WEBVTT 00:00:00.000 --> 00:00:03.680 Welcome back to CSE 316 — Data Communication and Networking. 00:00:03.730 --> 00:00:12.140 This is the detailed video version of Session 5, and it is a session with a warning attached: you will need a calculator. 00:00:12.190 --> 00:00:21.140 Here is the shape of it, in five sections, each ending with a checkpoint so you can tell whether to go on or go back. 00:00:21.530 --> 00:00:26.240 First, what the wire does to your signal — attenuation 00:00:26.290 --> 00:00:29.460 Second, distortion and noise, and SNR. 00:00:29.510 --> 00:00:33.640 Third, Nyquist's limit: for noiseless channel. 00:00:33.690 --> 00:00:37.960 Fourth, Shannon's limit: for noisy channel 00:00:38.010 --> 00:00:46.960 And fifth, how an engineer uses the two limits together, plus the answer to the question I am about to ask you. 00:00:47.095 --> 00:00:54.575 Before any formula, a question from ordinary life — or at least, from your parents' ordinary life. 00:00:54.625 --> 00:01:03.575 Dial-up modems — the boxes that carried the internet over ordinary telephone lines, screeching and whistling — stopped improving at about thirty-four kilobits per second. 00:01:06.295 --> 00:01:11.285 The best engineers at the richest companies on Earth tried to build a faster one. 00:01:11.335 --> 00:01:13.275 Nobody ever shipped one. 00:01:13.325 --> 00:01:17.855 Three suspects: the wire — maybe the copper was simply too old and too thin. 00:01:17.905 --> 00:01:23.235 The chip — maybe the electronics of the day were not clever enough. 00:01:23.285 --> 00:01:26.285 Or physics — maybe something deeper said no. 00:01:26.335 --> 00:01:28.395 Who killed the fast modem? 00:01:28.445 --> 00:01:32.665 Pause the video and commit to one of the three. 00:01:32.715 --> 00:01:37.615 Better, write it down somewhere. 00:01:37.665 --> 00:01:46.615 Section five settles it with a formula rather than an opinion: Shannon's capacity for a telephone line, computed to three significant figures. 00:01:47.325 --> 00:01:50.095 And that is what this session is. 00:01:50.145 --> 00:01:59.095 First we study what the wire does to your signal — it shrinks it, it bends it out of shape, and it buries it in noise — and we learn the decibel, the unit that keeps the books. 00:02:01.055 --> 00:02:10.005 Then we meet the two most important formulas in this half of the course: Nyquist's limit and Shannon's limit, the two speed limits of data communication. 00:02:11.235 --> 00:02:19.427 By the end, the modem question will have a one-word answer and a number attached. 00:02:19.477 --> 00:02:20.837 Section one. 00:02:20.887 --> 00:02:22.707 The imperfect wire. 00:02:22.757 --> 00:02:29.527 Up to now we have drawn signals as if the medium were a polite courier: what goes in comes out. 00:02:29.577 --> 00:02:30.807 It does not. 00:02:30.857 --> 00:02:39.807 Every medium takes its cut — and in this section we name the three ways it does, and then spend our time on the first one, because the first one is where the decibel lives. 00:02:45.244 --> 00:02:47.984 Transmission media are not perfect. 00:02:48.034 --> 00:02:53.554 What is sent is not what is received — and the imperfection is not random chaos. 00:02:53.604 --> 00:02:59.574 It goes wrong in exactly three ways, and the exam will ask you to tell them apart. 00:02:59.624 --> 00:03:02.884 One: the copy arrives smaller. 00:03:02.934 --> 00:03:04.524 That is attenuation. 00:03:04.574 --> 00:03:10.654 The signal spent energy fighting the medium, and that energy is gone — turned into heat. 00:03:10.704 --> 00:03:13.194 The wire literally warms up. 00:03:13.244 --> 00:03:16.684 Two: the copy arrives misshapen. 00:03:16.734 --> 00:03:18.144 That is distortion. 00:03:18.194 --> 00:03:27.144 The energy arrived, but the components of a composite signal were delayed by different amounts, and the shape that comes out is not the shape that went in. 00:03:27.504 --> 00:03:30.994 Three: the copy arrives with company. 00:03:31.044 --> 00:03:35.784 That is noise — signals you never sent, added to yours along the way. 00:03:35.834 --> 00:03:38.994 Energy on the line that carries no information. 00:03:39.044 --> 00:03:44.964 These are three different physical effects with three different fixes, and blurring them costs marks. 00:03:45.014 --> 00:03:50.064 Smaller, misshapen, with company — attenuation, distortion, noise. 00:03:50.114 --> 00:03:59.064 Today we measure the first one, understand the second, and put a number on the third. 00:03:59.558 --> 00:04:04.818 Impairment number one, attenuation — the medium takes its cut. 00:04:04.868 --> 00:04:13.218 A signal, simple or composite, pushes through copper by spending energy overcoming the resistance of the medium. 00:04:13.268 --> 00:04:14.888 Where does that energy go? 00:04:14.938 --> 00:04:15.708 Heat. 00:04:15.758 --> 00:04:24.368 A wire carrying signals gets warm, even hot — feel a charger cable that has been working hard and you are feeling attenuation. 00:04:24.418 --> 00:04:29.118 That lost energy is why long-haul cable runs need repeaters. 00:04:29.168 --> 00:04:32.978 Second, the loss is quoted per distance. 00:04:33.028 --> 00:04:39.008 Open any cable datasheet and it says something like minus nought point three dB per kilometer. 00:04:39.058 --> 00:04:41.668 Twice the cable, twice the loss. 00:04:41.718 --> 00:04:44.258 And look at the unit the datasheet chose. 00:04:44.308 --> 00:04:48.038 Not 'percent per kilometer'. dB per kilometer. 00:04:48.088 --> 00:04:54.168 That choice is not a fashion: slide 11 shows that a per-distance spec forces it. 00:04:54.218 --> 00:04:59.368 Third, we can fight back: amplifiers buy the loss back. 00:04:59.418 --> 00:05:04.978 Put an amplifier in the middle of a long run and the signal comes out stronger than it went in. 00:05:05.028 --> 00:05:13.978 The clip on the next slide shows what an amplifier can buy back — amplitude — and what it cannot: the signal-to-noise ratio. 00:05:16.121 --> 00:05:18.741 Here is the demonstration. 00:05:18.791 --> 00:05:27.741 A signal enters a cable that loses one dB every kilometer, with a single plus-seven-dB amplifier waiting at kilometer seven. 00:05:28.171 --> 00:05:34.611 Two counters run as it travels: the signal level, and the noise underneath it. 00:05:34.661 --> 00:05:43.611 Watch the green wave shrink, kilometer by kilometer — that is attenuation doing exactly what the datasheet promised. 00:05:46.161 --> 00:05:49.031 Now, kilometer seven. 00:05:49.081 --> 00:05:57.141 The amplifier fires, and the green wave jumps up — plus seven dB, bought and paid for. 00:05:57.191 --> 00:05:59.471 But look at the red band. 00:05:59.521 --> 00:06:01.081 It jumped too. 00:06:01.131 --> 00:06:10.081 The noise that had crept onto the line during those seven kilometers reached the amplifier along with the signal — and the amplifier, which cannot tell the difference, amplified both. 00:06:13.241 --> 00:06:15.971 The signal counter recovered. 00:06:16.021 --> 00:06:20.291 The signal-to-noise counter never did. 00:06:20.341 --> 00:06:25.071 Read the freeze frame out loud: you can buy back amplitude. 00:06:25.121 --> 00:06:26.921 You cannot buy back clean. 00:06:26.971 --> 00:06:34.100 Keep that sentence — it is the bridge to everything in section two. 00:06:34.150 --> 00:06:43.100 So attenuation needs a unit, and the unit is the decibel — dB, one tenth of a bel, named for Alexander Graham Bell. 00:06:43.240 --> 00:06:48.200 Here is the definition, and it stays on screen for the rest of the hour. 00:06:48.250 --> 00:06:54.340 dB equals ten times the base-ten logarithm of P two over P one. 00:06:54.390 --> 00:06:59.530 P one is the power before; P two is the power after. 00:06:59.580 --> 00:07:03.070 Divide, take the log, multiply by ten. 00:07:03.120 --> 00:07:06.410 Notice it is a pure ratio — the watts cancel. 00:07:06.460 --> 00:07:12.210 And the logarithm is what turns an awkward ratio into a small, human number. 00:07:12.260 --> 00:07:14.930 The sign carries meaning. 00:07:14.980 --> 00:07:20.700 If P two is smaller than P one, the ratio is below one and the log is negative: a loss. 00:07:20.750 --> 00:07:23.530 If P two is bigger, positive: a gain. 00:07:23.580 --> 00:07:25.270 Keep the sign. 00:07:25.320 --> 00:07:32.670 Everything we do with decibels later depends on the minus signs surviving your arithmetic. 00:07:32.720 --> 00:07:37.730 And one anchor to memorize — the way you know your own phone number. 00:07:37.780 --> 00:07:41.160 Minus three dB means half the power is gone. 00:07:41.210 --> 00:07:43.640 Plus three dB means it doubled. 00:07:43.690 --> 00:07:51.090 And this is not a rough figure: ten times the log of nought point five is minus three point nought one nought three. 00:07:51.140 --> 00:07:56.550 We round it to minus three, and the rounding error is one third of one percent. 00:07:56.600 --> 00:07:58.850 One caution before we practice. 00:07:58.900 --> 00:08:05.860 Some references use twenty times the log, because they work with voltage ratios instead of power ratios. 00:08:05.910 --> 00:08:09.220 This course, and this exam, work in power. 00:08:09.270 --> 00:08:10.810 The factor is ten. 00:08:10.860 --> 00:08:17.123 If you googled ahead and got confused — that is why. 00:08:17.173 --> 00:08:26.123 Let us calibrate the unit with Forouzan's examples three point two six and three point two seven — two tiny problems that give you your mental furniture. 00:08:27.193 --> 00:08:33.783 Case A: a signal travels through a medium and its power halves — P two is half of P one. 00:08:33.833 --> 00:08:36.143 What is the attenuation in dB? 00:08:36.193 --> 00:08:39.723 Case B: an amplifier multiplies the power by ten. 00:08:39.773 --> 00:08:41.663 What is the gain? 00:08:41.713 --> 00:08:50.003 Pause and do both — they take thirty seconds. 00:08:50.053 --> 00:08:59.003 Case A. dB equals ten log of nought point five, which is ten times minus nought point three nought one — minus three point nought one, and we say minus three dB. 00:09:00.983 --> 00:09:04.153 Losing half the power is minus three dB. 00:09:04.203 --> 00:09:11.343 That is the anchor fact from the last slide, now earned rather than asserted. 00:09:11.393 --> 00:09:13.883 Case B. dB equals ten log of ten. 00:09:13.933 --> 00:09:19.473 The log of ten is exactly one, so the gain is exactly plus ten dB. 00:09:19.523 --> 00:09:22.933 Multiplying power by ten is plus ten dB — the second anchor. 00:09:22.983 --> 00:09:28.853 And now watch what the two anchors do when you combine them. 00:09:28.903 --> 00:09:30.653 A hundred times the power? 00:09:30.703 --> 00:09:37.563 A hundred is ten times ten — and the decibels just add: ten plus ten, twenty dB. 00:09:37.613 --> 00:09:39.153 A thousand times? 00:09:39.203 --> 00:09:42.543 Ten times ten times ten: thirty dB. 00:09:42.593 --> 00:09:50.103 So plus thirty dB means a factor of one thousand, and you computed it without a calculator, by adding tens. 00:09:50.153 --> 00:09:59.103 You are already using the deep property of this unit: the logarithm turns multiplication into addition. Slide 11 makes it explicit. 00:10:02.714 --> 00:10:08.014 Now a real datasheet problem — Forouzan example three point three nought. 00:10:08.064 --> 00:10:12.314 A cable attenuates minus nought point three dB per kilometer. 00:10:12.364 --> 00:10:16.094 A two-milliwatt signal enters a five-kilometer run. 00:10:16.144 --> 00:10:21.634 Two questions: the total loss in dB, and the power that comes out the far end. 00:10:21.684 --> 00:10:26.554 Pause the video and try it. 00:10:26.604 --> 00:10:29.494 Ninety seconds, on paper, alone. 00:10:29.544 --> 00:10:32.814 This exact shape of question appears on the midterm. 00:10:32.864 --> 00:10:34.654 The loss first. 00:10:34.704 --> 00:10:43.654 Each kilometer contributes its own minus nought point three dB, so five kilometers is five times minus nought point three: minus one point five dB. 00:10:46.274 --> 00:10:52.044 Per-distance loss multiplies by the distance — because every kilometer adds its share. 00:10:52.094 --> 00:10:53.664 Then the power. 00:10:53.714 --> 00:11:00.794 To get back from decibels to watts, the dB goes into the exponent — divided by ten. 00:11:00.844 --> 00:11:09.794 P two equals two milliwatts times ten to the power of minus one point five over ten, which is ten to the minus nought point one five. 00:11:10.044 --> 00:11:12.334 That is nought point seven nought eight. 00:11:12.384 --> 00:11:17.784 Two milliwatts times nought point seven nought eight is about one point four milliwatts. 00:11:17.834 --> 00:11:22.164 So the cable kept seventy percent of the signal and took thirty percent as its cut. 00:11:22.214 --> 00:11:27.594 And here is the most common mistake on this question. 00:11:27.644 --> 00:11:36.594 They compute ten to the minus one point five — forgetting to divide by ten first — and get nought point nought six three milliwatts, a number more than twenty times too small. 00:11:38.814 --> 00:11:47.764 Forgetting the divide-by-ten is the most common dB slip there is. dB equals ten log; going backward, divide by ten before you exponentiate. 00:11:49.734 --> 00:11:53.719 Every time. 00:11:53.769 --> 00:11:58.849 Now the centerpiece of the section — Forouzan example three point two eight. 00:11:58.899 --> 00:12:01.549 Look at the chain across the top of the slide. 00:12:01.599 --> 00:12:04.749 A signal travels from point one to point four. 00:12:04.799 --> 00:12:08.759 Between one and two, a cable: minus three dB. 00:12:08.809 --> 00:12:13.079 Between two and three, an amplifier: plus seven. 00:12:13.129 --> 00:12:17.029 Between three and four, another cable: minus three. 00:12:17.079 --> 00:12:23.879 What is the total change from end to end — and does the signal arrive stronger or weaker than it left? 00:12:23.929 --> 00:12:32.059 The total from point one to point four is the sum of the three hops: minus three, plus seven, minus three. 00:12:32.109 --> 00:12:33.479 Plus one dB. 00:12:33.529 --> 00:12:38.959 No calculator and no multiplication — three signed numbers, summed. 00:12:39.009 --> 00:12:44.259 And plus one dB means the signal arrives stronger than it left. 00:12:44.309 --> 00:12:48.539 The ratio is ten to the one-tenth — one point two six. 00:12:48.589 --> 00:12:51.779 The amplifier more than repaid both cables. 00:12:51.829 --> 00:13:00.779 Now look at the same journey in raw ratios: nought point five nought one, times five point nought one, times nought point five nought one. 00:13:00.919 --> 00:13:09.869 Multiply those and you get the same one point two six — but that version is calculator territory, and the dB version was mental arithmetic. 00:13:10.839 --> 00:13:14.859 One more chain, to make sure the signs survive. 00:13:14.909 --> 00:13:17.719 A long cable: minus twelve dB. 00:13:17.769 --> 00:13:20.149 An amplifier: plus seven. 00:13:20.199 --> 00:13:22.429 A short cable: minus three. 00:13:22.479 --> 00:13:27.759 Sum: minus twelve plus seven minus three — minus eight dB. 00:13:27.809 --> 00:13:30.679 Negative, so this time it is a net loss. 00:13:30.729 --> 00:13:37.479 Ten to the minus nought point eight is about nought point one six — roughly a sixth of the power arrives. 00:13:37.529 --> 00:13:40.779 The amplifier helped, and it was not enough. 00:13:40.829 --> 00:13:49.779 A twenty-segment link works exactly the same way: keep the losses negative and the gains positive, and the total is one sum. 00:13:50.049 --> 00:13:58.999 Twenty segments as ratios is twenty multiplications; twenty segments in decibels is one addition. The next slide says why. 00:13:59.129 --> 00:14:04.879 The decibel-chain demo on the course page is the place to play with it. 00:14:04.929 --> 00:14:13.879 Load the preset called 'Forouzan 3.31 chain': it is this exact cascade, minus three, plus seven, minus three, drawn as blocks with a running total. 00:14:16.019 --> 00:14:24.789 The left readout shows the decibels adding; the right readout shows the power ratio multiplying — same physics, two languages. 00:14:24.839 --> 00:14:32.429 Then try 'Too much cable' and watch the bar sink below the dashed receiver-sensitivity line and the verdict turn red. 00:14:32.479 --> 00:14:37.659 Then 'Amplifier saves it' — one plus-seven block rescues the same chain. 00:14:37.709 --> 00:14:44.429 And then wreck it yourself: add cable blocks one at a time, and call out the running total before each click. 00:14:44.479 --> 00:14:51.975 When you can call the totals faster than you click, this section has done its job. 00:14:52.025 --> 00:14:54.835 And now the payoff on everything so far. 00:14:54.885 --> 00:15:01.695 Every unit you have ever used — meters, seconds, watts — combines by multiplication when effects cascade. 00:15:01.745 --> 00:15:05.635 Half the power, then a fifth of that, then half again. 00:15:05.685 --> 00:15:10.355 And yet every engineer on Earth measures loss in a unit they simply add. 00:15:10.405 --> 00:15:14.015 Why would anyone build such a unit? 00:15:14.065 --> 00:15:17.985 Here is the same three-segment link, written twice. 00:15:18.035 --> 00:15:26.175 In ratios: nought point five nought one, times five point nought one, times nought point five nought one. 00:15:26.225 --> 00:15:30.735 To combine them you multiply, and you reach for a calculator. 00:15:30.785 --> 00:15:34.015 In decibels: minus three, plus seven, minus three. 00:15:34.065 --> 00:15:39.785 To combine them you add, and you reach for nothing. 00:15:39.835 --> 00:15:47.405 The reason is one sentence of mathematics: the logarithm turns multiplication into addition. 00:15:47.455 --> 00:15:53.725 The log of a product is the sum of the logs — that is the defining property of every logarithm. 00:15:53.775 --> 00:15:57.135 Cascading segments multiplies power ratios; therefore it adds decibels. 00:15:57.185 --> 00:15:59.365 The convenience is not a habit. 00:15:59.415 --> 00:16:01.575 It has a theorem inside it. 00:16:01.625 --> 00:16:03.925 And there is a bonus. 00:16:03.975 --> 00:16:10.455 Logarithms compress absurd ranges into human numbers. 00:16:10.505 --> 00:16:18.095 On a long link, the end-to-end power ratio can be ten billion to one — ten with nine zeros after it. 00:16:18.145 --> 00:16:22.805 In decibels that unpronounceable number is simply one hundred dB. 00:16:22.855 --> 00:16:25.815 A factor of a thousand is just plus thirty. 00:16:25.865 --> 00:16:34.815 That is why every cable spec, every amplifier datasheet, every link budget on this planet is written in dB — the numbers add, and the numbers stay small. 00:16:36.705 --> 00:16:45.655 So when the datasheet said minus nought point three dB per kilometer, it was choosing the unit that makes 'per kilometer' work: each kilometer adds its share. 00:16:47.185 --> 00:16:53.075 A percent-per-kilometer spec would force you to multiply five times for five kilometers. 00:16:53.125 --> 00:17:02.075 The dB spec lets you multiply the rate by the distance, once, and you did exactly that three slides ago without noticing. 00:17:04.602 --> 00:17:11.622 I have just claimed that the decibel earns its place because it turns multiplication into addition. 00:17:11.672 --> 00:17:15.172 Here is that claim as a tool you can stack blocks in. 00:17:15.222 --> 00:17:24.062 The sender is on the left, the receiver on the right, and the red dashed line is the weakest signal the receiver can still decode. 00:17:24.112 --> 00:17:28.182 Nothing is in the chain yet, so the level sits at zero. 00:17:28.232 --> 00:17:33.712 Add a cable: minus three decibels, and the line steps down. 00:17:33.762 --> 00:17:39.202 Add a splitter: minus three and a half, and it steps down again. 00:17:39.252 --> 00:17:43.662 Add an amplifier: plus seven, and it steps up. 00:17:43.712 --> 00:17:45.452 Notice what I am not doing. 00:17:45.502 --> 00:17:53.762 I am not multiplying ratios, I am not taking logarithms, and I am not converting anything back to milliwatts between blocks. 00:17:53.812 --> 00:18:00.152 I am adding small signed numbers along a chain, which is the only reason this unit exists. 00:18:00.202 --> 00:18:08.852 Here is the chain from the worked example: minus three, plus seven, minus three — a net of plus one decibel. 00:18:08.902 --> 00:18:17.192 As a power ratio that is one point two six, and two point eight three milliwatts in becomes three point five six milliwatts out. 00:18:17.242 --> 00:18:22.232 Both languages, side by side, from the same three numbers. 00:18:22.282 --> 00:18:24.792 Now break it on purpose. 00:18:24.842 --> 00:18:33.732 Minus three, minus three, minus nought point five, minus three, minus three — twelve and a half decibels of loss. 00:18:33.782 --> 00:18:42.732 A power ratio of nought point zero five six, and only a hundred and fifty-nine microwatts arrive where the receiver needs three hundred and fifty-three. 00:18:43.092 --> 00:18:48.772 Dead link — and note how the tool phrases it: you are three and a half decibels short. 00:18:48.822 --> 00:18:51.972 Not 'some' short. Three and a half. 00:18:52.022 --> 00:19:00.452 Drop one amplifier into the same chain and the total goes from minus twelve and a half to minus five and a half. 00:19:00.502 --> 00:19:08.362 Seven hundred and ninety-seven microwatts arrive, and the link is decodable with a margin of three and a half decibels. 00:19:08.412 --> 00:19:14.497 That is link budgeting, and it is one addition long. 00:19:14.737 --> 00:19:16.057 First checkpoint. 00:19:16.107 --> 00:19:20.817 Pause the video and answer these three on paper, without scrolling back. 00:19:20.867 --> 00:19:22.077 One. 00:19:22.127 --> 00:19:26.397 A signal's power drops to one quarter of what was sent. 00:19:26.447 --> 00:19:28.767 What is the attenuation in dB? 00:19:28.817 --> 00:19:29.967 Two. 00:19:30.017 --> 00:19:34.887 A cable loses minus nought point five dB per kilometer. 00:19:34.937 --> 00:19:42.257 After eight kilometers, what is the total loss — and roughly what fraction of the power survives? 00:19:42.307 --> 00:19:43.117 Three. 00:19:43.167 --> 00:19:49.177 A chain: cable minus twelve dB, amplifier plus seven, cable minus three. 00:19:49.227 --> 00:19:54.817 What is the net change, and is the arriving signal stronger or weaker? 00:19:54.867 --> 00:19:59.297 Pause now. 00:19:59.347 --> 00:20:00.217 Answers. 00:20:00.267 --> 00:20:07.047 The first: ten log of nought point two five is minus six point nought two — call it minus six dB. 00:20:07.097 --> 00:20:15.507 And notice you did not need the calculator at all: a quarter is a half of a half, two halvings, minus three and minus three. 00:20:15.557 --> 00:20:20.367 The second: eight times minus nought point five is minus four dB. 00:20:20.417 --> 00:20:28.427 The fraction is ten to the minus nought point four, which is about nought point four — forty percent survives. 00:20:28.477 --> 00:20:33.687 And the third: minus twelve plus seven minus three is minus eight dB — weaker. 00:20:33.737 --> 00:20:42.687 Ten to the minus nought point eight, about nought point one six: a sixth of the sent power arrives. 00:20:43.573 --> 00:20:44.873 Section two. 00:20:44.923 --> 00:20:50.913 Two more ways the wire lies to you — and then one honest number that measures how badly. 00:20:50.963 --> 00:20:53.793 Attenuation made the copy smaller. 00:20:53.843 --> 00:21:00.390 Now: what makes it misshapen, and what makes it noisy. 00:21:00.440 --> 00:21:03.140 Impairment number two: distortion. 00:21:03.190 --> 00:21:07.690 This one is subtler than attenuation, because nothing is lost. 00:21:07.740 --> 00:21:11.120 The energy arrives — but the shape is wrong. 00:21:11.170 --> 00:21:16.180 And notice the sentence at the top of the slide: only composite signals distort. 00:21:16.230 --> 00:21:20.870 A lone sine wave can shrink — but a shrunken sine is still a sine. 00:21:20.920 --> 00:21:22.900 It has no shape to lose. 00:21:22.950 --> 00:21:26.050 Distortion needs a signal with parts. 00:21:26.100 --> 00:21:29.710 Here is the mechanism, in three steps. 00:21:29.760 --> 00:21:36.590 Recall from Session 4 that a composite signal is a sum of sine-wave components at different frequencies. 00:21:36.640 --> 00:21:40.760 Step one: the medium delays each frequency differently. 00:21:40.810 --> 00:21:49.760 Signal velocity through a real medium depends on frequency — so each component makes the journey at its own speed, and arrives with its own delay. 00:21:50.730 --> 00:21:56.790 Step two: a delay that is not a whole number of periods is a phase shift. 00:21:56.840 --> 00:22:03.440 If a component slides by exactly one of its own periods, it lines up again and nobody notices. 00:22:03.490 --> 00:22:09.750 If it slides by anything else, it arrives offset — out of phase with its neighbors. 00:22:09.800 --> 00:22:18.750 Step three: add up the same components, the same amplitudes, at the wrong offsets — and the sum draws a different picture. 00:22:18.930 --> 00:22:20.450 Nothing was removed. 00:22:20.500 --> 00:22:21.950 Everything arrived. 00:22:22.000 --> 00:22:24.420 And the shape is not the shape you sent. 00:22:24.470 --> 00:22:32.570 Suppose, a band where every musician plays the right notes at the right volume — but the drummer's sound reaches you a beat late. 00:22:32.620 --> 00:22:36.370 Same components, wrong alignment, ruined song. 00:22:36.420 --> 00:22:40.618 That is distortion. 00:22:40.668 --> 00:22:43.558 Impairment number three: noise. 00:22:43.608 --> 00:22:48.278 Four flavors, one effect — and the exam wants all four names. 00:22:48.328 --> 00:22:52.088 I will give you each one with a single image, and we move on. 00:22:52.138 --> 00:22:53.778 Thermal noise. 00:22:53.828 --> 00:23:02.778 The electrons in every wire jiggle, because the wire has a temperature — and jiggling charge is a signal, a signal nobody sent. 00:23:03.528 --> 00:23:05.888 You cannot switch thermal noise off. 00:23:05.938 --> 00:23:14.888 You can only cool the equipment, which is why the most sensitive receivers on Earth — radio telescopes, deep-space antennas — run at cryogenic temperatures, near 20 kelvin. 00:23:17.718 --> 00:23:19.608 Induced noise. 00:23:19.658 --> 00:23:26.078 Motors and appliances radiate, and an unshielded wire receives that radiation. 00:23:26.128 --> 00:23:33.018 Fluorescent lights, drills, refrigerator compressors: all of them transmit, and your cable listens. 00:23:33.068 --> 00:23:34.498 Crosstalk. 00:23:34.548 --> 00:23:38.448 One wire's signal induced onto its neighbor. 00:23:38.498 --> 00:23:44.518 Crosstalk is what you hear when a stranger's call is faintly audible on an old telephone line. 00:23:44.568 --> 00:23:52.008 Your wire and its neighbor are a sending antenna and a receiving antenna, lying side by side for kilometers. 00:23:52.058 --> 00:23:54.048 And impulse noise. 00:23:54.098 --> 00:23:59.208 Spikes — from lightning, power lines, switching surges. 00:23:59.258 --> 00:24:01.908 Enormous energy, tiny duration. 00:24:01.958 --> 00:24:10.908 The other three hiss continuously; impulse noise is brief and high-energy, and it destroys a contiguous run of bits rather than degrading all of them. 00:24:12.318 --> 00:24:17.558 Four sources, one ending: energy on the line that carries no information. 00:24:17.608 --> 00:24:23.078 The question that matters for engineering is — how much of it, compared to how much of you? 00:24:23.128 --> 00:24:27.716 That question has a number. 00:24:27.766 --> 00:24:33.856 Here is the number, and it is the most important quantity in the second half of this lecture. 00:24:33.906 --> 00:24:37.116 SNR — the signal-to-noise ratio. 00:24:37.166 --> 00:24:43.286 SNR equals the average signal power divided by the average noise power. 00:24:43.336 --> 00:24:52.286 And because it is a ratio of two powers, of course we quote it in decibels as well — same trick as an hour ago: SNR in dB is ten log of SNR. 00:24:54.876 --> 00:25:00.666 The two formulas on this slide have exactly the shape of the dB formula from section one. 00:25:00.716 --> 00:25:06.346 That visual rhyme is not an accident; it is the same idea wearing a different name. 00:25:06.396 --> 00:25:08.816 What does it mean? 00:25:08.866 --> 00:25:13.096 What you want, divided by what you didn't ask for. 00:25:13.146 --> 00:25:20.636 High SNR: the signal towers over the noise, and the received signal is barely corrupted — a clean line. 00:25:20.686 --> 00:25:25.746 Low SNR: the noise is a real fraction of what arrives — a swamp. 00:25:25.796 --> 00:25:30.426 Zero dB means a ratio of one: the noise is exactly as loud as you are. 00:25:30.476 --> 00:25:36.466 And we use averages of both powers, because both wobble from instant to instant. 00:25:36.516 --> 00:25:43.306 Now the warning, and today it stops being a pedantic remark and becomes deadly serious. 00:25:43.356 --> 00:25:45.356 SNR is a plain number. 00:25:45.406 --> 00:25:47.926 SNR-dB is its logarithm. 00:25:47.976 --> 00:25:55.256 They are the same fact in two languages — and exam questions love to hand you one when the formula wants the other. 00:25:55.306 --> 00:26:01.986 The conversion is the one you already know: SNR equals ten to the SNR-dB over ten. 00:26:02.036 --> 00:26:04.346 Forty dB is ten thousand. 00:26:04.396 --> 00:26:05.556 Not forty. 00:26:05.606 --> 00:26:14.556 In section four, a formula arrives that eats plain SNR — and feeding it decibels is the single most reliable way to lose marks in this chapter. 00:26:18.910 --> 00:26:21.940 Forouzan example three point three one. 00:26:21.990 --> 00:26:27.770 The power of a signal is ten milliwatts; the power of the noise is one microwatt. 00:26:27.820 --> 00:26:28.890 Find SNR, and SNR in dB. 00:26:28.940 --> 00:26:35.450 Pause and try it first — and watch your units. 00:26:35.500 --> 00:26:39.870 The units are where the marks are on this one. 00:26:39.920 --> 00:26:43.170 Step one, same units first. 00:26:43.220 --> 00:26:46.550 Ten milliwatts is ten thousand microwatts. 00:26:46.600 --> 00:26:55.550 Milli and micro differ by a factor of one thousand — and if you divide ten by one because you never converted, you get an SNR of ten instead of ten thousand. 00:26:57.180 --> 00:27:03.660 That is a factor of a thousand of error — thirty dB of wrongness, from one skipped line. 00:27:03.710 --> 00:27:06.490 Step two, the ratio. 00:27:06.540 --> 00:27:11.060 SNR equals ten thousand microwatts over one microwatt: ten thousand. 00:27:11.110 --> 00:27:15.600 A plain, honest, dimensionless number. 00:27:15.650 --> 00:27:19.130 The signal is ten thousand times the noise. 00:27:19.180 --> 00:27:22.210 Step three, the decibels. 00:27:22.260 --> 00:27:25.880 SNR-dB equals ten log of ten thousand. 00:27:25.930 --> 00:27:32.500 Ten thousand is ten to the fourth, so the log is exactly four, and ten times four is forty. 00:27:32.550 --> 00:27:34.060 Forty dB. 00:27:34.110 --> 00:27:39.970 Now look at both answers side by side: ten thousand, and forty dB. 00:27:40.020 --> 00:27:46.710 They are the same fact, spoken in two languages — the ratio language and the logarithm language. 00:27:46.760 --> 00:27:52.490 Practice flipping between them until it is automatic: forty dB, ten to the four. 00:27:52.540 --> 00:27:54.750 Twenty dB, one hundred. 00:27:54.800 --> 00:27:57.060 Thirty dB, one thousand. 00:27:57.110 --> 00:28:06.060 If those feel interchangeable now, the formula waiting in section four holds no terrors for you. 00:28:06.457 --> 00:28:09.857 So how good can the number get — and how bad? 00:28:09.907 --> 00:28:13.077 A high SNR is a clean line. 00:28:13.127 --> 00:28:21.777 Forty dB — ten thousand to one — means the noise is a ten-thousandth of the signal: the received waveform is barely corrupted. 00:28:21.827 --> 00:28:24.467 A good wired link lives up here. 00:28:24.517 --> 00:28:27.527 A low SNR is a swamp. 00:28:27.577 --> 00:28:36.087 At zero dB the ratio is one — the noise is exactly as strong as the signal, and half of what the receiver measures is garbage. 00:28:36.137 --> 00:28:40.487 Radio links in bad conditions fight in this territory. 00:28:40.537 --> 00:28:42.547 And the ideal? 00:28:42.597 --> 00:28:44.057 A noiseless channel. 00:28:44.107 --> 00:28:50.257 No noise power at all: SNR is the signal power divided by zero — infinity. 00:28:50.307 --> 00:28:52.087 Infinite dB, too. 00:28:52.137 --> 00:29:01.087 That is Forouzan's example three point three two, and it is an ideal, unreachable on any real wire, because thermal noise alone guarantees the denominator is never zero. 00:29:04.227 --> 00:29:06.827 The ideal is still worth holding on to. 00:29:06.877 --> 00:29:15.827 The next section opens with Nyquist, who assumes exactly that — a perfectly noiseless channel — and derives a speed limit from it. 00:29:16.077 --> 00:29:23.412 Section four then adds the noise back, and the ceiling drops. 00:29:23.462 --> 00:29:26.152 Before the checkpoint, four mistakes. 00:29:26.202 --> 00:29:30.292 Every one of them is real, and every one of them costs marks. 00:29:30.342 --> 00:29:31.572 One. 00:29:31.622 --> 00:29:40.572 'The attenuation is minus three dB, so I subtract three from the power in milliwatts.' No. dB is not a quantity of watts; it lives in the exponent. 00:29:44.402 --> 00:29:48.582 P two equals P one times ten to the dB over ten. 00:29:48.632 --> 00:29:53.172 Never add decibels to watts — they are not even the same kind of thing. 00:29:53.222 --> 00:29:54.382 Two. 00:29:54.432 --> 00:29:59.782 Dropping the minus sign on a loss, and then happily summing the gains. 00:29:59.832 --> 00:30:04.672 The cascade only works if losses stay negative and gains stay positive. 00:30:04.722 --> 00:30:09.222 Lose one minus sign and the total is wrong. 00:30:09.272 --> 00:30:09.882 Three. 00:30:09.932 --> 00:30:16.062 Plugging SNR-dB into a formula that wanted plain SNR. 00:30:16.112 --> 00:30:20.342 Convert first: SNR equals ten to the SNR-dB over ten. 00:30:20.392 --> 00:30:22.502 Forty dB is ten thousand — not forty. 00:30:22.552 --> 00:30:28.012 I will repeat this one in section four, where it costs the most. 00:30:28.062 --> 00:30:29.262 Four. 00:30:29.312 --> 00:30:34.272 Mixing milliwatts and microwatts inside one ratio. 00:30:34.322 --> 00:30:37.972 Same units, top and bottom, before you divide. 00:30:38.022 --> 00:30:42.662 Ten milliwatts over one microwatt is ten thousand — not ten. 00:30:42.712 --> 00:30:47.522 Four rows, one habit: slow down at the units and the signs. 00:30:47.572 --> 00:30:56.445 The formulas themselves are three lines of arithmetic; all the danger is in the bookkeeping. 00:30:56.495 --> 00:30:58.055 Second checkpoint. 00:30:58.105 --> 00:31:02.285 Same rule — pause, paper, no scrolling back. 00:31:02.335 --> 00:31:03.545 One. 00:31:03.595 --> 00:31:08.415 On an old phone line you faintly hear a stranger's conversation. 00:31:08.465 --> 00:31:10.505 Which flavor of noise is that? 00:31:10.555 --> 00:31:11.715 Two. 00:31:11.765 --> 00:31:18.325 Signal power twenty milliwatts, noise power two microwatts. 00:31:18.375 --> 00:31:20.105 Find SNR and SNR-dB. 00:31:20.155 --> 00:31:21.335 Three. 00:31:21.385 --> 00:31:26.635 Why does adding an amplifier in the middle of a link not improve the SNR? 00:31:26.685 --> 00:31:30.745 Pause now. 00:31:30.795 --> 00:31:32.215 Answers. 00:31:32.265 --> 00:31:37.735 The first is crosstalk — one wire's signal induced onto its neighbor. 00:31:37.785 --> 00:31:40.305 The second: same units first. 00:31:40.355 --> 00:31:45.825 Twenty milliwatts is twenty thousand microwatts; divided by two microwatts, the SNR is ten thousand. 00:31:45.875 --> 00:31:49.395 In decibels, ten log of ten thousand — forty dB. 00:31:49.445 --> 00:31:57.525 If you got ten, you skipped the unit conversion: milliwatts and microwatts differ by a factor of a thousand. 00:31:57.575 --> 00:32:00.405 And the third — remember the clip. 00:32:00.455 --> 00:32:08.385 The amplifier boosts everything that reaches it: the signal, and the noise that has already crept in. 00:32:08.435 --> 00:32:12.695 Both go up by the same factor, so their ratio does not move. 00:32:12.745 --> 00:32:14.605 You can buy back amplitude. 00:32:14.655 --> 00:32:19.285 You cannot buy back clean. 00:32:19.335 --> 00:32:20.215 Section three. 00:32:20.265 --> 00:32:22.345 The noiseless ceiling. 00:32:22.395 --> 00:32:31.345 We now allow ourselves the ideal from the last section — a channel with no noise at all, SNR infinite — and ask: on that perfect channel, is there any limit to how fast we can send data? 00:32:34.755 --> 00:32:43.705 The answer is yes, there is still a limit — and the man who wrote it down is Nyquist (Harry Nyquist). 00:32:45.166 --> 00:32:54.116 But before either formula, two words that must never blur together, because everything in the rest of this session hangs on the difference. 00:32:54.236 --> 00:32:57.276 Bandwidth, measured in hertz. 00:32:57.326 --> 00:33:00.816 Bandwidth is a property of the copper — of the channel. 00:33:00.866 --> 00:33:07.616 It is the width of the band of frequencies the channel actually passes, the B we established in Session 4. 00:33:07.666 --> 00:33:16.616 A telephone line passes roughly three hundred to three thousand three hundred hertz: a band three thousand hertz wide, so B is three thousand. 00:33:17.116 --> 00:33:20.346 You buy it, you inherit it, you are stuck with it. 00:33:20.396 --> 00:33:23.346 No cleverness at your desk changes B. 00:33:23.396 --> 00:33:26.866 Bit rate, measured in bits per second. 00:33:26.916 --> 00:33:32.196 Bit rate is a property of your scheme — of what you managed to squeeze through that channel. 00:33:32.246 --> 00:33:38.736 It is the thing you are trying to maximize, and it is the thing the two formulas of this session compute. 00:33:38.786 --> 00:33:44.636 Confusing these two is the single most expensive mistake in this chapter. 00:33:44.686 --> 00:33:53.636 'Three thousand hertz equals three thousand bits per second' is wrong in both directions at once — a channel of three thousand hertz can carry far more than three thousand bits per second, or far less, depending on the scheme and the noise. 00:33:59.846 --> 00:34:04.376 Hertz is the wire's property; bits per second is your result. 00:34:04.426 --> 00:34:11.394 The only bridges between them are the two formulas we now build. 00:34:11.444 --> 00:34:20.394 Nyquist (Harry Nyquist), Bell Labs, nineteen twenties — studying the telegraph, decades before the first computer network. 00:34:21.064 --> 00:34:26.794 Assume a perfectly noiseless channel of bandwidth B. Here is his first result. 00:34:26.844 --> 00:34:30.474 Bit rate equals two times B. 00:34:30.524 --> 00:34:39.384 The physical fact underneath it is this: a channel of bandwidth B can carry at most two B signal changes per second. 00:34:39.434 --> 00:34:48.384 A signal that changes level faster than that needs frequencies the channel does not pass — so the changes smear into each other at the far end, and the receiver cannot separate them. 00:34:50.564 --> 00:34:54.264 The change budget of a channel is two B, full stop. 00:34:54.314 --> 00:35:00.074 If each change carries one bit, the bit rate is two B bits per second. 00:35:00.124 --> 00:35:08.444 So put in the telephone line: B is three thousand hertz, one bit per change — six thousand bits per second. 00:35:08.494 --> 00:35:09.894 Six kilobits. 00:35:09.944 --> 00:35:14.944 That number assumes one bit per change, and that assumption is the lever. 00:35:14.994 --> 00:35:22.114 People downloaded music over telephone lines at rates well above it, so six kilobits is not the whole story. 00:35:22.164 --> 00:35:26.984 Look at the assumption we made silently: each change carries one bit. 00:35:27.034 --> 00:35:33.027 Nothing requires it, and the next slide drops it. 00:35:33.077 --> 00:35:35.757 Nobody said one change is one bit. 00:35:35.807 --> 00:35:38.027 That is the whole of step two. 00:35:38.077 --> 00:35:44.487 Bit rate equals two times B, times the base-two logarithm of L. 00:35:44.537 --> 00:35:50.927 L is the number of signal levels you choose to use — and 'choose' is the operative word. 00:35:50.977 --> 00:35:55.737 Use two levels and each change carries one bit: log two of two is one. 00:35:55.787 --> 00:36:03.277 Use four levels and each change picks one of four possibilities — that is two bits per change. 00:36:03.327 --> 00:36:05.507 Eight levels, three bits. 00:36:05.557 --> 00:36:07.877 Sixteen levels, four bits. 00:36:07.927 --> 00:36:12.097 Each doubling of L adds exactly one bit to every change. 00:36:12.147 --> 00:36:21.097 Note the fine print while we are here: L must be at least two — a single level never changes and carries nothing — and with B in hertz the formula answers in bits per second. 00:36:23.527 --> 00:36:31.107 And bits come whole where levels are counted: log two of L bits per change means we build L as a power of two. 00:36:31.157 --> 00:36:34.397 And look at what this formula did to history. 00:36:34.447 --> 00:36:40.347 The change budget, two B, is fixed by the copper — you cannot push changes faster. 00:36:40.397 --> 00:36:43.337 But the richness of each change is yours. 00:36:43.387 --> 00:36:51.657 The entire history of modem design is on this slide: engineers stopped trying to send faster, and started trying to send richer. 00:36:51.707 --> 00:36:58.087 Same wire, same three thousand hertz, more and more bits packed into every change. 00:36:58.137 --> 00:37:00.097 B is the wire's property. 00:37:00.147 --> 00:37:02.097 L is your design decision. 00:37:02.147 --> 00:37:11.097 Keep that division of ownership in mind. Slide 27 asks the greedy question about L, and the answer is the constraint Nyquist's formula leaves out. 00:37:15.924 --> 00:37:21.934 Forouzan examples three point three four and three point three five, plus one case for you. 00:37:21.984 --> 00:37:27.754 A noiseless channel with a bandwidth of three thousand hertz — the classic telephone line. 00:37:27.804 --> 00:37:33.804 Maximum bit rate with: a, two levels; b, four levels; c, sixteen levels. 00:37:33.854 --> 00:37:35.424 Case c is yours. 00:37:35.474 --> 00:37:38.204 Ninety seconds, on paper. 00:37:38.254 --> 00:37:45.224 Pause now, and come back for the reveal. 00:37:45.274 --> 00:37:48.754 Case a. 00:37:48.804 --> 00:37:52.864 Two times three thousand times log two of two. 00:37:52.914 --> 00:37:57.144 The log is one, so: six thousand bits per second. 00:37:57.194 --> 00:38:00.254 The number that disappointed us two slides ago. 00:38:00.304 --> 00:38:01.314 Case b. 00:38:01.364 --> 00:38:04.754 Two times three thousand times log two of four. 00:38:04.804 --> 00:38:09.414 Log two of four is two — four possibilities is two bits. 00:38:09.464 --> 00:38:11.684 Twelve thousand bits per second. 00:38:11.734 --> 00:38:15.104 Same copper, double the levels, double the rate. 00:38:15.154 --> 00:38:16.774 Case c, yours. 00:38:16.824 --> 00:38:20.474 Two times three thousand times log two of sixteen. 00:38:20.524 --> 00:38:26.914 Sixteen is two to the fourth, so the log is four: twenty-four thousand bits per second. 00:38:26.964 --> 00:38:30.354 If you wrote twenty-four kilobits, well done. 00:38:30.404 --> 00:38:33.364 Now stand back and see the pattern. 00:38:33.414 --> 00:38:40.704 Six, twelve, eighteen, twenty-four kilobits for two, four, eight, sixteen levels. 00:38:40.754 --> 00:38:46.754 Every doubling of L adds exactly one bit per change — a constant step in bit rate each time the levels double. 00:38:46.804 --> 00:38:48.794 The wire never changed. 00:38:48.844 --> 00:38:50.634 Only the scheme did. 00:38:50.684 --> 00:38:59.634 Which invites the obvious question: if doubling L keeps adding bit rate, why ever stop? 00:39:00.768 --> 00:39:02.388 So let us not stop. 00:39:02.438 --> 00:39:05.848 Nyquist's formula puts no limit on L. None. 00:39:05.898 --> 00:39:13.408 Let us be greedy with the same three-thousand-hertz telephone line: I want a gigabit per second. 00:39:13.458 --> 00:39:17.148 Rearrange the formula for the bits per change. 00:39:17.198 --> 00:39:26.148 Log two of L equals the bit rate over two B — ten to the ninth over six thousand — about one hundred sixty-six thousand, six hundred sixty-seven. 00:39:27.998 --> 00:39:32.918 Each signal change must carry a hundred and sixty-six thousand bits. 00:39:32.968 --> 00:39:40.758 So the number of levels is two raised to the one hundred sixty-six thousand six hundred and sixty-seventh power. 00:39:40.808 --> 00:39:45.548 That is not a big number; that is a number with about fifty thousand digits. 00:39:45.598 --> 00:39:48.578 And Nyquist's formula does not blink. 00:39:48.628 --> 00:39:51.668 It hands you the L and wishes you luck. 00:39:51.718 --> 00:39:57.178 One billion bits per second, on a three-thousand-hertz telephone line. 00:39:57.228 --> 00:39:59.908 Nyquist's formula says yes. 00:39:59.958 --> 00:40:03.108 No telephone line has ever carried a gigabit. 00:40:03.158 --> 00:40:07.228 So either the formula is wrong — or it left something out. 00:40:07.278 --> 00:40:09.538 It left something out. 00:40:09.588 --> 00:40:15.508 Look back at the very first words of its derivation: assume a perfectly noiseless channel. 00:40:15.558 --> 00:40:24.028 We borrowed the ideal from section two — SNR equals infinity — and there is no such channel anywhere in the universe. 00:40:24.078 --> 00:40:26.578 Thermal noise alone sees to that. 00:40:26.628 --> 00:40:33.818 Nyquist is not wrong; Nyquist answered the question he was asked, about a channel that does not exist. 00:40:33.868 --> 00:40:38.758 What does noise actually do to those two-to-the-hundred-sixty-six-thousand levels? 00:40:38.808 --> 00:40:46.092 That is section four, and it is the most important picture in this lecture. 00:40:46.142 --> 00:40:47.632 Third checkpoint. 00:40:47.682 --> 00:40:49.492 Pause and answer on paper. 00:40:49.542 --> 00:40:50.732 One. 00:40:50.782 --> 00:40:56.442 A noiseless channel has a bandwidth of four kilohertz and uses two levels. 00:40:56.492 --> 00:40:58.022 Maximum bit rate? 00:40:58.072 --> 00:40:59.232 Two. 00:40:59.282 --> 00:41:03.622 Same channel — you need thirty-two kilobits per second. 00:41:03.672 --> 00:41:05.052 How many levels? 00:41:05.102 --> 00:41:06.272 Three. 00:41:06.322 --> 00:41:11.852 Why can Nyquist's formula not give the ceiling of a real channel? 00:41:11.902 --> 00:41:15.952 Pause now. 00:41:16.002 --> 00:41:17.432 Answers. 00:41:17.482 --> 00:41:24.462 One: two times four thousand times log two of two — eight thousand bits per second. 00:41:24.512 --> 00:41:30.702 Two: you need log two of L equal to thirty-two thousand over eight thousand, which is four. 00:41:30.752 --> 00:41:34.272 So L is two to the fourth — sixteen levels. 00:41:34.322 --> 00:41:40.522 And three: because it assumes a noiseless channel, and real lines always have noise. 00:41:40.572 --> 00:41:49.522 Nyquist tells you what a scheme achieves on perfect copper; the ceiling of a real line belongs to the next section. 00:41:50.915 --> 00:41:52.305 Section four. 00:41:52.355 --> 00:41:53.725 The real ceiling. 00:41:53.775 --> 00:41:55.165 Noise gets a vote. 00:41:55.215 --> 00:42:04.165 Shannon (Claude Shannon) computed exactly how large a vote — and his answer is one of the very few genuinely unbreakable results in all of engineering. 00:42:08.381 --> 00:42:10.231 First, the picture. 00:42:10.281 --> 00:42:15.071 If you keep only one image from this entire session, keep this one. 00:42:15.121 --> 00:42:19.271 On the left: a clean channel running eight levels. 00:42:19.321 --> 00:42:22.221 Eight clean lines, comfortably separated. 00:42:22.271 --> 00:42:29.051 When the sender puts the signal at level five, the receiver measures level five — every single time. 00:42:29.101 --> 00:42:33.291 Three bits per change, honestly carried. 00:42:33.341 --> 00:42:37.901 On the right: the same eight levels, plus noise. 00:42:37.951 --> 00:42:46.311 Noise jitters every measurement up and down by a random amount — so each level is no longer a line, it is a smear, a fuzzy band. 00:42:46.361 --> 00:42:48.481 And look: the bands touch. 00:42:48.531 --> 00:42:54.771 A measurement in the overlap could have been level five or level six, and the receiver cannot tell. 00:42:54.821 --> 00:42:56.091 So it guesses. 00:42:56.141 --> 00:43:02.601 And a receiver that guesses is not carrying three bits per change — it is carrying noise. 00:43:02.651 --> 00:43:11.331 So say the verdict precisely, because the wrong mental model is very common: noise does not slow your signal down. 00:43:11.381 --> 00:43:14.031 The changes still arrive at the same rate. 00:43:14.081 --> 00:43:23.031 Noise makes each level fuzzy — it limits how finely you can slice your signal into levels, which is exactly the lever Nyquist left unguarded. 00:43:24.251 --> 00:43:32.251 Squeeze more levels into the same range and the bands overlap sooner; clean up the line and you can afford more levels. 00:43:32.301 --> 00:43:36.451 There is a rule in there, and section five derives it exactly. 00:43:36.501 --> 00:43:42.071 Open the channel-capacity demo on the course page and find it with your own hands first. 00:43:42.121 --> 00:43:48.281 It starts at B three thousand, four levels, SNR twenty dB — comfortably green. 00:43:48.331 --> 00:43:55.671 Drag the LEVELS slider up slowly and watch the bands close in on each other; they touch at about eleven levels. 00:43:55.721 --> 00:44:00.531 Notice what you did not touch: the wire, the bandwidth, the noise. 00:44:00.581 --> 00:44:05.291 Nothing physical changed — you simply asked for more than the noise permits. 00:44:05.341 --> 00:44:10.361 Now drag the SNR up and watch the same number of levels turn green again. 00:44:10.411 --> 00:44:14.791 Try the preset 'Push L too far' and read the red verdict. 00:44:14.841 --> 00:44:23.791 The relationship between the SNR and the number of levels the line tolerates is derived exactly in section five. 00:44:25.206 --> 00:44:34.156 In nineteen forty-four, Shannon (Claude Shannon) — the same Shannon who founded information theory — published the ceiling of a noisy channel: the highest data rate that any scheme, of any cleverness, can ever achieve on it. 00:44:37.086 --> 00:44:42.436 Capacity equals B, times the base-two logarithm of one plus SNR. 00:44:42.486 --> 00:44:48.976 What goes in: B, the bandwidth in hertz — the wire's property, from Session 4. 00:44:49.026 --> 00:44:56.126 And SNR — the signal power over the noise power, the number we built in section two. 00:44:56.176 --> 00:44:58.496 Notice whose properties these are. 00:44:58.546 --> 00:45:00.826 The bandwidth belongs to the line. 00:45:00.876 --> 00:45:02.986 The SNR belongs to the line. 00:45:03.036 --> 00:45:07.596 Your encoding scheme does not get a vote. 00:45:07.646 --> 00:45:16.596 And now the trap, stated as bluntly as I can, because it is the classic exam trap of this whole chapter. 00:45:18.566 --> 00:45:22.036 The SNR in this formula is the plain ratio. 00:45:22.086 --> 00:45:23.556 Never decibels. 00:45:23.606 --> 00:45:32.556 If a question hands you SNR in dB — and it will — convert first: SNR equals ten to the SNR-dB over ten. 00:45:32.666 --> 00:45:37.146 And while we are being careful: it is log two of one plus SNR, not log two of SNR. 00:45:37.196 --> 00:45:39.316 The one is part of the formula. 00:45:39.366 --> 00:45:42.496 Finally, notice what is missing. 00:45:42.546 --> 00:45:44.716 There is no L in this formula. 00:45:44.766 --> 00:45:53.486 Nyquist's lever — the number of levels, the thing the whole history of modem design pulled on — does not appear. 00:45:53.536 --> 00:45:57.776 Shannon does not care how many levels you picked. 00:45:57.826 --> 00:46:02.906 That absence is the entire point: this is not a formula about your scheme. 00:46:02.956 --> 00:46:10.066 It is a ceiling on every possible scheme — a limit that no amount of engineering cleverness gets past. 00:46:10.116 --> 00:46:13.066 Nyquist told you what your design achieves. 00:46:13.116 --> 00:46:18.576 Shannon tells you what the universe permits. 00:46:18.826 --> 00:46:27.536 Forouzan example three point three eight — and this is the calculation that pays off the hook, so do it with me, calculator in hand. 00:46:27.586 --> 00:46:36.376 A telephone line has a bandwidth of three thousand hertz — the band from three hundred to three thousand three hundred hertz, assigned for data. 00:46:36.426 --> 00:46:43.186 The signal-to-noise ratio of a normal telephone line is usually about three thousand one hundred sixty-two. 00:46:43.236 --> 00:46:50.166 Odd-looking number — it is simply thirty-five dB converted to a ratio: ten to the three point five. 00:46:50.216 --> 00:46:52.866 What is the capacity? 00:46:52.916 --> 00:47:01.866 First, the board trick, because most calculators have no log-base-two key: log two of x equals the natural log of x divided by the natural log of two. 00:47:03.536 --> 00:47:04.636 Write it down. 00:47:04.686 --> 00:47:08.276 Every log two in this course goes through that identity. 00:47:08.326 --> 00:47:09.916 Now the log. 00:47:09.966 --> 00:47:16.556 One plus three thousand one hundred sixty-two is three thousand one hundred sixty-three. 00:47:16.606 --> 00:47:25.306 The natural log of that is about eight point nought five nine; divide by nought point six nine three — eleven point six two seven. 00:47:25.356 --> 00:47:26.916 And the ceiling. 00:47:26.966 --> 00:47:35.916 C equals three thousand times eleven point six two seven — about thirty-four thousand, eight hundred and eighty bits per second. 00:47:36.396 --> 00:47:39.256 Call it thirty-four point nine kilobits. 00:47:39.306 --> 00:47:48.256 Forouzan rounds the log to eleven point six two and prints thirty-four thousand eight hundred sixty — same fact, honest rounding either way. 00:47:49.746 --> 00:47:53.006 Understand what kind of statement this is. 00:47:53.056 --> 00:47:54.196 This is a law. 00:47:54.246 --> 00:48:03.196 We have just proved that a telephone line never carries thirty-five kilobits per second — not with a better modem, not with a smarter scheme, not with a bigger budget. 00:48:04.626 --> 00:48:13.576 Keep that number — thirty-four point nine kilobits. Slide 40 settles the question from minute one with it. 00:48:14.867 --> 00:48:21.877 Before the payoff, let us walk into the trap once, deliberately, so you recognize the edges of the hole. 00:48:21.927 --> 00:48:27.027 A channel has B equal to one megahertz, and an SNR of twenty dB. 00:48:27.077 --> 00:48:28.657 What is its capacity? 00:48:28.707 --> 00:48:37.027 Pause and try it — the wording contains the whole danger. 00:48:37.077 --> 00:48:37.937 The wrong way. 00:48:37.987 --> 00:48:46.937 See the letters d and B, ignore them, and plug the twenty straight in: C equals ten to the sixth times log two of one plus twenty. 00:48:47.847 --> 00:48:54.437 Log two of twenty-one is about four point three nine — so 'four point three nine megabits per second'. 00:48:54.487 --> 00:48:57.477 A clean, confident, plausible number. 00:48:57.527 --> 00:48:59.647 No error message anywhere. 00:48:59.697 --> 00:49:01.097 And wrong. 00:49:01.147 --> 00:49:07.527 The conversion you skipped: SNR equals ten to the SNR-dB over ten. 00:49:07.577 --> 00:49:12.247 Twenty dB is ten to the two — a ratio of one hundred. 00:49:12.297 --> 00:49:13.397 Not twenty. 00:49:13.447 --> 00:49:14.417 The right way. 00:49:14.467 --> 00:49:20.437 C equals ten to the sixth times log two of one plus one hundred. 00:49:20.487 --> 00:49:24.327 Log two of one hundred one is about six point six six. 00:49:24.377 --> 00:49:30.857 So the capacity is about six point six six megabits per second — call it six point seven. 00:49:30.907 --> 00:49:36.857 Compare the two answers: four point three nine against six point six six. 00:49:36.907 --> 00:49:40.587 The wrong way threw away a third of the channel — silently. 00:49:40.637 --> 00:49:46.717 So build the habit: before anything else in a Shannon problem, check the units on the SNR. 00:49:46.767 --> 00:49:54.387 The letters 'dB' anywhere in the sentence — convert with ten to the dB over ten, first, every time. 00:49:54.437 --> 00:50:00.366 A plain ratio — straight into the formula. 00:50:00.416 --> 00:50:01.826 Fourth checkpoint. 00:50:01.876 --> 00:50:04.326 Pause, paper, calculator. 00:50:04.376 --> 00:50:05.586 One. 00:50:05.636 --> 00:50:12.996 A channel has B equal to two megahertz and SNR equal to sixty-three — a plain ratio. 00:50:13.046 --> 00:50:14.576 What is the capacity? 00:50:14.626 --> 00:50:15.776 Two. 00:50:15.826 --> 00:50:20.006 A question says 'the SNR is thirty dB'. 00:50:20.056 --> 00:50:22.606 What number goes into Shannon's formula? 00:50:22.656 --> 00:50:23.836 Three. 00:50:23.886 --> 00:50:28.976 What is absent from Shannon's formula — and why does that absence matter? 00:50:29.026 --> 00:50:33.066 Pause now. 00:50:33.116 --> 00:50:34.556 Answers. 00:50:34.606 --> 00:50:42.826 One: one plus sixty-three is sixty-four, which is two to the sixth, so the log is exactly six. 00:50:42.876 --> 00:50:47.716 Two times ten to the sixth times six — twelve megabits per second. 00:50:47.766 --> 00:50:54.456 Notice Forouzan's habit of choosing SNRs that land on powers of two; real exams share it. 00:50:54.506 --> 00:50:58.436 Two: ten to the thirty over ten — one thousand. 00:50:58.486 --> 00:51:02.926 Then the capacity is B times log two of one thousand and one. 00:51:02.976 --> 00:51:07.426 And three: L, the number of levels, is absent. 00:51:07.476 --> 00:51:14.736 Your scheme gets no vote — the capacity is a property of the line itself, and no choice of levels gets past it. 00:51:14.786 --> 00:51:20.639 That is what makes it a ceiling rather than a recipe. 00:51:20.689 --> 00:51:22.099 Section five. 00:51:22.149 --> 00:51:27.369 Two formulas now sit on your desk — so which one do you reach for, and when? 00:51:27.419 --> 00:51:31.079 The answer is: usually both, in a fixed order. 00:51:31.129 --> 00:51:35.289 Shannon sets the speed limit; Nyquist picks the gear. 00:51:35.339 --> 00:51:42.439 And once that is second nature, we pay off the question from minute one. 00:51:42.489 --> 00:51:50.699 In the demo I asked you to guess a rule: some relationship between how noisy the line is and how many levels you are allowed. 00:51:50.749 --> 00:51:56.809 Here is the derivation, four lines, and it ties the whole session together. 00:51:56.859 --> 00:52:05.809 The idea: my scheme's bit rate cannot exceed the line's capacity — so set Nyquist equal to Shannon and see what L that permits. 00:52:07.129 --> 00:52:08.299 Line one. 00:52:08.349 --> 00:52:14.479 Two B log two of L — Nyquist — equals B log two of one plus SNR — Shannon. 00:52:14.529 --> 00:52:15.909 Line two. 00:52:15.959 --> 00:52:19.829 B appears on both sides; divide it out. 00:52:19.879 --> 00:52:22.729 Two log two of L equals log two of one plus SNR. 00:52:22.779 --> 00:52:26.179 Notice what just happened: the bandwidth vanished. 00:52:26.229 --> 00:52:30.719 This rule will not care how wide the channel is. 00:52:30.769 --> 00:52:32.349 Line three. 00:52:32.399 --> 00:52:37.239 Two log two of L is log two of L squared — a log times two is the log of the square. 00:52:37.289 --> 00:52:42.389 Equal logs mean equal arguments: L squared equals one plus SNR. 00:52:42.439 --> 00:52:46.169 So: L equals the square root of one plus SNR. 00:52:46.219 --> 00:52:51.519 That is the rule the slider was following. 00:52:51.569 --> 00:52:58.569 The noise decides how many levels the line tolerates — and it decides it through a square root of the SNR. 00:52:58.619 --> 00:53:01.529 Check it against the numbers. 00:53:01.579 --> 00:53:10.069 SNR sixty-three: L is the square root of sixty-four — eight levels at most, three bits per change. 00:53:10.119 --> 00:53:19.069 And in the demo, at twenty dB — an SNR of one hundred — the bands closed at about eleven levels: the square root of one hundred one is just over ten. 00:53:22.409 --> 00:53:29.339 This derivation will be on the midterm, all four lines of it. 00:53:29.389 --> 00:53:35.289 Now the question an engineer actually gets asked — Forouzan example three point four one. 00:53:35.339 --> 00:53:37.619 Nobody says 'compute Shannon'. 00:53:37.669 --> 00:53:41.289 They say: here is a channel — design me a scheme. 00:53:41.339 --> 00:53:44.889 The channel has B equal to one megahertz and SNR equal to sixty-three. 00:53:44.939 --> 00:53:53.519 Three questions: the hard ceiling; a sensible working bit rate; and the number of signal levels to run it. 00:53:53.569 --> 00:54:00.429 Pause the video and work all three — four minutes. 00:54:00.479 --> 00:54:07.169 This is the exam's favorite shape of question, because it makes you use both formulas in the right order. 00:54:07.219 --> 00:54:11.229 Step one — Shannon first, for the ceiling. 00:54:11.279 --> 00:54:16.219 C equals ten to the sixth times log two of one plus sixty-three. 00:54:16.269 --> 00:54:20.869 Sixty-four is two to the sixth: the log is exactly six. 00:54:20.919 --> 00:54:22.769 Six megabits per second. 00:54:22.819 --> 00:54:26.499 That is the most this channel will ever do, for anyone. 00:54:26.549 --> 00:54:28.659 Step two — back off. 00:54:28.709 --> 00:54:37.659 The Shannon limit is an ideal you approach, not a setting you dial in; running at the ceiling requires infinitely elaborate coding. 00:54:38.159 --> 00:54:43.479 So choose a working rate comfortably below it — say four megabits per second. 00:54:43.529 --> 00:54:47.889 Step three — Nyquist, for the levels. 00:54:47.939 --> 00:54:56.889 Four times ten to the sixth equals two times ten to the sixth times log two of L. So log two of L is two: each change carries two bits, and L is four levels. 00:54:58.049 --> 00:55:06.999 A real, buildable scheme: one megahertz of bandwidth, two million changes per second, four levels, four megabits. 00:55:07.229 --> 00:55:16.179 And there is the division of labor in one sentence — the sentence to keep: Shannon tells you the speed limit. 00:55:16.619 --> 00:55:22.186 Nyquist tells you what gear to be in. 00:55:22.236 --> 00:55:31.186 Let me compress that into the rule you take into the exam hall, because the first thing to do with any Chapter 3 word problem is decide which formula it is asking for — and the problem tells you, through what it gives you. 00:55:38.976 --> 00:55:42.816 If the question gives you SNR, it wants Shannon. 00:55:42.866 --> 00:55:51.816 It is asking about a real, noisy line, and the ceiling of a real line is B log two of one plus SNR — converting from dB first if the sentence says dB. 00:55:52.636 --> 00:55:55.996 If the question gives you L, it wants Nyquist. 00:55:56.046 --> 00:56:03.096 It is asking what a particular scheme achieves, on the noiseless idealization: two B log two of L. 00:56:03.146 --> 00:56:07.576 And if it gives you both, it wants both — in that order. 00:56:07.626 --> 00:56:13.526 Shannon first for the ceiling, then Nyquist to pick the levels that run safely under it. 00:56:13.576 --> 00:56:16.176 Exactly what we did on the last slide. 00:56:16.226 --> 00:56:20.496 One line underneath the rule: Nyquist is a design. 00:56:20.546 --> 00:56:21.956 Shannon is a law. 00:56:22.006 --> 00:56:26.056 You can argue with a design — add levels, add cleverness. 00:56:26.106 --> 00:56:30.786 You cannot argue with a law. 00:56:31.316 --> 00:56:39.116 Two formulas, two jobs — and one tool that runs them side by side so you can see which one is binding. 00:56:39.166 --> 00:56:42.616 The picture at the top is the important part. 00:56:42.666 --> 00:56:48.626 Those horizontal bands are the levels as the receiver actually measures them, smeared by noise. 00:56:48.676 --> 00:56:52.806 While the bands are separate, the receiver can tell them apart. 00:56:52.856 --> 00:57:01.506 Here is the example from a few slides ago: three thousand hertz, two levels, thirty-five decibels of signal-to-noise. 00:57:01.556 --> 00:57:04.886 Nyquist asks for six kilobits a second. 00:57:04.936 --> 00:57:07.886 Shannon permits thirty-four point nine. 00:57:07.936 --> 00:57:16.716 Headroom, twenty-eight point nine kilobits — so the scheme is achievable, and there is room to be greedier. 00:57:16.766 --> 00:57:21.906 So let us be greedy, exactly the way the exam invites you to. 00:57:21.956 --> 00:57:27.406 Twenty thousand hertz, sixty-four levels, and a signal-to-noise ratio of only ten decibels. 00:57:27.456 --> 00:57:36.406 Nyquist says two hundred and forty kilobits per second, and Nyquist is not lying — that is genuinely what two B log two L computes. 00:57:36.796 --> 00:57:39.836 Now look at the picture. 00:57:39.886 --> 00:57:42.926 The bands have merged into one solid block. 00:57:42.976 --> 00:57:51.926 The receiver cannot tell those levels apart, and the tool says so in one line: this channel physically cannot carry more than sixty-nine point two kilobits, so drop to three levels or fewer. 00:57:55.856 --> 00:57:59.256 That is the whole relationship between the two formulas. 00:57:59.306 --> 00:58:02.336 Nyquist tells you what your scheme asks for. 00:58:02.386 --> 00:58:05.436 Shannon tells you what the channel will tolerate. 00:58:05.486 --> 00:58:14.436 And it works the other way too — clean the line up, and the bands separate again, the ceiling rises, and more levels become honest. 00:58:14.736 --> 00:58:23.686 Take it to the settings from the question I asked in minute one, and you will see exactly why nobody ever sold you a faster modem. 00:58:25.370 --> 00:58:26.290 Right. 00:58:26.340 --> 00:58:27.380 Minute one. 00:58:27.430 --> 00:58:34.710 Dial-up modems stopped improving at about thirty-four kilobits per second, and I asked you to name the killer: the wire, the chip, or physics. 00:58:34.760 --> 00:58:36.800 You wrote something down. 00:58:36.850 --> 00:58:38.390 Let us settle it. 00:58:38.440 --> 00:58:43.770 The math is the calculation you have already done. 00:58:43.820 --> 00:58:52.770 A telephone line: bandwidth about three thousand hertz, SNR about thirty-five dB — which is the plain ratio three thousand one hundred sixty-two. 00:58:53.960 --> 00:59:02.910 Shannon: three thousand times log two of three thousand one hundred sixty-three — about thirty-four thousand, eight hundred and eighty bits per second. 00:59:03.720 --> 00:59:08.150 The ceiling of a telephone line is just under thirty-five kilobits. 00:59:08.200 --> 00:59:17.150 And now look at where the modems stopped: the best analog modems ever shipped ran at thirty-three point six kilobits — parked as close under the ceiling as engineering could get. 00:59:18.990 --> 00:59:24.210 Nobody built a faster one because a faster one did not exist to be built. 00:59:24.260 --> 00:59:30.460 So the verdict: not the wire — the wire was carrying everything Shannon permitted. 00:59:30.510 --> 00:59:34.870 Not the chip — the chips were fine, and better chips changed nothing. 00:59:34.920 --> 00:59:36.630 Physics said no. 00:59:36.680 --> 00:59:41.400 But some of you have already objected: what about fifty-six k? 00:59:41.450 --> 00:59:46.060 The famous five-six modems — do they not break the ceiling we just computed? 00:59:46.110 --> 00:59:50.600 No — and the way they did not break it is the best lesson in this session. 00:59:50.650 --> 00:59:53.900 Engineers changed the physics, not the math. 00:59:53.950 --> 01:00:02.900 The downstream leg — from your internet provider toward your house — stopped being analog: it went digital at the telephone exchange, so the path had one noisy analog leg instead of two. 01:00:05.780 --> 01:00:09.720 Less noise means higher SNR — and a higher SNR moves the ceiling itself. 01:00:09.770 --> 01:00:13.010 Same formula, new inputs, new limit. 01:00:13.060 --> 01:00:22.010 Upstream, from your house, stayed fully analog — which is why upstream stayed at thirty-three point six forever, even on a fifty-six k modem. 01:00:22.530 --> 01:00:27.100 So the lesson of the whole session, in one line: you don't beat Shannon. 01:00:27.150 --> 01:00:30.640 You go and change the SNR. 01:00:30.690 --> 01:00:34.730 If your guess at minute one was 'physics' — take the credit. 01:00:34.780 --> 01:00:43.730 And if it was 'convenience of the engineers' or 'the chip' — you now own the formula that proves otherwise, which is worth more than the guess. 01:00:48.334 --> 01:00:55.664 Before the recap, the three mistakes of the second half — fast, ten seconds each, all of them real. 01:00:55.714 --> 01:00:56.984 One. 01:00:57.034 --> 01:01:04.664 Plugging SNR-dB straight into Shannon: C equals B log two of one plus thirty-six. 01:01:04.714 --> 01:01:12.764 No — thirty-six dB converts first: ten to the three point six, about three thousand nine hundred eighty-one. 01:01:12.814 --> 01:01:16.264 Then log two of three thousand nine hundred eighty-two. 01:01:16.314 --> 01:01:17.474 Two. 01:01:17.524 --> 01:01:23.074 Using Nyquist to find a real channel's ceiling — 'just add levels'. 01:01:23.124 --> 01:01:24.944 Real channels have noise. 01:01:24.994 --> 01:01:30.224 The ceiling is Shannon's; Nyquist only picks the L that runs under it. 01:01:30.274 --> 01:01:31.444 Three. 01:01:31.494 --> 01:01:39.734 Using hertz and bits per second interchangeably — 'three thousand hertz equals three thousand bits per second'. 01:01:39.784 --> 01:01:47.454 Hertz is the wire's property; bits per second is your result; the two formulas are the only bridge between them. 01:01:47.504 --> 01:01:55.728 If you can catch all three in someone else's work, you are ready for this week's online quiz. 01:01:55.778 --> 01:01:59.258 Let me name exactly what you should be able to do now. 01:01:59.308 --> 01:02:08.258 Tell the three impairments apart by what arrives: smaller — attenuation; misshapen — distortion; with company — noise. 01:02:09.288 --> 01:02:11.238 Work in decibels. 01:02:11.288 --> 01:02:20.238 Convert between a power ratio and dB in both directions, sum any chain of gains and losses in your head, and say why the unit adds: logarithms turn multiplication into addition. 01:02:24.978 --> 01:02:33.928 Compute SNR in both languages — plain ratio and dB — with the units matched first, and explain why an amplifier cannot improve it. 01:02:35.838 --> 01:02:44.788 Apply Nyquist: bit rate equals two B log two of L, on a noiseless channel — and say why that formula cannot cap a real one. 01:02:46.468 --> 01:02:55.418 And apply Shannon: capacity equals B log two of one plus SNR, with SNR as a plain ratio — and use the two together, ceiling first, then levels, including the bridge L equals the square root of one plus SNR. 01:03:03.868 --> 01:03:08.098 If any of those feel shaky, the section that covers it is still there. 01:03:08.148 --> 01:03:13.461 This is a video — use it as one. 01:03:13.511 --> 01:03:15.241 Five last questions. 01:03:15.291 --> 01:03:20.901 Pause and answer them on paper — calculator allowed, and needed. 01:03:20.951 --> 01:03:21.151 One. 01:03:21.201 --> 01:03:25.091 A cable loses minus two dB per kilometer. 01:03:25.141 --> 01:03:31.111 After four kilometers: the net dB, and roughly what fraction of the power survives? 01:03:31.161 --> 01:03:32.301 Two. 01:03:32.351 --> 01:03:36.711 An exam question says the SNR is thirty dB. 01:03:36.761 --> 01:03:39.311 What number goes into Shannon's formula? 01:03:39.361 --> 01:03:40.521 Three. 01:03:40.571 --> 01:03:45.971 A noiseless channel, B equal to four kilohertz, eight levels. 01:03:46.021 --> 01:03:47.591 Maximum bit rate? 01:03:47.641 --> 01:03:48.871 Four. 01:03:48.921 --> 01:03:51.531 B equal to one megahertz, SNR equal to fifteen. 01:03:51.581 --> 01:03:57.491 The capacity — and the number of levels that reaches it? 01:03:57.541 --> 01:03:58.811 Five. 01:03:58.861 --> 01:04:02.731 A question hands you both L and SNR. 01:04:02.781 --> 01:04:04.661 Which formula do you use? 01:04:04.711 --> 01:04:08.771 Pause now. 01:04:08.821 --> 01:04:11.891 Take your time on these. 01:04:11.941 --> 01:04:12.621 Answers. 01:04:12.671 --> 01:04:21.171 One — minus eight dB, and ten to the minus nought point eight is about nought point one six: roughly a sixth survives. 01:04:21.221 --> 01:04:25.481 Two — ten to the three: one thousand, never thirty. 01:04:25.531 --> 01:04:32.051 Three — two times four thousand times three: twenty-four thousand bits per second. 01:04:32.101 --> 01:04:41.051 Four — one plus fifteen is sixteen, log two is four, so four megabits per second — and L is the square root of sixteen: four levels. 01:04:42.391 --> 01:04:48.081 Five — both: Shannon first for the ceiling, then Nyquist for the levels. 01:04:48.131 --> 01:04:55.608 If you got all five, you are ready for anything Chapter 3 can ask you. 01:04:55.658 --> 01:04:57.168 So take this with you. 01:04:57.218 --> 01:04:59.448 The wire always takes its cut. 01:04:59.498 --> 01:05:05.648 The decibel is how you keep the books — and Shannon sets the ceiling that no cleverness moves. 01:05:05.698 --> 01:05:12.738 Everything numeric we did today was one trick wearing four costumes: take a ratio, take its log. 01:05:12.788 --> 01:05:19.118 Before the next session, read Forouzan section three point six — performance. 01:05:19.168 --> 01:05:24.408 Throughput, latency and the bandwidth-delay product are Session 6's whole story. 01:05:24.458 --> 01:05:32.848 Keep the calculator habit: every log two via natural log over natural log of two, until your hands do it without you. 01:05:32.898 --> 01:05:34.918 And try the two demos at home. 01:05:34.968 --> 01:05:40.448 Build a dB chain that loses exactly minus ten dB, three different ways. 01:05:40.498 --> 01:05:47.038 Then find the largest L a twenty-dB line will tolerate — and check it against the square root rule. 01:05:47.088 --> 01:05:49.088 And one date for the diary. 01:05:49.138 --> 01:05:57.958 Weekly Online Quiz A2 is live on the course website now — twenty minutes, one attempt, and it closes on Saturday at midnight. 01:05:58.008 --> 01:06:02.634 See you in Session 6.