1 00:00:00,000 --> 00:00:02,555 The following content is provided under a Creative 2 00:00:02,555 --> 00:00:03,650 Commons license. 3 00:00:03,650 --> 00:00:06,600 Your support will help MIT OpenCourseWare continue to 4 00:00:06,600 --> 00:00:10,030 offer high quality educational resources for free. 5 00:00:10,030 --> 00:00:12,810 To make a donation or to view additional materials from 6 00:00:12,810 --> 00:00:16,560 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:16,560 --> 00:00:17,810 ocw.mit.edu. 8 00:00:21,080 --> 00:00:27,350 PROFESSOR: I want to pretty much finish up today talking 9 00:00:27,350 --> 00:00:31,420 about modulation. 10 00:00:31,420 --> 00:00:34,250 Part of the reason we don't spend much time on this is, I 11 00:00:34,250 --> 00:00:39,360 think most of you have seen some kind of undergraduate 12 00:00:39,360 --> 00:00:42,260 class in communication or signal 13 00:00:42,260 --> 00:00:45,480 processing or something. 14 00:00:45,480 --> 00:00:50,070 Where you do a lot of this because a lot of this is just 15 00:00:50,070 --> 00:00:54,170 nice exercises and -- well, nice or not nice, depending on 16 00:00:54,170 --> 00:00:58,130 the way you about it -- in multiplying waveforms by 17 00:00:58,130 --> 00:01:02,350 cosines and by sines and by complex exponentials. 18 00:01:02,350 --> 00:01:06,770 And you keep doing this, and doing this, and doing this and 19 00:01:06,770 --> 00:01:09,330 you get these long expressions. 20 00:01:09,330 --> 00:01:13,180 And and it all means something. 21 00:01:13,180 --> 00:01:16,000 So, most of you have seen a good deal of this. 22 00:01:16,000 --> 00:01:20,230 You probably haven't seen the Nyquist criterion before. 23 00:01:20,230 --> 00:01:23,360 I'm sure you haven't seen the Nyquist criterion done 24 00:01:23,360 --> 00:01:28,680 carefully, because I've never seen it done carefully before. 25 00:01:28,680 --> 00:01:32,990 I don't think it is done carefully anywhere else. 26 00:01:32,990 --> 00:01:34,470 And I'm sure you don't care about this. 27 00:01:34,470 --> 00:01:37,820 But, at some point, if you deal with this stuff, you will 28 00:01:37,820 --> 00:01:38,570 care about it. 29 00:01:38,570 --> 00:01:41,070 Because at some point you will need it. 30 00:01:41,070 --> 00:01:46,220 But anyway, we were looking at pulse amplitude modulation, 31 00:01:46,220 --> 00:01:48,050 you remember. 32 00:01:48,050 --> 00:01:51,600 We were looking at the modulated waveform. 33 00:01:51,600 --> 00:01:56,980 And this is all done down at baseband, now, remember. 34 00:01:56,980 --> 00:02:02,430 And the interesting problems down at baseband are, first, 35 00:02:02,430 --> 00:02:05,280 how do you choose a signal set. 36 00:02:05,280 --> 00:02:09,530 Namely, how do you choose those quantities u sub k, out 37 00:02:09,530 --> 00:02:12,240 of some set of possible values. 38 00:02:12,240 --> 00:02:15,670 So they have an appropriate distance between them. 39 00:02:15,670 --> 00:02:19,400 And, next, how do you choose this waveform p of t, which is 40 00:02:19,400 --> 00:02:23,500 called the pulse, that you're using. 41 00:02:23,500 --> 00:02:26,410 And then, after you've chosen those two things, there's 42 00:02:26,410 --> 00:02:28,710 nothing more to be done. 43 00:02:28,710 --> 00:02:33,130 You simply form the modulated waveform as u of t equals the 44 00:02:33,130 --> 00:02:36,400 sum of the signals which are coming in regularly at 45 00:02:36,400 --> 00:02:37,900 intervals of t. 46 00:02:37,900 --> 00:02:42,300 You multiply each one of them by this waveform, p of t. 47 00:02:42,300 --> 00:02:47,150 Or, alternatively, you think of the waveform as being a sum 48 00:02:47,150 --> 00:02:48,850 of impulses. 49 00:02:48,850 --> 00:02:51,270 Each impulse weighted by u sub k. 50 00:02:51,270 --> 00:02:54,520 And you think of passing this string of impulses through a 51 00:02:54,520 --> 00:02:58,190 filter with impulse response p of t. 52 00:02:58,190 --> 00:03:00,890 Which is usually the way that it's implemented, except of 53 00:03:00,890 --> 00:03:03,900 course you're not using ideal impulses. 54 00:03:03,900 --> 00:03:08,550 You're using sharp pulses which have a flat spectrum 55 00:03:08,550 --> 00:03:11,950 over the bandwidth of the filter p of t. 56 00:03:11,950 --> 00:03:15,340 But anyway, somehow or other you implement this. 57 00:03:15,340 --> 00:03:18,790 And that is this fundamental step we've been talking about 58 00:03:18,790 --> 00:03:21,920 for quite a while now, of how do you turn 59 00:03:21,920 --> 00:03:24,020 sequences into waveforms. 60 00:03:24,020 --> 00:03:27,640 How do you turn waveforms into sequences, it's a fundamental 61 00:03:27,640 --> 00:03:28,860 piece of source coding. 62 00:03:28,860 --> 00:03:31,890 It's a fundamental piece of channel coding. 63 00:03:31,890 --> 00:03:35,350 Here we're doing the channel coding part of it. 64 00:03:35,350 --> 00:03:38,050 Then we did something kind of flaky. 65 00:03:38,050 --> 00:03:42,040 We said that when we received this waveform, what we were 66 00:03:42,040 --> 00:03:46,290 going to do is to first pass it through another filter, and 67 00:03:46,290 --> 00:03:48,180 then sample it. 68 00:03:48,180 --> 00:03:51,950 If you've done the homework yet, you will probably have 69 00:03:51,950 --> 00:03:55,850 looked at what happens when you take an arbitrary linear 70 00:03:55,850 --> 00:04:02,750 operation on this received signal to try to retrieve what 71 00:04:02,750 --> 00:04:05,090 these signals u sub k are. 72 00:04:05,090 --> 00:04:09,700 And you will have found that this filtering and sampling is 73 00:04:09,700 --> 00:04:16,170 in fact not a general linear operation, but it's the only 74 00:04:16,170 --> 00:04:20,060 general linear operation that you're interested in as far as 75 00:04:20,060 --> 00:04:23,550 retrieving these signals from that waveform. 76 00:04:23,550 --> 00:04:28,270 So, in fact, this is, in fact, more general than it looks. 77 00:04:28,270 --> 00:04:32,910 There a confusing thing here. 78 00:04:32,910 --> 00:04:42,900 If you receive u of t at the receiver, and your question 79 00:04:42,900 --> 00:04:50,400 is, how do we get this sequence of samples u sub k 80 00:04:50,400 --> 00:05:01,740 out of it, and suppose that this pulse p of t or the -- 81 00:05:01,740 --> 00:05:05,310 I mean, suppose for example that the pulse is a narrow 82 00:05:05,310 --> 00:05:10,870 bandwidth pulse, and there's just no way you can perform 83 00:05:10,870 --> 00:05:14,680 linear operations and get these signals out from it. 84 00:05:14,680 --> 00:05:18,000 Is it possible to do nonlinear operations and get the signals 85 00:05:18,000 --> 00:05:20,310 out from it? 86 00:05:20,310 --> 00:05:22,270 And you ought to think about this question a little bit 87 00:05:22,270 --> 00:05:25,020 because it's kind of an interesting one. 88 00:05:28,760 --> 00:05:34,180 If I don't tell you what the signals u sub k are, if I ask 89 00:05:34,180 --> 00:05:39,260 you to build something which extracts these samples u sub 90 00:05:39,260 --> 00:05:44,600 k, without any idea of what signal set they're taken from, 91 00:05:44,600 --> 00:05:46,080 then there's nothing you can do better 92 00:05:46,080 --> 00:05:48,890 than a linear operation. 93 00:05:48,890 --> 00:05:53,810 And, in fact, if this pulse p of t has a bandwidth that's 94 00:05:53,810 --> 00:05:57,340 too narrow, you're just stuck. 95 00:05:57,340 --> 00:06:01,440 If I tell you that these u sub k are drawn from a particular 96 00:06:01,440 --> 00:06:05,620 signal set -- for example, suppose they're binary -- then 97 00:06:05,620 --> 00:06:09,950 you have an enormous extra amount of information. 98 00:06:09,950 --> 00:06:14,220 And you can, in fact, given this waveform, even though p 99 00:06:14,220 --> 00:06:17,230 of t is a very, very narrow band, you can still in 100 00:06:17,230 --> 00:06:20,600 principle get these binary signals out. 101 00:06:20,600 --> 00:06:23,070 So what's the game we're playing here? 102 00:06:23,070 --> 00:06:27,040 I mean, we're doing kind of a phony thing. 103 00:06:27,040 --> 00:06:31,810 We're restricting ourselves to only linear operations. 104 00:06:31,810 --> 00:06:35,600 We are restricting ourselves to retrieving these signals 105 00:06:35,600 --> 00:06:40,380 without knowing anything about what the signal set is. 106 00:06:40,380 --> 00:06:43,230 Which is not really the problem that we're interested 107 00:06:43,230 --> 00:06:44,620 in looking at. 108 00:06:44,620 --> 00:06:47,270 So what are we really doing? 109 00:06:47,270 --> 00:06:49,690 What we're really doing is trying to look at this 110 00:06:49,690 --> 00:06:53,240 question of modulation before we look at 111 00:06:53,240 --> 00:06:54,490 the question of noise. 112 00:06:58,350 --> 00:07:01,500 And after we start looking at noise, the thing that we're 113 00:07:01,500 --> 00:07:04,900 going to find is that this received waveform is a 114 00:07:04,900 --> 00:07:08,650 received waveform plus a lot of noise on top of it. 115 00:07:08,650 --> 00:07:11,790 And if there's a lot of noise on top of it, these non-linear 116 00:07:11,790 --> 00:07:14,295 operations you can think of are not going 117 00:07:14,295 --> 00:07:17,720 to work very well. 118 00:07:17,720 --> 00:07:23,400 And I guess the problem is, you just have to take that on 119 00:07:23,400 --> 00:07:24,640 faith right now. 120 00:07:24,640 --> 00:07:28,240 And after we look at random processes, and after we look 121 00:07:28,240 --> 00:07:31,550 at how to deal with the noise waveforms, you will in fact 122 00:07:31,550 --> 00:07:35,180 see that these operations we're talking about here are 123 00:07:35,180 --> 00:07:37,770 exactly the things we want to do. 124 00:07:37,770 --> 00:07:39,650 Now, I don't know whether this is the right 125 00:07:39,650 --> 00:07:41,730 way to do it or not. 126 00:07:41,730 --> 00:07:44,670 Dealing with all this phony stuff that you see in 127 00:07:44,670 --> 00:07:48,650 elementary courses before dealing with the real stuff. 128 00:07:48,650 --> 00:07:51,370 I think it probably is, because for most of you this 129 00:07:51,370 --> 00:07:54,440 is sort of familiar and we'll come back later 130 00:07:54,440 --> 00:07:56,430 and make it all right. 131 00:07:56,430 --> 00:08:00,210 But who knows? 132 00:08:00,210 --> 00:08:01,460 Anyway. 133 00:08:04,090 --> 00:08:10,370 what we want to do then is to find some composite filter, g, 134 00:08:10,370 --> 00:08:19,450 which is what happens when you take the filter p, or the 135 00:08:19,450 --> 00:08:24,685 pulse p, pass it through a filter q of t, what you get is 136 00:08:24,685 --> 00:08:26,960 the convolution of p and q. 137 00:08:26,960 --> 00:08:31,390 And, therefore, what comes out after this filtering is done 138 00:08:31,390 --> 00:08:35,710 is a received waveform which is just a sum over k of u sub 139 00:08:35,710 --> 00:08:39,650 k times g of t minus k t. 140 00:08:39,650 --> 00:08:43,000 In other words, these two filters are not doing anything 141 00:08:43,000 --> 00:08:44,070 extra for you. 142 00:08:44,070 --> 00:08:46,890 All they are is a way of putting part of the filter at 143 00:08:46,890 --> 00:08:50,050 the transmitter, part of the filter at the receiver. 144 00:08:50,050 --> 00:08:53,090 When you look at noise you'll find out that there is some 145 00:08:53,090 --> 00:08:56,120 real difference between what's done at the transmitter and 146 00:08:56,120 --> 00:08:59,210 what's done at the receiver, because the noise comes in 147 00:08:59,210 --> 00:09:01,670 between the transmitter and the receiver. 148 00:09:01,670 --> 00:09:04,640 But for now, it doesn't make any difference so the only 149 00:09:04,640 --> 00:09:07,200 thing we're interested in is the properties of this 150 00:09:07,200 --> 00:09:10,370 composite waveform, g of t. 151 00:09:10,370 --> 00:09:15,160 And what we find is that if we receive r of t, which is this 152 00:09:15,160 --> 00:09:19,910 waveform sum of u k g of t minus k t, and if we want to 153 00:09:19,910 --> 00:09:24,580 retrieve the coefficient u sub k, it becomes duck soup to do 154 00:09:24,580 --> 00:09:29,830 so if in fact this waveform is like a sampling waveform. 155 00:09:29,830 --> 00:09:37,540 In other words, if g of t is equal to 1 at t equals 0, and 156 00:09:37,540 --> 00:09:41,900 it's equal to 0 at each other sample point, then all we have 157 00:09:41,900 --> 00:09:46,170 to do is take this waveform, simply sample it each capital 158 00:09:46,170 --> 00:09:48,930 T seconds, and we get these coefficients out 159 00:09:48,930 --> 00:09:50,840 automatically. 160 00:09:50,840 --> 00:09:55,280 Now, I should warn you at this point that in the notes the 161 00:09:55,280 --> 00:10:02,030 scaling business is not done quite right. 162 00:10:02,030 --> 00:10:09,110 Sometimes we talk about g of t as a filter whose shifts are 163 00:10:09,110 --> 00:10:10,930 orthonormal to each other. 164 00:10:10,930 --> 00:10:14,470 And sometimes as a filter whose shifts are orthogonal to 165 00:10:14,470 --> 00:10:15,720 each other. 166 00:10:15,720 --> 00:10:19,950 I advise you not to worry about that, because you have 167 00:10:19,950 --> 00:10:22,190 to make changes about five times in the notes 168 00:10:22,190 --> 00:10:24,160 to make it all right. 169 00:10:24,160 --> 00:10:26,600 And I will put up a new version of the notes on the 170 00:10:26,600 --> 00:10:28,590 web which in fact does this right. 171 00:10:28,590 --> 00:10:30,700 It's not important. 172 00:10:30,700 --> 00:10:35,310 It's just this old question of, do you use a sinc function 173 00:10:35,310 --> 00:10:37,960 when you're dealing with strictly band-limited 174 00:10:37,960 --> 00:10:42,490 functions, or do you multiply the sinc function by 1 over 175 00:10:42,490 --> 00:10:45,880 the square root of t to make it orthonormal. 176 00:10:45,880 --> 00:10:50,840 Or do you multiply it by 1 over t to make it -- 177 00:10:50,840 --> 00:10:53,570 I mean, you can scale it in a number of different ways. 178 00:10:53,570 --> 00:10:56,120 And, fundamentally, it doesn't make any difference. 179 00:10:56,120 --> 00:10:58,290 It's just that if you want to get the right answer you have 180 00:10:58,290 --> 00:11:01,130 to scale it the right way. 181 00:11:01,130 --> 00:11:04,880 And it's not quite right in the notes. 182 00:11:04,880 --> 00:11:09,780 So I'll change it around and send the thing out to you. 183 00:11:09,780 --> 00:11:12,830 Then it says that T-spaced samples of r then reproduce u 184 00:11:12,830 --> 00:11:15,470 sub k without intersymbol interference. 185 00:11:15,470 --> 00:11:20,490 The Nyquist criterion is different from this business 186 00:11:20,490 --> 00:11:23,220 of the pulse being ideal Nyquist. 187 00:11:23,220 --> 00:11:26,610 Ideal Nyquist is talking about the time domain. 188 00:11:26,610 --> 00:11:29,680 It simply says the trivial thing you, want a pulse which 189 00:11:29,680 --> 00:11:33,390 has 0's at every sample point except for the sample point 190 00:11:33,390 --> 00:11:34,790 you're interested in. 191 00:11:34,790 --> 00:11:38,700 The Nyquist criterion translates that ideal Nyquist 192 00:11:38,700 --> 00:11:43,550 property, in time, to a property in frequency. 193 00:11:43,550 --> 00:11:46,950 And it says that the frequency, that the Fourier 194 00:11:46,950 --> 00:11:50,830 transform of g of t has to satisfy this 195 00:11:50,830 --> 00:11:52,680 relationship here. 196 00:11:52,680 --> 00:11:56,950 And there's this added condition on g of f that it 197 00:11:56,950 --> 00:12:00,820 has to go to 0 fast enough as f goes to infinity. 198 00:12:00,820 --> 00:12:04,230 But we won't worry about that today. 199 00:12:04,230 --> 00:12:11,570 So the condition is this: the picture that I showed you last 200 00:12:11,570 --> 00:12:13,610 time is this. 201 00:12:16,110 --> 00:12:23,200 We defined the nominal band, the Nyquist band, as the base 202 00:12:23,200 --> 00:12:26,800 bandwidth w equals 1 over 2t. 203 00:12:26,800 --> 00:12:31,030 That's the bandwidth that a sinc pulse would have if you 204 00:12:31,030 --> 00:12:32,790 were using a sinc pulse. 205 00:12:32,790 --> 00:12:35,910 The actual baseband limit, b, should be close to w. 206 00:12:39,290 --> 00:12:42,230 And most of the work that people do trying to build 207 00:12:42,230 --> 00:12:45,240 filters and things like this, since what we're trying to do 208 00:12:45,240 --> 00:12:48,360 is find a waveform that we're trying to transmit. 209 00:12:48,360 --> 00:12:51,250 And we're stuck with the FCC and all these other things 210 00:12:51,250 --> 00:12:56,280 that say, you better keep this band-limited. 211 00:12:56,280 --> 00:12:59,690 What we're going to do is to make the actual band of the 212 00:12:59,690 --> 00:13:07,420 waveform close to the nominal bandwidth. 213 00:13:07,420 --> 00:13:10,515 So we're going to assume that it's less than twice the 214 00:13:10,515 --> 00:13:12,130 nominal bandwidth. 215 00:13:12,130 --> 00:13:14,130 In other words, you can have a little bit of slop, but you 216 00:13:14,130 --> 00:13:15,700 can't have too much. 217 00:13:15,700 --> 00:13:18,750 When you try to design a filter that way, and you 218 00:13:18,750 --> 00:13:23,160 satisfy the Nyquist criterion, which is talking about all of 219 00:13:23,160 --> 00:13:26,190 these bands all the way out to infinity, and the fact they 220 00:13:26,190 --> 00:13:30,310 have to add up in a certain way, if you only have this 221 00:13:30,310 --> 00:13:34,270 band here and part of the next band -- 222 00:13:34,270 --> 00:13:39,430 if this function has to go to 0 before you get out to 2w, 223 00:13:39,430 --> 00:13:42,530 then the only thing you have to worry about is what does 224 00:13:42,530 --> 00:13:44,370 this waveform look like here. 225 00:13:44,370 --> 00:13:47,740 You take this and you pick it up. 226 00:13:47,740 --> 00:13:50,200 And you put it over here, and you add it up. 227 00:13:50,200 --> 00:13:54,340 You take this, you put it over here and you add it up. 228 00:13:54,340 --> 00:14:00,460 And if the pulse, p of t is real, then what you have over 229 00:14:00,460 --> 00:14:05,030 here is the complex conjugate of what you have here. 230 00:14:05,030 --> 00:14:08,220 And if you make this real in frequency also -- in other 231 00:14:08,220 --> 00:14:11,470 words, you have symmetry in time and symmetry in 232 00:14:11,470 --> 00:14:16,190 frequency, then this band edge symmetry requirement is that 233 00:14:16,190 --> 00:14:20,110 when you add this to this, you get something which is an 234 00:14:20,110 --> 00:14:25,260 ideal rectangular pulse. 235 00:14:25,260 --> 00:14:30,430 Now, if you think of taking this waveform here -- now you 236 00:14:30,430 --> 00:14:34,040 only have to worry about the positive frequency part of it. 237 00:14:34,040 --> 00:14:39,140 And you take that point right there, which is halfway 238 00:14:39,140 --> 00:14:40,700 between 0 and t. 239 00:14:40,700 --> 00:14:42,800 In other words, this is at t over 2. 240 00:14:45,770 --> 00:14:49,200 And it's also at frequency w. 241 00:14:49,200 --> 00:14:56,080 And you rotate this thing around by 180 degrees, then 242 00:14:56,080 --> 00:14:58,050 this comes up there. 243 00:14:58,050 --> 00:15:00,650 And it fills in this little slot up here. 244 00:15:00,650 --> 00:15:02,500 That's what the band edge symmetry means. 245 00:15:02,500 --> 00:15:02,800 Yes? 246 00:15:02,800 --> 00:15:04,050 AUDIENCE: [UNINTELLIGIBLE] 247 00:15:15,500 --> 00:15:18,080 PROFESSOR: Ah, yes. 248 00:15:18,080 --> 00:15:21,740 We could, if we wanted to, put various 249 00:15:21,740 --> 00:15:25,240 notches in this filter. 250 00:15:25,240 --> 00:15:30,540 But we've defined the bandwidth, b, as the largest 251 00:15:30,540 --> 00:15:38,240 frequency, f, such that g hat of f is 0 beyond b. 252 00:15:38,240 --> 00:15:43,590 In other words, everywhere beyond b, g hat of 253 00:15:43,590 --> 00:15:45,360 f is equal to 0. 254 00:15:45,360 --> 00:15:54,560 Now, if g hat f cuts down to 0, say, back here, there's no 255 00:15:54,560 --> 00:15:57,060 way you can meet the Nyquist criterion. 256 00:15:57,060 --> 00:15:59,750 Because there's no way you can build this thing up with all 257 00:15:59,750 --> 00:16:03,260 these out-of-band components so that you get something 258 00:16:03,260 --> 00:16:05,930 which is flat all the way out to w. 259 00:16:08,530 --> 00:16:11,740 So you simply can't have a filter which is band-limited 260 00:16:11,740 --> 00:16:15,370 to a frequency less than w. 261 00:16:15,370 --> 00:16:19,860 What you need is to use these out-of-band frequencies as a 262 00:16:19,860 --> 00:16:25,030 way to help you construct this ideal rectangular pulse. 263 00:16:25,030 --> 00:16:26,800 Through aliasing. 264 00:16:26,800 --> 00:16:29,230 In other words, the point here is when we're doing 265 00:16:29,230 --> 00:16:33,400 transmission of data, we know what the data is. 266 00:16:33,400 --> 00:16:37,020 We know what the filter is, and we can use 267 00:16:37,020 --> 00:16:39,590 aliasing to our advantage. 268 00:16:39,590 --> 00:16:42,920 When we were talking about data compression, aliasing 269 00:16:42,920 --> 00:16:44,910 just hurt us. 270 00:16:44,910 --> 00:16:48,240 Because we were trying to represent this waveform that 271 00:16:48,240 --> 00:16:50,420 we didn't have any control over. 272 00:16:50,420 --> 00:16:56,200 And the out-of-band parts added into the baseband parts 273 00:16:56,200 --> 00:16:57,040 and they clobbered us. 274 00:16:57,040 --> 00:17:00,270 Because we couldn't get the whole thing back again. 275 00:17:00,270 --> 00:17:04,670 Here, we're doing it the other way. 276 00:17:04,670 --> 00:17:07,420 In, other words we're starting out with a sequence. 277 00:17:07,420 --> 00:17:10,160 We're going to a waveform, and then we're trying to get the 278 00:17:10,160 --> 00:17:13,880 sequence back from the waveform. 279 00:17:13,880 --> 00:17:17,370 So it's really the opposite kind of problem. 280 00:17:17,370 --> 00:17:21,230 And here, the whole game, namely, the thing that Nyquist 281 00:17:21,230 --> 00:17:26,050 spotted, back in 1928, was you could use these out-of-band 282 00:17:26,050 --> 00:17:29,440 frequencies to, in fact, help you to get rid of this 283 00:17:29,440 --> 00:17:31,900 intersymbol interference. 284 00:17:31,900 --> 00:17:35,000 Because all you need to do is make these things add up to 285 00:17:35,000 --> 00:17:38,590 this so that you have something rectangular. 286 00:17:38,590 --> 00:17:40,120 And then when you do the samples, you 287 00:17:40,120 --> 00:17:41,370 have to write samples. 288 00:17:46,580 --> 00:17:50,600 Now, the problem that a filter designer comes to when saying 289 00:17:50,600 --> 00:17:56,680 this is to say, OK, how do I design a frequency response 290 00:17:56,680 --> 00:18:03,070 which has the property that it's going to go to 0 291 00:18:03,070 --> 00:18:05,800 quickly beyond w. 292 00:18:05,800 --> 00:18:09,940 Because the FCC, when we translate this up to passband, 293 00:18:09,940 --> 00:18:13,010 is going to tell us we can't have much energy outside of 294 00:18:13,010 --> 00:18:15,000 minus w to plus w. 295 00:18:15,000 --> 00:18:17,880 And if we can't design a good filter, it means we have to 296 00:18:17,880 --> 00:18:19,300 make w smaller. 297 00:18:19,300 --> 00:18:22,730 So we can keep ourselves within this given bandwidth. 298 00:18:22,730 --> 00:18:25,760 And we don't want to do that because that keeps us from 299 00:18:25,760 --> 00:18:27,670 transmitting much data. 300 00:18:27,670 --> 00:18:31,930 And then we can't sell our product, so suddenly we have 301 00:18:31,930 --> 00:18:37,350 to design something which uses all of this bandwidth that we 302 00:18:37,350 --> 00:18:39,630 have available. 303 00:18:39,630 --> 00:18:44,480 So what we want to do, then, is to design something where b 304 00:18:44,480 --> 00:18:49,350 is just a little bit more than w, but where also we get from 305 00:18:49,350 --> 00:18:52,650 t down to 0 very quickly. 306 00:18:52,650 --> 00:18:57,290 We could just use the square pulse to start with. 307 00:18:57,290 --> 00:18:59,510 And what's the trouble with that? 308 00:18:59,510 --> 00:19:03,900 This rectangular pulse, its inverse Fourier transform is 309 00:19:03,900 --> 00:19:05,740 the sinc pulse. 310 00:19:05,740 --> 00:19:09,350 The sinc pulse, because a discontinuity in the Fourier 311 00:19:09,350 --> 00:19:15,510 transform, it can't go to 0 any faster than is 1 over t. 312 00:19:15,510 --> 00:19:18,120 And suddenly it goes to 0 as 1 over t goes to 313 00:19:18,120 --> 00:19:19,970 0 very, very slowly. 314 00:19:19,970 --> 00:19:26,270 In other words, you have enormous problems over time. 315 00:19:26,270 --> 00:19:28,130 And you have enormous delay. 316 00:19:28,130 --> 00:19:31,150 And since you have so many pulses adding up together, 317 00:19:31,150 --> 00:19:34,410 everything has to be done extraordinarily carefully. 318 00:19:34,410 --> 00:19:40,280 So what you want is a pulse which remains equal to t over 319 00:19:40,280 --> 00:19:42,320 a y bandwidth here. 320 00:19:42,320 --> 00:19:45,170 Which gets down to 0 very, very fast. 321 00:19:45,170 --> 00:19:47,880 So the problem is, how do you design something which gets 322 00:19:47,880 --> 00:19:50,310 from here down to here very, very 323 00:19:50,310 --> 00:19:53,420 quickly and very smoothly. 324 00:19:53,420 --> 00:19:57,020 You want it to go smoothly because if you have any 325 00:19:57,020 --> 00:20:01,030 discontinuities in g of f, you're back to the problem 326 00:20:01,030 --> 00:20:05,950 where g of t goes to 0 as 1 over t. 327 00:20:05,950 --> 00:20:09,150 If you have a slope discontinuity, g of t is going 328 00:20:09,150 --> 00:20:12,440 to go to 0 as 1 over t squared. 329 00:20:12,440 --> 00:20:15,030 If you have a second derivative discontinuity, it's 330 00:20:15,030 --> 00:20:18,580 going to go to 0 as 1 over t cubed. 331 00:20:18,580 --> 00:20:22,790 Now, 1 over t cubed is not bad, and filter designers sort 332 00:20:22,790 --> 00:20:24,430 of live with that. 333 00:20:24,430 --> 00:20:28,220 So they design these filters which are raised cosine 334 00:20:28,220 --> 00:20:35,110 filters, which over the band here -- someday I'll get a pen 335 00:20:35,110 --> 00:20:38,660 that works -- are flat. 336 00:20:38,660 --> 00:20:44,730 Over this band here, from here to here, this is a squared 337 00:20:44,730 --> 00:20:47,680 cosine, analytically. 338 00:20:47,680 --> 00:20:51,410 And a squared cosine is just the same as a cosine which you 339 00:20:51,410 --> 00:20:56,100 take and displace up so it's centered right there. 340 00:20:56,100 --> 00:21:03,280 Well, excuse me, it's the same as a sine 341 00:21:03,280 --> 00:21:05,010 which is centered there. 342 00:21:05,010 --> 00:21:07,470 Which is what you get when you square a cosine 343 00:21:07,470 --> 00:21:12,060 pulse as 1/2 plus. 344 00:21:12,060 --> 00:21:13,540 Anyway, it looks like this. 345 00:21:16,360 --> 00:21:19,430 So our problem is, how do you design a filter which gets 346 00:21:19,430 --> 00:21:24,930 from here to there quickly but where the inverse transform 347 00:21:24,930 --> 00:21:28,310 also goes to 0 relatively quickly? 348 00:21:28,310 --> 00:21:33,720 Now, if you want to do this and you also face the fact 349 00:21:33,720 --> 00:21:38,670 that in a Nyquist criterion, any part of g hat of f which 350 00:21:38,670 --> 00:21:42,660 is imaginary, the Nyquist criterion says that what do 351 00:21:42,660 --> 00:21:45,420 you have to do in-band and what you have to do out of 352 00:21:45,420 --> 00:21:48,100 band have to add up there also. 353 00:21:48,100 --> 00:21:51,460 That doesn't help you at all in getting from this real 354 00:21:51,460 --> 00:21:54,770 number t here down to 0. 355 00:21:54,770 --> 00:21:59,170 So anything you do as a complex part of g hat of f is 356 00:21:59,170 --> 00:22:00,770 just wasted. 357 00:22:00,770 --> 00:22:04,040 I mean, your problem is getting from t to 0 with a 358 00:22:04,040 --> 00:22:05,800 smooth waveform. 359 00:22:05,800 --> 00:22:10,330 You would like to make g hat of f strictly real. 360 00:22:10,330 --> 00:22:12,020 You would like to make it symmetric. 361 00:22:12,020 --> 00:22:15,100 Why would you like to make it symmetric? 362 00:22:15,100 --> 00:22:20,230 Because this thing down here and this thing up here are 363 00:22:20,230 --> 00:22:22,030 really part of the same problem. 364 00:22:22,030 --> 00:22:25,820 If you find a good way to make a function go to 0 quickly up 365 00:22:25,820 --> 00:22:29,980 here, you might as well use the same thing over here. 366 00:22:29,980 --> 00:22:33,460 So you might as well wind up with a function which is 367 00:22:33,460 --> 00:22:36,540 symmetric and real. 368 00:22:36,540 --> 00:22:39,860 That leads us into the next thing we want to look at. 369 00:22:39,860 --> 00:22:42,350 That's a slightly flaky argument. 370 00:22:42,350 --> 00:22:47,230 We're going to find a better argument as we go. 371 00:22:47,230 --> 00:22:50,840 What we've said is, the real part of g hat of f has to 372 00:22:50,840 --> 00:22:54,970 satisfy this band edged symmetry condition. 373 00:22:54,970 --> 00:22:59,860 Choosing the imaginary part unequal to 0 simply increases 374 00:22:59,860 --> 00:23:02,320 the energy outside of the Nyquist band. 375 00:23:02,320 --> 00:23:09,290 You don't get any effect on reducing delay out of that. 376 00:23:09,290 --> 00:23:12,910 Thus, we're going to restrict g of f to be real. 377 00:23:12,910 --> 00:23:16,600 And we're going to also restrict it to be symmetric. 378 00:23:16,600 --> 00:23:18,630 Although that's less important. 379 00:23:18,630 --> 00:23:22,040 Now, when we start to look at noise, we're going to find out 380 00:23:22,040 --> 00:23:22,840 something else. 381 00:23:22,840 --> 00:23:25,360 We're going to find out that we want to make the magnitude 382 00:23:25,360 --> 00:23:30,790 of p of f equal to the magnitude of g of f. 383 00:23:30,790 --> 00:23:34,100 Now, magnitude doesn't make any difference. 384 00:23:34,100 --> 00:23:37,560 So we want the frequency characteristic of p of f to be 385 00:23:37,560 --> 00:23:41,770 the same as the frequency characteristic of q of f. 386 00:23:41,770 --> 00:23:45,270 In other words, there's no point descending a p of f 387 00:23:45,270 --> 00:23:49,570 which is a perfect sinc function and then using a very 388 00:23:49,570 --> 00:23:51,620 sloppy q of f. 389 00:23:51,620 --> 00:23:53,440 Because that's kind of silly. 390 00:23:53,440 --> 00:23:57,100 There's no point to using it very sloppy p of f and then 391 00:23:57,100 --> 00:24:01,040 using a very sharp q of f, because somehow when you start 392 00:24:01,040 --> 00:24:05,530 looking at noise, you're going to lose everything. 393 00:24:05,530 --> 00:24:10,350 Because the noise gets added to this pulse that you're 394 00:24:10,350 --> 00:24:11,440 transmitting. 395 00:24:11,440 --> 00:24:15,350 So, what we're going to find when we look at this later, is 396 00:24:15,350 --> 00:24:17,310 we really want to choose the magnitudes 397 00:24:17,310 --> 00:24:20,970 of these to be equal. 398 00:24:20,970 --> 00:24:26,180 Since g hat of f is equal to this product, and since we've 399 00:24:26,180 --> 00:24:29,920 already decided we want to make this real, what this 400 00:24:29,920 --> 00:24:36,110 means is that q hat of f is going to be equal to p complex 401 00:24:36,110 --> 00:24:37,950 conjugate of f. 402 00:24:37,950 --> 00:24:41,590 What that means is the filter q of t should be equal to the 403 00:24:41,590 --> 00:24:44,860 complex conjugate of p of minus t. 404 00:24:44,860 --> 00:24:50,410 You take a p of t, and you turn it around like this. 405 00:24:50,410 --> 00:24:54,400 If p of t is real, this is called a matched filter. 406 00:24:54,400 --> 00:24:57,000 And it's a filter which sort of collects everything which 407 00:24:57,000 --> 00:25:01,560 is in p of t, and all brings it up to 1p, which is what we 408 00:25:01,560 --> 00:25:03,300 would like to do here. 409 00:25:03,300 --> 00:25:08,110 So, anyway, when we do this it means that g of t is going to 410 00:25:08,110 --> 00:25:10,850 be this convolution of p of t. 411 00:25:10,850 --> 00:25:14,200 And q of t, which we can now write as the integral of p of 412 00:25:14,200 --> 00:25:19,920 tau, times p complex conjugate of t minus tau, p tau. 413 00:25:19,920 --> 00:25:24,520 And what we're interested in is, is this going to be ideal 414 00:25:24,520 --> 00:25:25,272 Nyquist or not. 415 00:25:25,272 --> 00:25:26,990 And what does that mean if it is ideal Nyquist? 416 00:25:35,660 --> 00:25:40,680 If g of t is ideal Nyquist, it means that the samples of g of 417 00:25:40,680 --> 00:25:45,890 t, times k times this signaling interval, t, have to 418 00:25:45,890 --> 00:25:49,360 have the property that these samples are equal to 1 for k 419 00:25:49,360 --> 00:25:53,900 equals 0, and 0 for k unequal to 0. 420 00:25:53,900 --> 00:25:55,760 What does that mean? 421 00:25:55,760 --> 00:25:59,510 If you look at this, that kind of looks like these 422 00:25:59,510 --> 00:26:02,590 orthogonality conditions that we've been dealing with, 423 00:26:02,590 --> 00:26:04,210 doesn't it? 424 00:26:04,210 --> 00:26:07,460 So that what it says is that this set of functions, p 425 00:26:07,460 --> 00:26:09,220 of t minus k t. 426 00:26:09,220 --> 00:26:14,630 In other words, the pulse p of t and all of it shifts by t, 427 00:26:14,630 --> 00:26:17,310 2t, 3t, and everything else. 428 00:26:17,310 --> 00:26:21,970 This set of pulses all have to be orthogonal to each other. 429 00:26:21,970 --> 00:26:24,890 And the thing which is a little screwed up in the notes 430 00:26:24,890 --> 00:26:27,750 is whether these are orthogonal or 431 00:26:27,750 --> 00:26:29,750 orthonormal, or what. 432 00:26:29,750 --> 00:26:31,320 And you need to make a few changes to 433 00:26:31,320 --> 00:26:34,100 make all of that right. 434 00:26:34,100 --> 00:26:36,720 These functions are all real L2 functions. 435 00:26:40,250 --> 00:26:43,030 But we're going to allow the possibility of complex 436 00:26:43,030 --> 00:26:44,190 functions for later. 437 00:26:44,190 --> 00:26:46,440 In other words, if we're transmitting a baseband 438 00:26:46,440 --> 00:26:50,120 waveform on a channel, how do you transmit 439 00:26:50,120 --> 00:26:51,370 an imaginary waveform? 440 00:26:53,800 --> 00:26:57,910 Well, I've never seen anything in electromagnetics, or in 441 00:26:57,910 --> 00:27:00,590 optics, or anything else, that lets me transmit 442 00:27:00,590 --> 00:27:02,940 an imaginary waveform. 443 00:27:02,940 --> 00:27:05,110 These are not physical. 444 00:27:05,110 --> 00:27:07,320 We will often think of baseband 445 00:27:07,320 --> 00:27:09,880 waveforms that are imaginary. 446 00:27:09,880 --> 00:27:12,090 Real and imaginary complex. 447 00:27:12,090 --> 00:27:18,700 And when we translate them up to baseband, we'll find 448 00:27:18,700 --> 00:27:19,700 something real. 449 00:27:19,700 --> 00:27:21,720 But the actual waveforms that get 450 00:27:21,720 --> 00:27:25,240 transmitted are always real. 451 00:27:25,240 --> 00:27:27,430 There's no way you can avoid that. 452 00:27:27,430 --> 00:27:29,140 That's real life. 453 00:27:29,140 --> 00:27:30,550 Real life is real. 454 00:27:30,550 --> 00:27:31,990 That's why they call it real, I guess. 455 00:27:31,990 --> 00:27:34,300 That's why they call it real life. 456 00:27:34,300 --> 00:27:36,520 I don't know. 457 00:27:36,520 --> 00:27:41,720 I mean it's more real than something imaginary, isn't it? 458 00:27:41,720 --> 00:27:45,430 So, anyway, what gets transmitted is real. 459 00:27:45,430 --> 00:27:50,750 But we'll allow p of t to be complex just for when we start 460 00:27:50,750 --> 00:27:56,030 dealing with something called QAM, which is our next topic. 461 00:27:56,030 --> 00:27:59,860 So in vector terms, the integral of u of tau times q 462 00:27:59,860 --> 00:28:03,900 of k t minus tau is the projection of u 463 00:28:03,900 --> 00:28:08,100 of t onto this waveform. 464 00:28:08,100 --> 00:28:11,360 And partly for that reason, q of t it's called the matched 465 00:28:11,360 --> 00:28:12,540 filter to p of t. 466 00:28:12,540 --> 00:28:17,060 In other words, you use this waveform here as a way of 467 00:28:17,060 --> 00:28:21,950 selecting out the parts of this waveform u of tau, u of 468 00:28:21,950 --> 00:28:24,680 t, that we're interested in. 469 00:28:24,680 --> 00:28:29,900 So that any way you look at it, we're going to use a pulse 470 00:28:29,900 --> 00:28:36,400 waveform, p of t, which has this property that its shifts 471 00:28:36,400 --> 00:28:38,640 are all orthogonal to each other. 472 00:28:38,640 --> 00:28:42,285 When we start studying noise, you will be very thankful that 473 00:28:42,285 --> 00:28:44,040 we did this. 474 00:28:44,040 --> 00:28:46,780 Because when you use pulses that are orthogonal to each 475 00:28:46,780 --> 00:28:49,410 other, you can break up the noise into 476 00:28:49,410 --> 00:28:51,390 an orthogonal expansion. 477 00:28:51,390 --> 00:28:54,190 And what goes on at one place is completely independent of 478 00:28:54,190 --> 00:28:56,360 what goes on at every other place. 479 00:28:56,360 --> 00:28:59,190 And we'll find out about this as we go. 480 00:28:59,190 --> 00:29:03,450 But, anyway, we have the nice property now, that anytime we 481 00:29:03,450 --> 00:29:09,130 find a function g of t, that satisfies the Nyquist 482 00:29:09,130 --> 00:29:14,770 criterion And any time we choose p of t and g of t so 483 00:29:14,770 --> 00:29:19,100 that their Fourier transforms have the same magnitude, then 484 00:29:19,100 --> 00:29:23,330 presto, we have freely gotten a set 485 00:29:23,330 --> 00:29:26,200 of orthonormal functions. 486 00:29:26,200 --> 00:29:29,130 Which just comes out in the wash. 487 00:29:29,130 --> 00:29:34,210 Before we worked very hard to get these truncated sinusoid 488 00:29:34,210 --> 00:29:39,310 expansions and sinc weighted sinusoid expansions, and all 489 00:29:39,310 --> 00:29:41,900 of this stuff to generate different 490 00:29:41,900 --> 00:29:45,610 orthonormal sets of waveforms. 491 00:29:45,610 --> 00:29:48,600 Suddenly, we just have an orthonormal set of waveforms 492 00:29:48,600 --> 00:29:52,040 and a very large set of orthonormal waveforms popping 493 00:29:52,040 --> 00:29:54,840 up and staring us in the face here. 494 00:29:54,840 --> 00:29:57,610 And in fact these are the waveforms we're going to use 495 00:29:57,610 --> 00:30:00,230 for communication, so they're nice things. 496 00:30:00,230 --> 00:30:03,390 Nobody uses sinc functions for communication. 497 00:30:03,390 --> 00:30:05,570 Nobody uses rectangular functions. 498 00:30:05,570 --> 00:30:08,720 You can't use either one of them because rectangular 499 00:30:08,720 --> 00:30:12,680 functions have lousy frequency characteristics. 500 00:30:12,680 --> 00:30:15,760 Sinc functions have lousy time characteristics. 501 00:30:15,760 --> 00:30:20,330 These functions p of t are sort of nice compromises. 502 00:30:20,330 --> 00:30:21,760 And they're orthonormal, again. 503 00:30:26,110 --> 00:30:28,120 Let's go on to the rest of modulation. 504 00:30:30,810 --> 00:30:34,710 We've been talking about baseband modulation. 505 00:30:34,710 --> 00:30:37,490 And when we're thinking about PAM, pulse amplitude 506 00:30:37,490 --> 00:30:42,000 modulation, we are thinking in terms of this sequence of 507 00:30:42,000 --> 00:30:44,160 symbols coming in. 508 00:30:44,160 --> 00:30:46,590 The symbols being turned into signals. 509 00:30:46,590 --> 00:30:49,910 The signals been turned into waveforms. 510 00:30:49,910 --> 00:30:53,990 And what comes out here, then, is some baseband waveform. 511 00:30:53,990 --> 00:30:56,420 That's what the Nyquist criterion is designed for. 512 00:30:56,420 --> 00:31:01,240 How do you make a baseband waveform which is very sharply 513 00:31:01,240 --> 00:31:04,020 cut off in frequency? 514 00:31:04,020 --> 00:31:07,180 Usually what we want to transmit is something at 515 00:31:07,180 --> 00:31:10,310 passband, so we somehow want to take this baseband 516 00:31:10,310 --> 00:31:14,020 waveform, convert it up to passband. 517 00:31:14,020 --> 00:31:16,480 We're then going to transmit it on a channel. 518 00:31:16,480 --> 00:31:23,280 I mean, why do we have to turn it into passband anyway? 519 00:31:23,280 --> 00:31:25,860 Well, if you did everything at baseband, you wouldn't have 520 00:31:25,860 --> 00:31:28,330 more than one channel available. 521 00:31:28,330 --> 00:31:30,120 I mean, wireless, you know you have all 522 00:31:30,120 --> 00:31:31,880 these different channels. 523 00:31:31,880 --> 00:31:34,990 The way it's done today, you use something called CDMA, 524 00:31:34,990 --> 00:31:38,770 where you're not breaking into narrow channels. 525 00:31:38,770 --> 00:31:41,740 But you should understand how to break it into narrow 526 00:31:41,740 --> 00:31:45,930 channels before understanding how to look at it as 527 00:31:45,930 --> 00:31:47,920 co-division multiple access. 528 00:31:47,920 --> 00:31:50,600 If you're using optics, you want to send things in 529 00:31:50,600 --> 00:31:53,160 different frequency bands. 530 00:31:53,160 --> 00:31:57,060 Whether it's optics or electromagnetics simply 531 00:31:57,060 --> 00:32:00,020 determines the frequency band you're looking at anyway. 532 00:32:00,020 --> 00:32:03,540 You can't propagate things in every frequency band. 533 00:32:03,540 --> 00:32:06,910 Things don't propagate very well at baseband. 534 00:32:06,910 --> 00:32:10,130 So for all of these reasons, we want to convert these 535 00:32:10,130 --> 00:32:13,430 baseband waveforms to passband. 536 00:32:13,430 --> 00:32:15,900 Why don't we generate them originally at passband? 537 00:32:20,070 --> 00:32:23,800 Because things are changing too fast there. 538 00:32:23,800 --> 00:32:26,990 I mean, you want to do digital signal processing to massage 539 00:32:26,990 --> 00:32:29,110 these signals, to do all the filtering. 540 00:32:29,110 --> 00:32:31,720 To do most of the other things you want to do. 541 00:32:31,720 --> 00:32:34,590 And you can do those very easily at baseband. 542 00:32:34,590 --> 00:32:36,920 And it's hard to do them at passband. 543 00:32:36,920 --> 00:32:41,330 So the generic way things are done is to first 544 00:32:41,330 --> 00:32:42,970 take a signal sequence. 545 00:32:42,970 --> 00:32:45,410 Convert it into a baseband waveform. 546 00:32:45,410 --> 00:32:48,620 Take the baseband waveform, convert it up to passband. 547 00:32:48,620 --> 00:32:51,040 And the passband is appropriate to whatever the 548 00:32:51,040 --> 00:32:52,100 channel is. 549 00:32:52,100 --> 00:32:52,910 You send it. 550 00:32:52,910 --> 00:32:57,320 You take it down from passband back to baseband, and then you 551 00:32:57,320 --> 00:33:00,570 filter and sample and get the waveform back again. 552 00:33:00,570 --> 00:33:01,890 You don't have to do this. 553 00:33:01,890 --> 00:33:06,320 We could generate the waveform directly at passband. 554 00:33:06,320 --> 00:33:08,980 There's a lot of research going on now trying to do 555 00:33:08,980 --> 00:33:13,720 this, which is trying to make things a little bit simpler. 556 00:33:13,720 --> 00:33:15,870 Well, it's not really trying to make things simpler. 557 00:33:15,870 --> 00:33:18,770 It's really trying to trying to pull a fast one 558 00:33:18,770 --> 00:33:21,340 on the FCC, I think. 559 00:33:21,340 --> 00:33:23,930 But, anyway, this is being done. 560 00:33:23,930 --> 00:33:27,610 And it doesn't go through this two-step process on the way. 561 00:33:27,610 --> 00:33:31,670 So it saves a little extra work. 562 00:33:34,220 --> 00:33:39,500 So what we're going to do with our PAM waveform, them, we're 563 00:33:39,500 --> 00:33:43,530 going to take u of t, which is the PAM waveform. 564 00:33:43,530 --> 00:33:44,860 And I'm sure you've all seen this. 565 00:33:44,860 --> 00:33:46,720 I hope you've all seen it somewhere or other. 566 00:33:46,720 --> 00:33:49,370 Because everybody likes to talk about this. 567 00:33:49,370 --> 00:33:53,640 Because you don't have to know anything to talk about this. 568 00:33:53,640 --> 00:33:55,200 So you take u of t. 569 00:33:55,200 --> 00:33:59,400 We multiply it by e to the 2 pi i f c t. 570 00:33:59,400 --> 00:34:02,550 In other words, you multiply it by a sine wave. 571 00:34:02,550 --> 00:34:05,080 A complex sine wave. 572 00:34:05,080 --> 00:34:08,110 When you do this, this thing is complex. 573 00:34:08,110 --> 00:34:09,910 You can't transmit it. 574 00:34:09,910 --> 00:34:11,290 So what do we do about that? 575 00:34:11,290 --> 00:34:12,660 Well, this is real. 576 00:34:12,660 --> 00:34:14,280 This is complex. 577 00:34:14,280 --> 00:34:18,140 If we add the complex conjugate of this, this plus 578 00:34:18,140 --> 00:34:20,980 its complex conjugate is real again. 579 00:34:20,980 --> 00:34:26,270 So we transmit this times this complex sinusoid, plus this 580 00:34:26,270 --> 00:34:30,250 other complex sinusoid, which is the complex 581 00:34:30,250 --> 00:34:31,650 conjugate of this. 582 00:34:31,650 --> 00:34:37,010 And you get 2 u of t times the cosine of 2 pi f c t. 583 00:34:37,010 --> 00:34:39,110 Which is just what you would do if you were implementing 584 00:34:39,110 --> 00:34:39,770 this anyway. 585 00:34:39,770 --> 00:34:44,180 You take the waveform u of t, you multiply it by cosine wave 586 00:34:44,180 --> 00:34:46,010 at the carrier frequency. 587 00:34:46,010 --> 00:34:49,040 And bingo, up it goes to carrier frequency. 588 00:34:49,040 --> 00:34:50,960 This is real. 589 00:34:50,960 --> 00:34:52,690 This was real. 590 00:34:52,690 --> 00:34:55,030 And everybody's happy. 591 00:34:55,030 --> 00:34:59,760 And in frequency, what this looks like, since all we're 592 00:34:59,760 --> 00:35:03,840 doing here is just, by this shift formula that we have for 593 00:35:03,840 --> 00:35:10,070 Fourier transforms, the multiplying a time waveform by 594 00:35:10,070 --> 00:35:13,390 an exponential -- 595 00:35:13,390 --> 00:35:20,180 by a complex sinusoid, is simply is the same as shifting 596 00:35:20,180 --> 00:35:21,760 the frequency response. 597 00:35:21,760 --> 00:35:25,170 So the Fourier transform of this is u hat of 598 00:35:25,170 --> 00:35:26,850 that minus f c. 599 00:35:26,850 --> 00:35:30,440 The Fourier transform of u f t times this is u hat of 600 00:35:30,440 --> 00:35:32,160 f the plus f c. 601 00:35:32,160 --> 00:35:35,200 And you start out with this waveform, whatever 602 00:35:35,200 --> 00:35:38,720 that shape is here. 603 00:35:38,720 --> 00:35:42,000 This, I tried to draw to satisfy the Nyquist criteria, 604 00:35:42,000 --> 00:35:42,780 which it does it. 605 00:35:42,780 --> 00:35:46,230 Satisfies that band edge symmetry condition. 606 00:35:46,230 --> 00:35:48,060 So this gets shifted up. 607 00:35:48,060 --> 00:35:50,200 It also gets shifted down. 608 00:35:50,200 --> 00:35:57,400 And the transmitted waveform then exists in the band which 609 00:35:57,400 --> 00:36:03,130 I'll call b sub u. b sub u is the bandwidth of u of t. 610 00:36:03,130 --> 00:36:05,440 Namely, it's this baseband bandwidth that we've been 611 00:36:05,440 --> 00:36:06,810 talking about. 612 00:36:06,810 --> 00:36:11,740 But, unfortunately, when we do this thing shifted up and this 613 00:36:11,740 --> 00:36:15,820 thing shifted down, the overall bandwidth here is now 614 00:36:15,820 --> 00:36:18,870 twice as much as it was before. 615 00:36:18,870 --> 00:36:21,960 Now, every communication engineer in the world, I 616 00:36:21,960 --> 00:36:25,540 think, measures bandwidth in the same way. 617 00:36:25,540 --> 00:36:28,220 When you talk about bandwidth, you're always talking about 618 00:36:28,220 --> 00:36:30,430 positive bandwidth. 619 00:36:30,430 --> 00:36:35,570 Because, back a long time ago, communication engineers didn't 620 00:36:35,570 --> 00:36:37,350 know about complex sinusoids. 621 00:36:37,350 --> 00:36:42,000 So everything was done in terms of cosines and sines. 622 00:36:42,000 --> 00:36:44,770 Which was very good, because back then communication 623 00:36:44,770 --> 00:36:47,620 engineers didn't have much else to do. 624 00:36:47,620 --> 00:36:51,270 So they had to learn to write everything twice. 625 00:36:51,270 --> 00:36:54,410 And now, since we have so many other things to worry about, 626 00:36:54,410 --> 00:36:56,890 we want to use complex sinusoids and only write 627 00:36:56,890 --> 00:36:58,120 things once. 628 00:36:58,120 --> 00:37:01,610 Well, in fact we have to write it twice here, but we don't 629 00:37:01,610 --> 00:37:04,720 write it twice very often. 630 00:37:04,720 --> 00:37:08,720 But, anyway, when this thing, which exists for minus b u up 631 00:37:08,720 --> 00:37:12,820 to plus b u gets translated up in frequency, we have 632 00:37:12,820 --> 00:37:18,060 something which exists from f c minus b u to f c plus b u. 633 00:37:18,060 --> 00:37:21,310 And this negative band is down here. 634 00:37:21,310 --> 00:37:25,470 Now, we're going to assume everywhere, usually without 635 00:37:25,470 --> 00:37:30,840 talking about it, that when we modulate this up in frequency, 636 00:37:30,840 --> 00:37:35,840 that the bandwidth here, this b u here, is less than the 637 00:37:35,840 --> 00:37:37,580 carrier frequency. 638 00:37:37,580 --> 00:37:40,330 In other words, when we translate it up in frequency, 639 00:37:40,330 --> 00:37:44,300 this and this do not intersect with each other. 640 00:37:44,300 --> 00:37:46,910 If this and this intersected with each other, it would be 641 00:37:46,910 --> 00:37:49,220 something very much like aliasing. 642 00:37:49,220 --> 00:37:52,210 You just couldn't to sort out from this, plus 643 00:37:52,210 --> 00:37:55,540 this, what this is. 644 00:37:55,540 --> 00:37:59,120 Here, if I drew it on paper at least, if you tell me what 645 00:37:59,120 --> 00:38:04,100 this is, I can figure out what that is. 646 00:38:04,100 --> 00:38:08,520 Namely, demodulating it independent of how we design 647 00:38:08,520 --> 00:38:12,470 the demodulator is, in some sense trivial, too. 648 00:38:12,470 --> 00:38:14,720 You just take this and you bring it back 649 00:38:14,720 --> 00:38:17,520 down to passband again. 650 00:38:17,520 --> 00:38:21,200 Well, anyway, since communication engineers define 651 00:38:21,200 --> 00:38:25,410 bandwidth in terms of positive frequencies, the bandwidth of 652 00:38:25,410 --> 00:38:29,590 this baseband waveform is b sub u. 653 00:38:29,590 --> 00:38:33,220 The bandwidth of this waveform is 2 b sub u. 654 00:38:35,860 --> 00:38:38,420 You can't get away from that. 655 00:38:38,420 --> 00:38:42,200 You have doubled the bandwidth, and you wind up 656 00:38:42,200 --> 00:38:44,870 with this plus this. 657 00:38:44,870 --> 00:38:47,060 And this looks kind of strange. 658 00:38:47,060 --> 00:38:49,590 So let's try to sort it out. 659 00:38:54,300 --> 00:38:59,240 The baseband waveform is limited to b. 660 00:38:59,240 --> 00:39:01,620 If it's shifted up to passband, the passband 661 00:39:01,620 --> 00:39:05,270 waveform becomes limited to 2 b. 662 00:39:05,270 --> 00:39:09,320 Might as well put these little u's in here. 663 00:39:09,320 --> 00:39:13,530 Because putting in little u's here is a way of getting 664 00:39:13,530 --> 00:39:17,570 around the problem of talking about baseband 665 00:39:17,570 --> 00:39:19,150 waveforms for a while. 666 00:39:19,150 --> 00:39:21,800 And then talking about passband waveforms. 667 00:39:21,800 --> 00:39:24,810 And one of them is always twice the other one. 668 00:39:29,010 --> 00:39:37,420 If you filter out this lower band here, now, what's the 669 00:39:37,420 --> 00:39:41,420 lower sideband here? 670 00:39:41,420 --> 00:39:45,010 Who thinks the lower sideband is this? 671 00:39:45,010 --> 00:39:49,550 Who thinks the lower sideband is this little thing here? 672 00:39:49,550 --> 00:39:51,230 Well, you all should think that because 673 00:39:51,230 --> 00:39:52,600 that's what it is. 674 00:39:52,600 --> 00:39:55,350 So when people talk about sidebands, what they're 675 00:39:55,350 --> 00:39:59,880 referring to, it's not, this is one sideband and this is 676 00:39:59,880 --> 00:40:00,890 another sideband. 677 00:40:00,890 --> 00:40:04,890 What they're referring to is this is one sideband and this 678 00:40:04,890 --> 00:40:07,910 is one sideband. 679 00:40:07,910 --> 00:40:10,710 This is stuff you all know, I'm sure. 680 00:40:10,710 --> 00:40:12,180 But haven't thought about for a while. 681 00:40:15,510 --> 00:40:21,410 If you filter out this lower sideband, then this resulting 682 00:40:21,410 --> 00:40:29,470 waveform, which now runs only in this upper sideband here, 683 00:40:29,470 --> 00:40:32,270 and since it has to be real it has this accompanying lower 684 00:40:32,270 --> 00:40:36,160 sideband down here going with it, but you then have the 685 00:40:36,160 --> 00:40:41,350 frequency band b sub u like you had before. 686 00:40:41,350 --> 00:40:45,170 So, in principle, you can design a communication system 687 00:40:45,170 --> 00:40:49,010 by translating things up in frequency by the carrier, and 688 00:40:49,010 --> 00:40:52,440 then chopping off that lower sideband. 689 00:40:52,440 --> 00:40:55,450 And then you haven't gained anything in frequency. 690 00:40:55,450 --> 00:40:59,390 And everything is essentially the same as it was before. 691 00:40:59,390 --> 00:41:02,160 Now, this used to be a very popular thing to do with 692 00:41:02,160 --> 00:41:03,920 analog communication. 693 00:41:03,920 --> 00:41:07,440 Partly because communication engineers felt the only thing 694 00:41:07,440 --> 00:41:10,690 they had to study back then was, how do you change things 695 00:41:10,690 --> 00:41:13,020 in frequency and how do you build filters. 696 00:41:13,020 --> 00:41:14,300 So they're very good at this. 697 00:41:14,300 --> 00:41:15,720 They love to do this. 698 00:41:15,720 --> 00:41:17,300 And this was their preferred way of 699 00:41:17,300 --> 00:41:18,380 dealing with the problem. 700 00:41:18,380 --> 00:41:22,000 They just got rid of that sideband, sent this positive 701 00:41:22,000 --> 00:41:24,720 sideband, and then somehow they would get it 702 00:41:24,720 --> 00:41:27,100 back down to here. 703 00:41:27,100 --> 00:41:30,150 Single sideband is hardly ever used for digital 704 00:41:30,150 --> 00:41:31,510 communication. 705 00:41:31,510 --> 00:41:35,300 It's not the usual way of doing things. 706 00:41:35,300 --> 00:41:39,320 Partly because these filters become very tricky when you're 707 00:41:39,320 --> 00:41:42,820 trying to send data at a high speed. 708 00:41:42,820 --> 00:41:46,840 All sorts of noise in here when you try to do this and 709 00:41:46,840 --> 00:41:49,510 you don't do it quite right. 710 00:41:49,510 --> 00:41:54,320 Affects you enormously, and people have just seen over the 711 00:41:54,320 --> 00:41:58,410 years that those systems don't work as well as the systems 712 00:41:58,410 --> 00:42:01,940 which do something else that we'll talk about later. 713 00:42:01,940 --> 00:42:04,900 Namely, QAM, which is what we want to talk about. 714 00:42:08,040 --> 00:42:10,220 If you don't do this filtering, you call this 715 00:42:10,220 --> 00:42:14,840 system a double sideband pulse amplitude modulation system. 716 00:42:14,840 --> 00:42:17,520 Which is what happens when you use pulse amplitude 717 00:42:17,520 --> 00:42:18,830 modulation. 718 00:42:18,830 --> 00:42:22,410 Namely, this baseband choosing a baseband pulse, which is the 719 00:42:22,410 --> 00:42:23,530 thing we're interested in. 720 00:42:23,530 --> 00:42:26,290 Because that's where the Nyquist criterion and all this 721 00:42:26,290 --> 00:42:28,160 neat stuff comes in. 722 00:42:28,160 --> 00:42:30,520 And then you translate it up in frequency. 723 00:42:30,520 --> 00:42:33,640 And you waste half the available frequency. 724 00:42:33,640 --> 00:42:37,390 If you don't care about frequency management, this is 725 00:42:37,390 --> 00:42:38,650 a fine thing to do. 726 00:42:38,650 --> 00:42:40,580 Nothing wrong with it. 727 00:42:40,580 --> 00:42:42,360 You just waste some frequency. 728 00:42:42,360 --> 00:42:44,490 It's the cheapest way to do things. 729 00:42:44,490 --> 00:42:46,270 And there are lots of cheap communication 730 00:42:46,270 --> 00:42:47,720 systems which do this. 731 00:42:50,380 --> 00:42:55,320 But if you're trying to send data, if you're concerned 732 00:42:55,320 --> 00:43:00,630 about the frequency efficiency of the system, then you're not 733 00:43:00,630 --> 00:43:01,880 going to do this. 734 00:43:05,230 --> 00:43:11,060 So what we're going to do is do something called quadrature 735 00:43:11,060 --> 00:43:12,310 amplitude modulation. 736 00:43:15,090 --> 00:43:19,190 QAM, which is what quadrature amplitude modulation stands 737 00:43:19,190 --> 00:43:23,890 for, solves the frequency waste problem of double 738 00:43:23,890 --> 00:43:28,650 sideband amplitude modulation by using a complex baseband 739 00:43:28,650 --> 00:43:31,430 waveform u of t. 740 00:43:31,430 --> 00:43:35,990 Before, what we were talking about is these signals which 741 00:43:35,990 --> 00:43:37,910 were one-dimensional signals. 742 00:43:37,910 --> 00:43:40,120 We would use these one-dimensional signals to 743 00:43:40,120 --> 00:43:44,260 modulate this waveform p of t. 744 00:43:44,260 --> 00:43:46,830 And we wound up with a real waveform. 745 00:43:46,830 --> 00:43:50,600 Now what we're going to do is use complex signals, which 746 00:43:50,600 --> 00:43:53,500 then have two dimensions. 747 00:43:53,500 --> 00:43:57,420 Use them to modulate the same sort of 748 00:43:57,420 --> 00:44:00,500 pulse, p of t, usually. 749 00:44:00,500 --> 00:44:03,740 And wind up with a complex baseband waveform. 750 00:44:03,740 --> 00:44:06,600 And then we're going to take that baseband waveform, 751 00:44:06,600 --> 00:44:08,630 translate it up in frequency. 752 00:44:08,630 --> 00:44:11,290 So when we do this, what do we get? 753 00:44:11,290 --> 00:44:15,430 We need a waveform to transmit which is real. 754 00:44:15,430 --> 00:44:20,460 So we're going to take u of t, which is complex. 755 00:44:20,460 --> 00:44:23,110 Translate, shift it up in frequency 756 00:44:23,110 --> 00:44:25,070 by the carrier frequency. 757 00:44:25,070 --> 00:44:30,230 So we get u of t times e to the 2 pi i f c t. 758 00:44:30,230 --> 00:44:33,870 To make it real, we have to add all this junk down at 759 00:44:33,870 --> 00:44:36,530 negative frequencies, which we'd just as soon not think 760 00:44:36,530 --> 00:44:39,210 about if we didn't have to. 761 00:44:39,210 --> 00:44:40,850 But they have to be there. 762 00:44:40,850 --> 00:44:46,160 So our total waveform is x sub t equal this sum of things. 763 00:44:46,160 --> 00:44:50,160 When you look at this and you take the real part of this, 764 00:44:50,160 --> 00:44:53,295 the real part of this, the imaginary part of this, and 765 00:44:53,295 --> 00:44:56,860 the imaginary part of this, as I'm sure most of you have seen 766 00:44:56,860 --> 00:45:02,430 before, x of t becomes 2 times the real part of u of t. 767 00:45:02,430 --> 00:45:04,990 Times this complex exponential. 768 00:45:09,180 --> 00:45:13,140 Which is equal to 2 times the real part of u of t times this 769 00:45:13,140 --> 00:45:18,200 cosine wave minus 2 times the imaginary part of u of t times 770 00:45:18,200 --> 00:45:19,350 a sine wave. 771 00:45:19,350 --> 00:45:26,400 Which says, you take this real part of this baseband waveform 772 00:45:26,400 --> 00:45:27,660 you've generated. 773 00:45:27,660 --> 00:45:30,170 You multiply it by cosine wave. 774 00:45:30,170 --> 00:45:32,290 You take the imaginary part, and you multiply 775 00:45:32,290 --> 00:45:33,670 it by a sine wave. 776 00:45:33,670 --> 00:45:36,640 For implementation, is one thing going to be real and the 777 00:45:36,640 --> 00:45:38,550 other thing imaginary? 778 00:45:38,550 --> 00:45:38,800 No. 779 00:45:38,800 --> 00:45:42,510 You can't make things that are imaginary, so you just deal 780 00:45:42,510 --> 00:45:44,600 with two real waveforms. 781 00:45:44,600 --> 00:45:47,180 And you call one of them the real part of u of t. 782 00:45:47,180 --> 00:45:49,615 You call the other one the imaginary part of u of t. 783 00:45:49,615 --> 00:45:52,180 And the imaginary part of u of t is in fact a 784 00:45:52,180 --> 00:45:53,780 real waveform again. 785 00:45:53,780 --> 00:45:57,470 So all this imaginary stuff is just in our imagination. 786 00:45:57,470 --> 00:46:00,030 And the actual waveforms look like this. 787 00:46:00,030 --> 00:46:03,730 You take one waveform which is generated. 788 00:46:03,730 --> 00:46:05,320 Multiply it by cosine. 789 00:46:05,320 --> 00:46:07,350 Take another waveform. 790 00:46:07,350 --> 00:46:10,940 Multiply it by sine. 791 00:46:10,940 --> 00:46:14,800 What about these factors of two here? 792 00:46:14,800 --> 00:46:19,530 The factors of two are things that drive everybody crazy. 793 00:46:19,530 --> 00:46:23,330 Everyone I talk to, I ask them how they manage to keep all 794 00:46:23,330 --> 00:46:24,430 this straight. 795 00:46:24,430 --> 00:46:27,180 And they all give me the same answer: they say they can't 796 00:46:27,180 --> 00:46:27,930 keep it straight. 797 00:46:27,930 --> 00:46:30,680 It's just too hard to keep it straight, and after they're 798 00:46:30,680 --> 00:46:36,390 all done, they try to figure out what the answer should be 799 00:46:36,390 --> 00:46:39,310 by looking at energy or something else. 800 00:46:39,310 --> 00:46:44,310 Or by just fudging things, which is what most people do. 801 00:46:44,310 --> 00:46:48,680 And part of the trouble is, you can do this two ways. 802 00:46:48,680 --> 00:46:50,790 You can do it three ways, in fact. 803 00:46:50,790 --> 00:46:56,730 You can either I want to view x of t as being some real 804 00:46:56,730 --> 00:47:01,910 function times the cosine wave and leave out that 2. 805 00:47:01,910 --> 00:47:06,330 And some other function, imaginary part of u t times 806 00:47:06,330 --> 00:47:08,570 the sine, and leave out the 2 there. 807 00:47:08,570 --> 00:47:09,820 And many people do that. 808 00:47:12,790 --> 00:47:14,310 And would that be better? 809 00:47:14,310 --> 00:47:17,470 But when you put the 2 in with the cosines and the sines, you 810 00:47:17,470 --> 00:47:21,580 have to put a 1/2 in here and a 1/2 in here. 811 00:47:21,580 --> 00:47:24,650 Most people, when they think about these things for a long 812 00:47:24,650 --> 00:47:28,360 time, they find it's far more convenient to be able to think 813 00:47:28,360 --> 00:47:35,610 of this positive frequency part of x of t as just u of t 814 00:47:35,610 --> 00:47:37,100 translated up in frequency. 815 00:47:37,100 --> 00:47:40,050 In other words, they like this diagram here. 816 00:47:44,960 --> 00:47:46,540 Which says you take this. 817 00:47:46,540 --> 00:47:48,260 You translate it up. 818 00:47:48,260 --> 00:47:51,180 And after you translate it up, you create something else down 819 00:47:51,180 --> 00:47:53,540 here, to make the whole thing real. 820 00:47:53,540 --> 00:47:57,410 But what we think of is this going up to this all the time. 821 00:47:57,410 --> 00:47:58,830 So that's one way of doing it. 822 00:47:58,830 --> 00:48:01,720 The other way of doing it is thinking in terms of sines and 823 00:48:01,720 --> 00:48:05,370 cosines, removing that 2 here. 824 00:48:05,370 --> 00:48:10,190 And who can imagine what the third way of doing it is? 825 00:48:10,190 --> 00:48:11,680 Just split the difference. 826 00:48:11,680 --> 00:48:15,030 Which means you put a square root of 2 in. 827 00:48:15,030 --> 00:48:17,610 And, in fact, that makes a whole lot of sense. 828 00:48:17,610 --> 00:48:22,160 Because then when you take the waveform u of t, translate it 829 00:48:22,160 --> 00:48:23,110 up in frequency. 830 00:48:23,110 --> 00:48:26,710 Make it real, you have the same energy in the baseband 831 00:48:26,710 --> 00:48:29,960 waveform as you have in the passband waveform. 832 00:48:29,960 --> 00:48:30,980 I'm not going to show that. 833 00:48:30,980 --> 00:48:35,060 You can just figure it out relatively easily. 834 00:48:35,060 --> 00:48:41,040 I mean, you know that the power in a cosine wave is 1/2, 835 00:48:41,040 --> 00:48:43,250 the power in a sine wave is 1/2. 836 00:48:43,250 --> 00:48:48,660 So when you're multiplying things by 1/2 in here -- well, 837 00:48:48,660 --> 00:48:51,210 this has a power of 1/2. 838 00:48:51,210 --> 00:48:54,410 This has a power of 1/2. 839 00:48:54,410 --> 00:48:57,840 And when you start looking at power here, you find out that 840 00:48:57,840 --> 00:49:01,730 that has to be a square root of 2 rather than 2. 841 00:49:01,730 --> 00:49:03,450 So there are three ways of doing it. 842 00:49:03,450 --> 00:49:06,710 People do it any one of three different ways. 843 00:49:06,710 --> 00:49:09,240 It doesn't make any difference, because any paper 844 00:49:09,240 --> 00:49:12,900 you read will start out doing it one way and then, as they 845 00:49:12,900 --> 00:49:15,740 go through various equations, they will start doing it a 846 00:49:15,740 --> 00:49:16,780 different way. 847 00:49:16,780 --> 00:49:18,820 And these factors of 2 multiply and 848 00:49:18,820 --> 00:49:20,450 multiply and multiply. 849 00:49:20,450 --> 00:49:23,510 And in big complicated papers, sometimes I've found that 850 00:49:23,510 --> 00:49:27,370 these add up to a factor of 8 or 16 or something else. 851 00:49:27,370 --> 00:49:29,910 By the time people are all done. 852 00:49:29,910 --> 00:49:33,510 And we will explain later why, in fact, you don't really care 853 00:49:33,510 --> 00:49:36,140 about that very much. 854 00:49:36,140 --> 00:49:37,620 But you can't just totally ignore 855 00:49:37,620 --> 00:49:40,510 those factors, so anyway. 856 00:49:40,510 --> 00:49:43,270 This is the way we will do it. 857 00:49:43,270 --> 00:49:47,230 We will try to be consistent about this, and 858 00:49:47,230 --> 00:49:48,480 usually we will be. 859 00:49:55,130 --> 00:49:58,690 The way we want to think about this conceptually is that 860 00:49:58,690 --> 00:50:02,450 quadrature amplitude modulation is going to take 861 00:50:02,450 --> 00:50:06,830 this complex waveform u of t. 862 00:50:06,830 --> 00:50:11,450 It's going to shift it up in frequency to f sub c, and then 863 00:50:11,450 --> 00:50:13,650 we're going to add the complex conjugate. 864 00:50:13,650 --> 00:50:16,590 Add it to form the real x sub t. 865 00:50:16,590 --> 00:50:17,780 In other words, we're going to think of it 866 00:50:17,780 --> 00:50:20,040 as a two-stage operation. 867 00:50:20,040 --> 00:50:23,230 First you take waveform, you translate it up. 868 00:50:23,230 --> 00:50:27,510 Then you take the real part or something, or add the negative 869 00:50:27,510 --> 00:50:29,400 frequency part. 870 00:50:29,400 --> 00:50:33,160 And we're going to think both of this double operation of 1 871 00:50:33,160 --> 00:50:35,310 going from u of t to the positive 872 00:50:35,310 --> 00:50:36,900 frequency part of things. 873 00:50:36,900 --> 00:50:39,530 And then, of looking at the real waveform that 874 00:50:39,530 --> 00:50:42,740 corresponds to that. 875 00:50:42,740 --> 00:50:46,380 What we're going to be doing here in terms of all of this 876 00:50:46,380 --> 00:50:50,360 is, we're going to start out with binary data. 877 00:50:50,360 --> 00:50:54,190 From the binary data, we're going to go to symbols. 878 00:50:54,190 --> 00:50:56,400 And we're going to go to symbols by taking a number of 879 00:50:56,400 --> 00:50:58,600 binary data -- 880 00:50:58,600 --> 00:51:00,810 a number of binary digits. 881 00:51:00,810 --> 00:51:03,760 Framing then into b tuples. 882 00:51:03,760 --> 00:51:06,670 Each b tuple will correspond to a set 883 00:51:06,670 --> 00:51:08,330 of 2 to the b symbols. 884 00:51:08,330 --> 00:51:12,340 We're going to map these symbols into complex signals. 885 00:51:12,340 --> 00:51:15,750 We're going to map the complex signals into a baseband 886 00:51:15,750 --> 00:51:17,080 waveform u of t. 887 00:51:17,080 --> 00:51:20,270 We're going to map the baseband waveform u of t into 888 00:51:20,270 --> 00:51:23,940 this positive frequency waveform u of t times this 889 00:51:23,940 --> 00:51:25,540 complex sinusoid. 890 00:51:25,540 --> 00:51:28,580 And finally we're going to add on the negative frequency part 891 00:51:28,580 --> 00:51:30,400 to wind up with x of t. 892 00:51:30,400 --> 00:51:32,550 What do you think we do at the receiver? 893 00:51:32,550 --> 00:51:35,330 As always, we do just the opposite. 894 00:51:35,330 --> 00:51:38,470 Namely, one of the reasons for wanting to think about this 895 00:51:38,470 --> 00:51:42,000 this way, is we want to use this layering idea. 896 00:51:42,000 --> 00:51:44,700 And the layering idea says, you start out with the 897 00:51:44,700 --> 00:51:46,850 received waveform x of t. 898 00:51:46,850 --> 00:51:49,780 And later on we'll have to add the noise to it. 899 00:51:49,780 --> 00:51:53,050 You go from there to the positive frequency part. 900 00:51:53,050 --> 00:51:55,410 You go from the positive frequency part. 901 00:51:55,410 --> 00:51:57,420 You shift it down to u of t. 902 00:51:57,420 --> 00:52:00,120 How we got from here to there, I'll explain in a minute. 903 00:52:00,120 --> 00:52:03,100 You go from here down to baseband again. 904 00:52:03,100 --> 00:52:06,530 You go from the baseband to the complex signals, which 905 00:52:06,530 --> 00:52:09,760 we're going to do simply by filtering and sampling. 906 00:52:09,760 --> 00:52:13,170 We go from the complex signals to the symbols, which is in 907 00:52:13,170 --> 00:52:14,830 fact a trivial operation. 908 00:52:14,830 --> 00:52:16,820 It's just a look-up operation. 909 00:52:16,820 --> 00:52:20,640 And then from there we un-segment things into binary 910 00:52:20,640 --> 00:52:21,370 digits again. 911 00:52:21,370 --> 00:52:23,030 So that's the whole system. 912 00:52:23,030 --> 00:52:25,700 And it has all these different pieces to it. 913 00:52:25,700 --> 00:52:29,000 I couldn't draw it as our favorite kind of diagram, 914 00:52:29,000 --> 00:52:30,670 because it has too many blocks in it. 915 00:52:30,670 --> 00:52:33,730 So that has to do. 916 00:52:36,320 --> 00:52:38,700 What we're going to do now is look at each of these pieces 917 00:52:38,700 --> 00:52:40,670 one at a time. 918 00:52:40,670 --> 00:52:45,440 And the first part is the complex QAM signal set. 919 00:52:45,440 --> 00:52:49,210 And, just for some notation here, so we'll be on the same 920 00:52:49,210 --> 00:52:57,170 page, but we use r to talk about the bits per second at 921 00:52:57,170 --> 00:53:02,830 which data is coming into this whole system. 922 00:53:02,830 --> 00:53:05,400 That's the figure you're interested in, when 923 00:53:05,400 --> 00:53:06,670 everything is done. 924 00:53:06,670 --> 00:53:10,020 How many bits per second can you transmit? 925 00:53:10,020 --> 00:53:13,180 We're going to segment this into b bits at a time. 926 00:53:13,180 --> 00:53:16,350 So we're going to have a symbol set with 2 to the b 927 00:53:16,350 --> 00:53:18,850 elements in it. 928 00:53:18,850 --> 00:53:22,920 We're going to map these m symbols, which are binary b 929 00:53:22,920 --> 00:53:27,820 tuples, into elements from the signal set. 930 00:53:27,820 --> 00:53:31,570 The signal rate, then, is r sub s, which is r over b. 931 00:53:31,570 --> 00:53:35,270 This is the number of signals per second that we're sending. 932 00:53:35,270 --> 00:53:41,940 In other words, t, this signal interval that we've always 933 00:53:41,940 --> 00:53:44,390 been using in everything we've been doing, is 934 00:53:44,390 --> 00:53:46,740 one over r sub s. 935 00:53:46,740 --> 00:53:49,710 So t is the signal interval. 936 00:53:49,710 --> 00:53:53,090 Every t seconds, you've got to send something. 937 00:53:53,090 --> 00:53:56,170 If you didn't send something every t seconds, the way this 938 00:53:56,170 --> 00:54:01,350 stuff coming in from the source would start piling up 939 00:54:01,350 --> 00:54:05,450 and your buffers would overflow and it wouldn't work. 940 00:54:05,450 --> 00:54:08,420 The signals u sub k are complex numbers. 941 00:54:08,420 --> 00:54:10,300 Or real 2-tuples. 942 00:54:10,300 --> 00:54:14,610 So we can, when we're trying to decide what signal set 943 00:54:14,610 --> 00:54:18,890 we're using, we can just draw our signals on a plane. 944 00:54:18,890 --> 00:54:23,030 The signal set is a constellation, then, of m 945 00:54:23,030 --> 00:54:26,040 complex numbers or real 2-tuples. 946 00:54:26,040 --> 00:54:29,120 So the problem of choosing the signal set is, how do you 947 00:54:29,120 --> 00:54:32,870 choose m points on a complex plane. 948 00:54:32,870 --> 00:54:36,190 What problem is that similar to? 949 00:54:36,190 --> 00:54:41,060 It's similar to the quantization problem where we 950 00:54:41,060 --> 00:54:44,590 were trying to choose m representation points. 951 00:54:44,590 --> 00:54:47,240 And it's very close to that problem. 952 00:54:47,240 --> 00:54:49,490 It's a very similar problems. 953 00:54:49,490 --> 00:54:52,180 Has a few small differences, but not many. 954 00:54:56,370 --> 00:54:59,920 But, before getting into that, we want to talk about a 955 00:54:59,920 --> 00:55:03,420 standard QAM signal set. 956 00:55:03,420 --> 00:55:08,620 In a minute I'll explain why people do that. 957 00:55:08,620 --> 00:55:14,040 And, as you might imagine, a standard QAM is just a square 958 00:55:14,040 --> 00:55:16,270 array of points. 959 00:55:16,270 --> 00:55:20,340 It's the simplest thing to do, and sometimes the simplest 960 00:55:20,340 --> 00:55:21,700 thing is the best. 961 00:55:21,700 --> 00:55:25,770 So it's determined by some distance, d, that you want to 962 00:55:25,770 --> 00:55:28,770 have between neighboring points. 963 00:55:28,770 --> 00:55:31,800 And given that distance, d, you just create a 964 00:55:31,800 --> 00:55:33,370 square array here. 965 00:55:33,370 --> 00:55:37,220 The square array means that m has to have an 966 00:55:37,220 --> 00:55:38,690 integer square root. 967 00:55:38,690 --> 00:55:41,740 This is drawn for m equals 16. 968 00:55:41,740 --> 00:55:45,460 If you look at this, you see that the real part of this 969 00:55:45,460 --> 00:55:48,350 it's the standard PAM set. 970 00:55:48,350 --> 00:55:53,040 The imaginary part is a standard PAM set, which says 971 00:55:53,040 --> 00:55:56,340 you can deal with the real part and the imaginary part 972 00:55:56,340 --> 00:55:57,190 separately. 973 00:55:57,190 --> 00:55:59,970 You take half the bits coming in and you choose your real 974 00:55:59,970 --> 00:56:01,180 part signal. 975 00:56:01,180 --> 00:56:03,090 Take the other half of the bits coming in. 976 00:56:03,090 --> 00:56:05,560 You form your imaginary part signal, and 977 00:56:05,560 --> 00:56:07,310 bingo, you're all done. 978 00:56:07,310 --> 00:56:11,670 The energy per 2D signal, we can find the energy for 2D 979 00:56:11,670 --> 00:56:14,540 signal by looking at it this way. 980 00:56:14,540 --> 00:56:18,610 It's two PAM systems running in parallel to each other. 981 00:56:18,610 --> 00:56:24,580 For the PAM system, the energy in one of these dimensions is 982 00:56:24,580 --> 00:56:28,150 then d squared times the square root of n squared, 983 00:56:28,150 --> 00:56:31,210 minus 1 divided by 12. 984 00:56:31,210 --> 00:56:35,120 But now we want to look at the energy which we have in both 985 00:56:35,120 --> 00:56:37,410 the real part and the imaginary part. 986 00:56:37,410 --> 00:56:39,300 So we need this extra factor. 987 00:56:39,300 --> 00:56:41,710 Well, we need to add together two of these things. 988 00:56:41,710 --> 00:56:46,660 So we wind up with d squared times m minus 1 divided by 6. 989 00:56:46,660 --> 00:56:47,910 Big deal. 990 00:56:54,060 --> 00:56:57,460 Choosing a good signal set is similar to choosing a 2D set 991 00:56:57,460 --> 00:57:01,180 of representation points in quantization. 992 00:57:01,180 --> 00:57:04,610 If you like to optimize things, you see this problem 993 00:57:04,610 --> 00:57:08,530 and you say, gee, at least there's something I can put my 994 00:57:08,530 --> 00:57:11,410 teeth into here. 995 00:57:11,410 --> 00:57:15,820 What's the best way to choose a signal set? 996 00:57:15,820 --> 00:57:18,010 And we found that for quantization that wasn't a 997 00:57:18,010 --> 00:57:21,270 terribly nice problem, although at least we had 998 00:57:21,270 --> 00:57:26,340 things like algorithms to try to choose reasonable sets. 999 00:57:26,340 --> 00:57:29,140 And we then looked at entropy quantization and things like 1000 00:57:29,140 --> 00:57:32,120 this, and it was a certain amount of fun. 1001 00:57:32,120 --> 00:57:34,560 Here, this problem it's just ugly. 1002 00:57:34,560 --> 00:57:38,540 There's no other way to express it. 1003 00:57:38,540 --> 00:57:40,080 I had to be convinced of this. 1004 00:57:40,080 --> 00:57:43,440 I once spent an inordinate amount of time trying to find 1005 00:57:43,440 --> 00:57:46,910 the best signal set with eight points in it, in two 1006 00:57:46,910 --> 00:57:47,990 dimensions. 1007 00:57:47,990 --> 00:57:52,820 How do you put eight single points in two dimensions in 1008 00:57:52,820 --> 00:57:56,710 such a way that every point is distance at least d from every 1009 00:57:56,710 --> 00:57:58,980 other point, and you minimize the energy 1010 00:57:58,980 --> 00:58:02,100 of the set of points? 1011 00:58:02,100 --> 00:58:04,520 The answer is just absolutely ugly. 1012 00:58:04,520 --> 00:58:05,810 It has no symmetry. 1013 00:58:05,810 --> 00:58:07,400 Nothing nice about it. 1014 00:58:07,400 --> 00:58:10,000 You do the same thing for 16 points, and it's 1015 00:58:10,000 --> 00:58:11,180 just an ugly problem. 1016 00:58:11,180 --> 00:58:13,660 You do the same thing for any number of points. 1017 00:58:13,660 --> 00:58:14,680 Except for four points. 1018 00:58:14,680 --> 00:58:16,260 For four points, it's easy. 1019 00:58:16,260 --> 00:58:19,730 For four points, you use standard QAM and it's the best 1020 00:58:19,730 --> 00:58:21,220 thing to do. 1021 00:58:21,220 --> 00:58:23,460 And that problem is easy. 1022 00:58:23,460 --> 00:58:25,680 But you know, that's not much fun. 1023 00:58:25,680 --> 00:58:28,260 Because you say, bleugh. 1024 00:58:28,260 --> 00:58:32,650 So, partly for that reason, people use 1025 00:58:32,650 --> 00:58:36,020 standard signal sets. 1026 00:58:36,020 --> 00:58:39,370 Partly because you don't seem to be able to gain much by 1027 00:58:39,370 --> 00:58:42,860 doing anything else. 1028 00:58:42,860 --> 00:58:47,060 So that's about all we can say about standard -- oh, with 1029 00:58:47,060 --> 00:58:49,810 eight signals, you can't use a standard signal set. 1030 00:58:49,810 --> 00:58:52,510 That was one reason we had to worry about it. 1031 00:58:52,510 --> 00:58:56,480 Back a long time ago, we were trying to design a 7200 bit 1032 00:58:56,480 --> 00:58:58,440 per second modem. 1033 00:58:58,440 --> 00:59:03,210 Back in the days when people did 2400 bits per second. 1034 00:59:03,210 --> 00:59:06,250 And we managed to do 4800 bits per second by 1035 00:59:06,250 --> 00:59:09,120 using QAM, big deal. 1036 00:59:09,120 --> 00:59:12,320 And then we said, well, we can pile in an extra bit by using 1037 00:59:12,320 --> 00:59:17,350 three bits per two dimensions instead of two bits. 1038 00:59:17,350 --> 00:59:20,110 And spent this enormous amount of time trying to find a 1039 00:59:20,110 --> 00:59:21,490 sensible signal set. 1040 00:59:21,490 --> 00:59:23,580 I don't even want to tell you what it was, because it wasn't 1041 00:59:23,580 --> 00:59:25,810 interesting at all. 1042 00:59:25,810 --> 00:59:30,970 So, enough for signal sets. 1043 00:59:30,970 --> 00:59:33,860 The next thing is, how do you turn the signals 1044 00:59:33,860 --> 00:59:36,260 into complex waveforms? 1045 00:59:36,260 --> 00:59:39,420 Namely, how do you go from the signals in two dimensions, 1046 00:59:39,420 --> 00:59:45,100 complex signals into a baseband waveform u of t? 1047 00:59:45,100 --> 00:59:48,890 Well, fortunately, Nyquist's theory is exactly the same 1048 00:59:48,890 --> 00:59:52,380 here as it was when we were dealing with PAM. 1049 00:59:52,380 --> 00:59:55,370 Everything we said before works here. 1050 00:59:55,370 --> 00:59:58,270 The only difference is that you don't have to choose the 1051 00:59:58,270 --> 01:00:00,820 pulse p of t to be real. 1052 01:00:00,820 --> 01:00:03,240 But if you look back at what we did, we didn't assume that 1053 01:00:03,240 --> 01:00:05,810 p of t was real before, anyway. 1054 01:00:05,810 --> 01:00:09,120 We just said, you might as well choose it to be real. 1055 01:00:09,120 --> 01:00:13,460 But you don't have to choose it to be real. 1056 01:00:13,460 --> 01:00:19,680 Bandedge symmetry requires that g of t be real. 1057 01:00:19,680 --> 01:00:21,230 Anybody know why that is? 1058 01:00:24,300 --> 01:00:28,910 When you choose g of t to be real, the negative frequency 1059 01:00:28,910 --> 01:00:31,420 part is the complex conjugate of the 1060 01:00:31,420 --> 01:00:33,530 positive frequency part. 1061 01:00:33,530 --> 01:00:38,570 Which is why, when we took this out-of-band stuff at 1062 01:00:38,570 --> 01:00:42,070 negative frequencies, piled it into the positive frequencies, 1063 01:00:42,070 --> 01:00:45,530 we got the same thing as if we simply rotate it around on the 1064 01:00:45,530 --> 01:00:46,930 positive frequency. 1065 01:00:46,930 --> 01:00:50,660 So that bandedge symmetry condition really requires that 1066 01:00:50,660 --> 01:00:52,970 g of t be real. 1067 01:00:52,970 --> 01:00:56,330 The orthogonality of t a t minus k t, this set of 1068 01:00:56,330 --> 01:01:00,560 waveforms, requires g of t to be real. 1069 01:01:00,560 --> 01:01:03,060 Neither of these things require p of t to be real. 1070 01:01:03,060 --> 01:01:06,640 You can choose p of t to have any old phase characteristic 1071 01:01:06,640 --> 01:01:10,590 you want to, but if we're choosing p of t -- if we're 1072 01:01:10,590 --> 01:01:15,760 choosing p hat of f magnitude to be the square root of a 1073 01:01:15,760 --> 01:01:21,440 Nyquist waveform, then you can choose this phase to be 1074 01:01:21,440 --> 01:01:24,080 anything you want to make it. 1075 01:01:24,080 --> 01:01:29,970 But you're just restricted in, aside from the phase, you're 1076 01:01:29,970 --> 01:01:33,310 somewhat restricted in what p of t can be. 1077 01:01:33,310 --> 01:01:33,620 OK. 1078 01:01:33,620 --> 01:01:35,960 So we're going to make the nominal passband, Nyquist 1079 01:01:35,960 --> 01:01:37,640 band, with 1 over t. 1080 01:01:37,640 --> 01:01:40,360 Before we made the passband -- 1081 01:01:40,360 --> 01:01:44,590 before we made the baseband bandwidth 1 over 2t. 1082 01:01:44,590 --> 01:01:49,580 When we go up the passband we double the bandwidth so the 1083 01:01:49,580 --> 01:01:52,210 Nyquist bandwidth is now 1 over t. 1084 01:01:52,210 --> 01:01:54,800 The passband bandwidth is 1 over t. 1085 01:01:54,800 --> 01:01:57,270 That's the only thing that's changed. 1086 01:01:57,270 --> 01:02:00,150 Usually people design these filters, which they design at 1087 01:02:00,150 --> 01:02:05,340 baseband, to go 5-10% over the Nyquist band. 1088 01:02:05,340 --> 01:02:10,880 In other words, these filters are very, very sharp, usually. 1089 01:02:10,880 --> 01:02:13,390 I mean, once you design a filter, it 1090 01:02:13,390 --> 01:02:14,450 doesn't cost anything. 1091 01:02:14,450 --> 01:02:17,480 You put it on a chip and that's the end of it. 1092 01:02:17,480 --> 01:02:19,650 And the cost of it is zero. 1093 01:02:19,650 --> 01:02:22,850 So it's just the cost to design it, so you might as 1094 01:02:22,850 --> 01:02:24,100 well make it small. 1095 01:02:28,360 --> 01:02:31,630 So finally, we want to go to base, 1096 01:02:31,630 --> 01:02:33,540 from baseband to passband. 1097 01:02:33,540 --> 01:02:36,240 We talked about this a little bit. 1098 01:02:36,240 --> 01:02:40,990 In terms of these frequencies, the baseband frequency b sub 1099 01:02:40,990 --> 01:02:43,420 u, we want to assume that that's less 1100 01:02:43,420 --> 01:02:44,780 than the carrier frequency. 1101 01:02:44,780 --> 01:02:49,540 This is this condition that we needed to make sure that the 1102 01:02:49,540 --> 01:02:56,050 positive frequency part -- ah, here it is. 1103 01:02:56,050 --> 01:02:58,550 That's the condition that makes sure that this is 1104 01:02:58,550 --> 01:03:01,440 separated from that, and doesn't cause intersymbol 1105 01:03:01,440 --> 01:03:04,740 interference between the two. 1106 01:03:04,740 --> 01:03:08,110 So, everything we do, we'll make this assumption. 1107 01:03:08,110 --> 01:03:13,540 I mean, a part of this is, if you're going to modulate with 1108 01:03:13,540 --> 01:03:16,640 such a small carrier frequency, you might as well 1109 01:03:16,640 --> 01:03:17,630 not do it at all. 1110 01:03:17,630 --> 01:03:19,220 You might as well just generate the 1111 01:03:19,220 --> 01:03:21,180 waveform you want directly. 1112 01:03:21,180 --> 01:03:25,610 Because you don't gain that much by doing it at baseband. 1113 01:03:25,610 --> 01:03:30,150 Because you don't really have a baseband in that case. 1114 01:03:30,150 --> 01:03:34,480 So, u of t times e to the 2 pi i f c t is strictly in the 1115 01:03:34,480 --> 01:03:36,880 positive frequency band, then. 1116 01:03:36,880 --> 01:03:39,850 And these two bands don't overlap. 1117 01:03:39,850 --> 01:03:42,110 As I said before, we're going to view this as 1118 01:03:42,110 --> 01:03:43,670 two different steps. 1119 01:03:43,670 --> 01:03:46,490 The first step is, I'm going to take this complex 1120 01:03:46,490 --> 01:03:48,490 waveform, u of t. 1121 01:03:48,490 --> 01:03:51,410 Multiply it by a complex sinusoid, which shifts me up 1122 01:03:51,410 --> 01:03:52,720 in frequency. 1123 01:03:52,720 --> 01:03:56,650 Just going to call that u passband of t. 1124 01:03:56,650 --> 01:04:00,250 This is this passband signal that I want to think about. 1125 01:04:00,250 --> 01:04:02,480 The thing that's up in positive frequencies. 1126 01:04:02,480 --> 01:04:05,210 I'm going to ignore the thing at negative frequencies. 1127 01:04:05,210 --> 01:04:11,910 And then I'm going to form the actual waveform x sub t, as 1128 01:04:11,910 --> 01:04:14,640 this plus its conjugate. 1129 01:04:26,000 --> 01:04:29,640 If you think a little bit now, you can see that since these 1130 01:04:29,640 --> 01:04:33,770 two bands are separated, if you think in terms of complex 1131 01:04:33,770 --> 01:04:38,720 waveforms, how do you retrieve this band up here by a 1132 01:04:38,720 --> 01:04:42,750 waveform which has both bands in it? 1133 01:04:42,750 --> 01:04:46,270 Well, you filter out what's at negative frequencies, OK? 1134 01:04:48,800 --> 01:04:55,160 So we want to design a filter which filters out the negative 1135 01:04:55,160 --> 01:04:58,660 frequencies and x of t, and only leaves the positive 1136 01:04:58,660 --> 01:04:59,160 frequencies. 1137 01:04:59,160 --> 01:05:03,210 In other words, you want a filter whose frequency 1138 01:05:03,210 --> 01:05:08,470 response is just 1 for all positive frequencies, 0 for 1139 01:05:08,470 --> 01:05:11,070 all negative frequencies. 1140 01:05:11,070 --> 01:05:14,610 And that filter is called a Hilbert filter. 1141 01:05:14,610 --> 01:05:19,020 Have any of you ever heard of a Hilbert filter before? 1142 01:05:19,020 --> 01:05:21,460 I don't know of anybody that's ever built one. 1143 01:05:21,460 --> 01:05:24,520 And we'll see why they don't built them in a while. 1144 01:05:24,520 --> 01:05:26,420 But it's a nice idea. 1145 01:05:26,420 --> 01:05:28,750 I mean, if you try to build one you'll find that it's 1146 01:05:28,750 --> 01:05:35,820 harder to build -- 1147 01:05:35,820 --> 01:05:39,150 you'll find that you have to implement four real filters in 1148 01:05:39,150 --> 01:05:40,920 order to implement this filter. 1149 01:05:40,920 --> 01:05:43,160 So we'll find out that's not the thing to do. 1150 01:05:43,160 --> 01:05:46,400 But it's nice conceptually because it lets us study 1151 01:05:46,400 --> 01:05:51,260 things like energy, power, and linearity, and 1152 01:05:51,260 --> 01:05:52,510 all of these things. 1153 01:05:56,140 --> 01:06:01,490 So the transmitter then becomes this thing you start 1154 01:06:01,490 --> 01:06:04,990 out with a complex waveform of baseband. 1155 01:06:04,990 --> 01:06:07,860 You shift it up in frequency. 1156 01:06:07,860 --> 01:06:09,690 This gives you this high frequency 1157 01:06:09,690 --> 01:06:12,960 waveform, u sub p of t. 1158 01:06:12,960 --> 01:06:16,830 You then take 2 times the real part of that to 1159 01:06:16,830 --> 01:06:18,450 find the real waveform. 1160 01:06:18,450 --> 01:06:20,220 We won't worry about how to implement 1161 01:06:20,220 --> 01:06:23,050 this, you just do it. 1162 01:06:23,050 --> 01:06:26,440 This passband waveform, you then pass it through this 1163 01:06:26,440 --> 01:06:29,190 Hilbert filter, which just chops off the negative 1164 01:06:29,190 --> 01:06:31,000 frequency part of it. 1165 01:06:31,000 --> 01:06:34,120 Gives you a complex waveform again. 1166 01:06:34,120 --> 01:06:39,050 You multiply by e to the minus 2 pi i f c t, which takes this 1167 01:06:39,050 --> 01:06:41,260 positive frequency waveform, shifts it 1168 01:06:41,260 --> 01:06:43,610 back down to baseband. 1169 01:06:43,610 --> 01:06:46,490 So this is a nice convenient way of thinking about, how do 1170 01:06:46,490 --> 01:06:50,390 you go from baseband up to passband and passband down to 1171 01:06:50,390 --> 01:06:52,150 baseband again. 1172 01:06:52,150 --> 01:06:52,820 Now. 1173 01:06:52,820 --> 01:06:58,050 If you want to view these vectors here as vectors, want 1174 01:06:58,050 --> 01:07:03,980 to view u of t as a vector in L2, there's an important thing 1175 01:07:03,980 --> 01:07:05,660 here going on. 1176 01:07:05,660 --> 01:07:08,900 We'll have to talk about it a good deal later on. 1177 01:07:08,900 --> 01:07:11,410 This is a complex waveform. 1178 01:07:11,410 --> 01:07:14,970 You want to deal with it as a vector in complex L2. 1179 01:07:14,970 --> 01:07:19,580 In complex L2, when we're dealing with vectors, we have 1180 01:07:19,580 --> 01:07:23,390 scalars, which are complex numbers. 1181 01:07:23,390 --> 01:07:28,150 When we start dealing with real parts of these things, we 1182 01:07:28,150 --> 01:07:33,150 want to view the real parts as being elements of real L2. 1183 01:07:33,150 --> 01:07:36,370 Where the scalars are real numbers. 1184 01:07:36,370 --> 01:07:41,050 And what this says is that real L2 is not a subspace of 1185 01:07:41,050 --> 01:07:43,480 complex L2. 1186 01:07:43,480 --> 01:07:47,020 It's not a subspace because the scalars are different. 1187 01:07:47,020 --> 01:07:50,350 This might sound like mathematical nitpicking. 1188 01:07:50,350 --> 01:07:52,280 But, put it in the back of your mind. 1189 01:07:52,280 --> 01:07:54,650 Because at some point it's going to come 1190 01:07:54,650 --> 01:07:56,830 up and clobber you. 1191 01:07:56,830 --> 01:07:59,810 And at that point, you will want to think that in fact 1192 01:07:59,810 --> 01:08:04,950 real L2 is not a subspace of complex L2. 1193 01:08:04,950 --> 01:08:08,390 When we start thinking about orthonormal expansions for u 1194 01:08:08,390 --> 01:08:12,950 of t and orthonormal expansions for x of t, in fact 1195 01:08:12,950 --> 01:08:15,600 you have to be quite careful about this. 1196 01:08:15,600 --> 01:08:18,820 Because you take an orthonormal expansion here, 1197 01:08:18,820 --> 01:08:21,840 translate it up into frequency. 1198 01:08:21,840 --> 01:08:27,640 And you wind up with a bunch of complex waveforms. 1199 01:08:27,640 --> 01:08:29,910 And they aren't real waveforms. 1200 01:08:29,910 --> 01:08:32,430 And funny things start happening. 1201 01:08:32,430 --> 01:08:34,090 So we'll deal with all of that later. 1202 01:08:34,090 --> 01:08:37,100 This is just to warn you that we have to be 1203 01:08:37,100 --> 01:08:39,550 careful about that. 1204 01:08:39,550 --> 01:08:41,970 This is not the way people implement these things. 1205 01:08:41,970 --> 01:08:46,410 Because these Hilbert filters are in fact for real filters. 1206 01:08:49,020 --> 01:08:55,620 So the implementation is what you've seen. 1207 01:08:55,620 --> 01:08:57,350 I mean, the implementation is old. 1208 01:08:57,350 --> 01:09:00,940 And it's the way you want to build these things. 1209 01:09:00,940 --> 01:09:05,290 You start out with two real baseband waveforms. 1210 01:09:05,290 --> 01:09:07,940 One which we call the real part of u of t. 1211 01:09:07,940 --> 01:09:11,130 One which we call the imaginary part of u of t. 1212 01:09:11,130 --> 01:09:14,000 In this single diagram, one of them is the stuff 1213 01:09:14,000 --> 01:09:15,270 that goes this way. 1214 01:09:15,270 --> 01:09:17,880 And the other one is the stuff that goes this way. 1215 01:09:17,880 --> 01:09:21,940 And if, in fact, you're using a standard QAM signal set, the 1216 01:09:21,940 --> 01:09:25,490 two are completely independent of each other. 1217 01:09:25,490 --> 01:09:31,310 So the real part of u of t is just the sum of these shifted 1218 01:09:31,310 --> 01:09:34,160 pulses times the real parts. 1219 01:09:34,160 --> 01:09:36,600 This is the sum of the shifted pulses 1220 01:09:36,600 --> 01:09:39,590 times the complex parts. 1221 01:09:39,590 --> 01:09:43,050 The defined u sub k prime is a real part, and u sub k double 1222 01:09:43,050 --> 01:09:45,890 prime is the imaginary part. 1223 01:09:45,890 --> 01:09:53,320 In the notes, this and this are called a sub k and a sub k 1224 01:09:53,320 --> 01:09:55,170 double prime. 1225 01:09:55,170 --> 01:09:57,600 Which doesn't correspond to anything else. 1226 01:09:57,600 --> 01:10:00,430 So, this is the correct way of doing it. 1227 01:10:00,430 --> 01:10:03,490 But, anyway, when you get all done, x sub t is 2 times the 1228 01:10:03,490 --> 01:10:09,630 cosine of this low pass modulated PAM waveform. 1229 01:10:09,630 --> 01:10:13,530 Minus 2 times the sine of this low pass 1230 01:10:13,530 --> 01:10:16,240 PAM modulated waveform. 1231 01:10:16,240 --> 01:10:19,910 So, QAM, when you look at it this way, is simply two 1232 01:10:19,910 --> 01:10:23,160 different PAM systems. 1233 01:10:23,160 --> 01:10:26,410 One of them modulated on a cosine carrier, one of them 1234 01:10:26,410 --> 01:10:28,410 modulated on a sine carrier. 1235 01:10:36,220 --> 01:10:40,220 And the picture of that is this. 1236 01:10:40,220 --> 01:10:41,470 Aargh. 1237 01:10:46,970 --> 01:10:50,940 Can't keep my notation straight. 1238 01:10:50,940 --> 01:10:54,880 I'm sure it doesn't bother most of you that much, but it 1239 01:10:54,880 --> 01:10:57,330 bothers me. 1240 01:10:57,330 --> 01:10:59,410 All of those a's should be u's. 1241 01:10:59,410 --> 01:11:02,460 They were a's last year, but they don't make 1242 01:11:02,460 --> 01:11:05,290 any sense as a's. 1243 01:11:05,290 --> 01:11:12,640 So the thing we're going to do now is we start out with the 1244 01:11:12,640 --> 01:11:14,190 sequence of signals. 1245 01:11:14,190 --> 01:11:15,550 The real part of the signals and the 1246 01:11:15,550 --> 01:11:17,390 imaginary part of the signals. 1247 01:11:17,390 --> 01:11:21,530 This is why it's called double side band quadrature carrier, 1248 01:11:21,530 --> 01:11:23,050 because in fact we're doing two 1249 01:11:23,050 --> 01:11:26,760 different things in parallel. 1250 01:11:26,760 --> 01:11:31,010 We generate this as a pulse waveform. 1251 01:11:31,010 --> 01:11:32,930 We filter it by p of t. 1252 01:11:32,930 --> 01:11:36,850 We're thinking of p of t as a real waveform now. 1253 01:11:36,850 --> 01:11:39,840 If you want p of t to be complex you have to modify 1254 01:11:39,840 --> 01:11:40,950 this all a little bit. 1255 01:11:40,950 --> 01:11:46,720 But there's no real reason to make p of t complex anyway. 1256 01:11:46,720 --> 01:11:50,360 So when you get out of here, what you have is just this low 1257 01:11:50,360 --> 01:11:53,980 pass real waveform, real PAM waveform. 1258 01:11:53,980 --> 01:11:57,480 Here's another low pass real PAM waveform. 1259 01:11:57,480 --> 01:12:01,540 You module this up by multiplying by cosine of 2 pi 1260 01:12:01,540 --> 01:12:05,700 f c t, in fact, by 2 cosine of 2 pi f ct . 1261 01:12:05,700 --> 01:12:09,330 You modulate this up by multiplying by minus sine. 1262 01:12:09,330 --> 01:12:12,290 And you get the actual waveform that 1263 01:12:12,290 --> 01:12:13,880 you're going to transmit. 1264 01:12:13,880 --> 01:12:15,640 How do you demodulate this? 1265 01:12:15,640 --> 01:12:20,130 Well, again, I'm sure you've seen it in one of the 1266 01:12:20,130 --> 01:12:22,310 undergraduate courses you've taken. 1267 01:12:22,310 --> 01:12:25,840 Because if you take this waveform, which is the sum of 1268 01:12:25,840 --> 01:12:30,120 this and this, and you multiply this by cosine of 2 1269 01:12:30,120 --> 01:12:32,330 pi f c t, what's going to happen? 1270 01:12:34,960 --> 01:12:39,070 Taking this waveform and multiplying it by cosine is 1271 01:12:39,070 --> 01:12:41,770 going to take this cosine waveform. 1272 01:12:41,770 --> 01:12:45,560 Half of it goes up in frequency by f sub c. 1273 01:12:45,560 --> 01:12:50,440 The other half goes down in frequency by f sub c of t. 1274 01:12:50,440 --> 01:12:56,060 When you multiply by sine, the same thing happens. 1275 01:12:56,060 --> 01:13:01,640 And all of the stuff at this double frequency term all gets 1276 01:13:01,640 --> 01:13:02,940 filtered out. 1277 01:13:02,940 --> 01:13:04,730 I mean, you have enough filtering to 1278 01:13:04,730 --> 01:13:06,380 just wash that away. 1279 01:13:06,380 --> 01:13:10,260 And you wind up, just with this one waveform which is the 1280 01:13:10,260 --> 01:13:11,520 result of this. 1281 01:13:11,520 --> 01:13:14,920 Another waveform which is the result of this. 1282 01:13:14,920 --> 01:13:17,720 You can show that the two don't interfere at all, and 1283 01:13:17,720 --> 01:13:19,980 you just have to do the multiplication 1284 01:13:19,980 --> 01:13:20,810 to find this out. 1285 01:13:20,810 --> 01:13:23,940 It looks a little bit like black magic when you look at 1286 01:13:23,940 --> 01:13:24,730 it like this. 1287 01:13:24,730 --> 01:13:28,400 Because when you're multiplying by a cosine wave, 1288 01:13:28,400 --> 01:13:31,550 I mean it's easy to see what cosine squared does here. 1289 01:13:31,550 --> 01:13:33,010 But it's a little harder to see what 1290 01:13:33,010 --> 01:13:35,050 happens to all of this. 1291 01:13:35,050 --> 01:13:39,480 And when we look at it the other way, which was this 1292 01:13:39,480 --> 01:13:44,290 Hilbert filter kind of thing, when you look at in terms of 1293 01:13:44,290 --> 01:13:48,750 the Hilbert filter it's quite clear that you can filter out 1294 01:13:48,750 --> 01:13:51,550 the lower sideband and then you can just go back down to 1295 01:13:51,550 --> 01:13:52,480 baseband again. 1296 01:13:52,480 --> 01:13:55,550 So it's very clear that the whole thing works. 1297 01:13:55,550 --> 01:13:58,630 Except you wouldn't implement it this way. 1298 01:13:58,630 --> 01:14:02,300 Here you have to be more careful to see that works. 1299 01:14:02,300 --> 01:14:05,080 But in fact you would. 1300 01:14:09,160 --> 01:14:12,630 Well, after you get all done, then you get these baseband 1301 01:14:12,630 --> 01:14:15,250 PAM waveforms back again. 1302 01:14:15,250 --> 01:14:17,650 You sample them after filtering. 1303 01:14:17,650 --> 01:14:20,920 And you're all done. 1304 01:14:20,920 --> 01:14:26,630 With that, we are almost done with what we want to do with 1305 01:14:26,630 --> 01:14:31,720 modulating up to passband and down to baseband. 1306 01:14:31,720 --> 01:14:34,240 We'll spend a little bit of time reviewing a couple of 1307 01:14:34,240 --> 01:14:37,550 minor points on this next time. 1308 01:14:37,550 --> 01:14:40,660 Like, I guess, the main thing we have to talk about is how 1309 01:14:40,660 --> 01:14:45,090 do you do frequency recovery, which is kind of a neat thing. 1310 01:14:45,090 --> 01:14:48,000 And then we'll go on to talking about random processes 1311 01:14:48,000 --> 01:14:50,420 and how you deal with noise. 1312 01:14:50,420 --> 01:14:51,890 So. 1313 01:14:51,890 --> 01:14:57,040 If you want to read ahead, we will probably have the notes 1314 01:14:57,040 --> 01:15:01,820 on random processes on the web sometime tomorrow 1315 01:15:01,820 --> 01:15:03,370 afternoon or Sunday. 1316 01:15:03,370 --> 01:15:04,620 Thanks.