okay

every but i'm quite happy that i is that if

face

and and means the last they at its many people for already let so it's would to see that some

people and first

topic

a i don't then some work on optimal channel training a think that of mine was system

i'm sure most of you know this famous paper by a back of C B and bound to why on

how much training

is needed in

wireless fading links

you kind of

same work but in a network my setting with if you needing ingredients

um that get started directly

what we consider

as in some sense the distribute and ten system

so you have and a station to just give you didn't or

each base station is equipped with an intel

and this study a much a lexus channel from K single antenna user to i mean a

transmit to Z

E base stations

and see the stations do not to that its since for thing what's if receive

to a central process a which would do

all the joint decoding

the central

or

so is you is that the past

see

of of back calling link from one based they

and for station is see bits but channel use

i three you one may if you and quite schemes so this means we have and a have all frequency

sub-bands

but all use that time of you

the transmit on all of this up and

the same

um as i set

um Z base stations are of they can't to code then you of the user score work

and you also as you that need does use the time and it's not as the base station at any

channel state information

so the station it's the end

what estimate the channel

but then use this channel estimate to decode messages we see from more

the the motivation would be what is the optimal fraction of a coherent time of the korean style of the

channel we should use and this setting

for sending uplink pilot tones

for actually transmitting data

and since we have a limited back haul capacity we can also study what to see in impact of this

capacity on the optimal change set training time

um yeah i was system on is quite simple so on a

uh

yeah sub and a sub a and so we have at sub carriers

we have received signal back the of size be times and because we E base station each equipped with an

a

at the receive back the X K is and complex question where we assume that each user split it's of

i'll

available power key uniformly over all subcarriers

i

since a reasonable assumption because

use use time and have no channel information at all channels uh

the channel statistics on each subcarrier sir

same the splitting the power you you form me over the subcarriers carriers a

but we have some simple and noise

so far a specified how be one of the channel i would come to this something the

um um was base stations so the base station this that it back the right

i was what sea base station C

now

the base station a quantized observation and four but the quantized yep channel observation to the central station as a

central station would joint you all messages

um we are no

she yeah i have facing a distributed compression problem

so

quite

quite complex

solar

and especially

it's the base station

do not know what to the actual channel state you can't do any channel deep and compression scheme

yeah so you could too much but if you know

what is the extra standard state

you could do not and some since you're your quantization resolution

do actions channel state but since the base station do not have this information they can't do it

we can say a a sub you know

compression scheme

a which can be seen as simply adding a complex course noise to the observation

and of course the "'em" quantisation is

depends on the capacity of your channel

and i was on rate distortion theory

you can actually come

pressure for the um

for the quantisation noise variance

pa

um

uh uh uh uh uh but for channel

and this is nothing that's in something

as to of that you don't we made to the receive noise

how

or not to with the noise power plus the received signal power from or use of time in

and actually if you increase the capacity infinity at this balance with when image if it goes to zero rule

the quantization was are simply goes to

no how to be model of the channel or so we you want a rayleigh fading channel

a rayleigh block fading channel so we draw a random

realisation

based fixed for T channel uses that it changes independently from one block to the other

well a and of this big channel matrix of this the channel from all use that and all and at

all base station

actually have a different variances

yeah i J

and this variance

and on the past loss

from a a base station and tell a to user to i mean

since true that's a past most from a user turn the

to a and tell us of one base station is the same because the quite close together

it it a part was fact that at K and be multiplied by a and dimensional vectors so the actually

gets this balance profile of a watch and make sure

yeah so H is nothing it's in it

complex caution matrix which each

and a has a different variants yeah i

oh the channel estimation procedure quite standard

so we split the coherence time in tile

so that's for yeah for training and the rest is used for data transmission

or if you use the sort of the training so it's actually the base station these central station would estimate

a particular channel coefficients H I J

what this observation

you feel a training snr but depends of course of the length of your you're on training sequences you have

mouse

but use your have so that give quantization error

so the estimate the channel estimate would be an that

but the back wall

a few takes the ever the estimate of this channel you can decompose it in they estimate and in independent

noise term you computes the variances of C received signal back of the use for signal on a

channel energy and the energy of C estimation error

i see that the seems to see yeah quantisation of no uh variance of P

now if you consider that

received vector

and it's the same station

a connection prior to it

as the estimate channel H that might apply but a signal week which was sent

and

to which

which contains the contribution from some noise quantisation errors and channel estimation

of course

in the set to this isn't a few months to um this not depend of the signal you sent

so actually capacity of this channel is not known

are we use the them

yeah the same um rather have and on the true information as and the paper by have C

that's use you humour

that's the noise would be gosh an independent of few transmitted signal or with the covariance matrix K easy

i map and i is it by a number of a station and the number of antennas so the

a man about the true information per a given time

and this doesn't take into account the that we actually spent

for channel training data transmission so what we want to do is work to maximizing that about the achievable rate

was a simply please just about a good you but R T

apply by a discount factor

and would like to maximise this expression of was the conditions that can lead to have at least

can

a training symbols because you have use that time but but we can train or something could you

now if you think about it the quite tough problem

"'cause" you have in fact the expectation

i have a a a a a a complex course matrix of each and and

has a different variants so this a very um which was of to profile

and this is not known in closed-form

those was where you can calculate cd eigenvalue distribution of this matrix in closed form

what we do have a it is we use so to an approximation based to from the matrix re

so we assume your yeah was that we would have a many user turn it's

it and the about

of the number of base station and it's number of a tell us per base station it's some since the

total number of antennas goes to infinity

and as this

assumption

hmmm two information will converge to a deterministic quantity or we can find a deterministic approximations of the about but

which information

such that what the system was infinitely large

the difference between the approximation and exact result close to zero

this is actually results so the result for a channel a random matrix either the entries were but each with

a different variants was that a lapel of by by by tasha

two thousand seven

or simply applied to to i was set to is just one greedy and um

well it you

that's the using split so power of it a fink at men subcarriers

so reason for this is quite simple

if you system was infinitely large need to make sure that the energy in the system states finite

if you start spreading

just signal of an infinitely many sub carriers the energy per subcarrier goes to zero

but still the energy in the entire system states fixed

yeah and actually can computed for each

there are

can ever ever

a deterministic quantity

such as just different converters to zero i don't provide a on you because it doesn't provide a lot

oh have of the only thing we to compute since quality

is seen covariance matrix of single and of so interference in rows

and the variance profile of the estimated channel

and actually to see that's is makes L

i i i mean we consider an infinite large systems so what we actually looking at would be three base

station was we use that term that's in each base station is to tell us

and

that's what we consider him a smell medical example

so i have

like a screen

three corpora to base station seven three is a sum three different set

i drop some randomly

you you can that lot just but past was model and obvious we every every or over channel a realisation

actually is C

we this plot the

i got rate of

well as a cs so a

for a a system

what each base station has

two ten as

you have only one subcarrier

with have coherence time of a sound channel uses and B was optimize in was saying

we have a a training time of to

a lot for me that based on the asymptotic approximation

the um the not "'cause" of what got by simulations

and i mean for me is this look

as good as if you had C perfect um

that is and some since the asymptotic

approximation works better very well even for channel of size a six times three

no i i since for three different back haul capacities so the black the black light was corresponds to what

you would get was a back work that's your one

um

a channel was and you start increasing in of course you get

well i what we do see approach so

optimize of the train time is instead of optimising the the a got a rate but we can't which we

can't country treat a or

we try to optimize our

deterministic approximation

so we want to optimize a deterministic approximation of the mutual of the about get you the rate

for this still need to compute the first derivative

you need to show that it's called okay for once you have done this

a a simple line search what but you wish

and finds the optimal train at time

then

we show that

up to a lower value of have on

um asymptotic approximation converges to the optimal

so we real up to a result you would get

and if you can also concluded C optimal training time to you compute

converges to the optimal time

and lastly it is it remains to do was to better five as some of that are a some type

to optimal to try to to trust a is very close to the

to to what you would get if you could optimize to problems and some since we simply do mount to

colour based optimization we one many

many you see which training time maximise the or right

um

first of just to show that section a concave function

you see the uh got a get you rate as a function of the training length

would then first different back haul capacity is

so that and that i so i computed of the two most approximation it's to marcus simulation

and and makes sense if you that the so close together

but is you to look at a maximum point some but here it won't make a big difference whether to

optimize

like our approximation or not

and

when i had to all that is for a given

S and a a a a a of a coherence it's time of T one hundred

i so that this optimization problem this is a black man

that's a function of C "'em" S N

but that leads to to an exhaustive search opposed to meant to colour up to a um

yeah simulations and you see that the difference between these two values is actually legible

especially

well that exhaustive search i and to do some kind of a to just search because i current so for

vol

yeah a kind of an infant find groups of values

and that have a

you current to try to twenty point twenty five point three symbols you need to round at some point

okay uh just class to to look at C and of the back haul capacity on the optimal training length

you see that the optimal true as is actually of tobacco capacity

this is so cute

a each affect

but you compress it is no

man says of

the from was used C is actually due to quantisation rose

next right of a quantisation quantization have from a a a try to now

once or capacity bounds are back or lectures and infinite capacity

a train or depends and and so on

yeah on the on this is and nice you have an edge

a of course them just to point out the um

how bad actually are a sub optimal quantisation scheme was so so that's a lot to a

um can see the back haul capacity on the X

well as as the um

and and the go to go we're rate

and for

if you look at is value two point five

this was "'cause" is a um

relative power base station time teller

so if would multiplied by a two because we've to tell us

you would see to a Q chief as like

as a spectral efficiency of five bits per second per channel use

you that to have

twenty bits per second pitch and use of back or pass

yeah to conclude what um

we have used them

results from lot from the matrix you read to tech a you a and optimization problem in context

and have treated to channel is the variance profile of just you like a distributed and in a system was

um with the back haul capacity

uh yeah but we have to on the parents it's is asymptotic approximation but extremely rather for channel it's of

size three times we even to by two works quite well

a it's that it's a pack of back or limitation on the optimal training length

that's well that we can have the cross it's that's you and we work on distributed compression was in perfect

csi

so actually how do you compress when we don't know the channel

so i i i'm not aware of any paper which which problem so someone has some put that would what

have happy to get it

i

last as a question of how to decide whether but require parade or not

and some some that's the back back capacity we go to zero

i ever go to zero as well doesn't make sense because it's space station could at least decode individually

and then be treated like a set of B base station which live in the isolated you are so this

low in to the it's we haven't done it

um

i'd down so the extension a very straight but because when you do to take well some to come pilot

contamination

which happens

because can you to read like your sample training

um

last a just a few references so

we've of can subtract version of for our paper but just

going to be published in june

the transaction on signal processing

um like since

can is that for uh

right the as was about as but from bonnie touch and

of can see a classic paper or for for do that just just a good point a if you don't

know so much about that of my nose is over you paper by provide just back

and

yeah that that that you

yeah

okay

so we could have assumed from the beginning

that's a variance of the channel it's as one of a and

uh so as a classical assumption and people start it could be don't of

i i and on channel but each element like has the energy

you know just so from the big have an energy was i a channel was the a finite energy the

fresh start a is that it can't gain one more G you U and the back up some more energy

then work was actually sent right

so

we kind of either start to directly but making this assumption

a a some scale as a parallel

by a some so which goes to infinity that's the power up uh frequency band goes to zero

and the and it doesn't make a diff

yeah

yes you but infinite double T and it's a good i mean it's some point if you have but a

station it's a very far where you can let's this diversity

so that's is a reasonable assumption and for the scale in the N is C have used

we subcarrier

so in some since you didn't scale at all

and

so

much much Q do but to that mathematically correct you something new to

to this kind of scale the energy per uh

for channel and she was go to zero otherwise this doesn't work