oh i one the law my talk is went to i and you dimension however but this is just a subset of the implications of our work the talk will be mainly about i gender and mimo communication framework which we coral as divide and con so the system model that is something like this you mail be familiar with this but just to be sure of the assumptions let me describe it so we have uh transmit a having T transmit antennas and i D but having a track uh i receive and ten hours and you had the channel matrix which describes the gains between these and up is then a without loss of generality is assumed to be gaussian in nature and sorry that it's course in and without loss of generality it's as you to be zero mean and unit covariance so it's independent on all the receiver active and all the previous to find optimal value of precoder be uh a couple of assumptions here one is that this is a point to point communication system i the uh and second is that P is E mary nature so we are only looking at uh metrics values of B and the third is that uh the channel is known at both the transmitter and the C a some of the real let a real life situations but this model can see the picture are listed here mathematically speaking the problem can be written like this that that K function we have that then you by a generic objective function all and the notation signifies that when we calculate the value of all uh we know the value of H and we choose a value of P so the optimization problem becomes to optimize mice or what be this subject you function subject to a gaussian noise and a total power constraint which we had you know to do by draw and hands for will quite as as an and that can be get the constraints on other aspects of the system model like the channel matrix H the precoder matrix P and uh input constellation so you know but will basically look at how if we take to be simple as shins on edge B and X we can get pretty good oh formant uh in that respect in for as described the input then the strategy behind I solution and some results and discussion first the input lattices or a a this is basically uh gonna arrangement of points as that the points for i did to group among them themselves by definition a and M dimensional lack is is there did to group of all integer linear combination all of "'em" you nearly independent role but uh there are two a but this the two important it is for any given a this one is the minimum distance you uh for this example of uh this is a example of a two-dimensional dimensional as it's an integer that is and the minimum distance is one and the number of point uh which i at minimum distance from any given point uh i is known as the kissing number of the act which here it is for these two parameters a very important for decoding of that this is because example at low snr you know you want to have a low it number so that we do not confuse a transmit point with many it scene points and that has an we would want to have a high kissing number because i kissing number implies that you have a tightly packed lattice which means you preserving the power for example in two dimensions this would be the best let is to be used which is quite that it's not gonna that is which has a kissing number of so why do we use lattices uh one is that it has been traditionally used so five which move that it is easy to implement it is easy to address and easy to decode by easy i mean it's easy to decode and that is uh rather than taking points uh which optimize the power a power input to a system and recent sent uh in the past get it has all all so been proved that lattice codes i actually capacity it achieve so let's take an example if this is a system model uh and suppose we transmit as that for like is but two fifty six points which is which we can consider considered as a but in product of for independent time constellations we you have taken care of the X X aspect of the more what is the P and edge and what do we do with that so the strategy use is the divide and conquer strategy that's to at the divide part it's basically are trying to convert the given problem into a problem of balance sub channels so we use the singular value decomposition on the channel matrix H and uh which is given by you which lamp it's V H transpose but you at be B at a lot of normal matches as and i'm that is a diagonal matrix this diagonalization um actually uh and we also impose a diagonalization on the precoder as a so he a lamb that H T is the effect to be quoted in the are used parallel channel more so why do we use this out of the set first of all uh in the sense of capacity this won't lead to any loss as as was shown by out and even by shannon uh that are that means in the literature for example by the are at all have shown that for should can give object to functions the channel diagonalization structure it is all up to and for sure convex that's functions it is almost optimal in there the left uh eigen vectors of the precoder i don't do by this diagonalization by a one might have also shown recently uh similar results and actually in that but they have shown that for cost in signalling and low snr the the sing sing than the signal vectors of the precoder don't really play a big role in which case complete diagonalization is up and uh there is an is it's into two from a design point of so uh problem now is converted into this problem no what do we do about the objective function in all work we are assuming that the objective function actually that starting but the probability of a are using a maximum likelihood decoder but we well if few steps for that and try to find a good approximation to that objective function we also optimise i we normalize the inputs to have unit uh power in each dimension so uh let's let me describe the con curve part of the strategy that's suppose we choose a lattice this so when i when i a this i mean a lattice constellation that is the points chosen from the lattice um so in this picture you see that if it transmitter and is it is seen as i he had then the the probability that this happens is the probability that the noise takes a be uh to the right set of the by acting line so if E if we to this a bound to the pro of i don't we would have to consider all the inter uh point distances which is uh which becomes complex oh a computationally so luckily for it a major class of the lattices which uh corn root lattices uh as uh the it is a let is being a part of them uh the upper bound can be tight and by just considering the pairs of points which are at minimum distance from each of them so for i uh when we do that we in this up of or yeah at and is not exactly the kissing number but it's kind of like a i it's kissing number of the constellation so it it is the number of their so point which at minimum distance from me to the times two we we take more most step and uh of approximate the Q function band of their upper bound and this actually you to a much better mathematical solution so the justification for using the can part is the bone we all these bounds become type to high as and the problem is converted into a nice convex optimisation problem that there are some relationships that the bones on which information which and skip for uh this is the formulated problem statement using the objective function and this as one of the implications so when we use K T conditions to solve this of that so all the optimisation problem we get something like this um we can actually be or the the uh i'm sorry the sub sub script and every that means the and it sub channel and yep that N is the number of sub and so you know example it's for uh we can always hear in these sub is according to a channel is trend metric which is dependent on these parameters it is key and and you and and slowly only uh as as some that is increased uh the uh i strongest side the first some goes and the second uh the are known don't by uh one of the and all at D's as snrs so it is a simple expression so let's look at the performance is as um actually M node comparing but any well track no because uh uh a here on trying to prove that oh this method i to use up to move uh i is close to the performance is results when we try to optimize the actual error probably so here we can see for our our example that the results for a actual i don't rate and that bond uh become close for medium to high snr this is a a a a uh plot for and i have a lot of thousand channel realisation um we can extend a so one of the advantages of a work in that we can extend it easy lead to higher dimensions um instead of oh sending a like oh we'll one time sort we can send a two and a dimension lattice for example or what and time slots independently over it all the and subject so typically V choose and cross and space time slots here um which means we choose and lattice points in the higher dimensional act is as one symbol um the problem statement affective lit is the same or need the when the change changes that the purple part and scene changes um that some nice results when we look at the high snr regime um because in that in the power allocation actually tends to be equalization and then we can use the we can take advantage of the bossed the to in the past on the single input single output systems and just um we can extend the work to other aspects of minimizing power and uh and i don't rate um i at high snr the power location takes the form of like this and the objective function becomes something like this so we can see that them how mean of the channel as starts to play a role in here and this is actually the clerks in in the optimisation uh when we do bit loading um are the i the X i don't things that it can be extended uh is known only control probably signalling constellation shaping um all of these use and approximation called as approximation um just to give a of flavour of the results um so the blue line is the integer that is when B bit load as you can see that is a significant coding gain from the is uh they in to do act is used which is the red line uh you know a i as uh which was seen in the previous block and we can also compared it by choosing a uh and that a lattice and the a dimensions which is E eight here uh which is uh that type the that this which has the highest backing gain so know no we uh as in many approaches um the idea was that if we are trying to obtain the optimal precoder for any object you function it's like owning a white elephant uh in that the gains that are achieved by by trying to a a performance gains at you uh in time to obtain an optimal precoder is very likely to but compared to the uh as i'm shows that we take so uh uh for example in those well they're divide and conquer technique is a simple yet effective way of transmitting formation and because of the scalar isolation of the problem we can include all the other aspects uh that's some issues still uh with uh the to be dealt it like what if the coherence time is short a in which case we cannot use the and dimensional lattice as and to be fed across base transmission does you more optimal performance so how to relate the cost space and over what for four and that are issues in is decoding uh and that adjusting uh thank you i well we actually have quite a long time the questions is of the so use the last war um do you have any question tall okay one that's a take this opportunity to thank all of the speakers and i would not like to thank you a i has as the audience for being here to and uh i guess i get the opportunity to to say a hope you have a a had a uh i i i i right time here at icassp and i wish you a safe journey huh thanks for