of the 00:46:50.880 --> 00:46:54.810 output conditional on the knowledge of G sub 0. 00:46:54.810 --> 00:46:58.560 Which is just the Gaussian density shifted to 00:46:58.560 --> 00:47:00.380 a times G sub 0. 00:47:04.130 --> 00:47:11.580 You then look at the maximum likelihood estimate of G. So 00:47:11.580 --> 00:47:17.120 you're looking at the value you can put in to maximize 00:47:17.120 --> 00:47:25.560 this estimate which comes out here as a times G sub 2. 00:47:25.560 --> 00:47:29.020 And then at this appropriate time, you're looking at G sub 00:47:29.020 --> 00:47:32.690 2 here, plus a noise random variable. 00:47:32.690 --> 00:47:36.160 And since the noise is zero mean, this quantity here is in 00:47:36.160 --> 00:47:39.570 fact the best estimate in terms of the maximum 00:47:39.570 --> 00:47:41.260 likelihood that you can get. 00:47:41.260 --> 00:47:44.395 If you assume that this is a Gaussian random variable and 00:47:44.395 --> 00:47:47.520 this is a Gaussian random variable, you can solve a 00:47:47.520 --> 00:47:50.310 minimum mean square error estimation problem. 00:47:50.310 --> 00:47:55.210 It's much like the map problem except these random variables 00:47:55.210 --> 00:47:57.770 are all continuous here. 00:47:57.770 --> 00:48:01.250 But it's a little different from the map problem in the 00:48:01.250 --> 00:48:05.400 sense that you can't have equally likely inputs where 00:48:05.400 --> 00:48:07.570 you have a continuous random variable. 00:48:07.570 --> 00:48:09.700 You make them all equally probable. 00:48:09.700 --> 00:48:11.990 The only possible value you can have is zero. 00:48:11.990 --> 00:48:13.840 Because it has to extend forever. 00:48:13.840 --> 00:48:19.040 So anyway, maximum likelihood detection is just normalize 00:48:19.040 --> 00:48:22.820 what you get so that in the absence of Gaussian noise, you 00:48:22.820 --> 00:48:25.120 would get the variable you're looking for. 00:48:25.120 --> 00:48:26.720 And then ignore the Gaussian noise, 00:48:26.720 --> 00:48:28.470 and that's your estimate. 00:48:28.470 --> 00:48:29.050 OK. 00:48:29.050 --> 00:48:32.630 If you want to do this and you want to use the strategy, it 00:48:32.630 --> 00:48:34.860 looks like a very nice strategy. 00:48:34.860 --> 00:48:38.350 But what's the problem in it? 00:48:38.350 --> 00:48:41.940 If this sequence is somewhat longer, you need a whole lot 00:48:41.940 --> 00:48:45.200 of zeros in between each probing signal. 00:48:45.200 --> 00:48:48.590 And what that means is you're going to be using your energy 00:48:48.590 --> 00:48:52.550 and clumping it all up into the small number 00:48:52.550 --> 00:48:53.920 of degrees of freedom. 00:48:53.920 --> 00:48:57.590 Which means you're going to be sending a lot of energy at one 00:48:57.590 --> 00:48:59.990 instant of time and then nothing for a long period of 00:48:59.990 --> 00:49:03.100 time, then a very big signal for awhile then nothing for a 00:49:03.100 --> 00:49:06.630 long period of time, and so forth. 00:49:06.630 --> 00:49:11.010 If you do that, the FTC is really going 00:49:11.010 --> 00:49:12.610 to be down on you. 00:49:12.610 --> 00:49:15.970 Because you're not supposed to send too much energy in any 00:49:15.970 --> 00:49:19.570 small amount of time or any small amount of frequency. 00:49:19.570 --> 00:49:21.950 So you're supposed to spread things out a little bit. 00:49:21.950 --> 00:49:24.470 You say OK, that doesn't work too well. 00:49:24.470 --> 00:49:25.960 What am I going to do? 00:49:25.960 --> 00:49:30.100 How can I choose a sequence of input so they have relatively 00:49:30.100 --> 00:49:34.850 constant amplitude, but at the same time so that when I go 00:49:34.850 --> 00:49:38.700 through this kind of filter, I can sort out what's coming 00:49:38.700 --> 00:49:41.290 from here, and what's coming from here, and 00:49:41.290 --> 00:49:43.480 what's coming from here. 00:49:43.480 --> 00:49:46.020 Well it turns out that the answer to that question is to 00:49:46.020 --> 00:49:48.880 use a pseudonoise sequence. 00:49:48.880 --> 00:49:52.540 And the next thing I want to do is to give you some idea 00:49:52.540 --> 00:49:55.530 about why these pseudonoise sequences work. 00:49:58.310 --> 00:50:00.410 OK so we'll think in terms of vectors now. 00:50:04.520 --> 00:50:09.240 OK so we have a vector input, u sub 1, u sub 2, up to u sub 00:50:09.240 --> 00:50:11.210 n, a vector of length n. 00:50:11.210 --> 00:50:15.670 So we're putting these discrete signals in 00:50:15.670 --> 00:50:17.610 one after the other. 00:50:17.610 --> 00:50:19.960 We're passing them through this, which is a digital 00:50:19.960 --> 00:50:21.140 filter now. 00:50:21.140 --> 00:50:26.350 So what comes out here V prime is just a convolution of u and 00:50:26.350 --> 00:50:30.370 G. We then add the noise to it. 00:50:30.370 --> 00:50:32.670 I claim that what we ought to do is use the matched 00:50:32.670 --> 00:50:35.140 filter here to u. 00:50:35.140 --> 00:50:39.950 And if I use a matched filter to u here, that matched 00:50:39.950 --> 00:50:43.830 filter, if I'm using a pseudonoise sequence, is going 00:50:43.830 --> 00:50:48.820 to bingo give me the filter that I started out with, plus 00:50:48.820 --> 00:50:49.910 some noise. 00:50:49.910 --> 00:50:51.780 OK why is that? 00:50:51.780 --> 00:50:55.480 The property that pseudonoise sequences have, if I choose 00:50:55.480 --> 00:51:01.070 each of the inputs to have the magnitude of a, and think of 00:51:01.070 --> 00:51:04.150 it as being real plus or minus a, which is what people 00:51:04.150 --> 00:51:05.620 usually do. 00:51:05.620 --> 00:51:09.090 If you look at the correlation of this sequence, namely the 00:51:09.090 --> 00:51:14.560 correlation of u sub m with the complex conjugate of u sub 00:51:14.560 --> 00:51:20.540 m spaced a little bit, PN sequences have the property 00:51:20.540 --> 00:51:24.720 that this correlation function looks like an impulse. 00:51:24.720 --> 00:51:25.610 OK. 00:51:25.610 --> 00:51:29.290 Now how you find sequences that have that property is 00:51:29.290 --> 00:51:31.060 another question. 00:51:31.060 --> 00:51:32.630 But in fact they do exist. 00:51:32.630 --> 00:51:34.020 There are lots of them. 00:51:34.020 --> 00:51:35.570 They're easy to find. 00:51:38.730 --> 00:51:41.910 And they have this very nice property. 00:51:41.910 --> 00:51:46.550 Another way to say this is that is that the vector u has 00:51:46.550 --> 00:51:49.260 to be orthogonal to all of its shifts. 00:51:49.260 --> 00:51:51.800 That's exactly what this is saying. 00:51:51.800 --> 00:51:55.850 And another way of saying it is that u, if you pass it 00:51:55.850 --> 00:51:58.440 through the matched filter to u-- now remember what a 00:51:58.440 --> 00:52:02.440 matched filter is on an analog waveform. 00:52:02.440 --> 00:52:06.850 You take a waveform, you switch it around in time. 00:52:06.850 --> 00:52:09.200 You take the complex conjugate of it, and 00:52:09.200 --> 00:52:10.900 that's the matched filter. 00:52:10.900 --> 00:52:14.200 And when you convolve u with this matched filter, what it's 00:52:14.200 --> 00:52:18.490 doing is just exactly the same operation of correlation. 00:52:18.490 --> 00:52:22.110 OK in other words, convolution with one of the sequences 00:52:22.110 --> 00:52:26.380 turned around this time, is the same as correlation. 00:52:26.380 --> 00:52:29.460 And most of you have seen that I'm sure. 00:52:29.460 --> 00:52:34.070 So that if we take this matched filter where u tilde 00:52:34.070 --> 00:52:38.180 sub j is equal to the complex conjugate of u at time minus 00:52:38.180 --> 00:52:43.940 j, then I pass u through the filter G. Forget about the 00:52:43.940 --> 00:52:45.480 noise for the time being. 00:52:45.480 --> 00:52:49.900 I then pass it through the matched filter u tilde. 00:52:49.900 --> 00:52:53.870 What I'm going to get out, I claim, is G. And I'll show you 00:52:53.870 --> 00:52:56.270 why that is in just a minute. 00:52:56.270 --> 00:52:59.120 Let make caution you about something. 00:52:59.120 --> 00:53:01.970 Because you can get very confused with this picture. 00:53:01.970 --> 00:53:07.180 Because as soon as I take this input, u 1 up to u sub m. 00:53:07.180 --> 00:53:11.610 This matched filter is going to start responding at time u 00:53:11.610 --> 00:53:12.740 sub minus m. 00:53:12.740 --> 00:53:14.650 And it's going to finish responding a 00:53:14.650 --> 00:53:16.750 time u sub minus 1. 00:53:16.750 --> 00:53:19.580 So it responds before it's hit. 00:53:19.580 --> 00:53:23.310 Which again is this business of thinking of timing at the 00:53:23.310 --> 00:53:27.030 receiver being very much delayed from timing at the 00:53:27.030 --> 00:53:29.020 transmitter. 00:53:29.020 --> 00:53:30.920 Which is a trick that we've always played. 00:53:30.920 --> 00:53:33.710 Which is why we don't have to think of filters as being 00:53:33.710 --> 00:53:34.880 realizable. 00:53:34.880 --> 00:53:37.820 Still in this example, this becomes confusing. 00:53:37.820 --> 00:53:39.070 And I'll show you why in a minute. 00:53:44.280 --> 00:53:49.100 OK so I'm going to assume that I picked a good PN sequence. 00:53:49.100 --> 00:53:53.150 So when I convolve it with its matched filter I essentially 00:53:53.150 --> 00:53:58.170 get an impulse function, namely a discrete impulse. 00:53:58.170 --> 00:54:02.130 Which is the same as saying that u is orthogonal to all of 00:54:02.130 --> 00:54:03.370 its shifts. 00:54:03.370 --> 00:54:04.900 And that's exactly what you what to do. 00:54:04.900 --> 00:54:06.320 You want to think of turning it around 00:54:06.320 --> 00:54:07.490 and passing it through. 00:54:07.490 --> 00:54:10.660 And that's exactly what this is doing. 00:54:10.660 --> 00:54:16.880 OK so we have the output of this filter, which is u 00:54:16.880 --> 00:54:21.660 convolved with G. We're then convolving that with this 00:54:21.660 --> 00:54:24.170 matched filter u tilde. 00:54:24.170 --> 00:54:29.860 And now we use the nice property of convolution, which 00:54:29.860 --> 00:54:32.630 you probably don't think of very often. 00:54:32.630 --> 00:54:38.860 But the nice property that convolution has, is that it's 00:54:38.860 --> 00:54:41.670 both associative and commutative. 00:54:41.670 --> 00:54:42.180 OK. 00:54:42.180 --> 00:54:47.010 And therefore when we look at V prime times u tilde, it's 00:54:47.010 --> 00:54:51.510 the convolution of u with G-- that's what the prime is-- all 00:54:51.510 --> 00:54:54.450 convolved with the matched filter u. 00:54:54.450 --> 00:54:57.600 Because of the associativity and the commutativity, you can 00:54:57.600 --> 00:55:00.760 reverse these two things so you're taking the convolution 00:55:00.760 --> 00:55:03.080 of u with its matched filter. 00:55:03.080 --> 00:55:05.870 When you take the convolution of u with its matched filter, 00:55:05.870 --> 00:55:07.900 you get a delta function. 00:55:07.900 --> 00:55:11.350 And you take a delta function and pass it through G. And 00:55:11.350 --> 00:55:15.570 what comes out is a delta function weighted by a-squared 00:55:15.570 --> 00:55:19.240 n, which is just the energy of what we're putting in. 00:55:19.240 --> 00:55:23.050 OK so that says that if we can find pseudonoise sequences, 00:55:23.050 --> 00:55:23.940 all of this works. 00:55:23.940 --> 00:55:26.400 And it works just dandy. 00:55:26.400 --> 00:55:29.340 If you put noise in, what happens there? 00:55:29.340 --> 00:55:31.860 Well let's analyze the noise separately. 00:55:31.860 --> 00:55:35.890 The noise is going through this matched filter. 00:55:35.890 --> 00:55:41.030 Well if u is a pseudonoise sequence, if it has this nice 00:55:41.030 --> 00:55:45.300 correlation property and you flip it around in time, it's 00:55:45.300 --> 00:55:48.190 going to have the same nice correlation property. 00:55:48.190 --> 00:55:57.020 So that in fact u tilde is going to have the same 00:55:57.020 --> 00:56:01.240 property that it's orthogonal to all of its time shifts. 00:56:01.240 --> 00:56:05.490 If you now look at what happens when you take Z and 00:56:05.490 --> 00:56:10.290 send it through this filter, and you find the covariance 00:56:10.290 --> 00:56:14.270 matrix for Z passed through this filter, what that 00:56:14.270 --> 00:56:18.760 independence gives you is the correlation function is just 00:56:18.760 --> 00:56:22.510 all diagonal, all terms, all the same. 00:56:22.510 --> 00:56:27.770 Which says that all of these terms and this vector here are 00:56:27.770 --> 00:56:29.770 all white Gaussian noise variables. 00:56:29.770 --> 00:56:34.210 So what comes out is the filter plus white noise. 00:56:34.210 --> 00:56:36.710 Which is the same thing that happened when we put in a 00:56:36.710 --> 00:56:40.320 single input with zeros on both sides. 00:56:40.320 --> 00:56:42.500 OK. 00:56:42.500 --> 00:56:47.910 So using a PN sequence works in exactly the same way as 00:56:47.910 --> 00:56:51.380 this very special pseudonoise sequence, which just has one 00:56:51.380 --> 00:56:52.200 input in it. 00:56:52.200 --> 00:56:54.960 Which happens to be a pseudonoise sequence 00:56:54.960 --> 00:56:57.460 in this term also. 00:56:57.460 --> 00:57:00.300 OK so the output then, is going to be a maximum 00:57:00.300 --> 00:57:04.000 likelihood estimate of G. OK, this is the way that people 00:57:04.000 --> 00:57:06.420 typically measure channels. 00:57:06.420 --> 00:57:08.970 They use pseudonoise inputs. 00:57:08.970 --> 00:57:11.890 And the output that comes out, namely the 00:57:11.890 --> 00:57:14.360 output that comes out. 00:57:14.360 --> 00:57:18.510 When we put in a finite duration pseudonoise sequence, 00:57:18.510 --> 00:57:22.860 what we're going to look for is the output at the exact 00:57:22.860 --> 00:57:26.580 instant of the last digit as the input goes in. 00:57:26.580 --> 00:57:30.060 And the output then is G sub 0, followed by G sub 1, 00:57:30.060 --> 00:57:33.600 followed by G sub 2, and then silence. 00:57:33.600 --> 00:57:36.580 So you see nothing coming out until this big burst of 00:57:36.580 --> 00:57:44.760 energy, which is all digits of G. 00:57:44.760 --> 00:57:47.970 OK so now we want to put all of this together into 00:57:47.970 --> 00:57:50.070 something called a rake receiver. 00:57:50.070 --> 00:57:52.820 I wish I could spend more time on the rake receiver because 00:57:52.820 --> 00:57:55.110 it's a really neat thing. 00:57:55.110 --> 00:57:59.630 It was developed in the 50s about the same time that 00:57:59.630 --> 00:58:01.570 information theory was getting developed. 00:58:01.570 --> 00:58:06.770 But it was developed by people who were trying to do radar. 00:58:06.770 --> 00:58:09.110 And at the same time trying to do a little communication 00:58:09.110 --> 00:58:11.230 along with the radar. 00:58:11.230 --> 00:58:14.170 And this was one of the things they came up with. 00:58:14.170 --> 00:58:19.000 So they wanted to measure the channel and make decisions in 00:58:19.000 --> 00:58:21.870 transmitting data both at the same time. 00:58:21.870 --> 00:58:26.340 And the trick here is about the same as the trick we use 00:58:26.340 --> 00:58:30.700 in trying to measure carrier frequency, and make decisions 00:58:30.700 --> 00:58:31.930 at the same time. 00:58:31.930 --> 00:58:34.480 Namely you use the decisions you make 00:58:34.480 --> 00:58:36.000 to measure the frequency. 00:58:36.000 --> 00:58:38.190 You use the frequency that you've measured 00:58:38.190 --> 00:58:40.670 to make future decisions. 00:58:40.670 --> 00:58:43.510 And here, we're going to do exactly the same thing. 00:58:43.510 --> 00:58:44.770 We make decisions. 00:58:44.770 --> 00:58:48.680 We use those decisions as a way of measuring the channel. 00:58:48.680 --> 00:58:51.830 We then use the measurements of the channel to create this 00:58:51.830 --> 00:58:55.210 matched filter G tilde. 00:58:55.210 --> 00:58:58.940 And that's what we're going to use to make the decisions. 00:58:58.940 --> 00:59:03.860 OK if you have two different inputs, I mean here we'll just 00:59:03.860 --> 00:59:05.270 look at binary inputs. 00:59:05.270 --> 00:59:09.370 You take u sub 0 and u sub 1, and you look at what happens 00:59:09.370 --> 00:59:11.070 when you have those two inputs. 00:59:11.070 --> 00:59:14.830 This is just a vector white Gaussian noise problem that we 00:59:14.830 --> 00:59:17.570 looked at in quite a bit of detail when we were studying 00:59:17.570 --> 00:59:19.580 decision theory. 00:59:19.580 --> 00:59:24.460 What we want to do is to look at, I mean if these two 00:59:24.460 --> 00:59:28.540 signals are not antipodal to each other you want to look at 00:59:28.540 --> 00:59:30.650 the mean of them. 00:59:30.650 --> 00:59:34.065 And you'll want to look at u sub 0 minus that mean, and u 00:59:34.065 --> 00:59:37.280 sub 1 minus that mean as two antipodal signals. 00:59:37.280 --> 00:59:41.160 When you go through all of that, you find that the 00:59:41.160 --> 00:59:45.020 maximum likelihood decision is to take the real part of the 00:59:45.020 --> 00:59:53.490 output, of the inner product of the output, with u sub 0 00:59:53.490 --> 00:59:58.026 convolved with g, and the real part of v convolved with u sub 00:59:58.026 --> 00:59:59.510 1 convolved with G. 00:59:59.510 --> 01:00:02.520 OK in other words, what's happening here is that as far 01:00:02.520 --> 01:00:07.340 as anybody is concerned, we're not using u sub 0 and u sub 1 01:00:07.340 --> 01:00:09.330 in this making a decision. 01:00:09.330 --> 01:00:11.420 We know what the channel is. 01:00:11.420 --> 01:00:15.425 And therefore what exists right before the white noise 01:00:15.425 --> 01:00:20.485 is added, is these two signals u sub 0 convolved with g and u 01:00:20.485 --> 01:00:22.170 sub 1 convolved with g. 01:00:22.170 --> 01:00:24.480 So we're doing binary detection on 01:00:24.480 --> 01:00:26.750 those two known signals. 01:00:26.750 --> 01:00:29.610 And we're using the output to try to make the best choice 01:00:29.610 --> 01:00:30.500 between them. 01:00:30.500 --> 01:00:32.670 So this is the thing that we do. 01:00:32.670 --> 01:00:36.300 So we want to use a filter matched to u sub 0 01:00:36.300 --> 01:00:38.040 convolved with g. 01:00:38.040 --> 01:00:41.220 Now how do we build a filter matched to a convolution of 01:00:41.220 --> 01:00:43.140 two things? 01:00:43.140 --> 01:00:45.880 Well we convolve u sub 0 with g. 01:00:45.880 --> 01:00:48.080 And then we turn the thing around. 01:00:48.080 --> 01:00:50.340 And then we see that after turning it around what we've 01:00:50.340 --> 01:00:53.890 gotten is the turned around version of u convolved with 01:00:53.890 --> 01:00:56.460 the turned around version of g. 01:00:56.460 --> 01:00:57.770 I mean write it down and you'll see that 01:00:57.770 --> 01:00:59.850 that's what you have. 01:00:59.850 --> 01:01:08.120 So what you wind up with is the following figure. 01:01:08.120 --> 01:01:11.860 You either send u sub 0 or you send u sub 1. 01:01:11.860 --> 01:01:14.450 This is a way to send one binary digit. 01:01:14.450 --> 01:01:20.060 We're sending it by using these long PN sequences now. 01:01:20.060 --> 01:01:28.190 If u sub 0 goes through g, we got a V prime out, which is 01:01:28.190 --> 01:01:30.880 the output before noise is added. 01:01:30.880 --> 01:01:34.980 We then add noise so we get. 01:01:34.980 --> 01:01:38.610 And then we pass to try to detect whether 01:01:38.610 --> 01:01:40.530 this or this is true. 01:01:40.530 --> 01:01:48.000 We take this output V. We convolve it with u sub 1 01:01:48.000 --> 01:01:52.060 convolved with g, and with u sub 0 convolved with g. 01:01:52.060 --> 01:01:55.380 Now you'll all say I'm wasting stuff here. 01:01:55.380 --> 01:01:59.490 Because I could just put the g over here and then follow it 01:01:59.490 --> 01:02:02.850 with u sub 1 or u sub 0. 01:02:02.850 --> 01:02:04.090 Be patient for a little bit. 01:02:04.090 --> 01:02:05.390 I want to put both of them in. 01:02:05.390 --> 01:02:07.280 And I want to put them in this order. 01:02:07.280 --> 01:02:10.580 And then I make a decision here. 01:02:10.580 --> 01:02:14.520 OK well here comes the clincher to the argument. 01:02:14.520 --> 01:02:17.500 Look at what happens right there. 01:02:17.500 --> 01:02:24.120 If I forget about this and I forget about this, what I get 01:02:24.120 --> 01:02:26.440 here is u sub 0 coming in. 01:02:26.440 --> 01:02:28.900 It's going through the filter g. 01:02:28.900 --> 01:02:31.510 It has white noise added to it. 01:02:31.510 --> 01:02:35.520 It goes through the matched filter to u sub 0. 01:02:35.520 --> 01:02:38.770 And what comes out is a measurement of g. 01:02:38.770 --> 01:02:39.650 That's what we showed before. 01:02:39.650 --> 01:02:42.740 When we were trying to measure g, that was the way we did it. 01:02:42.740 --> 01:02:44.970 We started out with a PN sequence, go 01:02:44.970 --> 01:02:47.240 through g, add noise-- 01:02:47.240 --> 01:02:49.970 we can't avoid the noise-- go through the matched filter. 01:02:49.970 --> 01:02:53.260 That is a measurement of g at that point there. 01:02:53.260 --> 01:02:57.540 And if we send u sub 1, that's a measurement of g at that 01:02:57.540 --> 01:02:59.050 point there. 01:02:59.050 --> 01:03:06.790 So finally we have the rake receiver which does both of 01:03:06.790 --> 01:03:08.830 these things it once. 01:03:08.830 --> 01:03:11.670 You either send u sub 0 or u sub 1. 01:03:11.670 --> 01:03:12.810 You go through this filter. 01:03:12.810 --> 01:03:14.510 You add white noise. 01:03:17.240 --> 01:03:20.390 As far as making a decision is concerned, you do what we 01:03:20.390 --> 01:03:22.310 talked about before. 01:03:22.310 --> 01:03:25.070 You compare this output with this 01:03:25.070 --> 01:03:27.360 output to make a decision. 01:03:27.360 --> 01:03:31.270 After you make a decision you go forward in time, because 01:03:31.270 --> 01:03:34.500 we've done everything backwards in time here. 01:03:34.500 --> 01:03:39.290 And you take what is going to come out of here, which hasn't 01:03:39.290 --> 01:03:40.460 come out yet. 01:03:40.460 --> 01:03:44.550 And you use that to make a new estimate of g. 01:03:44.550 --> 01:03:48.180 You use that estimate of g turned around in time, to 01:03:48.180 --> 01:03:54.260 alter your estimate of the matched filter to g. 01:03:54.260 --> 01:03:57.090 And if you read the notes, the notes explains what's going on 01:03:57.090 --> 01:03:59.230 as far as the timing in here, a little bit 01:03:59.230 --> 01:04:00.940 better than I can here. 01:04:00.940 --> 01:04:02.970 But in fact, this is the kind of circuit that people 01:04:02.970 --> 01:04:07.060 actually use to both measure channels, and to send data at 01:04:07.060 --> 01:04:08.140 the same time. 01:04:08.140 --> 01:04:11.400 I want stop here because we're supposed to 01:04:11.400 --> 01:04:13.300 evaluate the class.