 Okay, so here we begin, this is lecture 31, okay, alright so we have been seeing linear equalizers and decision feedback, decision feedback equalizers with no constraint on implementation complexity in the sense that no constraint on number of tabs, okay, and no constraint and no concerns about being AIR anti-costal, so you can approximate it and that is the kind of thing we've been doing and so some lessons were hopefully learned from there as to how the receiver structure should look, right, should have a precursor filter and then the slicer, there should be a feedback loop around the slicer, should be a post-cursor equalizer in the slicer, you kind of assume that the slicer decisions are accurate in the design of your filters, okay, and the design for criteria are two types of criteria, one is the zero-forcing criteria, the other is the main square error criteria and we found the main square error criteria to be uniformly better than the corresponding zero-forcing criteria, okay, so in fact the, like I've been always saying the minimum main square error DFE, right, and a certain version of that minimum means, mean square error DFE which I'll call the unbiased DFE has been shown to be capacity approaching for some, some range, so there's something canonically good about the MMSE DFE structure, okay, so I'm going to talk now briefly about this mean square error and probability of error, particularly for the mean square error DFE equalizer structure, okay, so I've been always making this point that just because you minimize mean square error, if your noise is non Gaussian, it doesn't mean that you're necessarily minimizing probability of error, okay, so mean square error makes perfect sense optimally only for Gaussian, purely Gaussian ones, okay, so when you expect some symbols to be part of your noise just because you have lower mean square error does not necessarily mean you might have low probability of error, okay, so you'll see what the MSE DFE does in its quest to minimize mean square error, it'll do something which is not very optimal in terms of probability of error, first thing I want to do is to show you that then suggest a correction for that and then we'll just leave it at that, I won't do it very rigorously, just a notion of why it works, okay, so let's begin with this, so this title is basically called unbiased MMSE DFE, okay, so let's look once again at this model, SK going through H of Z then noise gets added, okay, I'm going to say the PSD of noise is SN which factors is gamma square MN, MN star, okay, and then you obtain ZK, right, and for the MMSE DFE, the optimal MMSE DFE, the precursor equalizer will work out to ES by gamma Z squared, right, H star MN inverse divided by MZ star, okay, so this was the structure and these corresponding quantities, well SC factors as gamma Z squared MZ, MZ star, okay, and then what is this ES? ES is expected value of mod SK squared, okay, so those are the various definitions and then you have the post cursor part which consists of a slicer and a feedback loop rounded and post cursor equalizer works out to MZ MN inverse minus 1, so it's hat K, I'm going to say this is approximately SK, okay, and I'll call this guy at the input of the slicer as XK, okay, so this is just the structure and so hopefully people saw the assignment that was posted, it was posted on Friday, I think, okay, so hopefully people saw that, it's got a whole bunch of problems on finding what the precursor equalizer is, post cursor equalizer is for simple single pole single zero systems and I think maybe even two zeros is there, just to get a feel for how it will work out in the positions of the poles and zeros change and all that, okay, so that's a good thing to work out, hopefully you get some intuition from that, okay, alright, so let's see, so before I proceed with the unbiased MMSE, I want to ask a few questions about some characteristics of these filters, okay, so look at the precursor filter, okay, what can you say about its impulse response, okay, in terms of causality, two-sidedness, one-sidedness, what can you say, making some canonical assumptions about H, okay, for instance, H won't have any poles outside the unit circle, right, so if you make such assumptions, what can you say about the impulse response of the precursor, so suppose it's white noise, okay, so MN inverse doesn't exist, forget about MN inverse, okay, it's not there, what can you say about the, see H star of 1 by Z star is going to have Z terms, right, so only Z terms will be in the numerator, what about MZ star, once again, just Z terms, so if you now expand it out and do the partial fraction expansion, you will see it will become IIR anti-causal, okay, so it will be one-sided in the sense that on the anti-causal side, it will go off to infinity, on the causal side, it will stop after a while, you see that, okay, stop after a while, okay, and if you do MN inverse, what happens? MN you expect to be causal, right, so in general, the precursor is going to roughly look like a lot of, both sided, okay, that's the main story, okay, so it's going to be two-sided, it's not going to be single-sided, definitely not, okay, what about the post-cursor, the precursor is going to be two-sided, what about the post-cursor, yeah, so you can expect it to be causal, for instance if you say MN inverse is not there, noise is white, and clearly it will be strictly causal in the sense that the first term won't be there, right, M of Z minus 1, it will go away, okay, even if MN inverse is there, it will go away, so it's the first term will not be there, okay, so it's strictly anti-causal in that sense, okay, so these are nice things to keep in mind, so when we go to constrain complexity and we want to limit the tabs, you should limit it suitably, you shouldn't say my precursor will not have a two-sided response, for instance, okay, so how will you implement a two-sided response in practice? You delay, okay, so as long as it's finite, you delay and then adjust for the delay in the latest section you can do, so that's the thing to keep in mind, anyway, so that's that comment, okay, the thing I want to do is try to get an expression for the slicer error, okay, what is the slicer error, okay, I'll call it E prime K, which is XK minus SK, okay, so this is the slicer error, what is the assumption I've already made once I say slicer error is, yeah, SAT is approximately S, okay, so I've made that assumption just to make my analysis simpler, if you don't make that, it just will go round and round with some non-linear processing, you'll never get to a quick answer, okay, to get a nice feel for it, I'm going to say SAT is approximately SK, so the slicer error becomes XK minus SK, okay, so now let's write XK, okay, so remember XK has is a difference of two signals, okay, one signal is what's coming from the channel and the other is from the feedback, okay, so you have to account for both, so if you write that very carefully, you'll see this will become, what do I write it, okay, so XK now I'm going to expand, I'm going to say it is SK convolved with, did I get that right, it seemed to be, okay, so I should be careful there, I should be careful there, so I want to say XK is the, okay, so let me be careful, so XK will have a noise component as well as a signal component, right, so I don't want to be worried about the noise component, okay, so I'm going to say, let me write down what XK is, so let me erase this part, this is not XK, let me be careful, okay, so let me do that, I'm going to say XK is SK convolved with HK, convolved with the impulse response of the precursor equalizer, okay, so I'll call it H pre-K, okay, the precursor equalizer, okay, minus what, SK convolved with the impulse response of the post-cursor equalizer, okay, so this is the signal component at the input of the slicer, okay, so I'm going to subtract this, from this I'm going to subtract SK to get the slicer symbol component error, okay, so I'm not worried about the noise, no, there'll be a noise term, let me just add to this, not worried about the noise, so this error E pre-K is just the slicer symbol error, as in the ISI, okay, ISI at the slicer, not, there's no noise, okay, so of course there's a minus SK, that's along here, is that clear, so this E pre-K does not include noise, only the ISI part, just for simplicity in the computation, okay, so of course I can now combine the two negative terms, okay, let me just do that real quick, HK convolved with H pre-K, we'll quickly move to the frequency domain, it'll become easier, but before that I just want to make sure I get the right part, okay, SK convolved with, how do you bring that in? You have to say H post K plus what? Delta K, okay, that's how you bring in that constant term, okay, so now let's move to the frequency domain, you do transforms, your favorite transform, DTFT or Z transform, you'll get E prime to be equal to, okay, okay, capital S, so capital S is the transform of SK, okay, so I'm going to call that capital S, okay, times what? H times, okay, the pre-equalizer, pre-KZ equalizer, the pre-KZ equalizer, the response I've been calling D times W, right, okay, so D was something, D times W, just to be consistent I'm doing that, minus, okay, the post-KZ equalizer plus Delta, so it will be simply W, okay, so you go in and substitute it, you will get E prime by S to be, to be H times, okay, so I'll do the substitution, you can go back and verify this, this is quite simple, gamma Z squared H star divided by MZ star, it's just the same substitution that I had before, M and inverse, so this is the pre-KZ equalizer, right, so this is the channel, this is the channel, this is the pre-KZ, from this I have to subtract the post-KZ plus 1, okay, which becomes MZ, M and inverse, okay, so what do we want to do now, okay, so the next thing to do is to look at this term, what is this? E S times mod H squared, so you can relate this to SZ, right, SZ is what? E S times mod H squared plus SN, okay, so you make that substitution in, play around with it, do some cancellations, finally you will get this very, very interesting expression, which becomes minus gamma N squared by gamma Z squared MN star divided by MZ star, okay, so you can play around with this, this is very simple, it's not too difficult to do this, this is E prime by capital S, remember what is capital S? Z transform with ZK, SK, I'm sorry, okay, and E prime, okay, once again capital E prime is Z transform with E prime K, which is the symbol component error in the slicer error, okay, so alright, okay, yeah, I'm going to come to it slowly, yes, I'm going to subtract that, okay, so now if you move back, so first thing is, what type of a filter is this? MN star divided by MZ star, okay, so it might have a lot of components, we don't care about it, but what will be the Z power zero term, okay, it will be one, okay, so you can, you can see that, so it will be monic, okay, so this will be monic filter divided by monic filter, so this is going to be monic, okay, so that's crucial, okay, so the Z power zero term is going to be monic, okay, so that's the only thing we'll use, we won't care about the other terms, okay, so Z power zero term is one, it's monic, so now if you go back to the time domain, you can write E prime K as a lot of other terms plus what will be the multiplying factor for SK minus gamma N squared by gamma Z squared times SK, right, so all the other terms will give you other multiplications, okay, but I don't care about it, but SK will have minus gamma N squared by gamma Z squared plus some other terms, okay, so this is what it's going to work out to, okay, so now if you look at the slicer input, okay, it's what, E prime K is XK, the slicer symbol component minus the slicer output, now if I want to look at the symbol component at the input of the slicer, I have to add SK to this, okay, so if I add SK to this, I get that my symbol component at slicer input is going to be other terms plus, which don't involve SK, plus SK minus gamma N squared by gamma Z squared times SK plus some other terms which don't involve SK, okay, so if you look at this carefully, it's going to be scaled by 1 minus gamma N squared by gamma Z squared, okay, all right, so when I slice or when I try to find S hat of K from the symbol component, my SK is not coming through without a bias, I'm multiplying the SK by some some number, okay, and then I'm slicing this, okay, so you first question to ask is how can, so this has to have a bad effect on the probability of it, okay, so that's the first observation, it's kind of loose, I'm not going to make it more rigorous than that, I'm going to say because of this multiplication, probability of error is going to be a little bit effect, okay, so it won't be as good as slicing just with SK, okay, all right, so basically, so the first question is, first point to make is slicer input is biased about SK, okay, so the slicer input you don't have SK plus something, you have constant times SK plus something, okay, and how are you going to slice? You're going to assume it's SK and slice, so if you slice assuming it's just K, then your probability of error will be poorer than assuming that you have a multiplication factor and then you slice, okay, so you have to adjust your receive constellation accordingly before you slice, okay, but remember only if you slice according to the original constellation, you get minimum mean square error, if you do this adjustment in the slicer, then your asset will change and you don't get necessarily minimum mean square error anymore, okay, because noise will also get multiplied by a suitable thing and you will get into trouble, okay, so while mean square error is a good quantity, it doesn't necessarily give you the lowest probability of error, in fact, if you adjust for this bias in your receiver before you slice, you will get a lower probability of error than the MMSE DFE, okay, but your mean square error will be larger, because while your symbol component is also being adjusted, your noise is also getting adjusted, so you have to be very careful about how the trade-off is going, if you just minimize your mean square error, you might get something which is not necessarily optimal with respect to noise, okay, well the trade-off, I'm not doing an exact analysis first of all, sir, so roughly, so what you should do, so, okay, so let me repeat what you should do, okay, so this is, so basically the MMSE DFE is biased, okay, that is the first observation, okay, so it's a good idea to unbiased it before you slice, okay, so an unbiased version would be something like this, okay, so you have, so you have SK coming in, H of Z, you add noise to it, then you have the same precursor as before, okay, so I'll call it the MMSE precursor, no change there, but before you slice, you multiply by gamma Z squared by gamma Z squared minus gamma N squared, okay, and then you slice, this is the same as adjusting your slicer also, you know, I mean, it's the same thing, whether you multiply before or you adjust the slicer suitably, you get the same thing, and then your feedback with the same MMSE post cursor, no problem there, okay, oh sorry, feedback goes to the plus and not to the multiplication, you do this, okay, you can verify for various examples if you want and even in other cases that such a structure will give you a lower probability of symbol error than the MMSE DFE, okay, it's likely to give and you can see the intuition for why it would give you a lower probability of error, because you're slicing properly, you're not doing slicing according to some other constellation, because you multiply it by something, right, however, if you compute the mean square error in this version, you will get a larger mean square error than the MMSE DFE, of course you have to, right, the MMSE was defined by minimizing the mean square error, okay, so this is an important lesson to learn when you minimize mean square error blindly, okay, when your error terms have non Gaussian components and you're just generally minimizing mean square error, you're never guaranteed naturally to get a minimum probability of symbol error also, okay, so these kind of things are called unbiased DFE's, okay, so if you learn an area called estimation detection theory, unbiased estimators are a very important class of estimators to see, okay, so the expected value of the estimator should be equal to the expected value before, if there is a scaling term then you don't necessarily have a good estimator in certain sense, okay, so all these things are interesting things to learn and unbiased estimators are always good to have, so the DFE that we had before is a biased estimator, it's not an unbiased estimator, so when you go to the unbiased estimator, things might improve and they tend to improve naturally, okay, so you can show out of all the unbiased estimators you have for such a channel, this estimator is the lowest MMSE unbiased estimator, you can show that also, okay, so this is a very good estimator, it's got some several canonical structures and you can show canonically its capacity achieving in some sense, okay, so you can read chapter 8 from Barry Lee and Mr. Smith to get more idea on this, I'm sorry, among the unbiased estimators, that's the lowest MMSE, if you bias it, you might get a lower MMSE, okay, but that doesn't mean you necessarily gain in terms of probability of some variable, so you have to be, so these are all some slightly advanced concepts in estimation theory, I don't want to just generally throw it at you, but I want to just point out that bias of the estimator is an important thing to keep track of, particularly when you slice at different points, okay, so since I've been mentioning about this mean square error, you shouldn't think that the mean square error is the best thing to do, okay, so because the calculations tend to imply that, you should also keep in mind that mean square error is only a rough indicator of what the actual problem is, particularly when the noise is non Gaussian, when you expect some signal components to be, okay, so bias is also important, okay, all right, so with this, we'll close the unconstrained complexity part, we'll move on to the constraint complexity, so if you have any questions on any of these things, now is a good time to ask, did you have a question, you seem to be not very happy with this, what? Which is better is unbiased MMSE DFE, if probability of symbol error is what you're looking at, right, it's better, any questions? So the assignment is already up, hopefully some of you saw it, there's a whole bunch of practice exercises and do it and make sure you stare at the final answer you get for the different situations and try to get some intuition out of it, okay, so that's the point of the assignment, it's got lots of questions, might sound repetitive to you, but don't just do it and forget the answer, okay, so look at the answer for a while, it'll give you some ideas, if your channel has a pole and if your noise also has a pole in the same part, same point, then some interesting things happen, if your channel has a zero and the noise also has a zero at the same point, some interesting things happen, lots of interesting things can happen with respect to noise being non-white, okay, and if noise is white, what happens when you have a pole in the channel, what happens when you have a zero in the channel, all that is also important, okay, so before I close, I just want to make a comment about Viterbi when you have in the general model, okay, so how do you run Viterbi when you have a pole or something in the channel is a question that's always complicated, so what you do is, after the precursor, what kind of a channel do you have, after the precursor you have a, usually you have a causal channel, right, so look if you think of after the precursor, you have a causal channel and it's also minimum phase in the sense that you equalize minimum phase, right, so you can approximate that and run a Viterbi at that point, so that's a structure is sometimes used, so you do a precursor equalizer to get your response to an approximate causal finite type version and then run a Viterbi on it, it's possible to do that also, okay, so I'm not talking about it because it's a point analyzing that, it's just a simulation based thing, so that's something you can try, after the precursor, your response actually becomes causal and minimum phase and nice, so you might want to run a Viterbi there, okay, so that's something that's done, okay, all right, so we'll stop there and move on to constraint complexity equalizes, okay, so we've been seeing a lot of things in this class and I've assumed a lot of background, right, so I've assumed that you know probability quite well, that you know DSP quite well, that you know the network systems part quite well, right, I don't know for those of you who are at least following the class or you feel scared of all the terms I'm using, you must surely know that all these things are being used in this course, everything is coming together in this course, right, only thing that I'm not using is probably Laplace transforms, okay, other than that everything else that you have learned has been extensively used in this course, okay, so now this constraint complexity equalizes, we're going to use one more area which you should have learned by now, okay, and that is fair amount of linear algebra, okay, so if you thought everything is not quite coming together, there are some things that you learned which are not being used in this course, if there is one final missing piece, okay, so if you're not, so one thing I'm going to urge you to do is I'll do it as you, as you, as I, as I keep going through the course, you'll probably notice a lot of things happening, so are enough, many of you doing the 515 course math methods, nobody's doing it, you're doing it, okay, so if you're doing that then you won't have any trouble, but if you've forgotten eigenvalues, eigenvectors, what is a positive definite matrix, what properties it has, all those things, it might be a good idea to go take a basic book in linear algebra and quickly refresh, okay, so I know a lot of people do basic matrices in your high school and for your JEE and after that you think linear algebra has nothing else, okay, and you happily sit back, relax and enjoy, okay, believe me, it's way more complicated than that, okay, so there are some very innocent looking problems in linear algebra which require a lot of theory to be solved, okay, so and we'll use some part of it here, fair amount we'll use, but this is a good opportunity for you to go back and revise linear algebra, okay, so it's a very useful tool, all right, so here's how our structure is going to look, okay, so I'm going to have SK which is my signal symbol, just going through a channel HC, okay, once again I'll assume a symbol rate sampled equivalent, okay, so that's a, so I'm going to have a front end which is like a low pass filter or something and I'm going to sample at symbol rate, so equivalently I would have a H of Z in the symbol rate world, okay, so that's my model for H of Z and then I have complex noise, okay, all these things are complex also, adding to it, okay, so I get my input ZK, symbol rate input ZK, okay, so to this I'm going to apply first a precursor filter C of Z which is of course going to be a finite type filter because I want to constrain my complexity, okay, and then I have a slicer, okay, so all these things are inspired by the previous constructions, okay, I have a slicer and then I have a post-cursor equalizer, okay, again finite type post-cursor equalizer, okay, so hopefully I get an estimate of SK at the output of the slicer, okay, so this is going to be my model, I'll call this the precursor equalizer, I'll call this the post-cursor equalizer, okay, so if I have, if I want to think of a linear equalizer, okay, what would I do? I would set the post-cursor to 0, okay, it doesn't exist, d of Z would be 0, so that would go away, okay, so both of them, both the DFE and the linear equalizer can be nicely modeled with just this one structure, okay, so if you want a linear equalizer only without a DFE, simply make d of Z non-existent, okay, so that goes away, all right, so again once again inspired by the previous constructions, we will select these tabs of C of Z carefully, okay, so I'll say C of Z has this form minus L to L Cm Z power minus M, okay, so I'm going to pick a two-sided response for C of Z, precursor as I expect, okay, and d of Z I'm going to pick as a strictly causal m equals 1 to L dm C, is that fine, okay, so that's how I'm going to pick, oh I had L for some other notation, really, I thought I always used size X or something, no, size X I took as L, oh L number of L, okay, okay, no, no, no, so maybe I need a different notation for it, I forgot about that, okay, so SK we said L, right, so this is not L, so I need another notation, so what can I use, capital M, is that, no, M we've been using all over the place, P is okay, okay, so minus P to P, seems like a safe, safe thing, I know I can do that, I know I can do that, so L doesn't really make sense, I might as well have L here, but since you're objected we'll have P, we also find it, okay, okay, so once again d of Z will be 0 for DFE, okay, so for linear equalizer I'm sorry, a linear equalizer and for the DFE it will be non-zero, so that's how we'll pick our Gaussian complexity equalizes, okay, any questions on the structure, the structure is fine, okay, so we'll pick, okay, so now, okay, so all right, so I said, so we're going to use linear algebra of course, right, so you'll see suddenly things will become vectors and matrices, that's the whole point in linear algebra, right, so I don't know if you're familiar with this convolution with the finite-tap filter can be conveniently represented as a matrix multiplication, okay, so that's useful and various things we'll see as we go along that that's useful, so that's the basic trick we'll use to move from signal domain to matrix domain, right, so we'll represent signals as vectors then convolution as matrix multiplication, so that's what I'm going to do next, okay, so keeping that in mind, the first thing I define is a vector zk which is a column vector which goes from zk plus l down to zk down to zk minus l, okay, okay p, so now there'll be trouble with me having to replace l with p everywhere, okay, so remember this is not a constant vector, the vector keeps changing with k for different k you have a different vector, okay, so how will you modify what is zk plus 1, push it down and then add the next k in the top, okay, so this vector one can imagine is useful for someone who's trying to implement the receiver, okay, if you're trying to actually implement the receiver you'll be getting this zk's, after your front end you'll be getting this zk's, so you put them in a in a register which is probably rotating or shifting that way, okay, and then you process only this vector, okay, the reason why you process only this vector is the output of the precursor is what? Output of precursor is you write typically zk convolved with ck, right, and then if you write it properly m equals minus l to l cm z k minus m, so you notice the kth output of the precursor depends on what? Depends on c of course on all values of c, but for z it depends only on zk, okay, on this vector zk, okay, so in fact you can write this very conveniently as c transpose zk, what would my c transpose be? c minus l c 0 c l, do you agree? So c in fact is a column vector, okay, c transpose is a row vector like this, okay, so the output of the precursor has this very convenient form, okay, so that's what the precursor is doing, okay, so you might have traditionally thought of it as a filter, the filter does nothing but multiplication by a row vector, finite type case, okay, so this is how we implement it, so this is a nice way of visualizing how to implement filters and practice, okay, so that's see remember all these things are complex, every one of these entries are complex in these vectors, okay, so it's not it's not scalar, it's not real, okay, okay, so what we're going to do now is first see linear equalizers, okay, so if I see linear equalizer then there's no feedback from the slicer, right, so d doesn't exist, okay, everything I want to optimize with respect to, is with respect to only c, okay, so I have to pick only c of z, so I'm going to look at linear equalizers first and now for linear equalizers there are several possible criteria, okay, there are two possible criteria that we saw, the first one is the zero forcing criteria, the second one was the mean square error criteria, okay, the zero forcing criteria is, we saw it's not as good as the mean square error criteria typically, so I'll do the mean square error criteria, okay, so that's what I'm going to do, okay, so we're going to see the mean square error linear equalizer, okay, so of course moment I say mean square error, you need to worry about the error itself, okay, so I'm going to define the error, okay, so let's see, let me just draw a picture before I proceed, okay, I'll call the output of the precursor as x and this is going to go through the slicer, there is no other filtering and you get this hat which is a cell, okay, that's fine, okay, so my error signal is going to be defined as xk minus sk, okay, so xk I know now can be written as c transpose zk, okay, well you have a minus sk, okay, so to move towards the mean square error, we're going to do expectation of mod e k square, okay, so now I have to think of everything as random process, okay, so first of all sk is going to be a random process, usually you assume sk to be the iid uniform random process over the alphabet x, okay, so maybe something else is also possible but we'll assume that to be random process, okay, so likewise here c is going to be a deterministic vector, okay, so there is no there's no randomness here, this is going to be constant, zk is going to be what, it's going to be a random process as well but it's a vector valued random process, okay, so it's slightly complicated, okay, so don't think of zk itself as a random thing but basically you think of this as a vector from the random process, what, zk, okay, so that's the best way to think about zk, the vector zk is basically a vector from the random process zk, how do you, how do you find zk, zk is nothing but sk convolved with hk plus nk, okay, so assuming I know hk and the statistics of nk, I can quite easily find the statistics for zk, okay, so but I have to know hk, if I don't know hk, I cannot find the statistics for zk, okay, so for now when we do the computation we'll assume we know hk and nk so that we can find the statistics of zk, so what are the statistics I might be interested in, typically you always make the white sense stationary assumption, so once you make the white sense stationary assumption out of filtering and computing mean square error the only statistic you'll need are first and second order, okay, so you'll need mean and auto correlation but be careful about auto correlation, these are complex valued processes, so auto correlation value is slightly defined slightly differently, I'll write it down as we go along, okay, so I'll assume white sense stationarity in fact jointly white sense stationarity, joint white sense stationarity of zk and sk and all that, okay, so all these things we'll assume sk and k are jointly white sense stationary, so that it's meaningful to define auto correlation and cross correlation etc, okay, so that's the thing, those are the things we'll be interested in, okay, so let me just quickly write down the expression for the mean square error and then we'll probably stop there is the expected value of mod ek squared, okay, so you're writing it in terms of vectors, so a lot of char is required, okay, so I'm going to write the conjugate term first, okay, so look at the first term, the first term is actually a real number, right, it's a product of two vectors but it's a real number, so when I conjugate and all that I can be happy, there's no problem, so I'll write c conjugate transpose zk conjugate minus sk conjugate, okay, so notice my notation here, star means conjugate, transpose means transpose, star transpose means what? Conjugate transpose, okay, so there's all kinds of notations possible here, different people use different notation, just to emphasize all of that I'm going to put this, okay, so remember that, okay, and then I'm going to do c transpose ck minus sk, all right, so that's the first step, we've written the mean square error in terms of these vectors, okay, so now you see what the modus operandi is going to be, okay, so you've written the mean square error in terms of c and you know the statistics for everything else, so when you pull the expectation inside, c will kind of nicely come out and you're going to optimize the mean square error with respect to c and find the best possible c, so that's the flow here, that's what I'm going to do, okay, so from here on it's just simple plain linear algebra, okay, there's nothing more to do, okay, so the intuition is from the channel and signal world has given us a nice simple linear algebra problem which we have to solve and find the best possible, okay, so but to do that it's algebra, okay, so you have to manipulate, play around with the terms, see what comes out, try to get some more intuition about the problem and then finally try to solve it, okay, so let's do the first step of simplification, okay, so the first product that I'm going to write down is the easiest term you can write down, expected value of mod sk squared, right, so that comes from the last, the two terms, the second terms multiplying, okay, and then you'll have a bunch of cross terms, okay, so I'll come to the cross terms later but before that I'll write the other square term, okay, so what is this, this product, c conjugate transpose zk star multiplied by c transpose zk, okay, so when you write product of several matrices you have to be very careful with brackets, okay, so you know this is a real, this is a number scalar, that is a number, so if you put brackets around each of these you can multiply but if I want to remove the brackets I have to be very, very careful, okay, for instance if I just write all the four matrices together it may not make much sense, okay, you have to be careful, okay, the way you write it, okay, so you should make sure that things are compatible and you write it nicely so that expressions become easy to evaluate, okay, so the way to write that very nicely is to pull c star conjugate out, keep the expectation inside and write zk star and zk transpose, okay, so remember this term is also equal to zk transpose c, okay, so the way I write it I do that and then I would get a c out here and then there would be cross terms, okay, cross terms I'll write in the next page, okay, there are cross terms so you have not written here, all right, at least the first two terms are easy to see and the cross terms will come in and we'll simplify, I think this is a good point to stop and pick up from here in the next class, okay, so we'll start simplifying things in the future, okay.