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

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

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

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

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

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 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