 So, this is lecture 35, looks like it is in the middle of several other lectures going on in parallel, okay, so like I said the next few things we will be talking about are practical things that are always done that you will always see in many receivers which are basically change of the structure of the implementation as opposed to the change in the implementation itself. All the filters and everything you derive according to the formulation that we have and then when you actually implement them you might want to change the structure a little bit so that it works out better for you, so that is motivated from a practical point of view. One of the first such structures is what is called fractionally spaced ecolysis, okay. So, so far what we have been looking at is you have a received signal R of t, it gets I guess it gets demodulated, okay, after that what you do is you do a match filtering, right. You are in complex baseband and you do a match filtering that is the ideal front end and then you do symbol rate sampling, okay, so this is that symbol rate. So we also saw when you design transmit pulses you want to get, you want to try to meet the Nyquist criteria, so typically people try to meet, try to design square root raise cosine as a transmit pulse, okay and to have the pulse die down very fast you also use some excess bandwidth beyond 1 by 2t, okay. So for 1 by t you know 1 by 2t is what is needed and then if you use a sink then 1 by 2t is good enough but sink does not die down very fast. So to make everything easier you want to make it die down fast so you use a little excess bandwidth, okay, so more than 1 by 2t, okay. So in theory there is no problem even though you are using more bandwidth than 1 by 2t, okay, so the transmit pulse basically uses, typically uses, uses excess bandwidth, okay. What do I mean by excess bandwidth? So roughly it is some 1 plus alpha by 2t for alpha greater than 0, okay. But remember what is, after filtering what is your sampling rate? It is 1 by t, okay, so this implies sampling rate for your signal is 1 by t, okay. So but you have adjusted the aliasing so carefully so that after aliasing you have a completely flat response, okay but nevertheless the sampling is going to introduce aliasing. If your channel was ideal then one can argue maybe you would not have any problems but if your channel is not ideal and your careful aliasing is still okay but still if the aliasing is, then it becomes a little bit out of control, you cannot control the aliasing very precisely, okay. You are doing h star of f but still the aliasing can cause trouble, okay, so in practice that turns out to be a problem, okay. So in general people who do process signals like to process at least Nyquist rate of the bandwidth. If you are occupying 1 plus alpha by 2t you want a sample so that you can reconstruct your signal if you want to, okay. You do not want to, you never want to sample a signal which you cannot reconstruct, right. So the way if the aliasing has worked out correctly you can recover what? You can recover the symbols properly and nicely with a nice detector but nevertheless you can never really reconstruct your signal because they have already caused aliasing and it has gone for a toss, okay. So it is useful at least at the front end to sample at slightly more than symbol rate and typical choice is to sample at twice the symbol rate, okay. So the reason is you always assume an excess bandwidth of alphas less than 1, right. So you always pick alpha between 0 and 1. So the maximum bandwidth you will ever go to is 1 by t, okay. So you sample at 2 by t so that your first sampling is Nyquist, okay. So typically that is where the fractionally spaced equalizers come in, okay. So equalizers that work with 2 times the symbol rate are called fractionally spaced equalizers. So within the symbol you are sampling twice, okay. So that is the first way of motivating fractionally spaced equalizers. But there is also one more problem which makes fractionally spaced equalizers much more practical than merely looking at Nyquist rate and being able to reconstruct the signal. One can argue why do you want to reconstruct the signal? I am only interested in the symbols. I may not want to reconstruct the signal, okay. So let me write down what the first motivation is. First motivation for 2 by t sampling is that 1 by t sampling cannot result in reconstruction. It cannot be used to reconstruct your signal, okay. So that is part of it. So you want to sample fast enough so you can do that. So but that is only part of the reason. The more important reason is that, yeah, so you may not want to reconstruct. Just giving you an idea. Whenever you want to do DSP you want to do sampling without any aliasing, right. So people might want to do that. That is one reason. The other real reason is if you are doing symbol rate sampling. So this is here is another problem with symbol rate sampling, okay. So you assume that your, the sampling time kt is exact, okay. So you always assume kt is precise, okay. So you know exactly where to sample. There is no delay, no other delay, etc. But in practice there will always be an unknown delay, okay. So and you will never know when to sample exactly, okay. So people are doing the lab can attest to that. So there will always be all kinds of delays and you cannot really do anything about it and you have to pretty much do trial and error, right. So when you have such kind of uncertainty in the timing, you might really want to sample fast enough so that you can reconstruct your signal and be precise about the timing later on, okay. So that is why the timing is the real issue, okay. So if kt is not precise, assume kt is precise. So if we assume, so okay, typically you assume kt is precise but this may not be true, okay. In practice this may not be true and if this is not true then you can never get back your timing. So I will show you some problems because of the kt not being precise. Suppose your h of t that your, your, your match filter is h star of minus t, right. Suppose the h of t which is coming in has been delayed by t not, okay. And you do not know the t not and you are continuing to sample at kt, okay. So you have not adjusted for the t not in your match filter here. Just for it in the match filter everything will be fine but if you do not adjust for it in the match filter, there is a delay in your h of t and you do not know for it but if you are sampling at kt, you will see the, the problem is to look at the folded spectrum, right. I want to look at the folded spectrum here. So my entire equalizer will adapt to the folded spectrum and if the folded spectrum is nice and flat without too many variations then I know my filter will adapt and I will be able to equalize reasonably okay. And if the folded spectrum has some nulls in the frequency or if it is varying a lot then I will never be able to adapt for one and maybe my noise enhancement will be so large that my performance will be very bad. All these things can happen in a, particularly in MMSC implementations of DFE, okay. So this can happen since MMSC DFE is what people may be using mostly. This such kind of problems can happen and the folded spectrum needs to be well behaved, okay. So in the previous case we knew the folded spectrum was nice and it was at least non-negative, real, etc. Okay, so here let's try to compute the folded spectrum. If you try to compute it, you get some interesting answers. So if you want to compute the folded spectrum there, e power j2 pi of t, what do I need? It's going to be 1 by t summation M equals minus infinity to infinity, okay. And then I need the spectrum at this point, what is the spectrum at that point? Yeah, so there is an e power minus j2 pi f t multiplied by h of f, multiplied by h star of f, okay. So I have to alias that, okay. That is the spectrum which is going to be alias. So I will have e power minus j2 pi f minus M by t t naught, right, times h of f minus M by t times h star of f minus M by t, okay. So this is going to be the folded spectrum, okay. And if you see this, you will have first of all e power minus j2 pi of t naught outside divided by t, summation M equals minus infinity to infinity, mod h of f minus M by t square times what? e power j2 pi M t naught by t, okay. So this is what the folded spectrum works out to. First of all, the folded spectrum now is not real anymore, okay. So it doesn't work out to be real because of course you are not matching properly. So it won't work out as real and definitely positive negative we can never say it can go to 0 in modulus, it can go to 0 we never know what is what is going to happen, okay. A useful way to look at it is to assume that your bandwidth is limited to 1 by t which will be which will happen mostly, okay. So even if you use 100% excess bandwidth, bandwidth is going to be limited to 1 by t. And then I am only interested in if I want to find S of e power j omega and if I am interested in omega between 0 and pi, f has to go from 0 and 1 by 2 t, right. 1 by 2 t is what I am interested in. In that interval I can quickly find the exact aliased signal, okay. So here is what happens if you do that. What happens is this, okay. So I am going to quickly write it down and let's see what happens. So it turns out if you do S of e power j omega from this, right. I put omega equals 2 pi f t and look at f between 0 and 1 by 2 t, okay. If I do that S of e power j omega becomes, well you have a e power minus j omega t naught by t times 1 by t should be 1 by t, right. Yeah, 1 by t is okay. It's not very relevant. You will have a modulus H of omega by 2 pi t squared, okay. And then you will have a term which looks something like this, 1 plus gamma e power minus j 2 pi f, 2 pi t naught by t, okay. And what is this gamma? So let me actually reproduce this on the next plot, on the next page, okay. So this is what you get for the folded spectrum for f between 0 and 1 by 2 t. You can check that. You will only have one aliasing term which will come from the one shift, okay. So in fact, gamma will depend on that. Gamma has nothing but modulus H of, well, f minus 1 by t squared divided by modulus H of f squared. Okay, so I have adjusted it so that this is f, okay. This is f. So I have pulled that one term out and I have written it down, okay. So this is what you get. And this term, okay, 1 plus gamma, so this is gamma is usually going to be, it is going to be between 0 and 1, okay. So usually the gamma is going to be less than 1, okay, so for most channels. So this 1 plus gamma times e power minus j 2 pi t naught by t is going to have a lot of fluctuations with, depending on t naught, okay. So it can really go down in magnitude depending on t naught. So I will show you how that works out. So if you look at modulus 1 plus gamma e power minus j 2 pi t naught by t, mod squared, this will work out to 1 plus gamma squared plus 2 gamma cosine 2 pi t naught by t, okay. So we can do the simple computation. So you see depending on t naught, for instance if t naught is t by 2, t by 2, then this becomes 1 minus gamma squared. So this is going to really cause a big dip in the way the frequency is going to behave, okay. So basically the thing is this is a function of t naught, okay. So the s of e power j omega varies according to t naught. And the variation cannot really be easily quantified. If t naught is t by 2, it is going to be really bad, okay. So it is going to go down significantly in magnitude. Otherwise it might change in different ways. So you cannot really control how this s of e power j omega is going to behave if you do not know t naught, okay. And even if your original channel did not have any nulls, because of this t naught, you might have reductions at various points, okay. So there might be fades and all that, okay. So depending on how the t naught adjusts, okay. So basically moral of the story is an unknown delay can cause bad spectrum, can result in a bad-folded spectrum, okay. And once you have a poor-folded spectrum, first of all the match filter bound itself might be bad, okay, which means your equalizer itself is going to be quite bad, okay. And then you cannot really, all your figures of merit will go for a toss if you are equalizing to a bad-folded spectrum, okay. So you remember your generalized model is trying to equalize to some folded spectrum, right. So if that itself is bad, you cannot do much. And secondly, if you are trying to adapt and if your folded spectrum has nulls, even that there is a problem, okay. So your eigenvalues might go up to zero, so you will have all kinds of instability problems, okay. So that is the main reason why symbol rate sampling, at least for the symbol rate sampling is not preferred immediately after match filter, okay. Because you have no control over the timing phase, you cannot be very precise in your timing and because of that you want to sample at least twice the symbol rate, so that you have an option of being able to figure out, so that this timing phase problem goes away, okay. So because in that case there is no aliasing, right. This extra term won't come in if you do two times the symbol rate sampling. So this one plus gamma times factors won't be there. So t0 won't really cause that significant a problem, okay. It will only cause a delay e power minus j omega t0 by t and that you can deal with if you have Nyquist rate sampling, your filter will roughly deal with it. You will know exactly where to do the sampling, okay. You can do some adjustments in your receiver for that, okay. But if you don't, if you allow the aliasing to happen, an unknown t0 will introduce factors like 1 plus gamma e power minus j t0 which is going to give you a lot of fluctuations in your spectrum and a lot of problems, okay. So how do you overcome the problem? You overcome the problem by doing what's called fractionally spaced equalizes, okay. So remember as far as performance metrics and structure is concerned, basically a fractionally spaced equalizer will exactly mimic any other symbol rate equalizer that you design for, okay. There is no change in the ideal case. The ideal case when you know t0 or something else, there is no change. Okay in practice when you don't know these things and you are adapting for an unknown t0 and you are adapting for an unknown channel, all these things will play a part, okay. So structurally an FSC will be exactly the same as a BFE or something. In fact, it will do the exact same filtering, conceptually it won't do any other change. But in practice when you adopt for unknown things, this is much more useful, okay. So here is how I am going to derive the FSC. I will start with the good old structure from before where you have a match filter followed by a symbol rate sampler followed by I will call it C1 of Z which is your symbol rate equalizer. In the case of linear equalizer, that's the only filter. In the case of MMSC, in the case of DFE, it's going to be the precursor filter, okay. So I will simply call it the precursor, the traditional precursor that you define that you derived using either what the ZFLE or the ZF DFE or the MMSCLE or the MMSC DFE. You would have derived a precursor, that's the precursor, okay. So remember in the precursor the assumption is what? Your sampling time is ideal and all that. So that assumption has already been made and in adaptive version you adapt, okay. So depending on what happens you get the, you get a filter, okay. So if you define designing without any constraints on complexity, you would assume ideal sampling, exact sampling, no delay, etc., delay has been compensated for and you do it, okay. So the first change, okay, first change which is very interesting that people make is in this moving towards fractally spaced equalizer is to push the precursor before the sampler, okay. So remember the precursor is the one that's going to be more sensitive to this sampling function problem, okay, because you do the, do most of the work. The post-cursor, if at all it's there, it's going to be a causal filter and it won't cause too much trouble. So the precursor is the one which is more sensitive to your sampling instant time. So you might, first thing you do is you push it to the left of the sampler, of course I have to sample again. I'm not going to do that in analog, okay. So that doesn't mean I'm going to implement the precursor in analog, okay, but I'm pushing it front and then I'll do twice the symbol rate sampling before so that I can comfortably do this design insensitive to timing phase. It's a very simple shift, but just follow the argument, okay. So r of t, once again, gets multiplied by e power minus j 2 pi fct and then you have h star of f, okay. So what will this filter be? I'm going to call it c1 e power j 2 pi ft, okay, followed by symbol rate sampling and the assumption is the post-cursor is not too sensitive to the, to errors in this sampling. So you can manage to do this filter properly. The post-cursor will not cause too much trouble, okay. All these spectral nulls and changes are not a big trouble for the post-cursor, okay. You can go back and check that, right. So now I want to implement these two filters at, at two times the symbol rate. The reason is then I won't cause any aliasing and my timing phase is no problem. I can, I don't have to worry about being exact about the timing phase when I do. It's not insensitive, it's not as sensitive as the precursor. Precursor is the one where you really do the, so if you go back and look at your MMC DFE design, you get the 1 by m stars and nasty stuff only in the precursor. Post-cursor, you'll mostly get causal, just nothing, the zero term is not there. It's just a causal filter and it will usually be stable. You can go back and look at many of the examples that you did. Usually it will be a simple nice filter, won't cause any problems, okay. So, so here's, here's what you do. So, so to make this insensitive the first, the whole thing insensitive to any timing problems, what, what usually is done is you do R of t, multiply by e power minus j 2 pi f c t, you put an anti aliasing low pass filter, okay. So this is pretty much an anti aliasing filter, okay. So maybe the bandwidth is 1 by t. Yeah, that's a good bandwidth to pick. And if you know, but if you know the excess bandwidth exactly, you can even pick 1 plus alpha by 2 t. So throw away everything else. And then you sample at kt by 2, okay. So twice the symbol rate. So you know you've done, you've done an anti aliasing filter and then you've done Nyquist rate sampling. So basically two times, more than two times the bandwidth is your sampling frequency. So you, you can have exact reconstruction of the signal that came in. And then you're doing filtering, but now I'll simply write it as one filter at e power j pi f t, okay, why, why don't I have the 2? Because now I'm a t by 2, okay. So I've adjusted that. So what will happen to your frequency when I do two times the samplings? Because of that, the, the, basically the axis will change, okay, before and after the symbol rate sampler. So you should adjust for that very carefully. And then you sample at kt, okay. So what is the c of e power j to, e power j pi f t? This will work out to h star of f times c1 e power j 2 pi f t, okay. So when you do two times, you'll have two versions of the lower, lower rate, right. So that's how the, that's how the thing will come. So you must have studied this is called the standard up sampling procedure, right. So you, so this equation is important, pay some attention to it. If you've not understood it, there is some reasonable amount of DSP involved here. Why do I get the pi of t for the c inside and j 2 pi of t for the c1, okay. So originally I designed the c1 of z, assuming symbol rate sampling. And then I'm moving it inside and then I'm actually sampling twice, okay. So, so what happens there is an important, important thing to pay attention to, okay. So, so basically from here to here, these two structures are exactly same in the ideal case. What is the ideal case when you sample, when your symbol rate sampling is exact and precise. There's no delay to work out for, no problem and the sampling phase is exact, then these two are exactly the same. But in the non-idl case when t0 is unknown, this fractionally spaced equalizer will work much, much better, okay. Because your spectrum after the anti-aliasing filter and if you do kt by 2 is well controlled. There is no aliasing from something else with an unknown t0 to push it down or push it up or do some crazy things, okay. It's very well controlled. You know exactly what you're doing and this filter is happily working at with no aliasing. So, there's no, it's not very sensitive to any timing problem. You can do a nice filtering and then you do a symbol rate sampling, okay. So, there's no problem, right. Now, remember this symbol rate sampling, you have to pick either the, see for every symbol there are two points that you have to pick. You have to either pick one or the other, okay. And then usually there is a timing recovery loop which will tell you which point to pick. If one point is not good enough, you push to the other. There's always a little slip that happens in timing recovery there. So, you can work with the two times, the two samples per symbol to decide which one you want to pick usually, okay. So, any questions on this? Okay, so this is a fractionally spaced equalizer, okay. The first point I want to point out is deriving a fractionally spaced equalizer is not any more complicated than all the derivations you've already done, okay. You've already, you already know given the folded spectrum, how to derive the zero forcing linear equalizer, zero forcing DFE, MMSC LE and MMSC DFE, okay. You just take the precursor from there, move it to the left and blindly use this formula, you will get the filter needed inside the, for the fractionally spaced equalizer, okay. That's the first point. So, no separate derivation is necessary. The same derivation will quickly tell you the answer for the fractionally spaced equalizer as well. That's the first point. Second thing is the fractionally spaced equalizer fuses the matched filter and the precursor into one filter, okay. And the analog front end is simply what? A low pass filter followed by Nyquist rate sampling, okay. So, you see why such a structure is very attractive in practice, okay. The analog front end usually has to be very standardized because you don't want to design that based on the channel, etc., etc. So, you want to standardize it to a low pass filter at a certain frequency, right. If it's that standard, then any other change you want to make, tomorrow if you want to change the transmit pulse, right. You don't have to change the analog front end, it's still the same. It's only a firmware level change if you want the technical language, right. There's no hardware change required. Anytime you need a hardware change, it's a complicated operation. Can't do anything else. But if you're only changing the transmit pulse from a, which is inside, well, usually these things are sitting inside some digital processor inside your board. And there's no problem, you just change the firmware on the processor, everything will work, okay. So that's why such a structure is more preferred today. So you just want to have a low pass filter, but you sample two times, you design a filter, it'll work, okay. So that's one of the many practical advantages of the fractionally spaced equalizer, okay. So any questions is okay. So unless actually, I think maybe if I give you a problem in the assignment and you actually work it out, at that time it'll be more clear, okay. But again, ideally, you will not expect to see any difference. For instance, a figure of merit will be exactly the same as what you had before. So the ideal analysis and everything will be exactly the same, nothing will change. The practice with an unknown T naught, when you're trying to design a system, two times the symbol rate is way better than one symbol rate, okay. So you'll see this, not knowing the delay will cause a lot of problem, okay. So you'll have to adjust for it a lot. Sometimes I think people are doing the lab, sometimes just start at the signal and manually do the adjustment, right. So you may not be able to do it in practice. But if you have two times the symbol rate sampling, and it's convenient to do it, okay. So that's the thing to do. Okay, so any other point I wanted to make? Yeah, so the next thing is about adaptation, okay. So adaptation is a bit of an issue. So remember, your adaptation happens only at symbol rate. Why does the adaptation happen only at symbol rate? Yeah, the error is calculated only for one symbol. You can't calculate for an imaginary symbol once again in the middle, okay. So the adaptation happens only at symbol rate. But one of your adapting coefficients is going to work at two times the symbol rate. So what do you do? Yeah, just drop every other sample that your precursor is outputting, okay. So don't adapt for every alternate sample. So only adapt whenever you're actually making a decision, okay. For the post cursor, though, there's no problem. Post cursor is only working at symbol rate. If you have a post cursor, post cursor is working at symbol rate. There's no problem. But the precursor is working at two times the symbol rate. So every alternate symbol, you won't adapt, okay. So you drop alternate sample, not symbol sample, okay, in your adaptation. So that's a very simple way to adapt. So that's the only modification. So remember the CK plus one is CK plus beta EK ZK star. That still remains the same, but your precursor is producing one extra sample. So you just simply drop it, take the only the symbol rate samples and adapt, okay. So the filter for the precursor will change only at symbol rate. Symbol rate won't change in any other way, okay. But there's a more subtle problem, okay. So because you did the anti-aliasing filter and aliased and your precursor is working with a very good spectrum, it might also work with, if your excess bandwidth is less than 100 percent, okay, and you're anti-aliasing down to 1 plus alpha by 2T and you're sampling at 2 by T, you will necessarily have zero for some part, okay. So your spectrum that the precursor has to adapt to will necessarily have zero. So what's the problem because of that? Well, the eigenvalues will go to zero, one of the eigenvalues will go to zero, okay. So because of that, there might be some instability, okay. So because of 2 by T sampling, this results in eigenvalues of phi going to zero, okay. So there's lots of theory here about how to do adaptive filter. So this is the problem in adaptive filter design, okay. So how do you adapt a filter when you know that this eigenvalues for phi are going to go to zero, this the correlation matrix. So that can cause some trouble, okay. So that's one thing to be aware of in the adaptation, okay. So there might be some instability because of that, but this is seriously not a problem. People can still deal with it, okay. There's one more problem from practice, not from adaptation, but from a complexity point of view that needs some attention, you have to pay some attention to, okay. So when you are doing, suppose N-tap filtering with symbol rate, okay. Now your precursor has to, now your fractionally spaced equalizer is working at 2 times the symbol rate, okay. So which means what? So what is the problem? If you once again stick to only N-tap for the precursor, you will be operating only over half the time duration, right. Your N-tap in the symbol rate would have worked over N times T time, over N times T time, okay. But when you stick to the same number of taps with the fractionally spaced equalizer, you are working with T by 2, so you will cut down to N-t by 2, okay. So but in spite of that, performance is comparable of these two things, okay. So that's one thing. So to keep complexity low, same number of taps for 2 by T and 1 by T sampled versions, precursors, okay. To keep the complexity the same, you want to do that, okay. Otherwise, your number of taps will go up, right and you may not be able to do it and this might cause some performance degradation, okay because you are not using, you are not, your tail is not long enough, right in your equalizer. But it's in practice people have done simulations because I don't think you can analyze it without simulations. So you do simulations and you see that it works out, you pick some suitable number. There are so many other advantages that FSE gives you that you are willing to live with this disadvantage, okay. So that's the other point to keep in mind, okay. All right, so the fractionally spaced equalizer is one thing which is done in practice. One more thing which is done is what's called pass band equalization, okay. So let me talk about that very briefly without going into great detail, okay. So often times what will happen is when your R of T comes in and you multiply by e power minus j 2 pi, so typically I assume it is FCT but you can never know that, right. You'll never know whether it is FCRF1. Usually there is something called a carrier recovery loop which will adjust for it, okay. So in your receiver, maybe I'll talk about it later but you'll adjust for it but in general you might want to adjust for the, maybe you might want to adjust for the delta F between your F1 and Fc. So if there is a delta F between your F1 and Fc. Maybe you want to adjust for the delta F inside your equalizer, okay. So there are several cases in which this might be useful, okay. So some cases what people do is the center frequency itself might be so high that you don't want to run a carrier recovery loop at that frequency. You want to run a carrier recovery loop at a lower frequency, okay. So for that you might do some intermediate frequency conversion but sometimes it's useful, maybe I'll justify it later when I draw the whole receiver, it's useful sometimes to put your carrier recovery, the carrier recovery at an intermediate frequency inside your equalization loop, okay. So I'll show you how that is done, okay. Hopefully I'll be able to reproduce this exactly, okay. So you have, so I'll show you just for the linear equalizer, the post cursor I won't show, only the precursor, okay. So the precursor is going through, okay, well, I'm sorry. The post cursor is completely missing in this picture. So remember that post cursor also exists but I'm just showing this real quick to motivate how this is going to work. So there's going to be an error calculated here and that's going to feed here for adapting, right. This is how your typical Equalizer design works. There's a loop here for adapting, okay. So now if there is a delta F, okay, and the slicer works without delta F, it's going to be big time confusion, okay. You see why there'll be confusion now, okay. So e part, so how will this show up, first of all? How will this show up here? So suppose this is some xk, right. And the, and xk you expect it to be around the transmitted constellation. Suppose the transmitted constellation was, say 4 QAM. The xk is going to fall, you expect somewhere around here, right. And that's why you're slicing for the 4 QAM to get your x hat, yeah, sk, sorry. S hat of k is what you're getting after the slicing, right. So now, if there is, if this, this, this would happen if delta F is 0. If delta F is not 0, what's going to happen? Xk is going to be this guy multiplied by e power j 2 pi delta Ft, okay. So you're going to get, this xk is going to get multiplied by your xk, xk is going to get multiplied by e power j 2 pi delta Ft, okay. In fact, even t is there, so, so t is going to be what, kt, I'm sorry. Should put kt here, because you've already sampled it, so kt, okay. So depending on k, what's going to happen? The rotation is also going to change. Admittedly, delta F will be small, it won't be very large, maybe some 0.001, but as you keep sending several symbols, they will slowly start rotating and you'll start getting points in a circle ideally or maybe even in an ellipse in some crazy situation, okay. So you'll start getting rotations and you have to compensate for this rotation in your slicer. Otherwise, your slicer error will go for a toss and your equalizer coefficients will also nicely rotate. So you'll see, they'll also follow some sine waves here and there in the complex domain, they'll also rotate. So everything will be rotating if your eigenvalues are not too bad and if you're catching up every time, you'll see the whole thing will keep on rotating if you don't adjust for it, okay. So this is something you can check. I think people in the lab should check this. If you find a delta F, see what happens. You'll see everything will rotate, everything will follow a sine wave, okay. So moral of the story is anytime you see a sine wave in base band, what's the culprit? There is a delta F somewhere, okay. So there's something as scripted, okay. So the way to adjust for it, okay, I'm running out of time. The way to adjust for it is what? If you know delta F, assuming you know delta F or you have some loop which is telling you what delta F is, okay, there is some loop which is telling you what delta F is, okay. If you know that, then what should you do? You should multiply here with e power minus j two pi delta Ft, oh, kt, I'm sorry, okay. Then what should you do after the slicer? Once again multiply because your error has to be calculated for the rotated version. Reason is your equalizer is working with the rotated version only. It's not working with the original one. So one more multiplication you put here and then you say this is getting multiplied by g power j two pi delta F, kt, okay. So this is something you'll have to put in a pass band equalizer where you expect the delta F, okay, so this delta F can be different depending on what your choice is. So this is a choice, so this is a practical choice that you might have to make some time depending on how your oscillators are in your receiver board and all these things. You might want to put some oscillators, not put some oscillators based on that, you might have to deal with the delta F inside your equalizer. In that case, you have what's called a pass band equalizer and you deal with it in this way, okay. There's an equivalent structure here where you can move the multiplication outside of the slicer loop, move it outside of the slicer, okay. So you'll see in later structures there's some use in moving it outside of the slicer because then your post cursor will come inside the slicer. Your post cursor is not going to be affected by this, all this confusion and multiplication, okay. So you might want to pull it out. The way to pull it out I believe is to take this multiplication, these two multiplications and put them, put one multiplication here, e power minus j two pi delta F kt and put the other multiplication after the error, e power j two pi delta F kt. Okay, so it's equivalent to what I had before except that I've pushed the multiplications outside of the slicer, move them far away from the slicer, okay. The slicer is not going to be affected that much better. Multiplications, so the post cursor will be largely okay. It won't function depending on the delta F. You don't have to worry about the post cursor, okay. So this equalizer structure is called a pass band equalizer. Okay, so I think I'm running out of time. I'll stop here. In the next class I'll start by drawing a very elaborate receiver which includes a fractionally spaced equalizer, includes the carrier recovery loop, includes the timing recovery loop and post cursor, pre cursor, everything. Show you one big structure and talk about that a little bit, okay. So we'll see you in the next class.