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