1 00:00:00,600 --> 00:00:04,10 Hi, and welcome to module nine of digital signal processing. 2 00:00:04,10 --> 00:00:06,770 This is the last module in our class, and this is really where it all comes 3 00:00:06,770 --> 00:00:09,691 together. In this module we will review the 4 00:00:09,691 --> 00:00:14,140 principles behind the success of digital communication systems. 5 00:00:14,140 --> 00:00:17,562 And we will look at different communication systems starting from the 6 00:00:17,562 --> 00:00:21,274 voice band modems that were popular a few years ago and that you can still hear 7 00:00:21,274 --> 00:00:24,928 when you use a fax machine, to the most recent incarnations like the ADSL box 8 00:00:24,928 --> 00:00:28,292 that you have in your home and that you're probably using to watch this 9 00:00:28,292 --> 00:00:34,708 video. Digital communication systems need no 10 00:00:34,708 --> 00:00:37,710 introduction. The amount of information that we consume 11 00:00:37,710 --> 00:00:42,90 and that we produce every day is staggering by an historical standard. 12 00:00:42,90 --> 00:00:45,482 And what is even more amazing is that we can access this wealth of information 13 00:00:45,482 --> 00:00:49,139 from basically anyway via a small device, like the smartphoen that you have in your 14 00:00:49,139 --> 00:00:52,944 pocket. There is actually a joke about that and 15 00:00:52,944 --> 00:00:55,700 suppose that someone from the Renaissance, like Leonardo, was 16 00:00:55,700 --> 00:00:59,667 teleported to today. And you'd have to explain to them what 17 00:00:59,667 --> 00:01:02,992 your smartphone does. Well you have to say this is a small 18 00:01:02,992 --> 00:01:06,473 device that allows me to access everything that has been done, written 19 00:01:06,473 --> 00:01:11,400 about, or were said by mankind since the beginning of history. 20 00:01:11,400 --> 00:01:14,120 And I use it mainly to look at pictures of cats. 21 00:01:14,120 --> 00:01:16,926 But jokes aside the truth remains that communications systems, digital 22 00:01:16,926 --> 00:01:20,440 communications systems. Are really the pinnacle achievment of 23 00:01:20,440 --> 00:01:23,981 digital signal processing. So in this module we'll start from the 24 00:01:23,981 --> 00:01:27,824 basic principles in module nine one and we'll see the kind of signals that we 25 00:01:27,824 --> 00:01:34,520 have to design in order to be able to transmit them over a physical channel. 26 00:01:34,520 --> 00:01:38,546 Now a physical channel whether it's a wireless channel, whether it's a piece of 27 00:01:38,546 --> 00:01:42,572 wire or an optical fiber will always impose two fundamental constraints on the 28 00:01:42,572 --> 00:01:47,310 kind of signal that can transit over the channel. 29 00:01:47,310 --> 00:01:50,460 The first one is a bandwidth constraint, which means that we will only have a 30 00:01:50,460 --> 00:01:55,330 certain range of frequencies over which we can send information. 31 00:01:55,330 --> 00:01:57,920 And the second constraint is a power constraint. 32 00:01:57,920 --> 00:02:01,640 It limits the amount of power that we can inject onto the channel. 33 00:02:01,640 --> 00:02:05,640 So in module 9.2, we will tackle the banther constraint, in detail. 34 00:02:05,640 --> 00:02:08,720 And in module 9.3, we will look at the power constraint. 35 00:02:08,720 --> 00:02:11,970 And we will see in the end how these two constraints limit the maximum amount of 36 00:02:11,970 --> 00:02:15,79 information that we can send over a channel. 37 00:02:16,320 --> 00:02:19,792 In Module 9.4, we will look at the modulation and demodulation techniques 38 00:02:19,792 --> 00:02:24,640 that are specially designed to transmit data over the telephone channel. 39 00:02:24,640 --> 00:02:28,204 And in Module 9.5, we will examine the several signal processes and tricks that 40 00:02:28,204 --> 00:02:31,390 are put in place to implement a receiver, which turns out to be much more 41 00:02:31,390 --> 00:02:35,62 complicated than the transmitter, because the receiver has to undo all the nasty 42 00:02:35,62 --> 00:02:40,752 things that happen to the signal. When it travels over the channel, 43 00:02:40,752 --> 00:02:44,674 including distortion and noise and so on. As a matter of fact, module 9.5 is like a 44 00:02:44,674 --> 00:02:48,37 teaser that will probably whet your appetite for more advanced signal 45 00:02:48,37 --> 00:02:54,560 processing techniques that you will be able to study in more advanced classes. 46 00:02:54,560 --> 00:02:59,20 And finally in module 9.6, we will study the ADSL protocol. 47 00:02:59,20 --> 00:03:03,170 Now it turns out that ADSL is just one big DFT. 48 00:03:03,170 --> 00:03:06,866 And so, the fact that we can implement it efficiently with the FFT algorithm, is 49 00:03:06,866 --> 00:03:10,226 really the reason behind the extraordinary commercial success, of the 50 00:03:10,226 --> 00:03:15,322 ADSL setup box. You will see that everything that we've 51 00:03:15,322 --> 00:03:18,182 studied so far really find it's place in the design of a sophisticated digital 52 00:03:18,182 --> 00:03:22,310 processing system. So we hope you have enjoyed this initial 53 00:03:22,310 --> 00:03:26,860 ride into the world of digital signal processing and hopefully we'll see each 54 00:03:26,860 --> 00:03:32,60 other again in more advanced classes in the future. 55 00:03:32,60 --> 00:03:36,752 Thank you. Hi and welcome to module 9.1 of Digital 56 00:03:36,752 --> 00:03:39,965 Signal Processing. In this module we will start to look at 57 00:03:39,965 --> 00:03:44,166 digital communication systems. In particular, we will look at the many 58 00:03:44,166 --> 00:03:49,520 incarnations that a signal will undergo from its source to its destination. 59 00:03:49,520 --> 00:03:51,830 This incarnations will travel through a variety. 60 00:03:51,830 --> 00:03:54,862 of different analog channel. And each channel will have a different 61 00:03:54,862 --> 00:03:58,540 set of constraints that the signal will have to submit itself to. 62 00:03:58,540 --> 00:04:02,131 And in this module we'll start to look how to design signals that fulfill the 63 00:04:02,131 --> 00:04:05,903 channel constraints. If you remember in the beginning of this 64 00:04:05,903 --> 00:04:09,167 class we gave you a little overview of the major improvements and through put 65 00:04:09,167 --> 00:04:12,660 for channels that we implicitly use every day. 66 00:04:12,660 --> 00:04:17,421 For instance, the transatlantic cables that allow telephoning from Europe to the 67 00:04:17,421 --> 00:04:21,837 Unites States have seen an improvement That went from five bits per second in 68 00:04:21,837 --> 00:04:28,52 1866 with the first cable to 60 terabytes per second last year. 69 00:04:28,52 --> 00:04:32,474 Similarly something you use every day at home, your modem that allows you to 70 00:04:32,474 --> 00:04:36,896 connect to the internet, has increased its data rate from 1,200 bits per second 71 00:04:36,896 --> 00:04:44,30 in the 50s to 24 megabits per second with the current incarnation of ADSL. 72 00:04:44,30 --> 00:04:47,34 Now what are the reasons behind this incredible success? 73 00:04:47,34 --> 00:04:51,370 Well, the first one clearly is the power of the DSP paradigm. 74 00:04:51,370 --> 00:04:55,330 The fact that DSP works with integers means that, for instance signals are very 75 00:04:55,330 --> 00:04:58,352 easy to regenerate. We have seen an example in the 76 00:04:58,352 --> 00:05:00,850 introduction, and we will see it again in a second. 77 00:05:00,850 --> 00:05:04,441 Also digital filters allow us to implement very precise phase control, and 78 00:05:04,441 --> 00:05:10,320 we will see how important phase is in the detection of a transmitted signal. 79 00:05:10,320 --> 00:05:16,280 And finally, we can seamlessly integrate adaptive algorithms into a DSP system. 80 00:05:16,280 --> 00:05:22,190 Adaptive algorithms are algorithmic procedures that adapt their behavior. 81 00:05:22,190 --> 00:05:25,454 As a function of the received signal. These are very hard things to do in 82 00:05:25,454 --> 00:05:28,950 analog hardware, but very easy to do in digital hardware. 83 00:05:28,950 --> 00:05:32,745 As a reminder of what happens when we use digital signals for communication, think 84 00:05:32,745 --> 00:05:38,0 of the problem of transmitting a string of binary digits over an analog channel. 85 00:05:38,0 --> 00:05:41,752 To do that, we build a very simple signal, an analog signal, where we 86 00:05:41,752 --> 00:05:46,380 associate the values plus 5 volts to the symbol 0. 87 00:05:46,380 --> 00:05:50,472 And minus 5 volts to symbol one. Now the signal is analog, but it encodes 88 00:05:50,472 --> 00:05:54,832 binary information, namely it encodes a string of integers. 89 00:05:54,832 --> 00:05:59,230 When we transmit this over wire, two things happen. 90 00:05:59,230 --> 00:06:03,640 The signal gets attenuated and noise gets added to the signal. 91 00:06:03,640 --> 00:06:07,918 So what we'll receive at the other end of the channel is The original signal 92 00:06:07,918 --> 00:06:12,610 attenuated by effect of G, summed to some random noise that corrupts the original 93 00:06:12,610 --> 00:06:17,361 signal. Now, if we want to regenerate the signal, 94 00:06:17,361 --> 00:06:21,20 the first thing we do is, undo the attenuation. 95 00:06:21,20 --> 00:06:25,50 So we multiply the received signal by, a gain factor, that is the reciprocal of 96 00:06:25,50 --> 00:06:28,881 the attenuation. So we multiply the signal by g, we obtain 97 00:06:28,881 --> 00:06:32,301 a signal that has, once again the amplitude of the original signal but in 98 00:06:32,301 --> 00:06:38,392 so doing we also amplified noise. And so, we have very unclean levels here, 99 00:06:38,392 --> 00:06:44,644 which could cause all sorts of problems. But since we know that signal is bi level 100 00:06:44,644 --> 00:06:49,772 all we need to do is threshold. This signal, and when we see that it's 101 00:06:49,772 --> 00:06:53,756 positive, we set it plus 5. And when we see that it's negative, we 102 00:06:53,756 --> 00:06:57,314 set it minus 5. This is easily accomplished in digital 103 00:06:57,314 --> 00:07:03,50 domain by taking the sign of the signal before undoing the attenuation factor. 104 00:07:03,50 --> 00:07:05,829 And this is the signal that we get at the other end of the transmission channel. 105 00:07:07,400 --> 00:07:12,20 And we can repeat this procedure as many times as we need and that explains why we 106 00:07:12,20 --> 00:07:16,360 can send so much information over very, very long cables that go all the way 107 00:07:16,360 --> 00:07:22,36 under the ocean. The second success factor for digital 108 00:07:22,36 --> 00:07:28,50 communications today comes from the algorithmic nature of DSP techniques. 109 00:07:28,50 --> 00:07:31,771 We have seen an example in image coding, in JPEG, where signal processing 110 00:07:31,771 --> 00:07:35,919 techniques such as the discreet cosign transform could be matched seamlessly to 111 00:07:35,919 --> 00:07:42,40 information theory techniques that involve the compression of bit streams. 112 00:07:42,40 --> 00:07:46,400 And this interplay between these two techniques from different domains. 113 00:07:46,400 --> 00:07:48,980 Creates such powerful compression algorithms. 114 00:07:48,980 --> 00:07:52,750 Other everyday examples can be found in CDs or DVDs. 115 00:07:52,750 --> 00:07:57,300 Where you have encoding of acoustic or video information matched to powerful 116 00:07:57,300 --> 00:08:02,232 error correcting codes. So that DVDs or CDs that are scratched or 117 00:08:02,232 --> 00:08:05,730 dusty still play. And in communications systems. 118 00:08:05,730 --> 00:08:09,6 Techniques such as trellis coded modulation and Viterbi decoding are used 119 00:08:09,6 --> 00:08:13,170 to exploit all the capacity of an analog communication channel. 120 00:08:13,170 --> 00:08:16,818 The third success factor for digital communications is related to hardware 121 00:08:16,818 --> 00:08:20,269 advancements. We can have today miniaturized devices 122 00:08:20,269 --> 00:08:24,238 that we can keep in our pocket, we can have general purpose platforms used to 123 00:08:24,238 --> 00:08:28,396 develop advanced communication systems, so we don't need to develop specific 124 00:08:28,396 --> 00:08:34,858 hardware for each different task. And communication devices have become 125 00:08:34,858 --> 00:08:39,280 very power efficient, so that we can have Large data centers, or central offices 126 00:08:39,280 --> 00:08:45,32 that process an enormous number of communication channels in peril. 127 00:08:45,32 --> 00:08:49,257 So let's have a look at what happens when you place a call from your mobile phone 128 00:08:49,257 --> 00:08:55,118 to someone that has their phone at home. The information is first sent over the 129 00:08:55,118 --> 00:08:59,343 air to the closest base station where it is now converted to a different format 130 00:08:59,343 --> 00:09:05,172 and sent over copper wires to a switch. The switch is designed to find the 131 00:09:05,172 --> 00:09:10,270 routing pattern that will send the information to the final destination. 132 00:09:10,270 --> 00:09:14,110 The switch will send information over what is going to most likely an optic 133 00:09:14,110 --> 00:09:17,910 fiber channel to the global telephone network. 134 00:09:17,910 --> 00:09:21,126 The telephone network will route your information to the central office that is 135 00:09:21,126 --> 00:09:25,369 closest to the person you'll calling. The central office will then send the 136 00:09:25,369 --> 00:09:28,968 same information in yet a different format over a coax cable to the switch 137 00:09:28,968 --> 00:09:32,980 that is closest to the telephone that is being called and finally from the closest 138 00:09:32,980 --> 00:09:39,227 switch to the phone in the house. There is what is called the last smile 139 00:09:39,227 --> 00:09:45,545 which is a longish piece of copper wire. So, you see at every change of channel 140 00:09:45,545 --> 00:09:51,376 many many things can happen. The signal can be converted to digital 141 00:09:51,376 --> 00:09:55,34 again and then back to analog. The modulation schemes and the signal 142 00:09:55,34 --> 00:09:58,466 formats that we will have to use on this different stretches of the channel will 143 00:09:58,466 --> 00:10:02,760 have to adopt to the physical characteristics of the medium. 144 00:10:02,760 --> 00:10:06,605 Every analog channel. Has two unescapable limits that we have 145 00:10:06,605 --> 00:10:10,700 to reckon with. The first is a bandwith constraint. 146 00:10:10,700 --> 00:10:14,354 The signals that we can send over an analog channel will have to be limited to 147 00:10:14,354 --> 00:10:18,8 a certain frequency band, and the second limit is the fact that we cannot use 148 00:10:18,8 --> 00:10:23,844 arbitrary power over that band. There will be limits on the power of the 149 00:10:23,844 --> 00:10:27,642 signal we can send. The maximum amount of informatin we will 150 00:10:27,642 --> 00:10:31,994 be able to send with the channel given this contraints is called a capicity of 151 00:10:31,994 --> 00:10:35,652 the channel. We will see a remarkable result of 152 00:10:35,652 --> 00:10:38,600 information theory later on that exactly quantifies the capcity of the channel 153 00:10:38,600 --> 00:10:41,830 given it's signal to noise ratio and it's bandwidth. 154 00:10:41,830 --> 00:10:46,110 As communication system engineers we are given the specifications of a chennel. 155 00:10:46,110 --> 00:10:51,120 And we want to design a system that sends as much information over this channel. 156 00:10:51,120 --> 00:10:56,485 And as reliably as possible give this unescapeable capacity constraint. 157 00:10:56,485 --> 00:11:00,565 Amount of information and reliability are concepts that are still a little fuzzy 158 00:11:00,565 --> 00:11:04,583 for the time being. They will become clearer later on but we 159 00:11:04,583 --> 00:11:08,800 can certainly look at the intuition behind this problem. 160 00:11:08,800 --> 00:11:11,712 For instance, if we look at the relationship between bandwidth and 161 00:11:11,712 --> 00:11:15,270 capacity, we can do this very simple thought experiment. 162 00:11:15,270 --> 00:11:19,214 Suppose we are going to transmit information encoded as a sequence of 163 00:11:19,214 --> 00:11:23,287 digital samples over a continuous time channel. 164 00:11:23,287 --> 00:11:27,503 So, what we do we take the samples we interpolate the samples with a certain 165 00:11:27,503 --> 00:11:31,719 sampling period Ts now if we make Ts very small it means that we can send more 166 00:11:31,719 --> 00:11:37,212 samples per second. But if we make Ts small we know that the 167 00:11:37,212 --> 00:11:40,932 bandwidth will grow as the reciprocal of Ts you remember the formula for 168 00:11:40,932 --> 00:11:47,298 interpolate signal. In the sampling theorem, it says that the 169 00:11:47,298 --> 00:11:53,94 analog spectrum will be zero outside of a band that goes from omega n to minus 170 00:11:53,94 --> 00:11:59,280 omega n. And omega n is Pi over Ts. 171 00:11:59,280 --> 00:12:02,829 If we make ts small the bandwidth will grow with 1 over Ts. 172 00:12:04,300 --> 00:12:08,268 So we see, that capacity, and the amount of information that we can send per 173 00:12:08,268 --> 00:12:13,660 second, are related in some way. Similarly, the relationship between the 174 00:12:13,660 --> 00:12:18,82 power constraint and capacity, can be appreciated, because we can never do away 175 00:12:18,82 --> 00:12:22,226 with noise. So, at the receiver, when we send the 176 00:12:22,226 --> 00:12:26,496 sequence of integers for instance, we will have to guess What has been set 177 00:12:26,496 --> 00:12:32,306 after it has been corrupted by noise. So suppose we have a channel that 178 00:12:32,306 --> 00:12:36,18 introduces a noise variance of 1 and suppose we are transmitting the integer 179 00:12:36,18 --> 00:12:40,372 between 1 and 10. If the variance is 1 lots of transmitted 180 00:12:40,372 --> 00:12:44,532 integers will have and error that will send them very close to the next integer 181 00:12:44,532 --> 00:12:48,480 in line. So suppose I'm sending the integers 182 00:12:48,480 --> 00:12:53,776 between 1 and 10. And so I'm sending say one but because of 183 00:12:53,776 --> 00:12:57,690 the noise the one will be 1.75 for instance. 184 00:12:57,690 --> 00:13:01,902 So I'm not really sure if what was sent was one or was two. 185 00:13:01,902 --> 00:13:05,900 And then the strategies say okay. Let's transmit only odd numbers. 186 00:13:05,900 --> 00:13:09,540 So instead of everything I will not just be at 0, we'll transmit 1 and then I will 187 00:13:09,540 --> 00:13:14,780 not transmit 2 but I will transmit 3. So I'm increasing the gap between 188 00:13:14,780 --> 00:13:19,655 possible symbols and so the noise that before Had probably me misguessing the 189 00:13:19,655 --> 00:13:24,455 transmission of 1, will still be small enough to bring me back to the original 190 00:13:24,455 --> 00:13:29,340 signal. Now it is rather intuitive that, all 191 00:13:29,340 --> 00:13:33,302 other things being equal. A signal with a wider range will have a 192 00:13:33,302 --> 00:13:36,572 larger power. So, if I want to keep the power constant, 193 00:13:36,572 --> 00:13:40,268 I will still have to send symbols between zero and 10, but now there are only half 194 00:13:40,268 --> 00:13:43,740 as many odd integers between zero and 10 that there are integers, and so the 195 00:13:43,740 --> 00:13:49,206 amount of information that I can send per unit of time. 196 00:13:49,206 --> 00:13:52,446 will be halved. Let's now look at some common 197 00:13:52,446 --> 00:13:57,70 communication channels and see what their power and bandwidth constraints are. 198 00:13:57,70 --> 00:14:01,690 Maybe the simplest communication channel that we're still familiar with, is the AM 199 00:14:01,690 --> 00:14:05,512 radio channel. AM stands for amplitude modulation, and 200 00:14:05,512 --> 00:14:08,480 indeed the radio transmitter is very simple. 201 00:14:08,480 --> 00:14:12,134 We take an analog signal, it can be voice or music, we do a low-pass filtering 202 00:14:12,134 --> 00:14:15,962 operation to limit its bandwidth, And then we do a very, very simple sinusoidal 203 00:14:15,962 --> 00:14:20,480 modulation with the cosine of a given carrier. 204 00:14:20,480 --> 00:14:23,286 The result in modulated signal, is simply put to an antenna, and it will be 205 00:14:23,286 --> 00:14:27,60 propogated in the radial spectrum. The radial spectrum is a very scarce 206 00:14:27,60 --> 00:14:29,629 resource. There's only one radial spectrum, 207 00:14:29,629 --> 00:14:33,216 everybody has to share it. Therefore, every frequency band in the 208 00:14:33,216 --> 00:14:38,600 spectrum, is strictly regulated by law. In the case of AM, the band is from 530 209 00:14:38,600 --> 00:14:43,987 kilohertz to 1.7 megahertz. This is divided into 8 kilohertz wide 210 00:14:43,987 --> 00:14:47,542 channels. And each radio station gets allocated a 211 00:14:47,542 --> 00:14:51,353 specific channel. The power is limited by law for a variety 212 00:14:51,353 --> 00:14:54,130 of reasons. The first is that the propagation 213 00:14:54,130 --> 00:14:58,760 patterns for AM waves is very different during the day, and during the night. 214 00:14:58,760 --> 00:15:02,258 In particular at night time, AM radio waves travel much further than during the 215 00:15:02,258 --> 00:15:05,406 day. So, they can create all source of 216 00:15:05,406 --> 00:15:09,380 interferences in distant places if the power is not limited. 217 00:15:09,380 --> 00:15:12,305 Also you don't want radio stations to use too much power because it wouldn't be 218 00:15:12,305 --> 00:15:15,185 healthy for people live in the vicinity of the transmitter and on the channel 219 00:15:15,185 --> 00:15:19,10 where all are familiar with is the telephone channel. 220 00:15:19,10 --> 00:15:22,178 The telephone network is more properly called the switched telephone network 221 00:15:22,178 --> 00:15:25,10 because instead of taking the combinatorial approach and having each 222 00:15:25,10 --> 00:15:28,620 phone connected to every other phone in the world. 223 00:15:28,620 --> 00:15:30,780 What happens is that when you call on other phone. 224 00:15:30,780 --> 00:15:34,724 Your phone is connected to the central office, and the central office determines 225 00:15:34,724 --> 00:15:38,436 which parts of the network have to be connected together so that your call can 226 00:15:38,436 --> 00:15:44,150 be routed to the destination phone. So, the piece of wire that connects you 227 00:15:44,150 --> 00:15:47,993 to the central office is up to, maybe say, a couple of kilometers long, and is 228 00:15:47,993 --> 00:15:52,795 called the last mile. The central office today is a bunch of 229 00:15:52,795 --> 00:15:57,20 digital switches, in the old days was mechanical rotary switches The network 230 00:15:57,20 --> 00:16:00,985 can be anything from optical fiber to satellite links to anything else in 231 00:16:00,985 --> 00:16:08,490 between, and here you have the symmetric part where you get to your destination. 232 00:16:08,490 --> 00:16:14,420 The telephone channel is conventionally limited from 300 hertz to 3,000 hertz. 233 00:16:14,420 --> 00:16:18,314 These are historical limits that depend on the kind of hardware that was used In 234 00:16:18,314 --> 00:16:22,210 the old days in central office and in the network. 235 00:16:22,210 --> 00:16:26,498 Today these limits are historical artifact but they are kept because anyway 236 00:16:26,498 --> 00:16:30,786 voice communications are perfectly intelligible within this band And with 237 00:16:30,786 --> 00:16:36,684 the reduced band, you can multiplex. Namely, you can put together very many 238 00:16:36,684 --> 00:16:41,262 communications on a wider channel. The power that you can send on a 239 00:16:41,262 --> 00:16:47,450 telephone wire is limited from 0.2 to 0.7 volts, or root mean square. 240 00:16:47,450 --> 00:16:50,736 And this a strictly enforced limit to make sure that you don't send signals 241 00:16:50,736 --> 00:16:54,155 that can burn the equipment at the central office. 242 00:16:54,155 --> 00:16:57,510 And the signal to noise ratio is rather good because the analog part of the 243 00:16:57,510 --> 00:17:00,590 telephone network operates in the bass band and there's not a lot of 244 00:17:00,590 --> 00:17:05,674 interference in the low frequencies. So let's how we're going to go about 245 00:17:05,674 --> 00:17:10,246 designing a communications system. Probably the most important concept here, 246 00:17:10,246 --> 00:17:13,80 is that we're going to adopt the all-digital paradigm. 247 00:17:13,80 --> 00:17:16,915 What this means is that, we will keep everything in the digital domain until we 248 00:17:16,915 --> 00:17:20,826 hit the physical channel. And if we were to describe this as a 249 00:17:20,826 --> 00:17:25,642 block diagram, it would look like this. We have a binary bit stream, can 250 00:17:25,642 --> 00:17:31,250 represent any sort of views or data. We have a transmitter that operates 251 00:17:31,250 --> 00:17:36,70 entirely in digital domain that generates a discreet time signal s of n. 252 00:17:36,70 --> 00:17:39,270 The last element in the transmission chain. 253 00:17:39,270 --> 00:17:42,930 Is a digital to analog converter operating at a given frequency, or at the 254 00:17:42,930 --> 00:17:46,470 given period as you prefer, that transforms this signal into an analog 255 00:17:46,470 --> 00:17:52,790 signal that we can send over the channel. So remember the channel constraints. 256 00:17:52,790 --> 00:17:55,210 Look a little bit like a filter design problem. 257 00:17:55,210 --> 00:18:00,40 We have a band width that is specified in terms of a maximum and minimum frequency. 258 00:18:00,40 --> 00:18:05,303 So we can only operate over this band. And then we have a power constraint that 259 00:18:05,303 --> 00:18:09,180 restricts the power associated with the signal that we produce. 260 00:18:09,180 --> 00:18:13,8 So if you want to convert this to our old digital paradigm the first thing to do is 261 00:18:13,8 --> 00:18:16,770 to convert the specs into discreet time specs. 262 00:18:16,770 --> 00:18:21,172 So we choose a frequency for the D2A converted, fs, this will be our niquist 263 00:18:21,172 --> 00:18:26,680 frequency, fs over 2, and with this we can convert the specs. 264 00:18:26,680 --> 00:18:30,574 Maximum frequency will be pi, and our minimum and maximum frequency bands will 265 00:18:30,574 --> 00:18:34,390 be omega min and omega max using the relation. 266 00:18:34,390 --> 00:18:41,994 Omega equal to 2 pi f over fs. And you can put here, f min or f. 267 00:18:41,994 --> 00:18:45,585 Now, here are some working hypotheses that are common to most transmission 268 00:18:45,585 --> 00:18:49,270 systems you will ever see. We start from a bitstream. 269 00:18:49,270 --> 00:18:52,420 And we will convert this bitstream into a sequence of symbols. 270 00:18:52,420 --> 00:18:56,100 For samples a of n, via something called a mapper. 271 00:18:56,100 --> 00:19:02,500 What the mapper does is associate group of bits to a specific symbol. 272 00:19:02,500 --> 00:19:05,611 Just to give you a concrete example assume we're going to map each group of 273 00:19:05,611 --> 00:19:10,199 bits to its decimal value. We want to model the sequence of symbols 274 00:19:10,199 --> 00:19:13,379 as a white random sequence and in order to do so, we have to assume that the 275 00:19:13,379 --> 00:19:17,28 bitstream is a completely random sequence. 276 00:19:17,28 --> 00:19:20,43 Now, this is not necessarily the case, for instance, imagine you're digitizing 277 00:19:20,43 --> 00:19:22,650 audio and you have long stretches of silence. 278 00:19:22,650 --> 00:19:26,120 This will result into a long sequence of zeros. 279 00:19:26,120 --> 00:19:29,840 And so, what we do is we put a scrambler in the line. 280 00:19:29,840 --> 00:19:33,365 What a scrambler does. It transforms a sequence of bits into a 281 00:19:33,365 --> 00:19:37,135 sequence that looks like a random sequence but this randomization is 282 00:19:37,135 --> 00:19:42,438 completely invariable at a receiver. So, we start with the sequence of zeroes 283 00:19:42,438 --> 00:19:45,176 for instance. We put into the scrambler, it's going to 284 00:19:45,176 --> 00:19:48,76 look like a completely random sequence of zeroes and one but it's done 285 00:19:48,76 --> 00:19:51,126 algorithmically so we can invert this randomization on the receiver and 286 00:19:51,126 --> 00:19:56,302 retrieve the original bitstream.. With this we can consider the sequence of 287 00:19:56,302 --> 00:20:00,450 symbol a of n as a wide sequence. And now we need to convert the sequence 288 00:20:00,450 --> 00:20:03,840 into a continuous time signal within the constraints. 289 00:20:03,840 --> 00:20:06,900 So here's the updated transmission scheme. 290 00:20:06,900 --> 00:20:11,572 User data goes into a scrambler. This is a random binary sequence. 291 00:20:11,572 --> 00:20:15,230 The mapper converts groups of bits to symbols. 292 00:20:15,230 --> 00:20:19,73 And then we have to decide what to do in here before converting this into an 293 00:20:19,73 --> 00:20:23,40 analog signal. The first problem is Fulfilling the 294 00:20:23,40 --> 00:20:27,106 bandwidth constraint. If we assume that the data is randomized 295 00:20:27,106 --> 00:20:31,264 and therefore the symbol sequence is a wide sequence, we know that the power 296 00:20:31,264 --> 00:20:35,554 spectral density is simply equal to the variance and so the power of the signal 297 00:20:35,554 --> 00:20:39,448 will be constant over the entire frequency band but we actually need to 298 00:20:39,448 --> 00:20:47,582 fit it into the small band here as specified by the bandwidth constraint. 299 00:20:47,582 --> 00:20:51,870 So, how do we do this. Well in order to do that we need to 300 00:20:51,870 --> 00:20:56,910 introduce a new technique called up sampling and we will see this in the next 301 00:20:56,910 --> 00:20:59,423 module.