WEBVTT 00:00:00.600 --> 00:00:04.100 Hi, and welcome to module nine of digital signal processing. 00:00:04.100 --> 00:00:06.770 This is the last module in our class, and this is really where it all comes 00:00:06.770 --> 00:00:09.691 together. In this module we will review the 00:00:09.691 --> 00:00:14.140 principles behind the success of digital communication systems. 00:00:14.140 --> 00:00:17.562 And we will look at different communication systems starting from the 00:00:17.562 --> 00:00:21.274 voice band modems that were popular a few years ago and that you can still hear 00:00:21.274 --> 00:00:24.928 when you use a fax machine, to the most recent incarnations like the ADSL box 00:00:24.928 --> 00:00:28.292 that you have in your home and that you're probably using to watch this 00:00:28.292 --> 00:00:34.708 video. Digital communication systems need no 00:00:34.708 --> 00:00:37.710 introduction. The amount of information that we consume 00:00:37.710 --> 00:00:42.900 and that we produce every day is staggering by an historical standard. 00:00:42.900 --> 00:00:45.482 And what is even more amazing is that we can access this wealth of information 00:00:45.482 --> 00:00:49.139 from basically anyway via a small device, like the smartphoen that you have in your 00:00:49.139 --> 00:00:52.944 pocket. There is actually a joke about that and 00:00:52.944 --> 00:00:55.700 suppose that someone from the Renaissance, like Leonardo, was 00:00:55.700 --> 00:00:59.667 teleported to today. And you'd have to explain to them what 00:00:59.667 --> 00:01:02.992 your smartphone does. Well you have to say this is a small 00:01:02.992 --> 00:01:06.473 device that allows me to access everything that has been done, written 00:01:06.473 --> 00:01:11.400 about, or were said by mankind since the beginning of history. 00:01:11.400 --> 00:01:14.120 And I use it mainly to look at pictures of cats. 00:01:14.120 --> 00:01:16.926 But jokes aside the truth remains that communications systems, digital 00:01:16.926 --> 00:01:20.440 communications systems. Are really the pinnacle achievment of 00:01:20.440 --> 00:01:23.981 digital signal processing. So in this module we'll start from the 00:01:23.981 --> 00:01:27.824 basic principles in module nine one and we'll see the kind of signals that we 00:01:27.824 --> 00:01:34.520 have to design in order to be able to transmit them over a physical channel. 00:01:34.520 --> 00:01:38.546 Now a physical channel whether it's a wireless channel, whether it's a piece of 00:01:38.546 --> 00:01:42.572 wire or an optical fiber will always impose two fundamental constraints on the 00:01:42.572 --> 00:01:47.310 kind of signal that can transit over the channel. 00:01:47.310 --> 00:01:50.460 The first one is a bandwidth constraint, which means that we will only have a 00:01:50.460 --> 00:01:55.330 certain range of frequencies over which we can send information. 00:01:55.330 --> 00:01:57.920 And the second constraint is a power constraint. 00:01:57.920 --> 00:02:01.640 It limits the amount of power that we can inject onto the channel. 00:02:01.640 --> 00:02:05.640 So in module 9.2, we will tackle the banther constraint, in detail. 00:02:05.640 --> 00:02:08.720 And in module 9.3, we will look at the power constraint. 00:02:08.720 --> 00:02:11.970 And we will see in the end how these two constraints limit the maximum amount of 00:02:11.970 --> 00:02:15.790 information that we can send over a channel. 00:02:16.320 --> 00:02:19.792 In Module 9.4, we will look at the modulation and demodulation techniques 00:02:19.792 --> 00:02:24.640 that are specially designed to transmit data over the telephone channel. 00:02:24.640 --> 00:02:28.204 And in Module 9.5, we will examine the several signal processes and tricks that 00:02:28.204 --> 00:02:31.390 are put in place to implement a receiver, which turns out to be much more 00:02:31.390 --> 00:02:35.620 complicated than the transmitter, because the receiver has to undo all the nasty 00:02:35.620 --> 00:02:40.752 things that happen to the signal. When it travels over the channel, 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 00:02:44.674 --> 00:02:48.370 teaser that will probably whet your appetite for more advanced signal 00:02:48.370 --> 00:02:54.560 processing techniques that you will be able to study in more advanced classes. 00:02:54.560 --> 00:02:59.200 And finally in module 9.6, we will study the ADSL protocol. 00:02:59.200 --> 00:03:03.170 Now it turns out that ADSL is just one big DFT. 00:03:03.170 --> 00:03:06.866 And so, the fact that we can implement it efficiently with the FFT algorithm, is 00:03:06.866 --> 00:03:10.226 really the reason behind the extraordinary commercial success, of the 00:03:10.226 --> 00:03:15.322 ADSL setup box. You will see that everything that we've 00:03:15.322 --> 00:03:18.182 studied so far really find it's place in the design of a sophisticated digital 00:03:18.182 --> 00:03:22.310 processing system. So we hope you have enjoyed this initial 00:03:22.310 --> 00:03:26.860 ride into the world of digital signal processing and hopefully we'll see each 00:03:26.860 --> 00:03:32.600 other again in more advanced classes in the future. 00:03:32.600 --> 00:03:36.752 Thank you. Hi and welcome to module 9.1 of Digital 00:03:36.752 --> 00:03:39.965 Signal Processing. In this module we will start to look at 00:03:39.965 --> 00:03:44.166 digital communication systems. In particular, we will look at the many 00:03:44.166 --> 00:03:49.520 incarnations that a signal will undergo from its source to its destination. 00:03:49.520 --> 00:03:51.830 This incarnations will travel through a variety. 00:03:51.830 --> 00:03:54.862 of different analog channel. And each channel will have a different 00:03:54.862 --> 00:03:58.540 set of constraints that the signal will have to submit itself to. 00:03:58.540 --> 00:04:02.131 And in this module we'll start to look how to design signals that fulfill the 00:04:02.131 --> 00:04:05.903 channel constraints. If you remember in the beginning of this 00:04:05.903 --> 00:04:09.167 class we gave you a little overview of the major improvements and through put 00:04:09.167 --> 00:04:12.660 for channels that we implicitly use every day. 00:04:12.660 --> 00:04:17.421 For instance, the transatlantic cables that allow telephoning from Europe to the 00:04:17.421 --> 00:04:21.837 Unites States have seen an improvement That went from five bits per second in 00:04:21.837 --> 00:04:28.520 1866 with the first cable to 60 terabytes per second last year. 00:04:28.520 --> 00:04:32.474 Similarly something you use every day at home, your modem that allows you to 00:04:32.474 --> 00:04:36.896 connect to the internet, has increased its data rate from 1,200 bits per second 00:04:36.896 --> 00:04:44.300 in the 50s to 24 megabits per second with the current incarnation of ADSL. 00:04:44.300 --> 00:04:47.340 Now what are the reasons behind this incredible success? 00:04:47.340 --> 00:04:51.370 Well, the first one clearly is the power of the DSP paradigm. 00:04:51.370 --> 00:04:55.330 The fact that DSP works with integers means that, for instance signals are very 00:04:55.330 --> 00:04:58.352 easy to regenerate. We have seen an example in the 00:04:58.352 --> 00:05:00.850 introduction, and we will see it again in a second. 00:05:00.850 --> 00:05:04.441 Also digital filters allow us to implement very precise phase control, and 00:05:04.441 --> 00:05:10.320 we will see how important phase is in the detection of a transmitted signal. 00:05:10.320 --> 00:05:16.280 And finally, we can seamlessly integrate adaptive algorithms into a DSP system. 00:05:16.280 --> 00:05:22.190 Adaptive algorithms are algorithmic procedures that adapt their behavior. 00:05:22.190 --> 00:05:25.454 As a function of the received signal. These are very hard things to do in 00:05:25.454 --> 00:05:28.950 analog hardware, but very easy to do in digital hardware. 00:05:28.950 --> 00:05:32.745 As a reminder of what happens when we use digital signals for communication, think 00:05:32.745 --> 00:05:38.000 of the problem of transmitting a string of binary digits over an analog channel. 00:05:38.000 --> 00:05:41.752 To do that, we build a very simple signal, an analog signal, where we 00:05:41.752 --> 00:05:46.380 associate the values plus 5 volts to the symbol 0. 00:05:46.380 --> 00:05:50.472 And minus 5 volts to symbol one. Now the signal is analog, but it encodes 00:05:50.472 --> 00:05:54.832 binary information, namely it encodes a string of integers. 00:05:54.832 --> 00:05:59.230 When we transmit this over wire, two things happen. 00:05:59.230 --> 00:06:03.640 The signal gets attenuated and noise gets added to the signal. 00:06:03.640 --> 00:06:07.918 So what we'll receive at the other end of the channel is The original signal 00:06:07.918 --> 00:06:12.610 attenuated by effect of G, summed to some random noise that corrupts the original 00:06:12.610 --> 00:06:17.361 signal. Now, if we want to regenerate the signal, 00:06:17.361 --> 00:06:21.200 the first thing we do is, undo the attenuation. 00:06:21.200 --> 00:06:25.500 So we multiply the received signal by, a gain factor, that is the reciprocal of 00:06:25.500 --> 00:06:28.881 the attenuation. So we multiply the signal by g, we obtain 00:06:28.881 --> 00:06:32.301 a signal that has, once again the amplitude of the original signal but in 00:06:32.301 --> 00:06:38.392 so doing we also amplified noise. And so, we have very unclean levels here, 00:06:38.392 --> 00:06:44.644 which could cause all sorts of problems. But since we know that signal is bi level 00:06:44.644 --> 00:06:49.772 all we need to do is threshold. This signal, and when we see that it's 00:06:49.772 --> 00:06:53.756 positive, we set it plus 5. And when we see that it's negative, we 00:06:53.756 --> 00:06:57.314 set it minus 5. This is easily accomplished in digital 00:06:57.314 --> 00:07:03.500 domain by taking the sign of the signal before undoing the attenuation factor. 00:07:03.500 --> 00:07:05.829 And this is the signal that we get at the other end of the transmission channel. 00:07:07.400 --> 00:07:12.200 And we can repeat this procedure as many times as we need and that explains why we 00:07:12.200 --> 00:07:16.360 can send so much information over very, very long cables that go all the way 00:07:16.360 --> 00:07:22.360 under the ocean. The second success factor for digital 00:07:22.360 --> 00:07:28.500 communications today comes from the algorithmic nature of DSP techniques. 00:07:28.500 --> 00:07:31.771 We have seen an example in image coding, in JPEG, where signal processing 00:07:31.771 --> 00:07:35.919 techniques such as the discreet cosign transform could be matched seamlessly to 00:07:35.919 --> 00:07:42.400 information theory techniques that involve the compression of bit streams. 00:07:42.400 --> 00:07:46.400 And this interplay between these two techniques from different domains. 00:07:46.400 --> 00:07:48.980 Creates such powerful compression algorithms. 00:07:48.980 --> 00:07:52.750 Other everyday examples can be found in CDs or DVDs. 00:07:52.750 --> 00:07:57.300 Where you have encoding of acoustic or video information matched to powerful 00:07:57.300 --> 00:08:02.232 error correcting codes. So that DVDs or CDs that are scratched or 00:08:02.232 --> 00:08:05.730 dusty still play. And in communications systems. 00:08:05.730 --> 00:08:09.600 Techniques such as trellis coded modulation and Viterbi decoding are used 00:08:09.600 --> 00:08:13.170 to exploit all the capacity of an analog communication channel. 00:08:13.170 --> 00:08:16.818 The third success factor for digital communications is related to hardware 00:08:16.818 --> 00:08:20.269 advancements. We can have today miniaturized devices 00:08:20.269 --> 00:08:24.238 that we can keep in our pocket, we can have general purpose platforms used to 00:08:24.238 --> 00:08:28.396 develop advanced communication systems, so we don't need to develop specific 00:08:28.396 --> 00:08:34.858 hardware for each different task. And communication devices have become 00:08:34.858 --> 00:08:39.280 very power efficient, so that we can have Large data centers, or central offices 00:08:39.280 --> 00:08:45.320 that process an enormous number of communication channels in peril. 00:08:45.320 --> 00:08:49.257 So let's have a look at what happens when you place a call from your mobile phone 00:08:49.257 --> 00:08:55.118 to someone that has their phone at home. The information is first sent over the 00:08:55.118 --> 00:08:59.343 air to the closest base station where it is now converted to a different format 00:08:59.343 --> 00:09:05.172 and sent over copper wires to a switch. The switch is designed to find the 00:09:05.172 --> 00:09:10.270 routing pattern that will send the information to the final destination. 00:09:10.270 --> 00:09:14.110 The switch will send information over what is going to most likely an optic 00:09:14.110 --> 00:09:17.910 fiber channel to the global telephone network. 00:09:17.910 --> 00:09:21.126 The telephone network will route your information to the central office that is 00:09:21.126 --> 00:09:25.369 closest to the person you'll calling. The central office will then send the 00:09:25.369 --> 00:09:28.968 same information in yet a different format over a coax cable to the switch 00:09:28.968 --> 00:09:32.980 that is closest to the telephone that is being called and finally from the closest 00:09:32.980 --> 00:09:39.227 switch to the phone in the house. There is what is called the last smile 00:09:39.227 --> 00:09:45.545 which is a longish piece of copper wire. So, you see at every change of channel 00:09:45.545 --> 00:09:51.376 many many things can happen. The signal can be converted to digital 00:09:51.376 --> 00:09:55.340 again and then back to analog. The modulation schemes and the signal 00:09:55.340 --> 00:09:58.466 formats that we will have to use on this different stretches of the channel will 00:09:58.466 --> 00:10:02.760 have to adopt to the physical characteristics of the medium. 00:10:02.760 --> 00:10:06.605 Every analog channel. Has two unescapable limits that we have 00:10:06.605 --> 00:10:10.700 to reckon with. The first is a bandwith constraint. 00:10:10.700 --> 00:10:14.354 The signals that we can send over an analog channel will have to be limited to 00:10:14.354 --> 00:10:18.800 a certain frequency band, and the second limit is the fact that we cannot use 00:10:18.800 --> 00:10:23.844 arbitrary power over that band. There will be limits on the power of the 00:10:23.844 --> 00:10:27.642 signal we can send. The maximum amount of informatin we will 00:10:27.642 --> 00:10:31.994 be able to send with the channel given this contraints is called a capicity of 00:10:31.994 --> 00:10:35.652 the channel. We will see a remarkable result of 00:10:35.652 --> 00:10:38.600 information theory later on that exactly quantifies the capcity of the channel 00:10:38.600 --> 00:10:41.830 given it's signal to noise ratio and it's bandwidth. 00:10:41.830 --> 00:10:46.110 As communication system engineers we are given the specifications of a chennel. 00:10:46.110 --> 00:10:51.120 And we want to design a system that sends as much information over this channel. 00:10:51.120 --> 00:10:56.485 And as reliably as possible give this unescapeable capacity constraint. 00:10:56.485 --> 00:11:00.565 Amount of information and reliability are concepts that are still a little fuzzy 00:11:00.565 --> 00:11:04.583 for the time being. They will become clearer later on but we 00:11:04.583 --> 00:11:08.800 can certainly look at the intuition behind this problem. 00:11:08.800 --> 00:11:11.712 For instance, if we look at the relationship between bandwidth and 00:11:11.712 --> 00:11:15.270 capacity, we can do this very simple thought experiment. 00:11:15.270 --> 00:11:19.214 Suppose we are going to transmit information encoded as a sequence of 00:11:19.214 --> 00:11:23.287 digital samples over a continuous time channel. 00:11:23.287 --> 00:11:27.503 So, what we do we take the samples we interpolate the samples with a certain 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 00:11:31.719 --> 00:11:37.212 samples per second. But if we make Ts small we know that the 00:11:37.212 --> 00:11:40.932 bandwidth will grow as the reciprocal of Ts you remember the formula for 00:11:40.932 --> 00:11:47.298 interpolate signal. In the sampling theorem, it says that the 00:11:47.298 --> 00:11:53.940 analog spectrum will be zero outside of a band that goes from omega n to minus 00:11:53.940 --> 00:11:59.280 omega n. And omega n is Pi over Ts. 00:11:59.280 --> 00:12:02.829 If we make ts small the bandwidth will grow with 1 over Ts. 00:12:04.300 --> 00:12:08.268 So we see, that capacity, and the amount of information that we can send per 00:12:08.268 --> 00:12:13.660 second, are related in some way. Similarly, the relationship between the 00:12:13.660 --> 00:12:18.820 power constraint and capacity, can be appreciated, because we can never do away 00:12:18.820 --> 00:12:22.226 with noise. So, at the receiver, when we send the 00:12:22.226 --> 00:12:26.496 sequence of integers for instance, we will have to guess What has been set 00:12:26.496 --> 00:12:32.306 after it has been corrupted by noise. So suppose we have a channel that 00:12:32.306 --> 00:12:36.180 introduces a noise variance of 1 and suppose we are transmitting the integer 00:12:36.180 --> 00:12:40.372 between 1 and 10. If the variance is 1 lots of transmitted 00:12:40.372 --> 00:12:44.532 integers will have and error that will send them very close to the next integer 00:12:44.532 --> 00:12:48.480 in line. So suppose I'm sending the integers 00:12:48.480 --> 00:12:53.776 between 1 and 10. And so I'm sending say one but because of 00:12:53.776 --> 00:12:57.690 the noise the one will be 1.75 for instance. 00:12:57.690 --> 00:13:01.902 So I'm not really sure if what was sent was one or was two. 00:13:01.902 --> 00:13:05.900 And then the strategies say okay. Let's transmit only odd numbers. 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 00:13:09.540 --> 00:13:14.780 not transmit 2 but I will transmit 3. So I'm increasing the gap between 00:13:14.780 --> 00:13:19.655 possible symbols and so the noise that before Had probably me misguessing the 00:13:19.655 --> 00:13:24.455 transmission of 1, will still be small enough to bring me back to the original 00:13:24.455 --> 00:13:29.340 signal. Now it is rather intuitive that, all 00:13:29.340 --> 00:13:33.302 other things being equal. A signal with a wider range will have a 00:13:33.302 --> 00:13:36.572 larger power. So, if I want to keep the power constant, 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 00:13:40.268 --> 00:13:43.740 as many odd integers between zero and 10 that there are integers, and so the 00:13:43.740 --> 00:13:49.206 amount of information that I can send per unit of time. 00:13:49.206 --> 00:13:52.446 will be halved. Let's now look at some common 00:13:52.446 --> 00:13:57.700 communication channels and see what their power and bandwidth constraints are. 00:13:57.700 --> 00:14:01.690 Maybe the simplest communication channel that we're still familiar with, is the AM 00:14:01.690 --> 00:14:05.512 radio channel. AM stands for amplitude modulation, and 00:14:05.512 --> 00:14:08.480 indeed the radio transmitter is very simple. 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 00:14:12.134 --> 00:14:15.962 operation to limit its bandwidth, And then we do a very, very simple sinusoidal 00:14:15.962 --> 00:14:20.480 modulation with the cosine of a given carrier. 00:14:20.480 --> 00:14:23.286 The result in modulated signal, is simply put to an antenna, and it will be 00:14:23.286 --> 00:14:27.600 propogated in the radial spectrum. The radial spectrum is a very scarce 00:14:27.600 --> 00:14:29.629 resource. There's only one radial spectrum, 00:14:29.629 --> 00:14:33.216 everybody has to share it. Therefore, every frequency band in the 00:14:33.216 --> 00:14:38.600 spectrum, is strictly regulated by law. In the case of AM, the band is from 530 00:14:38.600 --> 00:14:43.987 kilohertz to 1.7 megahertz. This is divided into 8 kilohertz wide 00:14:43.987 --> 00:14:47.542 channels. And each radio station gets allocated a 00:14:47.542 --> 00:14:51.353 specific channel. The power is limited by law for a variety 00:14:51.353 --> 00:14:54.130 of reasons. The first is that the propagation 00:14:54.130 --> 00:14:58.760 patterns for AM waves is very different during the day, and during the night. 00:14:58.760 --> 00:15:02.258 In particular at night time, AM radio waves travel much further than during the 00:15:02.258 --> 00:15:05.406 day. So, they can create all source of 00:15:05.406 --> 00:15:09.380 interferences in distant places if the power is not limited. 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 00:15:12.305 --> 00:15:15.185 healthy for people live in the vicinity of the transmitter and on the channel 00:15:15.185 --> 00:15:19.100 where all are familiar with is the telephone channel. 00:15:19.100 --> 00:15:22.178 The telephone network is more properly called the switched telephone network 00:15:22.178 --> 00:15:25.100 because instead of taking the combinatorial approach and having each 00:15:25.100 --> 00:15:28.620 phone connected to every other phone in the world. 00:15:28.620 --> 00:15:30.780 What happens is that when you call on other phone. 00:15:30.780 --> 00:15:34.724 Your phone is connected to the central office, and the central office determines 00:15:34.724 --> 00:15:38.436 which parts of the network have to be connected together so that your call can 00:15:38.436 --> 00:15:44.150 be routed to the destination phone. So, the piece of wire that connects you 00:15:44.150 --> 00:15:47.993 to the central office is up to, maybe say, a couple of kilometers long, and is 00:15:47.993 --> 00:15:52.795 called the last mile. The central office today is a bunch of 00:15:52.795 --> 00:15:57.200 digital switches, in the old days was mechanical rotary switches The network 00:15:57.200 --> 00:16:00.985 can be anything from optical fiber to satellite links to anything else in 00:16:00.985 --> 00:16:08.490 between, and here you have the symmetric part where you get to your destination. 00:16:08.490 --> 00:16:14.420 The telephone channel is conventionally limited from 300 hertz to 3,000 hertz. 00:16:14.420 --> 00:16:18.314 These are historical limits that depend on the kind of hardware that was used In 00:16:18.314 --> 00:16:22.210 the old days in central office and in the network. 00:16:22.210 --> 00:16:26.498 Today these limits are historical artifact but they are kept because anyway 00:16:26.498 --> 00:16:30.786 voice communications are perfectly intelligible within this band And with 00:16:30.786 --> 00:16:36.684 the reduced band, you can multiplex. Namely, you can put together very many 00:16:36.684 --> 00:16:41.262 communications on a wider channel. The power that you can send on a 00:16:41.262 --> 00:16:47.450 telephone wire is limited from 0.2 to 0.7 volts, or root mean square. 00:16:47.450 --> 00:16:50.736 And this a strictly enforced limit to make sure that you don't send signals 00:16:50.736 --> 00:16:54.155 that can burn the equipment at the central office. 00:16:54.155 --> 00:16:57.510 And the signal to noise ratio is rather good because the analog part of the 00:16:57.510 --> 00:17:00.590 telephone network operates in the bass band and there's not a lot of 00:17:00.590 --> 00:17:05.674 interference in the low frequencies. So let's how we're going to go about 00:17:05.674 --> 00:17:10.246 designing a communications system. Probably the most important concept here, 00:17:10.246 --> 00:17:13.800 is that we're going to adopt the all-digital paradigm. 00:17:13.800 --> 00:17:16.915 What this means is that, we will keep everything in the digital domain until we 00:17:16.915 --> 00:17:20.826 hit the physical channel. And if we were to describe this as a 00:17:20.826 --> 00:17:25.642 block diagram, it would look like this. We have a binary bit stream, can 00:17:25.642 --> 00:17:31.250 represent any sort of views or data. We have a transmitter that operates 00:17:31.250 --> 00:17:36.700 entirely in digital domain that generates a discreet time signal s of n. 00:17:36.700 --> 00:17:39.270 The last element in the transmission chain. 00:17:39.270 --> 00:17:42.930 Is a digital to analog converter operating at a given frequency, or at the 00:17:42.930 --> 00:17:46.470 given period as you prefer, that transforms this signal into an analog 00:17:46.470 --> 00:17:52.790 signal that we can send over the channel. So remember the channel constraints. 00:17:52.790 --> 00:17:55.210 Look a little bit like a filter design problem. 00:17:55.210 --> 00:18:00.400 We have a band width that is specified in terms of a maximum and minimum frequency. 00:18:00.400 --> 00:18:05.303 So we can only operate over this band. And then we have a power constraint that 00:18:05.303 --> 00:18:09.180 restricts the power associated with the signal that we produce. 00:18:09.180 --> 00:18:13.800 So if you want to convert this to our old digital paradigm the first thing to do is 00:18:13.800 --> 00:18:16.770 to convert the specs into discreet time specs. 00:18:16.770 --> 00:18:21.172 So we choose a frequency for the D2A converted, fs, this will be our niquist 00:18:21.172 --> 00:18:26.680 frequency, fs over 2, and with this we can convert the specs. 00:18:26.680 --> 00:18:30.574 Maximum frequency will be pi, and our minimum and maximum frequency bands will 00:18:30.574 --> 00:18:34.390 be omega min and omega max using the relation. 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. 00:18:41.994 --> 00:18:45.585 Now, here are some working hypotheses that are common to most transmission 00:18:45.585 --> 00:18:49.270 systems you will ever see. We start from a bitstream. 00:18:49.270 --> 00:18:52.420 And we will convert this bitstream into a sequence of symbols. 00:18:52.420 --> 00:18:56.100 For samples a of n, via something called a mapper. 00:18:56.100 --> 00:19:02.500 What the mapper does is associate group of bits to a specific symbol. 00:19:02.500 --> 00:19:05.611 Just to give you a concrete example assume we're going to map each group of 00:19:05.611 --> 00:19:10.199 bits to its decimal value. We want to model the sequence of symbols 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 00:19:13.379 --> 00:19:17.280 bitstream is a completely random sequence. 00:19:17.280 --> 00:19:20.430 Now, this is not necessarily the case, for instance, imagine you're digitizing 00:19:20.430 --> 00:19:22.650 audio and you have long stretches of silence. 00:19:22.650 --> 00:19:26.120 This will result into a long sequence of zeros. 00:19:26.120 --> 00:19:29.840 And so, what we do is we put a scrambler in the line. 00:19:29.840 --> 00:19:33.365 What a scrambler does. It transforms a sequence of bits into a 00:19:33.365 --> 00:19:37.135 sequence that looks like a random sequence but this randomization is 00:19:37.135 --> 00:19:42.438 completely invariable at a receiver. So, we start with the sequence of zeroes 00:19:42.438 --> 00:19:45.176 for instance. We put into the scrambler, it's going to 00:19:45.176 --> 00:19:48.760 look like a completely random sequence of zeroes and one but it's done 00:19:48.760 --> 00:19:51.126 algorithmically so we can invert this randomization on the receiver and 00:19:51.126 --> 00:19:56.302 retrieve the original bitstream.. With this we can consider the sequence of 00:19:56.302 --> 00:20:00.450 symbol a of n as a wide sequence. And now we need to convert the sequence 00:20:00.450 --> 00:20:03.840 into a continuous time signal within the constraints. 00:20:03.840 --> 00:20:06.900 So here's the updated transmission scheme. 00:20:06.900 --> 00:20:11.572 User data goes into a scrambler. This is a random binary sequence. 00:20:11.572 --> 00:20:15.230 The mapper converts groups of bits to symbols. 00:20:15.230 --> 00:20:19.730 And then we have to decide what to do in here before converting this into an 00:20:19.730 --> 00:20:23.400 analog signal. The first problem is Fulfilling the 00:20:23.400 --> 00:20:27.106 bandwidth constraint. If we assume that the data is randomized 00:20:27.106 --> 00:20:31.264 and therefore the symbol sequence is a wide sequence, we know that the power 00:20:31.264 --> 00:20:35.554 spectral density is simply equal to the variance and so the power of the signal 00:20:35.554 --> 00:20:39.448 will be constant over the entire frequency band but we actually need to 00:20:39.448 --> 00:20:47.582 fit it into the small band here as specified by the bandwidth constraint. 00:20:47.582 --> 00:20:51.870 So, how do we do this. Well in order to do that we need to 00:20:51.870 --> 00:20:56.910 introduce a new technique called up sampling and we will see this in the next 00:20:56.910 --> 00:20:59.423 module.