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