Okay, so this is lecture 19. Okay, so the last class we saw quite a few things and I think There might have been a few things which are quite totally new to you and maybe you're confused about it The only way to get rid of that confusion is to go back and read something more about DSP and speculative factorization All that so I'm gonna give a little bit of a glimpse into those things once again real quick to get you started on that okay, so So few few few ideas on speculative factorization before we proceed Okay, it's actually a very simple thing. It just sounds like a very major result, but it's it's quite simple.  Okay, so complex to complex is unique. There's no problem there Okay, so what happens in speculative factorization is you have to imagine that you're given the What are you given you're not given X of e power j omega you're given Only the magnitude of X of e power j omega as in so what what you're given the way it's described is you're given the Fourier transform of the Auto-correlation of X. Okay, suppose I do Rx Say Rxx of M and then do its DTFT. What will happen to that? You know, this is going to be this is what it's going to be. Okay from here. You cannot go back to a Unique Xn. Is that clear? Right clearly. I cannot go back to a unique Xn. Okay, so because only the magnitude of X of e power j omega is given the face can be Arbitrary, I can pick any face I want well arbitrary according to maybe some condition if X of n is real then the face has to satisfy some symmetry Etc. But if X of n is in general complex The face of X of e power j omega can be anything Okay, so there are an infinite number of face responses that we you can pick and you can go back from mod X of e power j omega 2, mod X of e power j omega 2 Technically an infinite number of X. Okay, so in some cases you might be interested in Something more specific than an infinite number of possibilities. Okay, so in that case a good criteria is to select this Minimum face Xn. Suppose I want a minimum face Xn then it turns out there's a unique answer So the way I wrote it down. There's a unique answer. That's what I gave you right in yesterday's class So if you if you want to go to the minimum face Xn, there is a unique answer. It has to be put a minimum face Constraint, it's a big Constraint right, so it has to be unique the way you can derive this in so many different ways And it has to be unique it works out to be unique in general. Okay, so that's the basic idea and spectral factorization Okay, so the way it showed up for us is a little bit different So it might not have been very obvious that this is what I was doing So this is the basic idea and spectral factorization the way it's used in practice might be in so many different ways Okay, so several times You might have come to x of e power j omega through some other route. Okay, maybe not a minimum face Xn, okay, which is what happened to us, right? So what did we do? We had H H of t Okay, from H of t we went to the Auto-correlation of H and from there we went to the PSD of H of t right, so that's what that's how we went but H of t may not me may not be Minimum face. Okay, so So if you want to go back just given the magnitude response You might want to go back to the minimum face thing and then we saw the advantage of having minimum face Okay, so once you once you split it up as a minimum phase factorization then you can do a suitable filtering to reduce The ISI to only the causal part that was a big advantage for instance, if it is not minimum phase if it has some zeros and poles outside or Things like that you can't do that filtering very easily. It's not clear how to do that filter Okay, once you do the minimum phase factorization, you can take the maximum phase part Do the reciprocal of that and get a meaningful filter once you get a meaningful stable filter you can run that and then get only the Causal part from a minimum phase filter as the ISI. Okay, so that's the idea that seems to Seems to make sense and hopefully to that to so many new ideas here. Hopefully some of it is Reasonable clear. Okay, so if there's any question lingering questions, let me write write down the structure then I'll ask you for questions Okay, so what's happening is? You have so remember this The symbol sequence. I'm thinking of it as actually a vector of length L from the alphabet, but I can also think of it as a Discrete time signal. There's no problem. It actually goes through a transmit filter GT and then after that a channel CT okay together these two I'm going to call as HT okay, then noise gets added Okay, and I got RT. Okay, so then I wanted to see how can I find S? Okay, one criteria seem to be maybe a distance Criteria based on S. So I define this JA which was some kind of a distance between R of T and Well summation AK H of T minus KT right Right, so this was my criteria in that criteria filtering by H star of minus T came out Naturally, okay, so I had to find this YK which is basically a filtering by H star of minus T and then Sampling okay, it came out in this criteria Okay, if you if you have chosen some other criteria something else might have happened I don't know. Okay, so since we chose this criteria the filtering by H star of minus T happened to occur Naturally, okay, and you sample at symbol rate to get YK Okay, so this YK we saw can be written in terms of SK in what fashion It's actually what SK convolved with Rho HK. Okay, so it's not just plus Some n prime K. The reason why I'm putting n prime is it's it's not really white. Okay, it's some it's got some PSD Okay, so notice. How did I get the autocorrelation? It's very crucial. Okay, the reason why I got autocorrelation is because I Did a match filtering. Okay, so if I did not do this match filtering here if this H star Was some other filter say a low-pass filter or some such thing then I won't get a autocorrelation Okay, so that's crucial case the fact that I know H of T to do H star of minus T Was crucial to get an autocorrelation if I chose some other arbitrary filter and sampled I won't get autocorrelation. I'll get what instead Some yeah, maybe cross correlation or some some some weird signal here Okay, what's the advantage in getting autocorrelation as opposed to any other signal? Then I can do spectral factorization. So you see how the spectral factorization flew into that picture so if you did not have spectral factorization and if you only had Yk and you had to compute this metric for each a There's no way you can do with linear computation. It'll become l square computation Maybe there are other cases where it works out But in general it'll be an l square computation which becomes pretty much impractical to do for any even a fixed a Okay, so all these things are important steps and they seem to come naturally and quickly when I do it But when you go back it won't should be clear in your head why it's coming through that way Okay, so how I got Yk and why is it that that resulted in the autocorrelation and why that results in a PSD? Which can be then spectrally factorized. Okay, so what's the big point in spectral factorization? Okay, so before I go there. What are the statistics for this? Okay, it's going to be it's it'll not be white Okay, but each So it's going to be Gaussian with PSD. What what will be the PSD sh of Well e part j 2 pi of t if you will Okay Is that clear? What is sh? That's the Fourier transform of rho h of k. Okay, how did I get that? Okay, it's actually h star of minus t multiplied by H of t itself that works out and you get the PSD. Okay, so this n prime k is not white Okay, that's another issue floating around.  Okay, so then you do a substitution. How do you do that substitution? You do it in a weird way you take the PSD Do a spectral factorization and get sh of e part j 2 pi of t as what? e part j 2 pi of t Okay, well it has to factor as magnitude square of something right, how did I pick that specific m? I picked it so that it became gamma square times m of e part j well 2 pi of t times m star of e part j 2 pi of t in addition If you do a DTFT of m of t you get a m of k which is Minimum phase. Okay, so this is well loosely minimum phase Okay, well like like I pointed out if you're given a PSD you can do this Factorization if you don't insist for minimum phase in an infinite variety of way Okay, so you can obviously right the magnitude response is fixed once you fix the magnitude response You can vary the phase in an infinite way and get any signal you want here, but that's not what I picked I took that one which was minimum phase. Okay, so once I did that it turns out It's meaningful to well. I also I might have written this in the z-transform in z-transform if you want to write it becomes what? SHZ is m square a gamma square m of z m star of 1 by z star. Okay, and if you want to write it in the time domain, what will happen? row h of k equals gamma square mk convolved with m star of minus k Okay, all these things are equivalent ways of writing the minimum phase factorization spectral factorization unique minimum phase spectral factorization Okay, so that uniqueness must have surprised you if it didn't surprise you before so once you say insist on minimum phase The factorization becomes unique. Okay, well It's also a convenience. Okay, any other type of face also make it unique. Yes, it's not a big deal But still it's minimum phase has a nice property that What is the nice property most of the energy is going to be concentrated in the first few taps of my mk Okay, so that that we will use later on in our in our decoder when we filter. Okay, so once we did this it turned out you take the yk and Do what remember? How did I get the yk? I took r of t filtered with sorry, this is r of t filtered with h star of minus t and sampled at Kt I got yk and then what do you do?  I have The thing works out, right? So that's the basic basic Okay Is that clear why this works out? Yes, think about it. Okay, so you get a zk and and then in terms of zk things become really really easy Okay, so once you write it in terms of zk it turns out this j a becomes equivalent to Another metric which I called j prime a which was what? norm of Okay, so let me not write norm. I'll just happily write down. What did I mean? I think I put a summation, right? Did I have a summation? Summation K equals 0 to infinity. Did I have 0 to infinity? Okay, it will be modulus zk minus what? ak convolved with mk square Okay, norm squared. There's a norm squared. Maybe it's not showing up properly in the Okay, so maybe maybe I should move this guy Oops Think didn't move Okay, so more than this filter being Anything implementable or worrying about that those are not the big deal. That's not the big deal. The big deal was Because I did this filtering My j a the metric became much simpler than before that's that is the big deal Okay, so forget about this filter and its properties may or may not have any of the properties I said that you don't care but because I did that special special substitution with the minimum phase Minimum phase m of z of this filtering with the minimum phase m it became The j a simplified to this okay You can go back and check it and use my the way I did that the minimum phase assumptions crucial there Otherwise, it won't won't work so nicely. Okay, so at least I believe that's what should happen Okay, so those are things you can check. Okay, so this is very nice to see Okay, so not only that there's one more thing that's nice nicely happens. What is the next thing that happens? Okay, what happens to zk if you want to write it in terms of sk? It's going to be sk convolved with mk plus Nk now what is Nk now? It's going to be white. Okay, so that's a big That's a big Advantage normal zero mean I mean variance and not by to gamma square Okay, so it's a simple derivation to do Okay, so once again you multiply you see sh of z on the top the denominator you have m star But then if you do the m star times m you once again get the sh and you'll have an extra gamma square floating around Okay, right? You have a gamma square in the denominator. You might say why did I put the gamma square? Maybe I don't want the gamma square. It's okay. Those are all constants which don't matter at the end Okay, so you put the gamma square there you get this guy Okay, so so several nice things happen when I do this Filtering okay, so you have to go back and ask yourself very closely the question of why why the spectral factorization is crucial Why can't I take some other? Instead of mk, what if I take some other factorization with some other phase necessarily not minimum phase, okay? So maybe most of these things you can repeat. Okay, but the biggest advantage which you'll never have is Most of the energy being concentrated in the first few times, okay Then you can eventually approximate m with only its first few taps and you'll see the first few taps will make a big difference to your Complexity as we go along. Okay, so maybe the crucial statement is in the minimum phase thing is most of the energy is concentrated in the first few Taps most of the other steps you might be able to repeat with any other factorization Okay, so it may not be a big deal but this crucial step of most of the energy being concentrated in the first few taps comes pretty much because of the Minimum phase criteria, but anyway go back and check those things and make sure you understand Where exactly this minimum unique thing is coming in what what are all the roles it's playing, okay? All right, so any questions it's fine People are happy with this picture Okay, so now based on this I was able to draw a nice picture of the entire receiver at least for the calculation of This j prime a okay, so what did I do? I took the zk Okay, remember should remember how I got the zk, okay So I got the zk through from r of t filter with h star of minus t sample at some simple rate and then 1 by gamma square m star of 1 by z star Okay, so we do that you get zk and then what do you do in computing j a prime? Okay, so well, let me do it differently Okay It's quite easy you send it through a summer difference with what? You send your ak through Mk or m of z capital M of z, right?  You sum square you will get your j prime a okay So in reality in practice if you want to implement anything like this one of the things you have to struggle with is delays Okay, so delay will never be obvious k zk is coming through so many Steps in your filtering there are so many delays that will be cost you might have delayed your cost for causality You might have delayed your transmit filter for causality. You might have to even delay or 1 by gamma square I don't know how you will implement your 1 by gamma square m star 1 basis So that might have been one more crazy implementation So the end there will be all kinds of delays Okay, and you have to adjust for the delay properly if this j a prime has to make any sense Okay, otherwise it will be something else and you'll be measuring something else and you have no business doing such things Okay, so I'm sure people who are doing the lab realize how important this delay and adjustment is measure at different points You get completely different things. This doesn't make any sense Okay, so very be very careful about delay when you if at all you're trying to implement such things in practice All right, so you do this you get j a prime So now I have a nice nice expression that I wrote down for the for the For my decoder. I'm going to minimize over All a j a prime Okay Right, so maybe the computation of j a prime is doable for one a one a but that's not where my problem ends Okay, so that's where my problem pretty much begins, right? So I have to then compute J a prime for each a and I have too many of those is case the number of such a is is size of X to the power L and that's one too many. Okay, so for any kind of Reasonable L you want and reasonable size of x even if it is too to power L is Too fastly growing an L and you can't hope to do anything Better than that. Okay, so that problem still remains to be solved Okay, so we've not solved that problem at and it turns out there is a wonderful algorithm called the Viterbi algorithm Which will solve that problem and make it still linearly grow with L assuming your mk is As doesn't doesn't have too many tabs. Okay, so you can do that at least as a finite number of tabs Okay, it's possible to solve that problem. We'll come to it later. But before that I want to do this neat and clean Autogonal projection and find an orthogonal basis for our set of signals and and see that we pretty much recover the same structure that we have now Okay, so we'll see the relationship between the two of them and then proceed along Okay, so so after at that point I'll make some general comments of what to do as we go along and then we'll stop there Okay, so any questions right now? Okay Okay, so the question is basically You're saying You're coming at this picture and saying why don't you first do a c inverse of t? Okay So the problem is okay, so I have to have to make a few comments I thought I'll make the comments after I show the orthogonal projection between even now It's a good time. Okay, so any filtering you do at the receiver It'll be nice if only the signal part went in through that and the noise part didn't go through that Okay, so but you can't control that. Okay, so the noise will also go through that same filter Okay, and c inverse of t may not be a good filter It might boost up your noise variance to a great level and that that's something you may not want Okay, so what you really want to do optimally is what? Do an orthogonal projection if you know the orthogonal projection, you know that's going to be Optimal, okay, so go to the signal space find an orthogonal basis Do an orthogonal projection that is the best thing to do. Okay, which is what I'm going to do now Okay, anything else you do like c inverse of t for instance is a heuristic Including this minimum distance is also a heuristic right now. Okay, it's a heuristic It may or may not give you the best probability of error Okay, but that's definitely doable and it's done. Okay, a lot of people do this It's called a linear equalization type So you do a c inverse of t get rid of it and then continue with your g of t it can be done in practice Maybe that's it's a good choice, but but it may need not be the ideal thing. Okay, any other question any other question Okay, so in practice also this this kind of a structure where you do a Uh, where you do two stages of equalization one stage for getting rid of the Anti-causal ISI and another stage for getting rid of the causal ISI Can be shown to be very good as in even theoretically one can show that can approach capacity So those kinds of constructions are usually preferred for that reason. Okay, but you can also do the c inverse of t It's okay. All right. So the next thing I want to do is this uh, orthonormal basis projection Let's let's figure out Okay, so I'll try to do this and make some Uh comments and then maybe we'll see how we are doing Okay. All right. So So what's our uh, what's our set of signals and what's an orthonormal basis for for the signals? Okay, so What you'll see is it's very interesting Okay, so what is our set of signals? Okay, set of signals at transmitter Okay, does it make sense to talk about some set of signals at transmitter? Okay, I should now I should never talk about Set of signals at transmitter, right? So I'm not doing I should not do orthogonal projection on that. What should I do? At the receiver signal component at the receiver for for that. I have to do the orthogonal projection. Okay, so always remember that I have to look at Set of signals at the receiver or in r of t. These are summation k equals 0 to l minus 1 a k H of t minus k t k for uh, a in Excel Okay, these are my set of signals. How many signals do I have? Size x to the power l. Okay, I have a lot of them now I want to find an orthonormal basis for This set of signals once I find it then I know the optimal thing to do is to project on to that orthonormal basis and then do a Minimum distance is here. Okay, so and if I do the orthonormal basis projection, I also know that my noise is going to be Just n naught by 2 and all that I know all those things. So it's very nice to do that Okay, so it's optimal in various ways and this will be the optimal receiver. Okay, so here's a claim Okay, which is an interesting claim and you'll see where it comes from Okay, suppose I take this h of t. Okay, let me write down a few properties for this h of t I'll come to the claim is slow enough Maybe it's Fourier transform as h f Okay, I'll let that be so and then Uh, I also want to do a few things before I'm able to define the orthonormal basis Okay, h of t convolved with h star of minus t Okay, if I do a Fourier transform, what will I get? mod h f square Okay, then if I sample this h of t convolved with h star of minus t At k t, what will I get? If I do a dt of t What will I get? It'll alias. No, I'll do a 1 by t aliasing. So 1 by t summation m equals minus infinity to infinity mod h of f minus m by t square. Okay, so this I will call sh e part j 2 pi f t Okay, so that will be my aliasing. Okay, so I'll take this sh e part j 2 pi f t. I'll do a spectral factorization on it. I know it's real and non-negative So spectral factorization will give me a unique m of e part j 2 pi f t. Same as before. I mean, there's nothing new here j 2 pi f t This is gamma square m of e part j 2 pi f t m star e part j 2 pi Okay, so this is spectral factorization All right Okay So I have all my notation and tools to define the orthonormal basis Okay, so first thing I'm going to do is I'm going to define a phi t which is The Fourier transform which is the inverse Fourier transform of phi f Which is defined to be h of f divided by gamma m e part j 2 pi f t So I'm wondering if there should be an m star or m. Let me just check if I'm contradicting myself somewhere Okay, it's only m Okay, okay, so this operation is clearly when defined. I take h of f and divide by gamma m of e part j 2 pi f t Okay Right, so so there's all kinds of interesting interplay between discrete time and continuous time here Okay Okay, so hopefully or it's all that is clear. Okay, so you can think of the denominator as the Fourier transform if you Forget I mean it's a periodic Fourier transform right, so it's a properly well-defined division. There's nothing Confusing there but on the top you now don't have sh of e part j 2 pi f t if you did you would only get m star of e part j 2 pi f t now you have simply h of f itself. Okay, which is actually a Fourier transform. It's not a Discrete time thing. Okay, but still I can adjust my axis so that both of these will match and then I can divide Okay, so it works out. All right, so make sure you Get a feel for the intricacies here. Okay, so here's the claim Okay, the claim is the following suppose I define a linear space as as the span of H of t minus k t What do I mean by span of h of t minus k t? Basically all the signals that I had will form belong to this and You do the same thing. Okay, so summation basically signals of the form Summation s k h of t minus k t Okay, k equals I don't know for all over k Okay, you can put so it's a span of the shifted versions of h of t. Okay so So I'm not guaranteed that h of t minus k t is an orthonormal basis. Okay, I'm not guaranteed. Okay So what I'm guaranteed on the other hand is that Fee of t minus k t Is an orthonormal basis for? Okay, so that's where the Interesting result comes through. Okay, so h of t minus k t need not be orthonormal But fee if you define it this way fee of t minus k t becomes an orthonormal basis for Any questions It's okay People are happy. How many of you can prove this? Okay, so think about proving it. So what's the way to prove it? How do you show a set of Vectors or entities as orthonormal? How do you do it? Yeah, so you have to compute the inner product and show it goes to Zero if there is a non-zero shift and it becomes one if there is a there's no shift And also, yeah, also that it spans case. That's also an important thing that I should show that this vector Spans the entire space That's spanned by h of t minus k t. So once they prove those two steps, then I'm done Okay, so I'll prove the first step that the inner product is uh goes to zero Okay All right, so how do I show that? So basically I have to find write down What is the inner product minus infinity to infinity fee of t? fee star of t minus k t Okay, I have to try to evaluate this inner product But once again, I'll use this way of evaluating inner products, which is what I will see that this is actually fee of t convolved with fee star of minus t which is sampled at k t Okay, so this allows me to use my knowledge of Fourier transforms and all that Okay, so if I do a Fourier transform here, what will I get? If I do a DTFT on the on this I will get 1 by t summation m equals minus infinity to infinity modulus fee of f minus m by t Square is that a question? Okay, so so the way the proof is what for each k? I'm not showing it is zero Okay, I'm looking at the entire sequence for all k I look at the sequence and I'm going to show that the entire sequence is equal to a delta Okay, that's what I'm showing both of them are the same For each non-zero. Okay, I'm showing you I'm ensuring the entire sequence is delta If the entire sequence is delta, what should happen to its DTFT? It should become one So basically I have to show that this DTFT is going to become one Okay, once I show that this DTFT is one, I know that this sequence for all k is going to become a delta Which is what I want to show Okay, so that's the way in which the proof will go and to show that this thing is one is very trivial Okay, so it's not a big deal.  Okay, that's all Okay, so the denominator now conveniently becomes independent of m. Okay, why is that? Where it will become okay multiplied through you get e power j 2 pi f t times Something which is one. Okay, so it goes away e power j 2 pi m and I can ignore that Okay, so I can pull that out. Okay, so what do I get 1 by Gamma squared mod m of e power j 2 pi f t Square times 1 by t Summation m equals minus infinity to infinity modulus H of f minus m by t square Okay So numerator and denominator this is both of these are exactly different ways of writing sh of e power j 2 pi f t do you notice that Okay, here I've written mod square, but what is this? What is this guy? It's m times m star. So it's also mod square only. Okay, so it's the same thing. So you notice I can write this as this k becomes F h of e power j 2 pi f t This is also What The same thing Both of them will cancel and you will get 1 Okay, so what I've shown is that the DTFT of my inner product sequence is 1 Which means the inner product sequence itself is a delta And phi of t minus kt is orthonormal. Okay, there's no problem here. Okay So I want you to go back and carefully check at what point I use the fact that m is minimum phase Go back and look at the derivation and tell me if I can repeat the same derivation with any other m which was Not minimum phase any other way of factoring sh of e power j 2 pi f t. Can I do that? Well, I still get an orthonormal basis At what point did I use the fact that m was minimum phase? I'm sorry No, but gamma square is just a convenience forget about gamma square.  Looks like I should right. I'm not doing anything there. Okay. There's nothing wholly about the minimum phase Okay, so you keep asking your question over and over again Okay, just because I write down something don't say I think that that's the only way of doing things Okay, so in fact if I chose the simple factorization h of f times h star of f also What'll happen? Will it work? Okay What'll happen if I did that? Okay, then it may not may not work. I don't know. Okay, so you have to worry so much about Uh several things pay attention to what you're doing. What'll happen if you did that? I would have got phi of f s 1 for all f which would mean phi of t became delta of t. Well, yeah, delta of t delta of t minus k d you can think of it as orthonormal basis if you want But it's it'll be of no use to you. Okay, so there might be problems later on. The second step is also equally Crucial Okay, so what's the second step? Okay, the first step seems to be okay. Okay, there seems to be several choices and we picked one choice and we showed it was Minimum it was orthonormal. There's no problem. What's the second step? The spanning step I have to show that I can obtain Any signal that I had before by a linear combination of phi of t minus k d Right, my signal space is what summation all linear combinations of h of t minus k d I have to now show that the phi of t minus k d Doing linear combinations on that I can get any other linear combination of h of t minus k d. How do I how do I show that? Think about it for a few minutes See if you can come up with some equations which tell you that it's always possible So, what's the problem? The problem is Suppose I have so this is for spanning Okay, suppose I have an s of t which is summation S k H of t minus k t Okay, how do I write this s of t as sum Summation Some bk phi of t minus k d. Can I write it? Okay, if I can write it then I know that It spans but can I write it is the question. How do you go about doing it? I don't want energy No, I want to be able to exactly write it. I have to find s of t at each t as a linear combination of phi of t minus and k t No, you mean you give me the bk. I think let me let me not be So I have to show To show that there exists bk such that s of t is summation bk phi of t minus k t In fact, you can explicitly construct a bk. Okay, what is that bk? Okay, so the the confusing thing is the interplay between Continuous time and discrete time. So how you are sampling here and actually you're getting a discrete time filter and continuous time filter It's not very easy to quickly deal with it. Okay So one way of doing it is to do a picture to just see at least what the complication really is Okay, so you have sk which you're thinking as A discrete time signal and then I suddenly put a ht and then what do I get? Another continuous time signal. Okay to keep everything consistent Let me make everything continuous time in that case. What does sk sk actually become? summation sk Delta t minus k t Do you see that if I think of it that way then here I get s t Okay, so that is fine. This is a picture. Okay, so now I know hf is what? phi of f divided by gamma m of e power j 2 pi ft is that how I defined it Oh f is on the left side. I'm sorry Yeah, it has to be on the left side Is that fine? Okay, so this is how I defined it. So now I have to be careful when I keep taking see I don't want to take inverse Fourier transform of m of e power j 2 pi ft Okay, I can take it. What what will I get if I take an inverse Fourier transform of m of j 2 pi ft Impulse train, right?  Okay, what will that impulse train be? Some impulse train with certain weights. Okay, that'll be mk, right? It'll be an impulse train with weight mk Okay, so that's the way to go back to continuous time if I want to go back to time picture here I can say h of t equals phi of t convolved with what? An impulse train defined by Well, there is a gamma. Don't forget the gamma mk delta t minus kt Okay, well it's getting ugly. So let me rewrite it Okay, so one needs to be careful about these impulse trains not to get too carried away and start differentiating or anything But as long as we write the equations like this, it's okay. It will be valid. Okay Do you see that that's a periodic Fourier transform and it will convert convert back an inverse Fourier transform to an impulse train That impulse train is given by the dtft if you freeze it down to one location It's all the same the Fourier series and all these things that they exactly match here. There's no problem Okay, so once I do this I'm done Right, am I done? Yeah, I'm really done. So all I have to do is take sk And Filter it in discrete time if I want even with mk Which is what is happening in the first step and then do what well multiply by gamma Okay, so multiplication by gamma is always there. So maybe I filter with gamma mk. Okay, and then I do phi t here And I would get st again Okay, so if I call this guy here bk I can write my st as summation bk phi of t minus kt and bk would turn out to be Gamma times sk convolved with Is that fine? So I've shown my Spanning property which worked out in a very simple way. Okay, so now the next question. What is the question? I'm going to ask At what point in this Proof did I use the fact that mk is minimum phase? It might be a trick question. I may not have used also Where did I use it? If I replace the denominator there with any other a of v pa j 2 pi of t as long as it has a proper inverse Discrete Fourier transform should it work? Is it fine? Do you see anything that is disturbing you there? Okay, it should work, right? That's what it seems like. Maybe there's maybe there is something between the lines, which is not very clear to me but But it seems to work.  Okay, maybe if it's minimum phase Is there a point? Yeah Yeah, but still the integrals and all will work Okay, so loosely minimum phase is not the problem But in the definition of phi f if you want phi f to be If you want phi f to be stable Maybe you want the zeros of The denominator a little to lie inside the unit circle But being minimum phase is more strict than that. You also want the poles So at least at least there is some justification for that In terms of stability. So this whole minimum phase thing is more for stability and those kinds of things as opposed to The requirement of orthonormality itself. Okay, so so that's something to keep in mind and minimum phase also is good because Later on if you want to approximate the Response by only the first few tabs, then minimum phase helps you because you know, that's the best Okay, so you might as well pick the minimum phase the utility of minimum phase in Is in a different sense as opposed to giving the basic orthonormality Okay, also basic orthonormality seems to come pretty much for free if you take anything But if you want stability for your filters and all that then you should need to do this Okay All right, so this seems fairly clean. Okay, so now I want to show some kind of an equivalence between The distance criteria we had before and the optimal criteria. Okay, so if you want to run an optimal receiver What should you do to r of t? Okay, what should you do? You filter with I'm sorry phi star of Minus t. Okay, so you do that you get your optimal filter and then sample at symbol rate k t Well, you're not done doing optimal stuff. Okay, so Okay, for instance, will you have isi now? You will have isi. Okay, don't imagine there'll be no isi Okay, phi of t need not satisfy the Nyquist criteria. It would be great if it did. Okay, if it's satisfied that It would be great case. Well, not just really phi of t. You need you probably need some some weird combination I don't know how it's going to work out. Okay, maybe there is some nyquist criteria possible here So that you could know I say but in general There will be isi so your detection problem is not at solve But you'll still be optimal. I mean, you're not going to lose any information You're not losing any information in the orthonormal projection. Okay, so what you get here will be will be what Okay, zk. So I don't want to call it zk because there is some constant missing. I'll call it z prime k Which is going to be what? gamma times sk convolved with mk plus nk and my noise now will be Normal with mean 0 and variance n naught by 2. It'll also be wide Gaussian Okay, this is completely compatible with the previous picture because there we multiplied everything divided everything by gamma Okay, so it's the same thing. It's not in any way different. Just the gamma has gone to something else Okay, don't be suddenly disturbed by the fact But clearly you have isi z prime of k It doesn't depend exactly on one sk, but it depends on several sk's sk convolved with mk, which was your minimum phase spectral factorization or that matter any other spectral factorization, but some related to the spectral factorization of sh Okay, so that's how it worked All right, so now you have the most important task of running this through the detector Some detector which we don't know how to build it. Okay Okay, we don't know how to efficiently build this detector But we know how to build it brute force How will you build it brute force? Right, what do you do? You run through all possibilities and find the minimum distance one Okay So remember now your transmit constellation was very nice. Okay, what's happening to your receive constellation if you just look at z prime All kinds of things will happen, right? You're doing filtering with mk for arbitrary sequences So your receive constellation will not will be very strange. Okay, one doesn't know how to do detection there Okay, but if you do exhaustive computation and then do minimum distance criteria, you should get the answer because after all You're in why it goes in noise. So there's nothing more complicated than that. Okay, so in theory the detector is Already given just do minimum detection. You'll minimum distance. You will get it. Okay But in practice we might have to do more things. Okay, so that's where the wittebi algorithm comes in Okay, so before we close up this the quick one minute. I want to take to show you the equivalence between this orthogonal projection and The h star of minus t sampled Followed by one by gamma square m star of z star. It's very easy to see actually Okay, so it's not a big deal. What is phi star of minus t if you do a Fourier transform? What do you get? h star of f divided by gamma m star of e part j 2 pi of t Okay, that's enough. I can pretty much stop there Okay, what did I do in the previous case? I took r of t I first filtered with h star of f whose impulse response is h star of minus t Okay, but then I sampled immediately and then filtered with one by gamma gamma square m star of one by z star. Okay, but that's okay I can sample either before or after because I know the denominator here is actually a Actually a discrete time filter so I can either sample before that or after that I won't lose any optimality Okay, so that's the way in which to quickly see the equivalence between this and that picture we have Okay, so so what's the final Selling point here. So you can reduce systems with isi to By orthogonal projection if you know h of t Okay, if you know h of t you can do the minimum phase factorization and in fact get In a in some way the least possible isi into your system Okay, when you do minimum phase you will get the least possible isi into your system, right? Because almost all your isi is concentrated in the first few tabs Only the first few symbols will affect you everything else will not affect you too much But still unless there are some special Nyquist criteria satisfied by a combination of things by sheer luck You will not be able to completely eliminate isi okay So later on we'll see some fancy multi-carrier modulation methods where people still eliminate isi But there again the nyquist criteria will be satisfied in some some way Okay, so it's possible to do that But then you'll have to do multi-carrier stuff which is more complicated with a single carrier It's clearly not possible in the general case Okay, so but it's possible to minimize the isi to the least possible effect and then you do Then you do something else like the witter be to do Okay, so before we close today, okay, maybe I'll make these comments on Monday more comments on this and how practical this Headstar of minus t and all this will will make


All right, another Wednesday, so that means another edition of everyone's favourite segment, Just the Facts. This evening, the second leg of the inaugural Concord Calf Caribbean Cup final between Jamaica's Cavalier FC and Surinam's Robin Hood will be played at the National Stadium in Kingston. The producers have decided to craft today's questions on the first staging of the competition. The pass mark, Ricardo, they say it's 6 out of 10. Are you ready? Mm-hmm. All right. Let's go. All right. Let's go. Oh, wow. The Concord Calf Champions Cup, a lot, three spots, a lot, three spots for the top three finishing Concord Calf Caribbean Cup teams. At what stage of the Concord Calf Champions Cup competition does the winner of the Concord Calf Caribbean Cup enter? Round of 16. All right. All right. You were ready. Let's go. How many teams participated in the inaugural Concord Calf Caribbean Cup 16? No. Ten. It was from 16 to 20. Okay. Okay. Ten. Fine. Name the clubs to have represented Trinidad and Tobago in the inaugural Concord Calf Caribbean Cup. Defense Force and AC Portas Fin. True or false? Jamaica had three representatives in the inaugural Concord Calf Caribbean Cup. True. Well, excuse me. This team suffered the biggest margin of defeat in the inaugural Concord Calf Caribbean Cup. Biggest margin defeat. Is it the Jamaican team? No. AC Portas Fin. Golden Lion. Golden Lion. Number six, Atlético Pantoja and this club from Trinidad and Tobago were the only clubs to not register a win in the inaugural Concord Calf Caribbean Cup. Defense Force. Well, it's between AC Portas Fin and Defense Force. You just said that. So pick one. Defense? I think no, it's AC. All right. AC. Yeah. Seven. Questions are represented in the Concord Calf Caribbean Cup. Jamaica, Trinidad and Tobago, Dominican Republic, Suriname, Martinique, five. No, Maricardo, you really studied for this test. Let's go. Come by each. No, I didn't actually. He is the leading goal scorer in the Concord Calf Caribbean Cup. Shaniel Thomas. Following the semi-finals of the 2023 Concord Calf Caribbean Cup, how many Cavalier players were named in the Caribbean Cup semi-final Best 11? Five. You say five. Cavalier. I feel like it was three. All right. Three. But maybe five could be right. Wait, some final to record. It's on plenty. I'm not sure, to be honest. Because it's 11 too. So yeah. Three? Yeah, let's go three. All right. This team has scored the most goals, so 16 in the inaugural Concord Calf. Cavalier. All right. So Ricardo had his smart hat on. Let's see how the results went. Round of 16. Yeah, 2021. Number three. Defense force and AC for Spain. Yeah. So that's three. Three from three. Four. That's true. Cavalier four from four. Cavalier done beholden. And half of you, the three Jamaican teams. The teams suffered the biggest. Ah, it was half of you. At the expense of Cavalier. Yeah. All right. Four. AC for the Spain. All right. So that's five out of six. How many nations? Five. That's five. It says five. Yeah. Five. So six out of seven. Yeah. Wow. Seven. Seven out of eight. Did you say six? No. I said three. So that's seven out of nine. Oh, I said five. Yeah. Seven and eight. And Cavalier. Eight out of ten. Wow. Congrats, Mario. Well done. No, no, no. Congrats to you. I like I didn't show up. But good job, Ricardo. You're here. That's showing off enough. Let's take a break. Good job, Ricardo. He did really well. Break time.


Is it safe for Asians to move to America right now or is the risk of danger just out of control? Welcome everybody to the hotpot boys David and Andrew here what are we talking about today?  Listen there was a lot of thoughtful and complex answers because the answer is spoiler alert not yes or no it depends but we have three main points that are going to help you think about this topic so if you're ready for this please hit that like button check out other episodes of the hotpot boys and please turn on your notification bell because we are dropping hot topics that are also important so David what's the first point the first point is that your safety is a conditional probability formula what do you mean by that basically there's a lot of factors in everybody's life I know in the Reddit thread some people were trying to say oh yeah should you move to the northeast should you move to the midwest should you move to the pacific northwest should you move to the south guys even within those states there's a ton of cities and even within those cities there's a ton of neighborhoods and even within those neighborhoods there's a ton of buildings and a ton of houses and a ton of streets it could vary so much and it even matters what you do for your job what's your risk exposure some people were saying oh well I lived in Houston my whole life and I'm just the IT guy that works from home my brother works in retail and he heard a hundred times thousand times more racist things than I ever did because he meets like 500 strangers every day who are mad at him yeah I mean to put it plainly I mean it's a fairly it looks like a simple mathematical equation but basically you just plug in the probability of different factors and they stack up on each other you multiply them and then you get the overall probability of something happening so it's like if yo is anything bad gonna happen to me if I move to New York City well obviously tons of people are safe but also yeah some bad things do happen so it kind of depends on where do you live where do you spend time at night are you walking the streets alone are you male are you female are you intimidating looking or are you not intimidating looking yeah there could be two buildings in New York City Andrew that are right across the street from each other and one building has a completely different risk exposure as an Asian living in it than the building across the street literally we're talking about like 100 feet separating a completely different life so I don't like it when people are trying to categorize like geographic regions that sounds like way oversimplified to me somebody could be super safe anywhere but yeah it's just different depending on like what life you live where you live what times you're out who you're seeing what kind are you having upside downside neutral interactions but listen guys it's all an equation and I guess Andrew to say that that is a more mathematical way of saying street smarts yeah have street smarts but street mathematics but but the street smarts stack on top of each other you know it's not just I don't want to just say oh be more street smart because it's like well how do you it's like listen if you want to think about things in a mathematical way guys your safety is an equation so just plug and chug those variables and you'll kind of know like nobody knows the exact numbers right oh you live in Upper West Side how much safer is that 40 or 80 percent safer is a 20 percent safer we don't know those exact numbers but you kind of know if it's positive or negative if it's high or it's low okay a shout out to the conditional probability formula this was brought up by somebody who was a mathematics statistics major point number two Andrew America is just a very complicated place that is sometimes very chaotic and very layered wait wait wait David David David you telling me that a country with different climates with different population different immigrant populations different freedoms the ability to have guns and even do drugs to a certain extent you know you're telling me that this country's complex yeah I don't believe it somebody in the comment section was from China and was like yeah it's pretty interesting because uh US is only 25 percent of the population of China but honestly it is probably like 1000 times more diverse in the lifestyle the type of people and their way of life I mean yeah I don't know if it's a thousand but it's it's way more diverse for sure guys and we know this because there's different like laws and systems that kind of make it more diverse I mean even just look at the whole abortion thing going on right now it's like insane dude state to state capitalism does uh you know allow for a lot of diversity because there's a lot of different socioeconomic and by the way that's why America is such an interesting fascinating textured place to live that in my eyes is so interesting but man it can get really crazy and we just spent a bunch of time in Canada less layered less chaotic for sure I don't want to say Canada is less interesting I mean we were in Toronto I had a great time and the people are really nice but it is less diverse and a little bit more in the middle everything's like kind of pushed down from the top and pulled up way more social compression less poor people less rich I will say this the homeless people in Canada they seem less erratic they seem a little bit more smiley and I think it's because they get universal health care somebody was saying yeah you know it's crazy sometimes you meet somebody that tells you go back to China and then the next white person you meet their kids study Mandarin and love Chinese culture yeah and he was just like yeah it's just person to person it took a very so incredibly much uh I used to work retail very azn job I used to sell telecom wireless in college and Andrew in the zone I was working in sometimes I would meet 200 people a day in terms of me being Asian I had people not want to shake my hand because they thought I had SARS some people saw I don't want to do business with that guy get somebody else to sell me a Blackberry he's not going to give me a good deal he's an Asian to other people saying oh give me the Asian he's going to get me in and out of here the most efficient build a Toyota Lexus style way possible and I had other people a lady who worked at Microsoft with a bunch of Asians go why are you doing this job you need to be moving up the corporate ladder at Verizon instead of working retail with all these other bums she almost tried to like save me when I was like hey I'm just an undergrad in college right now it was a good job for me right now I mean whatever your job is listen a lot of people they don't face racism during the weekday but during the weekend you go to different places you're out on the street more it could be more dangerous and I will say this one thing that they didn't mention in the reddit post Andrew sometimes you need to activate your agent like let's say for example you're a regular Asian on the street uh yeah people would have to be pretty racist it happens though obviously we've all seen the cases of this extremity the extremity imagine if you were an Asian cop or an Asian politician something now you have two charged identities on you the layers are there and that's like a double whammy now I would imagine that Asian cops Asian judges Asian politicians where they have to knock door to door being like I'm running for city council I'm trying to represent you and solve your problems person who's completely different from me they might hear a hundred thousand percent more crazy things about their race you know good and bad yeah um okay moving on to our third point that you gotta remember David what is it Asians just need to make sure that they're not only becoming safer in a dangerous environment but also being proactive to make that environment just overall safer yeah so I think it's kind of like about playing defense and offense you know I think a lot of Asians that I know know how to survive they know how to how to avoid Matisse all defense the offense yeah but they don't know how to make things safer and I do think that obviously in America as an immigrant group whether you're one g2g3g which is referring to first gen second gen third gen shout out to the reddit guy who used that I think that's a kind of cool way to refer to it but depending on what level you're at you feel different levels of ownership over the country and I do think that people who feel more confident and ownership over how they live in their identities they're more willing to step out and like do like I don't want to say like mini police work in their community but just like be more proactive in their safety for sure like like talking to people asking people keeping an eye out or asking kids like oh where your parents at or even if you're a bunch of like buff Asian guys or just you're talking about you're referring to just being a good steward and having camaraderie within a community yeah because I think a lot of immigrants who feel like oh this isn't my country I don't have any ownership I don't want to get involved but that doesn't make your area any safer I'm not saying I don't understand that feeling but at some point you need to be able to step up yeah I think that Asians um are getting to that point more and more Asian Americans that are born here but I think that obviously anytime you notice something like sinophobia which is very more on a macro media level and then the anti-Asian hate which is more visceral because you could see that and feel that depending on you know your own risk exposure and who you've come across um I think it's easy to look at a baseline threshold situation and go oh if I increase my cautiousness or my scaredness I will be able to control my probabilities but there also has to be to be said something for care taking and improving the situation playing offense I'm not saying everybody gotta be Batman guys I'm talking about Robert Pattinson Batman but not everybody should be this guy in Batman the Asian guy in the beginning of the movie who was getting picked on by everybody here's my last little thing listen I know and this is for Asian guys out there because we're always wondering like what more can we do within reason right without becoming like all MMA fighters and getting like a tactical vest on and training like how police people do whatever I'm just saying like uh you know as as as healthy young Asian guys if you're in a group together like you have to understand that you essentially also double not just as a friend group but as security detail for other Asians or anybody for that matter so you just have to know as a group of guys together especially at a time like this there is a little bit of an expectation that maybe you guys are thinking about not just your own safety but other people's safety a little bit too right at least in a radius zone yeah like you know like a radar like if you like hanging outside of your friends bars like sometimes we hang out outside of our friends restaurants and whatever but I'm not gonna lie when it's New York City and I'm outside I I do have my head on a swivel I'm looking like uh you know who's like is everything gonna be safe are we safe is other people safe like I'm not saying I'm I'm a save the day and I'm a superhero right but it's on my mind yeah so I think that that's something that we would like to put out into the ether into the universe hopefully people you know the more they hear it that at least the more they think about it come to your own conclusions I'm not here to tell anybody how to live their life I couldn't control you even if I wanted to so again to run down the points and recap guys your safety is a conditional probability formula the long story short Andrew that's just street IQ but providing I'm just using a book smart IQ way to describe street IQ point number two America is a very complicated layered place with a lot of complex and potentially chaotic layers to it that's what people like about it that's what people hate about it this is the country that we live in it's a great place for a lot of opportunities and opportunities that are positive and a lot of bad opportunities too you need to know how you interface with that chaos or do you just want to remove yourself from and move to Irvine California point number three Asians just need to make sure they are training offensively as well as defensively you got to play both sides of the ball guys you can't just be a pure offensive player like buddy healed and you can't just be a pure defensive player like Matisse Thible both of those guys getting benched right now guys it's the playoffs both guys unplayable in the playoffs hey man don't just be a buff guy for the IG and the EDC guys be a buff guy for your community so anyway like we said Reddit thread a lot of really interesting stuff in there people were talking about microaggressions people were talking about this that we chose to derive these three main points from it we'll link you to the Reddit thread below please let us know what you think in the comment section below I know there has been an uptick in crime nationally but is it safe for Asians to move to America I think it's all about probabilities and how you manage them hopefully you get to a point where you can manage them for other people a little bit as well hey and one person said it's kind of sad that we have to say it depends as an answer to that question but that's where we're at guys please let us know like subscribe turn on your notifications until next time it's the hot pop boys we out peace


I am progressing. I can move this this hand and arm. I've now got movement in my right arm and my left leg a lot of it. Just a small amount of movement in my right leg. And so that's the way this is. It's one small step at a time. I'm working every day as hard as I know to work. I will do as much as I can do and that God will allow me to do. I didn't know that I would come back to work after that, but I've got to the point where I'm afraid not to come back to work now, so I'm back. Today is my first day back.


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Hey guys, it's Nate, aka The Foot Accountant. Welcome back to the channel. We're gonna talk Future Stars day number one, talking about everything that happened today, the awesome new objective cards in Odegaard, the SPC from Moise Keen, the player selection, the cards and packs, tons of crazy stuff today, and even a new season with new, the icon objectives are all out, the icon swaps are all out, and all sorts of stuff. So we got a ton of new content today. We're gonna talk a little bit about it here, and then we're also gonna talk about some of the EA mess up stuff that happened earlier today as well. So we're gonna be taking a look at a lot of the pages. I wanna look at the team first because I think this is what we noticed first about today. We'll talk about the objective. I think the objective Odegaard is like the biggest win on FIFA that we've had in a long time because it's something new, it's something that we've seen in other EA titles like Madden, and they finally brought it to foot. And the concept just springs so much more possibility into the future basically. But batch one, so this again guarantees we're getting two sets of future stars, and it makes sense, right? Because we don't have guys like Haland in this group. A lot of people thought that he was getting in, and among others that we don't have in this group. But I think this is a pretty solid batch one, but I also see that there's gonna be, I think batch two will be pretty good as well. I wonder if they'll make it actually two full weeks instead of like one week and then a half a week. We'll have to see. But Jal Felix, Rodrigo Tonali. So these are the guys on the loading screen from yesterday, which we did end up guessing and end up guessing correctly. Greenwood is a nice one, the striker position. That card's pretty expensive right now. Martinelli got a skill move and a week foot upgrade, I believe, he's three star, three star. Dominguez, Mason Mount, Kabak. Don't know that guy, but that's a nice looking card right there. Diabi, Martinez, Kamara, Musa Dembele, who we've seen before with some special cards. And then Max Aarons and Emerson. I'm actually really pumped about both of these guys because they kinda, you know, Max Aarons isn't an insane card, but it's still pretty solid. It's a 90 paced right back that is English in the Prem so a lot of people will be interested in that card. And then we finally have a Brazilian right back that we can use, right? We always used to have Dani Alves, and that was like your Brazilian right back that you could use. Now we have Emerson, 89 paced, 89 physical, it looks like a pretty solid card. And I wanna make a point about this, Lissandra Martinez. So many people thought we were getting La Taro Martinez, future star. And I think EA kinda used this one here when they were putting into the code, you know, to kinda like play a trick on the people that like to leak cards that we get every week, because, you know, an Argentinian Martinez, people probably thought of La Taro right away, but it was actually Lissandra Martinez. So I thought that was kinda interesting. Overall, I think it's a really solid set of cards. This card, Tonali, if I had to pick up one card from this entire set as the best card that they released today, it would either be Giao Felix or Tonali. This Tonali card, right away when I look at it, it kinda reminds me of Balak. It kinda looks like a Balak-esque player. I don't know what his build like is in game, but his stats just look incredible. So this is gonna be maybe like Alwar's future start last year. This card's gonna be used for the rest of the game because that is an end game card. Giao Felix is end game. Rodrigo is end game. So a lot of these cards are really good. Even Mason Mount has some pretty good looking stats. I think he's only got three star skills, but four star weak foot, that's nice. So a lot of good cards that we did get today. I'll talk about Odegard again in a second. We'll look at that in the game. And then, of course, I wanna cover these frequently asked questions. These are basically the same as last year. Somebody who is 23 under and has high potential, but has not yet become an established global star, or has not been released in the past future stars campaign. So going forward, we know that we're not going to be getting duplicates, basically. So the whole Calvert Lewin thing with the birthplace or whatever with the coordinates in the code are on the loading screens. Maybe that's not going to be the case. It doesn't look like based on what this says, it doesn't look like a past future star is gonna be getting a card. So that's interesting to point out. How are the ratings determined? It's all about what they could achieve at the peak of their career. And that's kind of the whole thing with the objective, the upgradable objective players as well. So kind of what you do is you, you have that dedicated objective, you earn that version, then you submit that card, and you do the other objective. So I wanna go talk about that Odegaard right now. We're gonna go into objectives and talk about everything that we saw in here today. And of course, we did get the icon swaps. I wanna cover those first and we'll look at the season. I think they did another. This is pretty nice, by the way, the no room for racism stuff. That's really cool. Gigi's the E on that. They did say that it was gonna be coming in the future and they have put it in now. A stadium theme, a TFO, and I think a badge, no room for racism, which is a pretty cool. That's pretty cool, honestly. And a lot of people are gonna get that cause it's really early on in the levels. So that's pretty awesome from EA there. Objective storyline players, Kalu, which is a pretty solid looking right mid, Nigerian, not Nigerian. Is that Nigerian? Am I right? Am I close? Where's the nationality? Nigeria, I was right, thank you. I was thinking of Kocha. Four star, four star, high medium. Not a bad card. Yamaguchi, which is a pretty good looking card, a center mid for Vissel Kobe. Not a terrible looking card. Japan links right there. And then El Shirawi, I think a lot of people might end up taking this card just for the, you know, the FIFA feels. This guy just hits you in the FIFA feels if you've played FIFA for a long time. Those, a set of three cards, more TFOs, some packs in here, some celebrations, but at the end, I think at the end of the season, which you need a lot of XP for, you need 100,000 XP, but Akanshi, Malikovic Savage, and Kovacic are all pretty solid options. I think if you're gonna pick one of these cards, I think I might have to go with Akanshi, because this card looks really, really good. A lot of you guys have that in Babu. This is a card you can stick right next to him. Malikovic Savage is a pretty solid card as well. Hasn't gotten any forums or special cards this year that I can think of off the top of my head. Four star, four star. He's a FIFA favorite from years past. And then Kovacic, this is like the highest rated card I think Kovacic has ever had, at least from what I've seen playing FIFA. So this is a pretty solid looking card as well. If you get to that level 30, I'm really curious to see who you guys are thinking about taking as of right now. Of course, it's gonna be a while for you. Get there. This season's a long one. 41 days or 42 days is this entire season. And that again does mean that icon swaps are out and that all of the swaps are attainable. So you can kinda grind at your own pace if you're going all the way for the Essian. That's something that you can get on right now. If you're gonna do the mid icon pack, that's something that you can start grinding towards as well. So the second set of 12 swaps have been released. And I also think that this gives them the okay and the ability to start putting out or to start thinking about putting out the base or the mid icon upgrade SBC. So since we have this pack available through icon swaps now, since all the tokens are out, I think they're gonna allow themselves to put that SBC out here in the next couple of weeks for sure. So again, these tokens really quick. It's basically the same. You gotta win a lot of games. I'll win 12 footchamps games, five rivals, five rivals, four rivals. Though they're pretty tough again. The online matches are pretty tough and the offline ones, you can kinda do what it is, but you need some legendary difficulty there, which is interesting. I don't really grind that stuff to be honest. I kinda stay away. We have an Henri Tifo, which is cool. And then let's talk Oda Guard. So this is how the Oda Guard thing goes, right? You kinda have to work your way down this list and you kinda just kinda work at these, right? And you get the different cards. So you have the 87 Oda Guard, which is the Foot Future Star Academy. It's like the name of this card. Rise through the ranks with Oda Guard and earn untradable rewards.  You do this, score three goals using midfielders and you submit this Oda Guard. Or you get this 80 rated Oda Guard. Then you have to use him, score three goals using 80 overall Future Stars Academy Oda Guard. You get a one player pack. Boom, you keep going. You gotta score, assist two goals with him. And you gotta score using the 80 rated card in four separate matches during the 84. So you have to score three goals, assist two and score in four separate matches. Is this say what, game mode? It doesn't say in any game mode. Interesting, so you can actually do this in like online friendlies or like in squad battles if you really wanted to is what I'm seeing. That's actually really cool. I did not see that at first. Assist two through balls using the 84 overall Future Stars and arrivals. Okay, now we have the specific instructions. But then what you do is kinda, you get that 80 rated card and that unlocks the 84. You get that 84. And then you continue down, you get the 85. And you finish these objectives while getting the 85. And then play 20 games with the Oda Guard and you're starting lineup.  And then of course we have the plus one which is score in division four above. And you get a 50K pack. So you get a 50K pack if you're in division four above. So there's a little bit more incentive for those higher rated division grinders which is pretty nice. And then there's an SBC over here that kinda goes along with this, right? Is it Future Stars Challenge? No, it's this one. Big Transfer Oda Guard. Exchange all versions of the Foot Future Stars Academy Oda Guard. So you in here, you turn in the 80, the 84. Is that what it is? You turn in the 84, you turn in the 80, the 85 and the 87. Really, so you can turn in the 87 and get a 50K pack? Interesting. Exchange all versions of Foot Futures Academy Oda Guard. So if you don't want the card, you can grind the objective and turn in all of these to get a 50K pack. You get 200 coins for each of these and you put in, interesting. That is very interesting. I think that's how it goes, right? Am I mistaken by this? I think that's how it goes. That's very intriguing. Yeah, so you get up to the 87 and you can either keep it or put it in this SBC for a 50K pack. So that's kind of cool that if you don't want the card, you can still grind up to it and turn in for a 50K pack if you really want to. That's intriguing. Anyways, but this concept is massive. Honestly, this is the picture over here on Twitter. Play them, upgrade them is kind of the way that EA has marketed this. For the first time ever in FIFA Ultimate Team, you can upgrade Select Future Stars by completing dedicated milestone objectives. And again, they said this year that they were gonna really hone down and put a lot of stuff into objectives. Think about it. Icons, SBCs that we had last year, all those are attainable through objectives this year. We've got seasons and season progress, milestone players, storyline players, and now we have upgradable Foot Future Star Academy players. So this is all like an objective grind, trying to get people to grind the game and maybe try to make it a different aspect, like more grindable side of things and the more like pay-to-win side of teams. So today we got a grindable objective and then the cards and packs as well. So whatever it is, if you like grinding objectives, I might even go for this just because it's the very first one and it doesn't seem too terribly hard right out the bat. I do feel like they could have made it easier though instead of having the 84 or the 85, why don't we just go 80 to 84 to 87 and make it a three part, you know? Just make it a three part thing and then instead of having the 85, just keep that out of there and make it even easier because I mean, this card is pretty good, but is it gonna be like that super end game and that super amazing? I don't know. One thing to note is, one thing to definitely note is that this card could be getting a winter upgrade because right now on Sophipha, his card, I'm gonna pull up this tweet actually. I tweeted about it earlier today because I thought it was pretty important information. Right here's the tweet. Odegaard is getting a plus four from his 78 rated base gold card right here to his 82, which is on Sophipha. That's a plus four rating, right? And of course he does have the 80 to 84, 85 and 87 and if they go by last year's rules, right? The stats that are on this website have to be higher than this 82 rated inform, which they are. He's got 78 pace on the inform on Sophipha. I think he has like 80 pace or something like that. So that means he should be getting the upgrade to the special cards, but is that going to affect the Foot Future Star Academy cards? And in the 87, is it actually gonna end up affecting those cards? We just really don't know yet if it is or not, but I think that it should because they upgrade other objective cards or like SBCs, like last year with Footmist SBCs, they upgrade those when those players get winter upgrades. So Odegaard should go to an 82 for his base card, moving this inform to an 84, which I then think would move this card to like an 84 and this one to an 86. This one to an 87. I think this 87 would go to an 88. So I do think that he could get another plus one boost through winter upgrades, which might give him 90 pace. And then we're talking about a really, really sick card right here. So if that does happen, this could be a pretty mental card. Obviously he's not gonna fit into a ton of teams because he's Norwegian and he's from Real Sociedad. So the links aren't really there, but this is the first of hopefully many that we get at the Foot Future Stars Academy or just the grindable objectives upgrading through different progressions, as EA words it, progressing through their career as their cards as they get older and as they continue to progress in their career. So again, the really, really cool thing behind this is just the thought process, right? The thought process that stuff like that could be coming throughout this game, throughout the rest of the year. And then many other ways to come, right? That's the first time we've ever had that though. It just opens the doors to so many possibilities. Like I said, I think it's in Madden already. They have like a grindable and upgradeable players by just playing the game. So it's a GG that we finally got those in FIFA. We got a Moise Keane SBC today as well. 83 and 84 rated squads, no in-forms. This card is just fine. I mean, it's okay.  Go for it. He's only out for two days though. So unless we get upgrade packs tomorrow, I don't think a lot of people would do this unless people are trying to grind some upgrade packs to pack a future star or so. And then we also got a mega pack from the future stars challenge SBC today as well. This one really threw me a loop first time. Like I don't know what the heck was going on with this SBC, but you can turn in all four versions of that Odegard if you want the 50K pack if you really want to. I would be curious to see if people do that. Like how rigged that pack is because sometimes we know like objective packs. I've gotten a lot of good things out of the objective packs. I wonder if a pack like this would be pretty rigged too. Like have special odds on it or something. I don't know if you can do that, but that is something to think about. And then of course, Trent Alexander Arnold's Player of the Month, there was more Player of the Month voting that came out today before the promo started. So that is something that we have to keep in mind as well. We'll talk about market stuff in just a second. I'm still trying to sell some things. As you guys know, they dropped the new team today, but there were some massive, massive mixups from EA in terms of server stuff. And I want to look at this for a second over on Twitter. And you guys were probably aware of this. You had some issues, especially on the PlayStation 4. There were issues with people connecting. They disabled match creation. And people like literally weren't able to log onto the game for like two and a half hours. The lighting rounds didn't sell out. There were like open bids expiring, people getting stupidly good deals because some people were on the game, on the web app. Some people were not on the game. And it just kind of was a mess for EA today in this promo. Because they weren't making the money that they would have made if the lighting rounds would have sold out, which the first one, the 50K lighting round, sold out in the flash. But then we had the server issues. And the second and third lighting rounds with the 100 and 125K packs, they didn't even sell out both times. So that kind of hurt EA's pocket today. So hopefully they figure out how to fix their freaking servers so that they know that this doesn't happen to them again or they can fix whatever happened today. It doesn't have to happen again, because it's just hurting their selves out of money when that kind of stuff happens. So I'm sure they're not happy about that, but it is what happened today. So that really threw this market for a loop, right? Because these guys usually would have gotten more supply. These cards are pretty rare right now. Felix was 2.4 million coins right away when the packs came out. And then he boomed right back up. He's up to 2.9. He's actually down to 2.7. He was selling at 2.9 for a moment. And this is basically just due to nobody was on the game and there was no supply for these. There was not packs being opened as people couldn't access the game. Once the game came back online, there weren't as many lighting rounds or people were just like not gonna open packs. EA did release a later lighting round, which I thought was gonna happen. They did end up doing it, but it was 125K pack lighting round kind of later in the night. So I'm really curious to see what happens with these cards tomorrow. Obviously, a lot of people were looking at these cards and thinking, yo, these might be really, really good for overnight flips, because the pack weight on these is really similar to headliners and the first week of headliners, the overnight flips from one night into the second were really good. So I was really hoping to do some of that with these cards, but since the whole server issues today, I'm not gonna be touching that. These cards are really high. Let them drop please. They're gonna get packed more tomorrow in my opinion. Will you see them rise up in the morning? Maybe a slight bit, but honestly, I wouldn't expect a too terrible amount of a rise on them. Maybe a guy like Martinelli rises up a little bit just because these are the type of cards that everybody wants to try, right? Brazilian left mid. You can dump him in your team pretty easily. In English, right back from the prem, you can put him in pretty easily. A center back that's French. You can throw him in pretty easily. So a lot of these guys, people are probably just gonna try to buy, throw in their teams, but I do expect these guys to go down a little bit as we continue through the week. And that's what I would say. If you're looking to buy one of these, we'll kind of look at the situation more tomorrow and see what happens with more pack weight, but I do think they're gonna drop down a decent amount. Speaking of pack weight, 2.9% odds of packing one of these from a 100,000 coin pack. So these cards are very similar to the headliners from last week. I think the headliners were like around 3% out of 100K and I think it's like three and a half or 4% from the 125K packs for these cards. So these cards are pretty rare. They honestly are pretty rare and that's good for us is a trading sense for a trading out of packs. When these cards go out of packs, they will make them more tradable and more investable and stuff like that. So again, if you want to buy one of these cards, start to like think about later in the week as a possible timeframe to get in on some of these. I wanna look at tonally right now. You know, it's easy to search these cards too because they don't have any of their special items, at least most of them. You don't have to put like a minimum buy now on any of these. You just kind of search up the special and boom, they're there. So tonally right now is 1186. Again, this card looks ridiculous, ridiculous. I'm really curious to see what the reviews are for some of these cards, how they play in game, how they feel and how people rate them. But yeah, today was an interesting day of content, right? We got some of the most content and like best content that we've had especially with the like the upgradable players through the grinding through objectives and stuff like that. I guess one thing that I should mention as well as the week in league got extended. It was honestly tempting me to play, but I don't know if I'm gonna end up getting that done. It's kind of like, do I wanna work on Ode of Guard this weekend with the little time that I have or do I wanna try to play week in league and get some packs for this promo and maybe if there's gonna be a good stuff happening this weekend with team of the week type players. That sort of thing but I'm probably just gonna stay away from it. We'll see what happens. You guys will see in the next video tomorrow basically but that's kind of my thoughts for this weekend. You do have an extra day though. So that's the only thing that's tempting, right? If I don't get my games done on a Saturday or Sunday like I normally would, I would have Monday evening to be able to grind some of this if I wasn't gonna grind Ode of Guard. So just one thing to think about with that right there. Again, we have a little bit of headliner stuff that we have to talk about as well. I know you're like, dude, why are we talking about headliners? Because we just got a new promo but there are more games this weekend and that means that these cards are impacted. Headliner Salah. I think Liverpool play Southampton today, probably a few hours after this video goes live and Liverpool, they've been winning, man. Liverpool has been playing exceptionally well. I think they're on a three game win streak in the Prem which is what people are really interested with this and of course this is what we're really looking for in terms of the streak, right? So Liverpool right now are on a three game win streak in the Prem. That's what Sofa score is telling me right now. So does that mean Salah is going to be getting upgraded if they win tomorrow? Liverpool beat Manchester United on the 19th which was their first game after Salah came out. Then they beat Wolverhampton on the 23rd so there's two in a row and then they also beat West Ham on the 29th. So those are all Premier League games. So Salah, if they win this game against Southampton today, that means he's getting an upgrade. As far as I'm concerned, and I don't hear a ton of people talking about this but this is something that we need to think about and to talk about because I think Salah is gonna be the first person to get an upgrade. He's 1.96 million coins right now. Wow, that's nuts. If you have this card, and again, I've talked about this a lot. Since these guys are really inflated almost two million coins right now, I would sell this card pre-game. If you have this in your squad, he's going up plus one, right? He's gonna get the 80 physical, it's a benchmark stat. He's not gonna, he might hit 90 passing as well. That would be another benchmark stat. But this card at two million coins, do I think that he's worth it? I don't know, man. That's a lot of coins for this card. If I were you, since knowing that he was like a 1.5, 1.6 earlier, is this one fresh? One owner, 25 games played. Interesting, cheapest one on the market right now. Man, this is crazy, dude. 1.96 mil, this guy was 1.2 first day. He was 1.2 mil first day, two weeks ago when they were in packs. I would take the safe route and take the coins on a card like this. Because if you compare him to like one of the new cards, like what if we compare him to Rogerigo, Rogerigo. How do you say his name? Let's compare over here and flip in really quick. This new Rogerigo card. Let's compare these two. 93 versus a 91 rated. He's 600K less in price.  Skill move and weak foot are the same. So yeah, Sala does have a pretty solid advantage in terms of total stats, a head by 19. But in-game stats, he's only a head by 24. So, and he's Brazilian. So easy, a little bit more easy to link, but of course Sala is a live item. Still, that doesn't make me change my thought process behind these. If you have a Sala, if he's gonna get upgraded, it's probably gonna take 48 hours, that's what they said, to upgrade these cards. So I would take the coins. If you can get a sale at two million coins on this card, I think I would take it because there's so many new cards in the game that people might wanna go and try out, and I think these are gonna act like road to the finals will.  So, if you have a Sala on your squad, I would try to get it out at like two million coins-ish. Maybe a little bit more than that. If people start to realize tomorrow, hey, he's gonna get upgraded, but I feel like that upgrade to the 94 rated card is kinda built in his price because people knew Liverpool is gonna win four in a row, so. That's just something I wanted to mention. I got a sale, which is nice, who was that, and then we have one more thing to talk about. Another Fabinho, man. The Fabinos, they're selling like hotcakes right now. This is a GG. I bought some delis, I bought some Fabinos today, some Rashford's, and then some of the other headliners that I had. Just some flips, right? I got a Gretzger at 130, he was really low this morning. I think this card is definitely more than 150K worth it. There's just not as many center mids or CDMs. He's a box-to-box center mid, he's Bayern, he's German, and his road to the final card's like 700 something K. So, this card being around like 200,000 coins is what I feel like he should be in this market being inflated as it is. So, him getting panics all the day down to 130 was pretty nuts, and yeah, we'll kinda be following that card in the next few days as well. But quickly at the end of this video, it's been way too long already. There's just been so much to talk about. High-rated's are getting down to a very reasonable spot because especially today, we didn't have like a crazy SBC. We had Moise Keen, but like other than that, eh, you know? I do believe we'll get another SBC today on Saturday though, so if you're looking to invest in this stuff, try to get some on bid. Again, 86 is around 14,000 coins, 85's at the low eights, 84's under 4K for the really good linking ones, 87-rated's around 21K. Those are getting low, 88's try to get 26 by seven, and then the 89's try to get like middle to lower 30's too. But we're in a club stock category, right? Because we are probably not too far away from a mid-icon SBC, and just more SBCs coming out during this promo. So, all it takes is one big Future Stars SBC, and you're seeing a lot of these guys go up in price. So, a club stock this weekend, and I would say even a club stock before the Saturday lightning rounds or Saturday content drop, I would feel it would be a safe move, because I do feel like they're gonna give you a lot of SBCs that come out during this promo. So, let's see if we can snag like a Parejo here on bid under 15,000 coins-ish. Looks like a lot of people are bidding on them. Maybe like Isco or something like that would be a decent one too. But if you're somebody who wants to stock the club, it looks like there's a lot of bid competition in these cards right now, but maybe try to do a club stock, buy a couple of them at least, you know? Because they could go down one or two K more tomorrow at the most if we don't get an SBC, but I feel like we will get an SBC, and I keep saying tomorrow, I mean Saturday, today on foot. This video's been way too long already. Thank you guys for watching. Hopefully you're hyped about the promo and the new stuff we got today. The Oda Guard is by far the number one thing, of course, just with the upgradeable, grindable cards with like the different tiers, right? They have the 80 rated, then the 84, then the 85. I just think that stuff is flat cool. So I'm so glad that EA put this out. It just, again, the potential for the future is crazy with this type of objective and this type of upgradeable objective. So I'm super duper pumped about that. If you enjoyed the video, smash the thumbs up on it. Comment down below if you have any questions and subscribe to the channel. If you're new, it's been Nate, the Foot Accountant, catch you guys later. Peace out.