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