1 00:00:00,400 --> 00:00:03,690 >> Welcome to module one of Digital Signal Processing. 2 00:00:03,690 --> 00:00:06,830 In this module we're going to see what signals actually are. 3 00:00:06,830 --> 00:00:09,550 We're going to through history, see the earliest of examples 4 00:00:09,550 --> 00:00:13,970 of discreet time signals, actually it goes back to Egyptian times. 5 00:00:13,970 --> 00:00:16,715 Then through this history see how digital signals for 6 00:00:16,715 --> 00:00:21,520 example, with the telegraph signals, became important in communications. 7 00:00:21,520 --> 00:00:25,840 And today, how signals are pervasive in many applications 8 00:00:25,840 --> 00:00:27,860 in every day life objects. 9 00:00:27,860 --> 00:00:29,900 For this we're going to see what the signal is, 10 00:00:29,900 --> 00:00:33,320 what a continuous time analog signal is, what the discreet 11 00:00:33,320 --> 00:00:38,230 time continuous amplitude signal is and how these signals 12 00:00:38,230 --> 00:00:42,750 relate to each other and are used in communication devices. 13 00:00:42,750 --> 00:00:45,000 We are not going to have any math in this first module, 14 00:00:45,000 --> 00:00:49,330 it's more illustrative and the mathematics will come later in the class. 15 00:00:51,670 --> 00:00:54,480 >> Hi, welcome to our Digital Signal Processing class. 16 00:00:54,480 --> 00:00:56,660 In this introduction we would like to give you 17 00:00:56,660 --> 00:01:00,040 an overview of what digital signal processing is all about. 18 00:01:00,040 --> 00:01:05,210 And perhaps the best way to do that is to consider in turn what we mean 19 00:01:05,210 --> 00:01:09,360 when we use the word signal, when we use the word processing or the word digital. 20 00:01:09,360 --> 00:01:13,090 And you will see that digital signal processing is 21 00:01:13,090 --> 00:01:17,530 really an intermediate point in a reflection about physics, 22 00:01:17,530 --> 00:01:21,580 about math and about the reality around us that started 23 00:01:21,580 --> 00:01:24,140 a very long time ago and continues to this day. 24 00:01:25,720 --> 00:01:29,250 So let's consider the concept of signal to begin with. 25 00:01:29,250 --> 00:01:34,060 In general a signal is a description of the evolution of a physical phenomenon. 26 00:01:34,060 --> 00:01:35,770 This is best understood by example. 27 00:01:35,770 --> 00:01:37,490 Take the weather for instance, the weather is a 28 00:01:37,490 --> 00:01:41,300 physical phenomenon that we usually measure in terms of temperature. 29 00:01:41,300 --> 00:01:42,650 So temperature becomes 30 00:01:42,650 --> 00:01:45,180 a signal that evolves overtime and that 31 00:01:45,180 --> 00:01:48,560 represents a measurement of the underlying physical phenomenon. 32 00:01:48,560 --> 00:01:50,160 We could've chosen another variable. 33 00:01:50,160 --> 00:01:52,702 For instance, we could've chosen rainfall. 34 00:01:52,702 --> 00:01:54,964 that would constitute another signal related 35 00:01:54,964 --> 00:01:57,250 to the same underlying physical phenomenon. 36 00:01:58,380 --> 00:02:01,300 Another example, easy to understand is sound. 37 00:02:01,300 --> 00:02:03,430 Sound can have very many origins take for 38 00:02:03,430 --> 00:02:06,660 instance some musical instrument or a person singing. 39 00:02:06,660 --> 00:02:08,030 Now when you measure sound 40 00:02:08,030 --> 00:02:10,670 with a microphone for instance, what you're measuring is 41 00:02:10,670 --> 00:02:15,190 the pressure, the air pressure at the point of measurement. 42 00:02:15,190 --> 00:02:17,910 The microphone translates the air pressure into 43 00:02:17,910 --> 00:02:21,540 an electrical signal that represents the sound. 44 00:02:21,540 --> 00:02:25,770 Now if you want to record the sound on a magnetic tape for instance, you will have 45 00:02:25,770 --> 00:02:28,150 to convert this electrical signal to a magnetic 46 00:02:28,150 --> 00:02:31,810 deviation that can be impressed over the magnetic tape. 47 00:02:31,810 --> 00:02:33,220 And again, these are different 48 00:02:33,220 --> 00:02:36,910 representations of the same underlying physical phenomenon. 49 00:02:36,910 --> 00:02:39,830 Taking a photograph is a very similar operation. 50 00:02:39,830 --> 00:02:44,290 In this case, we're mapping the light intensity of a scene onto gray levels, in 51 00:02:44,290 --> 00:02:45,300 the case of a black and white 52 00:02:45,300 --> 00:02:48,830 photograph, that can be recorded by photographic paper. 53 00:02:48,830 --> 00:02:51,270 The only difference is that in this case we're 54 00:02:51,270 --> 00:02:54,180 mapping the signal over space rather than over time. 55 00:02:55,260 --> 00:02:58,310 To make things more tangible let's go back to an 56 00:02:58,310 --> 00:03:01,660 experiment that most likely you carried out in elementary school when you 57 00:03:01,660 --> 00:03:05,090 first learned about experimental procedures and 58 00:03:05,090 --> 00:03:07,050 analysis of the world around you. 59 00:03:07,050 --> 00:03:11,070 You were probably asked to map the daily temperature for say, 60 00:03:11,070 --> 00:03:14,730 a period of a month, and to chart it over graph paper. 61 00:03:14,730 --> 00:03:18,530 And so you dutifully looked at the thermometer every morning and then at 62 00:03:18,530 --> 00:03:22,020 the end of the month you probably ended up with a graph like this. 63 00:03:22,020 --> 00:03:24,180 So here we have two concepts 64 00:03:24,180 --> 00:03:27,670 that are fundamental to digital signal processing. 65 00:03:27,670 --> 00:03:31,090 The first concept is that the temperature measurements are taken at 66 00:03:31,090 --> 00:03:35,410 discrete moments in time and they constitute a finite countable set. 67 00:03:36,850 --> 00:03:39,440 And the second similar observation is that 68 00:03:39,440 --> 00:03:42,540 the range of temperature is actually subdivided into 69 00:03:42,540 --> 00:03:46,060 a finite number of possible values which are 70 00:03:46,060 --> 00:03:49,410 determined by the resolution of the graduating scale 71 00:03:49,410 --> 00:03:50,520 on the thermometer. 72 00:03:50,520 --> 00:03:52,910 So, we look at the height of the mercury column 73 00:03:52,910 --> 00:03:55,550 and we chose the tick that is closest to that level. 74 00:03:55,550 --> 00:03:58,599 But, nonetheless, the number of ticks that we can chose from is finite. 75 00:04:00,160 --> 00:04:03,250 So the two fundamental concepts here are the 76 00:04:03,250 --> 00:04:06,470 discretization of time due to the fact that we 77 00:04:06,470 --> 00:04:12,260 take observations regularly but not continuously and the discretization 78 00:04:12,260 --> 00:04:14,460 of amplitudes due to the fact that our measuring 79 00:04:14,460 --> 00:04:16,600 device has a finite resolution. 80 00:04:17,950 --> 00:04:22,490 Now the discretization of amplitude is usually treated as a precision problem. 81 00:04:22,490 --> 00:04:27,640 We can use more sophisticated instruments and achieve a better precision. 82 00:04:27,640 --> 00:04:30,650 But the discretization of time is a veritable 83 00:04:30,650 --> 00:04:34,900 paradigm shift in the way we think about reality. 84 00:04:34,900 --> 00:04:39,540 So much so that the problem appeared for the first time over 2500 85 00:04:39,540 --> 00:04:44,830 years ago when the great Greek philosophers started to think about 86 00:04:44,830 --> 00:04:48,100 why is it that we perceive reality the way we do. 87 00:04:48,100 --> 00:04:51,130 And the first character in the story here 88 00:04:51,130 --> 00:04:56,280 is Pythagoras, who in 500 BC, maintained that 89 00:04:56,280 --> 00:04:58,470 most of reality, if not all of reality, 90 00:04:58,470 --> 00:05:01,640 could be described in terms of numbers and measurements. 91 00:05:01,640 --> 00:05:04,490 Think for instance of the Pythagorean theorem, 92 00:05:04,490 --> 00:05:06,800 if one draws a right triangle, one can 93 00:05:06,800 --> 00:05:10,380 verify experimentally with the ruler that the square 94 00:05:10,380 --> 00:05:14,286 built on the hypotenuse is equal to the sum of the squares built on the sides. 95 00:05:14,286 --> 00:05:18,660 But what Pythagoras said, is that this is a universal 96 00:05:18,660 --> 00:05:23,440 property that applies to the abstract class of all right triangles. 97 00:05:23,440 --> 00:05:26,070 And this was really a major change in 98 00:05:26,070 --> 00:05:30,000 the way people started to think about abstract concepts. 99 00:05:30,000 --> 00:05:32,650 Pretty much the same time Parmenides, 100 00:05:32,650 --> 00:05:35,770 another character in our story, brought this 101 00:05:35,770 --> 00:05:41,040 line of reasoning more into the waters of metaphysics by planting the seed of 102 00:05:41,040 --> 00:05:44,770 a fundamental dichotomy, a fundamental difference between 103 00:05:44,770 --> 00:05:46,870 the reality that we can experience with 104 00:05:46,870 --> 00:05:52,060 our senses and an ideal reality that we will never be able to know. 105 00:05:52,060 --> 00:05:55,040 This of course was later developed by Plato into 106 00:05:55,040 --> 00:05:58,420 a full fledged philosophical theory of the ideals. 107 00:05:58,420 --> 00:06:01,690 So that even today when we talk about the Platonic ideal, we refer 108 00:06:01,690 --> 00:06:08,110 to some form of perfect reality that lies beyond the veil of appearances. 109 00:06:08,110 --> 00:06:11,000 But more interestingly for us is the fact 110 00:06:11,000 --> 00:06:14,850 that right when this idea started to appear in 111 00:06:14,850 --> 00:06:17,890 the global consciousness of the time, there were philosophers 112 00:06:17,890 --> 00:06:21,090 that were ready to point out the potential pitfalls 113 00:06:21,090 --> 00:06:26,180 of this new abstract models of reality. And the leader of the pack, so to speak, 114 00:06:26,180 --> 00:06:31,790 was Zeno of Elea whose paradoxes have survived to this day. 115 00:06:31,790 --> 00:06:37,790 One of Zeno's most famous paradoxes is the paradox of the arrow, which states that if 116 00:06:37,790 --> 00:06:39,760 you shoot an arrow from point A to 117 00:06:39,760 --> 00:06:43,970 point B, the arrow will never reach its destination. 118 00:06:43,970 --> 00:06:46,000 And the reasoning goes like so, 119 00:06:46,000 --> 00:06:50,580 well if we modeled reality with the concepts of geometry then 120 00:06:50,580 --> 00:06:55,258 we know that any segment can be divided into smaller segments. 121 00:06:55,258 --> 00:06:59,970 So what Zeno said was that the arrow, after leaving point a, and before 122 00:06:59,970 --> 00:07:04,640 reaching point b, will have to travel through the mid-point between a and b. 123 00:07:06,650 --> 00:07:08,490 Let's call this point c. 124 00:07:08,490 --> 00:07:11,770 But now after it has reached the midpoint between A and B 125 00:07:11,770 --> 00:07:16,400 it will also have to pass through the midpoint between C and B. 126 00:07:16,400 --> 00:07:18,310 Let's call this point D. 127 00:07:18,310 --> 00:07:24,300 And so on so forth, for every interval you can always find an extra midpoint 128 00:07:24,300 --> 00:07:26,110 and the arrow will have to cross 129 00:07:26,110 --> 00:07:29,550 all of this midpoints before reaching it's destination. 130 00:07:29,550 --> 00:07:31,900 But because of the geometric modelling 131 00:07:31,900 --> 00:07:37,590 of reality there is an infinite number of midpoints and so Zeno said, well in order 132 00:07:37,590 --> 00:07:43,330 to cross an infinite number of points, you will need an infinite amount of time. 133 00:07:43,330 --> 00:07:46,570 Of course today we rebut such an argumentation 134 00:07:46,570 --> 00:07:49,670 by saying simply that we can express the length 135 00:07:49,670 --> 00:07:52,230 of the segment as the sum of all the 136 00:07:52,230 --> 00:07:55,470 sub segments and that this sum converges to one. 137 00:07:55,470 --> 00:07:57,120 But this is a false answer to the problem 138 00:07:57,120 --> 00:07:59,910 because the problem was never with computing the sum. 139 00:07:59,910 --> 00:08:02,580 The problem was with a model of reality in 140 00:08:02,580 --> 00:08:05,900 which the infinite and the finite are at odds. 141 00:08:05,900 --> 00:08:08,900 And it took over 2,000 years of mathematical and philosophical 142 00:08:08,900 --> 00:08:13,070 research to amend that model and come to today's model. 143 00:08:13,070 --> 00:08:15,490 A model in which the sum of an infinite number 144 00:08:15,490 --> 00:08:20,750 of terms can indeed converge to a finite quantity without contradictions. 145 00:08:20,750 --> 00:08:22,630 Now you see the relevance of this problem 146 00:08:22,630 --> 00:08:28,090 to digital signal processing where we are measuring physical quantities at regular 147 00:08:28,090 --> 00:08:31,110 intervals in time while assuming that 148 00:08:31,110 --> 00:08:35,140 the underlying physical quantity is actually continuous. 149 00:08:35,140 --> 00:08:37,380 One of the reasons why arriving at this better 150 00:08:37,380 --> 00:08:40,700 model of reality took over 2,000 years is that in 151 00:08:40,700 --> 00:08:42,845 the Middle Ages, as you can see from this picture, 152 00:08:42,845 --> 00:08:49,460 people were concerned with much more mundane tasks than refining 153 00:08:49,460 --> 00:08:54,600 a mathematical model of reality. But progress did come in the end and the 154 00:08:54,600 --> 00:09:01,000 two towering figures of the 17th century, in this sense, are Galileo and Descartes. 155 00:09:01,000 --> 00:09:05,930 Descartes, the inventor of the Cartesian plane, started by put a name to think. 156 00:09:05,930 --> 00:09:09,120 So if you have a point on the plane like so, Descartes said, well, 157 00:09:09,120 --> 00:09:14,470 if I use a coordinate system around this point I can give a name 158 00:09:14,470 --> 00:09:17,320 to this point and I can use algebraic formulas 159 00:09:17,320 --> 00:09:22,130 to describe geometrical entities and perform operation on them. 160 00:09:22,130 --> 00:09:25,720 So for instance, a line would map to a first degree equation. 161 00:09:25,720 --> 00:09:30,510 This allowed Descartes to solve algebraically, geometric problems that had 162 00:09:30,510 --> 00:09:34,770 baffled Greeks, such as for instance the trisection of the angle. 163 00:09:34,770 --> 00:09:39,610 Much more importantly for us is the fact that the Cartesian Plane is the grand 164 00:09:39,610 --> 00:09:42,820 daddy of all vector spaces and you will see how 165 00:09:42,820 --> 00:09:47,810 useful vector spaces are in the context of digital signal processing. 166 00:09:47,810 --> 00:09:50,550 Well, Europe in the 17th century was flush with 167 00:09:50,550 --> 00:09:53,855 money and Europeans had two things on their minds. 168 00:09:53,855 --> 00:09:56,770 Finding new markets and winning the war of 169 00:09:56,770 --> 00:10:01,230 conquest that came with the appropriation of new markets. 170 00:10:01,230 --> 00:10:04,882 Calculus, that was invented in those years, purported to 171 00:10:04,882 --> 00:10:08,910 provide a new answer to both problems, in the sense that you could use calculus 172 00:10:08,910 --> 00:10:12,000 to find optimal ship routes around the 173 00:10:12,000 --> 00:10:17,110 globe and to find optimal trajectory for cannonballs. 174 00:10:17,110 --> 00:10:20,760 Galileo, in particular, worked on the cannonball problem. 175 00:10:20,760 --> 00:10:25,710 And operated by running a series of experiments in which 176 00:10:25,710 --> 00:10:31,230 the trajectory of balls thrown by a cannon was experimentally determined. 177 00:10:31,230 --> 00:10:33,630 And then working backwards to derive an 178 00:10:33,630 --> 00:10:37,290 ideal Platonic model of the balls trajectory. 179 00:10:37,290 --> 00:10:41,730 That is given by this equation where the initial velocity, expressed as a vector in 180 00:10:41,730 --> 00:10:48,020 the Cartesian plane, is coupled with the pull of gravity to give a parabolic shape. 181 00:10:48,020 --> 00:10:51,890 So the way science proceeded was by starting from set of 182 00:10:51,890 --> 00:10:56,590 experimental data points and then work backwards to find the description 183 00:10:56,590 --> 00:11:01,180 of the underlying phenomenon in the form of a perfect algebraic equation. 184 00:11:02,190 --> 00:11:06,100 This usually worked very well for astronomy, which was a main concern 185 00:11:06,100 --> 00:11:11,430 in those days, because the trajectories of the planets are perfect conic curves. 186 00:11:11,430 --> 00:11:16,670 The invention of calculus and the availability of models for reality 187 00:11:16,670 --> 00:11:21,830 based on functions of real variables, led naturally to what we call 188 00:11:21,830 --> 00:11:23,860 continuous time signal processing. 189 00:11:23,860 --> 00:11:26,490 So if you have a function like this which is 190 00:11:26,490 --> 00:11:29,760 for instance, a temperature function, you can compute the average, 191 00:11:29,760 --> 00:11:33,630 in continuous time, by taking the integral of the function 192 00:11:33,630 --> 00:11:36,660 over its support and dividing by the length of the support. 193 00:11:36,660 --> 00:11:39,170 Without calculus what you would have to do 194 00:11:39,170 --> 00:11:42,470 is take daily measurements say of the temperature. 195 00:11:42,470 --> 00:11:46,930 And to compute the average you would just sum this values together and then divide 196 00:11:46,930 --> 00:11:48,308 by the number of days. 197 00:11:48,308 --> 00:11:53,390 Now the question is what is the relation between these two averages? 198 00:11:53,390 --> 00:11:58,900 What is the error I incur if I use experimental data rather 199 00:11:58,900 --> 00:12:04,090 than finding the ideal function behind the data and then computing the integral. 200 00:12:04,090 --> 00:12:08,740 And even if I can do that for certain signals that appeared to be smooth 201 00:12:08,740 --> 00:12:12,990 and slow like this one, can I do the same if the signal is fast? 202 00:12:12,990 --> 00:12:15,010 In other words, if I have a set 203 00:12:15,010 --> 00:12:18,510 of measurements for something that appears to move quite 204 00:12:18,510 --> 00:12:21,720 quickly, do I have any chance even at 205 00:12:21,720 --> 00:12:25,230 recovering the ideal function that lies underneath the data. 206 00:12:26,380 --> 00:12:28,190 Well it took a long time since the 207 00:12:28,190 --> 00:12:31,190 17th century to answer this question because one 208 00:12:31,190 --> 00:12:36,810 of the missing pieces of the puzzle was how to measure this speed of the signal. 209 00:12:36,810 --> 00:12:39,040 The answer came from Joseph Fourier, 210 00:12:39,040 --> 00:12:40,070 the inventor of Fourier analysis. 211 00:12:40,070 --> 00:12:44,910 That showed us how to decompose any physical phenomenon, any 212 00:12:44,910 --> 00:12:49,480 description of a physical phenomenon, into a series of sinusoidal components. 213 00:12:49,480 --> 00:12:55,140 A sinusoidal component is like a wave, and a wave is parameterized by its 214 00:12:55,140 --> 00:13:01,140 frequency, which is really a way of measuring how fast the wave oscillates. 215 00:13:01,140 --> 00:13:04,060 You can see an example of this sinusoidal decomposition of 216 00:13:04,060 --> 00:13:08,220 the signal in the spectral analyzer of your MP3 player. 217 00:13:08,220 --> 00:13:11,388 By splitting a signal into frequency components you 218 00:13:11,388 --> 00:13:14,230 can see where the energy of the signal is. 219 00:13:14,230 --> 00:13:19,090 And if it is in the high frequencies then the signal will be moving very fast. 220 00:13:19,090 --> 00:13:22,670 Whereas if it is in the low frequencies it will be moving very slow. 221 00:13:24,150 --> 00:13:29,240 The next piece of the puzzle that completes the path from continuous 222 00:13:29,240 --> 00:13:33,780 reality to discreet reality was given by Nyquist and 223 00:13:33,780 --> 00:13:38,315 Shannon, two researchers at Bell Labs in the 50s. 224 00:13:38,315 --> 00:13:41,130 Their sampling theorem is really the bridge that 225 00:13:41,130 --> 00:13:44,790 connects the analog world to the digital world. 226 00:13:44,790 --> 00:13:48,440 The formula is like so, and it looks pretty complicated right now but 227 00:13:48,440 --> 00:13:51,350 you will be very familiar with it by the end of the class. 228 00:13:51,350 --> 00:13:55,010 If you just have a tiny look at it, you will see that 229 00:13:55,010 --> 00:13:59,802 the formula relates a continuous time function, the Platonic ideal 230 00:13:59,802 --> 00:14:05,080 we were talking about, to a set of discrete time measurements. 231 00:14:05,080 --> 00:14:08,020 And this sum really is a weighted sum where 232 00:14:08,020 --> 00:14:12,450 for each discrete sample we associate a special shape. 233 00:14:12,450 --> 00:14:14,460 Graphically it looks like so. 234 00:14:14,460 --> 00:14:20,240 If this is our continuous time function, the ideal function, and we have a set 235 00:14:20,240 --> 00:14:22,520 of measurements that we indicate with these red 236 00:14:22,520 --> 00:14:26,531 dots, we can reconstruct the original functions starting 237 00:14:26,531 --> 00:14:29,920 from the samples just by associating what we 238 00:14:29,920 --> 00:14:33,860 call a sinc function to each of the points. 239 00:14:33,860 --> 00:14:38,980 So you scale copies of the same function at each measured interval and 240 00:14:38,980 --> 00:14:42,880 then when you sum them all together you obtain the original function back. 241 00:14:42,880 --> 00:14:45,560 All you need for this magic trick to happen is 242 00:14:45,560 --> 00:14:48,430 that the original function is not too fast, in 243 00:14:48,430 --> 00:14:52,860 the sense that it doesn't contain too many high frequencies. 244 00:14:52,860 --> 00:14:55,380 We will see this in more detail during the class. 245 00:14:55,380 --> 00:14:58,260 If we now go back to the beginning of our overview, you 246 00:14:58,260 --> 00:15:00,170 remember the second fundamental ingredient in 247 00:15:00,170 --> 00:15:03,340 digital signals is the discrete amplitude. 248 00:15:03,340 --> 00:15:09,340 What that means is that we have two forms of discretization of an ideal function. 249 00:15:09,340 --> 00:15:10,990 Take for example this 250 00:15:10,990 --> 00:15:12,510 sine wave. 251 00:15:12,510 --> 00:15:18,260 The first discretization happens in time and we get a discrete set of samples. 252 00:15:18,260 --> 00:15:22,370 And then the second discretization happens in amplitude, where each sample 253 00:15:22,370 --> 00:15:28,130 can take values only amongst a predetermined set of possible levels. 254 00:15:28,130 --> 00:15:32,210 The very important consequence of the discretization independently of the 255 00:15:32,210 --> 00:15:36,930 number of levels is that the set of levels is countable. 256 00:15:36,930 --> 00:15:41,590 So we can always map the level of a sample to an integer. 257 00:15:41,590 --> 00:15:44,570 If our data is just a set of integers now, it 258 00:15:44,570 --> 00:15:46,690 means that its representation is 259 00:15:46,690 --> 00:15:50,080 completely abstract and completely general purpose. 260 00:15:50,080 --> 00:15:54,070 This has some very important consequences in three domains. 261 00:15:54,070 --> 00:15:56,610 Storage becomes very easy because any memory 262 00:15:56,610 --> 00:15:59,860 support that can store integers can store signals. 263 00:15:59,860 --> 00:16:03,660 And computer memory comes to mind as a first candidate. 264 00:16:03,660 --> 00:16:08,840 Processing becomes completely independent on the nature of the signal because 265 00:16:08,840 --> 00:16:12,120 all we need is a processors that can deal with the integers. 266 00:16:12,120 --> 00:16:15,230 And again, CPUs are general purpose processors 267 00:16:15,230 --> 00:16:17,520 that can deal with integers very, very well. 268 00:16:17,520 --> 00:16:20,960 And finally transmission, with digital signal we 269 00:16:20,960 --> 00:16:23,300 will be able to deploy very effective 270 00:16:23,300 --> 00:16:28,040 ways to combat noise and transmission errors as we will see in a second. 271 00:16:28,040 --> 00:16:28,820 As far as storage 272 00:16:28,820 --> 00:16:32,920 is concerned, just consider the difference between attempting to store 273 00:16:32,920 --> 00:16:37,600 an analog signal, which requires a medium dependent support for 274 00:16:37,600 --> 00:16:40,740 each kind of application, and the task of storing a 275 00:16:40,740 --> 00:16:44,010 digital signal, which requires just a piece of computer memory. 276 00:16:45,040 --> 00:16:51,110 In analog storage the medium evolved as technology evolved. 277 00:16:51,110 --> 00:16:53,980 And for instance when it came to sound we had wax 278 00:16:53,980 --> 00:16:57,465 cylinders in the beginning, and then you had vinyl, and then 279 00:16:57,465 --> 00:17:01,570 reel-to-reel tapes, and then compact cassettes, and so on and so forth. 280 00:17:01,570 --> 00:17:05,330 Each medium was incompatible with its predecessor 281 00:17:05,330 --> 00:17:09,640 and required specialized hardware to be reproduced. 282 00:17:09,640 --> 00:17:14,990 Today everything is stored in general purpose support systems, like memory card 283 00:17:14,990 --> 00:17:19,080 or a hard drive, and is completely independent on the type of 284 00:17:19,080 --> 00:17:21,420 content that is recorded. 285 00:17:21,420 --> 00:17:24,720 If you consider for instance the evolution of memory 286 00:17:24,720 --> 00:17:28,380 supports, this is a famous picture from the internet. 287 00:17:28,380 --> 00:17:33,840 Just one microSD card will contain all the information 288 00:17:33,840 --> 00:17:39,740 that was contained in countless floppy discs and CDs from just a few years back. 289 00:17:39,740 --> 00:17:44,390 But what does not change, although the support changes and the capacity improves 290 00:17:44,390 --> 00:17:47,150 with time, what does not change is the format 291 00:17:47,150 --> 00:17:50,780 of the data which will remain the same across media. 292 00:17:50,780 --> 00:17:52,430 When it comes to processing again, the 293 00:17:52,430 --> 00:17:54,710 fact that the representation of the data is 294 00:17:54,710 --> 00:17:57,870 completely decoupled from the origin of the data 295 00:17:57,870 --> 00:18:00,720 will allow us to use general purpose machines. 296 00:18:00,720 --> 00:18:04,780 Here on the left you have three three analog processing devices. 297 00:18:04,780 --> 00:18:10,290 A thermostat on top with its temperature sensitive coil, 298 00:18:10,290 --> 00:18:12,740 you have a set of gears that for instance can be 299 00:18:12,740 --> 00:18:17,540 used to measure movement or time, and a discrete electronics amplifier. 300 00:18:17,540 --> 00:18:20,580 Each of these devices had to be designed and 301 00:18:20,580 --> 00:18:24,800 built to process just one type of analog signal. 302 00:18:24,800 --> 00:18:27,060 Conversely, on the right, you see just a 303 00:18:27,060 --> 00:18:30,850 piece of C code that implements a digital filter. 304 00:18:30,850 --> 00:18:36,420 Now this filter can be used to process a temperature signal or a sound signal and 305 00:18:36,420 --> 00:18:40,130 its structure or its implementation will not change. 306 00:18:40,130 --> 00:18:41,980 Finally, let's consider the problem of data 307 00:18:41,980 --> 00:18:44,310 transmission which is probably the domain where 308 00:18:44,310 --> 00:18:49,420 digital signal processing has made the most difference in our day to day life. 309 00:18:49,420 --> 00:18:54,280 So if you have a communication channel and you try to send information from 310 00:18:54,280 --> 00:18:58,680 a transmitter to a receiver you are faced with a fundamental problem of noise. 311 00:18:58,680 --> 00:19:00,840 So let's see what happens inside the channel. 312 00:19:00,840 --> 00:19:02,560 You have a signal 313 00:19:02,560 --> 00:19:06,080 that will be put into the channel. The channel will introduce an attenuation. 314 00:19:06,080 --> 00:19:08,610 It will lower the volume of the signal, so to speak. 315 00:19:08,610 --> 00:19:13,360 But it will also introduce some noise, indicated here as sigma of t. 316 00:19:13,360 --> 00:19:15,660 And what you will receive at the end is 317 00:19:15,660 --> 00:19:19,370 an attenuated copy of your original signal, plus noise. 318 00:19:19,370 --> 00:19:22,140 This is just facts of nature that you cannot escape. 319 00:19:22,140 --> 00:19:27,560 So, if this is your original signal what you will get at the end is an attenuated 320 00:19:27,560 --> 00:19:31,170 copy scaled by a factor of G, plus noise. 321 00:19:32,330 --> 00:19:35,610 So how do you recover the original information? 322 00:19:35,610 --> 00:19:39,650 Well, you try to undo the effects introduced by the 323 00:19:39,650 --> 00:19:43,690 channel but the only thing you can undo is the attenuation. 324 00:19:43,690 --> 00:19:47,340 So you can try and multiply the received signal by a gain 325 00:19:47,340 --> 00:19:52,480 factor that is the reciprocal of the attenuation introduced by the channel. 326 00:19:52,480 --> 00:19:59,260 So if you do that you introduce a gain here at the receiver and what you get is, 327 00:19:59,260 --> 00:20:02,800 let's start again with the original signal, attenuated copy, 328 00:20:02,800 --> 00:20:06,910 some noise added and then let's undo the attenuation. 329 00:20:06,910 --> 00:20:11,500 Well what happens unsurprisingly is that the gain factor has 330 00:20:11,500 --> 00:20:15,240 also amplified the noise that was introduced by the channel. 331 00:20:15,240 --> 00:20:18,280 So you get a copy of the signal that is yes, 332 00:20:18,280 --> 00:20:21,710 of a comparable amplitude to the original signal, but 333 00:20:21,710 --> 00:20:24,520 in which the noise is much larger as well. 334 00:20:24,520 --> 00:20:27,920 This is typical situation that you get in second 335 00:20:27,920 --> 00:20:31,490 generation or third generation copies of say a tape. 336 00:20:31,490 --> 00:20:34,371 Or if you try and do a photocopy of a photocopy, just 337 00:20:34,371 --> 00:20:38,700 to give you an idea of what happens with this noise amplification problem. 338 00:20:38,700 --> 00:20:40,910 Now, why is this very important? 339 00:20:40,910 --> 00:20:43,320 This is important because if you have a very long 340 00:20:43,320 --> 00:20:46,610 cable, so for instance, if you have a cable that goes from Europe 341 00:20:46,610 --> 00:20:51,290 to the United States, and you try to send a telephone conversation over there. 342 00:20:51,290 --> 00:20:55,160 What happens is that you have to split the channel into several 343 00:20:55,160 --> 00:21:00,670 chunks and try to undo the attenuation of a chunk in sequence. 344 00:21:00,670 --> 00:21:03,840 So you actually put what are called, repeaters along the 345 00:21:03,840 --> 00:21:08,365 line that regenerated the signal to the original level every say, 346 00:21:08,365 --> 00:21:10,630 ten kilometers of cable or so. 347 00:21:10,630 --> 00:21:14,630 But unfortunately the cumulative effect of this chain of receiver, is that some 348 00:21:14,630 --> 00:21:20,230 noise gets introduced at each stage and gets amplified over and over again. 349 00:21:20,230 --> 00:21:24,646 So for instance, if this is our original signal which again, gets 350 00:21:24,646 --> 00:21:30,800 attenuated and gets corrupted by noise in the first segment of the cable. 351 00:21:30,800 --> 00:21:33,310 After amplication you would get this, 352 00:21:33,310 --> 00:21:34,375 which is in that before. 353 00:21:34,375 --> 00:21:37,610 Then this signal is injected into the second section of the cable. 354 00:21:37,610 --> 00:21:38,820 It gets attenuated. 355 00:21:38,820 --> 00:21:40,990 New noise gets added to it and when 356 00:21:40,990 --> 00:21:44,750 you amplify it you get double the amplified noise. 357 00:21:44,750 --> 00:21:49,910 And after N sections of the cable you have N times the amplified noise. 358 00:21:49,910 --> 00:21:51,160 This can lead very quickly to a 359 00:21:51,160 --> 00:21:55,670 complete loss of intelligibility in a phone conversation. 360 00:21:55,670 --> 00:21:58,330 Let's now consider the problem of transmitting a digital 361 00:21:58,330 --> 00:22:00,850 signal over the same trans oceanic cable. 362 00:22:01,910 --> 00:22:05,025 Now, a digital signal as we said before, 363 00:22:05,025 --> 00:22:08,496 is composed of samples whose values belong to a 364 00:22:08,496 --> 00:22:11,700 countable finite set of levels and so their 365 00:22:11,700 --> 00:22:15,420 values can be mapped to a set of integers. 366 00:22:15,420 --> 00:22:19,193 Now transmitting a set of integers means that we can encode 367 00:22:19,193 --> 00:22:23,659 these integers in binary format and therefore we end up transmitting 368 00:22:23,659 --> 00:22:28,560 basically just a sequence of zeros and ones, binary digits. 369 00:22:28,560 --> 00:22:32,940 We can build an analog signal associating say the level plus five 370 00:22:32,940 --> 00:22:37,250 volt to the digit zero and minus five volt to the digit 371 00:22:37,250 --> 00:22:40,530 one and we will have a signal and we will have a 372 00:22:40,530 --> 00:22:45,710 signal that will oscillate between these two levels as the digits are transmitted. 373 00:22:46,880 --> 00:22:49,220 What happens on the channel is the same 374 00:22:49,220 --> 00:22:54,440 as before, we will have an attenuation, we will have the addition of noise and 375 00:22:54,440 --> 00:22:59,360 we will have an amplifier at each repeater that will try to undo the attenuation. 376 00:22:59,360 --> 00:23:02,570 But on top of it all, we will have what is called a 377 00:23:02,570 --> 00:23:05,460 threshold operator that will try to reconstitute 378 00:23:05,460 --> 00:23:08,380 the original signal as best as possible. 379 00:23:08,380 --> 00:23:10,160 Let's see how that works. 380 00:23:10,160 --> 00:23:11,870 If this is what we transmit, say an 381 00:23:11,870 --> 00:23:14,360 alternation of zero and one mapped to these two 382 00:23:14,360 --> 00:23:17,750 voltage levels, the attenuation and the noise 383 00:23:17,750 --> 00:23:20,740 will reduce the signal to this state. 384 00:23:20,740 --> 00:23:25,940 The amplification will regenerate the levels and will amplify the noise. 385 00:23:25,940 --> 00:23:28,050 So, the noise is much larger than before. 386 00:23:28,050 --> 00:23:30,840 But now we can just threshold and say, if the 387 00:23:30,840 --> 00:23:35,310 signal value is above zero we just output five volts. 388 00:23:35,310 --> 00:23:39,690 And vice versa, if it's below zero we will output minus five volts. 389 00:23:39,690 --> 00:23:43,270 So the thresholding operator will reconstruct a signal like so. 390 00:23:43,270 --> 00:23:46,840 So you can see that at the end of the first repeater we actually 391 00:23:46,840 --> 00:23:51,260 have an exact copy of the transmitted signal and not a noise corrupted copy. 392 00:23:52,376 --> 00:23:55,550 The effectiveness of the digital transmission schemes 393 00:23:55,550 --> 00:23:58,560 can be appreciated by looking at the evolution 394 00:23:58,560 --> 00:24:00,520 of the throughput, the amount of information 395 00:24:00,520 --> 00:24:03,640 that can be put on a transatlantic cable. 396 00:24:03,640 --> 00:24:04,400 In 1866, 397 00:24:04,400 --> 00:24:08,440 the first cable was laid down and it had a capacity of 398 00:24:08,440 --> 00:24:13,812 eight words per minute which corresponded to approximately five bits per second. 399 00:24:13,812 --> 00:24:18,470 In 1956 when the first digital cable was laid down on the ocean floor 400 00:24:18,470 --> 00:24:23,030 the capacity all of the sudden sky rocketed to three mega bits per second. 401 00:24:23,030 --> 00:24:25,530 So, ten to the power of six. 402 00:24:25,530 --> 00:24:29,218 Six order of magnitudes larger than the analog cable. 403 00:24:29,218 --> 00:24:33,432 And in 2005 when a fiber cable was laid down another six 404 00:24:33,432 --> 00:24:39,800 order of magnitude were added for a capacity of 8.4 terabits per second. 405 00:24:39,800 --> 00:24:42,660 Similarly and literally closer to home, we can look 406 00:24:42,660 --> 00:24:47,620 at the evolution of the throughput for in-home data transmission. 407 00:24:47,620 --> 00:24:51,670 In the 50s the first voice-band modems came out of Bell Labs. 408 00:24:51,670 --> 00:24:54,280 Voice-band meaning that they were devices designed 409 00:24:54,280 --> 00:24:57,420 to operate over a standard telephone channel. 410 00:24:57,420 --> 00:25:02,218 Their capacity was very low, 1200 bits per second, and they were analog devices. 411 00:25:02,218 --> 00:25:07,300 With the digital revolution in the 90s digital modems 412 00:25:07,300 --> 00:25:11,770 started to appear and very quickly reached basically the ultimate 413 00:25:11,770 --> 00:25:15,770 limit of data transmission over the voice-band channel which was 414 00:25:15,770 --> 00:25:19,290 56 kilobits per second at the end of the 90s. 415 00:25:19,290 --> 00:25:26,380 The transition to ADSL pushed that limit up to over 24 megabits per second in 2008. 416 00:25:26,380 --> 00:25:28,690 Now, this evolution is of course partly due 417 00:25:28,690 --> 00:25:32,310 to improvements in electronics and to better phone lines. 418 00:25:32,310 --> 00:25:36,150 But fundamentally its success and its affordability are 419 00:25:36,150 --> 00:25:39,530 due to the use of digital signal processing. 420 00:25:39,530 --> 00:25:44,740 We can use small yet very powerful and cheap general purpose processors 421 00:25:44,740 --> 00:25:48,110 to bring the power of error correcting codes 422 00:25:48,110 --> 00:25:53,080 and data recovery even in small home consumer devices. 423 00:25:53,080 --> 00:25:54,928 In the next few weeks we will study 424 00:25:54,928 --> 00:25:58,300 signal processing stun, starting from the ground up. 425 00:25:58,300 --> 00:26:03,350 And by the end of the class will have enough tricks in our bag to fully 426 00:26:03,350 --> 00:26:09,890 understand how an ADSL modem works. And so after this very far reaching and 427 00:26:09,890 --> 00:26:14,130 probably rambling introduction, it's time to go back to basics and we will 428 00:26:14,130 --> 00:26:18,070 see you in module two to discover what discreet time signals are all about.