0:00:13 | okay |
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0:00:13 | every but i'm quite happy that i is that if |

0:00:16 | face |

0:00:17 | and and means the last they at its many people for already let so it's would to see that some |

0:00:21 | people and first |

0:00:22 | topic |

0:00:23 | a i don't then some work on optimal channel training a think that of mine was system |

0:00:29 | i'm sure most of you know this famous paper by a back of C B and bound to why on |

0:00:33 | how much training |

0:00:35 | is needed in |

0:00:36 | wireless fading links |

0:00:38 | you kind of |

0:00:38 | same work but in a network my setting with if you needing ingredients |

0:00:43 | um that get started directly |

0:00:45 | what we consider |

0:00:46 | as in some sense the distribute and ten system |

0:00:49 | so you have and a station to just give you didn't or |

0:00:53 | each base station is equipped with an intel |

0:00:56 | and this study a much a lexus channel from K single antenna user to i mean a |

0:01:01 | transmit to Z |

0:01:02 | E base stations |

0:01:04 | and see the stations do not to that its since for thing what's if receive |

0:01:08 | to a central process a which would do |

0:01:10 | all the joint decoding |

0:01:12 | the central |

0:01:13 | or |

0:01:14 | so is you is that the past |

0:01:16 | see |

0:01:17 | of of back calling link from one based they |

0:01:19 | and for station is see bits but channel use |

0:01:22 | i three you one may if you and quite schemes so this means we have and a have all frequency |

0:01:27 | sub-bands |

0:01:28 | but all use that time of you |

0:01:30 | the transmit on all of this up and |

0:01:33 | the same |

0:01:34 | um as i set |

0:01:36 | um Z base stations are of they can't to code then you of the user score work |

0:01:41 | and you also as you that need does use the time and it's not as the base station at any |

0:01:45 | channel state information |

0:01:47 | so the station it's the end |

0:01:49 | what estimate the channel |

0:01:51 | but then use this channel estimate to decode messages we see from more |

0:01:57 | the the motivation would be what is the optimal fraction of a coherent time of the korean style of the |

0:02:03 | channel we should use and this setting |

0:02:05 | for sending uplink pilot tones |

0:02:07 | for actually transmitting data |

0:02:10 | and since we have a limited back haul capacity we can also study what to see in impact of this |

0:02:15 | capacity on the optimal change set training time |

0:02:19 | um yeah i was system on is quite simple so on a |

0:02:23 | uh |

0:02:24 | yeah sub and a sub a and so we have at sub carriers |

0:02:28 | we have received signal back the of size be times and because we E base station each equipped with an |

0:02:34 | a |

0:02:35 | at the receive back the X K is and complex question where we assume that each user split it's of |

0:02:41 | i'll |

0:02:41 | available power key uniformly over all subcarriers |

0:02:45 | i |

0:02:46 | since a reasonable assumption because |

0:02:48 | use use time and have no channel information at all channels uh |

0:02:52 | the channel statistics on each subcarrier sir |

0:02:54 | same the splitting the power you you form me over the subcarriers carriers a |

0:02:59 | but we have some simple and noise |

0:03:01 | so far a specified how be one of the channel i would come to this something the |

0:03:07 | um um was base stations so the base station this that it back the right |

0:03:13 | i was what sea base station C |

0:03:15 | now |

0:03:15 | the base station a quantized observation and four but the quantized yep channel observation to the central station as a |

0:03:22 | central station would joint you all messages |

0:03:25 | um we are no |

0:03:27 | she yeah i have facing a distributed compression problem |

0:03:31 | so |

0:03:31 | quite |

0:03:32 | quite complex |

0:03:33 | solar |

0:03:34 | and especially |

0:03:35 | it's the base station |

0:03:36 | do not know what to the actual channel state you can't do any channel deep and compression scheme |

0:03:42 | yeah so you could too much but if you know |

0:03:44 | what is the extra standard state |

0:03:46 | you could do not and some since you're your quantization resolution |

0:03:50 | do actions channel state but since the base station do not have this information they can't do it |

0:03:55 | we can say a a sub you know |

0:03:57 | compression scheme |

0:03:59 | a which can be seen as simply adding a complex course noise to the observation |

0:04:04 | and of course the "'em" quantisation is |

0:04:07 | depends on the capacity of your channel |

0:04:10 | and i was on rate distortion theory |

0:04:12 | you can actually come |

0:04:14 | pressure for the um |

0:04:15 | for the quantisation noise variance |

0:04:17 | pa |

0:04:18 | um |

0:04:19 | uh uh uh uh uh but for channel |

0:04:22 | and this is nothing that's in something |

0:04:23 | as to of that you don't we made to the receive noise |

0:04:26 | how |

0:04:27 | or not to with the noise power plus the received signal power from or use of time in |

0:04:32 | and actually if you increase the capacity infinity at this balance with when image if it goes to zero rule |

0:04:38 | the quantization was are simply goes to |

0:04:44 | no how to be model of the channel or so we you want a rayleigh fading channel |

0:04:49 | a rayleigh block fading channel so we draw a random |

0:04:52 | realisation |

0:04:53 | based fixed for T channel uses that it changes independently from one block to the other |

0:04:59 | well a and of this big channel matrix of this the channel from all use that and all and at |

0:05:04 | all base station |

0:05:06 | actually have a different variances |

0:05:08 | yeah i J |

0:05:09 | and this variance |

0:05:11 | and on the past loss |

0:05:13 | from a a base station and tell a to user to i mean |

0:05:17 | since true that's a past most from a user turn the |

0:05:20 | to a and tell us of one base station is the same because the quite close together |

0:05:24 | it it a part was fact that at K and be multiplied by a and dimensional vectors so the actually |

0:05:30 | gets this balance profile of a watch and make sure |

0:05:33 | yeah so H is nothing it's in it |

0:05:35 | complex caution matrix which each |

0:05:37 | and a has a different variants yeah i |

0:05:41 | oh the channel estimation procedure quite standard |

0:05:44 | so we split the coherence time in tile |

0:05:47 | so that's for yeah for training and the rest is used for data transmission |

0:05:52 | or if you use the sort of the training so it's actually the base station these central station would estimate |

0:05:57 | a particular channel coefficients H I J |

0:05:59 | what this observation |

0:06:01 | you feel a training snr but depends of course of the length of your you're on training sequences you have |

0:06:07 | mouse |

0:06:07 | but use your have so that give quantization error |

0:06:10 | so the estimate the channel estimate would be an that |

0:06:13 | but the back wall |

0:06:15 | a few takes the ever the estimate of this channel you can decompose it in they estimate and in independent |

0:06:20 | noise term you computes the variances of C received signal back of the use for signal on a |

0:06:26 | channel energy and the energy of C estimation error |

0:06:30 | i see that the seems to see yeah quantisation of no uh variance of P |

0:06:36 | now if you consider that |

0:06:38 | received vector |

0:06:40 | and it's the same station |

0:06:42 | a connection prior to it |

0:06:44 | as the estimate channel H that might apply but a signal week which was sent |

0:06:48 | and |

0:06:49 | to which |

0:06:50 | which contains the contribution from some noise quantisation errors and channel estimation |

0:06:57 | of course |

0:06:58 | in the set to this isn't a few months to um this not depend of the signal you sent |

0:07:03 | so actually capacity of this channel is not known |

0:07:05 | are we use the them |

0:07:07 | yeah the same um rather have and on the true information as and the paper by have C |

0:07:12 | that's use you humour |

0:07:14 | that's the noise would be gosh an independent of few transmitted signal or with the covariance matrix K easy |

0:07:22 | i map and i is it by a number of a station and the number of antennas so the |

0:07:26 | a man about the true information per a given time |

0:07:30 | and this doesn't take into account the that we actually spent |

0:07:34 | for channel training data transmission so what we want to do is work to maximizing that about the achievable rate |

0:07:40 | was a simply please just about a good you but R T |

0:07:43 | apply by a discount factor |

0:07:45 | and would like to maximise this expression of was the conditions that can lead to have at least |

0:07:49 | can |

0:07:50 | a training symbols because you have use that time but but we can train or something could you |

0:07:56 | now if you think about it the quite tough problem |

0:07:58 | "'cause" you have in fact the expectation |

0:08:01 | i have a a a a a a complex course matrix of each and and |

0:08:05 | has a different variants so this a very um which was of to profile |

0:08:09 | and this is not known in closed-form |

0:08:12 | those was where you can calculate cd eigenvalue distribution of this matrix in closed form |

0:08:16 | what we do have a it is we use so to an approximation based to from the matrix re |

0:08:22 | so we assume your yeah was that we would have a many user turn it's |

0:08:26 | it and the about |

0:08:27 | of the number of base station and it's number of a tell us per base station it's some since the |

0:08:32 | total number of antennas goes to infinity |

0:08:35 | and as this |

0:08:36 | assumption |

0:08:38 | hmmm two information will converge to a deterministic quantity or we can find a deterministic approximations of the about but |

0:08:44 | which information |

0:08:46 | such that what the system was infinitely large |

0:08:48 | the difference between the approximation and exact result close to zero |

0:08:54 | this is actually results so the result for a channel a random matrix either the entries were but each with |

0:09:01 | a different variants was that a lapel of by by by tasha |

0:09:04 | two thousand seven |

0:09:05 | or simply applied to to i was set to is just one greedy and um |

0:09:10 | well it you |

0:09:11 | that's the using split so power of it a fink at men subcarriers |

0:09:16 | so reason for this is quite simple |

0:09:18 | if you system was infinitely large need to make sure that the energy in the system states finite |

0:09:23 | if you start spreading |

0:09:25 | just signal of an infinitely many sub carriers the energy per subcarrier goes to zero |

0:09:29 | but still the energy in the entire system states fixed |

0:09:34 | yeah and actually can computed for each |

0:09:37 | there are |

0:09:37 | can ever ever |

0:09:39 | a deterministic quantity |

0:09:41 | such as just different converters to zero i don't provide a on you because it doesn't provide a lot |

0:09:47 | oh have of the only thing we to compute since quality |

0:09:50 | is seen covariance matrix of single and of so interference in rows |

0:09:55 | and the variance profile of the estimated channel |

0:09:59 | and actually to see that's is makes L |

0:10:01 | i i i mean we consider an infinite large systems so what we actually looking at would be three base |

0:10:06 | station was we use that term that's in each base station is to tell us |

0:10:11 | and |

0:10:11 | that's what we consider him a smell medical example |

0:10:15 | so i have |

0:10:16 | like a screen |

0:10:17 | three corpora to base station seven three is a sum three different set |

0:10:21 | i drop some randomly |

0:10:24 | you you can that lot just but past was model and obvious we every every or over channel a realisation |

0:10:30 | actually is C |

0:10:32 | we this plot the |

0:10:34 | i got rate of |

0:10:36 | well as a cs so a |

0:10:38 | for a a system |

0:10:39 | what each base station has |

0:10:41 | two ten as |

0:10:42 | you have only one subcarrier |

0:10:44 | with have coherence time of a sound channel uses and B was optimize in was saying |

0:10:49 | we have a a training time of to |

0:10:52 | a lot for me that based on the asymptotic approximation |

0:10:57 | the um the not "'cause" of what got by simulations |

0:11:00 | and i mean for me is this look |

0:11:02 | as good as if you had C perfect um |

0:11:05 | that is and some since the asymptotic |

0:11:07 | approximation works better very well even for channel of size a six times three |

0:11:13 | no i i since for three different back haul capacities so the black the black light was corresponds to what |

0:11:19 | you would get was a back work that's your one |

0:11:21 | um |

0:11:22 | a channel was and you start increasing in of course you get |

0:11:26 | well i what we do see approach so |

0:11:28 | optimize of the train time is instead of optimising the the a got a rate but we can't which we |

0:11:35 | can't country treat a or |

0:11:38 | we try to optimize our |

0:11:40 | deterministic approximation |

0:11:42 | so we want to optimize a deterministic approximation of the mutual of the about get you the rate |

0:11:47 | for this still need to compute the first derivative |

0:11:50 | you need to show that it's called okay for once you have done this |

0:11:53 | a a simple line search what but you wish |

0:11:56 | and finds the optimal train at time |

0:11:58 | then |

0:11:59 | we show that |

0:12:02 | up to a lower value of have on |

0:12:04 | um asymptotic approximation converges to the optimal |

0:12:08 | so we real up to a result you would get |

0:12:11 | and if you can also concluded C optimal training time to you compute |

0:12:15 | converges to the optimal time |

0:12:19 | and lastly it is it remains to do was to better five as some of that are a some type |

0:12:23 | to optimal to try to to trust a is very close to the |

0:12:28 | to to what you would get if you could optimize to problems and some since we simply do mount to |

0:12:32 | colour based optimization we one many |

0:12:35 | many you see which training time maximise the or right |

0:12:39 | um |

0:12:40 | first of just to show that section a concave function |

0:12:43 | you see the uh got a get you rate as a function of the training length |

0:12:47 | would then first different back haul capacity is |

0:12:49 | so that and that i so i computed of the two most approximation it's to marcus simulation |

0:12:56 | and and makes sense if you that the so close together |

0:12:58 | but is you to look at a maximum point some but here it won't make a big difference whether to |

0:13:03 | optimize |

0:13:04 | like our approximation or not |

0:13:06 | and |

0:13:08 | when i had to all that is for a given |

0:13:10 | S and a a a a a of a coherence it's time of T one hundred |

0:13:14 | i so that this optimization problem this is a black man |

0:13:18 | that's a function of C "'em" S N |

0:13:21 | but that leads to to an exhaustive search opposed to meant to colour up to a um |

0:13:25 | yeah simulations and you see that the difference between these two values is actually legible |

0:13:30 | especially |

0:13:31 | well that exhaustive search i and to do some kind of a to just search because i current so for |

0:13:36 | vol |

0:13:37 | yeah a kind of an infant find groups of values |

0:13:40 | and that have a |

0:13:41 | you current to try to twenty point twenty five point three symbols you need to round at some point |

0:13:48 | okay uh just class to to look at C and of the back haul capacity on the optimal training length |

0:13:54 | you see that the optimal true as is actually of tobacco capacity |

0:13:59 | this is so cute |

0:14:00 | a each affect |

0:14:01 | but you compress it is no |

0:14:03 | man says of |

0:14:04 | the from was used C is actually due to quantisation rose |

0:14:08 | next right of a quantisation quantization have from a a a try to now |

0:14:12 | once or capacity bounds are back or lectures and infinite capacity |

0:14:16 | a train or depends and and so on |

0:14:19 | yeah on the on this is and nice you have an edge |

0:14:24 | a of course them just to point out the um |

0:14:27 | how bad actually are a sub optimal quantisation scheme was so so that's a lot to a |

0:14:32 | um can see the back haul capacity on the X |

0:14:36 | well as as the um |

0:14:37 | and and the go to go we're rate |

0:14:40 | and for |

0:14:42 | if you look at is value two point five |

0:14:44 | this was "'cause" is a um |

0:14:46 | relative power base station time teller |

0:14:49 | so if would multiplied by a two because we've to tell us |

0:14:52 | you would see to a Q chief as like |

0:14:54 | as a spectral efficiency of five bits per second per channel use |

0:14:59 | you that to have |

0:15:00 | twenty bits per second pitch and use of back or pass |

0:15:06 | yeah to conclude what um |

0:15:07 | we have used them |

0:15:09 | results from lot from the matrix you read to tech a you a and optimization problem in context |

0:15:15 | and have treated to channel is the variance profile of just you like a distributed and in a system was |

0:15:19 | um with the back haul capacity |

0:15:22 | uh yeah but we have to on the parents it's is asymptotic approximation but extremely rather for channel it's of |

0:15:28 | size three times we even to by two works quite well |

0:15:31 | a it's that it's a pack of back or limitation on the optimal training length |

0:15:37 | that's well that we can have the cross it's that's you and we work on distributed compression was in perfect |

0:15:43 | csi |

0:15:44 | so actually how do you compress when we don't know the channel |

0:15:47 | so i i i'm not aware of any paper which which problem so someone has some put that would what |

0:15:51 | have happy to get it |

0:15:53 | i |

0:15:54 | last as a question of how to decide whether but require parade or not |

0:15:58 | and some some that's the back back capacity we go to zero |

0:16:01 | i ever go to zero as well doesn't make sense because it's space station could at least decode individually |

0:16:07 | and then be treated like a set of B base station which live in the isolated you are so this |

0:16:12 | low in to the it's we haven't done it |

0:16:16 | um |

0:16:16 | i'd down so the extension a very straight but because when you do to take well some to come pilot |

0:16:21 | contamination |

0:16:22 | which happens |

0:16:23 | because can you to read like your sample training |

0:16:27 | um |

0:16:28 | last a just a few references so |

0:16:30 | we've of can subtract version of for our paper but just |

0:16:32 | going to be published in june |

0:16:34 | the transaction on signal processing |

0:16:37 | um like since |

0:16:38 | can is that for uh |

0:16:40 | right the as was about as but from bonnie touch and |

0:16:44 | of can see a classic paper or for for do that just just a good point a if you don't |

0:16:48 | know so much about that of my nose is over you paper by provide just back |

0:16:53 | and |

0:16:54 | yeah that that that you |

0:17:21 | yeah |

0:17:28 | okay |

0:17:28 | so we could have assumed from the beginning |

0:17:31 | that's a variance of the channel it's as one of a and |

0:17:35 | uh so as a classical assumption and people start it could be don't of |

0:17:39 | i i and on channel but each element like has the energy |

0:17:43 | you know just so from the big have an energy was i a channel was the a finite energy the |

0:17:48 | fresh start a is that it can't gain one more G you U and the back up some more energy |

0:17:54 | then work was actually sent right |

0:17:56 | so |

0:17:57 | we kind of either start to directly but making this assumption |

0:18:00 | a a some scale as a parallel |

0:18:03 | by a some so which goes to infinity that's the power up uh frequency band goes to zero |

0:18:09 | and the and it doesn't make a diff |

0:18:13 | yeah |

0:18:17 | yes you but infinite double T and it's a good i mean it's some point if you have but a |

0:18:22 | station it's a very far where you can let's this diversity |

0:18:25 | so that's is a reasonable assumption and for the scale in the N is C have used |

0:18:30 | we subcarrier |

0:18:32 | so in some since you didn't scale at all |

0:18:35 | and |

0:18:35 | so |

0:18:36 | much much Q do but to that mathematically correct you something new to |

0:18:41 | to this kind of scale the energy per uh |

0:18:44 | for channel and she was go to zero otherwise this doesn't work |