 So, here we begin, this is lecture 27 and the previous lecture we saw I gave you a simple example of took a very simple case of the channel and we derived all the three types of equalizers we have seen so far, which was the MLSD, linear equalizer and DFE, zero forcing both of them and then we calculated the figure of merit, looked at what the structure was and then try to conclude something that is what we tried Saturday morning since the vast majority was not here we missed out on it maybe you should go back and look at the lecture okay. So, anyway so the final I have one final comment before we proceed the MLSD and the match filter bound actually were equal and that worked out to 4 times 1 plus alpha squared by sigma square right. So, the channel I took you remember was what? 1 plus alpha Z inverse okay so it is the first order single zero model the simplest model you can have for FIR okay and then the zero forcing linear equalizer worked out as what? 4 times 1 minus alpha squared by sigma square okay and then the DFE worked out as 4 by sigma square okay. So, three different ways of dealing with ISA and we got three different figures of merit so the question now is how do you define SNR okay so that is an important question so what do we think of SNR always signal to noise ratio some signal power in the numerator and noise power at the denominator look at the match filter bound and MLSD okay so if you look at the transmit power what is the transmit signal power expected value of the BPSK constellation okay so that works out to something like 2 or 1 okay so which is some constant okay but if you look at the receive signal power how does the receive signal power work out roughly as yeah 1 plus alpha squared right so it's not equal to the transmit signal power okay so there is some confusion there okay so there's extra 4 times 1 plus alpha squared causes some confusion in the definition of SNR right if you define as just 1 by sigma square then it looks like ISA will be better than the non-ISA just BPSK is no ISI case right there the figure of merit was 1 by sigma square suddenly you're getting 4 times 1 plus alpha square the numerator okay so it looks like there is some confusion there so you should be careful about how you compare ISI and no ISI case okay so the trick is to define a suitable SNR for the ISI case which nicely captures both the situations okay you also don't want to do receive power as the signal that's one possibility but turns out that also is not the ideal case there's something else you have to do and maybe as you learn more digital communication in the future if at all you do will learn more about how to define SNR properly for this and then for comparison across ISI and non-IS okay so that's the point I wanted to make before we proceed okay so so far we've been always looking at a channel model which is SK going through an MZ and then noise being added you get you get Z case and you go ahead and process ZK with your detector okay so this we assumed was monic causal loosely minimum phase and NK we assumed was right quite Gaussian noise okay so far we made that assumption all these assumptions were made possible by the crucial choice of the front end okay what was that front end front end which is kind of invisible in this picture this is an equivalent model is what the whitening match filter okay first you had a match filter which is which is a filter whose response is H star of minus t just completely exactly matched to H of t which is the received signal right and then you have a whitening which relies on spectral factorization of the power spectrum of H of t okay so if you can do both of that well the folded spectrum okay so you do both of that then you get to a model like this okay so for this we know how to do at least three different types of equalizers of varying complexity and maybe we can be happy about that okay so but in practice hardly people people hardly ever use such models okay so why why do you think something like that is happening I'm sorry okay so time varying is a problem suppose I say time varying is not a problem then what do you do okay time varying is definitely a problem because in most systems things change with temperature day or time of the day except for so many things change right so you can't can't expect to hold everything a constant supposing maybe I can deal with it differently any other problems okay so first problem is knowing H of t okay so that's got to do with time varying but even otherwise knowing H of t exactly knowing the channel exactly maybe you don't want to know the channel exactly okay so you want to put some other front that's one thing other other issue is several cases hit star of minus t may not be that implementable okay so if you think of a rational Z transform for H of t right you expect H of t to be causal and stable right right I mean right so what will happen to hit star of minus t yeah it's going to be anti-causal and IIR that too right because even if it has zeros it will only have a finite number of zeros and once you do H star of minus t it's going to go anti-causal and IIR okay so there's no way you can implement it exactly you're going to be only able to implement an approximation to them okay so because anti-causal ISI IIR filters you cannot you cannot implement it doesn't work okay even if you delay it doesn't work that well I mean ultimately you have to stop okay so that's one problem okay so those are the problems you want to highlight with the white and match filter front in first problem is with MF itself okay hit star of minus t may not be known at the receiver or maybe it's varying with with time and you you can't change it very easily and maybe it's not implementable for so many of those reasons okay so these this this can cause problems okay so this that's what's one level the other problem is the whitening filter so so in cases where you can spectrally factorize maybe you whiten okay so maybe there'll be cases where you cannot spectrally factorize and you can't keeping keep doing that over time right every time HFT changes you can't change your spectrally factorization and all that all those things might be difficult so whitening is also an issue okay so both of these guys are serious sometimes serious issues okay that's one thing the other thing is usually if your front end is going to be analog okay at some level you want to keep it as simple as possible you can't expect it to suddenly change response and all these things are tough to do right now so you want to keep the front end as simple as possible and do everything else in the digital domain as much as possible okay so that's why you might want to simply have what what's the simplest front end you can think of what would be the simplest possible front end you can think of okay some low pass filter right so the your band of interest you have a low pass filter that's it you don't want to do anything more as far as front end is concerned you do that and then you sample then do whatever you want in the digital side slowly okay so that's what that's what you might want so you might want to deal with an equalization model with different front ends equalization model with a non WF front end okay so that's what that's the first step we're going to go we're going to do to move towards a more general equalization model as opposed to just a WMF front end okay suppose it's not that it's more general what do you okay but I don't want to do it in gory detail with continuous time and all that I want to still have a symbol time model okay symbol rate sample model okay I don't want to have anything which is non symbol rate sample okay just to keep things simple maybe later on we'll have a better model but we want to still keep it symbol rate sample okay so this will give you enough flexibility in the design you can study various type of structures and then come up with more general equalization methods for a different situation than when you assume a minimum phase MOC okay that may not be always possible the other other criteria is other other other important thing is zero forcing criteria okay the zero forcing criterion may not be the best okay so we're already seeing some trouble because of that at least in linear equalizer we saw that the zero forcing criteria causes a lot of problems in case there is a zero for your sir for your response you just can't it's not stable right it doesn't work okay so so some serious problems with the zero forcing criteria and also it causes noise enhancement okay so your variance of your noise at the slicer is going up okay so we saw it in the example also how it's how it's causing problems okay so because of that the zero forcing criteria may not be important so you might want to have other criterion for deciding for designing your equalizer filter even in the linear equalizer the filter instead of trying to knock out all the ISI maybe should do something else okay so those are other criteria that you might want to do okay so all these things we'll try to explore in a more general equalization model setup which is what we're going to do next okay so let's try to move towards that such a model it's quite simple and similar to what we did before except that I won't do spectral factorization I won't assume my my receive filter is matched okay so those are the two things which you won't do let's see how to come up with the model for okay so it's not too difficult so so let's see what we have okay so at the transmitter end we still have a sequence of symbols that you want to transmit it goes through let's say a general transmit filter okay so I'll say gt of t okay this is the transmit filter okay and then you go through a channel c of t of course this is a complex baseband equivalent and then you have the complex baseband equivalent noise n of t okay and then I have the received signal r of t okay so I'm going to imagine a situation where the receiver maybe doesn't know h of t or doesn't want to do matched filtering okay so in that case the most general thing that one can do is put a general receive filter g r of t okay so for instance a low pass filter okay let's put a low pass filter just in general okay and then you sample at signal rate so this is my model okay at symbol rate you sample okay and you get I'm gonna say you get zk here okay this is what you get and I want to build a discrete time model from sk to zk that's what I want to do okay so all these things of continuous time is what actually happens I want to have a discrete time model from sk to zk okay so see what's again it's easy to do the first thing that suggests itself is to look at the convolution of all these guys and call it h of t okay so look at what I'm calling h of t this is very different from the h of t we had before right we stopped here for the previous h of t now I'm going to simply convolve the received guy also and then call the overall response h of t okay so if you want to think in terms of frequency domain h f is going to be g t f c f g r f that's the Fourier transform okay okay so before the sampler here how can I write the symbols that are coming out okay so it'll have two components right one component that comes from the signal and the other component which comes from noise okay so if you want to write the signal here it's going to be this noise going through the receive filter and then the signal signal how am I thinking of the signal in continuous time it's an impulse train at every t seconds right so that gets convolved with the overall response h of t okay so delayed in impulse train when convolved is going to be h of t delayed okay so after that you sample at k t okay so so if you put all that together I can write z k very easily as sk h of t minus m t okay so remember this is my signal component before the sampler right now what am I going to do I'm going to sample this at k t okay and to this some noise gets added right remember this is n of t going through a receive filter and then being sampled okay so I'll call that some nk okay so I have to figure out some psd for this nk we'll do it and it's not a big deal but it's some noise process nk okay remember all of these guys are complex it's complex processes okay so it's quite simple okay so let's start with the signal part okay for the signal part right so if I substitute t equals k t here right it actually becomes sk h of k minus m t right so I can write that as a can I write that as a convolution I've written it happily as a convolution okay so I made a mistake here it's sm no sorry sorry it's what I thought this problem okay so sm here h of k minus m t okay so now if I if I think of my h of t sampled at k t as a discrete time signal h k okay then I can write z k as convolution of sm and sk and hk okay so it's always confusion a discrete time convolution of sk and hk plus nk all right so you can see that very easily here so you put k t here it becomes k minus m t okay so once a sample k minus m t actually becomes h square brackets k minus m so it's sm h k minus m which is exactly the convolution you have between sk and hk okay so so well that's in short my symbol rate model okay so I have a symbol rate model now okay so the question is what is this hk okay with hk now we'll imagine is a general discrete time response okay so both both sides because both the causal side and the anti-causal side it's got zeros poles all over the place okay so that's how we'll imagine hk to be okay so so so let's let's do a little bit more work here just to make sure we understand what this hk is in terms of hf okay so let me write this model once again okay so remember finally we have z k going through hz okay and then some noise getting added to it to obtain zk oh sorry seemed to be very sharp today so let me see all right so what is this hz now okay I have to think so I have to take hk and do z transform to get hz right right so if I think of dtft or evaluating the h of z on the unit circle then I can write that as the alias spectrum of hf that I had before okay so that's a nice way of relating everything getting it together okay so h of e power j2 pi of t okay also h of z evaluated on the unit circle is going to be what the 1 by t alias spectrum m minus infinity to infinity of hf okay so because I sampled the aliasing will happen okay there'll be no square term here okay so I can't say h of f is going to be non negative real right previously I did a match filtering so I got non non negative real response at the input to the sampler and then when it aliased I definitely knew I'm going to get non negative real now that assumption we cannot make about this h of z h of z restricted to the unit circle is not going to be non negative real okay it's going to be in general some complex quantity okay so I don't know what it is okay so it depends on hf what it is okay so that's that's one thing we can't say okay what about nk what can we say about the noise process nk just fall what is the distribution will it be it'll be Gaussian right it'll definitely be Gaussian see it's it's white white Gaussian noise continuous time noise that's what we assumed it's going through a filtering definitely it's going to be then sam it's going to be Gaussian there's no problem but in general we cannot assume it is white right when can we assume it's white okay so when the receive filter satisfies Nyquist criteria with rate 1 by t if it satisfies that then I know my nk will be white right that's the thing because g of t g r of t and g r of t minus kt will be orthogonal so and it becomes an orthogonal projection for noise okay but we may not want to even make that assumption why why make that assumption we need not even make the assumption we'll say it's a general low pass filter in which case we can only provide a psd what will be the psd here okay once again it'll be 1 by t aliased the reason for the 1 by t aliasing is because of the sampling okay and then the original psd is what the input psd multiplied by mod g r of f squared okay input psd is n0 okay the reason why I don't put the by 2 is because it's complex I'm going to say it's n0 so it'll become n0 1 by t aliased mod g of f minus m by t squared does that sound okay all right so that's my psd okay so in general I'll say my nk has psd which is say Sn of z and its restriction to the unit circle is Sn e power j theta okay so this will be a psd some power spectral density and my filter is h of z which is a general generic if you want the rational model you can always take rational models so without too much of a problem you can take rational h of z okay so this will be our general model for equalization equalization so you notice couple of important changes from the minimum phase model which was h of z would be minimum phase in that case and n of k would be white okay and you know that's an optimal way of doing things here we don't know anything about optimality it's just doing it because we want something that will work okay without too much of detail that's to work so we'll assume all right any questions before we proceed so notice the subtle change in the in the way we have modeled it is not nothing anything great but one thing I want you to be convinced of is when if at all you actually get to build the communication system and you're doing a symbol rate sampling you can always model your samples as the transmitted symbols convolved with some discrete time impulse response plus nk that's always true it's nothing that will go wrong there okay so it's always true so that's what we're doing it makes sense even at that level so that's what's happening okay so the h of z now it's like I said it's taken to be rational typically so if you draw a pole zero plot okay remember it's an actual physical response of actual devices that exist and actual physical channels so so it can't be crazy can't be crazy in the sense it can't have poles on the unit circle it will become unstable you don't expect real actual systems to be unstable right so if you draw the unit circle and draw poles and zeros okay one more assumption you can make is real systems are not going to be anti-causal again okay so if you're building something you can also expect that it'll be causal okay if at all there are things outside the unit circle it'll only be zeros and there'll be only a finite number of them okay so remember once again let me remind you quickly about this ROC and all that stability you always need ROC to contain the unit circle and if you have poles inside the unit circle your ROC is going to be ring including the unit circle and outside okay the largest pole and everything outside if you have poles outside and you want the system to be stable then you can't have a causal thing so you have to stop somewhere and then kind of come back in so it'll give you both sided right and poles outside so they're going to give you both sided response so that's the picture you have to keep in mind okay so when I say H of z is a real actual response that you see in practice mostly the poles and zeros will well zeros can occur outside maybe okay a finite number of zeros can occur outside but the poles will definitely occur inside the unit circle okay so maybe there'll be zeros on the unit circle okay but definitely there won't be poles on the unit circle okay so these are very realistic assumptions that you can make about H of z okay and and your equalizer or any signal processing you do beyond that has to respect that kind of a model you can't build something which won't work within this okay so this will be roughly the kind of thing we'll be looking at so basically poles are inside the unit circle of course you'll also have this pole at infinity okay where does this pole at infinity come from right everything in terms of z inverse you put z equals infinity things can go to zero if you don't have anything at z equals zero right so poles at infinity always happen likewise zero poles at zero also sometimes happens because of where you do it okay those those things I'll generally not worry about because that doesn't cause any causality or stability problem okay zeros can be both inside or outside unit cell okay so in any case whatever model you have the ROC has to include the unit circle so if you have poles outside it's going to be IAR anti-costal which is very very unreasonable to expect in a real system okay so this is the this is the picture okay so if you want to write H of z so another thing I'll do is I'll drop this H of z the reason is sometimes I'll want to talk about the z transform and equally well it'll apply when you restrict the z transform to the unit circle right so if I just write H it can be both H of z or H of e pa j theta sometimes you'll only define it on the unit circle okay you don't care about what happens outside you can do analytic extensions and all that if you are interested okay so so in general we'll take H to be okay so you can say you can argue out that it'll have to have a this a form like this hit zero which is some complex constant times some z power r which is some general delay to adjust for all these things times a minimum phase component okay the minimum phase component is going to include all the zeros and poles on the units inside the unit circle and then the maximum phase component which is going to include all the poles and zeros outside the unit circle likewise I mean I'm going to say obviously this will not be too many poles there won't be any poles outside the unit circle but anyway let the H max times hit zero which is going to be the zeros on the unit circle okay so this this is how we can split it these two terms you can conveniently not worry about a complex constant and the delay is not going to affect any of the signal processing you do later okay so this is a minimum phase component includes poles and zeros on the unit circle okay this is going to include zeros outside unit circle okay so it's going to be what so once you have only zeros outside unit circle that's going to be so this is going to be an FIR part okay there are no poles it's only numerator okay so it's going to be FIR and then this is the zeros on unit circle okay so in general H can be written in this form okay so about the noise spectrum we won't make too many assumptions simply SN of Z is the noise spectrum okay okay so so so so so so so so so in general this will be omitted okay so we'll omit this okay without any problem if at all it causes any problem it's only a delay adjustment in your receiver you can always do that so several samples of delay and even the constant will really ignore sometimes it can be there may not be there it's not a problem okay so this is our model okay so for this model one of the first things to try to do is the zero forcing linear equalize okay so that's the first thing we'll try to build we'll try to build a zero forcing linear equalize okay so it's very easy to build a zero forcing linear equalizer because you know exactly what to do what should you do okay so what's the model for this you have noise adding then you have ZK what does the zero forcing linear equalizer do you put a filter here I call it some D of Z such that the output of the filter does not contain any ISI it has contributions only from the kth term kth symbol and then you slice okay you make decisions okay this is what you do okay no ISI here so clearly what's the choice for D of Z 1 by h0 okay so that's going to be 1 by h max h min h0 okay right so you can see I've omitted some something so I'll put a tilde here okay so if you have to implement the zero forcing linear equalizer right what are the various filters you have to implement you have to invert a minimum phase filter is that easy yeah that's true so it will work there's no problem holes and zeros inside you invert it everything is perfectly fine you can nicely do it how about inverting h0 it won't work it's going to be unstable right once you put 1 by h0 it's got zeros on the unit circle and 1 by h0 is going to have poles on the unit circle so for the Z of le to be stable h0 has to be equal to 1 okay you cannot have anything else okay so for stability clearly you need equals 1 okay so no zeros on the unit circle the moment you have zeros on the unit circle there's zero forcing linear equalizer goes out of the picture okay so you don't want to implement okay and then what are the problems can you imagine h max inverse is also going to cause problems for you okay the reason is h max is a fir filter with zeros outside the unit circle so when you do 1 by h max for stability you have to make it anti-causal but what IIR okay so what does anti-causal IAR mean I mean you can't do it this infinite delay right unless you do infinite delay you can't do it okay and since that these two are I mean these are zeros that are going to be anywhere so you can't really control how your impulse response will be you can only approximate h max inverse okay so h max inverse is a problem and you can't implement very easily okay right you can see the we see it's got zero h max has got zeros outside the unit circle so when you do 1 by h max all those things will become poles okay and your ROC is going to include the unit circle which means for any possibility of implementing ROC has to include the unit circle so so if you do that you'll see you'll get anti-causal IIR filter anti-causal IIR is pretty much ruled okay it's tough to do these things okay so you can't implement h max inverse okay so so in the one the only way in which zero forcing linear equalizer becomes implementable is when h is minimum phases okay or maybe loosely minimum phase here and there but even there is a problem so it's only minimum phase okay so you go back into your old picture okay so you don't you're not able to get out of the old picture okay so that's one problem on top of that even in the old picture you're going to have noise enhancement right so the noise that you see at the input of the sampler the variance is going to be larger by the arithmetic mean of 1 by the power spectrum that you had okay so that's a problem power spectrum of the receive filter okay so that's a problem you have to face okay so both of those are serious problems with the zero forcing linear equalize okay so what we're going to do next is to change the criteria okay instead of doing zero forcing and expecting zero ISI at the input of the slicer we'll say we'll allow for some ISI and but we'll modify the criteria I'll try to minimize it in different in a different way instead of only throwing ISI to zero and living with all the enhanced noise we'll try to minimize something else which is a combination of both ISI and noise okay and maybe we'll get an implementable filter okay maybe we'll get better figure of merit okay those are the things that we'll go after okay so that criteria happens to be mean squared error okay so our msc it's called okay so there's lots of reasons why mean squared error is really really popular okay many people use this in so many different areas as a criteria okay so you might say wise why look at things like mean squared error why can't we just look at the important metric that we are ever worried about what's the only metric we're really worried about ultimately probability of error right so that's what you care about you don't care about mean squared error between any of these things or anything like that you only were care about one thing which is the probability of error at the output of the slicer I don't care about anything else okay but but we'll see probability of error is not really that calculatable on the other hand mean squared error can be calculated very easily okay so I'll point out so that's why there's lots of practical advantages to it and it's really really popular okay so let's see how how this is defined it's a very simple definition let me remind you what our model is once again a model is sk going through h of z and noise gets added okay so we are still in the linear equalizer model so I'm going to say I want to put some filter here and then get an xk and run it through the slicer okay I'm still going to do that as had k okay but instead of trying to design d of z such that xk has no isi for every term I'll try to relax that trend and I'll change the constraint the constraint I'll be looking at is this mean squared error or m s e which I'm going to define as the expected value of sk minus xk modulus square of sk minus xk so it makes sense right so the output you want the mean squared error between the input to the slicer and sk has to be as small as possible okay so if there's no noise then this will be no noise no is i this will be 0 okay xk will agree exactly with sk so it works out perfectly right okay but one thing I want to quickly remind you is this is not the same as minimizing probability of error right probability of error is some other expression some nonlinear not even square some weird expression q and all is there right so it's not a very direct simple expression okay you don't know how to think of that but this is just mean square error which is a which is may might be an equally good metric okay so this is the first thing I want to talk about the next thing is where did this expectation come from okay so expectation should be over a certain distribution right okay what are the random things that I have here okay so I'm going to say both sk and xk I will think of as random processes okay so far in the equalization problem I've been thinking of you have l symbols which you are actually transmit and you receive all your zk's and you're trying to process it and decode a s hat so you might have thought sk is a fixed lentil sequence but I'm going to in the model to calculate expectation I'm going to say the expectation xk and sk will be also a random process okay random process but each sk is actually a random variable with alphabet what alphabet x it's the same x won't change all that but I'll simply say it's a random process in fact the most standard assumption to make it is what IED uniform okay so we'll say IED uniform is the most in most cases but watch out for this I'm just throwing it to you as the simplest possible assumption in fact later on if you do some advanced studies you'll see this IED uniform assumptions will be relaxed okay for some reason you won't see it but but for ISI it turns out you might want to relax that assumption you might want to put some further filtering to condition your symbols but that's different we'll for our purposes mostly in this course we'll assume IED uniform for our sk model okay and xk of course I mean once you have nk itself xk becomes a random process it's no problem but in addition to nk sk also is the random process so xk becomes a it's a random process just by definition okay so it's I'll say it's an induced random random process by all these other things and the definition okay all right so the expectation is over the joint distribution of all these case yes that's that's the way you do it you'll see in practice it'll work out in a much easier way don't don't don't be too too confused about okay so that's the main thing okay so to simplify this we'll define an error process which is what what is the error random process ek which is sk minus xk okay so I think it's usually defined as xk minus sk let me be careful here yeah it's defined as xk minus sk so let me be consistent okay this is my error process okay so so the msc in terms of the error process has a simpler and nicer interpretation okay expected value of mod e of k squared okay so if I think of it in the spectrum domain how will I define msc it's the area under the power spectrum of ek okay so if you think of ek has having power spectrum sc e pa j theta okay msc becomes what the arithmetic mean of s is that clear okay so once I do that msc becomes the arithmetic mean of s okay so usually if at all you want to accurately estimate s c the msc you're going to use the spectrum domain okay so you can't try to find the joint pdf and you can find it it's more more confusing more integrations involved okay so if you do the spectrum approach it might be much easier all right so this is ek okay so so our goal is to minimize the area under sc of e pa j theta okay so you want to design your d of z so that this is minimized okay so I want to comment I will comment more about how to go about doing it and all that before that but before that I want to make an important comment okay so this msc is computable at the receiver okay so that's why it's it's most useful so how would you compute it at the receiver any ideas some approximation only I mean not the exact computation but an approximate version of the msc can be computed at the receiver how will you compute an approximate version of the msc s hat of k right so that will be a very simple and nice way of computing ek so I'll put a question mark here because this is not guaranteed to be equal to ek all the time but assuming your system is kind of working reasonably it's going to be equal to ek most of the time and so if you compute square some square of ek here that's going to be a good estimate of your mean square error okay so whether or not you know sk you can compute the msc at the receiver this is this is a very crucial reason why msc is used all the time or some version of difference and some error notion is used all the time because you can keep track of this gay and see every time this error is blowing up what does it mean something is wrong with your equalization so you'll know all these things so these kind of things are very crucial at the receiver now instead of msc if I had said probability of error how do you compute probability of error at the receiver it's not that easy to keep track of you should know sk okay so usually in the in communication systems there's a training phase where you know what sk is in that training phase you can keep track of probability of error in fact in the training phase you can keep track of the exact msc also even probability of error can be kept track of but if you are in the decision directed phase which is after the training phase everything has to be controlled by what decisions you make at the receiver okay so actually it turns out in today's complex systems there are some decision directed ideas which can help you keep track of how well you're doing even without msc and all that but at the heart of it is all assumptions like this you compute your decision directed error then use msc mode square and all that without probability of error you can nicely keep track of how well you're doing okay so those are the practical advantages you wanted to quickly comment on these things before we proceed looking at how to minimize the error okay so let's see so let's go about doing it so it's a question of just finding sc okay you find sc and look at the error and try to minimize sc that's what we're going to do okay see remember sc is non-negative real on the unit circle so if you minimize sc at each point you minimize the arithmetic mean as well so that's what we'll be doing okay so it's a very it's a straightforward thing to do except that the expressions are a little bit nasty okay so what contributes to se suppose they have to evaluate sc okay so let me cut and paste my model here so that I have a picture of the model ready oops so here's a model so so ek is what ek is xk minus sk okay so if I want to compute the power spectrum density of s e there are two different components to xk right okay so so okay so let me go slowly okay so this is the this is going to be power spectrum density of did I get it right okay so let me write this let me do the simplification first then I'll come to it and just looking at the time when I'm getting trying to figure out if I can do this properly or not okay so let's write xk as what sk convolved with what xk is what sk convolved with hk convolved with dk okay and then what plus okay so I'm going to say nk convolved with dk it's people usually don't write such things just writing it because I want to do the computation minus sk itself okay so if you want to write this carefully I can write it as sk convolved with hk convolved with dk minus what minus 1 for all k okay so I'm going to call that 1 plus nk convolved with dk all right so so you make the assumption now that sk and nk are what independent random processes okay for all k they are independent okay so once you do that you'll see this random process is going to be independent of this random process so if I have to find the power spectral density of e I can find the psd of this separately and then the nk process separately and then add the two psd because the correlations will vanish okay so whatever correlation terms you get you go off to zero because I'm assuming the cross correlation terms go to zero okay so that's a reasonable assumption to make did you have a question minus what oh delta oh sorry sorry sorry sorry okay okay sorry it's okay I mean the psd is one anyway doesn't matter so now if you find psd right this is not too bad to find so if you do if you do Fourier transform here you get what hd minus one okay and mod square of that is hd minus one squared so you do that you find your psd very easily okay so this becomes the power spectral density of sk what is the power spectral density of s I'm going to assume it is iid uniform right so power spectral density will be flat okay it will be equal to the variance of 1 sk okay so that I'll call energy in the signal s okay so what is this energy in the signal s it's expected value of sk squared assuming it is iid uniform right so it's basically what you've been doing average square of your constellation modulus hd minus one squared plus what power spectral density of noise modulus d squared okay okay is that okay so this filter as responds hd minus one so when the symbol process goes through that it becomes es times mod hd minus one squared that's the psd and the psd here is power spectral density of noise going through d which is mod d squared okay so this is the psd okay so you see the power spectral density of the error has two distinct components one component which comes from the signal another component which comes purely from noise and they are additive okay so remember this is not just one number okay so each of these guys are functions of e pa j theta okay so keep that in mind so don't think of this as each just one number so this is actually I've dropped that e pa j theta for convenience okay so each of these guys are component for is it okay so now the problem is a simple algebra problem it's a question of finding that d which minimizes s e okay and you've written s e in terms of d and h and es and sn are constants for you they're given to you you can't change that you can only change d to find the best possible s e or the lowest possible s e it's enough if I minimize s e for each theta right the reason is I'm trying to minimize the arithmetic mean and it's non-negative real so this is no problem I can just minimize this thing for each theta and I'll get the overall minimum okay so the trick to doing that is to do completing the square here okay so this is some square time plus some other square you go through and complete the square but remember all these guys are complex quantities it's a little bit tricky if you're not used to this but I'm going to give you the final answer the final answer works out to okay so after you do the minimization after completing of the square the final answer will work out to maybe we'll come back and derive this s z okay so final answer will involve z okay so I'm going to call this zk no no not this oh my god what is zk this is zk okay the final answer will involve sz mod d minus es sz inverse h star okay so h star square plus es sn sz inverse what is sz sz is the power spectral density of z power spectral density of z here will have two components very easy to write down it's going to be sn plus es times mod h square okay the reason I wanted to leave you with this is try to try to prove that the c we had before is the same as the c here okay make an attempt when we'll pick up from here in the next class