 okay so this is okay okay this is lecture what's the number 39 okay all right so some reason the volume seems to be a little bit low and I'm worried about that but I think it should be okay okay so so we're going to begin to see OFDM which is which is related to what's called multi-carrier modulation okay I said it's related to this reason why I said it's related to which it is not multi-carrier modulation in the strict sense of the word but it's kind of related to it gives you the flexibility that provides okay so so far what is it that we've been seeing we've been seeing a situation where you produced symbols from a certain alphabet right at a certain rate okay and then you sent it through a transmit filter okay which was then up converted right it did a up conversion you multiplied by a 2 pi fct there was only one up conversion then maybe there's a real part here okay this went through the channel okay to this noise was added of course at the receiver noise gets added whatever you do and then this goes through a processing first I'm going to call it front end which is presumably analog or very highly sampled okay and then you have a equalizer stroke slicer okay the reason why I'm not writing equalizer and slicer differently is there are configurations where the slicer is built into the equalizer and use feedback okay so I'm saying equalizer slicer and and and then you're able to make an estimate of SK at the output of the slicer this might be a hard estimate or it can also be a soft estimate in case you're producing LLR here so this is the kind of system that we've been looking at so essentially what will happen is if you if you see from here to here somewhere in the middle here so here here I mean this is the transmit constellation so say some some four points and here if I look at the input to the slicer you will get what's what could be called the received constellation okay so if all your processing is going well nothing very dangerous is happening what you will get that the input to the slicer will be you'll get on average maybe the points of the input constellation and then but actually it's going to be bunched up around that okay right this is how the whole picture looks and you know what each of these components are and how they're going to be designed okay so ideally if the variance of this received constellation is equal to the variance of the additive noise right roughly okay maybe it should be slightly different because the channel might introduce some advantages for you right if you're using all the multipath to the channel or several things to the channel properly you might get something slightly better like what the match filter bound does right it gives you some distance from the channel also if you can gather everything properly then you would have a certain variance for these noise so maybe the variance is equal to the noise that was added in fact it should be slightly better than that because the channel is giving you some maybe giving you something okay so if it's as good as that then there's no noise enhancement and you've done the best possible thing that can be done okay but we saw the way we did the calculations unless the channel is flat you're not going to get anywhere close to that kind of performance if the channel shows a lot of dips and if it goes to zero then you will have noise enhancement your MSE calculations involve the arithmetic and geometric mean of the received spectrum in some way and it translates into the channel spectrum also and if it shows a lot of dips and changes the means are going to fall they will not be very good numbers okay so we saw there that the channel does something non-trivial then you are going to suffer because of the channel itself and not just because of the additive noise and you're not going to be able to gain a lot of mileage from the way you're doing the processing okay so so this is this is a traditional single carrier system that we've but we saw so far okay so so if you if you have to summarize in in a few lines if channel is flat then there is a good performance the channel varies very widely then a lot of things that you lose in this okay so this is what we saw that's one thing another thing is today you keep getting larger and larger bandwidth to use around your carrier okay your carrier is going to the gigahertz range okay so which means your bandwidth is going to be at least in the megahertz range okay so you're getting large bandwidths around a central carrier which is moving further and further to the right okay so the channel will never be flat or such a large frequency range so many things are happening to it to do the so many circuitries and so many multipaths and that all those things get added so there's no way the channel is going to remain flat over such a large frequency range okay and you know your receiver processing is going to be simple and efficient and good performs well only when the channel is flat okay so a nice alternative is to do multiple carriers okay so you have a bunch of bandwidth and you think of it as one carrier instead of thinking of it as one carrier you think of it as several carriers but with small bandwidth around them right so instead of dealing with the whole channel as one chunk of frequencies and having one center center frequency and worrying about a channel which is fluctuating a lot I'm going to limit my bandwidth but I'm going to increase my carriers okay so then I can hope since I'm using very small bandwidth around each carrier that the channel will be very flat and I'll get good performance okay so that's the motivation behind moving from single carrier to multi-carrier okay so towards if you want to move towards multiple carriers the advantages are okay channel is flat in in a small bandwidth around each carrier okay so that's the motivation and you know a receiver processing is going to become simpler everything is going to become nice if you do that that's what it's that's the first point that the other point is from a capacity point of view right from a capacity point of view it turns out if you have a large range of frequencies with you but whichever very varying response right channel frequency is very different it turns out what you have to do is you have to do different things in different bands that's what capacity tells you okay so from an information theory point of view if you look at capacity of such channels you learn that you have to do different things in different brands you have to put more power in certain bands you have to put less power in certain other bands okay right so that's what information theory tells you I know you have not studied that much of it but that's just believe believe it when I tell you okay so if you have a lot of frequencies a lot of variations information theory tells you that you have to do different things in different channel regions okay and if you're doing one carrier then the only control you have over putting different power and different frequencies is what which is the point in this block diagram where you can do that control where you can change the transmit filter right that's the only thing you can do okay so it turns out if you want to do that you have to keep adapting the transmit filter in a in strange ways of the channel changes right so there are so many things that are not very easy to do according to the channel in one transmit filter okay so if you do multi-carriers then naturally what do you have you have naturally different frequencies and different transmit filters so to speak okay so in fact in multiple carriers usually people choose a rectangular pulse the reason why you can choose a rectangular pulse is what such a small bandwidth then you spend sending a very large time so you can choose as well a rectangular pulse okay so the one so just by adjusting the amplitude of the rectangular pulse you can get nice control over the overall spectrum that you're using over the band okay so that's the other point in this multiple carriers so it gives you better control over subbands okay and this is required from a information theoretic point of view okay so so these two advantages are are very good in multiple carriers but there are a lot of disadvantages also okay so we'll slowly go towards the disadvantages and then how to make sure all the disadvantages go away is also what we'll see okay so so let me briefly talk about multi-carrier modulation how such a block diagram looks and then we'll talk about some disadvantages of doing multiple carriers over just one carry okay so but these two things are very important okay you get better control over the subbands that are available to you okay so instead of looking at the entire band then each subband is very easy to equalize and process it's going to be flat and your transmit filter can be just a rectangular pulse you don't have to do anything fancy there so so many things are nice in this in this kind of a thing okay well I have to be careful when I say rectangular pulse maybe it's not quite a rectangular pulse anymore but anyway we'll see we'll see how it's done later okay so don't get carried away by the rectangular pulsing okay so many maybe you don't want to do that okay so we'll see this as we go along okay so this is how a multi-carrier modulation picture would look so if you look at multi-carrier modulation basically here instead of doing one up conversion you imagine doing several up conversions and let me write down carefully how this is going to look okay so I have several brands and several transmit filters okay so I'll write down okay I'll do three of them and then put dot dot to show I have several of them so how many of them do I have first one takes a symbol s1 of k second one is s2k takes a symbol sequence right s1 of k s2 of k so until maybe I'll call this snc of k so nc denotes the number of carriers okay so so one can imagine each signal each symbol sequence coming from a different alphabet okay need not be from the same alphabet you can imagine different alphabets so I'll just put x1 can you see this x1 x2 okay so maybe just about you can see it okay x1 x2 xnc okay so these are the alphabets from which these symbols are coming okay they can all be the same for instance in most cases maybe they are all the same but they can also be different if you want to change some things okay and then I'm going to multiply by a carrier I'm going to use a special case here so I'm going to say the first carrier is e power j 2 pi n1 delta f t okay the next carrier is e power j 2 pi n1 plus 1 delta f t so until the last carrier is going to be e power j 2 pi n2 t delta f t I'm sorry okay so my nc carriers are all multiples of a small frequency delta f okay n1 delta f 2 and 2 delta f so nc equals what n2 minus n1 plus okay so have this many carriers and delta f you think of as a small frequency okay so delta f is actually the what the bandwidth of each subband okay so this is there's also another way of viewing it previously you had one channel and one carrier now you have a bunch of parallel channels okay so you have nc parallel channels and each channel you deal with separately okay and all the channels can exist simultaneously because in bandwidth or in frequency they don't overlap okay so you make sure they don't overlap in frequency and then you can use them simultaneously it turns out when you do OFDM you'll actually let them overlap in frequency also okay so so keep that in mind also a clever baseband processing you might even be able to overlap it but first I'm going to talk about a situation where you don't allow these different bands to overlap in frequency okay so then you do what then you do the standard real part conversion here and then you add them all up okay you have a summation okay and you produce your pass band transmit signal X of t okay so this is going to go through a channel then noise is going to get added okay and it signal goes into the receiver okay so this is how a general multi-carrier modulation picture looks and and and and and what okay so if you look at it from a spectrum point of view suppose you had a bandwidth that was initially available to you okay so this is the like the frequency line okay maybe zeroes here okay so maybe you had a bunch of bandwidth available to you which went it went from n1 delta f all the way to n2 delta f centered at that and so the total frequency that was available to you was maybe nc times delta f okay so this was the total frequency that was available to you maybe centered from n1 delta f to n2 delta f okay what you did was you separated this into nc bands okay of delta f each okay so nc bands of delta f each this is delta f and you want to treat each of this delta f hertz of frequency as a separate parallel channel okay and you don't want these channels to interfere or cause trouble to each other okay one way of doing them doing it is to make sure that they don't overlap in frequency okay so which is called frequency division multiplexing okay so that's the first approach which seems like the simple simple very very simple approach you can do something called frequency division multiplexing what you do in this case is basically treat each channel individually okay so if you have a frequency of delta f under your control what would you do the first thing to do is to okay so if you just look at the first first thing n1 delta f is my center frequency delta f is the bandwidth that is available okay so I'm going to first choose some transmit filter to occupy the delta f frequency I might choose a sync transmit filter but that might cause problems so maybe I choose a trace cosine with some excess bandwidth so the bandwidth the data rate that I'll be actually using will be 1 by t equals some fs which is less than delta f okay so each channel then each channel symbol rate on each channel would be what per channel or per carrier will be some 1 by t which is an fs which is less than delta f you won't be doing 1 by delta f which is the maximum possible you're doing some I'll put less than or equal to if you want to use full bandwidth also you can use it okay so that's the problem okay so this this is this is one reason but if you use full bandwidth there is another problem if you want to treat each of these guys as separate okay and you're using the full bandwidth there is another problem what is the problem okay what do you do at the receiver is the question so what do you do at the receiver you have to filter out the bands how can you filter out the band if you're using the entire frequency it's not possible you need some guard interval between them so typically fs is also going to use I mean some of some some reduction here is needed for the guard band as well so fs will definitely be less than delta f okay so the frequency that you'll be really using here will only be something smaller like this okay there will be a so if you use something like this there will be a guard here okay some guard is needed here okay so once you have a guard band you can define a designer band pass filter as a nice frontend for your receiver so for each channel you have a different band pass filter okay so if you have NC carriers at the transmitter you need NC up converters okay a bank of up converters and then at the receiver you need a bank of band pass filters at different center frequencies to bring out just extract your channel and then process just like before your processing will definitely be simpler because you expect your channel to be very flat and maybe a one tap or a two tap equalizer will take care of the equalization and you won't get any noise enhancement it'll be very nice okay so all that is fine but you need a guard band and the data rate goes down okay so this is frequency division multiplexing hopefully it's clear it's just like bunch of different channels working together simultaneously to transmit differently and you separate them at the receiver and you work okay so what are the possible advantages and disadvantages here okay so first thing you notice your symbol period for each transmitter is very long okay so your delta f is small even if your overall bandwidth was large your delta f is small and your symbol period is very very long so that gives you a lot of advantages in processing those are the advantages that I've been talking about okay so let's let's just try to rest down some advantages and disadvantages okay so so first thing is receiver processing is okay equalization is simple let me be very careful let me not say receiver processing okay simple equalization okay so maybe a one tap equalizer will do in most cases okay so you may not need anything more than one tap maybe two tap okay so then there's no noise enhancement well I'm saying no noise enhancement of course very minimal noise enhancement because the channel is very flat okay with any resummel processing you will get good performance okay and then the third point is about the symbol time okay so your symbol is really really long right so delta f can be very small and your symbol is very very long so because of that if you had some noise which was impulsive in time okay like for instance what you have in wireless channels right so it's gonna be a noise or channel fluctuations which are very impulsive in time it's gonna be a deep fade and then you're going to come back to the regular strength before so if you have that since your symbol duration is really really long and when you integrate things out right you're going to have an integrate and dump type equalizer right so as much as that okay so then all those time impulsive behavior will be will not affect your performance that much okay so it's immune to time impulsive distortions these are the advantages that we would like to keep okay so from multi-carrier modulation okay particularly in wireless communications this is very very useful okay so that's the first thing so disadvantages the first thing is guard bands result in a in wastage of bandwidth okay so and this is very very critical okay whatever you do your filters you're going to require a significant amount of guard band and from your precious in the NC times delta f bandwidth you're dropping something at the rate of NC even if you require a small guard band gets multiplied by NC okay and it plays a very significant factor into your whole trade-off right for in your trade-off you want your delta f to be small so you want your NC to be large and when you want your NC to be large the amount of bandwidth you're wasting also goes through the roof okay so that's a significant problem nobody will be willing to do this for that reason okay so if you have to put guard bands and frequency is so expensive nobody is going to nobody's money investment that's the first thing second problem is if you look at the transmitter and the receiver it looks like you need a bank of up converters and a bank of bandpass filters and down converters etc okay so these things are also extremely expensive in hardware okay so anything you want to build in the R of domain and analog is going to be extremely complex so maybe you can do some smart design so that you avoid the power amplifiers and each thing separately okay but still ultimately there is enough circuitry in there which is going to be very complex because of the bank of up converters and bandpass filters okay so that is a significant disadvantage okay so more things that I want to add to the advantages you get control over power of sub bands okay and this is an advantage like I said we didn't show why it was an advantage but from a capacity point of view there is an advantage here okay so all these things are nice advantages but there are two significant disadvantages need for a guard band and need for a bank of up converters and down converters okay the solution to this this whole problem well this is not really a problem you have four advantages and two disadvantages there is a technique which takes these disadvantages I mean gets rid of these disadvantages and keeps all the advantages okay so in that technique is what's called orthogonal frequency division multiplexing okay contrary to the definition it is not frequency division multiplexing there is no frequency division multiplexing that actually happens here you happily allow all the sub bands to overlap okay so that's the main principle there but still it gives you enough control and gets rid of the guard bands and the bank of converters and down converters needed okay so that's where the OFDM comes in okay so basically it gives you the best of both worlds okay but even though FDM is part of the acronym like I said it's not frequency division multiplexing you do something else you actually allow the bands to overlap significantly okay there is a very significant overlap between the bands okay which causes some trouble in implementation but all of that can be overcome but it gives you all gets rid of all the disadvantages in multi-carrier modulation gives you the advantages okay so so what's the idea here first of all okay so if you want to get rid of the guard bands what should your FSB your symbol rate 1 by T is to be equal to delta F right so that's the first thing that has to happen so you have to take that as the delta F okay so that's the first step in OFDM you set 1 by T which is a symbol rate per carrier to be equal to delta F the second and very surprising thing which may not be that that obvious is your transmit filter is said to be a rectangular pulse of width T okay your transmit filter becomes a rectangular pulse okay so so clearly when your transmit filter is a rectangular pulse with width 1 by delta F what's what's going to be its frequency response it's going to be a sink and it's going to overlap into the neighboring carriers okay so it's going to overlap in quite a significant fashion with every next carrier so on okay so you might say how do I get rid of it okay it turns out you can do very clever baseband processing to separate these carriers even though they are they get they start interfering with each other okay so that's the main idea okay so you do baseband processing to separate carriers okay so maybe you do this but still one of the problems that still remain is having these bank of up converters and band pass filters it turns out well this is not really because of OFDM even in the multi-carrier modulation world people knew it can be it could be done okay it turns out you can get rid of the bank of up converters and BPFs by doing by converting them into clever FFT and IFFT operation okay so that's possible like I said it was known even before OFDM came along but maybe in the context of OFDM it becomes much much more precise and nice okay so you use FFT IFFT to simplify transmitter and this here okay so so the whole I mean at least in my belief and belief for so many others the entire area of single processing rests on one algorithm okay which is the FFT and the efficiency of the FFT okay the fact that it can be done in N log N as opposed to N squared okay so the entire single processing area rests on that if that were not true single processing would have gone out of the window log back yes the entire thing is made possible because of the efficiency of and here is one more illustration of why this comes about okay so who who found out that could be done in N log N first time I'm sorry well that's all recent who did it prehistoric days it was known who was the first guy yeah came from Gauss again so okay so the whole of electrical engineering completely rests on Gauss' shoulders nobody else who played a role okay so even today is a very important character to get to know okay so all right so let's let's move on and see how this is done in so many so many different ways okay so the first thing I'm going to show you is how to convert these bank of up converters into an IFFT okay and then we'll see how FFT gets rid of gets rid of that okay so that's the first thing we'll see okay so how does an IFFT enter the transmitter okay so once again remember this picture this is my multi-carrier transmitter okay so I have each thing each transmitter each channel kind of producing an independent symbol it gets encoded but my transmit filter is going to be I'm going to set it to be a rectangular pulse and my delta f I mean one by t okay so one by t here is going to be equal to delta f okay so those are the things that I will use and my frequencies are all multiples of a given delta f n1 delta f 22 delta f okay so there's going to be a whole bunch of notation and I might get confused with the notation hopefully hopefully we'll see how to write okay so the signal x of t which was obtained by adding all the different channels okay so if you do that well after the real part also okay so I've done the real part also so it's going to be real part of summation i equals 1 to nc okay so don't worry too much about the subscripts and superscripts just come up with something a little change as we go along okay it's going to be sik e power j 2 pi n1 plus i minus 1 delta f t is that okay this is my entire thing okay so the transmit filter plays no role right it's a it's a rectangular pulse so sik just multiplies the transmit filter so this will be from for what time 0 to t which is one by delta f okay so that's the time okay so keep that in mind and this t is going to be large so this so okay so so looking at this before I proceed and show you all the simplification the first difference that should pop up into your head is what we did before was what we took each si of k then converted into kind of multiplied by a rectangular pulse and then converted into a carrier and then transmitted so here what's actually happening is you collect nc si of k okay so you don't send each sa of k separately you collect nc sa of k then multiply each of them by a separate carrier then add it up and send okay so this is an example of a block modulation scheme you don't do symbol level modulation each of them independently you take a block of symbols together and then you do one common modulation to it okay so right now it seems like there are several different carriers so but I'll pull the common term out and you'll see one carrier will come out so this is an example of a block modulation so OFDM is block modulation you don't do memory less symbol by symbol modulation you collect a bunch of symbols together and modulate them at one go and that has advantages particularly when your channel is behaving very differently that has an advantage okay so it's all natural in so many different intuitive ways the first time you see it it's always confusing okay definitely the notation is very confusing so try to stay with me we'll see how it goes okay so I'm going to what I'm going to try is from this n1 delta f to n2 delta f I'm going to pick a center frequency a convenient center frequency it doesn't matter what I pick okay so it's just for convenience I'm going to pick one center frequency and then I'm going to pull it out and see the remaining base band signal complex base band signal okay so that's what I'm going to do and like I said you can pick any center frequency you can even pick it to be equal to n1 delta f or n2 delta f I'm going to pick n1 plus n2 by 2 into delta f so you can pick anything you want okay it only affect it only it'll only introduce a phase factor and that will anyway get married to your carrier and you don't know the face of your carrier anyway so it doesn't really matter in in any significant way okay so that the center frequency I'm going to pick I'm going to write this as real part of okay I'm going to pull out a e power j 2 pi nc delta f t then write it as a summation I'll change my indices here I think just to make sure I don't run into any trouble later I'm going to write here as l plus 1 okay once again don't pay attention to the subscripts the reason why I'm putting l plus 1 is what I shifted from 1 to nc to 0 to nc minus 1 so I just wanted some confirmation there now I have to do e power j what 2 pi l minus what okay I have to I have to subtract something okay so that something turns out to be I think the way I choose it nc minus 1 by 2 like I said it doesn't really matter what this is don't worry too much about that okay the choice I've made for nc is n1 plus n2 by 2 okay so because of that this thing works out okay so that's a capital nc so like I said it's not really a great thing to worry too much about I could have made a minor mistake also but it doesn't really matter so this becomes your carrier okay and this you can call it x tilde of t is like your complex base band signal okay so you see why it's the complex bin I've just taken e power j nc delta of t and I've written that multi-carrier modulation signal as a complex envelope of e power j 2 pi nc delta of t okay so what you know about complex envelopes this nc is just an arbitrary choice you can change it but you'll still get something else the only thing that will change is any power j fixed phase term which anyway can be pulled into the carrier it's not a big deal don't worry about it maybe there is an extra delta f also okay don't don't worry about that center frequency it won't change what's inside very significantly you'll see soon enough okay so now when I want to generate this x of t at the transmitter all I have to do is generate this complex base band signal and use one up converter followed by a real part to get to my x of t I don't have to do each of them separately okay so this is clear even from multi-carrier modulation you don't need any OFDM or anything but it has been helped by the fact that you're doing rectangular pulse okay because you're doing rectangular pulse there's no strange factor showing up here okay so it's all very simple and straightforward because of that rectangular pulse okay so there are ways to get around that also but anyway the rectangular pulse is good enough okay so so now let's let's draw a brief picture to illustrate this okay so it will give you a nice sense of what's actually happening okay so you have in your frequency you had n1 delta f to n2 delta f okay so I've picked my nc delta f somewhere here okay so previously you were using the band from here to here okay so now I've pulled out this e power j nc delta f so this entire thing is nc times delta f so your origin will shift to nc delta f I have a past complex base band signal of bandwidth what x tilde of t has bandwidth which is equal to nc delta f by 2 okay well it's not quite nc delta f by 2 why sorry depends on yeah yeah the way I have chosen nc it's going to be nc delta f by 2 okay that's fine I've chosen it to be the center so it's nc delta f by 2 but there is one more thing which is missing here remember my I had a rectangular pulse shape so what will happen the bandwidth that's being used will be a sync on both sides so I have to bring in some extra lobes of sync on both sides so when I write nc delta f by 2 I'll also have some extra bandwidth on both sides for the some extra lobes of sync plus some extra sync lobes only then I'll be able to accurately represent my x of t in base band properly so maybe a few more lobes maybe some three or four on both sides you might want to pick to make sure the entire bandwidth that you're actually using gets nicely captured in base band also okay this is because of the rectangular pulse shape that I chose okay so that's going to spill over into into the as a side lobes okay is that okay it's fine right so you keep getting so a few multiples of delta f on both sides it's not to not be a huge bandwidth it'll be a multiples of delta f because my time is one by delta f right so you'll have a few lobes on both sides and that has to be brought in to accurately give my x tilde of t okay is there a question what's the question that's good to ask oh no no it'll be just some few sync lobes I mean the for instance the rectangular pulse right in the middle may not be you may not need extra sync lobes it's only for the edge a few sync lobes a few delta f's on both sides you will have to pick okay so now we have to generate this fancy looking base band signal okay which has the e power j term and all that so when you have something so fancy okay you don't want to keep multiplying with complex exponentials you don't want to do that but you know it's a base band signal what is a very easy way of generating such signals okay what's the principle we have used in generating base band signals we don't generate them in analog what do we do sample them over sample them generate them in digital sample them at Nyquist rate whatever rate is needed generate them in digital and then rely on your d to a interpolation to give you the continuous time signal okay so don't generate base band signals in analog you never do that right so you want to sample it at Nyquist rate generate the base band signal in digital and then rely on your d to a to give you the final conversion to analog okay so maybe you do any kind of interpolation d to a maybe you do a sync proper interpolation or you do just a point to point just linear interpolation maybe okay so maybe you do a sync interpolation that's that's actually better but you rely on your d to a to give you the actual analog signal okay so that's what we're going to do next I'm going to take this base band signal and sample it at Nyquist rate okay whatever rate is needed okay and then that will give me a discrete time base band signal and you'll see that is naturally generated by a by an IFF okay so that's what we're going to okay so we're going to sample x tilde of t at what rate okay I have bandwidth but it's not enough if you do NC right NC delta F the reason is you want to bring in the extra side loops of the sync so you have to put some larger number so I'm going to say I'm going to sample it at a frequency which is some n t o t so maybe I'll just call it n here okay maybe it's n t o t it's okay some larger number okay so this n t o t is going to be greater than NC okay so reasonably large so that so that you'll cover the entire thing okay so if I do this if I do this what is my x tilde of n going to be which is x tilde of n times what n times some t sampling the t sampling is going to be 1 by n total into delta F if you do that what will you get you get summation l equals 0 to NC minus 1 yes l plus 1 k e power j 2 pi well you have this l minus NC minus 1 all these things are not so crucial 2 times what delta F times n times t sam right so remember delta F is 1 by t okay so if you do all that you'll get simply n by n total okay so this is these are the samples that I have to generate okay what is the rate of these samples what is the separation in time between two of these samples t by n total okay so the samples are separated in time actually by t by n total t is your overall time which is 1 by delta F which is very large but these samples are coming out at t by n total not not that t okay so this is separated by t by n total so when it goes into the d2a it should go at with the clock which is n total times the clock that you are using for your symbol rate it cannot be the same clock okay so your over sampled it right because it has to fit into the bandwidth you're getting somewhere you have to pay you can't use a large bandwidth and happily skip sampling at t it won't happen so that's what you're doing okay so now I want to convert this this nasty looking expression into a nice ifft type expression okay so that's what I'm going to do next okay so for that for doing that you notice here you only have nc data symbols and your division is here by n total which is larger okay and you know in your ifft these two things have to match so what do you do you put zeros up on top and down below okay so you take your sl of k then add some zeros on top add some zeros below to make it equal to n total okay so this is a standard thing if you look at any OFDM standard today they will always put some zeros up on top and down below this is to bring in the things that you have okay so that's what we're going to do what we'll do next is we'll take we will set basically nc to be equal to n total to be equal to some n okay and then what about sl of k sl of k will be equal to zero for l equals some n1 minus delta 1 so on till n1 minus 1 and then it will be equal to the transmit symbols no okay so I don't know wait wait wait wait okay so maybe you're getting confused by the notation here so let me not well I can I can do this it's not a big deal I'll show you how I'm doing it it's okay it's not a big deal just wait for wait for a few minutes it's equal to the transmitted symbol for n equals n12 and 2 see notice my sl of k is zero for those things so it's okay right so I'm adjusting my terminology if you don't like it just don't use it it's okay I'm not transmitting anything for n1 minus delta 1 to n1 minus 1 those extra things will actually be zero I'm not using those bandwidths okay so don't worry about this nc being equal to n total okay and then again once again set it to be equal to zero for l equals n2 plus 1 so on till n2 plus some delta 2 okay so you have have your actual carriers n2 minus n1 plus 1 which is right in the middle which are actually going to be your data carriers which are carrying something you want to send and then the initial few carriers you say you're actually using them but you only send zero and likewise in the last last few carriers you say you're using them but you're sending zero on it okay the reason is you're actually using those carriers okay so you can't say I'm not using that bandwidth I'm using that bandwidth okay so I'm sending nc to be that okay so it's a bit of a bad notation I know the notation is bad here I just want to show you how this works okay all right so once I do this the zero padding okay you'll see the whole thing will simplify so x tilde of n up to some phase term okay there'll be a phase term e power j some n tot minus 1 by n tot or some such thing there'll be a phase term up to a phase term x tilde of n will become l equals 0 to n minus 1 slk okay once again don't worry about the l and l plus 1 and all that okay we did this go here all right let's just hide it slk e power j 2 pi ln by n okay so it'll become that there'll be a phase term I know I'm not accounting for the phase term here the reason is ultimately the phase term can go and get absorbed in the carrier okay I don't care about the phase term when I produce my x tilde of n okay and my equalizer will take care of any phase term anyway so I'm going to have a one tap equalizer so it'll take care of my first term so I don't have to worry about any phase problems so up to a phase term this will be equal to this quantity x tilde of n okay so remember I'm going to now this the sample separation here is capital T by n okay so this will last for capital T seconds okay so it will still last for capital T seconds but I have to do a interpolation here with the d2a before I take my up conversion with the nc okay so this becomes an ifft okay so what will be a choice for capital n typically yeah some power of 2 so 250 6 5 12 10 24 whatever so one of those choices you can do whatever fft can be implement okay so there might also be a 1 by root 10 if you want to normalize your ifft anyway all these scaling things don't really matter in real life right so you just adjust them however you want finally all right so I'm going to come back and draw a picture to show you how this works in the next class but anyway so this conversion is what matters any questions okay all right so so what's the so so so what's coming next is to show you what do you do on the receiver okay so we've let the carriers overlap massively at the transmitter so what do you do at the receiver is not clear but even though I've left the even though I let the carriers overlap advantage 4 still remains okay the overlap but the significant lobe is still going to be at each of these things and that power you can control by your constellation so you do have control over the subband to an extent you haven't lost it at all okay right but they overlap you have to come up with a problem of dealing with them okay that's the one thing and then equalization is a problem you can only do suboptimal equalizes if you want to do optimal equalizes you'll once again have all these problems but suboptimal equalizes are very very very easy to build for these things so that's where the main strength comes okay suboptimal equalizes which don't lose too much on the optimal equalizes okay so that's the way it goes all right so let's stop