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Dialogue: 0,0:00:00.01,0:00:03.49,Default,,0000,0000,0000,,>> Welcome to module one of Digital Signal Processing.
Dialogue: 0,0:00:03.49,0:00:06.51,Default,,0000,0000,0000,,In this module we are going to see what signals actually are.
Dialogue: 0,0:00:06.51,0:00:11.27,Default,,0000,0000,0000,,We are going to go through a history, see the earliest example of these discrete-time signals.
Dialogue: 0,0:00:11.27,0:00:13.73,Default,,0000,0000,0000,,Actually it goes back to Egyptian times.
Dialogue: 0,0:00:13.73,0:00:17.28,Default,,0000,0000,0000,,Then through this history see how digital signals,
Dialogue: 0,0:00:17.28,0:00:21.09,Default,,0000,0000,0000,,for example with the telegraph signals, became important in communications.
Dialogue: 0,0:00:21.09,0:00:25.65,Default,,0000,0000,0000,,And today, how signals are pervasive in many applications,
Dialogue: 0,0:00:25.65,0:00:27.60,Default,,0000,0000,0000,,in every day life objects.
Dialogue: 0,0:00:27.62,0:00:29.91,Default,,0000,0000,0000,,For this we're going to see what the signal is,
Dialogue: 0,0:00:29.91,0:00:32.63,Default,,0000,0000,0000,,what a continuous time analog signal is,
Dialogue: 0,0:00:32.65,0:00:36.53,Default,,0000,0000,0000,,what a discrete-time, continuous-amplitude signal is
Dialogue: 0,0:00:36.53,0:00:42.33,Default,,0000,0000,0000,,and how these signals relate to each other and are used in communication devices.
Dialogue: 0,0:00:42.33,0:00:45.03,Default,,0000,0000,0000,,We are not going to have any math in this first module.
Dialogue: 0,0:00:45.03,0:00:49.36,Default,,0000,0000,0000,,It is more illustrative and the mathematics will come later in this class.
Dialogue: 0,0:00:50.68,0:00:54.72,Default,,0000,0000,0000,,This is an introduction to what digital signal processing is all about.
Dialogue: 0,0:00:54.72,0:00:57.95,Default,,0000,0000,0000,,Before getting going, let's give some background material.
Dialogue: 0,0:00:57.95,0:01:03.07,Default,,0000,0000,0000,,There is a textbook called Signal Processing for Communications by Paolo Prandoni and myself.
Dialogue: 0,0:01:03.07,0:01:08.17,Default,,0000,0000,0000,,You can have a paper version or you can get the free PDF or HTML version
Dialogue: 0,0:01:08.17,0:01:11.24,Default,,0000,0000,0000,,on the website here indicated on the slide.
Dialogue: 0,0:01:11.24,0:01:16.84,Default,,0000,0000,0000,,There will be quizzes, there will be homework sets, and there will be occasional complementary lectures.
Dialogue: 0,0:01:16.84,0:01:19.76,Default,,0000,0000,0000,,What is actually a signal?
Dialogue: 0,0:01:19.76,0:01:23.85,Default,,0000,0000,0000,,We talk about digital signal processing, so we need to define what the signal is.
Dialogue: 0,0:01:23.85,0:01:28.12,Default,,0000,0000,0000,,Typically, it's a description of the evolution over physical phenomenon.
Dialogue: 0,0:01:28.12,0:01:32.81,Default,,0000,0000,0000,,Quite simply, if I speak here, there is sound pressure waves going through the air
Dialogue: 0,0:01:32.81,0:01:36.43,Default,,0000,0000,0000,,that's a typical signal.\NWhen you listen to the speech, there is a
Dialogue: 0,0:01:36.43,0:01:40.35,Default,,0000,0000,0000,,loud speaker creating a sound pressure\Nwaves that reaches your ear.
Dialogue: 0,0:01:40.35,0:01:44.12,Default,,0000,0000,0000,,And that's another signal.\NHowever, in between is the world of
Dialogue: 0,0:01:44.12,0:01:48.95,Default,,0000,0000,0000,,digital signal processing because after\Nthe microphone it gets transformed in a
Dialogue: 0,0:01:48.95,0:01:51.88,Default,,0000,0000,0000,,set of members.\NIt is processed in the computer.
Dialogue: 0,0:01:51.88,0:01:54.65,Default,,0000,0000,0000,,It is being transferred through the\Ninternet.
Dialogue: 0,0:01:54.65,0:01:59.24,Default,,0000,0000,0000,,Finally it is decoded to create the sound\Npressure wave to reach your ears.
Dialogue: 0,0:01:59.24,0:02:04.18,Default,,0000,0000,0000,,Other example are the temperature\Nevolution over time, the magnetic
Dialogue: 0,0:02:04.18,0:02:09.40,Default,,0000,0000,0000,,deviation for example, L P recording , is\Na grey level on paper for a black and
Dialogue: 0,0:02:09.40,0:02:13.38,Default,,0000,0000,0000,,white photograph,some flickering colors on\NTV screen.
Dialogue: 0,0:02:13.38,0:02:17.50,Default,,0000,0000,0000,,Here we have a thermometer recording\Ntemperature over time.
Dialogue: 0,0:02:17.50,0:02:22.71,Default,,0000,0000,0000,,So you see the evolution And there are\Ndiscrete ticks and you see how it changes
Dialogue: 0,0:02:22.71,0:02:26.27,Default,,0000,0000,0000,,over time.\NSo what are the characteristics of digital
Dialogue: 0,0:02:26.27,0:02:28.97,Default,,0000,0000,0000,,signals.\NThere are two key ingredients.
Dialogue: 0,0:02:28.97,0:02:33.32,Default,,0000,0000,0000,,First there is discrete time.\NAs we have seen in the previous slide on
Dialogue: 0,0:02:33.32,0:02:38.35,Default,,0000,0000,0000,,the horizontal axis there are discrete\NEvenly spaced ticks and that corresponds
Dialogue: 0,0:02:38.35,0:02:42.58,Default,,0000,0000,0000,,to discretisation in time.\NThere is also discrete amplitude because
Dialogue: 0,0:02:42.58,0:02:47.14,Default,,0000,0000,0000,,the numbers that are measured will be\Nrepresented in a computer and cannot have
Dialogue: 0,0:02:47.14,0:02:51.57,Default,,0000,0000,0000,,some infinite precision.\NSo what amount more sophisticated things,
Dialogue: 0,0:02:51.57,0:02:56.36,Default,,0000,0000,0000,,functions, derivative, and integrals.\NThe question of discreet versus
Dialogue: 0,0:02:56.36,0:03:01.74,Default,,0000,0000,0000,,continuous, or analog versus discreet,\Ngoes probably back to the earliest time of
Dialogue: 0,0:03:01.74,0:03:04.74,Default,,0000,0000,0000,,science, for example, the school of\NAthens.
Dialogue: 0,0:03:04.74,0:03:09.68,Default,,0000,0000,0000,,There was a lot of debate between\Nphilosophers and mathematicians about the
Dialogue: 0,0:03:09.68,0:03:14.84,Default,,0000,0000,0000,,idea of continuum, or the difference\Nbetween countable things and uncountable
Dialogue: 0,0:03:14.84,0:03:17.79,Default,,0000,0000,0000,,things.\NSo in this picture, you see green are
Dialogue: 0,0:03:17.79,0:03:23.17,Default,,0000,0000,0000,,famous philosophers like Plato, in red,\Nfamous mathematicians like Pythagoras,
Dialogue: 0,0:03:23.17,0:03:28.01,Default,,0000,0000,0000,,somebody that we are going to meet again\Nin this class, and there is a famous
Dialogue: 0,0:03:28.01,0:03:33.09,Default,,0000,0000,0000,,paradox which is called Zeno's paradox.\NSo if you should narrow will it ever
Dialogue: 0,0:03:33.09,0:03:37.19,Default,,0000,0000,0000,,arrive in destination?\NWe know that physics Allows us to verify
Dialogue: 0,0:03:37.19,0:03:42.30,Default,,0000,0000,0000,,this but mathematics have the problem with\Nthis and we can see this graphically.
Dialogue: 0,0:03:42.30,0:03:47.00,Default,,0000,0000,0000,,So you want to go from A to B, you cover\Nhalf of the distance that is C, center
Dialogue: 0,0:03:47.00,0:03:52.24,Default,,0000,0000,0000,,quarters that's D and also eighth that's E\Netc will you ever get there and of course
Dialogue: 0,0:03:52.24,0:03:57.01,Default,,0000,0000,0000,,we know you gets there because the sum\Nfrom 1 to infinity of 1 over 2 to the n is
Dialogue: 0,0:03:57.01,0:04:02.33,Default,,0000,0000,0000,,equal to 1, a beautiful formula that we'll\Nsee several times reappearing in this.
Dialogue: 0,0:04:02.33,0:04:07.29,Default,,0000,0000,0000,,Unfortunately during the middle ages in\NEurope, things were a bit lost.
Dialogue: 0,0:04:07.29,0:04:12.56,Default,,0000,0000,0000,,As you can see, people had other worries.\NIn the 17th century things picked up
Dialogue: 0,0:04:12.56,0:04:15.69,Default,,0000,0000,0000,,again.\NHere we have a physicist and astronomer
Dialogue: 0,0:04:15.69,0:04:20.59,Default,,0000,0000,0000,,Galileo, and the philosopher Rene\NDescartes, and both contributed to the
Dialogue: 0,0:04:20.59,0:04:26.14,Default,,0000,0000,0000,,advancement of mathematics at that time.\NDescartes' idea was simple but powerful.
Dialogue: 0,0:04:26.14,0:04:30.47,Default,,0000,0000,0000,,Start with a point, put it into a\Nco-ordinate system Then put more
Dialogue: 0,0:04:30.47,0:04:34.46,Default,,0000,0000,0000,,sophisticated things like lines, and you\Ncan use algebra.
Dialogue: 0,0:04:34.46,0:04:39.31,Default,,0000,0000,0000,,This led to the idea of calculus, which\Nallowed to mathematically describe
Dialogue: 0,0:04:39.31,0:04:43.76,Default,,0000,0000,0000,,physical phenomenon.\NFor example Galileo was able to describe
Dialogue: 0,0:04:43.76,0:04:49.81,Default,,0000,0000,0000,,the trajectory of a bullet, using infinite\Ndecimal variations in both horizontal and
Dialogue: 0,0:04:49.81,0:04:54.36,Default,,0000,0000,0000,,vertical direction.\NCalculus itself was formalized by Newton
Dialogue: 0,0:04:54.36,0:04:59.57,Default,,0000,0000,0000,,and Leibniz, and is one of the great\Nadvances of mathematics in the 17th and
Dialogue: 0,0:04:59.57,0:05:02.77,Default,,0000,0000,0000,,18th century.\NIt is time to do some very simple
Dialogue: 0,0:05:02.77,0:05:07.81,Default,,0000,0000,0000,,continuous time signal processing.\NWe have a function in blue here, between a
Dialogue: 0,0:05:07.81,0:05:10.80,Default,,0000,0000,0000,,and b, and we would like to compute it's\Naverage.
Dialogue: 0,0:05:10.80,0:05:15.46,Default,,0000,0000,0000,,As it is well known, this well be the\Nintegral of the function, divided by the
Dialogue: 0,0:05:15.46,0:05:19.03,Default,,0000,0000,0000,,length's of the interval, and it is shown\Nhere in red dots.
Dialogue: 0,0:05:19.03,0:05:23.49,Default,,0000,0000,0000,,What would be the equivalent in this\Ndiscreet time symbol processing.
Dialogue: 0,0:05:23.49,0:05:27.17,Default,,0000,0000,0000,,We have a set of samples between say, 0\Nand capital N minus 1.
Dialogue: 0,0:05:27.17,0:05:32.44,Default,,0000,0000,0000,,The average is simply 1 over n, the sum\NWas the antidote terms x[n] between 0 and
Dialogue: 0,0:05:32.44,0:05:36.04,Default,,0000,0000,0000,,N minus 1.\NAgain, it is shown in the red dotted line.
Dialogue: 0,0:05:36.04,0:05:41.54,Default,,0000,0000,0000,,In this case, because the signal is very\Nsmooth, the continuous time average and
Dialogue: 0,0:05:41.54,0:05:45.18,Default,,0000,0000,0000,,the discrete time average Are essentially\Nthe same.
Dialogue: 0,0:05:45.18,0:05:49.92,Default,,0000,0000,0000,,This was nice and easy but what if the\Nsignal is too fast, and we don't know
Dialogue: 0,0:05:49.92,0:05:54.35,Default,,0000,0000,0000,,exactly how to compute either the\Ncontinuous time operations or an
Dialogue: 0,0:05:54.35,0:05:59.57,Default,,0000,0000,0000,,equivalent operation on samples.\NEnters Joseph Fourier, one of the greatest
Dialogue: 0,0:05:59.57,0:06:04.77,Default,,0000,0000,0000,,mathematicians of the nineteenth century.\NAnd the inventor of Fourier series,
Dialogue: 0,0:06:04.77,0:06:09.68,Default,,0000,0000,0000,,Fourier analysis which are essentially the\Nground tools of signal processing.
Dialogue: 0,0:06:09.68,0:06:13.47,Default,,0000,0000,0000,,We show simply a picture to give the idea\Nof Fourier analysis.
Dialogue: 0,0:06:13.47,0:06:17.99,Default,,0000,0000,0000,,It is a local Fourier spectrum as you\Nwould see for example on an equalizer
Dialogue: 0,0:06:17.99,0:06:21.54,Default,,0000,0000,0000,,table in a disco.\NAnd it shows the distribution of power
Dialogue: 0,0:06:21.54,0:06:26.91,Default,,0000,0000,0000,,across frequencies, something we are going\Nto understand in detail in this class.
Dialogue: 0,0:06:26.91,0:06:31.77,Default,,0000,0000,0000,,But to do this quick time processing of\Ncontinuous time signals we need some
Dialogue: 0,0:06:31.77,0:06:35.50,Default,,0000,0000,0000,,further results.\NAnd these were derived by Harry Niquist
Dialogue: 0,0:06:35.50,0:06:38.78,Default,,0000,0000,0000,,and Claude Shannon, two researchers at\NBell Labs.
Dialogue: 0,0:06:38.78,0:06:44.61,Default,,0000,0000,0000,,They derived the so-called sampling\Ntheorem, first appearing in 1920's and
Dialogue: 0,0:06:44.61,0:06:48.98,Default,,0000,0000,0000,,formalized in 1948.\NIf the function X of T is sufficiently
Dialogue: 0,0:06:48.98,0:06:54.83,Default,,0000,0000,0000,,slow then there is a simple interpolation\Nformula for X of T, it's the sum of the
Dialogue: 0,0:06:54.83,0:07:00.61,Default,,0000,0000,0000,,samples Xn, Interpolating with the\Nfunction that is called sync function.
Dialogue: 0,0:07:00.61,0:07:05.71,Default,,0000,0000,0000,,It looks a little but complicated now, but\Nit's something we're going to study in
Dialogue: 0,0:07:05.71,0:07:10.72,Default,,0000,0000,0000,,great detail because it's 1 of the\Nfundamental formulas linking this discrete
Dialogue: 0,0:07:10.72,0:07:13.62,Default,,0000,0000,0000,,time and continuous time signal\Nprocessing.
Dialogue: 0,0:07:13.62,0:07:18.72,Default,,0000,0000,0000,,Let us look at this sampling in action.\NSo we have the blue curve, we take
Dialogue: 0,0:07:18.72,0:07:24.27,Default,,0000,0000,0000,,samples, the red dots from the samples.\NWe use the same interpolation.
Dialogue: 0,0:07:24.27,0:07:28.72,Default,,0000,0000,0000,,We put one blue curve, second one, third\None, fourth one, etc.
Dialogue: 0,0:07:28.72,0:07:33.49,Default,,0000,0000,0000,,When we sum them all together, we get back\Nthe original blue curve.
Dialogue: 0,0:07:33.49,0:07:37.00,Default,,0000,0000,0000,,It is magic.\NThis interaction of continuous time and
Dialogue: 0,0:07:37.00,0:07:41.14,Default,,0000,0000,0000,,discrete time processing is summarized in\Nthese two pictures.
Dialogue: 0,0:07:41.14,0:07:44.47,Default,,0000,0000,0000,,On the left you have a picture of the\Nanalog world.
Dialogue: 0,0:07:44.47,0:07:49.39,Default,,0000,0000,0000,,On the right you have a picture of the\Ndiscrete or digital world, as you would
Dialogue: 0,0:07:49.39,0:07:54.42,Default,,0000,0000,0000,,see in a Digital camera for example, and\Nthis is because the world is analog.
Dialogue: 0,0:07:54.42,0:07:58.69,Default,,0000,0000,0000,,It has continuous time continuous space,\Nand the computer is digital.
Dialogue: 0,0:07:58.69,0:08:03.48,Default,,0000,0000,0000,,It is discreet time discreet temperature.\NWhen you look at an image taken with a
Dialogue: 0,0:08:03.48,0:08:06.92,Default,,0000,0000,0000,,digital camera, you may wonder what the\Nresolution is.
Dialogue: 0,0:08:06.92,0:08:11.68,Default,,0000,0000,0000,,And here we have a picture of a bird.\NThis bird happens to have very high visual
Dialogue: 0,0:08:11.68,0:08:16.48,Default,,0000,0000,0000,,acuity, probably much better than mine.\NStill, if you zoom into the digital
Dialogue: 0,0:08:16.48,0:08:21.69,Default,,0000,0000,0000,,picture, after a while, around the eye\Nhere, you see little squares appearing,
Dialogue: 0,0:08:21.69,0:08:27.38,Default,,0000,0000,0000,,showing indeed that the picture is digital\NBecause discrete values over the domain of
Dialogue: 0,0:08:27.38,0:08:32.36,Default,,0000,0000,0000,,the image and it also has actually\Ndiscrete amplitude which we cannot quite
Dialogue: 0,0:08:32.36,0:08:37.23,Default,,0000,0000,0000,,see here at this level of resolution.\NAs we said the key ingredients are
Dialogue: 0,0:08:37.23,0:08:41.52,Default,,0000,0000,0000,,discrete time and discrete amplitude for\Ndigital signals.
Dialogue: 0,0:08:41.52,0:08:46.76,Default,,0000,0000,0000,,So, let us look at x of t here.\NIt's a sinusoid, and investigate discrete
Dialogue: 0,0:08:46.76,0:08:49.93,Default,,0000,0000,0000,,time first.\NWe see this with xn and discrete
Dialogue: 0,0:08:49.93,0:08:53.32,Default,,0000,0000,0000,,amplitude.\NWe see this with these levels of the
Dialogue: 0,0:08:53.32,0:08:59.23,Default,,0000,0000,0000,,amplitudes which are also discrete ties.\NAnd so this signal looks very different
Dialogue: 0,0:08:59.23,0:09:02.72,Default,,0000,0000,0000,,from the original continuous time signal x\Nof t.
Dialogue: 0,0:09:02.72,0:09:08.31,Default,,0000,0000,0000,,It has discrete values on the time axes\Nand discrete values on the vertical
Dialogue: 0,0:09:08.31,0:09:11.90,Default,,0000,0000,0000,,amplitude axis.\NSo why do we need digital amplitude?
Dialogue: 0,0:09:11.90,0:09:16.73,Default,,0000,0000,0000,,Well, because storage is digital, because\Nprocessing is digital, and because
Dialogue: 0,0:09:16.73,0:09:20.79,Default,,0000,0000,0000,,transmission is digital.\NAnd you are going to see all of these in
Dialogue: 0,0:09:20.79,0:09:23.97,Default,,0000,0000,0000,,sequence.\NSo data storage, which is of course very
Dialogue: 0,0:09:23.97,0:09:27.72,Default,,0000,0000,0000,,important, used to be purely analog.\NYou had paper.
Dialogue: 0,0:09:27.72,0:09:30.76,Default,,0000,0000,0000,,You had wax cylinders.\NYou had vinyl.
Dialogue: 0,0:09:30.76,0:09:37.74,Default,,0000,0000,0000,,You had compact cassettes, VHS, etcetera.\NIn imagery you had Kodachrome, slides,
Dialogue: 0,0:09:37.74,0:09:42.75,Default,,0000,0000,0000,,Super 8, film etc.\NVery complicated, a whole biodiversity of
Dialogue: 0,0:09:42.75,0:09:46.20,Default,,0000,0000,0000,,analog storages.\NIn digital, much simpler.
Dialogue: 0,0:09:46.20,0:09:51.27,Default,,0000,0000,0000,,There is only zeros and ones, so all\Ndigital storage, to some extent, looks the
Dialogue: 0,0:09:51.27,0:09:53.97,Default,,0000,0000,0000,,same.\NThe storage medium might look very
Dialogue: 0,0:09:53.97,0:09:58.80,Default,,0000,0000,0000,,different, so here we have a collection of\Nstorage from the last 25 years.
Dialogue: 0,0:09:58.80,0:10:03.76,Default,,0000,0000,0000,,However, fundamentally there are only 0's\Nand 1's on these storage devices.
Dialogue: 0,0:10:03.76,0:10:07.64,Default,,0000,0000,0000,,So in that sense, they are all compatible\Nwith each other.
Dialogue: 0,0:10:07.64,0:10:10.73,Default,,0000,0000,0000,,Processing also moved from analog to\Ndigital.
Dialogue: 0,0:10:10.73,0:10:15.82,Default,,0000,0000,0000,,On the left side, you have a few examples\Nof analog processing devices, an analog
Dialogue: 0,0:10:15.82,0:10:19.98,Default,,0000,0000,0000,,watch, an analog amplifier.\NOn the right side you have a piece of
Dialogue: 0,0:10:19.98,0:10:22.80,Default,,0000,0000,0000,,code.\NNow this piece of code could run on many
Dialogue: 0,0:10:22.80,0:10:27.08,Default,,0000,0000,0000,,different digital computers.\NIt would be compatible with all these
Dialogue: 0,0:10:27.08,0:10:30.42,Default,,0000,0000,0000,,digital platforms.\NThe analog processing devices Are
Dialogue: 0,0:10:30.42,0:10:35.22,Default,,0000,0000,0000,,essentially incompatible with each other.\NData transmission has also gone from
Dialogue: 0,0:10:35.22,0:10:38.44,Default,,0000,0000,0000,,analog to digital.\NSo lets look at the very simple model
Dialogue: 0,0:10:38.44,0:10:42.70,Default,,0000,0000,0000,,here, you've on the left side of the\Ntransmitter, you have a channel on the
Dialogue: 0,0:10:42.70,0:10:46.97,Default,,0000,0000,0000,,right side you have a receiver.\NWhat happens to analog signals when they
Dialogue: 0,0:10:46.97,0:10:50.57,Default,,0000,0000,0000,,are send over a channel.\NSo x of t goes through the channel, its
Dialogue: 0,0:10:50.57,0:10:55.40,Default,,0000,0000,0000,,first multiplied by 1 over G because there\Nis path loss and then there is noise added
Dialogue: 0,0:10:55.40,0:10:59.83,Default,,0000,0000,0000,,indicated here with the sigma of t.\NThe output here is x hat of t.
Dialogue: 0,0:10:59.83,0:11:04.32,Default,,0000,0000,0000,,Let's start with some analog signal x of\Nt.
Dialogue: 0,0:11:04.32,0:11:08.85,Default,,0000,0000,0000,,Multiply it by 1 over g, and add some\Nnoise.
Dialogue: 0,0:11:08.85,0:11:13.75,Default,,0000,0000,0000,,How do we recover a good reproduction of x\Nof t?
Dialogue: 0,0:11:13.75,0:11:22.19,Default,,0000,0000,0000,,Well, we can compensate for the path loss,\Nso we multiply by g, to get xhat 1 of t.
Dialogue: 0,0:11:22.19,0:11:26.83,Default,,0000,0000,0000,,But the problem is that x1 hat of t, is x\Nof t.
Dialogue: 0,0:11:26.83,0:11:34.92,Default,,0000,0000,0000,,That's the good news plus g times sigma of\Nt so the noise has been amplified.
Dialogue: 0,0:11:34.92,0:11:41.46,Default,,0000,0000,0000,,Let's see this in action.\NWe start with x of t, we scale by G, we
Dialogue: 0,0:11:41.46,0:11:47.90,Default,,0000,0000,0000,,add some noise, we multiply by G.\NAnd indeed now, we have a very noisy
Dialogue: 0,0:11:47.90,0:11:51.60,Default,,0000,0000,0000,,signal.\NThis was the idea behind trans-Atlantic
Dialogue: 0,0:11:51.60,0:11:57.92,Default,,0000,0000,0000,,cables which were laid in the 19th century\Nand were essentially analog devices until
Dialogue: 0,0:11:57.92,0:12:02.58,Default,,0000,0000,0000,,telegraph signals were properly encoded as\Ndigital signals.
Dialogue: 0,0:12:02.58,0:12:08.22,Default,,0000,0000,0000,,As can be seen in this picture, this was\Nquite an adventure to lay a cable across
Dialogue: 0,0:12:08.22,0:12:13.62,Default,,0000,0000,0000,,the Atlantic and then to try to transmit\Nanalog signals across these very long
Dialogue: 0,0:12:13.62,0:12:17.55,Default,,0000,0000,0000,,distances.\NFor a long channel because the path loss
Dialogue: 0,0:12:17.55,0:12:23.32,Default,,0000,0000,0000,,is so big, you need to put repeaters.\NSo the process we have just seen, would be
Dialogue: 0,0:12:23.32,0:12:27.72,Default,,0000,0000,0000,,repeated capital N times.\NEach time the paths loss would be
Dialogue: 0,0:12:27.72,0:12:32.31,Default,,0000,0000,0000,,compensated, but the noise will be\Namplified by a factor of n.
Dialogue: 0,0:12:32.31,0:12:38.11,Default,,0000,0000,0000,,Let us see this in action, so start with x\Nof t, paths loss by g, added noise,
Dialogue: 0,0:12:38.11,0:12:44.66,Default,,0000,0000,0000,,amplification by G with the amplification\Nthe amplification of the noise, and the
Dialogue: 0,0:12:44.66,0:12:48.58,Default,,0000,0000,0000,,signal.\NFor the second segment we have the pass
Dialogue: 0,0:12:48.58,0:12:54.63,Default,,0000,0000,0000,,loss again, so X hat 1 is divided by G.\NAnd added noise, then we amplify to get x
Dialogue: 0,0:12:54.63,0:12:59.75,Default,,0000,0000,0000,,hat 2 of t, which now has twice an amount\Nof noise, 2 g times signal of t.
Dialogue: 0,0:12:59.75,0:13:05.15,Default,,0000,0000,0000,,So, if we do this n times, you can see\Nthat the analog signal, after repeated
Dialogue: 0,0:13:05.15,0:13:07.47,Default,,0000,0000,0000,,amplification.\NIs mostly noise.
Dialogue: 0,0:13:07.47,0:13:10.68,Default,,0000,0000,0000,,And that becomes problematic to transmit\Ninformation.
Dialogue: 0,0:13:10.68,0:13:13.81,Default,,0000,0000,0000,,In digital communication, the physics do\Nnot change.
Dialogue: 0,0:13:13.81,0:13:16.61,Default,,0000,0000,0000,,We have the same path loss, we have added\Nnoise.
Dialogue: 0,0:13:16.61,0:13:20.30,Default,,0000,0000,0000,,However, two things change.\NOne is that we don't send arbitrary
Dialogue: 0,0:13:20.30,0:13:23.84,Default,,0000,0000,0000,,signals but, for example, only signals\Nthat[INAUDIBLE].
Dialogue: 0,0:13:23.84,0:13:29.94,Default,,0000,0000,0000,,Take values plus 1 and minus 1, and we do\Nsome specific processing to recover these
Dialogue: 0,0:13:29.94,0:13:33.21,Default,,0000,0000,0000,,signals.\NSpecifically at the outward of the
Dialogue: 0,0:13:33.21,0:13:37.90,Default,,0000,0000,0000,,channel, we multiply by g, and then we\Ntake the signa operation.
Dialogue: 0,0:13:37.90,0:13:41.69,Default,,0000,0000,0000,,So x1hat, is signa of x of t, plug g times\Nsigma of t.
Dialogue: 0,0:13:41.69,0:13:46.100,Default,,0000,0000,0000,,Let us again look at this in action.\NWe start with the signal x of t that is
Dialogue: 0,0:13:46.100,0:13:49.36,Default,,0000,0000,0000,,easier, plus 5 or minus 5.\N5.
Dialogue: 0,0:13:49.36,0:13:55.22,Default,,0000,0000,0000,,It goes through the channel, so it loses\Namplitude by a factor of g, and their is
Dialogue: 0,0:13:55.22,0:13:59.22,Default,,0000,0000,0000,,some noise added.\NWe multiply by g, so we recover x of t
Dialogue: 0,0:13:59.22,0:14:04.64,Default,,0000,0000,0000,,plus g times the noise of sigma t.\NThen we apply the threshold operation.
Dialogue: 0,0:14:04.64,0:14:10.68,Default,,0000,0000,0000,,And true enough, we recover a plus 5 minus\N5 signal, which is identical to the ones
Dialogue: 0,0:14:10.68,0:14:15.08,Default,,0000,0000,0000,,that was sent on the channel.\NThanks to digital processing the
Dialogue: 0,0:14:15.08,0:14:18.88,Default,,0000,0000,0000,,transmission of information has made\Ntremendous progress.
Dialogue: 0,0:14:18.88,0:14:23.55,Default,,0000,0000,0000,,In the mid nineteenth century a\Ntransatlantic cable would transmit 8 words
Dialogue: 0,0:14:23.55,0:14:26.36,Default,,0000,0000,0000,,per minute.\NThat's about 5 bits per second.
Dialogue: 0,0:14:26.36,0:14:30.34,Default,,0000,0000,0000,,A hundred years later a coaxial cable with\N48 voice channels.
Dialogue: 0,0:14:30.34,0:14:36.44,Default,,0000,0000,0000,,At already 3 megabits per second.\NIn 2005, fiber optic technology allowed 10
Dialogue: 0,0:14:36.44,0:14:41.32,Default,,0000,0000,0000,,terabits per second.\NA terabit is 10 to the 12 bits per second.
Dialogue: 0,0:14:41.32,0:14:47.44,Default,,0000,0000,0000,,And today, in 2012, we have fiber cables\Nwith 60 terabits per second.
Dialogue: 0,0:14:47.44,0:14:52.86,Default,,0000,0000,0000,,On the voice channel, the one that is used\Nfor telephony, in 1950s you could send
Dialogue: 0,0:14:52.86,0:14:56.65,Default,,0000,0000,0000,,1200 bits per second.\NIn the 1990's, that was already 56
Dialogue: 0,0:14:56.65,0:15:00.56,Default,,0000,0000,0000,,kilobits per second.\NToday, with ADSL technology, we are
Dialogue: 0,0:15:00.56,0:15:05.94,Default,,0000,0000,0000,,talking about 24 megabits per second.\NPlease note that the last module in the
Dialogue: 0,0:15:05.94,0:15:11.72,Default,,0000,0000,0000,,class will actually explain how ADSL The\Nworks using all the tricks in the box that
Dialogue: 0,0:15:11.72,0:15:16.80,Default,,0000,0000,0000,,we are learning in this class.\NIt is time to conclude this introductory
Dialogue: 0,0:15:16.80,0:15:19.66,Default,,0000,0000,0000,,module.\NAnd we conclude with a picture.
Dialogue: 0,0:15:19.66,0:15:25.04,Default,,0000,0000,0000,,If you zoom into this picture you see it's\Nthe motto of the class, signal is
Dialogue: 0,0:15:25.04,0:15:25.89,Default,,0000,0000,0000,,strength.