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