Přepis řeči - PROBABILISTIC SINR CONSTRAINED ROBUST TRANSMIT BEAMFORMING: A BERNSTEIN-TYPE INEQUALITY BASED CONSERVATIVE APPROACH

0:00:13	uh my is the weight down from a nation which in wireless didn't i one
0:00:16	and this that's and it's to work with my
0:00:19	uh the member
0:00:20	when you one and uh my of my
0:00:23	a original and G at an much more in
0:00:25	and uh with
0:00:26	it's so we can model and the province and and so and that's when is you most new home
0:00:33	oh this work
0:00:34	with that screen how see the uh
0:00:35	then the model use of owning since the
0:00:38	there's a just me to want to communicate with a set of
0:00:41	single can use uses using chance may people
0:00:45	do to maybe beamforming uh
0:00:46	just meet that has to know the channel state information
0:00:49	already used
0:00:50	so you wisest as
0:00:52	and i can T D system is then
0:00:55	to never transmitted to obtain channel state information the user has to to uh bring the chain
0:01:00	or you may have a dc since then users have to feed back their channel state information to adjust meat
0:01:05	in our in
0:01:06	i don't case
0:01:08	no it doesn't
0:01:09	like in
0:01:10	you need is that the
0:01:11	that channel estimation maybe be perfect or in F T D distance
0:01:15	you to find a feedback the channel estimation may also be the train no information it just a meeting
0:01:20	may also be of
0:01:21	so the core over this work is try to decide a
0:01:24	we mean vectors like best night okay and
0:01:27	a low passed okay and thirty C as i
0:01:31	so we you know a D S the number just me obtain a set just meet and
0:01:35	case than um uh pay
0:01:37	a the number of them go to not use a
0:01:40	it you i is the i-th uses channel that to
0:01:42	and uh you i
0:01:44	involving that the used to chat me information a
0:01:47	i you
0:01:48	so
0:01:49	but a of or use i K is given by T equation here
0:01:53	the you rate is the single in power and that this is called no interference
0:01:57	and the signal my scores uh
0:01:59	noise power
0:02:00	oh
0:02:01	if we assume that just meta has a
0:02:03	perfect csi
0:02:04	then a a you to uh
0:02:06	it design for is given by is power minimization problem
0:02:10	which i to minimize the which powers
0:02:12	a such a that such a a a a subject to the constraint is
0:02:15	that each are we use that can achieve
0:02:17	that design snr requirement
0:02:19	come i
0:02:21	so this problem has been we'll start
0:02:23	a a you can use like
0:02:24	a the counting your eighteen you can use them at that do dictation all
0:02:29	you can be formulated problem
0:02:30	signal to calm program and it's be efficient
0:02:34	so it's i say that all this work we can see that that that
0:02:37	just need to either
0:02:38	as the perfect i
0:02:39	so
0:02:40	like we you not edge i S the a channel a able to is meet
0:02:45	then this edge i've are what be if will found the true channel H I and that the difference
0:02:50	yeah i is the csi M
0:02:53	or this work we soon the
0:02:55	i are are a and then in the following in the complex gaussian distribution with zero mean
0:03:00	and variance
0:03:02	combines matches the i
0:03:03	which is a positive semidefinite
0:03:07	so in the presence of a C is i ever as the receiver the user is going to
0:03:11	are going to suffer from a a a it's an all T
0:03:15	so
0:03:17	the core of the people mean he the people and idea to try to decide is W I i
0:03:21	such a i'd the
0:03:23	each of the receivers can achieve that is an hour requirement with a very high probability
0:03:28	so
0:03:29	that's a common to say in i'd the we want each of a use can achieve a high it's than
0:03:33	not if for collision probability so one this one
0:03:36	this
0:03:37	this probably probability in because to one
0:03:40	so that's but look at of what's going
0:03:43	what's the to be call and you call set if occlusion probability of number of past design i
0:03:49	this is a simulation example
0:03:51	we can see the a and D a cape with a three and the noise covariance matches all one or
0:03:55	two
0:03:56	which is uncorrelated and see my score is four point no one
0:04:00	then
0:04:01	a a this is the result
0:04:02	all of he spoke when of P and P recall and snr
0:04:05	setting fiction probably
0:04:07	is the full call mike a fine
0:04:09	do you can eve only it's speaker for most of the cases
0:04:12	no achieve a a a chip it's i not significant in probability is best ten
0:04:16	or point fine which implies that more than fifteen percent that
0:04:20	that users will suffer from a snr outage
0:04:23	so if the what this to a should make it even was
0:04:26	if we can see the if we're is use require required
0:04:29	even higher is not required
0:04:31	the goal here is try to design a beamformer
0:04:34	to move these pointed to
0:04:36	a a a like in as close to one
0:04:38	like if want a a a each use that to what
0:04:41	a a nineteen percent sinr i do not puppy eighteen
0:04:44	then want to these points to move to a and sum up
0:04:50	so uh
0:04:52	that uh uh as you know that no i it's uh next mark of voice an outage probability for use
0:04:57	uh
0:04:58	then i'll can see that design formulation is
0:05:01	yeah
0:05:01	we want to minimize just medium power and the subject to a constraint that
0:05:05	each user will have a a a a as i not satisfaction probability of each user a
0:05:11	is an or less than one minus low i
0:05:14	but is problem is
0:05:15	if typical
0:05:16	because but so you can write to this problem is probably reading you court in a sort of a you
0:05:21	and the U see died to the argument of these probability functions
0:05:24	which is
0:05:25	not convex
0:05:27	with was that respect to that i
0:05:28	this first
0:05:29	point
0:05:30	second is so to the best of our knowledge we can uh find a a close form inspiration
0:05:34	for is probability
0:05:37	oh also but existing works
0:05:39	they focus on of a uh
0:05:41	two
0:05:42	uh folks on
0:05:43	some approximation that's S
0:05:45	like
0:05:46	the one by my question or the even is be in two thousand a
0:05:49	they use the probabilistic a it's are not constrained as all she problem
0:05:53	as a approximation
0:05:55	our previous work we use a was case robust be from design as an approximation
0:06:00	is to methods up score coding brown to a
0:06:03	a console if that's S which means the up ten approximate solutions
0:06:07	okay in to be feasible
0:06:09	the original design problem is would in price these probably be it's i'm not sitting fixation probably
0:06:15	is getting to be set you five
0:06:18	or these work are we are going to present a a a a a a new mass says that we
0:06:22	be shown to perform better than
0:06:23	no previous work
0:06:26	so the first stage that we use "'em" at that in addition
0:06:29	we try to remove it this non-convex in arguments of these probably in found
0:06:34	in that used as the a will be price each of the rank one mention is W i that information
0:06:39	by a cost use them at that frame match kept a copy i we sought out any
0:06:43	true
0:06:43	so we end up with this
0:06:45	probably a bit in culture
0:06:47	so a rate at in your life in these
0:06:50	oh argument that does not help too much because
0:06:52	is a probability of changed the old
0:06:54	a has not from
0:06:55	it's
0:06:57	oh next they or try to approximate
0:07:00	you can see that these E I S complex comes so these probably in
0:07:05	a training can be a
0:07:07	uh a an S it is up to form that
0:07:09	is the court form
0:07:11	of complex comes the the both
0:07:13	is the pop been you quality of called ready for complex culture of in the right
0:07:18	the want to find approximation to this
0:07:20	yeah quote
0:07:21	uh
0:07:23	all that's that's based on these they
0:07:25	we
0:07:25	is the name of a died uh if we we have a
0:07:28	how but cost to the better
0:07:30	back to and uh we have Q are it's up to but
0:07:33	therefore for any you know which you
0:07:34	braun to one to going to one
0:07:36	and
0:07:37	for these these right is
0:07:39	the probability of these quite return grading equal to T
0:07:42	is given by Q
0:07:43	is
0:07:44	oh is no less than one may scroll so these is going to host to it's it's that in this
0:07:48	T
0:07:48	is a
0:07:49	it is a
0:07:51	that you right
0:07:52	at that thing are functions of these low R S is given parameter is and the this low
0:07:56	is
0:07:57	a Q R S P given problem in in this
0:07:59	all uh two probability
0:08:02	a the problem you quality in these the post and type in you quality so but this person typing court
0:08:08	uh that's
0:08:09	you do will refer to at least you in up was time watching
0:08:13	means that they are you qualities not problems that to probability team
0:08:17	of of sound of random variables be eighteen functions mean like
0:08:21	the
0:08:22	mark of in you quarantine
0:08:23	could be shipping quarter know that one of buttons and train the open to leave
0:08:28	but typing court
0:08:30	so actually is can help us a lot
0:08:33	using the above a i can show like that these conditions is going to be a sufficient condition for achieving
0:08:38	these probability quality
0:08:40	so the the same for in you court
0:08:42	these can you question here is that
0:08:45	this is deterministic and no
0:08:47	probability in this in which
0:08:51	so we can use this equation in you
0:08:53	equation as an approximation
0:08:55	to these probability in you quality
0:08:58	so
0:08:59	so the think was is not as if we you you can see I D in quality may a are
0:09:03	quite about the which is not convex
0:09:06	but that a crucial observation so actually
0:09:09	and and use uses some slack variables
0:09:11	you can
0:09:12	before my
0:09:13	these can you quality S is well come S and constraints
0:09:17	which in price we can use these for a convex constraints that's a approximation
0:09:21	to these probability inequality
0:09:23	so we can right this idea to each of the
0:09:26	uh i an a certification probably give users then we
0:09:30	are up with these
0:09:31	how mix problem
0:09:32	so it you can check each the file of D function all constraints all
0:09:36	so we can stop with a very efficient
0:09:40	uh because we use S T are so one probably is that the up that we should may not be
0:09:44	rank one
0:09:46	and in that case we have to use some additional as was an approximation procedures like
0:09:50	comes in the might agents
0:09:52	to obtain a rank one approximate solution
0:09:54	but a quite surprisingly all some racial results if fall the knight's very where
0:09:58	it does it six have look solutions is very rare
0:10:01	to obtain a hiring solution
0:10:05	so like the preset
0:10:06	some to measure results we consider a a a a simple case of speech as me a you know as
0:10:10	we use as
0:10:11	but
0:10:12	a channel estimate that just is uh
0:10:14	a complex gaussian distribution
0:10:16	the
0:10:17	all to probability
0:10:18	or point one and a noise variance of point one
0:10:22	the csi comments metrics is or one two which is as window the errors that are uncorrelated this
0:10:28	especially
0:10:30	so we first look go drink once that which
0:10:33	you we say a a a a few ice rank one face
0:10:36	the ratio of the largest eigenvalue of of be one
0:10:39	to a ways to is it's great and you go to
0:10:42	or point nine nine so which you
0:10:44	in bright light that uh
0:10:45	that may have mike can batteries are around a to times larger than T a race of a very
0:10:51	a and we say i the the problem will people link one some J for W I they all say
0:10:56	to find his condition
0:10:58	so these this that's to the first lois estimation results
0:11:01	for low like what to a point one which missed nineteen percent out
0:11:05	is a not to defect region probably it
0:11:07	and the teacher or to here that you know had to
0:11:09	means that non ball
0:11:11	uh be
0:11:12	feed the about channel is we test it and that the you might take here use a
0:11:17	the number or channel or i'd H for which are bank once so which is up to a
0:11:21	so you can see from here to for or phone got mike to one two
0:11:25	the fifteen
0:11:26	fifteen
0:11:27	that
0:11:27	that's a one two percent that
0:11:29	we all of ten one solution
0:11:31	but if if we
0:11:33	that no i point open or one which implies that more demanding the phones requirement
0:11:38	then we encode the case
0:11:41	a a column i with see
0:11:42	three db and is one problem channel died vision that the so which change not bring one
0:11:47	but if we do in
0:11:49	for the you'll find a i'd uh for this particular case
0:11:52	the the these ratio show a lot or point a or a two to or series of which that's is
0:11:58	also quite close to a link so which is so if you use
0:12:01	got the minimization best going you obtain quite
0:12:04	quite good performance
0:12:06	the
0:12:07	so that they can we want to check if the proposed approximation formulation can
0:12:13	so you find a it's an set if probably due not
0:12:16	yeah we also compare with the mess by my question or time the in them is an it is what
0:12:21	you
0:12:21	and this is our previous work
0:12:23	is is the proposed to that's in this
0:12:26	in this work
0:12:27	the we can see that
0:12:28	or or next we S can see defined a desired like ninety percent
0:12:33	certificates fiction would be at but you can see that
0:12:36	a this one seems to kinds about if
0:12:38	because we only one ninety percent
0:12:40	it gives you want to per
0:12:42	is once is and this white
0:12:44	the proposed a given
0:12:45	is given beta
0:12:48	so
0:12:49	that's so we want to compare a transmission power
0:12:52	and here the recognise uh
0:12:54	one by mac channel that that this one is by
0:12:57	a previous work in this one is the proposed will
0:12:59	the problem lies uh
0:13:01	just mission power not low design so which can
0:13:04	so as a benchmark for
0:13:06	these robots robust
0:13:07	again we can see these figures i full
0:13:10	from got got model in to find one to nineteen is that's going in the
0:13:14	propose a have its most power you fish
0:13:18	you want to compare the computational complexity in the stream S
0:13:22	yeah we can pick uh will compute the
0:13:25	that i
0:13:26	the time of C V X
0:13:27	solving the form a nation is of each for the mess on the test
0:13:31	then you can see that a a going to of proposed method more
0:13:35	a computationally efficient then that one
0:13:38	proposed
0:13:39	by i'm sure that and but it's
0:13:42	more computational
0:13:43	expensive it in our previous work
0:13:45	so this is a a a cup and the performance trade
0:13:48	but to in uh
0:13:50	the work proposed in this
0:13:51	the the the mess the proposed in his work and our previous work
0:13:56	so in summary uh we have a pretty sent uh
0:14:00	new approximation for a probably probably if the
0:14:03	it's i not constrained
0:14:05	robust beamforming problem
0:14:06	and that the to all mess up a some to in great the to in gradient
0:14:11	the first it's segment that and relaxation
0:14:13	that's second is a
0:14:14	but and typing you according which you so so
0:14:17	convex
0:14:18	comes of better approximation to a probability you cost straight
0:14:23	then estimation result have shown that the proposed method based quite
0:14:26	a of X six in message
0:14:28	thank
0:14:36	i we asked or sometimes all questions uh and questions from the audience
0:14:40	oh in a single question because i one of the corpus
0:14:56	you have to ask
0:14:57	with the money
0:14:58	microphone for simple
0:15:02	okay so i was here just out of it to i apologise to that
0:15:05	um but it is your you approximation a conservative one or or is that the
0:15:11	you can so what he is it is
0:15:13	okay
0:15:14	that's what i would of to the beginning thanks
0:15:16	think
0:15:17	but actual one have a few words so is
0:15:20	restriction and and them relaxation
0:15:23	relaxation this semi-definite relaxation and restriction is with the pop up this the constraint
0:15:32	yeah
0:15:35	so that run realisation when you get to to rank to solution
0:15:39	can be sure that to
0:15:40	wasn't and rank one solutions to
0:15:43	sorry
0:15:44	a a in this scenario when you a the uh will find a rank two solution
0:15:49	it could to be rank one solution will so
0:15:53	to me
0:15:54	mean and you to the X the france one or of the this the set of solution perhaps
0:15:59	could to be used at
0:16:00	some under solution that's rank one
0:16:03	i i i not sure if i'll
0:16:05	correct
0:16:06	you mean when you to
0:16:08	so you me vibe obtain a drink two solution
0:16:11	then
0:16:12	that even if you obtain the rank
0:16:14	two solution the might excess
0:16:16	my one solution that you
0:16:17	C doesn't happen to find
0:16:19	uh i
0:16:22	uh
0:16:23	this is the response is one strange
0:16:26	oh is this
0:16:28	so yeah that you calm down brand
0:16:31	when one that were then when one solution
0:16:33	in in nineteen nine nine nine nine percent of cases is just run what
0:16:37	well do what would you mind can scroll back to
0:16:40	a the situation now or or or do the the back here yeah at the situation where we see round
0:16:46	one
0:16:46	well here we you very strange shape see a one room a situation
0:16:51	but this right and you can use some other tricks to get around the
0:16:58	i don't questions
0:17:02	okay can you just a normal one that's past the speaker

PROBABILISTIC SINR CONSTRAINED ROBUST TRANSMIT BEAMFORMING: A BERNSTEIN-TYPE INEQUALITY BASED CONSERVATIVE APPROACH

Beamforming and MIMO

Přednášející: Kun-Yu Wang, Autoři: Kun-Yu Wang, Tsung-Hui Chang, National Tsing Hua University, Taiwan; Wing-Kin Ma, Anthony Man-Cho So, The Chinese University of Hong Kong, Hong Kong SAR of China; Chong-Yung Chi, National Tsing Hua University, Taiwan