0:00:13oh i one
0:00:13the law my talk is
0:00:16went to i and you dimension
0:00:18however but this is just a subset of the implications of our work
0:00:22the talk will be mainly about
0:00:24i gender and mimo communication framework
0:00:27which we
0:00:28coral as divide and con
0:00:31so the system model that is something like this
0:00:34you mail be familiar with this but just to be sure of the assumptions
0:00:38let me describe it so we have
0:00:41uh transmit a having T transmit antennas
0:00:45i D but having a track uh i receive and ten hours
0:00:48and you had the channel matrix which describes the gains between these and up is
0:00:54then a without loss of
0:00:57is assumed to be gaussian in nature and
0:01:00sorry that it's course in and without loss of generality it's
0:01:04as you to be zero mean and unit covariance
0:01:08it's independent
0:01:10on all the receiver active
0:01:12and all the previous to find optimal value of precoder be
0:01:16uh a couple of assumptions here
0:01:18one is that this is a point to point communication system
0:01:22i the uh
0:01:24and second is that
0:01:25P is E mary nature so we are only looking at
0:01:29metrics values of B
0:01:31and the third is that
0:01:33uh the channel is known at both the transmitter and the C
0:01:38a some of the real let a real life situations but this model can see the picture are listed here
0:01:43mathematically speaking the
0:01:45problem can be written like this
0:01:48that that K function we have that then you by a generic
0:01:51objective function all and the notation signifies that when we calculate the value of all
0:01:57uh we know the value of H and we choose a value of P
0:02:01so the optimization problem
0:02:03becomes to optimize mice or what be this subject you function
0:02:06subject to
0:02:09a gaussian noise and a total power constraint
0:02:12which we had you know to do by draw and hands for will quite as as an
0:02:17and that can be get the constraints on
0:02:20other aspects of the system model like the channel matrix H the precoder matrix P and uh input constellation
0:02:27so you know but will basically look at how if we take to be simple as shins on edge
0:02:33B and X
0:02:35we can get pretty good
0:02:36oh formant
0:02:38uh in that respect in
0:02:40for as described the input
0:02:42then the strategy behind I solution and
0:02:45some results and discussion
0:02:48first the input
0:02:51or a a this is basically uh gonna arrangement of points as that the points for i did to group
0:02:57among them themselves
0:02:59by definition
0:03:00a and M dimensional lack is is there did to group of all integer linear combination
0:03:05all of "'em" you nearly independent role but uh
0:03:09there are two
0:03:11a but this the two important it is for any given a this one is the minimum distance you
0:03:18for this example of uh this is a example of a two-dimensional dimensional as
0:03:23it's an integer that is
0:03:25the minimum distance is one
0:03:27and the number of
0:03:30uh which i at minimum distance from any given point uh
0:03:34i is known as the kissing number of the act
0:03:37here it is for
0:03:38these two parameters a very important
0:03:40for decoding of that this is because
0:03:44example at low snr
0:03:46you know you want to have a low it number
0:03:48so that we do not confuse a transmit point with many it scene points
0:03:53and that has an we would want to have a high kissing number because
0:03:57i kissing number implies that
0:03:59you have a tightly packed lattice
0:04:01which means you preserving the power
0:04:05for example in two dimensions this would be the best let is to be used which is quite that it's
0:04:10not gonna that is which has a kissing number of
0:04:13so why do we use lattices
0:04:15uh one is that it has been traditionally used so five which move that it is easy to implement
0:04:21it is easy to address
0:04:24easy to decode by easy i mean it's easy to decode and that is uh rather than taking
0:04:31uh which optimize the power
0:04:34a power input to a system
0:04:36and recent sent uh in the past get it has all all so been proved that lattice codes
0:04:41i actually capacity it achieve
0:04:45so let's take an example
0:04:47if this is a system model
0:04:51suppose we transmit as that for like is but two fifty six points which is which we can consider considered
0:04:56as a
0:04:57but in product of for independent time constellations
0:05:01we you have taken care of the X
0:05:03X aspect of the more
0:05:05what is the P and edge
0:05:07and what do we do with that
0:05:09so the strategy use is the divide
0:05:12and conquer strategy
0:05:13that's to at the divide part
0:05:15it's basically
0:05:16are trying to
0:05:18convert the given problem into a problem of balance sub channels
0:05:22so we use the singular value decomposition on the channel matrix H
0:05:29uh which is given by you which lamp it's V H transpose but you at be B at a lot
0:05:33of normal
0:05:34matches as and i'm that is a diagonal matrix
0:05:37this diagonalization um actually
0:05:42uh and we also impose a diagonalization on the precoder
0:05:46as a
0:05:46so he a lamb that H T is
0:05:50the effect to be quoted in the are used
0:05:52parallel channel
0:05:56so why do we use this
0:05:58out of the set
0:05:59first of all uh in the sense of capacity
0:06:02this won't lead to any loss as as was shown by out and
0:06:06even by shannon
0:06:08uh that are that means
0:06:10in the literature for example
0:06:11by the are
0:06:13at all have shown that for
0:06:14should can give object to functions the channel diagonalization structure it is all up to
0:06:19and for sure convex that's functions it is almost optimal in there the
0:06:24uh eigen vectors of the precoder
0:06:27i don't do by this diagonalization
0:06:30by a one might have also shown recently uh similar results and actually in that but they have shown that
0:06:37for cost in signalling
0:06:38and low snr
0:06:40the the sing sing than the signal vectors of the precoder don't really play a big role
0:06:45in which case
0:06:47complete diagonalization is up
0:06:50and uh there is an is
0:06:52it's into two
0:06:53from a design point of
0:06:56so uh problem now
0:06:57is converted into this problem
0:07:00no what do we do about the objective function in all work we are assuming that the objective function
0:07:05actually that starting but the probability of a are using a maximum likelihood decoder but we
0:07:10well if few steps for that and try to find
0:07:13a good approximation to that objective function
0:07:16we also optimise
0:07:18i we normalize the inputs to have unit
0:07:22uh power in each dimension
0:07:26so uh let's
0:07:27let me describe the con curve part of the strategy that's suppose
0:07:30we choose a lattice this
0:07:33so when i when i a this i mean a lattice constellation that is the points chosen from the lattice
0:07:41so in this picture you see that
0:07:44it transmitter and is
0:07:46it is seen as i he had
0:07:51the probability that this happens is
0:07:53the probability that the noise takes a be uh to the right set of the by acting line
0:07:59so if E
0:08:00if we to this a bound to the pro of i don't we would have to consider all the inter
0:08:06uh point distances which is
0:08:08uh which becomes complex
0:08:10oh a computationally so
0:08:13luckily for it
0:08:14a major class of the lattices which
0:08:17uh corn root lattices
0:08:18uh as
0:08:20uh the it is a let is being a part of them
0:08:24the upper bound can be tight and by just considering the pairs of points which are at minimum distance from
0:08:29each of them
0:08:30so for i uh when we do that we in this up of or
0:08:33yeah at and is not exactly the kissing number
0:08:36but it's kind of like a i it's kissing number of the constellation
0:08:40so it it is the number of their so point
0:08:44which at minimum distance from me to the
0:08:46times two
0:08:49we we take more most step and
0:08:52uh of approximate the Q function band of their upper bound
0:08:56and this actually
0:08:58you to a much better mathematical solution
0:09:03so the justification for using the can part is the bone
0:09:07we all these bounds become type to high as
0:09:11the problem is converted into a nice convex optimisation problem
0:09:15that there are some relationships that the bones on which information which and skip for
0:09:21uh this is the formulated problem statement using the objective function
0:09:26and this as one of the implications so when we use K T conditions to solve this of that
0:09:31so all the optimisation problem we get
0:09:34something like this
0:09:36we can actually be or the the
0:09:39uh i'm sorry the sub
0:09:41sub script and
0:09:42every that means the and it sub channel
0:09:45and yep that N is the number of sub and so you know example it's for
0:09:49uh we can always hear in these
0:09:52sub is according to a channel is trend metric which is dependent on these parameters it is key and and
0:09:58you and
0:09:59slowly only uh as as some that is increased
0:10:03uh the uh i
0:10:04strongest side
0:10:05the first
0:10:06some goes and the second
0:10:09uh the are known don't by uh
0:10:12one of the and all at D's as snrs
0:10:15so it is a simple expression
0:10:17so let's look at the performance is as
0:10:20um actually M node comparing but
0:10:23any well track no because
0:10:25uh uh a here on trying to prove that
0:10:29this method i to use up to move uh
0:10:31i is close to
0:10:33the performance is results
0:10:34when we try to optimize the actual error probably
0:10:38so here we can see for our our example
0:10:41that the results for a actual i don't rate and
0:10:44that bond
0:10:45uh become close
0:10:47for medium to high snr
0:10:49this is a a a a uh plot for
0:10:52and i have a lot of thousand channel realisation
0:10:57um we can extend
0:10:58a so one of the advantages of a work in that we can extend it easy lead to higher dimensions
0:11:03um instead of
0:11:06oh sending a like
0:11:07oh we'll one time sort we can send
0:11:10a two and a dimension lattice for example or what and time slots independently over it all the and subject
0:11:17so typically V choose and cross and space time slots here
0:11:23which means we choose and lattice points in the higher dimensional act is
0:11:27as one symbol
0:11:30um the problem statement
0:11:32affective lit is the same or need the when the change changes that the purple part and scene changes
0:11:38um that some nice results when we look at the high snr
0:11:42um because in that in
0:11:46the power allocation actually tends to be equalization
0:11:49and then we can use
0:11:50the we can take advantage of the bossed the to in the past on the single input single output systems
0:11:57and just um
0:11:59we can extend the work
0:12:01to other aspects of minimizing power and uh and
0:12:05i don't rate
0:12:09at high snr
0:12:10the power location takes the form of like this and the objective function becomes something like this so we can
0:12:16see that
0:12:17them how mean of the channel as
0:12:19starts to play a role in here
0:12:21and this is actually the clerks
0:12:23in in the optimisation
0:12:25uh when we do bit loading
0:12:28um are the
0:12:30i the X
0:12:31i don't things that it can be extended
0:12:33uh is known only control probably signalling constellation shaping
0:12:37um all of these use
0:12:40and approximation called as approximation
0:12:44um just to give a of flavour of the results um
0:12:48so the blue line is the integer that is
0:12:51when B
0:12:52bit load
0:12:53as you can see that is a significant coding gain from the
0:12:56is uh
0:12:57they in to do act is used which is the red line
0:13:00uh you know a
0:13:01i as uh which was seen in the previous block
0:13:05we can also compared it by
0:13:07choosing a uh and that a lattice and the a dimensions which is
0:13:11E eight here
0:13:13uh which is uh
0:13:17the that this which has the highest
0:13:19backing gain
0:13:25know no we uh as in many approaches um
0:13:30the idea was that
0:13:31if we are trying to
0:13:33obtain the optimal precoder for any object you function it's like owning
0:13:38a white elephant
0:13:39uh in that the gains that are achieved by
0:13:43by trying to a a performance gains at you
0:13:48in time to obtain an optimal precoder
0:13:52very likely to but compared to the
0:13:54uh as i'm shows that we take
0:13:57so uh uh for example in those well they're divide and conquer technique
0:14:01is a simple yet effective way of transmitting formation
0:14:04and because of the scalar isolation of the problem we can include
0:14:07all the other aspects
0:14:09uh that's some issues still
0:14:12uh with
0:14:13uh the to be dealt it like
0:14:16what if the coherence time is short
0:14:18a in which case we cannot use
0:14:20the and dimensional lattice as
0:14:24to be fed across base transmission does
0:14:28more optimal performance so
0:14:30how to relate the cost space
0:14:32and over what for four
0:14:34and that are issues in is decoding
0:14:37uh and that adjusting
0:14:39uh thank you
0:14:45well we actually have quite a long time
0:14:47the questions
0:14:49is of the so use the last war
0:14:51um do you have any question
0:15:05okay one that's a take this opportunity to thank all of the speakers and i would not like to thank
0:15:12a i has as the audience for being here to
0:15:15and uh
0:15:16i guess i get the opportunity to to say a hope you have a a had a
0:15:20uh i i i i right time here at icassp and i wish you a safe journey huh
0:15:25thanks for