 So let's begin. This is lecture 40. So the last thing we were saying was basically OFTM and how the transmitter can be implemented efficiently within FFT. So how does the block look? So I'm going to draw a block diagram of how the whole thing would look. So you have a series of symbols, I mean the sequence of symbols coming in. So you have NC, say NC symbols, some smaller number. You have to zero pad it top and the bottom. So that's what you do and evaluate an endpoint IFFT. So you have, so I'll call it S1k, S2k, so until say SN minus delta k. Here you have zeros and here you have zeros being input as well. So in total you have an endpoint FFT that's evaluated. After you evaluate the endpoint FFT, simply get X tilde 0, X tilde 1. I'll call it X tilde, X tilde n minus 1. So this is your, the result of the endpoint IFFT. So then you do a serial to, a parallel to serial conversion. So it's called next block, P stroke S to get your X tilde k. Remember you wanted to generate an X tilde of t, how are we going to generate it? We're going to sample the X tilde of t and then send that samples through a D2A converter to get a required analog signal. That's what we are going to do. And when we found, when we tried to sample the X tilde of t, we found that those samples were nothing but the IFFT of the symbols that we wanted to transmit. So you do the IFFT first and then convert them into a series of samples. The timing is important. So this needs to be shifted out at what? t by n. So clock will be t by n. You shift them out at t by n. Then this goes through. And then what do you do? You have a D2A converter. So the ideal D2A converter is going to do a sync interpolation. You're going to multiply its sample with a sync located at that point. You do that. So in practice, many D2A converters would do sample and hold and then do a low pass filter. The low pass filter is also a sync. So that's also the same thing. So some version of that. So in effect, you can assume that sync interpolation is being done. It's not a very bad approximation. So that's being done. And then what do you do? You get your x tilde of t, you do upconversion. So you upconvert and then you transmit. Take the real part. You're going to get your signals, your pass band, transmit sync. So this is how a transmitter looks when you're doing OFTM. So one of the points you want to make, this is basically a block modulation scheme. That's the first point I want to make. So the overall pulse at base band occupies how much time? A total of t seconds. In t seconds, you're sending n symbols, but not really n. It's n minus delta because it's a minor zero problem. But anyway, let's think of it as n symbols. You're sending n symbols in capital t seconds with a very, very long time domain waveform. The waveform corresponding to each signal, what actually happens is there are several waveforms corresponding to each signal. They're all added together. Remember that's what's happening roughly in the multi-carrier waveform. Each symbol gets multiplied by a e power j 2 pi n delta f t. And then with the rectangular pulse shape that lasts from 0 to t. So the signal corresponding to each symbol will last the entire t seconds and all of these guys are being added up. So it's a block modulation scheme and it works in that fashion. So if you want to relate it to serial PAM, if you want to relate it to serial PAM and see an equivalent system, the serial PAM will have to work at a symbol rate of t by n. Well, n by t. Symbol rate would be n by t. Symbol period, the symbol would come out every n by t. But you would have only one transmit filter for all of them. Here you have different transmit filters for each of these guys. Is that clear? So that's the way it works. So one of the advantages is since the signal corresponding to each symbol now lasts for a long time, right? Time impulsive noise or fading will be not that serious a problem in OFDM, in the block modulation scheme. You are letting the signal corresponding to each symbol last for a really long time. In serial PAM you don't do that, right? Maybe the signal corresponding to one symbol will last for three or four symbols. And if it's t by n and it's only three t by n or four t by n, it's not the entire t. Here the signal corresponding to each symbol is lasting for such a long time. So if you have deep fades or if you have time impulsive noise, you are a little bit immune, okay? So that's an advantage in this block modulation scheme. Might happen. So it happens actually in several cases in telephone lines it happens because there might be something else happening nearby, right? So these telephone cables are not kept in isolation from anything else. They might be next to some power circuitry which is being turned on out of a sudden that might give you an impulsive noise in time. So these things happen in telephone circuitry a lot. In wireless the deep fading is one thing. So once in a while in wireless communications because of the way the sinusoid are adding will suddenly have a big drop in amplitude. It'll recover fast but you'll have a drop in amplitude. So for those kind of things OFDM is certain, okay? So this is a transmitter, okay? So the transmitter is easy enough but you have to be able to demodulate on the other side, right? It's not clear how we're going to extract all the symbols. In the FDM case, all of the symbols were separated in frequency nicely with the guard band. So we could put a bank of band pass filters and separate them out, okay? The signals corresponding to each symbol were in fact orthogonal, okay? They were not overlapping in frequency. If the frequencies don't overlap then in time domain also they are orthogonal, right? When you do the dot product you can do the dot product in the frequency domain also. That'll go to zero. It's very clear that they are orthogonal, okay? It turns out even in OFDM the signals are orthogonal, okay? But they don't, they're not distinct in frequency, okay? So they actually overlap significantly in the frequency domain but still the two signals are orthogonal, okay? So that's the next point which is slightly not very obvious. We'll prove it as we go along, okay? The signals are orthogonal, okay? So that's why people loosely call it orthogonal frequency division multiplexing. So it's actually not frequency division multiplexing but it's still orthogonal so you can separate it, okay? So that's what we're going to see next. We're going to see how to build a demodulator for this, okay? So if you go back to first principles, how do you build demodulators? What's the ideal way of building a receiver? You look at the set of all possible signals, build a signal space. How do you build a signal space? You do grams with orthonormalization or figure out the orthonormal basis for your signal space which will span your signal space and then what should you do? Correlate with the basis. Then look at the results of the correlation and slice the results of the correlation to get your transmitted signal, okay? So that's the readymade procedure and that procedure will not change. Whatever modulation you do, it's the exact same procedure that you have to follow, okay? So the first thing to figure out is what is the signal space, okay? So for that, we'll look at the signal here. Okay? If I have a sequence of symbols that I'm sending, I'm going to look at the signal here as the signal corresponding to that set of symbols. Then I'll list out all possible signals and then see which is the basis for my signal space and then go on and, okay? So you'll see there'll be a very simple basis because the signals will be orthogonal, okay? Alright, so that's what we're going to do now. So I'm going to call this signal x tilde t, okay? So after the d to a, x tilde of t is going to be, okay, I'm going to assume ideal sync interpolation in my d to a. There might be some approximation here but anyway, it's a good enough assumption when you're trying to study, okay? It's going to be x tilde of t, n equals 0 to n minus 1. The samples are going to be multiplied by shifted syncs. So the syncs have to be shifted by t by n and what is this p of t? Okay? So in its normalized form, it'll look like this sine pi nt by t divided by pi t, okay? So if you're wondering what is this, if you do a Fourier transform here, you will get a rect between minus n by t, n by t, okay? So that's the, it's a Fourier transform of that to get a sync interpolation, okay? So this is what you hope to do in an ideal d to a. It'll give you a very accurate, complete interpolation for your samples to get your x tilde of t, okay? All right. So what we're going to do now, see this is x tilde of t in terms of x tilde of n. I want to find the signal corresponding to a sequence of symbols that I have, not necessarily something in the middle that is generated. So I have to basically substitute x tilde of n in terms of s, okay? If I do that, then I'll get my symbols corresponding to the signal corresponding to the input symbols that I had, okay? So that's what I'm going to substitute. When I do that, x tilde of t becomes summation n equals 0 to n minus 1. Remember, x tilde of n is the inverse Fourier transform of, or d of t, or f of t or whatever you want to call it, slk e power j 2 pi ln by n times p of t minus nt by n, okay? So this is the x tilde of t, all right? I'm sorry? Sn, no? I think I might have, yeah, I know L is the running variable but I think in the way I drew the block diagram, I'm thinking of the running variable being in the subscript for s, okay? It's just the way I wrote it. If you want, you can think of that as sl also. It's okay. That's also fine. You can replace this with s of l, okay? If you're confused, that's clearer to you, that's s of l. It's fine. The reason why I wanted to bring in K is people usually think of an OFDM, what's called an OFDM symbol? What's an OFDM symbol? All of them together becomes OFDM symbol, okay? So because one signal corresponds to one OFDM symbol, which is one block symbol, right? So that's why this K, this K can index that OFDM symbol, that's why I have the K but I think this K is completely relevant for us. We're not going to look at multiple OFDM symbols. We will look at it but mostly only one OFDM symbol we'll look at. Anyway, it's also fine. So now I'm going to interchange the order of summation to pull my symbols out. I want to have finally a form where my symbols are multiplying. So let me just use sl, okay? Summation n equals 0 to n minus 1, okay? It's also customary to put a 1 by root 10 if you want for the IFFT, okay? What do you want to put the 1 by root 10? Make sure you don't add energy when you do IFFT, right? So when you do 1 by root 10, it becomes a proper unitary transformation. Nothing else will happen to it, okay? So but anyway, that's just a constant. I might ignore it. I might have it but maybe we'll keep it late, okay? Once again, that's important only for the exam. So keep that in mind. e power j 2 pi ln divided by n p t minus nt by n, okay? There you go. So my signal that's actually being transmitted, the baseband signal that's actually being transmitted is a linear combination of several sums of sinks, right? Summs of shifted sinks or well maybe something like that, okay? The multiplication is by some exponential and finally the linear combination is with the symbols itself, S of l, right? These are my signals. How many possible signals will I have? Yeah, it's the size of x power n and if you have different size of x for each of these things, product of the size of the alphabets, right? The only thing that's varying here is the symbol. The symbol is different but what is showing up inside here is the same, whatever your symbol is, okay? So these things will change. So clearly this is my signal space. These things form my signal space. I have to find an orthogonal basis for my signal space and it turns out magically enough that these guys for different l form an orthonormal set of signals, okay? So once that comes, you see immediately directly you can identify what your orthonormal basis is for the signal space and you know what your constellation is. You can go ahead and do your reception very easily, okay? So that's what I'm going to write down. So I'm going to call this as, what am I going to call it? I'm going to call it as glT, okay? So this is G subscript lT, okay? So maybe I'll put glT, okay? Just to be sure about that. This is going to be glT, okay? So that is my, that signal, okay? And my claim is, okay, so this is okay, right? So my claim is this set glT of T for l running from 0 to n minus 1 is orthonormal, okay? You can prove this. It's not too difficult. We will not prove it in this class. You can easily prove it, okay? So it's not a problem, okay? So once that becomes an orthonormal basis, the orthonormal set of signals, you can make that the basis of your signal space. So this becomes the basis of signal space. So that directly tells you how to do the reception, okay? So you'll have to correlate with each of these glT of T, then look at the result of the correlation and then slice it, okay? So if you go back and look at how the signal is being formed, S of l multiplies glT of T. So if you correlate with glT of T, no, glT of T, what will happen? Only S of l will remain, right? That correspond, okay? So I should be careful, say glT of T, only that S of j will remain. Everything else will disappear, okay? So the slicing you can do independently for each channel. You don't have to do a common slicing between all of them because of the way the signals are formed, okay? It's orthogonal, each signal is, each symbol is multiplying an orthogonal signal. So you simply do a correlation, then whatever result you get, you slice for that channel alone separately. It doesn't have to relate to the other channel. So that's another nice thing, okay? Even though you're doing a block modulation together at the slicer, you don't have to do a block slicing. You just do individual slicing, okay? So that's a nice thing about OFDM again, okay? So that's one thing that's being suggested. Okay, so but I want to draw a few plots of this glT of T just to give you an idea of how these things will look, okay? Okay, it might seem a little bit complicated when you look at glT of T. So if you see glT of T, the way we wrote it down, what did I write it down? Okay, so it's going to be summation e power j n equals 0 to n minus 1 j 2 pi ln by n, that might be a normalization, p of T minus nT by, well, nT by n, sorry, nT by n, okay? So it looks and seems like a complicated picture to draw, right? First of all, how can you draw it? This is actually a complex valued signal, okay? You can't draw it very directly. So you'll have to draw the real part and the imaginary part. But you expect the real part and imaginary part to be quite related. It'll be kind of sinusoidal something, okay? So how do you see that this is a sinusoid? Basically, you can define another signal which I'll call glT, which is say e power j 2 pi lT by T, okay? And then you do a rectangular window between 0 and capital T, okay? U of T is the unit step. So U of T minus U of T minus T is the rectangular window from 0 to T. You define this as glT, okay? You can plot the real and imaginary part of this. What is the real and imaginary part of this? Sine and cos, okay? Cos and sine. So you can easily draw it, okay? How do you obtain glT from gl of T? You sample gl of T at T by n and then do a sink interpolation. Put it through a d to a, okay? So you sample it at T by n and put it through a d to a. You get that thing, okay? So if you are sampling and you are interpolating, you should get back the original picture, right? Roughly. At least between 0 and T, you should get definitely, if you do proper sink interpolation, you should get back the overall thing, okay? So that's what you will get. Between 0 and T, you will get a sinusoid for the imaginary part and a cosine for the real part, okay? Well, phase pi by 2 shifted sinusoid. But it will also extend beyond 0 and after T. Why? That's what the sink interpolation will do, okay? If you want to get the proper sink interpolation, you will get some extension before 0 and after T, okay? But in this signal, when you do OFTM transmission, you are not going to send before 0 and after T. So you will truncate that, okay? So that's a bit of a suboptimality there as well, okay? Okay? Is that clear? Okay? So think about it. If you did not truncate, you will be limited in bandwidth. In bandwidth, you will be between N by 2T and minus N by 2T. But because of your truncation, you will spill over beyond N by 2T and minus N by 2T, which is adjusted with your zeros in the IFFT, okay? So everything will work out properly. But you have to remember that the sink interpolation is a bit of a non-ideality here. And the way this plot will look also should be clear to you. Between 0 and T, this will be a sinusoid. So that's how the plot will look. Sinusoids of different frequencies. Depending on L, if L is 0, it will be a it will be something. Yeah, it will be constant. Well, as 1, it will be a sinusoid. As 2, it will be one more higher frequency sinusoid. So basically, sinusoids of different frequencies are your, is your GL tilde of T, okay? That's why they become orthogonal as well. Sinusoids of multiple frequencies, they become orthogonal, okay? So pretty much the same reason will work out, okay? So this is an important thing to keep in mind. This signal, even though it's an ideal version of that signal, it's closely related to the actual transmit signal, okay? So all these things are some nice, nice points to keep in mind, okay? So let's move on along and see the receiver picture, okay? So the first assumption we'll make is that the channel is benign, okay? So we'll say a channel doesn't do, adds noise, but it doesn't do any filtering, okay? So it's delta, okay? Channel is a delta of T and then you add noise to it. So that's the first receiver we'll build. That's what we did even for PAM, okay? So serial PAM, first we assume there is no linear distortion in the channel, then we built a receiver and then we'll see what happens when there is ISI or channel does something, okay? So that's the first receiver we'll build. So this is the receiver for no ISI case. In the no ISI case, right? Even at the receiving point, all your signals, the same orthonormality will hold, okay? Because what you receive will be the same transmit signal plus noise and your signal space is still the same. It's not been changed by any convolution, okay? It's still the same. So the same orthonormality holds. So the optimal receiver is to do correlation with gl tilde of T, okay? So I should be careful here. It's gl tilde conjugate, okay? So it's a dot product. So you have to do correlation with that, okay? So previously, we never dealt with a complex signal. We always had real signal, so these things didn't matter, but here you have to put a conjugate carefully, okay? So let's do the correlation. So correlation is going to be integral minus infinity to infinity r of T, which is my received signal. Remember r of T is the baseband received signal x tilde of T plus n of T, n of T is once again the baseband noise, okay? So that's my model, r of T gl star of T dt, okay? This is the lth correlator. I'm going to call the result of the lth correlator as y tilde l, okay? So you have to do this for what l? All of them, okay? 0 to n minus 1. So now let's substitute this and you'll see a nice, one more, once more a nice simplification happens, okay? Summation l equals 0 to n minus 1. You'll have e power minus j 2 pi n l by n p of T minus l T by n. So you get that right? No, n, no, okay. Okay. So what should I do here? n is down below, right? It says, okay. Well, it's all the same. Anyway, we'll do it. All I have to do is change l to n there, right? It's okay. This is fine, no? This is okay. Okay. So I do this and then I do a dt, okay? So now once again, we'll interchange the summation and the integration and see if something nice comes out. You see summation n equals 0 to n minus 1 e power minus j 2 pi n l by n times what? Integral minus infinity to infinity r of T p of T minus nT by n dt, okay? All right? So even this correlation with gl star of T is going to simplify, okay? So how is it going to simplify? You don't have to do this operation for every l repeatedly. Why? Because that operation is independent of l, okay? You do it only for only once to the r of T and then you do something with respect to l inside, okay? So what is that something? So this I am going to call rn. Of course, this depends only on r. It doesn't depend on anything else. It's rn, okay? What is this rn? If you look at it very closely, it's nothing but filtering of r of T with a sink, okay? A shifted sink. Well, but still it's just a filtering of r of T with the sink and sampled at T by n. Am I right? So you can write it differently in a different way as filtering of r of T with a sink, sampled at T by n. So what is filtering with a sink? Same as a low pass filter. So if you take r of T, put an anti-aliasing low pass filter between minus n by 2T and n by 2T and then sample it at n by T frequency, you get r of n, okay? So how do you get r of n? r of n is obtained from r of T by doing a low pass filter which is an anti-aliasing low pass filter between minus n by 2T and n by 2T and then you sample at, sample at what? T by n, okay? So n T by n, you get rn, okay? So this is how you obtain rn, okay? So once I have rn, how am I going to write my y tilde of l? It's going to be summation n equals 0, no. And n equals 0 to n minus 1, am I right? Yeah, r of n e power minus j 2 pi nl by n. What is this? This is the f of T of the sequence rn from evaluated at l, okay? So this is nothing but the f of T of r0 through r n minus 1 at l. So that's it. So now the receiver, the entire receiver becomes much, much more simpler. First you do a low pass filter followed by a sampling at T by n and then what do you do? You do a serial to parallel conversion, okay? So you do a serial to parallel conversion to collect r of 0, r of 1, so on till r n minus 1. And then what do you do? You do a, I am sorry, n point f of T. Once you do an n point f of T, what will you get? You get your y tilde of 0, y tilde of 1, y tilde of n minus 1. Now what can you do? Okay? A block slicing is not necessary because we saw once you correlate, only the subcarrier remains, the single corresponding subcarrier remains. So you can do independent slicing to obtain each of your received signals, okay? So behind this ugly looking picture is a very neat simplification of the signal processing involved. All you are doing is low pass filtering followed by sampling, then putting it together, run it through an f of T, you get your blocks to slice, okay? So even though OFDM looks like a very complicated modulation scheme, much more complicated than serial PAM, the signal processing involved involves pretty much the same thing as serial PAM except for the IFFT and FFT blocks, okay? Which seem like extra added things which are not really necessary, right? If you are doing serial PAM on a benign channel, why do you want to do an FFT and IFFT, okay? But it turns out when you have a ISI channel, it gives you several possibilities of having suboptimal equalizers with very, very cheap receivers. The cheap receivers as in assuming FFT is easy to implement, okay? Very cheap receivers are possible even for ISI channels, okay? So that is the great thing about OFDM, okay? So if you do serial PAM, you have to pay for the equalization you are doing, okay? Something extra you have to pay in OFDM, there are some very many great simplifications, okay? Any questions on this? So maybe I should, I am going to draw next the overall picture once again, but in the discrete time model, so far we have been thinking of continuous time and all these things, but eventually ultimately we want to throw away the D2As and the A2Ds, right? And then only draw a picture that involves discrete time signals, okay? So that is what I am going to draw finally, okay? So the final discrete time picture for the no ISI case is going to look like this, okay? So you have symbols coming in at what, okay? So whatever it is, okay? So this is a vector of signals which are coming in at 1 by T, okay? So I do a serial to parallel conversion, okay? I should do a large serial to parallel conversion block to obtain my S of 0, S of 1, so on till S of n minus 1. Remember this is a vector, this is, these are the components of the vector, S k. So if k denotes the kth OFDM symbol, okay? So as the, it is a vector of several symbols, okay? k changes, the next 2 OFDM symbol comes in, okay? I am going to first run it through a IFFT, okay? Then what happens? I have a parallel to serial conversion, I get my what I called as x tilde n, okay? So this is going to go through a channel which is once again can be made discrete time equivalent, okay? Well, some noise and I get my Rn, okay? So remember the channel is not there because it is a delta channel, that is my assumption, okay? So then you do a serial to parallel conversion, okay? Basically put R0 through Rn minus 1 and then you do a, you do a FFT, you get, okay? So I am going to run out of room, let me try. You get your decision variables, y tilde of 0, y tilde of n minus 1, okay? Then all you have to do is put it through a parallel to serial converter and you slice. So you get your sequence out. Yeah, if you have multiple slices, then it is okay. If you want to have one slicer, like if everything depends on how you build it. Many of these serial parallel and all, I am just putting it there for convenience just to take one input and make it multiple inputs, multiple outputs, right? So that is all, okay? So this is how a discrete time OFTM looks. In comparison, serial PAM, okay? Serial PAM would have a transmit filled out, would have basically all of the signals going through without any OFTM and all that to noise. But the signals would come in at, you would imagine them coming in at n by t as opposed to 1 by t, okay? So it is the same thing but it is being done differently, okay? So this is OFTM and the advantage just like I mentioned gives you several advantages, okay? So what I am going to write next is a matrix model for exactly what is happening here because that matrix model is very useful when we go along and consider ISI, right now there is no ISI so it seems very simple. The next thing I am going to introduce is I am going to put x tilde of n through a channel, a discrete time equivalent channel, baseband channel, okay? Once it goes through that, then all these assumptions will not be true anymore, okay? Your transmit signal will not be the orthogonality assumption you made will not be orthogonal anymore, it is getting convolved with the H of t and your signal space would have changed, okay? Even if it is orthogonal, you do not know if it will span your signal space, you do not know anything about your signal space. So only thing you can do is suboptimal reception, okay? So there this matrix picture will help, okay? So this is a matrix equivalent of what I am going to write. I am going to write basically at the transmitter, the x tilde of 0s, the x tilde of 0 to x tilde of n minus 1 is basically obtained as a matrix which is an FFT matrix, okay? Which I will call FFTn, okay? What is the ijth entry of the FFT matrix? Well, I do not know, maybe ikth entry of the FFT matrix, e power j, e power minus j 2 pi, right? j, whatever, ik divided by capital N, okay? So it is in fact an IFFT, so I will write it as FFT inverse, okay? So you can imagine that this FFT matrix is normalized by 1 by root 10 if you want a unitary behavior, okay? So this multiplies s of 0, s of 1 through s of n minus 1, okay? The initial and final symbols will have to be 0 to make sure things fit, but it is okay to put that, okay? So this is your, this we might call as the vector x, this we might call as the vector s, okay? So the vector r which I am going to call r0 to rn minus 1s, okay? That is going to be the vector r, see that is going to be what? It is basically FFTn inverse s plus a noise vector, okay? So this matrix has a lot of good properties, one of them is that it is unitary, okay? So basically it is rotation, okay? Angles are not changed, magnitudes are not changed, it is a pure rotation in a very large dimension, okay? It is a unitary matrix, okay? Whatever n you choose, it is unitary, okay? So it turns out in such a problem, optimal way of decoding s is to simply multiply throughout by FFTn, okay? So it is very clear, it is unitary. So if you multiply n by a unitary matrix, what will happen? Nothing will happen, okay? No statistics will change, it will remain exactly the same, there will be no noise enhancement, nothing will really happen. But if this matrix were not unitary, then you are in trouble, if you do that, noise will get enhanced very severely, okay? So this is a similar model. So when you receive, it is okay to simply multiply by FFTn r gives you s plus FFTn times noise and this has the same statistics as n, okay? Basically there is no noise enhancement. So simply do FFT and then you do the same, same slicer works, okay? So that is the discrete time picture, okay? So now we are going to look at the case bar, there is ISI, okay? So ISI and OFDn, okay? We look at only the discrete time model because that is simpler, that is simple. But if you want to go back to the continuous time model, imagine a continuous time filter then it is going through a same A to D in the sampler, okay? So eventually between the X tilde of n and R of n, you will always have a discrete time picture, okay? That is the discrete time picture that I am having here, okay? So I am going to say my X tilde of n is going through a channel which is H of n to which noise gets added and then you receive Rn, okay? So what is H of n? I am going to say H of n is basically H of 0, H of 1, so on till H, L minus 1, okay? So it is an L tap channel, okay? L or L minus 1 or anything you want to call it, okay? So basically finally you can write R of n is X tilde of n convolved with H of n plus, well I have to write, I will simply write noise, okay? Is it okay? So this is what is your ISI model. So in the ISI model your GL tilde of t will change, it will not be spanning the signal space anymore at the receiver, because the receiver got convolved with some H of t, some channel, we do not know what the channel is. So we cannot hope to do optimal reception at the receiver unless you figure out what the signal space is, okay? So that is not what the main advantage of OFDM is, it turns out within there is a suboptimal receiver, suboptimal way of dealing with ISI which is really, really cheap for the receiver, okay? So you have a very cheap and easy way of dealing with ISI in a suboptimal way and that is by what is adding what is called a cyclic prefix, okay? So what happens in a cyclic prefix is you change your transmission, you do not simply send X tilde of n, you send something more, you incur a penalty in your transmission, okay? But you send it so that this linear convolution becomes a circular convolution, okay? So that is the idea, basic idea behind adding a cyclic prefix is you do not just send your X tilde of n, you send something more, basically you repeat a part of the X tilde of n, you repeat it in such a way that this linear convolution becomes a circular convolution. The reason why circular convolution is nice is F of t is very well behaved with circular convolution, okay? Not linear convolution, okay? So remember cyclic prefix is a huge penalty, okay? So you are sending some extra symbols for which you are wasting some time and power, okay? Maybe not bandwidth, but you are wasting both time and power. Those symbols need not really be transmitted, right? If you can decode them in the receiver, but they simplify your receiver drastically. Essentially you have the same receiver whether or not you have ISI, the exact same receiver will work, okay? So that is the basic advantage, okay? So how do you do that? Let me just quickly go through that picture and then we will see how it works, okay? So this is what you do, you have S of 0 to S of n minus 1, you are doing a, see your transmission has to change, remember that is not just a deception, you do an IFFT, okay? You get X tilde 0 to X tilde n minus 1, okay? And then you have to add cyclic prefix, okay? So this box becomes slightly bigger, okay? What you do is you keep this part unchanged, you send X tilde of 0 to X tilde n minus 1. So in the previous L minus 1 symbols you add X tilde n minus L minus 1 all the way down to X tilde n minus 1, okay? So this is a way you simulate circular convolution through the channel, okay? And then you do a parallel to serial conversion, then this goes through your channel hn, then noise gets added and you receive Rf. So instead of simply sending X tilde of 0 to X tilde of n minus 1, you take the last L minus 1 from your X tilde and then put it before this block. And this has the effect of simulating circular convolution from where to where? From, yeah, only from the received values corresponding to X tilde of 0 to X tilde of n minus 1. The previous things and all won't be circular convolution, it will be something else with, something else which is different, okay? But from the received values corresponding to X tilde of 0 to X tilde of n minus 1, you will be doing circular convolution. And so what you do with the receiver is you throw away everything else, only keep the circular convolution part, then happily repeat your FFT and you will see that has a simplifying effect, okay? I want to show you how that works, okay? So if you do that transmission, remember what is your R of n? It is summation m equals 0 to L minus 1 hm X tilde n minus m plus noise, right? Okay? So if I were to write down the vector R which is, okay, so now the vector R I have to be very careful. The vector R will have a lot of coordinates, you will have, so I already be careful where I start with 0, okay? So I am going to start with 0 corresponding to X tilde of 0. So let me draw, let me write down X tilde also. So the X tilde vector is going to be written as X tilde n minus L plus 1, so on till X tilde n minus 1, then X tilde 0, so on till X tilde n minus 1. This is my transmission for X tilde. The received values corresponding to these things, I am going to write as, I am going to start with R tilde 0 here and this is going to become R tilde of minus 1 for me, okay? So this will be R tilde of minus L plus 1. This is R tilde of 0 to R tilde of n minus 1, okay? So this is how I am going to write my received value R. You will have more because you are transmitting more, but I am going to only concentrate on this part. So this is my new vector R. So I want to write R tilde of 0 all the way down to R tilde of n minus 1 in terms of some matrix here times X tilde of 0 to X tilde of n minus 1. It turns out you will have a, what is called a circle and matrix here, okay? So we will do this in the next class, but I just want to get you thinking about it. Try to fill out this matrix. You will have a circle and matrix which actually corresponds to a circular convolution between the column of the circle and matrix and the X tilde of 0 to X tilde of n minus 1 and that is nothing but if you take FFT now, it will be FFT of that column times the FFT of X tilde and that will give you a very simple equalization which cannot be done in serial PIM or anything like that, okay? So we will stop here for now and we will continue from here in the next class.