0:00:00.012,0:00:03.491 >> Welcome to module one of Digital Signal Processing. 0:00:03.491,0:00:06.511 In this module we are going to see what signals actually are. 0:00:06.511,0:00:11.274 We are going to go through a history, see the earliest example of these discrete-time signals. 0:00:11.274,0:00:13.732 Actually it goes back to Egyptian times. 0:00:13.732,0:00:17.276 Then through this history see how digital signals, 0:00:17.276,0:00:21.094 for example with the telegraph signals, became important in communications. 0:00:21.094,0:00:25.651 And today, how signals are pervasive in many applications, 0:00:25.651,0:00:27.603 in every day life objects. 0:00:27.619,0:00:29.914 For this we're going to see what the signal is, 0:00:29.914,0:00:32.630 what a continuous time analog signal is, 0:00:32.646,0:00:36.530 what a discrete-time, continuous-amplitude signal is 0:00:36.531,0:00:42.327 and how these signals relate to each other and are used in communication devices. 0:00:42.327,0:00:45.026 We are not going to have any math in this first module. 0:00:45.026,0:00:49.361 It is more illustrative and the mathematics will come later in this class. 0:00:50.683,0:00:54.722 This is an introduction to what digital signal processing is all about. 0:00:54.722,0:00:57.952 Before getting going, let's give some background material. 0:00:57.952,0:01:03.068 There is a textbook called Signal Processing for Communications by Paolo Prandoni and myself. 0:01:03.068,0:01:08.170 You can have a paper version or you can get the free PDF or HTML version 0:01:08.170,0:01:11.235 on the website here indicated on the slide. 0:01:11.235,0:01:16.835 There will be quizzes, there will be homework sets, and there will be occasional complementary lectures. 0:01:16.835,0:01:19.764 What is actually a signal? 0:01:19.764,0:01:23.851 We talk about digital signal processing, so we need to define what the signal is. 0:01:23.851,0:01:28.117 Typically, it's a description of the evolution over physical phenomenon. 0:01:28.117,0:01:32.807 Quite simply, if I speak here, there is sound pressure waves going through the air 0:01:32.807,0:01:36.429 that's a typical signal.[br]When you listen to the speech, there is a 0:01:36.429,0:01:40.350 loud speaker creating a sound pressure[br]waves that reaches your ear. 0:01:40.351,0:01:44.124 And that's another signal.[br]However, in between is the world of 0:01:44.124,0:01:48.952 digital signal processing because after[br]the microphone it gets transformed in a 0:01:48.952,0:01:51.876 set of members.[br]It is processed in the computer. 0:01:51.876,0:01:54.650 It is being transferred through the[br]internet. 0:01:54.650,0:01:59.236 Finally it is decoded to create the sound[br]pressure wave to reach your ears. 0:01:59.237,0:02:04.176 Other example are the temperature[br]evolution over time, the magnetic 0:02:04.176,0:02:09.405 deviation for example, L P recording , is[br]a grey level on paper for a black and 0:02:09.405,0:02:13.378 white photograph,some flickering colors on[br]TV screen. 0:02:13.378,0:02:17.504 Here we have a thermometer recording[br]temperature over time. 0:02:17.504,0:02:22.710 So you see the evolution And there are[br]discrete ticks and you see how it changes 0:02:22.710,0:02:26.266 over time.[br]So what are the characteristics of digital 0:02:26.266,0:02:28.973 signals.[br]There are two key ingredients. 0:02:28.973,0:02:33.322 First there is discrete time.[br]As we have seen in the previous slide on 0:02:33.322,0:02:38.354 the horizontal axis there are discrete[br]Evenly spaced ticks and that corresponds 0:02:38.354,0:02:42.582 to discretisation in time.[br]There is also discrete amplitude because 0:02:42.582,0:02:47.136 the numbers that are measured will be[br]represented in a computer and cannot have 0:02:47.136,0:02:51.570 some infinite precision.[br]So what amount more sophisticated things, 0:02:51.570,0:02:56.359 functions, derivative, and integrals.[br]The question of discreet versus 0:02:56.359,0:03:01.741 continuous, or analog versus discreet,[br]goes probably back to the earliest time of 0:03:01.741,0:03:04.742 science, for example, the school of[br]Athens. 0:03:04.742,0:03:09.676 There was a lot of debate between[br]philosophers and mathematicians about the 0:03:09.676,0:03:14.835 idea of continuum, or the difference[br]between countable things and uncountable 0:03:14.835,0:03:17.789 things.[br]So in this picture, you see green are 0:03:17.789,0:03:23.171 famous philosophers like Plato, in red,[br]famous mathematicians like Pythagoras, 0:03:23.171,0:03:28.007 somebody that we are going to meet again[br]in this class, and there is a famous 0:03:28.007,0:03:33.086 paradox which is called Zeno's paradox.[br]So if you should narrow will it ever 0:03:33.086,0:03:37.189 arrive in destination?[br]We know that physics Allows us to verify 0:03:37.189,0:03:42.301 this but mathematics have the problem with[br]this and we can see this graphically. 0:03:42.301,0:03:47.004 So you want to go from A to B, you cover[br]half of the distance that is C, center 0:03:47.004,0:03:52.240 quarters that's D and also eighth that's E[br]etc will you ever get there and of course 0:03:52.240,0:03:57.014 we know you gets there because the sum[br]from 1 to infinity of 1 over 2 to the n is 0:03:57.014,0:04:02.328 equal to 1, a beautiful formula that we'll[br]see several times reappearing in this. 0:04:02.329,0:04:07.286 Unfortunately during the middle ages in[br]Europe, things were a bit lost. 0:04:07.286,0:04:12.556 As you can see, people had other worries.[br]In the 17th century things picked up 0:04:12.556,0:04:15.694 again.[br]Here we have a physicist and astronomer 0:04:15.694,0:04:20.592 Galileo, and the philosopher Rene[br]Descartes, and both contributed to the 0:04:20.592,0:04:26.142 advancement of mathematics at that time.[br]Descartes' idea was simple but powerful. 0:04:26.142,0:04:30.468 Start with a point, put it into a[br]co-ordinate system Then put more 0:04:30.468,0:04:34.456 sophisticated things like lines, and you[br]can use algebra. 0:04:34.456,0:04:39.313 This led to the idea of calculus, which[br]allowed to mathematically describe 0:04:39.313,0:04:43.758 physical phenomenon.[br]For example Galileo was able to describe 0:04:43.758,0:04:49.806 the trajectory of a bullet, using infinite[br]decimal variations in both horizontal and 0:04:49.806,0:04:54.362 vertical direction.[br]Calculus itself was formalized by Newton 0:04:54.362,0:04:59.570 and Leibniz, and is one of the great[br]advances of mathematics in the 17th and 0:04:59.570,0:05:02.768 18th century.[br]It is time to do some very simple 0:05:02.768,0:05:07.810 continuous time signal processing.[br]We have a function in blue here, between a 0:05:07.810,0:05:10.796 and b, and we would like to compute it's[br]average. 0:05:10.796,0:05:15.464 As it is well known, this well be the[br]integral of the function, divided by the 0:05:15.464,0:05:19.033 length's of the interval, and it is shown[br]here in red dots. 0:05:19.033,0:05:23.487 What would be the equivalent in this[br]discreet time symbol processing. 0:05:23.487,0:05:27.170 We have a set of samples between say, 0[br]and capital N minus 1. 0:05:27.170,0:05:32.441 The average is simply 1 over n, the sum[br]Was the antidote terms x[n] between 0 and 0:05:32.441,0:05:36.043 N minus 1.[br]Again, it is shown in the red dotted line. 0:05:36.043,0:05:41.543 In this case, because the signal is very[br]smooth, the continuous time average and 0:05:41.543,0:05:45.179 the discrete time average Are essentially[br]the same. 0:05:45.179,0:05:49.925 This was nice and easy but what if the[br]signal is too fast, and we don't know 0:05:49.925,0:05:54.349 exactly how to compute either the[br]continuous time operations or an 0:05:54.349,0:05:59.573 equivalent operation on samples.[br]Enters Joseph Fourier, one of the greatest 0:05:59.573,0:06:04.772 mathematicians of the nineteenth century.[br]And the inventor of Fourier series, 0:06:04.772,0:06:09.677 Fourier analysis which are essentially the[br]ground tools of signal processing. 0:06:09.677,0:06:13.470 We show simply a picture to give the idea[br]of Fourier analysis. 0:06:13.470,0:06:17.990 It is a local Fourier spectrum as you[br]would see for example on an equalizer 0:06:17.990,0:06:21.540 table in a disco.[br]And it shows the distribution of power 0:06:21.540,0:06:26.910 across frequencies, something we are going[br]to understand in detail in this class. 0:06:26.910,0:06:31.767 But to do this quick time processing of[br]continuous time signals we need some 0:06:31.767,0:06:35.498 further results.[br]And these were derived by Harry Niquist 0:06:35.498,0:06:38.776 and Claude Shannon, two researchers at[br]Bell Labs. 0:06:38.776,0:06:44.606 They derived the so-called sampling[br]theorem, first appearing in 1920's and 0:06:44.606,0:06:48.977 formalized in 1948.[br]If the function X of T is sufficiently 0:06:48.977,0:06:54.827 slow then there is a simple interpolation[br]formula for X of T, it's the sum of the 0:06:54.827,0:07:00.609 samples Xn, Interpolating with the[br]function that is called sync function. 0:07:00.609,0:07:05.707 It looks a little but complicated now, but[br]it's something we're going to study in 0:07:05.707,0:07:10.723 great detail because it's 1 of the[br]fundamental formulas linking this discrete 0:07:10.723,0:07:13.625 time and continuous time signal[br]processing. 0:07:13.625,0:07:18.720 Let us look at this sampling in action.[br]So we have the blue curve, we take 0:07:18.720,0:07:24.266 samples, the red dots from the samples.[br]We use the same interpolation. 0:07:24.266,0:07:28.720 We put one blue curve, second one, third[br]one, fourth one, etc. 0:07:28.720,0:07:33.488 When we sum them all together, we get back[br]the original blue curve. 0:07:33.488,0:07:37.000 It is magic.[br]This interaction of continuous time and 0:07:37.000,0:07:41.140 discrete time processing is summarized in[br]these two pictures. 0:07:41.140,0:07:44.474 On the left you have a picture of the[br]analog world. 0:07:44.474,0:07:49.394 On the right you have a picture of the[br]discrete or digital world, as you would 0:07:49.394,0:07:54.415 see in a Digital camera for example, and[br]this is because the world is analog. 0:07:54.415,0:07:58.693 It has continuous time continuous space,[br]and the computer is digital. 0:07:58.693,0:08:03.476 It is discreet time discreet temperature.[br]When you look at an image taken with a 0:08:03.476,0:08:06.919 digital camera, you may wonder what the[br]resolution is. 0:08:06.919,0:08:11.685 And here we have a picture of a bird.[br]This bird happens to have very high visual 0:08:11.685,0:08:16.479 acuity, probably much better than mine.[br]Still, if you zoom into the digital 0:08:16.479,0:08:21.693 picture, after a while, around the eye[br]here, you see little squares appearing, 0:08:21.693,0:08:27.381 showing indeed that the picture is digital[br]Because discrete values over the domain of 0:08:27.381,0:08:32.358 the image and it also has actually[br]discrete amplitude which we cannot quite 0:08:32.358,0:08:37.234 see here at this level of resolution.[br]As we said the key ingredients are 0:08:37.234,0:08:41.516 discrete time and discrete amplitude for[br]digital signals. 0:08:41.516,0:08:46.755 So, let us look at x of t here.[br]It's a sinusoid, and investigate discrete 0:08:46.755,0:08:49.926 time first.[br]We see this with xn and discrete 0:08:49.926,0:08:53.325 amplitude.[br]We see this with these levels of the 0:08:53.325,0:08:59.228 amplitudes which are also discrete ties.[br]And so this signal looks very different 0:08:59.228,0:09:02.722 from the original continuous time signal x[br]of t. 0:09:02.722,0:09:08.308 It has discrete values on the time axes[br]and discrete values on the vertical 0:09:08.308,0:09:11.903 amplitude axis.[br]So why do we need digital amplitude? 0:09:11.903,0:09:16.727 Well, because storage is digital, because[br]processing is digital, and because 0:09:16.727,0:09:20.787 transmission is digital.[br]And you are going to see all of these in 0:09:20.787,0:09:23.966 sequence.[br]So data storage, which is of course very 0:09:23.966,0:09:27.718 important, used to be purely analog.[br]You had paper. 0:09:27.718,0:09:30.764 You had wax cylinders.[br]You had vinyl. 0:09:30.764,0:09:37.736 You had compact cassettes, VHS, etcetera.[br]In imagery you had Kodachrome, slides, 0:09:37.736,0:09:42.748 Super 8, film etc.[br]Very complicated, a whole biodiversity of 0:09:42.748,0:09:46.195 analog storages.[br]In digital, much simpler. 0:09:46.195,0:09:51.271 There is only zeros and ones, so all[br]digital storage, to some extent, looks the 0:09:51.271,0:09:53.974 same.[br]The storage medium might look very 0:09:53.974,0:09:58.795 different, so here we have a collection of[br]storage from the last 25 years. 0:09:58.796,0:10:03.764 However, fundamentally there are only 0's[br]and 1's on these storage devices. 0:10:03.764,0:10:07.641 So in that sense, they are all compatible[br]with each other. 0:10:07.641,0:10:10.727 Processing also moved from analog to[br]digital. 0:10:10.727,0:10:15.825 On the left side, you have a few examples[br]of analog processing devices, an analog 0:10:15.825,0:10:19.975 watch, an analog amplifier.[br]On the right side you have a piece of 0:10:19.975,0:10:22.797 code.[br]Now this piece of code could run on many 0:10:22.797,0:10:27.076 different digital computers.[br]It would be compatible with all these 0:10:27.076,0:10:30.422 digital platforms.[br]The analog processing devices Are 0:10:30.422,0:10:35.216 essentially incompatible with each other.[br]Data transmission has also gone from 0:10:35.216,0:10:38.435 analog to digital.[br]So lets look at the very simple model 0:10:38.435,0:10:42.705 here, you've on the left side of the[br]transmitter, you have a channel on the 0:10:42.705,0:10:46.969 right side you have a receiver.[br]What happens to analog signals when they 0:10:46.969,0:10:50.574 are send over a channel.[br]So x of t goes through the channel, its 0:10:50.574,0:10:55.404 first multiplied by 1 over G because there[br]is path loss and then there is noise added 0:10:55.404,0:10:59.833 indicated here with the sigma of t.[br]The output here is x hat of t. 0:10:59.833,0:11:04.318 Let's start with some analog signal x of[br]t. 0:11:04.318,0:11:08.853 Multiply it by 1 over g, and add some[br]noise. 0:11:08.853,0:11:13.751 How do we recover a good reproduction of x[br]of t? 0:11:13.751,0:11:22.191 Well, we can compensate for the path loss,[br]so we multiply by g, to get xhat 1 of t. 0:11:22.191,0:11:26.833 But the problem is that x1 hat of t, is x[br]of t. 0:11:26.834,0:11:34.917 That's the good news plus g times sigma of[br]t so the noise has been amplified. 0:11:34.917,0:11:41.455 Let's see this in action.[br]We start with x of t, we scale by G, we 0:11:41.455,0:11:47.898 add some noise, we multiply by G.[br]And indeed now, we have a very noisy 0:11:47.898,0:11:51.605 signal.[br]This was the idea behind trans-Atlantic 0:11:51.605,0:11:57.924 cables which were laid in the 19th century[br]and were essentially analog devices until 0:11:57.924,0:12:02.575 telegraph signals were properly encoded as[br]digital signals. 0:12:02.575,0:12:08.225 As can be seen in this picture, this was[br]quite an adventure to lay a cable across 0:12:08.225,0:12:13.620 the Atlantic and then to try to transmit[br]analog signals across these very long 0:12:13.620,0:12:17.549 distances.[br]For a long channel because the path loss 0:12:17.549,0:12:23.317 is so big, you need to put repeaters.[br]So the process we have just seen, would be 0:12:23.317,0:12:27.717 repeated capital N times.[br]Each time the paths loss would be 0:12:27.717,0:12:32.307 compensated, but the noise will be[br]amplified by a factor of n. 0:12:32.307,0:12:38.108 Let us see this in action, so start with x[br]of t, paths loss by g, added noise, 0:12:38.108,0:12:44.663 amplification by G with the amplification[br]the amplification of the noise, and the 0:12:44.663,0:12:48.580 signal.[br]For the second segment we have the pass 0:12:48.580,0:12:54.629 loss again, so X hat 1 is divided by G.[br]And added noise, then we amplify to get x 0:12:54.629,0:12:59.751 hat 2 of t, which now has twice an amount[br]of noise, 2 g times signal of t. 0:12:59.751,0:13:05.151 So, if we do this n times, you can see[br]that the analog signal, after repeated 0:13:05.151,0:13:07.472 amplification.[br]Is mostly noise. 0:13:07.472,0:13:10.680 And that becomes problematic to transmit[br]information. 0:13:10.680,0:13:13.810 In digital communication, the physics do[br]not change. 0:13:13.810,0:13:16.606 We have the same path loss, we have added[br]noise. 0:13:16.606,0:13:20.299 However, two things change.[br]One is that we don't send arbitrary 0:13:20.299,0:13:23.840 signals but, for example, only signals[br]that[INAUDIBLE]. 0:13:23.840,0:13:29.936 Take values plus 1 and minus 1, and we do[br]some specific processing to recover these 0:13:29.936,0:13:33.211 signals.[br]Specifically at the outward of the 0:13:33.211,0:13:37.902 channel, we multiply by g, and then we[br]take the signa operation. 0:13:37.902,0:13:41.689 So x1hat, is signa of x of t, plug g times[br]sigma of t. 0:13:41.689,0:13:46.998 Let us again look at this in action.[br]We start with the signal x of t that is 0:13:46.998,0:13:49.362 easier, plus 5 or minus 5.[br]5. 0:13:49.362,0:13:55.218 It goes through the channel, so it loses[br]amplitude by a factor of g, and their is 0:13:55.218,0:13:59.222 some noise added.[br]We multiply by g, so we recover x of t 0:13:59.222,0:14:04.639 plus g times the noise of sigma t.[br]Then we apply the threshold operation. 0:14:04.639,0:14:10.675 And true enough, we recover a plus 5 minus[br]5 signal, which is identical to the ones 0:14:10.675,0:14:15.076 that was sent on the channel.[br]Thanks to digital processing the 0:14:15.076,0:14:18.881 transmission of information has made[br]tremendous progress. 0:14:18.881,0:14:23.549 In the mid nineteenth century a[br]transatlantic cable would transmit 8 words 0:14:23.549,0:14:26.361 per minute.[br]That's about 5 bits per second. 0:14:26.361,0:14:30.338 A hundred years later a coaxial cable with[br]48 voice channels. 0:14:30.339,0:14:36.439 At already 3 megabits per second.[br]In 2005, fiber optic technology allowed 10 0:14:36.439,0:14:41.316 terabits per second.[br]A terabit is 10 to the 12 bits per second. 0:14:41.316,0:14:47.444 And today, in 2012, we have fiber cables[br]with 60 terabits per second. 0:14:47.444,0:14:52.862 On the voice channel, the one that is used[br]for telephony, in 1950s you could send 0:14:52.862,0:14:56.649 1200 bits per second.[br]In the 1990's, that was already 56 0:14:56.649,0:15:00.559 kilobits per second.[br]Today, with ADSL technology, we are 0:15:00.559,0:15:05.936 talking about 24 megabits per second.[br]Please note that the last module in the 0:15:05.936,0:15:11.716 class will actually explain how ADSL The[br]works using all the tricks in the box that 0:15:11.716,0:15:16.797 we are learning in this class.[br]It is time to conclude this introductory 0:15:16.797,0:15:19.661 module.[br]And we conclude with a picture. 0:15:19.661,0:15:25.035 If you zoom into this picture you see it's[br]the motto of the class, signal is 0:15:25.035,0:15:25.890 strength.