WEBVTT 00:00:00.000 --> 00:00:02.210 NARRATOR: The following content is provided under a 00:00:02.210 --> 00:00:03.640 Creative Commons license. 00:00:03.640 --> 00:00:06.540 Your support will help MIT OpenCourseWare continue to 00:00:06.540 --> 00:00:09.970 offer high quality educational resources for free. 00:00:09.970 --> 00:00:12.810 To make a donation or to view additional materials from 00:00:12.810 --> 00:00:16.830 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:16.830 --> 00:00:19.260 ocw.mit.edu. 00:00:19.260 --> 00:00:24.870 PROFESSOR: Today, I want to spend a fair amount of time, 00:00:24.870 --> 00:00:28.750 to start out with, talking about this business of, how do 00:00:28.750 --> 00:00:33.300 you go from pass band to a baseband in a wireless system 00:00:33.300 --> 00:00:39.970 and some of these questions about time reference and all 00:00:39.970 --> 00:00:40.730 of these things. 00:00:40.730 --> 00:00:45.400 When we were dealing with ordinary channels that didn't 00:00:45.400 --> 00:00:49.190 have fading in them and didn't have motion or anything like 00:00:49.190 --> 00:00:52.370 that, it was a fairly straightforward thing to think 00:00:52.370 --> 00:00:57.000 of a transmitter having its time reference and the 00:00:57.000 --> 00:01:01.740 receiver locking in on its time reference, which had a 00:01:01.740 --> 00:01:03.310 certain amount of delay from what the 00:01:03.310 --> 00:01:05.610 transmitter was doing. 00:01:05.610 --> 00:01:08.390 As soon as you start having multiple paths, each with 00:01:08.390 --> 00:01:12.610 different delays in them, this starts to get confusing. 00:01:12.610 --> 00:01:15.060 If you try to write it down carefully, 00:01:15.060 --> 00:01:18.090 it gets doubly confusing. 00:01:18.090 --> 00:01:21.760 If these path lanes are changing dynamically, it gets 00:01:21.760 --> 00:01:23.620 even more confusing. 00:01:23.620 --> 00:01:26.170 So I wanted to spend a little bit of time at the beginning 00:01:26.170 --> 00:01:29.560 of the hour trying to sort that out. 00:01:29.560 --> 00:01:33.040 It's not sorted out that well in the notes. 00:01:33.040 --> 00:01:36.050 As you'll see, it doesn't get sorted out too well here 00:01:36.050 --> 00:01:40.600 either because the problem is, whenever you write everything 00:01:40.600 --> 00:01:42.990 down, it becomes a nightmare. 00:01:42.990 --> 00:01:49.040 So this is just another effort at trying to do this. 00:01:49.040 --> 00:01:52.420 Let's start out by just looking at a single path. 00:01:52.420 --> 00:01:55.880 This single path, we're going to assume, has a time varying 00:01:55.880 --> 00:02:00.990 attenuation, which we'll call beta of t and a time varying 00:02:00.990 --> 00:02:03.590 delay, which we'll call tau of t. 00:02:03.590 --> 00:02:06.290 Namely, this is the same thing that we did before. 00:02:06.290 --> 00:02:08.460 We started out with a fixed 00:02:08.460 --> 00:02:11.190 transmitter and a fixed receiver. 00:02:11.190 --> 00:02:15.810 Now, since we're allowing both the attenuation and the delay 00:02:15.810 --> 00:02:19.510 to vary, but only having one path. 00:02:19.510 --> 00:02:22.760 What we're essentially thinking is, that is the 00:02:22.760 --> 00:02:26.490 receiver, which is in a vehicle for example, which is 00:02:26.490 --> 00:02:31.680 moving away from or towards the transmitter. 00:02:31.680 --> 00:02:36.480 So these are the, or in a sense an easy way of keeping 00:02:36.480 --> 00:02:40.740 track of that physical situation. 00:02:40.740 --> 00:02:46.490 So the response at passband to a real waveform, x of t, it's 00:02:46.490 --> 00:02:52.600 going to be just beta sub t times x of t minus tau of t. 00:02:52.600 --> 00:02:55.020 In other words, here's the delay here. 00:02:55.020 --> 00:02:57.040 Here's the attenuation here and we just 00:02:57.040 --> 00:02:58.290 have a single path. 00:03:00.770 --> 00:03:03.390 Now if you start out with a baseband waveform, if you 00:03:03.390 --> 00:03:07.680 start out with a complex baseband waveform, say u of t, 00:03:07.680 --> 00:03:12.230 you're going to shift this up first to a positive frequency 00:03:12.230 --> 00:03:17.040 band, which we'll call u sub p of t, which is u of t e to the 00:03:17.040 --> 00:03:22.950 2 pi ifct and you're going to transmit it as x of t, which 00:03:22.950 --> 00:03:27.410 is equal to two times a real part of that. 00:03:27.410 --> 00:03:29.900 Half the time we've been dealing with going through 00:03:29.900 --> 00:03:34.340 this modulation from baseband to passband in terms of 00:03:34.340 --> 00:03:36.160 cosines and sines. 00:03:36.160 --> 00:03:39.210 Half the time we've been doing it in terms of complex 00:03:39.210 --> 00:03:40.980 exponentials. 00:03:40.980 --> 00:03:43.780 One of the problems that appears as soon as you get 00:03:43.780 --> 00:03:48.320 into wireless is that if you really like to do things in 00:03:48.320 --> 00:03:52.120 terms of cosines and sines, you really run into a lot of 00:03:52.120 --> 00:03:57.460 trouble here because with phases varying all over the 00:03:57.460 --> 00:04:02.890 place, it just becomes very, very difficult to track what's 00:04:02.890 --> 00:04:07.280 going on with all of the cosine terms and all of the 00:04:07.280 --> 00:04:08.520 sine terms. 00:04:08.520 --> 00:04:13.670 So what we sort of have to wind up with here is this, in 00:04:13.670 --> 00:04:17.010 a sense, simpler but more abstract viewpoint where we 00:04:17.010 --> 00:04:20.820 think of modulation as being a two step process, where we 00:04:20.820 --> 00:04:24.750 first take the take the low pass waveform, move it up in 00:04:24.750 --> 00:04:29.340 frequency and then we add on the negative frequency part as 00:04:29.340 --> 00:04:32.185 an afterthought and then at the receiver, what we're going 00:04:32.185 --> 00:04:35.930 to do is have a Hilbert filter, which you would never 00:04:35.930 --> 00:04:36.780 build, of course. 00:04:36.780 --> 00:04:39.500 You would always build it in terms of cosines and sines, 00:04:39.500 --> 00:04:42.670 but conceptually have a Hilbert filter which blocks 00:04:42.670 --> 00:04:45.670 off that negative frequency part and then you just shift 00:04:45.670 --> 00:04:48.020 things down in frequency again. 00:04:48.020 --> 00:04:52.320 So that's the way we're thinking about this here, so 00:04:52.320 --> 00:04:55.500 we're going to have u of t shifted up and then 00:04:55.500 --> 00:05:00.570 transmitted as 2 times the real part of up of t and 00:05:00.570 --> 00:05:02.610 therefore, it's going to get received -- 00:05:02.610 --> 00:05:06.880 the received waveform at the passband. 00:05:06.880 --> 00:05:10.810 We're still using the timing at the transmitter. 00:05:10.810 --> 00:05:14.460 That's the purpose of going through all this analysis, to 00:05:14.460 --> 00:05:17.520 look at what's going on at transmit time and what's going 00:05:17.520 --> 00:05:20.150 on at receive time. 00:05:20.150 --> 00:05:24.340 So in terms of transmit time, this received waveform at 00:05:24.340 --> 00:05:30.510 passband is just the attenuation term times x of t 00:05:30.510 --> 00:05:35.040 minus tau of t, which is two times the real part of theta 00:05:35.040 --> 00:05:37.990 of t times this positive frequency part. 00:05:37.990 --> 00:05:41.280 Now when you write it out in all of its glory, it's 2 times 00:05:41.280 --> 00:05:45.410 the real part of the attenuation times the input 00:05:45.410 --> 00:05:51.750 you started with, but now this has been delayed by tau of t 00:05:51.750 --> 00:05:57.320 and e to the 2 pi ifct, but this was the carrier that was 00:05:57.320 --> 00:06:01.030 put on at the transmitter, physically at the transmitter 00:06:01.030 --> 00:06:04.130 and now we're at the receiver, so that carrier has gotten 00:06:04.130 --> 00:06:07.980 delayed by a factor of tau sub t. 00:06:07.980 --> 00:06:10.890 The question is, how do you demodulate this? 00:06:10.890 --> 00:06:14.660 How do you come down again from transmitter to receiver? 00:06:14.660 --> 00:06:17.830 What we've thought all along and what makes a lot of sense 00:06:17.830 --> 00:06:21.150 until you start putting these delays in, which are really a 00:06:21.150 --> 00:06:24.150 pain in the neck, is that what we're going to do is just 00:06:24.150 --> 00:06:27.750 multiply this positive frequency waveform by e to the 00:06:27.750 --> 00:06:31.860 minus 2 pi i f sub ct, but that's not what we're going to 00:06:31.860 --> 00:06:35.440 do because the receiver sitting there and the receiver 00:06:35.440 --> 00:06:38.260 has to recover timing and it has to 00:06:38.260 --> 00:06:41.760 recover carrier frequency. 00:06:41.760 --> 00:06:44.770 So what's it going to do? 00:06:44.770 --> 00:06:48.700 It's going to figure out what the timing is and the timing 00:06:48.700 --> 00:06:53.480 that it wants is at time 0, it wants to be seeing what the 00:06:53.480 --> 00:06:56.730 transmitter transmitted at transmit time 0. 00:06:56.730 --> 00:07:00.250 In other words, if you look at what the transmitter was doing 00:07:00.250 --> 00:07:03.530 at time 0, it was sending, say, the first bit that was 00:07:03.530 --> 00:07:05.040 going to be sent. 00:07:05.040 --> 00:07:08.500 In receiving this, we want to have our timing so we're 00:07:08.500 --> 00:07:10.920 looking at that first bit sent. 00:07:10.920 --> 00:07:15.340 So our timing is going to be shifted from what it was then. 00:07:15.340 --> 00:07:17.140 So we're going to take this new timing. 00:07:17.140 --> 00:07:21.900 The receiver clock time is now going to be t prime, which is 00:07:21.900 --> 00:07:25.890 t minus this delay term. 00:07:25.890 --> 00:07:28.910 If you get confused between the minuses and the pluses 00:07:28.910 --> 00:07:33.470 here, don't worry about it. 00:07:33.470 --> 00:07:36.610 I get confused about them too and the only way I can 00:07:36.610 --> 00:07:40.410 straighten them out as to put down one or the other and then 00:07:40.410 --> 00:07:44.720 spend ten minutes looking at it and try to figure out -- 00:07:44.720 --> 00:07:46.500 after I write it down -- 00:07:46.500 --> 00:07:49.150 whether it should really be a plus or a minus and I think 00:07:49.150 --> 00:07:54.115 these sines are right, but I'm not going to try to argue why 00:07:54.115 --> 00:07:56.140 in realtime. 00:07:56.140 --> 00:08:00.820 But anyway, this received waveform now, as a function of 00:08:00.820 --> 00:08:06.440 received time, is going to be the received waveform and in 00:08:06.440 --> 00:08:10.150 terms of transmit time -- 00:08:10.150 --> 00:08:15.100 and in place of t now, I'm going to have t prime so that 00:08:15.100 --> 00:08:20.090 since you t prime is equal to t minus tau of t, this is why 00:08:20.090 --> 00:08:22.290 it's t prime plus tau of t. 00:08:22.290 --> 00:08:25.590 In other words, what I'm looking at now, at time 0 at 00:08:25.590 --> 00:08:28.560 the receiver is what was being transmitted a 00:08:28.560 --> 00:08:29.910 little while ago. 00:08:29.910 --> 00:08:34.590 So that's this term and that's going to be the real part of 00:08:34.590 --> 00:08:39.070 this, where I've taken account of all of these shift terms. 00:08:39.070 --> 00:08:43.260 Now if you look at this carefully, you see that u of t 00:08:43.260 --> 00:08:47.170 minus t of t is turned into tau of t prime. 00:08:47.170 --> 00:08:53.420 This term has turned into 2 pi ifct prime, so everything is 00:08:53.420 --> 00:08:54.700 fine there. 00:08:54.700 --> 00:08:58.920 What I should have done with this term is to compensate for 00:08:58.920 --> 00:09:04.010 the fact that I'm now looking at it in a different time and 00:09:04.010 --> 00:09:08.520 at this point, what I'm going to do is to say, this quantity 00:09:08.520 --> 00:09:11.310 is changing so slowly with time that I 00:09:11.310 --> 00:09:13.090 don't care about that. 00:09:13.090 --> 00:09:17.920 If you really try to adjust this to make it right, it's a 00:09:17.920 --> 00:09:20.000 terrible mess. 00:09:20.000 --> 00:09:23.430 So we're just not going to worry about it. 00:09:23.430 --> 00:09:26.280 So what we receive then is approximately equal to this 00:09:26.280 --> 00:09:29.900 where the approximation is due to the fact that I'm not 00:09:29.900 --> 00:09:34.070 evaluating this attentuation term at exactly the right 00:09:34.070 --> 00:09:37.170 time, because it's a pain in the neck to do so. 00:09:41.350 --> 00:09:45.550 All of this stuff with wireless, you simply have to 00:09:45.550 --> 00:09:48.150 make approximations all over the place. 00:09:48.150 --> 00:09:50.930 You have to make crazy modeling assumptions all over 00:09:50.930 --> 00:09:53.470 the place, because the physical medium is so 00:09:53.470 --> 00:09:57.740 complicated that you can't do much else and the whole 00:09:57.740 --> 00:10:02.180 question is trying to make the right approximations and get 00:10:02.180 --> 00:10:04.960 some sense of which thing's very fast and which thing's 00:10:04.960 --> 00:10:06.900 very slowly. 00:10:06.900 --> 00:10:08.620 So that's the equation that we had. 00:10:08.620 --> 00:10:13.800 What we receive now at passband, but at the received 00:10:13.800 --> 00:10:18.050 time scale is this quantity here. 00:10:18.050 --> 00:10:20.150 So what's the receiver going to do? 00:10:20.150 --> 00:10:24.640 The receiver, at its time is going to demodulate by 00:10:24.640 --> 00:10:28.890 multiplying by e to the minus 2 pi ifct prime. 00:10:28.890 --> 00:10:32.090 First, we're going to take away this 2 times the real 00:10:32.090 --> 00:10:35.960 sine by going through the Hilbert filter, which just 00:10:35.960 --> 00:10:37.210 removes this. 00:10:39.930 --> 00:10:42.810 At that point, we have a complex positive frequency 00:10:42.810 --> 00:10:46.390 waveform and then we're going to multiply that positive 00:10:46.390 --> 00:10:52.315 frequency waveform by e to the minus 2 pi ifct prime, where t 00:10:52.315 --> 00:10:55.860 prime is the only thing the receiver knows anything about. 00:10:55.860 --> 00:10:58.650 So after you shift the baseband with a recovered 00:10:58.650 --> 00:11:03.820 carrier in receiver time, what you get in receiver time is v 00:11:03.820 --> 00:11:07.980 of t prime as equal to beta of t prime; namely, the 00:11:07.980 --> 00:11:11.820 attenuation at t prime -- or this is the approximate part 00:11:11.820 --> 00:11:14.080 of it -- times what was actually set. 00:11:18.270 --> 00:11:23.300 So now, think of this as saying, suppose this delay 00:11:23.300 --> 00:11:26.210 term is actually changing with time. 00:11:26.210 --> 00:11:32.780 Namely, it's some tau 0 minus a velocity term, vt divided by 00:11:32.780 --> 00:11:35.120 the velocity of light. 00:11:35.120 --> 00:11:40.590 In other words, the time delay that you incur, where the 00:11:40.590 --> 00:11:44.480 change in the time delay that you incur, is due to the time 00:11:44.480 --> 00:11:47.620 that it takes light to travel from where the receiver was to 00:11:47.620 --> 00:11:50.220 where the receiver is now. 00:11:50.220 --> 00:11:55.650 We can also write that as tau 0 minus the Doppler shift 00:11:55.650 --> 00:12:01.120 times the time divided by the carrier frequency. 00:12:01.120 --> 00:12:04.030 Now here's another approximation, because what 00:12:04.030 --> 00:12:06.180 we're sending -- 00:12:06.180 --> 00:12:09.380 I mean, this quantity here is exact. 00:12:09.380 --> 00:12:13.180 When you try to convert from the velocity, which you're 00:12:13.180 --> 00:12:16.100 going to, for what it does in frequency, which is the 00:12:16.100 --> 00:12:20.090 Doppler shift, it's a function of the frequency. 00:12:20.090 --> 00:12:23.120 Here what we're assuming is that all the frequencies we're 00:12:23.120 --> 00:12:26.480 dealing with are so close to the carrier frequency that we 00:12:26.480 --> 00:12:28.500 can just ignore everything else. 00:12:28.500 --> 00:12:38.170 So we're just writing this as the Doppler shift divided by 00:12:38.170 --> 00:12:40.170 this carrier frequency. 00:12:40.170 --> 00:12:45.510 Now if you view the passband waveform, going into a 00:12:45.510 --> 00:12:50.220 baseband of modulation and you think of doing this in 00:12:50.220 --> 00:12:53.200 transmit time, what are you going to do? 00:12:53.200 --> 00:12:58.100 In terms of transmit time, you've got to get rid of not 00:12:58.100 --> 00:13:02.220 this term, but this thing here. 00:13:02.220 --> 00:13:06.070 So in terms of transmit time, you're going to be multiplying 00:13:06.070 --> 00:13:10.520 by e to the minus 2 pi i times carrier frequency minus the 00:13:10.520 --> 00:13:13.740 Doppler shift. 00:13:13.740 --> 00:13:16.460 So the thing that's happening is that time at the 00:13:16.460 --> 00:13:22.300 transmitter gets spread out a little bit at the receiver. 00:13:22.300 --> 00:13:26.480 In other words, as this receiving antenna is moving 00:13:26.480 --> 00:13:30.910 away from the transmitter, a period of time like this at 00:13:30.910 --> 00:13:35.210 the transmitter is a period of time like this at the receiver 00:13:35.210 --> 00:13:38.000 and therefore, a certain number of cycles of carrier at 00:13:38.000 --> 00:13:41.110 the transmitter looks like a different number of cycles of 00:13:41.110 --> 00:13:43.110 carrier at the receiver. 00:13:43.110 --> 00:13:46.450 Therefore, if you're dealing with transmit time, you're 00:13:46.450 --> 00:13:48.830 taking account of this Doppler shift. 00:13:48.830 --> 00:13:52.240 You're multiplying not by the carrier frequency, but by the 00:13:52.240 --> 00:13:54.660 carrier frequency minus the Doppler shift. 00:13:54.660 --> 00:14:00.020 If you do it in receiver time, you're just multiplying by the 00:14:00.020 --> 00:14:11.450 by this carrier frequency, because the trouble is, your 00:14:11.450 --> 00:14:14.230 clock is running a little slow. 00:14:14.230 --> 00:14:16.880 Because your clock is running a little slow, you think 00:14:16.880 --> 00:14:20.410 you're demodulating at the actual carrier frequency, 00:14:20.410 --> 00:14:22.720 whereas in terms of what the transmitter thinks, you're 00:14:22.720 --> 00:14:25.180 doing something else. 00:14:25.180 --> 00:14:28.440 If you all got confused when you studied relativity, this 00:14:28.440 --> 00:14:32.550 is exactly the same problem you got confused about there. 00:14:32.550 --> 00:14:36.120 There's nothing very mysterious about it or hard 00:14:36.120 --> 00:14:39.230 about it, except that to most of us, time is a very 00:14:39.230 --> 00:14:41.850 fundamental quantity and you don't like 00:14:41.850 --> 00:14:43.690 to monkey with that. 00:14:43.690 --> 00:14:46.780 If you can't count on time being what it's supposed to 00:14:46.780 --> 00:14:48.990 be, you're really in deep trouble. 00:14:48.990 --> 00:14:53.290 When you try to write equations that do that, it 00:14:53.290 --> 00:14:55.450 makes things very, very tough also. 00:14:55.450 --> 00:15:02.210 So anyway, the result after we look at this in both ways, is 00:15:02.210 --> 00:15:05.700 that you get the same answer each way, but the receiver 00:15:05.700 --> 00:15:10.340 sees the carrier frequency and sees this carrier frequency in 00:15:10.340 --> 00:15:13.390 transmitter time and it sees this 00:15:13.390 --> 00:15:16.480 frequency in received time. 00:15:16.480 --> 00:15:19.960 Of course, the receiver only sees received time because the 00:15:19.960 --> 00:15:23.260 receiver sitting there trying to measure from the waveform 00:15:23.260 --> 00:15:27.420 coming in what that carrier frequency is and what that 00:15:27.420 --> 00:15:28.670 time base is. 00:15:32.100 --> 00:15:34.190 So now it gets to be fun. 00:15:34.190 --> 00:15:37.580 We look at multiple paths. 00:15:37.580 --> 00:15:47.460 When you look at it in transmit time, what gets 00:15:47.460 --> 00:15:50.710 received at the receiver, according to the 00:15:50.710 --> 00:15:52.760 transmit clock -- 00:15:52.760 --> 00:15:54.990 so here we are at the transmitter and we're peering 00:15:54.990 --> 00:16:03.270 at what's going on at this distant receiver and you now 00:16:03.270 --> 00:16:05.770 have the thing that we've talked about in the notes 00:16:05.770 --> 00:16:08.390 where you have all these different paths. 00:16:08.390 --> 00:16:12.660 Each path is associated with a certain attenuation and each 00:16:12.660 --> 00:16:17.370 path is associated with a time delay. 00:16:17.370 --> 00:16:20.950 This can be written in terms of what has gotten modulated 00:16:20.950 --> 00:16:26.170 up from baseband as 2 times the real part of this sum -- 00:16:26.170 --> 00:16:29.870 of beta j of t times the positive frequency part of 00:16:29.870 --> 00:16:34.500 what got transmitted and then it now has this delay in it. 00:16:34.500 --> 00:16:37.420 This can be rewritten in the same way that we did before, 00:16:37.420 --> 00:16:40.580 but now we just have all of these paths in there instead 00:16:40.580 --> 00:16:41.830 of one path. 00:16:45.520 --> 00:16:48.490 If you look at this expression then, we have all of these 00:16:48.490 --> 00:16:50.530 propogation delay terms. 00:16:50.530 --> 00:16:54.660 We have this baseband input, which has been delayed in 00:16:54.660 --> 00:16:59.880 different ways for each path and you have have this 00:16:59.880 --> 00:17:04.260 exponential, which has been delayed also. 00:17:04.260 --> 00:17:08.310 So at this point, the receiver is in a real pickle because if 00:17:08.310 --> 00:17:10.630 the receiver is smart enough to see -- yes? 00:17:10.630 --> 00:17:17.680 AUDIENCE: [UNINTELLIGIBLE PHRASE] 00:17:17.680 --> 00:17:20.450 PROFESSOR: The receiver doesn't know anything. 00:17:20.450 --> 00:17:22.600 It's like you're driving in your car, you're talking on 00:17:22.600 --> 00:17:25.930 your cell phone and your cell phone -- 00:17:25.930 --> 00:17:28.960 I mean, the only way your cell phone knows that you're moving 00:17:28.960 --> 00:17:33.380 is that the cell phone is getting some waveform coming 00:17:33.380 --> 00:17:38.190 in and it has to tell from that waveform what's going on. 00:17:38.190 --> 00:17:41.400 So it's going to do things like the carrier tracking that 00:17:41.400 --> 00:17:45.510 we were talking about before and it's not going to have too 00:17:45.510 --> 00:17:48.560 much trouble because all of these changes are taking place 00:17:48.560 --> 00:17:52.190 relatively slowly, relative to this very high -- 00:17:55.080 --> 00:17:58.100 either one gigabits or two gigabits or four gigabits, or 00:17:58.100 --> 00:18:00.350 what have you, carrier frequency. 00:18:00.350 --> 00:18:03.890 So all these changes look like almost nothing, but at the 00:18:03.890 --> 00:18:06.830 same time, they're fairly important because they keep 00:18:06.830 --> 00:18:09.820 changing faces. 00:18:09.820 --> 00:18:15.950 But anyway, in terms of the transmitter clock, this is 00:18:15.950 --> 00:18:19.960 what you actually receive. 00:18:19.960 --> 00:18:24.220 So at this point, what the receiver has to do is it has 00:18:24.220 --> 00:18:28.190 to retrieve clock time and the clock time that it retrieves 00:18:28.190 --> 00:18:29.460 is going to be -- 00:18:38.260 --> 00:18:42.690 receiver clock time is -- 00:18:42.690 --> 00:18:47.170 is t minus tau 0 sub t -- 00:18:47.170 --> 00:18:52.830 so it's t prime equals t minus tau 0 sub t. 00:18:52.830 --> 00:18:56.330 That's the same thing it was before, except that this tau 00:18:56.330 --> 00:18:58.830 zero of t here doesn't really amount to 00:18:58.830 --> 00:19:00.830 anything physical anymore. 00:19:00.830 --> 00:19:04.140 It's just whatever the circuitry that we have trying 00:19:04.140 --> 00:19:07.870 to recover carrier and trying to recover timing -- 00:19:07.870 --> 00:19:11.220 it's just whatever that happens to come up with, so 00:19:11.220 --> 00:19:15.520 with this best sense of the timing it should use if it's 00:19:15.520 --> 00:19:17.870 going to try to detect what the signal is. 00:19:17.870 --> 00:19:21.630 So it's some arbitrary value and again, this 00:19:21.630 --> 00:19:23.600 is varying in time. 00:19:23.600 --> 00:19:28.060 So again what we have is now this expression becomes really 00:19:28.060 --> 00:19:36.150 messy, because instead of having a receiver time 00:19:36.150 --> 00:19:42.710 variation, which cancels out what the actual variation in 00:19:42.710 --> 00:19:47.110 path length is, you really have two terms that are 00:19:47.110 --> 00:19:51.320 different and you have the same thing in terms of this 00:19:51.320 --> 00:19:53.170 phase, which is changing and this is the 00:19:53.170 --> 00:19:55.740 more important term. 00:19:55.740 --> 00:19:58.020 But anyway, you have that. 00:19:58.020 --> 00:19:59.840 You can then demodulate. 00:19:59.840 --> 00:20:01.920 What do you do when you demodulate? 00:20:01.920 --> 00:20:06.840 You take this thing in receiver time and you multiply 00:20:06.840 --> 00:20:11.360 by e to the 2 pi i, e to the minus 2 pi i -- 00:20:11.360 --> 00:20:14.890 this carrier frequency, what you think it is, times this 00:20:14.890 --> 00:20:17.960 time reference, what you think that is -- 00:20:17.960 --> 00:20:22.320 So that term has disappeared and what we then wind up with 00:20:22.320 --> 00:20:26.190 is the carrier frequency times these two terms here. 00:20:26.190 --> 00:20:27.780 That's to the left there. 00:20:27.780 --> 00:20:32.810 These two terms turn out to have both Doppler shift terms 00:20:32.810 --> 00:20:37.190 in them and also time spread terms. 00:20:37.190 --> 00:20:42.600 So when we write that out in terms of Doppler shift, what 00:20:42.600 --> 00:20:49.420 we wind up with is the input, which is then delayed by some 00:20:49.420 --> 00:20:52.590 arbitrary amount of time, where this is our best 00:20:52.590 --> 00:20:56.010 estimate of the right time to use and this is the actual 00:20:56.010 --> 00:20:57.910 time on that path. 00:20:57.910 --> 00:21:01.080 The thing that's happening here, which you don't see in 00:21:01.080 --> 00:21:06.560 the notes unfortunately, in the notes, it pretty much 00:21:06.560 --> 00:21:10.440 looks at things as far as transmit time is concerned. 00:21:10.440 --> 00:21:14.070 Therefore, what you see is expressions for impulse 00:21:14.070 --> 00:21:18.710 response where the impulse response is at some time much 00:21:18.710 --> 00:21:20.570 later than time 0. 00:21:20.570 --> 00:21:21.620 You have a -- 00:21:21.620 --> 00:21:25.030 I mean, you put in an impulse of time 0, what you see is 00:21:25.030 --> 00:21:29.230 stuff dribbling out over some very tiny interval, but a good 00:21:29.230 --> 00:21:32.340 deal later than what got put in. 00:21:32.340 --> 00:21:36.510 What we want to do now is to shift that so that in fact, 00:21:36.510 --> 00:21:39.120 when we look at filtering and things like that, we have a 00:21:39.120 --> 00:21:43.190 filter which starts a little bit before 0 and ends a little 00:21:43.190 --> 00:21:48.540 bit after 0 and that's the whole purpose of this. 00:21:48.540 --> 00:21:52.310 If you don't think my purposes just to confuse you, it really 00:21:52.310 --> 00:21:57.440 isn't, although I realize this is very confusing, but you 00:21:57.440 --> 00:21:59.970 sort of have to go through with it to see why these time 00:21:59.970 --> 00:22:02.950 shifts appear in the places where they do. 00:22:02.950 --> 00:22:06.910 But anyway, up here in the phase, you wind up with the 00:22:06.910 --> 00:22:10.050 Doppler shift terms and down here, when you're worrying 00:22:10.050 --> 00:22:14.450 about the input waveform, you're worrying about these 00:22:14.450 --> 00:22:15.520 time shift terms. 00:22:15.520 --> 00:22:18.950 We'll see we'll see how they come in a little later. 00:22:24.210 --> 00:22:27.990 To make the expression a little bit easier, let me look 00:22:27.990 --> 00:22:33.310 at tau j prime of t as tau j minus tau 0 and let me look at 00:22:33.310 --> 00:22:37.480 the Doppler differential as the Doppler that actually 00:22:37.480 --> 00:22:41.040 occurs on path j minus what the receiver thinks the 00:22:41.040 --> 00:22:42.860 Doppler is. 00:22:42.860 --> 00:22:47.010 This is this receiver term that we've demodulated by and 00:22:47.010 --> 00:22:50.220 therefore, we've really gotten rid of this term. 00:22:50.220 --> 00:22:54.460 Then when we rewrite this received baseband waveform -- 00:22:54.460 --> 00:22:59.110 and I've now gotten rid of the t primes because everything 00:22:59.110 --> 00:23:02.980 we've done up until now, we've just automatically assumed 00:23:02.980 --> 00:23:06.060 that the receiver timing and the transmit timing, we didn't 00:23:06.060 --> 00:23:09.120 have to be careful about it and it wasn't until we got to 00:23:09.120 --> 00:23:11.370 wireless that we do have to be careful about it. 00:23:11.370 --> 00:23:15.420 So now we're throwing it out again and what we have is 00:23:15.420 --> 00:23:19.360 these attenuations, which are very slowly varied. 00:23:19.360 --> 00:23:22.990 These terms, which are now delayed by this difference 00:23:22.990 --> 00:23:27.650 between the actual delay and what the receiver is assuming 00:23:27.650 --> 00:23:28.750 that the light ought to be. 00:23:28.750 --> 00:23:32.600 So these are really just differential delays relative 00:23:32.600 --> 00:23:34.560 to each other. 00:23:34.560 --> 00:23:37.770 These are Doppler shifts relative to each other. 00:23:37.770 --> 00:23:40.940 So that's the whole receive baseband waveform then. 00:23:46.820 --> 00:23:50.300 If you look at this, it's not apparent where the carrier 00:23:50.300 --> 00:23:54.770 frequency is coming in and the carrier frequency is coming in 00:23:54.770 --> 00:23:59.030 because these Doppler shifts have the carrier 00:23:59.030 --> 00:24:00.140 frequency in them. 00:24:00.140 --> 00:24:04.250 Namely, the Doppler shift is the velocity times the carrier 00:24:04.250 --> 00:24:06.520 frequency divided by the speed of light. 00:24:09.110 --> 00:24:11.710 One of the things you have to be very careful with in the 00:24:11.710 --> 00:24:14.990 wireless business, is that everybody thinks it's a neat 00:24:14.990 --> 00:24:17.210 idea to be able to be able to go up to higher and higher 00:24:17.210 --> 00:24:19.570 frequencies. 00:24:19.570 --> 00:24:23.440 When you go from one gigahertz up to four gigahertz, one of 00:24:23.440 --> 00:24:25.540 the things that you have to deal with is that every 00:24:25.540 --> 00:24:29.020 Doppler shift you're dealing with is now four times higher 00:24:29.020 --> 00:24:30.270 than it was before. 00:24:33.880 --> 00:24:37.650 So there's one big disadvantage in operating at 00:24:37.650 --> 00:24:39.530 extremely high frequencies. 00:24:39.530 --> 00:24:44.080 It's not necessarily a bad thing, but it's there and if 00:24:44.080 --> 00:24:47.540 you have a bunch of different paths which are all acting in 00:24:47.540 --> 00:24:50.520 different ways, you suddenly wind up 00:24:50.520 --> 00:24:53.330 with some real problems. 00:24:53.330 --> 00:24:58.050 The received baseband waveform on path j is going to change 00:24:58.050 --> 00:25:00.270 completely over an interval -- 00:25:00.270 --> 00:25:08.380 1 over 4 times the magnitude of this 00:25:08.380 --> 00:25:10.680 differential Doppler shift. 00:25:10.680 --> 00:25:13.600 The Doppler shift might be positive or negative, but if 00:25:13.600 --> 00:25:17.050 you look at this expression up here, that's the amount of 00:25:17.050 --> 00:25:20.020 time that it takes for this quantity to 00:25:20.020 --> 00:25:23.120 change by pi over 2. 00:25:23.120 --> 00:25:27.940 When this changes by pi over 2, this exponential goes from 00:25:27.940 --> 00:25:32.580 its maximum one down to zero or it goes from zero to minus 00:25:32.580 --> 00:25:39.430 one or it goes from minus one to zero or from zero up to 00:25:39.430 --> 00:25:42.530 plus one, and that's a pretty major change. 00:25:47.910 --> 00:25:51.940 So the interval which is required to do that is 1 over 00:25:51.940 --> 00:25:53.600 4 times that Doppler shift. 00:25:56.370 --> 00:26:00.490 The Doppler spread now is defined as the difference 00:26:00.490 --> 00:26:04.490 between the maximum Doppler shift and the 00:26:04.490 --> 00:26:08.390 minimum Doppler shift. 00:26:08.390 --> 00:26:13.580 If we choose this Doppler shift D sub 0 that we had, 00:26:13.580 --> 00:26:16.280 because of what the receiver was doing, seeing all these 00:26:16.280 --> 00:26:18.740 different Doppler shifts and trying to come up with 00:26:18.740 --> 00:26:24.960 something in the middle, that's pretty close to what 00:26:24.960 --> 00:26:30.330 the receiver thinks it should be using as far as receive 00:26:30.330 --> 00:26:31.600 time is concerned. 00:26:31.600 --> 00:26:35.860 So it's the maximum plus the minimum divided by 2 and then 00:26:35.860 --> 00:26:40.060 each of these differential Doppler shifts is going to be 00:26:40.060 --> 00:27:00.430 somewhere between that overall Doppler spread minus Doppler 00:27:00.430 --> 00:27:03.970 spread over 2 and plus Doppler spread over 2. 00:27:03.970 --> 00:27:05.220 It goes from -- 00:27:09.000 --> 00:27:17.130 minus D over 2 up to plus D over 2 and this difference in 00:27:17.130 --> 00:27:25.130 here, is strangely enough, D. So what we've done is by 00:27:25.130 --> 00:27:27.770 adjusting our received timing -- 00:27:27.770 --> 00:27:31.520 and slowing it down or speeding it up if necessary, 00:27:31.520 --> 00:27:34.850 is we wind up with Doppler shifts that are sort of spread 00:27:34.850 --> 00:27:39.580 around from negative Doppler shifts to plus Doppler shifts. 00:27:39.580 --> 00:27:43.520 What that means is, if you look at this expression, it 00:27:43.520 --> 00:27:46.660 takes a period of time, about -- 00:27:49.570 --> 00:27:55.780 here we had 1 over 4 times the differential Doppler, which is 00:27:55.780 --> 00:28:02.000 now 1 over 2 times this overall Doppler spread. 00:28:02.000 --> 00:28:07.270 So what that says is, there's a coherence time on the 00:28:07.270 --> 00:28:12.150 channel of about 1 over 2 times the Doppler spread. 00:28:12.150 --> 00:28:16.090 That's important because that says the channel is going to 00:28:16.090 --> 00:28:21.570 look like the same thing for a period of time, which is about 00:28:21.570 --> 00:28:26.330 1 over 2 times this Doppler spread term. 00:28:26.330 --> 00:28:31.270 I think this will be D instead of D 0. 00:28:31.270 --> 00:28:34.050 So you find out what the Doppler spread is and the 00:28:34.050 --> 00:28:36.970 reason you want to find out what the Doppler spread is is 00:28:36.970 --> 00:28:41.920 that that tells you how long the channel remains stable. 00:28:41.920 --> 00:28:43.400 Why do you want to know how long the 00:28:43.400 --> 00:28:45.460 channel remains stable? 00:28:45.460 --> 00:28:47.590 If you're going to do detection or anything like 00:28:47.590 --> 00:28:50.170 that, if you're going to filter the received waveform 00:28:50.170 --> 00:28:54.040 to try to find out what the input waveform is, you would 00:28:54.040 --> 00:28:55.780 like to know what the channel is doing. 00:28:55.780 --> 00:28:59.060 You'd like to be able to measure the channel. 00:28:59.060 --> 00:29:02.450 If you're going to measure the channel, you really want to 00:29:02.450 --> 00:29:06.100 know how long it stays the same before you have to 00:29:06.100 --> 00:29:07.970 measure it again. 00:29:07.970 --> 00:29:10.300 If you look at all these cellular systems and you look 00:29:10.300 --> 00:29:14.360 at every other wireless system in the world, all of them do 00:29:14.360 --> 00:29:17.780 one of one of two things: either periodically they 00:29:17.780 --> 00:29:21.760 measure what the channel is so they can use it and the other 00:29:21.760 --> 00:29:28.040 thing they do is every once in awhile, they send a pilot tone 00:29:28.040 --> 00:29:32.760 and they use the pilot tone to try to figure out what kind of 00:29:32.760 --> 00:29:33.930 channel they have. 00:29:33.930 --> 00:29:37.510 This is the thing which tells you how often you have to send 00:29:37.510 --> 00:29:39.290 that pilots tone. 00:29:39.290 --> 00:29:42.700 If you know about estimation theory, it's also the thing 00:29:42.700 --> 00:29:45.210 that tells you whether you have any chance or not of 00:29:45.210 --> 00:29:47.930 estimating what the channel is, because it takes you a 00:29:47.930 --> 00:29:52.110 certain amount of time in the presence of noise to estimate 00:29:52.110 --> 00:29:54.410 something and if that something you're trying to 00:29:54.410 --> 00:29:58.520 estimate it's changing faster than you can estimate it, then 00:29:58.520 --> 00:30:00.030 you don't have a prayer of a chance. 00:30:02.930 --> 00:30:04.490 So yes, this is important. 00:30:04.490 --> 00:30:05.620 This is the prime coherence. 00:30:05.620 --> 00:30:08.860 It's one of the main parameters of a wireless 00:30:08.860 --> 00:30:11.330 channel you want to understand. 00:30:11.330 --> 00:30:13.900 It tells you how long the channel will remain 00:30:13.900 --> 00:30:16.400 essentially the same before it changes and 00:30:16.400 --> 00:30:18.390 becomes something different. 00:30:20.960 --> 00:30:24.420 Both of these quantities are very approximate. 00:30:24.420 --> 00:30:27.680 If anyone wants to tell you that it's twice what I say it 00:30:27.680 --> 00:30:31.060 is here, I certainly wouldn't quarrel with them. 00:30:31.060 --> 00:30:34.060 I mean, it's like talking about the bandwidth of a 00:30:34.060 --> 00:30:35.740 waveform them that you send. 00:30:35.740 --> 00:30:38.090 I mean, you look at the spectrum of what you send and 00:30:38.090 --> 00:30:40.900 it's going to be something which sort of tails off. 00:30:40.900 --> 00:30:43.400 If you want to call it the maximum out there where it's 00:30:43.400 --> 00:30:47.940 going to zero or the minimum, where it starts to go to zero 00:30:47.940 --> 00:30:50.590 or what have you, it doesn't make much difference. 00:30:50.590 --> 00:30:52.790 Here, it's a worse situation because you 00:30:52.790 --> 00:30:54.590 don't shape this waveform. 00:30:54.590 --> 00:30:58.210 It's just given to you and you find something which is going 00:30:58.210 --> 00:30:59.800 to be tailing off slowly. 00:30:59.800 --> 00:31:06.040 It's going to have little bits of nothingness way out there. 00:31:06.040 --> 00:31:09.890 All this is is just one very approximate number, which 00:31:09.890 --> 00:31:13.600 tells you what that Doppler spread is and therefore, which 00:31:13.600 --> 00:31:17.740 tells you how long the channel is going to remain stable and 00:31:17.740 --> 00:31:21.030 yes, in half of that time the channel starts to change and 00:31:21.030 --> 00:31:24.300 in twice that time, the channel might be still sort of 00:31:24.300 --> 00:31:25.450 like it was before. 00:31:25.450 --> 00:31:28.570 So this is only a very approximate way of looking at 00:31:28.570 --> 00:31:32.000 these things, but it gives you an idea about what you can do 00:31:32.000 --> 00:31:33.380 and what you can't do. 00:31:33.380 --> 00:31:36.170 It gives you an idea, for example, if you say, can I 00:31:36.170 --> 00:31:41.550 take this wireless system I've been using and use it at a 00:31:41.550 --> 00:31:44.360 frequency that's 10 times as high -- 00:31:44.360 --> 00:31:47.480 and you know when you use it at a frequency 10 times as 00:31:47.480 --> 00:31:51.910 high that the stability of the channel is going to be 10 00:31:51.910 --> 00:31:53.650 times as small. 00:31:53.650 --> 00:31:55.930 You know if you were having trouble measuring the channel 00:31:55.930 --> 00:31:58.930 before, you know you don't have a prayer of a chance when 00:31:58.930 --> 00:32:02.030 you go up to that higher frequency, so it so it tells 00:32:02.030 --> 00:32:03.280 you things like that. 00:32:05.660 --> 00:32:08.390 There are two other sort of related quantities, which are 00:32:08.390 --> 00:32:13.210 called multipath spread and coherence frequency. 00:32:13.210 --> 00:32:17.130 So we have Doppler spread on the one hand, which is a 00:32:17.130 --> 00:32:23.840 physical phenomena and that gives rise to coherence time, 00:32:23.840 --> 00:32:27.720 which says how long the channel will stay the same and 00:32:27.720 --> 00:32:29.110 in the opposite domain -- 00:32:29.110 --> 00:32:34.360 and these are almost dual quantities of each other, you 00:32:34.360 --> 00:32:38.220 have something called multipath spread, which -- 00:32:38.220 --> 00:32:40.800 and multipath spread is pretty much what it 00:32:40.800 --> 00:32:41.690 sounds like it is. 00:32:41.690 --> 00:32:44.290 You have a bunch of different paths. 00:32:44.290 --> 00:32:48.690 You send a very short duration waveform and things start 00:32:48.690 --> 00:32:53.070 trickling in on the shortest path and then you got a big 00:32:53.070 --> 00:32:58.650 bunch of stuff in all the paths which are sort of the 00:32:58.650 --> 00:33:03.480 same lanegth and then they sort of dribble away again. 00:33:03.480 --> 00:33:06.570 So the waveform that you see coming in is spread out from 00:33:06.570 --> 00:33:09.830 what you send because of these multiple paths and the 00:33:09.830 --> 00:33:13.570 multiple paths having different delays on them. 00:33:13.570 --> 00:33:19.560 Because you have all of these different delay paths, you're 00:33:19.560 --> 00:33:22.330 going to have an impulse response for this channel, 00:33:22.330 --> 00:33:25.580 which has a certain duration to it. 00:33:25.580 --> 00:33:29.210 When you take the Fourier transform of that, you find a 00:33:29.210 --> 00:33:34.480 certain duration for Fourier transform. 00:33:34.480 --> 00:33:37.240 As we're going to see in just a minute, that's the thing 00:33:37.240 --> 00:33:41.550 that tells you how stable this channel is in frequency. 00:33:41.550 --> 00:33:47.620 If you measure it at one frequency, how far do you have 00:33:47.620 --> 00:33:50.960 to go in frequency before you find something which is 00:33:50.960 --> 00:33:55.460 totally separate from what you measure down here? 00:33:55.460 --> 00:33:59.650 This is something the cuts both ways. 00:33:59.650 --> 00:34:03.520 Having a large frequency coherence and having a large 00:34:03.520 --> 00:34:07.460 time coherence can be good or it can be bad. 00:34:07.460 --> 00:34:10.000 If you have a very large time coherence and a very large 00:34:10.000 --> 00:34:12.910 frequency coherence, you can measure the channel once and 00:34:12.910 --> 00:34:16.290 you can use it for a long time and you can use it over a 00:34:16.290 --> 00:34:19.360 broadband width and everything looks very nice. 00:34:19.360 --> 00:34:22.290 If the channel happens to be faded over that long period of 00:34:22.290 --> 00:34:25.440 time and over that long interval of frequency, you're 00:34:25.440 --> 00:34:27.480 really in the soup. 00:34:27.480 --> 00:34:30.500 On the other hand, if you have a very small time coherence 00:34:30.500 --> 00:34:33.540 and a very small frequency coherence, if the channel 00:34:33.540 --> 00:34:36.030 doesn't look good there, you move to a different time 00:34:36.030 --> 00:34:39.710 interval or you moved to a different frequency interval 00:34:39.710 --> 00:34:42.440 and you can transmit again. 00:34:42.440 --> 00:34:46.940 So both of these quantities have good aspects to them and 00:34:46.940 --> 00:34:50.900 bad aspects to them, but if you want to evaluate what a 00:34:50.900 --> 00:34:55.550 physical wireless channel is and somebody lets you ask two 00:34:55.550 --> 00:35:00.510 questions, the two questions you ought to ask are, what is 00:35:00.510 --> 00:35:03.450 the time coherence and what is the frequency coherence? 00:35:03.450 --> 00:35:05.570 Those are the first things you want to find out. 00:35:05.570 --> 00:35:10.080 Everything else is sort of secondary. 00:35:10.080 --> 00:35:14.500 So trying to claim that the frequency coherence here 00:35:14.500 --> 00:35:18.160 almost has a matter duality, as 1 over 00:35:18.160 --> 00:35:20.440 twice the time spread. 00:35:20.440 --> 00:35:26.170 Let's take a slide to see why that is. 00:35:29.700 --> 00:35:38.160 If I take the impulse response at baseband of this channel, 00:35:38.160 --> 00:35:41.100 which I wrote down here -- 00:35:41.100 --> 00:35:48.310 here's the impulse response and you get that directly from 00:35:48.310 --> 00:35:52.130 the baseband response for giving input waveform. 00:35:52.130 --> 00:35:55.600 With all these different paths, what this impulse 00:35:55.600 --> 00:36:02.400 response is at a particular time t -- namely, this is the 00:36:02.400 --> 00:36:08.770 response at time t to an impulse tau seconds earlier. 00:36:08.770 --> 00:36:11.520 In other words, when I'm using this notation here, I'm really 00:36:11.520 --> 00:36:15.400 thinking of t as a receiver time and t minus tau as 00:36:15.400 --> 00:36:17.490 transmit time. 00:36:17.490 --> 00:36:19.530 But anyway, this is what it is. 00:36:19.530 --> 00:36:25.920 You take the Fourier transform of this for a particular t -- 00:36:25.920 --> 00:36:27.240 in other words, we talked about this a 00:36:27.240 --> 00:36:29.070 little bit last time. 00:36:29.070 --> 00:36:34.090 You have a function of two parameters here and at any 00:36:34.090 --> 00:36:38.420 given t at the receiver, what you see is a spread of 00:36:38.420 --> 00:36:41.650 different inputs coming in from the transmitter. 00:36:41.650 --> 00:36:45.540 What we're interested in now is at that particular time t 00:36:45.540 --> 00:36:49.870 at the receiver, what's the Fourier transform, when you 00:36:49.870 --> 00:36:53.440 take the Fourier transform on tau to see what kinds of 00:36:53.440 --> 00:36:56.100 frequencies are coming in? 00:36:56.100 --> 00:37:03.090 We take the Fourier transform of a unit delta function and 00:37:03.090 --> 00:37:04.990 you just get an exponential function. 00:37:04.990 --> 00:37:09.820 So the transform of that delta is just this e to the minus 2 00:37:09.820 --> 00:37:14.080 pi if times tau sub j prime of t. 00:37:14.080 --> 00:37:17.820 This is this differential delay that we're faced with 00:37:17.820 --> 00:37:20.860 and we still have the Doppler shift over here. 00:37:20.860 --> 00:37:24.100 So when we look at it this way, we have both the Doppler 00:37:24.100 --> 00:37:29.640 shift sitting there and we have this delay 00:37:29.640 --> 00:37:31.840 shift sitting up here. 00:37:31.840 --> 00:37:35.700 Now the question we ask is, how much does the frequency 00:37:35.700 --> 00:37:39.280 have to change before this quantity is going to change 00:37:39.280 --> 00:37:41.380 materially? 00:37:41.380 --> 00:37:44.330 The answer is the same as it was before. 00:37:44.330 --> 00:37:49.590 When this quantity here changes by pi over 2, you have 00:37:49.590 --> 00:37:52.330 a totally different result than you had before. 00:37:52.330 --> 00:37:55.970 So the question is, how much does a frequency have to 00:37:55.970 --> 00:38:05.650 change in order for f times tau j prime of t to the change 00:38:05.650 --> 00:38:10.260 by a factor of one-fourth. 00:38:10.260 --> 00:38:14.780 So the answer that we come up with is, this has to go from 00:38:14.780 --> 00:38:23.740 minus L over 2 to plus L over 2 and this is where we get the 00:38:23.740 --> 00:38:25.630 result that we were just talking about, that the 00:38:25.630 --> 00:38:34.290 frequency coherence is 1 over 2 times the multipath spread 00:38:34.290 --> 00:38:35.540 of the channel. 00:38:44.580 --> 00:38:49.180 The next thing we want to do in this long process -- 00:38:49.180 --> 00:38:53.860 it's a process that we've sort of been doing all along. 00:38:53.860 --> 00:39:00.440 The whole purpose of looking at waveforms and looking at 00:39:00.440 --> 00:39:04.220 digital communication is to be able to go from sequences to 00:39:04.220 --> 00:39:06.060 waveforms -- 00:39:06.060 --> 00:39:08.190 and what do you think the next thing we're going to do is? 00:39:08.190 --> 00:39:10.140 We're going to go from these waveforms -- 00:39:10.140 --> 00:39:13.245 we've now got them down to baseband waveforms and we want 00:39:13.245 --> 00:39:17.840 to go back to looking at sequences. 00:39:17.840 --> 00:39:23.650 So what we would like to do is to take this input waveform, 00:39:23.650 --> 00:39:26.170 view it in terms of the sampling theorem -- 00:39:26.170 --> 00:39:29.880 better thing to do would be to view it in terms of input 00:39:29.880 --> 00:39:34.400 pulses, but that's too complicated for now, so we'll 00:39:34.400 --> 00:39:38.950 just take the input waveform and view it as a sequence of 00:39:38.950 --> 00:39:41.390 data that we're sending -- the u sub k's 00:39:41.390 --> 00:39:45.560 times these sinc waveforms. 00:39:45.560 --> 00:39:47.070 Same thing we've been doing all along. 00:39:47.070 --> 00:39:51.190 We take a sequence of numbers, we put little sine x over 00:39:51.190 --> 00:39:54.160 x-hats around them and that gives us a 00:39:54.160 --> 00:39:57.600 band limited waveform. 00:39:57.600 --> 00:40:00.760 The baseband output is then -- 00:40:00.760 --> 00:40:02.210 you can look at the same way. 00:40:02.210 --> 00:40:04.380 The baseband output is going to be 00:40:04.380 --> 00:40:07.630 limited to the same frequency. 00:40:07.630 --> 00:40:11.880 Question here: You have these Doppler shifts. 00:40:11.880 --> 00:40:16.550 So when I send a frequency which is right up at the limit 00:40:16.550 --> 00:40:20.990 of w over 2, what I get back from that is going to be 00:40:20.990 --> 00:40:23.320 spread out a little bit from that. 00:40:23.320 --> 00:40:27.720 So if I send a waveform that's limited to w over 2, I'm going 00:40:27.720 --> 00:40:30.120 to get back a waveform which is a little 00:40:30.120 --> 00:40:31.750 bit bigger than that. 00:40:31.750 --> 00:40:35.230 What do I do about that? 00:40:35.230 --> 00:40:37.130 I worried about it in the notes a little bit and I'm 00:40:37.130 --> 00:40:41.960 probably going to take it out from the notes, because as a 00:40:41.960 --> 00:40:46.400 practical matter, the amount of Doppler shift that you get 00:40:46.400 --> 00:40:54.380 is such a negligible fraction of the kind of bandwidth you 00:40:54.380 --> 00:40:57.020 would be using for transmission that you just 00:40:57.020 --> 00:40:59.430 want to forget about that. 00:40:59.430 --> 00:41:02.670 So we are going to forget about it and what's coming in 00:41:02.670 --> 00:41:07.090 is again a band limited function with maybe a little 00:41:07.090 --> 00:41:09.930 bit of squishiness on it. 00:41:09.930 --> 00:41:14.870 We get this baseband output, which we can then view in 00:41:14.870 --> 00:41:16.840 terms of the sampling theorem. 00:41:16.840 --> 00:41:20.320 We can sample the output and what do we get 00:41:20.320 --> 00:41:22.020 when we're all done? 00:41:22.020 --> 00:41:28.390 Here's the nice thing that we can jump to: The output at 00:41:28.390 --> 00:41:32.920 this free time m is then the sum -- 00:41:32.920 --> 00:41:35.780 namely, it's the convolution, it's the discrete convolution 00:41:35.780 --> 00:41:42.210 of what went in at time m, at time m minus 1, m minus 2, m 00:41:42.210 --> 00:41:45.840 plus 1, m plus 2 -- 00:41:45.840 --> 00:41:51.320 and what we're doing is, we're designing this recovery time 00:41:51.320 --> 00:41:56.870 so the main peak is about a time 0, so k goes from minus 00:41:56.870 --> 00:42:00.890 something to plus something here and then we wind up with 00:42:00.890 --> 00:42:06.020 these channel terms, which are again, sampled at this point 00:42:06.020 --> 00:42:09.690 because we can sample them, because it's only these low 00:42:09.690 --> 00:42:13.850 frequency components of the channel response that make any 00:42:13.850 --> 00:42:15.000 difference. 00:42:15.000 --> 00:42:19.300 They're the only things that give us this output here and 00:42:19.300 --> 00:42:23.280 this output has already been filtered 00:42:23.280 --> 00:42:26.540 down to this low frequency. 00:42:26.540 --> 00:42:29.880 When I look what these components are, these are big 00:42:29.880 --> 00:42:31.080 messes again. 00:42:31.080 --> 00:42:37.370 They're sums over all of these different paths. 00:42:37.370 --> 00:42:45.050 So what we're winding up with is a function -- 00:42:45.050 --> 00:42:52.680 if I sketch g of tau and t as a function of tau, what I'm 00:42:52.680 --> 00:42:55.120 going to find is a bunch of different terms. 00:43:08.760 --> 00:43:13.420 Now I'm going to filter this so every one of these turns 00:43:13.420 --> 00:43:16.730 into a little sinc x over x and then I'm 00:43:16.730 --> 00:43:18.590 going to sample it. 00:43:18.590 --> 00:43:22.960 So what I wind up with then is this goes into something which 00:43:22.960 --> 00:43:26.130 looks like this. 00:43:26.130 --> 00:43:29.065 It's going to look like something which goes up and 00:43:29.065 --> 00:43:32.340 comes down again, and I'm just interested in what it's sample 00:43:32.340 --> 00:43:38.140 values are and I might wind up with five or six sample values 00:43:38.140 --> 00:43:39.840 and that expresses the whole thing. 00:43:44.100 --> 00:43:46.030 I mean, this is something you have to think about a little 00:43:46.030 --> 00:43:49.490 bit because it's something that we've done about 10 times 00:43:49.490 --> 00:43:54.120 already, but now when we do it here it looks very different 00:43:54.120 --> 00:43:55.780 from what we've done before -- 00:43:58.280 --> 00:44:00.970 except this is what it is analytically. 00:44:00.970 --> 00:44:11.570 This is what it is in pictures and that's where it comes out. 00:44:17.470 --> 00:44:20.830 The thing that's going on here then is we started to look at 00:44:20.830 --> 00:44:23.690 this physical modeling in terms of ray tracing and 00:44:23.690 --> 00:44:36.170 things like that and what we're doing at this point is 00:44:36.170 --> 00:44:40.180 we're modeling what's going on at the frequency bandwidth 00:44:40.180 --> 00:44:41.840 that we're using -- 00:44:41.840 --> 00:44:45.300 namely, not the carrier frequency anymore, but the 00:44:45.300 --> 00:44:47.550 baseband bandwidth. 00:44:47.550 --> 00:44:51.380 We're viewing this filter as having filtered these things 00:44:51.380 --> 00:44:56.650 out for us so we wind up with a baseband discrete filter 00:44:56.650 --> 00:45:00.420 that filters these impulses and aggregates them under 00:45:00.420 --> 00:45:02.270 discrete caps. 00:45:02.270 --> 00:45:07.780 So at this point, when we look at this filter for what the 00:45:07.780 --> 00:45:11.630 channel is doing, instead of seeing a bunch of impulses, we 00:45:11.630 --> 00:45:14.120 see a rather smooth waveform. 00:45:14.120 --> 00:45:18.110 If we look at what contributes to each tap, we see a large 00:45:18.110 --> 00:45:20.775 number of different paths, which all 00:45:20.775 --> 00:45:24.720 contribute to the same tap. 00:45:24.720 --> 00:45:28.710 If the multipath spread is not too large 00:45:28.710 --> 00:45:31.230 relative to 1 over w -- 00:45:31.230 --> 00:45:35.600 if if the multipath spread is small relative to 1 over w, 00:45:35.600 --> 00:45:40.510 then you can really represent this whole thing in one path. 00:45:40.510 --> 00:45:42.790 In other words, if the multipath spread is very, very 00:45:42.790 --> 00:45:47.500 small and the sampling time interval is big, everything 00:45:47.500 --> 00:45:51.980 just appears under one tap and that's called flat fading. 00:45:51.980 --> 00:45:56.490 When you have flat fading, the output that comes out is 00:45:56.490 --> 00:46:00.360 really just a faded version of the input and there isn't any 00:46:00.360 --> 00:46:01.710 time dispersion on it. 00:46:01.710 --> 00:46:02.960 It all just -- 00:46:06.730 --> 00:46:10.420 it's an undistorted version of what got sent, except it's 00:46:10.420 --> 00:46:16.640 very slowly fading and coming back up again and fading again 00:46:16.640 --> 00:46:17.950 and coming back up again. 00:46:21.600 --> 00:46:25.790 If you have a slightly wider bandwidth, then you need 00:46:25.790 --> 00:46:28.660 multiple taps. 00:46:28.660 --> 00:46:32.610 If you look at most of the of the systems that are being 00:46:32.610 --> 00:46:37.110 built today, the largest number of taps they ever used 00:46:37.110 --> 00:46:40.780 in this kind of implementation is three to five or something 00:46:40.780 --> 00:46:44.310 in that order, so it's not a huge number of taps that we're 00:46:44.310 --> 00:46:47.560 talking about, because the bandwidths are not huge. 00:46:53.990 --> 00:47:03.740 But anyway, statistically what we wind up with is that if the 00:47:03.740 --> 00:47:09.090 bandwidth that we're using is relatively narrow and it's not 00:47:09.090 --> 00:47:17.430 too many times what this term 1 over L is -- 00:47:17.430 --> 00:47:20.870 namely, this multipath spread, then we wind up with a small 00:47:20.870 --> 00:47:22.910 number of paths. 00:47:22.910 --> 00:47:26.440 We can sort of assume that we have a large number of paths 00:47:26.440 --> 00:47:30.130 coming in on each tap and then what we're doing is that each 00:47:30.130 --> 00:47:34.810 tap strength is going to be a sum of a bunch of terms which 00:47:34.810 --> 00:47:38.250 are essentially random. 00:47:38.250 --> 00:47:40.670 If you add up a bunch of complex terms that are 00:47:40.670 --> 00:47:42.480 essentially random, what do you get? 00:47:45.240 --> 00:47:48.850 If you add up a very humongous number of them, you get 00:47:48.850 --> 00:47:50.260 something which is Gaussian. 00:47:54.890 --> 00:47:58.460 If you were modeling these things statistically, you 00:47:58.460 --> 00:48:00.250 would like them to be Gaussian. 00:48:00.250 --> 00:48:01.870 So what do you do? 00:48:01.870 --> 00:48:04.610 You assume that they're Gaussian, which is what 00:48:04.610 --> 00:48:07.180 everyone does. 00:48:07.180 --> 00:48:13.240 As a slight excuse for that, what we're really interested 00:48:13.240 --> 00:48:17.900 in if you're designing cellular systems, you're 00:48:17.900 --> 00:48:21.720 trying to build cellular receivers which will deal with 00:48:21.720 --> 00:48:24.460 all of the different circumstances that they come 00:48:24.460 --> 00:48:28.850 up against and as you walk around or drive around using 00:48:28.850 --> 00:48:34.360 these things, if you look at the ensemble of different 00:48:34.360 --> 00:48:39.150 situations that these phones are faced with, over that 00:48:39.150 --> 00:48:43.180 ensemble of things, each of these taps is pretty much 00:48:43.180 --> 00:48:47.090 going to look like a Gaussian random variable. 00:48:47.090 --> 00:48:51.270 If you look at one sample path of one little cellular phone, 00:48:51.270 --> 00:48:54.430 then no, it won't look like that -- but as far as a design 00:48:54.430 --> 00:48:58.300 tool, it looks like something which you can model as 00:48:58.300 --> 00:49:04.040 Gaussian because you're going to view each of these taps in 00:49:04.040 --> 00:49:08.390 this filter, in this time varying filter, the sum of 00:49:08.390 --> 00:49:12.430 lots of unrelated components and therefore, we're going to 00:49:12.430 --> 00:49:18.040 take the density of these taps and a statistical model as 00:49:18.040 --> 00:49:19.410 being jointly Gaussian -- 00:49:22.320 --> 00:49:25.700 a proper Gaussian random variable for each of them. 00:49:25.700 --> 00:49:29.430 What happens if different taps is jointly Gaussian? 00:49:29.430 --> 00:49:35.030 So what we wind up with is a model for wireless channels 00:49:35.030 --> 00:49:41.820 then, where the channel itself is a Gaussian random process 00:49:41.820 --> 00:49:46.050 and as a Gaussian random process both in tau and in t, 00:49:46.050 --> 00:49:52.460 it's easier to look at if we look at it in terms of these 00:49:52.460 --> 00:49:53.490 discrete samples. 00:49:53.490 --> 00:49:58.770 Namely, we're going to look at the channel now, not in terms 00:49:58.770 --> 00:50:00.880 of a wavelength type of thing. 00:50:00.880 --> 00:50:04.920 We're just going to look at it as a bunch of taps, which the 00:50:04.920 --> 00:50:10.720 receiver has to add over to get to the received waveform. 00:50:10.720 --> 00:50:13.660 We're going to model each of those taps as being Gaussian. 00:50:13.660 --> 00:50:17.810 We're going to model in time t as being stationary. 00:50:17.810 --> 00:50:23.450 We're going to model them in time tau as coming up and 00:50:23.450 --> 00:50:27.180 going back down again and existing over a multipath 00:50:27.180 --> 00:50:30.260 spread that's about L in duration. 00:50:33.000 --> 00:50:39.430 So the phase is going to be uniform with this density. 00:50:39.430 --> 00:50:43.270 This is this two dimensional independent Gaussian density 00:50:43.270 --> 00:50:47.010 that we've looked at so many times and if you look at this, 00:50:47.010 --> 00:50:50.560 it has circular symmetry to it, which means if you look at 00:50:50.560 --> 00:50:54.940 it in an amplituding phase, the phase is random, uniform 00:50:54.940 --> 00:50:57.790 between zero and 2 pi. 00:50:57.790 --> 00:51:03.700 The energy in it is exponential and the magnitude 00:51:03.700 --> 00:51:07.050 is this, which comes from that -- 00:51:07.050 --> 00:51:11.310 which is what you call a Rayleigh density. 00:51:11.310 --> 00:51:14.840 So Rayleigh fading channels are simply channels which you 00:51:14.840 --> 00:51:18.820 have modeled as having taps, which are 00:51:18.820 --> 00:51:21.590 really random variables. 00:51:21.590 --> 00:51:28.030 Real and imaginary parts, each of them have the same 00:51:28.030 --> 00:51:29.200 distribution. 00:51:29.200 --> 00:51:32.320 I mean, you'd be very very surprised if you looked at 00:51:32.320 --> 00:51:37.640 these taps after the way we've derived them and you didn't 00:51:37.640 --> 00:51:40.260 find the same distribution on the real part as 00:51:40.260 --> 00:51:41.510 the imaginary part. 00:51:48.260 --> 00:51:51.900 If you look at the real part and imaginary part, they're 00:51:51.900 --> 00:51:56.100 simply coming as an arbitrary phase, which comes from some 00:51:56.100 --> 00:51:59.800 arbitrary demodulating frequency that you've used 00:51:59.800 --> 00:52:02.670 with a phase in there that doesn't have any physical 00:52:02.670 --> 00:52:04.540 connotation at all. 00:52:04.540 --> 00:52:07.650 I mean, what is real is what you choose to call real and 00:52:07.650 --> 00:52:11.490 what is imaginary is what you choose to call imaginary. 00:52:11.490 --> 00:52:15.650 If you made your time reference just a little bit, a 00:52:15.650 --> 00:52:18.870 smidgen away from where your time reference is, what is 00:52:18.870 --> 00:52:21.840 real would become imaginary and what is imaginary would 00:52:21.840 --> 00:52:25.250 become real again and you can't expect the channel to 00:52:25.250 --> 00:52:29.140 know what time reference you've chosen. 00:52:29.140 --> 00:52:32.350 I mean, this is sort of a big philosophical argument, but 00:52:32.350 --> 00:52:36.400 you certainly wouldn't expect these random variables to have 00:52:36.400 --> 00:52:38.900 a distribution which is anything other 00:52:38.900 --> 00:52:41.220 than uniform in phase. 00:52:41.220 --> 00:52:45.750 So that's the distribution we get and what we now have is we 00:52:45.750 --> 00:52:53.170 changed this awful physical situation into a very simple 00:52:53.170 --> 00:52:58.070 analytical situation where what we're doing is modeling 00:52:58.070 --> 00:53:03.350 this complex channel as just a bunch of taps in time, at some 00:53:03.350 --> 00:53:07.190 bandwidth that we're using and each of these taps is now 00:53:07.190 --> 00:53:14.820 Gaussian and the output is going to be the discrete sum 00:53:14.820 --> 00:53:18.710 as we take a discrete sequence of inputs, pass it through 00:53:18.710 --> 00:53:23.100 this discrete filter and then we add up these outputs and at 00:53:23.100 --> 00:53:25.840 that point, we have to worry about how do we detect things 00:53:25.840 --> 00:53:27.730 from what's coming out? 00:53:27.730 --> 00:53:31.040 So they say here -- this is a really flaky modeling 00:53:31.040 --> 00:53:36.370 assumption and it's only partly justified by looking at 00:53:36.370 --> 00:53:40.890 lots of different situations that a given cellular phone 00:53:40.890 --> 00:53:44.310 would be placed in. 00:53:44.310 --> 00:53:47.790 If we look at this whole ensemble, it makes a little 00:53:47.790 --> 00:53:56.380 bit of sense, but really what we're doing here is saying 00:53:56.380 --> 00:54:00.280 what we want is a vehicle to let us understand how to 00:54:00.280 --> 00:54:02.010 receive these waveforms. 00:54:02.010 --> 00:54:04.350 We know there is going to be fading. 00:54:04.350 --> 00:54:07.290 Since we know there is going to be fading, we have to find 00:54:07.290 --> 00:54:10.520 some kind of statistical model for it if we're going to find 00:54:10.520 --> 00:54:13.100 sensible things to do and this is the one 00:54:13.100 --> 00:54:14.350 we're going to pick. 00:54:16.800 --> 00:54:20.530 It's very common instead of using a Rayleigh fading model 00:54:20.530 --> 00:54:22.870 to use a Rician fading model. 00:54:22.870 --> 00:54:28.620 A Rician fading model makes sense because if you have a 00:54:28.620 --> 00:54:34.790 line of sight directly to a base station, the waveform you 00:54:34.790 --> 00:54:38.090 get from the base station is going to be kind of strong and 00:54:38.090 --> 00:54:41.790 all of these waveforms you get reflected from other obstacles 00:54:41.790 --> 00:54:45.320 and so forth are going to be rather weak. 00:54:45.320 --> 00:54:48.300 So in that case, you're going to have one strong received 00:54:48.300 --> 00:54:52.290 waveform, which is one big peak and everything else is 00:54:52.290 --> 00:54:53.840 going to be very weak. 00:54:56.910 --> 00:54:59.490 The thing that that says is the kind of statistics that 00:54:59.490 --> 00:55:04.020 you want are not a sum of a lot of little tiny things, but 00:55:04.020 --> 00:55:07.910 the sum of one big thing and lots of tiny things. 00:55:07.910 --> 00:55:09.910 That's not a good model either. 00:55:09.910 --> 00:55:13.340 What you would really like is a sum of a bunch of medium 00:55:13.340 --> 00:55:17.420 sized things most of the time, but about as far as anybody 00:55:17.420 --> 00:55:26.280 gets talking about statistical models for fading channels is 00:55:26.280 --> 00:55:30.090 talking about Rician model and talking about Rayleigh models 00:55:30.090 --> 00:55:32.820 and then worrying about what happens the different paths of 00:55:32.820 --> 00:55:34.970 these filters. 00:55:34.970 --> 00:55:38.590 The trouble with Rician random variables is they're very 00:55:38.590 --> 00:55:39.710 complicated. 00:55:39.710 --> 00:55:42.050 They have vessel functions in them. 00:55:42.050 --> 00:55:47.100 If you read the notes I passed out today, you find that when 00:55:47.100 --> 00:55:51.690 you try to detect things, when they've gone through this kind 00:55:51.690 --> 00:55:57.780 of Rician fading, everything is just a bloody mess. 00:55:57.780 --> 00:56:00.690 That's the way it is. 00:56:03.210 --> 00:56:07.850 Since we are talking now about a statistical model for these 00:56:07.850 --> 00:56:12.463 channels, we would like to have something called a tap 00:56:12.463 --> 00:56:19.040 gain correlation function, which is going to tell us how 00:56:19.040 --> 00:56:22.540 each of these taps vary in time. 00:56:25.650 --> 00:56:33.020 That's just the expected value of Gkm and G of k prime n. 00:56:33.020 --> 00:56:36.050 What we're going to assume, and what is really a very good 00:56:36.050 --> 00:56:40.220 assumption, is if you look at the things that add up as taps 00:56:40.220 --> 00:56:45.870 under a particular value of k and under some other value of 00:56:45.870 --> 00:56:50.040 k -- in other words, all of the paths that are coming in 00:56:50.040 --> 00:56:53.710 in one range of delays and all the paths that come in on 00:56:53.710 --> 00:56:56.450 another range of delays, those things are pretty much 00:56:56.450 --> 00:56:58.190 independent of each other. 00:56:58.190 --> 00:57:05.080 So almost always, when you start calculating tap gain 00:57:05.080 --> 00:57:07.840 correlation functions, you assume that this is 00:57:07.840 --> 00:57:10.370 independent when k is unequal to k prime. 00:57:10.370 --> 00:57:13.340 So really the only thing you're concerned about here is 00:57:13.340 --> 00:57:18.030 how these quantities vary in time. 00:57:18.030 --> 00:57:20.830 We've already talked about how they vary in time. 00:57:20.830 --> 00:57:26.860 In other words, that was why we talked about all of this 00:57:26.860 --> 00:57:30.520 business about time coherence and Doppler shifts. 00:57:30.520 --> 00:57:33.610 The thing which is causing these things to change in time 00:57:33.610 --> 00:57:36.000 is these Doppler shifts. 00:57:36.000 --> 00:57:39.040 We have already decided that the amount of time the channel 00:57:39.040 --> 00:57:46.040 remain stable is about 1 over 2 times this Doppler spread. 00:57:46.040 --> 00:57:50.390 Here we have an analytical, statistical way of doing the 00:57:50.390 --> 00:57:52.570 same thing. 00:57:52.570 --> 00:57:56.840 In other words, you can look at this correlation function 00:57:56.840 --> 00:58:00.710 here and you can see how long it takes before it drops 00:58:00.710 --> 00:58:02.800 essentially to zero. 00:58:02.800 --> 00:58:08.270 So you can define the time coherence in terms of this 00:58:08.270 --> 00:58:12.900 duration here just as well as the way we've already done. 00:58:12.900 --> 00:58:17.410 In other words, how big does n have to get before this thing 00:58:17.410 --> 00:58:19.660 gets small? 00:58:19.660 --> 00:58:22.080 What's the advantage of doing it this way instead of in 00:58:22.080 --> 00:58:25.060 terms of Doppler shifts? 00:58:25.060 --> 00:58:27.160 If you want to measure it, you don't have to go out and 00:58:27.160 --> 00:58:29.210 measure where these paths are. 00:58:29.210 --> 00:58:34.670 If you want to measure it, you just take a cellular phone and 00:58:34.670 --> 00:58:37.490 you measure where these things are coming in and how they 00:58:37.490 --> 00:58:41.380 change in time and that gives you a direct view of what the 00:58:41.380 --> 00:58:43.970 time coherence is. 00:58:43.970 --> 00:58:49.360 Why that W was there, there's a problem in the problem set 00:58:49.360 --> 00:58:51.720 that will give you some insight into why it's there, 00:58:51.720 --> 00:58:55.720 but otherwise it's just arbitrary. 00:58:55.720 --> 00:58:58.290 I want to spend the rest of the time talking about 00:58:58.290 --> 00:59:03.440 Rayleigh fading, because Rayleigh fading is something 00:59:03.440 --> 00:59:05.420 you can analyze easily. 00:59:10.320 --> 00:59:12.280 We're going to do something even simpler. 00:59:12.280 --> 00:59:15.110 We're going to do Rayleigh fading where you have 00:59:15.110 --> 00:59:17.380 a single tap model. 00:59:17.380 --> 00:59:21.040 In other words, where the bandwidth that you're using is 00:59:21.040 --> 00:59:28.090 small enough that all these different paths all fall in 00:59:28.090 --> 00:59:29.630 the same delay range. 00:59:29.630 --> 00:59:37.690 In other words, it's where the multipath spread, L, is small 00:59:37.690 --> 00:59:42.370 relative to 1 over W. That's what's called flat fading, so 00:59:42.370 --> 00:59:45.550 we're assuming flat fading and we're assuming that it changes 00:59:45.550 --> 00:59:48.240 in time according to a Rayleigh model. 00:59:50.980 --> 00:59:59.030 Suppose that we do something really simple, like trying the 00:59:59.030 --> 01:00:02.060 antipodal signalling that we've been using all along, 01:00:02.060 --> 01:00:06.130 which is really neat when you have Gaussian noise. 01:00:06.130 --> 01:00:14.390 It's just about the best kind of signaling that you can use. 01:00:14.390 --> 01:00:18.410 It doesn't work at all here and it doesn't work at all 01:00:18.410 --> 01:00:23.360 because since this channel is Rayleigh fading, you don't 01:00:23.360 --> 01:00:27.570 know what the phase is of the channel, so you send a bit and 01:00:27.570 --> 01:00:31.180 what comes in is something which, sure, it has an 01:00:31.180 --> 01:00:34.880 amplitude to it, but it has a completely random phase to it, 01:00:34.880 --> 01:00:36.530 so you can't tell the difference between 01:00:36.530 --> 01:00:38.450 one and minus one. 01:00:38.450 --> 01:00:42.350 The only way you can tell is to send first one signal and 01:00:42.350 --> 01:00:45.620 then another signal, but if you send one signal and 01:00:45.620 --> 01:00:49.910 another signal, you might as well choose a single pattern 01:00:49.910 --> 01:00:52.720 which lets you do this sensibly. 01:00:52.720 --> 01:00:55.840 So the particular thing that we'll look at is something 01:00:55.840 --> 01:00:58.880 called post position modulation. 01:00:58.880 --> 01:01:01.850 This is essentially the same as about ten other schemes and 01:01:01.850 --> 01:01:06.580 we'll talk about that a little more later, but what you're 01:01:06.580 --> 01:01:10.630 going to use is two degrees of freedom, which is two 01:01:10.630 --> 01:01:13.060 different time instants. 01:01:13.060 --> 01:01:15.930 In the first instant, you're going to send an a. 01:01:15.930 --> 01:01:19.680 In the second instant, you're going to send a zero where 01:01:19.680 --> 01:01:22.850 conversely, you're going to send a zero in the first 01:01:22.850 --> 01:01:26.750 degree of freedom and an a in the second degree of freedom. 01:01:26.750 --> 01:01:30.280 So you can do this either with two different frequencies, you 01:01:30.280 --> 01:01:32.570 can do it with two different time instants. 01:01:32.570 --> 01:01:34.960 You can do it in all sorts of different ways. 01:01:34.960 --> 01:01:39.660 Any time you have two degrees of freedom, you either use one 01:01:39.660 --> 01:01:42.940 degree of freedom or you use the other degree of freedom. 01:01:42.940 --> 01:01:45.870 It's not like the situation we've talked about before, 01:01:45.870 --> 01:01:49.320 because these degrees of freedom are now complex 01:01:49.320 --> 01:01:50.910 degrees of freedom. 01:01:50.910 --> 01:01:55.420 If I send the one, it's going to come in as some complex 01:01:55.420 --> 01:02:01.140 number, which is going to have uniform phase. 01:02:01.140 --> 01:02:09.240 So my hypotheses now are one hypothesis is a and zero and 01:02:09.240 --> 01:02:13.260 the other hypothesis is zero and a. 01:02:13.260 --> 01:02:18.550 What's going to happen with these hypotheses is that 01:02:18.550 --> 01:02:21.440 first, the channel is going to do its thing to them. 01:02:21.440 --> 01:02:25.430 In other words, if I send this, the channel is going to 01:02:25.430 --> 01:02:27.470 take that a. 01:02:27.470 --> 01:02:31.780 It's going to multiply it by a circularly symmetric Gaussian 01:02:31.780 --> 01:02:37.150 random variable, so it's going to come as some blob and in 01:02:37.150 --> 01:02:41.410 the other dimension, what I send is a zero, so what comes 01:02:41.410 --> 01:02:43.170 in there is nothing. 01:02:43.170 --> 01:02:47.890 I'm going to add noise to this, so when I send a zero, 01:02:47.890 --> 01:02:49.530 I'm going to get -- 01:02:49.530 --> 01:02:51.950 from the channel, I get a blob in this 01:02:51.950 --> 01:02:53.490 first degree of freedom. 01:02:53.490 --> 01:02:55.440 The noise adds another blob, which is 01:02:55.440 --> 01:02:58.110 also circularly symmetric. 01:02:58.110 --> 01:03:01.600 What comes in the other degree of freedom is zero on the 01:03:01.600 --> 01:03:04.120 channel and a blob of noise. 01:03:04.120 --> 01:03:08.660 So I'm going to base my detection on either a big blob 01:03:08.660 --> 01:03:12.310 or a little blob and on the other hypothesis, I got a 01:03:12.310 --> 01:03:15.610 little blob here and a big blob there. 01:03:15.610 --> 01:03:21.210 So the question is, you have to look at this pair of blobs 01:03:21.210 --> 01:03:23.380 and decide from it what you think was 01:03:23.380 --> 01:03:25.840 the most likely signals. 01:03:25.840 --> 01:03:34.000 Sounds hard, but it turns out that it's easy, because look, 01:03:34.000 --> 01:03:39.820 under the hypothesis H 0, what you're going to receive -- 01:03:39.820 --> 01:03:43.800 and these are a times a complex 01:03:43.800 --> 01:03:47.310 variable, G 0 plus Z 0. 01:03:47.310 --> 01:03:50.090 Both of these are Gaussian random variables, both zero 01:03:50.090 --> 01:03:55.630 mean, both circularly symmetric and V 1 is going to 01:03:55.630 --> 01:03:57.630 be just Z 1. 01:03:57.630 --> 01:04:01.340 So here I have a sum of two Gaussian random variables. 01:04:01.340 --> 01:04:04.840 Here I have one Gaussian random variable. 01:04:04.840 --> 01:04:17.190 V 0 is complex Gaussian and its variance is a squared plus 01:04:17.190 --> 01:04:21.550 N 0 times W, because that's what the noise is. 01:04:21.550 --> 01:04:27.320 So I look at the probability density of these two complex 01:04:27.320 --> 01:04:31.330 Gaussian random variables, sample value v 0 and v 1, 01:04:31.330 --> 01:04:33.350 conditional on H 0. 01:04:33.350 --> 01:04:35.700 What I have is some constant here -- 01:04:35.700 --> 01:04:37.370 I don't even care about it -- 01:04:37.370 --> 01:04:41.470 times e need to the minus v 0 squared divided by a 01:04:41.470 --> 01:04:45.070 squared plus wn 0. 01:04:45.070 --> 01:04:49.700 This is the variance of V 0, because V 0 was a 01:04:49.700 --> 01:04:51.820 squared times G 0. 01:04:51.820 --> 01:04:54.000 That's the a squared there. 01:04:54.000 --> 01:05:00.170 Variance is Z 0 WN 0 and for the other random variable -- 01:05:00.170 --> 01:05:03.260 that's the little blob random variable -- 01:05:03.260 --> 01:05:06.660 probability density of that is minus v 1 squared 01:05:06.660 --> 01:05:11.510 divided by WN 0. 01:05:11.510 --> 01:05:14.640 For the alternative hypothesis, I get the same 01:05:14.640 --> 01:05:17.470 constant out front, which I haven't even bother to write 01:05:17.470 --> 01:05:23.480 down, times e to the minus v 0 squared over WN 0. 01:05:23.480 --> 01:05:26.430 Here are the other variables: The one where we have the big 01:05:26.430 --> 01:05:32.060 blob, so that's the variance a squared plus WN 0. 01:05:32.060 --> 01:05:36.980 I take the log likelihood ratio, which is to take the 01:05:36.980 --> 01:05:42.400 logarithm of the ratio of these two hypotheses, of these 01:05:42.400 --> 01:05:48.450 two likelihoods, and when I take this quantity and divide 01:05:48.450 --> 01:05:52.890 it by this quantity, these are both up in the exponents, so 01:05:52.890 --> 01:05:56.590 I'm just taking this and subtracting this. 01:05:56.590 --> 01:06:00.450 When I subtract this quantity from this quantity and then 01:06:00.450 --> 01:06:05.000 subtract this quantity from this quantity, what I get is 01:06:05.000 --> 01:06:10.110 just this quantity here. 01:06:10.110 --> 01:06:14.480 I mean, you can see it's simple because this difference 01:06:14.480 --> 01:06:16.970 is going to be the same as this difference, except they 01:06:16.970 --> 01:06:18.670 have a different sign. 01:06:18.670 --> 01:06:22.060 So it's just algebra to find out what happens when you when 01:06:22.060 --> 01:06:25.190 you take this minus this and then you take 01:06:25.190 --> 01:06:27.980 minus this plus this. 01:06:27.980 --> 01:06:30.660 This is what you get. 01:06:30.660 --> 01:06:34.400 Then you look at that for a bit and you say, how do I find 01:06:34.400 --> 01:06:37.580 the probability of error from this? 01:06:37.580 --> 01:06:41.210 Before we find the probability of error, the first thing we 01:06:41.210 --> 01:06:47.050 have to do is say, what's the maximum 01:06:47.050 --> 01:06:50.500 likelihood rule to use here? 01:06:50.500 --> 01:06:54.530 Maximum likelihood rule is you take the log likelihood ratio 01:06:54.530 --> 01:06:57.730 and if it's positive, you choose v0. 01:06:57.730 --> 01:07:01.770 If it's negative, you choose v1 and if it's 0 -- if 0 was 01:07:01.770 --> 01:07:05.390 zero probability, it doesn't matter what you do. 01:07:05.390 --> 01:07:09.140 So what we're interested in now is, what is the 01:07:09.140 --> 01:07:13.800 probability that this quantity here is going to 01:07:13.800 --> 01:07:17.460 be bigger than zero? 01:07:17.460 --> 01:07:24.720 But you see, this isn't hard because when you look at the 01:07:24.720 --> 01:07:29.430 difference between v 0 squared and v 1 squared, v 0 squared 01:07:29.430 --> 01:07:32.430 is an exponential random variable. 01:07:32.430 --> 01:07:37.490 It's just the energy of a complex Gaussian. 01:07:37.490 --> 01:07:41.210 So we're taking one exponential random variable, 01:07:41.210 --> 01:07:44.710 subtracting off another exponential random variable 01:07:44.710 --> 01:07:49.790 and we're just looking at it over on the region where these 01:07:49.790 --> 01:07:53.170 things are positive. 01:07:53.170 --> 01:07:59.470 When you integrate that, this is the answer that you get. 01:07:59.470 --> 01:08:00.875 I mean, it's not hard to integrate it. 01:08:00.875 --> 01:08:02.125 I just don't -- 01:08:04.800 --> 01:08:07.850 and if you look at the notes, the notes carry it out in all 01:08:07.850 --> 01:08:12.900 of its gory detail, but it really is about two steps. 01:08:12.900 --> 01:08:17.350 The thing we want to look at is this result here, because 01:08:17.350 --> 01:08:21.370 it looks so different from all of the error probability 01:08:21.370 --> 01:08:23.610 results that we've gotten before. 01:08:23.610 --> 01:08:26.880 I mean, before, error probabilities go down 01:08:26.880 --> 01:08:33.670 exponentially as a square of the signal amplitudes that 01:08:33.670 --> 01:08:37.350 we're using and they go down exponentially with the energy 01:08:37.350 --> 01:08:38.890 that we're using. 01:08:38.890 --> 01:08:46.230 Here, this crazy thing is going down as 1 over 2 plus 01:08:46.230 --> 01:08:48.270 energy to noise ratio. 01:08:48.270 --> 01:09:02.390 This is 1 over 2 plus Ed over N 0. 01:09:02.390 --> 01:09:04.880 So it really goes down slowly. 01:09:04.880 --> 01:09:07.440 So what's going on? 01:09:07.440 --> 01:09:10.500 I mean, why is it that with all of these Gaussian random 01:09:10.500 --> 01:09:14.440 variables where things go down so fast, we're really in such 01:09:14.440 --> 01:09:15.690 deep trouble here? 01:09:18.810 --> 01:09:23.010 The trouble is, every time the fading is significant and the 01:09:23.010 --> 01:09:27.340 fading is significant whenever -- 01:09:27.340 --> 01:09:32.630 under hypothesis 0, if you don't have much 01:09:32.630 --> 01:09:34.190 of a channel -- 01:09:34.190 --> 01:09:37.110 namely, if both v 0 and v 1 are both very -- 01:09:41.480 --> 01:09:45.940 if both the real part of G 0 and the imaginary part of G 0 01:09:45.940 --> 01:09:47.750 are both very small -- 01:09:47.750 --> 01:09:51.160 which happens with reasonable probability -- 01:09:51.160 --> 01:09:59.280 then you don't have a chance of trying to decode correctly 01:09:59.280 --> 01:10:02.740 because the two signals are both going to look the same, 01:10:02.740 --> 01:10:06.660 because both of them are just Gaussian random variables. 01:10:06.660 --> 01:10:12.220 So as a result of that, when you're dealing with Rayleigh 01:10:12.220 --> 01:10:17.580 fading, you really have to find something else to do in 01:10:17.580 --> 01:10:19.700 order to make this work. 01:10:19.700 --> 01:10:22.000 I mean, if you have Rayleigh fading and you just try to 01:10:22.000 --> 01:10:26.320 communicate in the way that we tried to communicate before -- 01:10:26.320 --> 01:10:30.700 just sending individual bits and hoping for the best, 01:10:30.700 --> 01:10:32.890 you're in very deep trouble. 01:10:32.890 --> 01:10:36.660 So next time we'll start to talk about ways of dealing 01:10:36.660 --> 01:10:41.170 with this using diversity, using channel measurement, 01:10:41.170 --> 01:10:44.400 using all sorts of other things, but -- 01:10:44.400 --> 01:10:46.190 I don't know what happened to the person who was supposed to 01:10:46.190 --> 01:10:53.380 evaluate the class, but maybe they will come in next week. 01:10:53.380 --> 01:10:54.900 I don't Know.