0:00:13every but i'm quite happy that i is that if
0:00:17and and means the last they at its many people for already let so it's would to see that some
0:00:21people and first
0:00:23a i don't then some work on optimal channel training a think that of mine was system
0:00:29i'm sure most of you know this famous paper by a back of C B and bound to why on
0:00:33how much training
0:00:35is needed in
0:00:36wireless fading links
0:00:38you kind of
0:00:38same work but in a network my setting with if you needing ingredients
0:00:43um that get started directly
0:00:45what we consider
0:00:46as in some sense the distribute and ten system
0:00:49so you have and a station to just give you didn't or
0:00:53each base station is equipped with an intel
0:00:56and this study a much a lexus channel from K single antenna user to i mean a
0:01:01transmit to Z
0:01:02E base stations
0:01:04and see the stations do not to that its since for thing what's if receive
0:01:08to a central process a which would do
0:01:10all the joint decoding
0:01:12the central
0:01:14so is you is that the past
0:01:17of of back calling link from one based they
0:01:19and for station is see bits but channel use
0:01:22i three you one may if you and quite schemes so this means we have and a have all frequency
0:01:28but all use that time of you
0:01:30the transmit on all of this up and
0:01:33the same
0:01:34um as i set
0:01:36um Z base stations are of they can't to code then you of the user score work
0:01:41and you also as you that need does use the time and it's not as the base station at any
0:01:45channel state information
0:01:47so the station it's the end
0:01:49what estimate the channel
0:01:51but then use this channel estimate to decode messages we see from more
0:01:57the the motivation would be what is the optimal fraction of a coherent time of the korean style of the
0:02:03channel we should use and this setting
0:02:05for sending uplink pilot tones
0:02:07for actually transmitting data
0:02:10and since we have a limited back haul capacity we can also study what to see in impact of this
0:02:15capacity on the optimal change set training time
0:02:19um yeah i was system on is quite simple so on a
0:02:24yeah sub and a sub a and so we have at sub carriers
0:02:28we have received signal back the of size be times and because we E base station each equipped with an
0:02:35at the receive back the X K is and complex question where we assume that each user split it's of
0:02:41available power key uniformly over all subcarriers
0:02:46since a reasonable assumption because
0:02:48use use time and have no channel information at all channels uh
0:02:52the channel statistics on each subcarrier sir
0:02:54same the splitting the power you you form me over the subcarriers carriers a
0:02:59but we have some simple and noise
0:03:01so far a specified how be one of the channel i would come to this something the
0:03:07um um was base stations so the base station this that it back the right
0:03:13i was what sea base station C
0:03:15the base station a quantized observation and four but the quantized yep channel observation to the central station as a
0:03:22central station would joint you all messages
0:03:25um we are no
0:03:27she yeah i have facing a distributed compression problem
0:03:32quite complex
0:03:34and especially
0:03:35it's the base station
0:03:36do not know what to the actual channel state you can't do any channel deep and compression scheme
0:03:42yeah so you could too much but if you know
0:03:44what is the extra standard state
0:03:46you could do not and some since you're your quantization resolution
0:03:50do actions channel state but since the base station do not have this information they can't do it
0:03:55we can say a a sub you know
0:03:57compression scheme
0:03:59a which can be seen as simply adding a complex course noise to the observation
0:04:04and of course the "'em" quantisation is
0:04:07depends on the capacity of your channel
0:04:10and i was on rate distortion theory
0:04:12you can actually come
0:04:14pressure for the um
0:04:15for the quantisation noise variance
0:04:19uh uh uh uh uh but for channel
0:04:22and this is nothing that's in something
0:04:23as to of that you don't we made to the receive noise
0:04:27or not to with the noise power plus the received signal power from or use of time in
0:04:32and actually if you increase the capacity infinity at this balance with when image if it goes to zero rule
0:04:38the quantization was are simply goes to
0:04:44no how to be model of the channel or so we you want a rayleigh fading channel
0:04:49a rayleigh block fading channel so we draw a random
0:04:53based fixed for T channel uses that it changes independently from one block to the other
0:04:59well a and of this big channel matrix of this the channel from all use that and all and at
0:05:04all base station
0:05:06actually have a different variances
0:05:08yeah i J
0:05:09and this variance
0:05:11and on the past loss
0:05:13from a a base station and tell a to user to i mean
0:05:17since true that's a past most from a user turn the
0:05:20to a and tell us of one base station is the same because the quite close together
0:05:24it it a part was fact that at K and be multiplied by a and dimensional vectors so the actually
0:05:30gets this balance profile of a watch and make sure
0:05:33yeah so H is nothing it's in it
0:05:35complex caution matrix which each
0:05:37and a has a different variants yeah i
0:05:41oh the channel estimation procedure quite standard
0:05:44so we split the coherence time in tile
0:05:47so that's for yeah for training and the rest is used for data transmission
0:05:52or if you use the sort of the training so it's actually the base station these central station would estimate
0:05:57a particular channel coefficients H I J
0:05:59what this observation
0:06:01you feel a training snr but depends of course of the length of your you're on training sequences you have
0:06:07but use your have so that give quantization error
0:06:10so the estimate the channel estimate would be an that
0:06:13but the back wall
0:06:15a few takes the ever the estimate of this channel you can decompose it in they estimate and in independent
0:06:20noise term you computes the variances of C received signal back of the use for signal on a
0:06:26channel energy and the energy of C estimation error
0:06:30i see that the seems to see yeah quantisation of no uh variance of P
0:06:36now if you consider that
0:06:38received vector
0:06:40and it's the same station
0:06:42a connection prior to it
0:06:44as the estimate channel H that might apply but a signal week which was sent
0:06:49to which
0:06:50which contains the contribution from some noise quantisation errors and channel estimation
0:06:57of course
0:06:58in the set to this isn't a few months to um this not depend of the signal you sent
0:07:03so actually capacity of this channel is not known
0:07:05are we use the them
0:07:07yeah the same um rather have and on the true information as and the paper by have C
0:07:12that's use you humour
0:07:14that's the noise would be gosh an independent of few transmitted signal or with the covariance matrix K easy
0:07:22i map and i is it by a number of a station and the number of antennas so the
0:07:26a man about the true information per a given time
0:07:30and this doesn't take into account the that we actually spent
0:07:34for channel training data transmission so what we want to do is work to maximizing that about the achievable rate
0:07:40was a simply please just about a good you but R T
0:07:43apply by a discount factor
0:07:45and would like to maximise this expression of was the conditions that can lead to have at least
0:07:50a training symbols because you have use that time but but we can train or something could you
0:07:56now if you think about it the quite tough problem
0:07:58"'cause" you have in fact the expectation
0:08:01i have a a a a a a complex course matrix of each and and
0:08:05has a different variants so this a very um which was of to profile
0:08:09and this is not known in closed-form
0:08:12those was where you can calculate cd eigenvalue distribution of this matrix in closed form
0:08:16what we do have a it is we use so to an approximation based to from the matrix re
0:08:22so we assume your yeah was that we would have a many user turn it's
0:08:26it and the about
0:08:27of the number of base station and it's number of a tell us per base station it's some since the
0:08:32total number of antennas goes to infinity
0:08:35and as this
0:08:38hmmm two information will converge to a deterministic quantity or we can find a deterministic approximations of the about but
0:08:44which information
0:08:46such that what the system was infinitely large
0:08:48the difference between the approximation and exact result close to zero
0:08:54this is actually results so the result for a channel a random matrix either the entries were but each with
0:09:01a different variants was that a lapel of by by by tasha
0:09:04two thousand seven
0:09:05or simply applied to to i was set to is just one greedy and um
0:09:10well it you
0:09:11that's the using split so power of it a fink at men subcarriers
0:09:16so reason for this is quite simple
0:09:18if you system was infinitely large need to make sure that the energy in the system states finite
0:09:23if you start spreading
0:09:25just signal of an infinitely many sub carriers the energy per subcarrier goes to zero
0:09:29but still the energy in the entire system states fixed
0:09:34yeah and actually can computed for each
0:09:37there are
0:09:37can ever ever
0:09:39a deterministic quantity
0:09:41such as just different converters to zero i don't provide a on you because it doesn't provide a lot
0:09:47oh have of the only thing we to compute since quality
0:09:50is seen covariance matrix of single and of so interference in rows
0:09:55and the variance profile of the estimated channel
0:09:59and actually to see that's is makes L
0:10:01i i i mean we consider an infinite large systems so what we actually looking at would be three base
0:10:06station was we use that term that's in each base station is to tell us
0:10:11that's what we consider him a smell medical example
0:10:15so i have
0:10:16like a screen
0:10:17three corpora to base station seven three is a sum three different set
0:10:21i drop some randomly
0:10:24you you can that lot just but past was model and obvious we every every or over channel a realisation
0:10:30actually is C
0:10:32we this plot the
0:10:34i got rate of
0:10:36well as a cs so a
0:10:38for a a system
0:10:39what each base station has
0:10:41two ten as
0:10:42you have only one subcarrier
0:10:44with have coherence time of a sound channel uses and B was optimize in was saying
0:10:49we have a a training time of to
0:10:52a lot for me that based on the asymptotic approximation
0:10:57the um the not "'cause" of what got by simulations
0:11:00and i mean for me is this look
0:11:02as good as if you had C perfect um
0:11:05that is and some since the asymptotic
0:11:07approximation works better very well even for channel of size a six times three
0:11:13no i i since for three different back haul capacities so the black the black light was corresponds to what
0:11:19you would get was a back work that's your one
0:11:22a channel was and you start increasing in of course you get
0:11:26well i what we do see approach so
0:11:28optimize of the train time is instead of optimising the the a got a rate but we can't which we
0:11:35can't country treat a or
0:11:38we try to optimize our
0:11:40deterministic approximation
0:11:42so we want to optimize a deterministic approximation of the mutual of the about get you the rate
0:11:47for this still need to compute the first derivative
0:11:50you need to show that it's called okay for once you have done this
0:11:53a a simple line search what but you wish
0:11:56and finds the optimal train at time
0:11:59we show that
0:12:02up to a lower value of have on
0:12:04um asymptotic approximation converges to the optimal
0:12:08so we real up to a result you would get
0:12:11and if you can also concluded C optimal training time to you compute
0:12:15converges to the optimal time
0:12:19and lastly it is it remains to do was to better five as some of that are a some type
0:12:23to optimal to try to to trust a is very close to the
0:12:28to to what you would get if you could optimize to problems and some since we simply do mount to
0:12:32colour based optimization we one many
0:12:35many you see which training time maximise the or right
0:12:40first of just to show that section a concave function
0:12:43you see the uh got a get you rate as a function of the training length
0:12:47would then first different back haul capacity is
0:12:49so that and that i so i computed of the two most approximation it's to marcus simulation
0:12:56and and makes sense if you that the so close together
0:12:58but is you to look at a maximum point some but here it won't make a big difference whether to
0:13:04like our approximation or not
0:13:08when i had to all that is for a given
0:13:10S and a a a a a of a coherence it's time of T one hundred
0:13:14i so that this optimization problem this is a black man
0:13:18that's a function of C "'em" S N
0:13:21but that leads to to an exhaustive search opposed to meant to colour up to a um
0:13:25yeah simulations and you see that the difference between these two values is actually legible
0:13:31well that exhaustive search i and to do some kind of a to just search because i current so for
0:13:37yeah a kind of an infant find groups of values
0:13:40and that have a
0:13:41you current to try to twenty point twenty five point three symbols you need to round at some point
0:13:48okay uh just class to to look at C and of the back haul capacity on the optimal training length
0:13:54you see that the optimal true as is actually of tobacco capacity
0:13:59this is so cute
0:14:00a each affect
0:14:01but you compress it is no
0:14:03man says of
0:14:04the from was used C is actually due to quantisation rose
0:14:08next right of a quantisation quantization have from a a a try to now
0:14:12once or capacity bounds are back or lectures and infinite capacity
0:14:16a train or depends and and so on
0:14:19yeah on the on this is and nice you have an edge
0:14:24a of course them just to point out the um
0:14:27how bad actually are a sub optimal quantisation scheme was so so that's a lot to a
0:14:32um can see the back haul capacity on the X
0:14:36well as as the um
0:14:37and and the go to go we're rate
0:14:40and for
0:14:42if you look at is value two point five
0:14:44this was "'cause" is a um
0:14:46relative power base station time teller
0:14:49so if would multiplied by a two because we've to tell us
0:14:52you would see to a Q chief as like
0:14:54as a spectral efficiency of five bits per second per channel use
0:14:59you that to have
0:15:00twenty bits per second pitch and use of back or pass
0:15:06yeah to conclude what um
0:15:07we have used them
0:15:09results from lot from the matrix you read to tech a you a and optimization problem in context
0:15:15and have treated to channel is the variance profile of just you like a distributed and in a system was
0:15:19um with the back haul capacity
0:15:22uh yeah but we have to on the parents it's is asymptotic approximation but extremely rather for channel it's of
0:15:28size three times we even to by two works quite well
0:15:31a it's that it's a pack of back or limitation on the optimal training length
0:15:37that's well that we can have the cross it's that's you and we work on distributed compression was in perfect
0:15:44so actually how do you compress when we don't know the channel
0:15:47so i i i'm not aware of any paper which which problem so someone has some put that would what
0:15:51have happy to get it
0:15:54last as a question of how to decide whether but require parade or not
0:15:58and some some that's the back back capacity we go to zero
0:16:01i ever go to zero as well doesn't make sense because it's space station could at least decode individually
0:16:07and then be treated like a set of B base station which live in the isolated you are so this
0:16:12low in to the it's we haven't done it
0:16:16i'd down so the extension a very straight but because when you do to take well some to come pilot
0:16:22which happens
0:16:23because can you to read like your sample training
0:16:28last a just a few references so
0:16:30we've of can subtract version of for our paper but just
0:16:32going to be published in june
0:16:34the transaction on signal processing
0:16:37um like since
0:16:38can is that for uh
0:16:40right the as was about as but from bonnie touch and
0:16:44of can see a classic paper or for for do that just just a good point a if you don't
0:16:48know so much about that of my nose is over you paper by provide just back
0:16:54yeah that that that you
0:17:28so we could have assumed from the beginning
0:17:31that's a variance of the channel it's as one of a and
0:17:35uh so as a classical assumption and people start it could be don't of
0:17:39i i and on channel but each element like has the energy
0:17:43you know just so from the big have an energy was i a channel was the a finite energy the
0:17:48fresh start a is that it can't gain one more G you U and the back up some more energy
0:17:54then work was actually sent right
0:17:57we kind of either start to directly but making this assumption
0:18:00a a some scale as a parallel
0:18:03by a some so which goes to infinity that's the power up uh frequency band goes to zero
0:18:09and the and it doesn't make a diff
0:18:17yes you but infinite double T and it's a good i mean it's some point if you have but a
0:18:22station it's a very far where you can let's this diversity
0:18:25so that's is a reasonable assumption and for the scale in the N is C have used
0:18:30we subcarrier
0:18:32so in some since you didn't scale at all
0:18:36much much Q do but to that mathematically correct you something new to
0:18:41to this kind of scale the energy per uh
0:18:44for channel and she was go to zero otherwise this doesn't work