 okay so this is lecture what what's the number okay at least some people are keeping track okay everybody's ready good so so I want to quickly summarize what we've been doing and then get along I think there was some notation and even in the definition there might have been some inconsistencies about what I called SNR for QAM and PAM I want to just make sure we consistently define it then we'll proceed for cases be a quick summary so far we've been looking at ideal channels and hopefully people would be comfortable people would be comfortable dealing with ideal channels okay so maybe it's useful to distinguish between two cases that we considered one case is baseband PAM okay and then we saw passband we saw QAM in the passband it's a passband PAM baseband PAM and it's typical in this case in the ideal case to assume you have a bandwidth between minus W and W which is flat okay and you want to use this and do your transmission okay so we found if the channel is ideal one can eliminate ISI okay so one can do your some filtering at the transmitter and careful filtering at the receiver and completely eliminate ISI by choosing your pulse shape to be what okay so you avoid ISI by by choosing a Nyquist pulse shape okay what do I mean by a Nyquist pulse shape okay what's an example that you can take square root raise cosine okay so you do that square root raise cosine is an example and if you do that let's say a roll-off factor beta you see that your symbol rate then becomes 1 by t equals 2W by 1 plus beta okay for beta between 0 and 1 okay so this is your symbol rate okay so what's the largest possible symbol rate 2W okay so if you go in anything about 2W then you cannot avoid ISI that's all I mean if you're willing to deal with ISI you can go larger than that that's okay so but you can't avoid so typically 2W is taken as the largest symbol rate nobody will exceed 2W okay so that's what it means okay so when you do baseband PIM okay your your constellation has to be real right so you have a real constellation okay maybe I'll call it XR okay to emphasize that and so from here one can define what I'm going to call as let's say bits per dimension okay so the reason is I want to keep this R as bits per dimension even when I go to QAM okay so there I'll have to divide by 2 right so you'll have a complex constellation so you'll have if you do log base 2 of size of the constellation those that many bits you send in two dimensions okay to normalize everything we'll say R as bits per dimension in this case here it works out better so once you come up with this you can also define very easily what the bit rate will be okay so maybe RB this is going to be log base 2 XR divided by T if you want R by T if you want we can write it as 2W log base 2 size XR okay so well that is R right divided by 1 plus beta okay so there's a closely related quantity called spectral efficiency which is defined as the actual bit rate divided by the amount of positive bandwidth that you're using okay so that's the spectral efficiency so you have bit rate RB here what is the positive bandwidth you're using W so you divide by this and you get your spectral efficiency okay so so maybe there's not too much room here so maybe I'll go to the next page and write down spectral efficiency for for baseband PAM okay so typically it's denoted mu RB divided by W and it works out to 2 times log base 2 divided by 1 plus beta okay so as you increase your roll-off factor what's happening to the spectral efficiency it's going down and that's to be expected right so you're kind of wasting that roll-off bandwidth no just for you're not using it for the symbol rate at all you already said beta equals 0 using all the bandwidth for for your symbol rate in a way and that you get maximum spectral efficiency okay so that's these are nice things to remember this is a very good thing to keep in mind okay so when when you do most practical designs this is an equation you'll use all the time to determine your parameters okay so typically there will be a target bit rate and a given bandwidth okay so you divide RB by W you get your spectral efficiency and then from there you select a suitable beta and a size of your constellation okay so there might be multiple possibilities and you have to pick a nice thing which works out works out for you all right so that's I think there were a series of problems on this in the tutorial the solutions based on this and also even in the exam there was a question based on this and I think some people did not answer that questions maybe maybe you should be careful when you prepare for the exam the next time all right so so the last piece of thing to tie it down is the SNR calculation see all this is fine but ultimately of course I'm interested in probability of error versus eb over and not right so that's my real goal that's that's complete that's what completely characterizes that so for that the SNR in the waveform side we found to be equal to power divided by and not W I did this computation and showed this will be equal to in the baseband case ES by and not by 2 okay right so this is a closely related quantity which is eb over and not which works out for PIMS SNR divided by 2R okay so you can see all this will work out properly I did this computation the last class okay so typically this eb over and not is preferred to SNR as the x-axis the reason is it's normalized with respect to rate okay so you can compare BPSK to 16 PIM or 32 PIM in a fair way because you're normalizing with respect to rate right if somebody is doing 32 PIM okay and achieving a certain probability of error at a certain eb over and not okay it will seem like he needs a huge eb over and not while somebody when he's doing BPSK seems like he's needing a much lesser eb over and not but not eb over and not as lesser SNR okay but that's because you're comparing two different things once you're normalize with respect to rate you see the eb over and not will approximately be the same for both case if you're achieving if they're achieving similar points okay so it kinds of gets rid of these kind of differences in rate by normalizing there's another way to normalize SNR which is the SNR norm so instead of dividing by R 2 times R you divide by m squared minus 1 that's also another way of doing it and we saw that's a very good equalizer right even for BPSK as well as any large M PIM you got very similar curves once you shifted to normalize SNR in the x axis okay so that's also a way to do it but today most people do eb over and not and that's accepted so you can do eb over and not as alright so that's the that's the that's the baseband PIM there's any question that's disturbing people it's a good time to ask any definition it was not clear okay alright so if you want probability of error there's an expression that you can do if you restrict your constellation to M PIM okay so probability of error works out to well a simple error works out to roughly okay two times Q okay if you want to write it in terms of eb over and not it will work out to 6 log M base 2 divided by m squared minus 1 eb over and not okay so that's the formula think in terms of SNR it works out to root 3 SNR by m squared minus 1 and all these things are things you can quickly work out alright so that's baseband PIM and this much should be clear so given an ideal chunk of bandwidth and baseband you should be able to actually design these things okay so it's very easy to do very quickly you can come up with the transmit filter and receive filter and it's no big deal alright so if you another thing to keep in mind is the passband QAM picture and it's very similar to this except that you'll you'll now deal with the the complex symbols okay so even in baseband PAM right I can think of my frequency W hertz as a passband frequency with the center frequency of W by 2 so I could do if I wanted a passband modulation even with a baseband okay so but usually you'll see the spectral efficiencies will be okay you can as well do real symbols will be okay okay so we'll see see the calculation one needs to be careful okay so what is the situation here here you have a frequency of W hertz available around the center frequency okay so this is what is given to you okay it's okay to deal with these two things as distinct quantities but you should see the relationship between passband QAM and baseband PAM you'll see the very closely related at least in the ideal case they are very similar there's no problem okay so instead of dealing with passband QAM we typically do what we do this down conversion and treat everything in as complex baseband okay so this becomes completely similar to a picture where you have a bandwidth of minus between minus W by 2 and W by 2 you shifted by FC right but then you have to imagine sending complex symbols okay because when you move this out your transmits spectrum need not be symmetric around zero okay when you wanted real pair baseband it had to be symmetric around zero okay so now you have it need not be symmetric so you can actually send passband I mean non symmetric thing which is complex okay so now in this complex baseband picture you can easily apply the avoiding ISI rule okay so how do you avoid ISI once again by choosing Nyquist pulses which will give you W by 2 equals 1 plus beta by 2t where beta is the roll-off factor okay so your symbol rate becomes okay what does your symbol rate become 1 by t becomes W by 1 plus beta it seems to be half of the real case real baseband case but but you're doing complex so it's pretty much the other degree of freedom enters the picture and you get the same thing okay and so this is a way of studying both together and people usually when you think of passband QAM you think of a positive frequency of W hertz and baseband PAM think of again a positive frequency of W hertz but it works out differently the way you look at okay so this is the symbol rate and this is symbols per second okay so now you can do a similar calculation with all the other things okay so you'll have a complex constellation say maybe chi C okay so from here if you want to go to R okay so this is going to give you an R which is half log base 2 mod chi C bits per what dimension okay you can also define a rate which is bits per symbol which would be what this factor of half will be gone okay it'll be simply log base 2 okay so it depends on how you view it okay but usually since capacity is specified for bits per dimension it's it's better to do this for bits per dimension as well okay so that's the that's a nice thing to do okay so from here once again can easily go to the bit rate okay so the bit rate RB will work out as well I'll do the direct calculation C divided by 1 plus beta okay so once again if you move to spectral efficiency okay mu which is again defined as RB by W here because amount of positive frequency as still W okay so this works out to log base 2 chi C plus 1 plus beta okay so artificially it looks like you have lost spectral efficiency okay but that's not true because typically everything else remaining the same if you can afford the same SNR and all that you'll see size of chi C can be the square of size of chi R okay everything else being the same at the same point you'll be able to afford that in QAM okay so it is the same spectral efficiency both ways okay so that's my point so now if you do SNR I think here is where I might have been a little bit inconsistent in the way I defined SNR for PAM and QAM okay in the continuous case we'll get P by N0W this will work out to be ES by N0 okay so I might have defined ES by N0 by 2 as the SNR at some point but I think we'll go back and revise it I think ES by N0 is a better definition for QAM okay so this will be the definition for QAM because energy is over two dimensions right so noise also it's good to add up both the dimensions and make it N0 by 2 plus N0 by 2 in the denominator okay so if you do this with your definitions EB by N0 will still work out to SNR by 2R okay so you can keep that normalization the same okay right so remember R you have to keep as bits per dimension okay somewhere you have to take care of that half it's just some factor of two just be careful with these things so that we are consistent at least in this course okay alright so that's the SNR definition and now we can do probability of error I think I did it for M squared QAM okay the probability of error will roughly be 4 times Q okay so I think if you express it in terms of EB over N0 it'll work out to something very similar to before okay M squared minus 1 EB over okay so I think these two are going to be finally similar all right so all right is that fine okay so these are all trade-offs you can do so suppose for instance you want to decide you want a certain probability of error okay so you go back and figure out how much EB over N0 you want and I figured out what power you need what bandwidth you need and then you go back and you know M also so you know what the spectral efficiency is all these things you can nicely compute with these simple expressions so it's not a very difficult computation okay so roll off also plays a very simple role okay so hopefully there are no questions as far as these things are concerned okay any questions okay I know this is quiz week so presumably you're not spending any time looking at 419 since the quiz is over okay so maybe maybe this is a time for you to think about all these things if you had any questions about how these things are defined it's a good time to ask okay all right so all right so one more last thing last piece is to simply draw the whole picture and then of the actual system and then we'll move on okay so we go ahead and see okay so once again I want to remind you of this picture because this picture is very important you can quickly forget it and go through but that's probably not a very good idea okay so what's our what's our motor software and you up to now okay so we have some bits we're going to do a mapping and convert it into a set of let me say complex symbols okay so I'll restrict myself to complex baseband because that's general enough right there's no restriction complex baseband also includes real baseband and real passband okay so I'll restrict I'll always stick to complex baseband from now on okay so I'm producing complex symbols okay and this goes through a transmit filter okay then I do an up conversion okay it goes through my channel noise gets added and then I do a down conversion and then I do a receive filter and sample okay once I do that what am I sure of okay sample at symbol rate right yes where some two I'm missing dangerous man where am I missing a two I think there'll be a d squared by 12 or something no okay so let's okay so there's some dispute about the actual probability of error term I wrote down so maybe we'll talk about it after class and just make sure we're on the right page okay so they can be all factors of two all over the place one needs to be careful and there are ways of doing it to get rid of them okay so this channel now is a passband channel right centered at the frequency in which you are converting and down converting remember this is a symbol rate sampler okay what do I mean by symbol rate sampler you produce one sample per symbol okay so that's what you're doing and you know for sure that if you design your transmit filter and receive filter to avoid ISI satisfying the Nyquist criterion okay then what will happen this will be symbol plus noise okay and then you do your detection to produce estimates of bits okay you produce bits hat okay so this has been our picture so far okay initially we started out by saying we will occupy a very small bandwidth or increase our symbol rate so large so that what can be my transit filter simply a hold okay I don't have to do anything else in that case the receive filter will only be a integrate and dump it's a very simple design but the problem is you're not occupying all of the ideal bandwidth that you have so then we said we can in fact go further and further and we figured out what's the maximum symbol rate that you can use so that you still have no ISI okay so you don't want one symbol to contribute to the sampling time see when you sample for corresponding to a particular symbol you want no other symbol to contribute noise can contribute but you want no other symbol to contribute that was the definition of ISI right so that's the that's something that we wanted and we saw we could do that only up to a certain symbol rate and then you do a roll off to adjust for certain practical aspects there for the sync signal to died on fast and all that and those things will mean in practice some delays in the transmit filter some delays in the receive filter left account for all that all that is fine okay so you do all that finally you get symbol plus noise and you can detect all right so there's also a complex baseband view of this whole thing okay so this whole up conversion down conversion instead of viewing it in this form okay one can view it equivalently as simply a complex baseband channel okay so this channel is going to be complex but it'll be baseband okay so in this class I never did the baseband equivalent for noise okay so it turns out you can do a very similar baseband equivalence even for noise okay noise in the band one can do it so this noise also I can think of as complex noise in baseband okay so there is something I'm not going to do it and it's not too important because once you filter and sample it's easy to picture it but within this picture one can also imagine noise and then everything becomes baseband for you there's no passband and from now on I will happily only consider complex baseband okay so in this complex channel the frequency response I mean the channel the impulse response is complex which means there is no symmetry around zero and the amplitude okay it can have asymmetry but still it's okay alright so this is the equivalent that we will be looking at okay so this receive filter okay so plays a crucial role okay so the way we the way I justified it or introduced it this what they're saying I'm projecting on to the orthogonal signal space okay so as we go along there will be several heuristic interpretations for these receive filters okay so one one heuristic interpretation is to say this filter rejects out of band noise okay which is something that you have to do so noise itself will occupy a huge bandwidth and you don't want to incur all of that so it rejects out of band noise so in some systems you might even have a low pass filter before this receive filter okay which rejects everything that is out of the band okay so that's the that's one thing to keep in mind so there are several interpretations for this one of them is that this filter rejects out of band noise another interpretation is that it provides some symbol rate statistics for you to do detection right after this receive filter you only have symbol rate statistics what do we mean by symbol rate statistics you get one value per symbol and that is enough for your detection in the ideal case okay so we'll later on maybe we'll see something non-ideal something approximate even there we will use some approximations so so one more one more heuristic interpretation for the receive filter is people will say it maximizes the SNR at that time when you sample okay so people will design the receive filter so that some SNR maximization happens so all these things are very heuristic simple ways of thinking about the receive filter so you might have a receipt filter in future for rejecting out of band noise or maximizing SNR at the time when we when you sample okay so those are all heuristic interpretations which have a lot of meaning you'll see when you read books several books talk about these kind of things and this nice ways of understanding what the receipt filters doing okay but theoretically the good way of viewing it is it's doing correlations and producing correlated outputs which are important for your detector okay so that's how that's how you view it okay so any questions on this picture okay was this an important picture okay because if at all you're actually implementing a communication system you'll have to think of these blocks in this fashion okay so it's very very important that you understand this picture so many elements here all the assumptions of this picture okay the coherent assumption of FC being available exactly the timing assumption of capital T being known exactly okay so if you're familiar with these circuits oscillators are never stable the temperature changes the oscillator changes right so many things changes okay so many things change so one needs to be careful about these assumptions so in practice there'll be lots of circuits around these things to fix these problems okay so you should anticipate those problems as well okay so I think we have to move ahead and moving ahead we're going to now slowly relax the assumption of so that's summary okay so that's what we did so far so the moving ahead we're going to relax the assumption of what what is the assumption that we will relax sorry ideal channel assumption okay so we're going to relax the ideal channel assumption okay so that's what we'll do so first question you might ask is why would you ever want to do something like that why can't we just sit inside a ideal flat channel in fact maybe it's possible to view any spectrum as several parallel channels of small small bandwidth each where it's ideal in fact that's done today people view it that way also but initially we're going to see what what do you do the channel is not ideal okay so that's the that's a good thing to study as well and in that case we'll see you can't avoid ISI so you'll have ISI and you'll deal with so this is what's coming further ahead okay non-ideal channels okay which means ISI okay okay so we still want to do symbol rate statistics okay we still want to have only one sample per symbol and all that and then we'll deal with ISI in a different ways you'll see there's so many so many heuristic ideas here now okay so if you want to do the optimal thing with ISI it's very difficult usually write down the whole thing it becomes very painful there's no intuition and you can't start building things immediately okay to be able to build we'll do a lot of heuristics we'll say okay this seems like an interesting thing to try so what is it need what is needed there okay eventually I'll also show that one such construction is optimal okay but but for now we'll just simply build on some some simple heuristics and try to come up with some receiver structure which you know we can build okay at least and at some level of assumptions which we know we can build and then we'll study its properties more and more and maybe simplify it to get it to a very practical level okay so that's my approach there are several different approaches for dealing with ISI channels my approach is going to be try to get some system which you can build okay and then study its properties or maybe study its optimality etc etc okay so that's how I'm going to approach this okay so I'll closely follow section 5.4 in Barry Lee and Messerschmitt for this okay so for in this lecture and maybe the next one we will follow section 5.4 so if you have the book you should probably go and take a look at it okay so what's our picture now okay so what what does our picture become when I said non-ideal channels what does what does it become so you still have bits okay and I'm still going to map okay so I'm going to still do a mapping to some constellation okay to produce a set of symbols okay so I'll call it s which is say s0 through sl minus 1 each of this is a complex valued symbol okay so I'm still going to think of it that way sometimes instead of thinking of it as a vector I'll think of it as a discrete time complex signal okay in that case my notation for this will become sk okay so that's a standard notation there okay and then I'm still going to do a transmit filter here okay I'll say my transmit filter impulse responses some g of t okay so when I do that my signal is going to become s of t equals what summation k equals 0 to l minus 1 what sk g of t minus kt remember that was my picture I'm going to think of the symbol sequence as a impulse sequence and then I'm going to convolve with g of t so everything will become like this okay capital t is my symbol time okay so 1 by t is the symbol rate okay so 1 by t is the symbol rate all right so so you might choose g of t using various considerations maybe you can still choose it to be a square root okay it's okay all right so but what's going to happen now my channel is going to do something to each of these things okay maybe my channel response is c of t okay remember this is complex baseband okay c of t is going to be complex but its frequency is going to be in baseband okay frequency spectrum is going to do this and then you have noise added to it and you have to process this in baseband remember there's an up conversion and the down conversion which I'm swallowing into a complex baseband channel and the complex baseband equivalent of noise okay so I'm doing all of that okay so this r of t is going to become okay so you have to convolve s of t with c of t and there is a delay and you can also incorporate that delay very easily and ultimately you will get an expression of this form okay k equals 0 to l minus 1 sk h of t minus kt okay plus and t what is this h of t what will this h of t be yeah g of t convolved with c of t okay so that's what it's going to be okay so h of t is g of t convolved with c of t okay so it's the same picture okay right g of t and c of t are occurring in sequence obviously in series so you can look at it as convolved and one thing happened so far the way we chose g of t we were able to assume that what h of t was equal to g of t itself in that way c of t was ideal we restricted ourselves in that band okay so maybe now you want to increase that and allow for h of t to be different from g of t okay so that's the basic issue here okay so right now I'm no longer guaranteed h of t is same as g of t when I know it when I know it is same as g of t I am in this simple I know the orthonormal basis okay then I can happily project onto the orthonormal basis I won't lose anything there's no problem but when I don't know that I have to do something else okay so I cannot do the same as what I had before and you have you will have ISI in in any configuration and you see when I when you when I I'll consider several configurations from now on and whenever I do sampling at symbol red I will see in several of those configurations I will have ISI okay so each each sample will not correspond to only one symbol multiple symbols will play a role okay so I haven't come to that ad is there a question yeah but the point is you don't know h of t you don't know or you cannot control h of t okay well c of t okay okay so let me be careful okay so right now so okay so you're saying design g of t so that h of t becomes orthonormal h of t and h of t minus kt become orthonormal yeah that could be a way of doing things okay but but that will be slightly impractical in the sense that then you can't build a transmitter which works for different channels you'll have to do something more right so if you want to just take the transmitter and plug it into another channel maybe you have to do some more training there and then the filter will change it it might be possible I'm not saying no actually today's system are adaptable in that way but I want to consider a situation where you don't want to alter your g of t too much and you want to deal with any c of t that might be there okay that's one issue other issue is even when you have a fixed channel with time things will change okay tomorrow the temperature might be a little bit higher okay so because of that something else might do some some non-linearity or some such thing and your channel might slightly change okay somebody might might be doing something else so you know things change like that and you don't want to hang your entire design on the fact that g of t convolved with c of t is still going to give me a Nyquist pulse okay I don't know for sure okay and so many things have to change okay not only g of t right if c of t changes h of t changes which means your match filter here will have to change when you can't tie it to one specific h of t very easily so those are practical issues which you might have to deal with so maybe it's a good idea to do something else but having said that my first configuration is going to assume I know h of t at the receiver okay at the receiver not at the transmitter okay just at the receiver okay so in that case you can't really go back and adjust your g of t okay you have to live with whatever you got at the receiver okay so that's the those are the kind of assumptions that we'll make to make it more practical all right so I have to figure out what to do here okay I don't know what to do here but I know what I want out of it what do I want out of it I want s hat okay an estimate for my symbol sequence or s hat of k okay that's what I want okay so so like I said we're going to consider simple heuristic approaches to come up with something that we can build and then we'll study how optimal it is okay so one can obviously take this problem in its full generality and do the theoretical derivation of what the optimal receiver is I'm sure there are good books which do that I'm sure that I've got people who can do that but I'm going to take a simple approach and come up with some I'm going to follow what Barry Lee and Mishra Smith do it's a simple approach and I like that because it introduces a lot of nice ideas at the right time okay so so what are we going to do okay so here is a principle that I'm going to use okay I have r of t which contains information about l symbols which have collected into a vector s okay I'm going to simply take so first thing is I'm going to assume that h of t is known at the receiver okay so it's known at the receiver and I'm going to define some kind of a minimum distance criteria between r of t and a possible s of t and then pick based on that okay so that's what I'm going to do and that's turns out to have a lot of general principles in it okay so here's the heuristic minimum distance receiver okay so what am I going to do it works in this principle s hat it will set to be equal to argument of the minimum of a belonging to chi power l what do I mean by chi power l vectors of length l with components from this chi okay so the x script x okay so basically vectors of length l from my alphabet from my constellation okay so that's okay what am I going to do I'm going to minimize some distance between r of t and okay remember I know h so I'm going to say I'm going to minimize distance between this n summation small l equals 0 to l minus 1 a sub l h of t minus kt I'm sorry okay so maybe I'll just use k want to be careful here okay all right some kind of distance okay so I have to define distance between two continuous time signals right so r of t which is a signal I got and some signal that I have to generate so how many such signals do I have to generate I'm sorry mod x power l okay so if you're looking at say 16 q am and l equals 10 okay this becomes huge okay so or if you want to fail becomes thousand it becomes really huge okay so it's slightly impractical in that respect maybe we'll make it more practical as we go along but I don't want to worry about it at least from a conceptual point of view I want to be able to do it okay so one thing you might say okay well this looks very optimal to me okay so some distance you define but I've not done that yet okay so how do I go to an optimal receiver I have to start with something I have to start with probability of error okay and then minimizing the probability of error okay that's how you do the optimal depart okay I'm not done any MAP or ML or anything yeah I'm not done anything just come to the distance directly okay so I don't know if it's optimal or not but at least it gives me a nice way of writing something down and thinking about the decoding okay otherwise it'll just keep going round and round it's not very easy okay so right now it doesn't seem that practical but at least it seems like a good start okay and notice I'm assuming h of t is known okay right if it's not known at the receiver transmit I need not do it but the receiver has to know it okay all right so this this guy this distance I will call it a sum metric which I will say j sub a okay well it'll be the l2 distance between these two waveforms what will be the l2 distance integral from minus infinity to infinity model is r of t minus summation k equals 0 to capital l minus 1 okay a k so I should be careful how I write a k right so I'll write a k here h of t minus k t okay mod square dt okay so remember once again I should write a k carefully so I'm thinking of it as a vector but also it is a simple sequence as in it's a signal okay so remember r of t is complex okay the a k's are complex h of t is also complex I mean all these things are complex quantities so when I take the modulus it's really a modulus of a complex number okay so it's not just absolute value of positive okay so we'll start with this and write down is there a question I'm sorry which one j a is a number it's actually a real number j a is a real number what's inside the the modulus what's inside the modulus is a continuous time signal continuous time complex signal is there a question r of t is a continuous time complex signal what do you mean by a complex signal basically you have two wires and both signals matter to you the real part and the imaginary part it's got a i channel and a q channel okay what do you get on the i channel what do you get on the q channel you do that plus j this you get the complex space band equal what's the question which is a vector there's no vector a is a complex vector you can think of a as a complex vector if you want but I'm also seeing that as a discrete time complex signal okay so once you do this substitution this will this these two will actually be this will be a complex continuous time signal okay so I mean don't worry about how to evaluate these things no I mean just let's write it down eventually when you simplify we'll come up with the discrete time version of the evaluation okay you're not going to definitely evaluate this in continuous time it's very difficult to make it work in continuous time okay so we will come up with some simple rate evaluations in discrete time okay so that's the it's just a start I want to show that we can do it slowly okay surely you don't want to try and implement this maybe you can try I don't know today's technology goes in crazy ways okay so you can't predict it maybe you'll sample it totally over sample it some hundred times and then try to implement it sounds like a bad idea okay so now uh how do we write mod square for a complex number for each t it's actually a complex number so how do you write mod square number times number conjugate and we have to do a series of simplifications which I'm going to try and quickly do and cut through and finally write the final answer okay so after I can write it down slowly but it just uh it's too painful and it's not really very illustrative but the final expression is very important okay so you do that you'll get this you'll get this the first term will be mod r of t squared dt okay what is this this is like the energy in r of t okay so you can see why it comes right r of t minus this times it's conjugate so the first term will be mod r of t squared the next term the middle terms the mixed terms you can join together and write it as two times real part okay so if you're used to this you'll see where it comes from minus two times real part of okay I'll push the summation outside of the integral okay summation k equals 0 to l minus 1 ak star k star k integral minus infinity to infinity r of t h star of t minus kt dt okay so this whole thing is inside the summation okay how did I get two times real part you'll see you'll get this whatever is inside plus it's conjugate okay so that will become two times real part okay of one of those things and I'm choosing this you can choose the other one also I'm choosing this for convenience okay and then the last term will be product of two summations and then an integral I can again pull the summations out and carefully rewrite the integral to get this form okay so whenever you do this you have to use two different dummy variables okay these are all standard things which you might be familiar with from a long time of doing these kind of computations okay ak aj star okay and then what you have inside will be a integral from minus infinity to infinity h of t minus kt times h star of t minus j capital t dt and that's okay so this whole thing is inside the summation okay so I know I didn't prove it line by line but this will work out to be the same thing okay so now the so you see what's the point no this is a better form than the first form I had okay why oh there is summations at symbol rates outside and then what's inside is it's almost like a filtering okay right you it's almost like filtering of r of t and h of t right integral minus infinity r of t h star of t minus kt is nothing to be very scared about it's an integration it's a filter followed by a symbol rate sampler same thing is happening with h okay so you see already it's becoming better okay so you can quickly transition from the continuous time into a symbol rate sample version just by doing this simple thing okay so the crucial interchange of summation and integration and how all that worked out is fairly important okay so so what do we do next next we try to write down those symbol rate samplers and filters separately and separately and then try to rewrite this expression okay so that's what we're going to try next and I have about five minutes I'm going to quickly take it up okay so the first thing I'm going to define is a signal y of k which is integral minus infinity to infinity r of t h star of t minus kt it's clear why I'm defining it okay so you take r of t you filter with h star of minus t and do a simple rate sampler symbol rate sampler you'll get y k okay so it's a discrete time signal the other thing is what I'm going to call rho h k which is kind of like a sample autocorrelation of h of t integral minus infinity to infinity h of t h star of t minus kt remember all of these case can be complex okay so once I do these two definitions you see I can quickly write ja in this form okay the first term I will write as er what is er it's the energy in r okay so fill in what this is it's a very simple expression and I don't have to worry so much about that first term because yeah it's independent of a okay so I don't have to worry too much about those kind of terms and the next term is two times real part of summation k equals 0 to l minus 1 a k star y k okay and then the last term is a double summation k equals 0 to l minus 1 j equals 0 to capital l minus 1 a k a j star rho h of j minus k okay check that this is this matches with the previous thing it's easy to do it it'll be I mean the j minus k the reason it comes is the way refined rho h of k is h of t h star of t minus kt there I had h of t minus kt and h star of t minus jt so the difference is what matters you'll get you'll get a j minus k from there okay so that's your that's your j a expression okay so already I think many of us will agree that it's beginning to look better okay maybe not all that better but it's beginning to look better than what we had before okay so now it's about two or three minutes left so what can we calculate at the receiver how do we go about calculating it rho h of k assuming you know h of t can be pre computed right what type of a signal will rho h of k be okay so it will be actually an autocorrelation function right it is an autocorrelation function sampling of an autocorrelation function of h of t okay so so that's one property you know about rho h so what will happen if I do a Fourier trans discrete temporary transform of rho h of k I will get a non-negative real value discrete temporary transform okay so all those things you know about rho h how will you find yk okay so how do you find yk you take r of t and then match filter with yeah basically filter with h star of minus t which is the match filter corresponding to h of t not g of t okay it's g of t convolved with c of t okay so match filter corresponding to h of t and then what do you do you do a symbol rate sampling and you get yk okay right so this is what we write down in theory in practice there will be a t equals kt plus delta what is the delta coming from all the delays you have to incorporate okay so there'll be some delays I mean your filter will delays channel will delay there'll be so many delays from all over the place okay channel phase response will never be zero it'll be a some delay factor so all these things will happen so you have to adjust for all these things in practice there will be a delta in theory when I write it down I'm just going to write it as kt assuming that the delay has been swallowed in the model and in the adjustments okay so for those of you who are doing 471 when you actually implement this these delays will be important okay so you'll have to adjust for those delays maybe even manually okay so all these things you'll have to do okay pay attention to this when I write down some things like this it's not just kt it's there is a delay okay okay so you do that you get yk okay so y of k can be easily computed at the receiver assuming h star of minus t is an implementable filter okay even if it is not implementable see you can expect it to be causal right so h star of minus t maybe it's going to be anti-causal okay so how do you do it you delay okay but it's going to be finite energy and all that because it's a real channel so it has to be finite energy and all that so it'll be an implementable filter it won't be too bad okay so if you know it you can implement it okay so y of k can be found and I'll talk more about rho h of k in the next class it's more important it's very important that you understand rho h of k much more than yk seems like a simple thing okay so we'll see that more detail in the next class okay are there any questions about how I wrote this down okay so remember once again it's a heuristic condition I'm not justified it rigorously based on optimality later on we'll see there is an optimal structure which is very similar to this okay so but that's it this is a heuristic to get started about and how to think of your receiver and all right