 Okay, so this is lecture one for EE419 which is digital communications, okay so there was a room change I guess all of you know about it so you are here and can you see it from the very last, can you see what is being written here, it is fine, if not there are some seats available up here also, okay so that is the choice, okay so let us start so a couple of things before I begin there are actually one course which is listed as prerequisite for this course which is 356, right all of you here have done it, okay so I'll need two other prerequisites, okay so I think both of those are core for you so it should be okay. The first one is signals and systems, I don't know the course number so I don't know it's called network systems, okay so I'm going to say signals and systems, whatever that course number is and the next one would be the digital signal processing which is again something you have done, I think the course is called analog and digital signal processing and depending on who does it you get some version of it which is fine and the next thing I would put down is I would say probability and random processes, okay so these are the three prerequisites that I will assume, okay so I will briefly state some results in these areas mainly for notation just to fix notation for us nothing more, okay so I will not go into detail and I'm going to assume that you are you're fairly familiar with this right now, okay you might have read it when you did your second semester or third semester and you might have forgotten that is the case my strong suggestion is please go back and revise, okay so I will not do that revision for you here and this course will go way over your head if these those concepts are not clear in your head, okay so I would say good familiarity with these things, okay so any questions on the prerequisite something that somebody has not done, okay for instance there was a student who came and said he didn't do 356 but he knew probability and random process I asked him two or three questions about Gaussian random variables and he didn't know, okay so if you don't know enough about Gaussian random variables you're going to be in big trouble in this class, okay so all those things are emphasized a lot so I'm not going to repeat those things here in terms of revision or anything like that, okay so I'm going to assume you know enough about Gaussian random variables as far as probability and random process concern pretty much that's the only thing you need there then in DSP Z transforms or Z transforms and all those things should be very clear DTFT should be very clear to you, okay signals and systems in general you should know, okay so those are the things we'll require, no questions on prerequisites, okay it's fine so that's 419 and there's also there's also a course that's running in parallel with that which is the lab course for dual degree students, right how many of you here are dual degree, okay so I think there's supposed to be 18 and class strength is supposed to be 52 it's the list I have but not many people are here, okay so that's I think 471, okay so I don't know the name I think it's called advanced communications lab so I'm going to run it along with this course and it'll pretty much run in parallel, okay so I might in fact do most of the lectures in this course and then do the corresponding lab there and students who are not dual degree are also welcome to do the lab most of the lab information will be up on the websites so you might be able to do it it's you don't require any equipment all you need is a PC and a few other things if you have that you can try it you're welcome to try the lab the only thing I won't do is I won't be grading everybody okay so I'll be only grading the dual degree students on the lab and that's a lab that runs along with this, okay so the lab will not start at least for the next two weeks, okay so it will start only after two weeks for the first two weeks it's only lectures, okay so the 419 lectures are part of the lab okay so don't think of that as something else and this is some other course it's part of the lab, okay if you don't attend the 419 lectures you just show up for the lab you'll probably fail the lab, okay I don't know if you know the story about the last time I ran this lab course, okay a lot of people must have maybe spoken to you about it people got very bad grades, okay so I insist on individual work and if you don't do individual work I divide by the number of students who did the work, okay so it's you can get very bad grades in the lab if you do such things, okay so that's happened last time so one more warning to dual degree students particularly make sure you keep up to speed with 419 because if you don't do this the lab will also suffer, okay so that's as far as the introductions is concerned we'll have some TAs we're not here right now so we'll have some TAs for tutorial classes and other things and for the lab as well, okay so as far as grading is concerned it's pretty much going to be quiz one plus quiz two plus finals, okay so this will be 25% 25% 50% okay so typically I give an option here I'll come to it later whenever we get there so there might be an option which involves some programming assignment type thing instead of quiz two later on, okay only for only for selected students okay it's usually not open to everybody only students who I think have shown enough promise so far in the course I've attended enough lectures to do this will be allowed to do an option for quiz two, okay so that's otherwise just typical finals there will be some tutorials and homeworks which I will assign periodically and I will strongly suggest that you do it because most of my quizzes and finals there is a certain fraction which shows up just from the tutorials so if you just do the tutorials it's a good chance you'll do well in the exams as well, okay so as far as textbooks are concerned the main textbook I'll use is digital communications by okay I'll write down the last name of the three authors the three authors are Barry Lee and and Mesha Smith okay so I'll just write down Mesha dot dot dot okay so Mesha Smith it's it's available in cheap editions so if you go to Tata book house or any other bookstore that you like you can buy this book it costs about I think 300 rupees or so it's not so expensive not too expensive I would strongly suggest that you buy it if you for all your interested in digital communications and you want to do it for a long time in your life this is a good book to buy okay so keep that in mind there are a couple of other books I'll just write down the authors here okay so not the entire book because those are not the main books for this course the other two books are the book by Prohacus okay and then there's a book by Mado Upamanya Mado okay those are the two books that I will use as well for the course but you don't have to buy it okay so any questions things that I missed out anything that's usually said in the first class that you want here fine okay so so that's it so one more thing is we meet we meet in a slot okay okay so this a slot is okay or more or less except for the Monday morning 8 a.m. class okay so I think what are the other times I think it's Monday 8 a.m. and Tuesday 11 a.m. and Wednesday 9 a.m. and then Thursday what 1 p.m. right am I right okay so how many of you doing labs on Thursday what labs on Thursday haven't been decided okay so I guess it's something we can't avoid okay so in case it turns out that none of you are doing any lab on Thursday I'm okay with extending the Thursday are from 1 to 3 and skipping the Monday morning 8 a.m. class okay if that happens but I don't want you to push for it or anything like that okay so if it happens that way then we can do that otherwise we'll just we'll just meet at 8 a.m. on Monday morning okay so that's sorry what time 2 o'clock usually it's around 3 or something find out if it's free then we can do it okay otherwise it's not possible okay and that's the only time all right so of course attendance and all these things are there so I'm going to pass out pass around a book which has your names here find your name and then I have to write down the date okay so maybe the first day you can write the date on top okay then put a tick mark next to your name okay so I think that's it I don't have anything more to say the quizzes will be at the regular time final will be at the regular time I'm okay beyond all right so let's begin and this lecture will mostly be overall high level introduction of the kind of things that we'll see in this course and to give you a feel for how this course is probably ties up a lot of the things that you've been learning so far it takes you towards description of communication okay so introduction is pretty much the main topic of this lecture okay so there's no real need to motivate digital communication today all of you have pretty much grown up with the I mean you knew the internet as you grew up so okay so you know digital communication is part and parcel of most of your lifestyles today okay so including cell phones and all these things so there's no need to motivate digital communication and maybe maybe what you what you need motivation for is the need to understand the basics of what's going on there okay so there is there is some there is some maturity to this field in the sense that today you have a lot of these systems well-built right you're able to use internet without knowing anything about digital communication okay so why should you know okay that's the question that you might ask and the answer to that maybe it's because it's not really clear okay you've already opted for the course which means you want to learn about it okay so we'll be dealing with digital communication at a much more fundamental level than internet okay internet is supposed to be a very high level at work where you transmit packets receive packets programs can do these things we'll be going much lower okay to the what's called the physical layer pretty much we'll deal with how to physically send bits from one one part one point in us in a network to another point okay so it's we'll be dealing with physical layer so that's a that's much more mathematical and much more intense than you can imagine and the internet can become okay so the setting roughly in most cases will be the following okay so you have a transmitter okay which wants to send a sequence of bits okay so how do I think of a sequence of bits it's just a sequence of bits right so 1 0 0 1 1 0 okay so it's a sequence of bits the transmitter wants to send and what are the resources it has access to one can imagine it has access to electronic circuitry right it can build some electronic circuitry and then there's a pair of wires that connects it to a receiver on the other side I'm going to start with a pair of wires okay so in future maybe we'll think of other other channels so to speak other than pair of wires but pair of wires is a very fundamental nice thing to start with so there's a pair of wires that connects transmitter to a receiver okay so you have to imagine that these guys want to build electronic circuits and all that so that they can send these bits using this pair of wires from the transmitter to the receiver okay so there are some goals what are the goals what would you like on the receiver okay very simple goal to think of is all the bits should be accurately reproduced right so it should be no error okay so you want no errors as your goal okay so all these things are very clear and obvious but still I would like to emphasize these notions okay so if you go back to the very classic early papers and digital communication these processes were clearly defined okay so for instance communication was defined as a process of transmitting information in a way so that there's no error and all those things okay so just for completion I'm defining this okay so how would you go about doing it if if this weather this were an issue how would you go about doing it you would imagine that this pair of wires is really really long okay so it's not like one feet it's one feet then there's really no point in communication okay so it's very very long and you want to communicate at a reasonable speed you don't want to wait for a long long time okay so how would you go about doing it the first thing that comes to mind is you want to convert the bits into some circuit quantity right and then convert send that circuit quantity in along using the wire to the other side okay the circuit quantities you're familiar with are voltage or current okay so we'll use both maybe I will never refer to voltage and circuit as we go along in the future but roughly think of it that way okay I want to represent everything I have as voltage and current and then transmit that using the pair of wires to the received side okay so so that's the first task okay so one way of doing it on the transmitter side a very simple first cut elementary way of doing it without any without bothering about too too many things is to say I'll take bit one okay and then represent it as a say a 5 volt voltage level okay so something like 5 volts this 5 can change it can be 3 volts 3.3 volts 2.5 volts whatever number you want okay so 5 volts I'll say bits 0 is say 0 volts okay so maybe maybe this is not a very good choice maybe you want plus 5 and minus 5 okay why would you want plus 5 and minus 5 as opposed to plus 5 and 0 any reason why you might want that okay so think about these kind of things as you go along maybe maybe I'll emphasize these things later okay so maybe you want this suppose you want this then suppose you do this then what happens your sequence of bits becomes sequence of voltage levels okay so now we have to hold that voltage level at 5 volts for a reasonable amount of time because if you don't hold it for a reasonable amount of time there's no chance that it'll ever propagate far away to some distance point on a pair of wires okay so so so there's a time period involved for each bit okay so I'm gonna say bit 1 is 5 volts for say t seconds okay and then bits 0 is 0 volts for again t seconds okay so once I do it I get actually a voltage waveform corresponding to my bit sequence okay so this process is pretty much the first step that one can visualize in doing a communication of bits from one side to the one from a transmitter to the receiver okay so I'm gonna call that as they say step 1 okay what do I do in step one I convert bits to it's too noisy I don't know I don't have access to the switches but I need this fan okay yeah I think your fans you can switch off okay okay okay so so converts bit convert bits to I'll say signal okay some signal okay so signal what is the signal signal is it's like a voltage waveform or a current waveform okay so that's so you think about it for instance if I have this bit sequence 1 0 0 1 1 0 and I use this kind of a conversion right bit 1 going to 5 volts and bit 0 going to 0 volts right so I will how will the waveform look how will the waveform look it's quite trivial right it's going to be some kind of a rectangular waveform okay so I'll start with time 0 so I will get 2 t here 2 t here 3 t here 4 t here so on okay and this is 5 volts so I can I can set this as my signal that I want to transmit okay so that's the first step and this the step seems simple enough right but there are a lot of lot of careful questions you have to ask at this point okay so when you build electrical systems there are so many things you have to worry about a signal okay right when you when you think of signal there are so many so many ways to quantify it and think about it you have to be very careful right what are the things you might want to worry about in a signal what are the things you might worry about I'm sorry yes okay what happens at the receiver is the question but even before going to the receiver at the transmitter itself when you look at the signal you should be worried about something when you build a circuit to support such a signal what are the things you're worried about I'm sorry switching okay so okay first thing to be worried about is power levels right current and voltage and all these things are significant power is very significant right depending on what power it is you may or may not be able to build it in a certain type of circuit right so you have to worry about power okay so power in a signal is very important okay so that's the first thing you might worry about in the signal oftentimes you'll be power limited since the total amount of power available to you at the transmitter for transmission towards the receiver might be limited okay it may not be like thousands of megawatts right so you may not have access to that kind of power okay there'll be some limitations on how much power you can use for communications the governments might limit limited even otherwise some other basic fundamental physics might limited okay depending on the depending on the system okay so power will usually be something of a constraint you will be constrained by it so what happens typically is when you want to drive signals over a long distance you have to boost their powers up right so you'll use something called a power amplifier at the very end okay which will push your signal to a high enough power so that like he pointed out it can get to it can get to the receiver at some level okay otherwise the power at the receiver will be almost zero and then you can't detect it at all okay so to do that you'll push it up and these power amplifiers are typically very very expensive okay to build a what's called a linear power amplifier over a range of frequencies very difficult it's very expensive so based on your budget also you might be limited so power in general is a very precious commodity in communications even at the transmitter side so so whenever you build a signal like this you might worry about power okay that's the first thing what else will you worry about in a signal what are the attribute of the signal you might you might want to control the transmitter anything else the time period of the signal okay so in effect what you're trying to say is I'm sorry the rate yeah okay yeah the rate is another point right so but that's yeah that's connected to the signal as well you might want a rate that is as high as possible you don't want to keep transmitting at a very low rate then your bits will not go through at a fast enough rate so what's the rate here in this picture 1 by T bits per second right so the rate is something that you're worried about rate of information transfer which is in this case 1 by T bits per second okay so anything else okay so all these things are captured by one notion of signals what is that the bandwidth and the frequency representation good looks like signals and systems has gotten through somewhere okay so anytime you have a signal like this you're worried about its frequency content okay what is the frequency content of the signal what bandwidth does it occupy what shape does it have in that bandwidth why should you be worried about such things okay so many systems you build react differently to different signals based on what their frequency content is okay for instance this power amplifier right typically when you build a power amplifier you think the amplifier is a transfer function which looks like which is a constant over all frequencies it's very it's pretty much impossible to build such amplifiers okay it'll be only constant over certain frequencies okay so all everything you build depends on your frequency content of this signal okay so clearly you're worried about the what what shall I say it's not the frequency the spectrum of the signal okay so so maybe you're convinced already but I'll give you more reasons why for why why you should be worried about the spectrum because so right now it's a pair of wires and whatever bandwidth you want to use is available to you okay but if you imagine a band of radio frequencies and you're using the RF range for transmission then you don't necessarily own the spectrum who owns the spectrum sorry the government owns the spectrum okay so and what do they do they lease it out to companies and whoever wants to use it there are some spectrum which is free and open but there are lots of regulations for that and on top of that additional spectrum which is leased to companies and you can't use any spectrum that you want okay any RF transmission should be licensed suitably okay once you're licensed for a certain spectrum there is a certain mask that people will expect they'll put on top of it and expect you to adhere to that mask as far as spectrum is concerned okay your spectrum should go down to a certain db power at certain frequencies there'll be some dependency so your bandwidth of the signal that you're putting out needs to be carefully controlled otherwise it may not even be legal okay you might end up in jail okay so it's a serious problem okay so bad spectrum is something you're to be very very worried about okay so how do you find spectrum for this signal are you aware of tools can do you know how to find the spectrum of a signal like this first of all what kind of a signal is this is it a okay so is it the thing I'm looking for is deterministic versus random okay so can you say it's a deterministic signal no why yeah because the bits are random right if the bits are deterministic then there's really no no point in communicating them even the receiver might know it okay so the bits are random coming into the transmitter and you don't know what they are and depending on what the bits are you will get different signals out okay so there's no point in computing one spectrum for one set of bits okay typically what do you do okay this is what you might have learned in 356 you do probability and random process how to compute spectrum for yeah some well random processes defined in a careful way so you can define this to be a certain kind of random process for which spectrum meaningfully exists okay what is the kind I'm talking about okay I think you guys are going to have some serious trouble okay go back and at least read the 356 notes once again okay so you need the random process to be what's called stationary in the in the weak sense okay and in fact this this will not be this can be shown to be cyclo stationary and all these things and then you then you find spectrum for it you can define what's called the autocorrelation which is an average of certain things and then then you define the spectrum for that okay so it's possible to think of a spectrum for this and the spectrum will be closely connected to the spectrum of what what do you think the spectrum will be connected to yeah well ultimately that will be what shape sync right why is it a sync it'll it'll be the spectrum of the rect of the rect signal 0 to t it'll be related to that it'll be a scaled version of that it's not not be significantly different from that this is what you must have seen in 356 if you do not see it go back and look at this very closely okay so these bits are random which makes the signal itself random and to find its spectrum carefully you need some knowledge of how the spectrum is defined okay what what how do you define these things okay but it's possible to find it hopefully you have tools for doing that okay so couple of relationships I want to point out okay one relationship between what are the relationship between these three quantities power rate and spectrum okay power and rate maybe it's not clear right now okay we'll go back and relate it later on I'll give you a relationship which is very nice but rate and spectrum you can quickly see a relationship can you see I'm sorry yeah so it'll be kind of linearly kind of relative if you want more rate what will you have to use we'll have to use more spectrum right so your t is going to come down which means your sync is going to expand more and more on the on the spectrum side okay so so that at least this relationship is clear based on basic signals and systems and you have to use a lot of advanced communication theory as I've so to speak to derive the other relationship what's the relationship between power and rate and spectrum and all this it's a little bit more complicated but maybe we'll see it as we go along okay so so as you see the first step itself which is probably the simplest step that you can think of is has a lot of complications okay so it's not very clear how to how to do it in a perfect way okay suppose I say my spectrum has to look in a certain way okay how can I do this conversion from pitch to signal it's not clear okay right now the spectrum is going to be a sink what if I don't want to sink okay the sink there is a problem right sink dies very slowly right it dies as one by frequency you may not want that okay maybe you want something which dies faster okay but then you have to do something else okay so all these things all these things we'll see as we go along in the first so that's the first step and this is typically called what okay at least in this course we'll call it modulation okay all right so that's the that's the first part okay all right so so the next step next step after you get a signal is so basically it's the transmission step okay so you transmit the signal and basically on the pair of wires to get a received signal what I call a received signal okay so this is where you have to you have to use a lot of okay this is where really a power of probabilistic modeling and system modeling comes in okay so this is where the big leap is made which kind of gives us a lot of freedom and flexibility in the design process okay so I'll describe it it will sound very simple to you initially but later on you'll appreciate how this model beautifully captures the entire system and at the same time gives you a lot of flexibility and design okay so this is actually done on done for most channels but a very nice way of modeling a pair of wires is to say it is an LTI system what is an LTI system how do you expand LTI linear and time invariance system okay so seems like a simple thing to do but if you've learned enough about these long pair of wires you'll know it's not quite LTI okay there's lots of lots of things going on there there are reflections and to worry about all kinds of crazy stuff but but we'll say more or less it's a pair it's a LTI system there's no problem okay so once you have an LTI system how would you characterize how it behaves it's enough to specify something called the the impulse response or the frequency response okay so you specify the the impulse response okay so which I'll say maybe it's h of t okay which will in turn give rise to a frequency response represented as h of f okay so suppose I give you a pair of wires how will you go about finding h of f I'm sorry okay so that's one way of doing it but the sharp pulse is almost like a delta right so you want to keep putting more and more voltage into it you don't want to burn up the pair of wires okay so anything else which may or may not burn up the pair of wires sinusoid measurement right so it's the simplest thing you can do you can directly measure h of f you send a sinusoid and what do you expect on the other side yeah it's going to be scaled by the the magnitude of h of f for something okay so you may not worry about the phase it's possible to also find the phase response using some clever measurements but let's not worry about it at least the spectrum at the magnitude is important to us we'll assume linear phase and then kill the phase problem okay so mostly we'll be so that'll be a constant in this course we'll mostly worry only about the magnitude response okay the phase response will be assumed to be linear and even otherwise it's okay so once in a while I'll talk about it but usually the magnitude magnitude response will be our main problem at least in this course okay so so that's fine so I'm going to represent my think of my pair of wires as a LTI system okay so in general most communication channels can be very nicely modeled as LTI systems okay so that's that's that's a bit of a liberation you know I mean previously if you think of communications in the good old days physics was tied to it very strongly okay if you think of optics and people are worried about photons and all these things if you think of RF then people are worried about electromagnetics Laplace equation all these things those things are very difficult okay they're not that easy but once you abstract all the doubt and say I put a transmit signal what I get out on the other side is the transmit signal convolved with a certain impulse response it completely becomes a mathematical description okay and you're liberated from all the physics of the situation so you can develop a theory with a certain mathematical description and it'll apply equally well to several systems just depending on how your impulse response works out okay so that's the power of this model okay so you as you see as we go along you'll see this is this is a tremendous simplification it enables a lot of lot of smart things to happen around communications okay so that's the first simplification so that's the first thing okay so what's going to happen to your transmit signal as it goes along the pair of wires is it's going to get convolved with an impulse response that represents the pair of wires okay the LTI system model basically okay so but that's not the only thing that will happen okay if that were the only thing that happens then usually communications is not so serious the serious problem is because of noise okay so any electronics you build okay as long as it operates at a non-zero temperature will always produce noise okay there will be some noise that is produced by any electronics that you build okay so the at the receiver what are you going to have you're going to have some electronics to receive the signal okay so that's going to also produce noise okay so so that's the next problem the receiver noise okay okay so how do we so the pair of wires the impulse response we modeled as a convolution we'll usually model receiver noise as additive okay additive as in you have a transmit signal it's going to get convolved with the impulse response and then the noise is going to add to that okay that's how we model there are systems where noise doesn't add it there are systems where it multiplies and all that those are really crazy systems which people typically don't design too much for so we'll only design pretty much for additive noise at least in this course okay so putting these two together my model is going to look like this okay this is a very powerful equation it's very simple it looks very obvious maybe to most of you but it's very very powerful our model is going to look like this okay so let me box this it's an important enough equation okay this is my transmit signal this is my channel response okay i'll call it channel response channel impulse response this is my additive receiver noise okay all right so this will be our model pretty much for the whole of this course in fact we'll simplify this further at least as in the first few lectures but then later on we'll we'll try to attack this model we'll want to design communication systems pretty much under this model okay and like i said there are a lot of physical systems which will fit into this model people have used this model in pretty much all the communication systems that are out there and it has worked out very well okay so you can believe in this model you don't have to be very afraid of it okay you can feel you can happily use it okay so that's the model okay so let me talk a little bit more about this model just to give you a feel and let you think more about it first of all the transmit signal we talked about enough okay right it's it's going to be a random signal right and it's going to it's going to be a result of the modulation of the bits okay so that's the transmit signal what about the channel response what will it be okay can pretty much be anything right so it's difficult to say what the channel response is okay so yeah maybe some assumptions you can make but usually this will be at least in this course taken to be deterministic okay so we won't look at a random thing later on maybe when you learn about more complicated communication systems particularly wireless systems there'll be a reason why that will have to be random as well okay but for now we will at least in this course take it to be deterministic okay the issue then is who knows h of t okay who needs to know h of t and who knows h of t right you have two parties right you have the transmitter and then you have the receiver who do you think can know h of t receiver okay but receiver can know it only with some cooperation from the transmitter right so maybe then they can send it back and forth and all these things so typically the assumptions vary depending on what the channel is but depending on our convenience we'll say it's known at the transmitter as well as the receiver or maybe at the receiver but definitely at the receiver we'll assume you know h of t okay so those are assumptions that are made typically in systems and today you can measure it right you can send a sinusoid on one side and see what's what comes out on the other side and you can measure it so it's possible to measure it so at least the receiver can definitely know okay and there are systems where people assume that even the transmitter knows h of t and that's also feasible it's not too bad to imagine that you first have a training phase where you exchange all this information and then communicate at a really really fast rate initially you transmit communicate very slowly make sure you pass all this information around and then you do fast okay so those that's kind of known known everywhere what about the additive noise what do you do with the additive noise okay so you have to model it somehow right so you have to come up with a model okay so typically it's modeled as a random process okay random process with a with a certain distribution what do you think you'll assume this most natural distribution Gaussian random process okay so typically this is assumed as a Gaussian random process and we will make this assumption pretty much throughout this course in fact we will make a stronger assumption we'll say it's a white Gaussian random process what's white yeah well I'm going to say n of t1 and n of t2 are uncorrelated or independent okay so can I say both uncorrelated implies independent or not for the Gaussian case yes okay so it implies so I can say it's it's uncorrelated between time okay at whatever time interval you choose over close it is it's going to be uncorrelated set a reasonable assumption one can question it but in most cases it's reasonable because you're going to be working with a much smaller bandwidth than the bandwidth of this noise signals bandwidth of this noise signal is really really huge comes from some very basic physics and it's very very huge and you're working with a very short bandwidth and within your bandwidth and the time intervals at which you're sampling this will be independent pretty much you can assume assume that without any without any problem okay so that will be an assumption that we will make okay so I'll qualify these things as we go along and we'll be more precise but roughly this is how the system looks so how does the system look you have a certain bits that you want to transmit you're converting it into a certain transmit signal that signal is going to go through a channel which we model as a convolution and then some noise is going to get added to it and y of t is available at your receiver okay so this is your received signal okay okay so we'll make a lot of assumptions simplifying assumptions about h of t and n of t to make our process to make our at least study easy in the first few lectures and then we'll slowly relax those assumptions and study more and more general cases okay so that's how we'll proceed in this course but but this is the model this is the model that we'll use okay so this model has a name it's called the linear Gaussian channel linear additive Gaussian channel or linear Gaussian channel okay you can see why it's linear and why it's Gaussian it's easy enough to see okay so so the problem that we'll be dealing with this class can be given in this course can be given a very specific definition okay what is my definition I have a sequence of bits say some n bits I have to convert it into a certain x of t so that my receiver can recover those n bits error free from y of t which is x of t convolved with h of t plus n of t okay and I have to do that at as fast a rate as possible that's my problem okay so what's my problem okay communicate okay so communicate I'll put it within codes because bits from okay to two criteria as fast as possible if you can error free okay and the model is this okay the model is so so like you like you see I mean it's not it's it sounds like a very specific course in the sense that it's got a real target right it's a very simple course in the sense that you have just one problem in the whole course right x of t equals x of t convolved with h of t plus n of t there's nothing more to it right very simple sounding thing but there's lots of theory behind it as you read along you'll see one that's lots of theory behind this okay but the essentially it's simple we're not doing something we're not doing too many things at least we're doing very few things in this course okay so that's the that's the setup let me talk a little bit about the simplifying assumptions we'll make okay so so what's the ideal h of t that you can have okay so what's the ideal h of t you can have what would you like h of t to be ideally delta right so that's what you would like okay so but the problem with ideal ideal delta is what the spectrum then is infinite okay so then h of f has infinite bandwidth okay so so that's the case sometimes we'll consider okay once in a while we'll say suppose you have infinite bandwidth what would you do I mean I might ask that question somewhere maybe in your quiz right so some something like that we'll consider but typically it's not the case okay always you know you'll never have infinite bandwidth okay so so it's usually not possible so that's not a case it's good to consider but there's a nice nicer assumption to start off with which captures all the theory very nicely okay and that's the assumption we'll start off with usually okay so this is the simplifying assumption we'll use which is not as hypothetical as the delta case but it's not as complicated as having an arbitrary h of t which is what which is what we're looking for the simplifying assumption which is really good is the following okay so I'm going to say my h of f okay is going to be flat from minus w hertz to plus w hertz okay this is my h of f well it's I'm showing only the magnitude response okay so whenever I draw h of f and draw a one-dimensional graph it means obviously I'm showing only the magnitude okay the face might be something which I'm not showing okay and then maybe it dies down like this okay I really don't care okay but within minus w and plus w I'll assume h of f is flat okay and I'll assume the level is a some a actually we'll show later on this a is not very relevant to us we'll we'll kind of get rid of it usually we'll assume a is one okay it's it's it's you'll see later on it's not a significant problem to have a non-zero a as a non-unity a okay but we'll assume h of f is flat between minus w and plus w and we'll say my x of t will be such that what x of f is zero for outside w so I'm going to restrict my transmit signal to a bandwidth which is within the bandwidth for which my channel response is flat okay that's what I'm doing okay that's my simplifying assumption to start off with okay so there is there is something I'm giving up here what am I giving up here okay you can get a feel for it I'll give up something on rate right I'm going to say my transmit signal is going to be constrained on bandwidth I'm not going to allow it to have more and more bandwidth which means there is a certain maximum rate that I'm willing to transmit at that's fine okay we'll start with that assumption and then we'll see later on how to relax them okay but once you make this assumption things become very nice what happens once you make this assumption what will happen what will happen to my model h of t drops out right you see why it drops out once I say my x of t is within a bandwidth minus w to plus w and my h of f is flat there and I multiply by h of f I'm not doing anything okay it's just flat okay so it drops out and my model pretty much becomes what y of t equals x of t plus n of t of course this x of t has to satisfy this condition and h of t is actually present in that way okay it is telling you that your bandwidth of x of t has to be only between minus w and w and that's it after that it drops out okay what about the a the a I'm going to pretty much say is one okay I'm going to drop it out you'll see later on it doesn't matter okay even if it is not there can imagine some low noise amplifier amplifying your signal to whatever okay if you want to be something say something very specific in fact you don't even need that you'll see later on okay so this is this is a simplifying assumption and and it doesn't doesn't sound doesn't don't doesn't sound too bad okay so you're not assuming infinite bandwidth you're just saying bandwidth is within the region where my channel response is flat okay so that's my that's my assumption okay so this is a model which we will start off with and pretty much maybe at least for the first month of the course we'll work with this model okay maybe once in a while I'll go back to the other model but usually we'll stick to this model for the rest of this course okay so so you should also know what it means this assumption means that I'm giving up on a certain maximum rate okay maybe I can go faster but I'm not doing that I'm saying I'll restrict myself to what's available in the channel okay all right so once you have this let's move on to the last step what does the receive receiver do with y of t okay so that's step three okay when can the receiver go wrong in a very simple way at least for starting off with then we'll make it more rigorous as we go along okay so you can you can think of what can happen right so if your x of t is something like this okay I can't remember what my think I did this okay so something like this say this is your x of t okay so how can your y of t look how can your y of t look okay I'm going to draw two different possibilities for y of t okay one will be what's called the low noise case what is the low noise case when the power in n of t is very low compared to the power in x of t okay so n of t is a very very low amplitude signal compared to the x of t that you're sending out okay so this will be the low noise case how will it look typically okay it's going to look something like this right I mean it's it's random and then something like this it's going to look okay right so that's how your y of t is going to look okay it doesn't sound too bad okay even on seeing on a scope you might be able to read off a few bits okay who knows it's not doesn't seem too bad okay but when you have high noise or noise which is comparable to x of t when I say low and high it's always relative to x of t okay so the power of the noise is comparable to the power of x of t say for instance you have same amplitude okay which is called a 0 dB signal to noise ratio okay so high in the high noise case what will happen okay so it's going to be some crazy type signal okay maybe you'll see some high and down but it's clearly it won't have a very nice pattern right so it's powers are roughly the same all kinds of things are going to happen okay there's going to be a lot of a lot of things okay so this is how your y of t is going to look and now the question to ask is what should the receiver do okay one of the simplest things you can do right away without worrying too much is to sample this y of t and make based on the sample maker decision okay to sample what do you need you need two things well one thing okay first thing is the same clock should be available at the receiver okay the same capital t the exact same capital t okay the reason why it should be exactly same as if you go to very small t so small error can build up very quickly okay so you need very very exact same capital t so we'll make that assumption we'll say the same clock is available to the receiver and also they are in sync in sync in the sense kind of the receiver knows when the transmitter started okay so after a delay it will know where the bits are starting okay so both of those we'll assume we'll assume synchronous transmission and we'll assume the clock is available okay so those two things we'll assume if you assume those two a nice thing to do is to sample at t by 2 3t by 2 5t by 2 so on okay to get to get some values right so maybe i'll call this y1 you call this y2 i'll call this y3 so on okay how will you decide on the bits yeah greater than 2.5 you say it is bit 1 if it is less than 2.5 you say it is bit 0 right seems like a very natural way of doing things we'll justify this more rigorously later on but for now it seems seems clear enough so you so what you do is you get it or threshold it at 2.5 right so i'm going to say some such thing so it's 2.5 so you threshold it at 2.5 maybe i drew the thresholding wrongly okay so if it's greater than yi is greater than 2.5 the bit bi that was transmitted we'll say is 1 so if it is less than 2.5 we'll say bit bi is i'll say bi cap okay this is my estimate of what was transmitted okay so bit sequences this is a 1 and 0 okay but like i like i showed you in the high noise case this can cause errors okay so if the noise was really high there can be errors okay the sense that bi cap and bi you have to compare and see if they are the same if they're the same there's no error if they're different there can be an error okay so we're out of time today so i'll stop here we'll pick up from here the next class and we'll carry on with what happens now to quantify errors and how to relate it to the other quantities we saw before okay so the so the so so so the things to remember what we saw is that the parameters in the problem right the problem that we're dealing with that are important are power at the transmitter rate of transmission bandwidth and errors okay well error rate okay the rate at which you're going to make an error okay if you transmit thousand bits how many errors will you get error rate rate at which you're making errors okay it should be these four are very important quantities and hopefully i gave you a feeling for what these things are and where they come from and we'll see there is a wonderful and great relationship at least for our model y of t equals x of t plus n of t there's a beautiful relationship between all these four which was found long long back okay so we'll try to we'll try to get there in the next class