Přepis řeči - GRASSMANNIAN PREDICTIVE CODING FOR LIMITED FEEDBACK MULTIUSER MIMO SYSTEMS

0:00:13	i was speak
0:00:15	very
0:00:15	so
0:00:18	i just thinking um i mean my first conference presentation
0:00:21	every gave a member was a our ten a member of like
0:00:25	i spend the whole session
0:00:27	hammering like all the speakers of questions of my presentation was that the and
0:00:30	course after after my presentation they'll hammered me together so
0:00:34	so my plan is i'm an speak the whole time go five minutes extra low we no time for questions
0:00:38	and the will i be very quickly
0:00:41	um
0:00:42	it was so i i guess we're gonna get started here i mean you know a B being a professor
0:00:46	i can basically fill and whatever time is required i have like fifty back up so i i can do
0:00:51	this and an hour a can do than twenty minutes a do five
0:00:54	um so this is this is joint work with my P H D student to count you know we he's
0:00:57	a national instruments now
0:00:59	and uh
0:01:01	i
0:01:01	work that he did for P H T actually about a year ago
0:01:05	something about one students graduated becomes a lot harder to get them to actually submit the work on time for
0:01:10	some reason
0:01:11	uh so anyway as time but to be here presenting a here so the the title is many and predictive
0:01:16	coding for limited feedback multiuser mimo system
0:01:19	and i'm glad about that the especially the previous presentation
0:01:23	because uh it gives me a nice introduction that i was expecting to get in this uh sessions i don't
0:01:28	have to tell you what is multiuser mimo set
0:01:30	so the the system that i'm in a consider in this talk is
0:01:33	so multiuser user mimo communication system with limited feedback
0:01:36	um i'm going to make
0:01:38	the uh
0:01:39	the main assumption that
0:01:41	i i guess we just found on the previous talk wasn't good which is i'm assume the that the quality
0:01:45	information is perfect
0:01:46	and i'm to focus on quantizing
0:01:49	the uh direction information and so you the multi jim i'm most set of is
0:01:53	uh we're gonna do in a in a case and do zero forcing precoding at the transmitter we have a
0:01:57	single receive even of the objective at
0:01:59	each user is we're gonna quantized that direction vector we're gonna send a back of the limit feedback laying
0:02:04	we to put all those together or design trans the beam formers and
0:02:08	everything works just like in that the previous talk
0:02:11	and uh that so this is a set of that we're considering
0:02:14	and uh the basic
0:02:16	uh challenge here
0:02:17	is as follows here so if if you been following the the field for a long time you know that
0:02:21	you know we started off doing this was single user beamforming
0:02:24	um we were looking at
0:02:26	codebooks books on the order of three four five six bits for up to four antennas
0:02:31	in three G P B ended up taking four bats
0:02:33	so so you don't need that much feedback it's
0:02:36	that's great
0:02:36	now the problem is you know
0:02:38	leveraging this work my general you find out that
0:02:41	generally speaking
0:02:43	the to achieve a constant gap from performance with the optimal um
0:02:47	there were some rate that you chi with the ah with um perfect channel state information
0:02:51	you need to have a codebook that scales with the number of users
0:02:55	and all i also scales with the the snr so
0:02:58	the the application here is that you you tend to need um high resolution otherwise are gonna be interference limited
0:03:04	and and exactly like that was the nice
0:03:06	a plot before
0:03:07	you know you at at some point at high snr
0:03:09	you wanna do multiuser mimo all
0:03:12	and the reason is that we don't have enough resolution to become quantisation
0:03:15	interference
0:03:16	limited and um
0:03:18	performances
0:03:19	is the problem and then we're we're also seeing this right now a base station coordination we're people are applying
0:03:24	multiuser mimo to distributed in a settings
0:03:27	and we need just as much
0:03:28	feedback back there if if not
0:03:31	so the main thing here is that um yeah as we look at more sophisticated transmission techniques we we really
0:03:35	need higher code-books we bigger code-books
0:03:38	and
0:03:39	i becomes or problem "'cause" you have more feedback
0:03:41	so
0:03:42	um
0:03:43	of the other issues is that
0:03:44	in addition the quantisation i mean we also have delay and mobility me if you you have a large codebook
0:03:49	size
0:03:50	the channel mobile you have to send a lot of bits back very quickly is that means the net feedback
0:03:54	to have that link is
0:03:56	is very high
0:03:58	so
0:03:58	you know for channel the the transmitters not won't point about the channel and
0:04:01	um this is a problem so
0:04:03	well we're gonna proposes a predictive coding framework
0:04:06	that uh will allow us to
0:04:08	um take advantage of correlation in the channels so that when the channels not moving very quickly
0:04:13	we can get effectively larger codebook sizes then we otherwise what have
0:04:18	so i'll i'll explain a little bit about the key idea and a mention some some prior work so
0:04:23	the predictive coding concept the mean it
0:04:25	anyone working and source coding
0:04:27	uh ah this would be probably the first thing that you would do and
0:04:30	um
0:04:31	the idea as as follows here's base if you have a you have a source score over time
0:04:36	we do is um you you find or solve a good predictor this might be uh
0:04:40	and i'm a C predictor for example or or you have a favour you could plug it in
0:04:44	and use use that predictor
0:04:46	to predict the next value of the source and then instead of quantizing
0:04:49	the noose you know the the new value of the source you quantise the error between the predicted value of
0:04:54	the source and the source that comes at
0:04:56	this is used to it
0:04:57	with great success in speech and image i mean it's is
0:05:01	use all over the place and the main things you need here i mean you need
0:05:04	a a a predictor
0:05:06	you need to be a quantized the error you need to be able to compute the difference between the predicted
0:05:11	and the true
0:05:12	and then you need to be a to track your um
0:05:15	pretty good sequence
0:05:16	and then um
0:05:17	the decoder essentially what it does is a
0:05:20	it also implements this pretty cut with this um predictor so takes the quantized there updates and that keeps implement
0:05:26	the prediction and that gives a
0:05:28	it's you keep updating the
0:05:29	the decoder over time
0:05:31	and so for example um
0:05:33	what the traditional like use like a first-order one step predictor
0:05:36	you might have to weights here you take a linear you're combination of the previous two symbols that becomes your
0:05:41	predicted value and
0:05:43	you know you can optimize as weights using
0:05:45	mmse and
0:05:46	um
0:05:46	and then if you wanna design the code but probably the typical way to do this is with the lloyd
0:05:50	algorithm there is also a of structure techniques tree structure and so on
0:05:54	and uh this this is
0:05:55	this is known as widely deployed a mean people the doing this for
0:05:59	at least twenty thirty years
0:06:01	so
0:06:01	um the the problem here is that
0:06:04	as follows in the limited feedback beamforming case are the information that we wanna quantise that normalized vector a normalized
0:06:10	vector space invariant it's actually
0:06:13	um represented by a point on the grass and of
0:06:16	in this case
0:06:17	uh
0:06:18	the space of beamforming vectors are considered be points on
0:06:21	G and come one so this is the set of sub-spaces of
0:06:25	in N dimensional space with one to mention
0:06:27	and you can write it like
0:06:29	points on a sphere that's
0:06:31	more of an illustration not exactly correct but gives you
0:06:34	kind of the idea here
0:06:35	and so it effectively the problem that we have with them if you back beamforming is that
0:06:40	we want to do um
0:06:41	predictive quantisation of a source that lives on this matter
0:06:45	and um the and so that's that's the basic idea
0:06:48	now let's let's figure out what the problem is we white why are we talking about this now after we've
0:06:52	be doing them if you back for
0:06:54	eight or nine
0:06:56	well the problem as follows your so first all we have a subspace source
0:06:59	that's going click
0:07:00	coder
0:07:01	i we need to generate the error
0:07:03	and the problem with working on
0:07:05	it manifold special the grassmann manifold here is that
0:07:08	simple operations like um
0:07:10	adding up to point
0:07:13	not necessarily well the finite
0:07:15	so like for example at if if i'm looking at those two lines there
0:07:18	what is it need to add those two lines up
0:07:20	you got another line or a me what we or getting points on a sphere
0:07:24	you get a point that's not on the sphere ones you're done if you "'em" an euclidean space
0:07:28	so
0:07:28	i
0:07:29	you need to do something special just a add these things that so the net of that is that generating
0:07:33	a a a a a of the concept of a error it's actually
0:07:35	not obvious
0:07:36	um
0:07:38	well you also need to predict things on the subspace "'em" if you really wanna take this manifold full structure
0:07:42	into account we shouldn't be predict euclidean space we should be predicting on on this manifold a cell so we
0:07:47	need a we need a manifold predictor but
0:07:49	then
0:07:50	wow
0:07:51	what is an mmse manifold predictor or mean what what is multiplying what is weights to me all these things
0:07:56	the they're not um very well the fine
0:07:59	and so that's why it's be very hard to come up with even
0:08:03	you know extremely simple um examples of
0:08:06	of doing
0:08:07	predictive coding here's because
0:08:09	these operations are hard what that means is that
0:08:11	in the predictive coder
0:08:12	the generating the error is not straightforward want in the errors not straightforward
0:08:17	predicting the the sequence not straightforward an updating the predictors not straight for
0:08:22	and so
0:08:23	um
0:08:24	each of these proposed blocks you need to have something you here
0:08:27	and so there there has been a a lot of work on this i mean
0:08:30	it certainly my group another
0:08:32	i mean everybody wants to exploit the temporal correlation the channel mean it it's it's the right thing to do
0:08:36	a mean we know the multi to my what doesn't work well as the channels very quickly and
0:08:40	you know this temporal correlations just sitting there begging to be exploited
0:08:43	so in trying to do that
0:08:45	uh i mean prior the earliest work this is um
0:08:47	and a certain as either and i need two thousand two they have some nice papers with
0:08:51	with gradient based update
0:08:52	there's some dynamic codebook approaches where you
0:08:55	you kind of adaptively
0:08:57	select a subset of a codebook depending on how fast the channels moving if it's very slow you end up
0:09:02	with a
0:09:03	very directional codebook for correlation if is going fast you and of a kind of a grassmannian codebook
0:09:08	there's a these progressive or successive refinement techniques where you zoom and on the channel estimate depending on how a
0:09:16	fast or slow it's going that's more like a tree kind of quantisation
0:09:19	uh there there is
0:09:20	there been some work using would euclidean prediction you could use the euclidean predictor in font size it and depending
0:09:25	on how you do it sometimes like an actually work very well but sometimes
0:09:28	you lose the um
0:09:30	the phase and variance all the structure that we were trying to exploit the grassmann manifold first what
0:09:34	so i can be a problem um
0:09:36	probably the best
0:09:38	or to seen is a different role approach
0:09:40	like a at all and this is related to something that they've actually proposed
0:09:45	i the three G P P standard where it's essentially you look at a difference between the last vector in
0:09:49	the next vector here
0:09:50	and that that approach um
0:09:52	works reasonably well but it's
0:09:54	um
0:09:55	does a really use anything
0:09:56	stochastic and then there's also some rating expression ray compression techniques
0:10:00	so i mean the main message here as i mean there there's definitely like a lot of work on this
0:10:04	topic but
0:10:05	there's not really a comprehensive framework for solving the problem that i just told you about doing predictive a quantisation
0:10:10	the grass aggressive mouth
0:10:12	so what i'm gonna do is i'm it's tell you about
0:10:15	our approach to solving this problem and fortunately
0:10:18	uh we do have a general solution i mean the general solution with you know general manifolds at many points
0:10:23	is still an open problem on it so you about the solution that we have
0:10:26	um um that we're proposing for
0:10:28	the case where we have a a two points
0:10:30	in so one a do is i'm a to build i'm a explain some of the mathematical concepts that we
0:10:34	need to use of i've got the
0:10:36	the equations here
0:10:38	but i'm i'm a really focus on the the picture "'cause" the point is to get the intuition of what's
0:10:41	happening the picture and then all to you how we
0:10:44	yeah
0:10:45	so operating on
0:10:46	um
0:10:47	in these kinds of manifolds here you can the fine
0:10:50	uh several cards as one of them is
0:10:51	is this notion of a tangent vector so
0:10:54	this tangent vector is of i have two point X one and X two that live on the manifold here
0:10:58	i can the fine a tangent vector which are you want to
0:11:03	which is a vector that pointing
0:11:05	in a direction
0:11:07	of X two from X one
0:11:08	and so far gone all to X one
0:11:11	and member X one is a
0:11:12	is a point on the grass of manifold we can represent as a unit vector so E the E is
0:11:17	if vector that's orthogonal to X one but it's not a unit vector it has a link that's link is
0:11:23	dependent on the court of distance between X one and X two
0:11:26	and you can go through there's um so mice paper is it'll minutes miss that have
0:11:30	some very general descriptions of the notions of tangents and geodesic spot but those are
0:11:35	um if you look through which you're fine that
0:11:38	because everything is so general actually very hard to pick it out and so one of
0:11:41	the contributions here is that we simplified everything down for the beamforming case said it turns out this that the
0:11:46	equation simplifies dramatically and there is actually a lot of intuition and equations old
0:11:51	but through all
0:11:52	but basically the tended vector here you can decompose an the two pieces one which has to do with the
0:11:56	link
0:11:57	which is the arc link between X one and X two
0:12:00	a second which is the
0:12:01	unit tangent direction and then this is a function of
0:12:04	the inner product between X one and X two and it's a function of a coral this
0:12:08	and
0:12:09	so we're gonna use the tangent vector
0:12:11	uh to give us a a quick one notion of air
0:12:15	okay so some concept we use the stewardess so a geodesic this this is
0:12:19	the curve that's between X one and X two that's the shortest path between these two
0:12:24	you can come up with an equation for um for that's curve that can sit that's a function of you
0:12:30	can write as a function of X one and X two but becomes more convenient be write is a function
0:12:33	of X one in the tangent vector
0:12:35	you can write it like this here
0:12:37	you've actually got a um
0:12:38	X one times the cosine of this thing
0:12:40	plus
0:12:41	um
0:12:42	sign of this thing in the T equal zero gives you X one T one gives you X two it
0:12:47	turns out that the this
0:12:49	and this orthogonal
0:12:50	you can see that because if you remember that the tangent vector is orthogonal to X one
0:12:54	do you this nice if orthogonal decomposition
0:12:56	so we're gonna use this geodesic together something that looks like an addition
0:13:01	now the sort thing that we need is
0:13:03	we we want to predict so we wanna try to
0:13:05	figure out where the sequence of points on the grassmann manifold
0:13:09	and um
0:13:10	we so we wanna get some function that's got some or we can optimize over it and
0:13:14	so for this work what we did this
0:13:16	oh
0:13:16	where we're gonna take a really something very simple
0:13:18	really use this concept of parallel transport of the parallel transport is essentially a way of
0:13:23	uh taking this
0:13:25	tangent vector X one
0:13:27	and mapping it over to X two
0:13:30	a remember that
0:13:31	that's change a vector has to be orthogonal to the point you have to do this mapping in such a
0:13:35	way that the orthogonality is maintained to the new vector
0:13:39	and so you can do that and it turns out that
0:13:41	it it actually looks something like the negative of the previous tangent back
0:13:45	so we're gonna use the parallel transport concept to fine
0:13:48	a predicted value
0:13:52	okay so now i'm point how we use these mathematical operations to do press ready and
0:13:56	pretty quantisation so the first thing here is let's generated error was suppose we have a predict sequence the predicted
0:14:01	sequences
0:14:02	is gonna be denoted by
0:14:04	X
0:14:04	X still to here
0:14:05	so this X had is the state
0:14:07	this is um
0:14:09	this is gonna be known at both the uh transmitter receiver this is the predicted value here
0:14:14	which just from the state and then read take the difference between the predicted now sir
0:14:18	so what happens is we're
0:14:19	this is the predicted value here that we know the receiver and transmitter this is a new observation
0:14:24	we want compute this vector
0:14:26	tangent vector that'll take us from here to here
0:14:29	and so this is
0:14:30	or really do with that the error tangent
0:14:32	there we're gonna use the
0:14:34	a value the new value to generate a error
0:14:38	the second thing that really do here run a quantized a tangent vector
0:14:41	now the the D first um reaction at least at me looking at this problem is yes
0:14:46	nother grass many quantisation problem that's great the problem is is not actually grass many quantisation problem because a change
0:14:51	of vectors not you or
0:14:53	it has a a structure its orthogonal to X one
0:14:57	but otherwise it lives in the space orthogonal to that space so
0:15:00	it's a slightly different quantisation problem so we're gonna quantise that tangent vector in this paper here what we did
0:15:06	is we actually decompose the changing two pieces one that's a addressed many quantisation one that's a
0:15:11	not to again it's a complex king
0:15:13	quantisation is kind of like a quality and then a direction
0:15:17	so how we propose a codebook for that
0:15:20	and
0:15:20	that's where we used to quantized
0:15:22	ten
0:15:24	and the third thing is the state update here
0:15:26	so this is how we actually
0:15:28	update we add that a to the um predicted value three use the geodesic for that's so what we're gonna
0:15:34	do is we're gonna take the um
0:15:36	take the X still like here
0:15:38	we're gonna add the um
0:15:40	this
0:15:41	a parallel trance
0:15:42	as a problem transfer error vector here
0:15:45	take a full step and that's what we're gonna get for the uh updated
0:15:49	eek what's here yeah okay right so this is the
0:15:51	state update were add the error on and in this is the predicted value
0:15:55	so the project
0:15:56	what we're gonna do is we're gonna take a um
0:15:59	the difference between these two previous state factors
0:16:01	and then we're gonna kind of like keep going in the same direction
0:16:05	add that
0:16:06	on to the X
0:16:07	had here
0:16:08	goal
0:16:09	a right here so that's gonna be are are predicted value
0:16:11	and i should point out here that um
0:16:13	it in this paper we've simplified everything down counts were taking a full step for were
0:16:18	can probably realise that going of false step it may not be the right thing to do and so in
0:16:23	some of our other work
0:16:24	we have actually optimized the step size and and you can optimize that as of uh over time in there's
0:16:29	some cool things related to that and that that of here to uh i T A
0:16:33	a that result
0:16:34	does is our proposed predictor here
0:16:36	and so that's a basically the idea here i mean i taking
0:16:39	the geometric tools
0:16:40	for for doing things on the grass manifold and i'm gonna write these interesting equations to you
0:16:46	to get notions of prediction and
0:16:48	to get notions of error update quantized error as i'm i use all of that now to quantise a correlate
0:16:53	sequence over time
0:16:55	uh for this spring for the simulations in this paper really use a
0:16:58	uh autoregressive model
0:17:00	which is which is rather standard though
0:17:02	you could do better if you use to i a a uh
0:17:05	clark gone slight model which is band limited has better prediction problem so this is actually more of a worst
0:17:09	case
0:17:10	we're gone you consider the sum rate without of so we're gonna use or for a multi to my mow
0:17:15	and we compare with using ran a vector quantisation which is essentially a good way of finding a a fixed
0:17:19	codebook
0:17:21	or different sizes
0:17:22	so that the first um
0:17:24	result as i just wanted to point out that
0:17:26	uh indeed using this the using this our of um it has result in effectively high resolution so this is
0:17:33	i i i forgot the the parameters the simulation here but this is basically
0:17:36	a nine to codebook
0:17:38	the sequences varying over time in this is you know we keep quantizing it over time here and and it
0:17:42	the coral dozens fluctuates because sometimes you're quantized values close to the true channel sometimes as far as fluctuating
0:17:48	and then this is our approach down here also with nine bits
0:17:52	so we're getting about
0:17:54	vector five down here
0:17:55	and our approach is um
0:17:57	because we're tracking as over of things are changing over time we're not converging to zero because it's very
0:18:02	so
0:18:03	um but this is just to show you that and it does actually
0:18:06	decrease the error
0:18:07	terms of average word of this
0:18:10	now uh let's look at the
0:18:12	uh sum rate comparison
0:18:13	and
0:18:15	so the blue curve here
0:18:16	this is a perfect csi as for users
0:18:19	for in as we don't have any more the reverse is we didn't select the that's for users we just
0:18:23	picked randomly for users and
0:18:25	they have the same average snr so there
0:18:27	there are like sitting on the same sort if you like
0:18:30	so here's this the blue is the
0:18:32	you know what we're trying to achieve here
0:18:34	that's perfect csi
0:18:36	and
0:18:37	this uh uh what colour that is
0:18:39	kind of a mustard looking over that
0:18:41	is a grassmannian i D
0:18:43	go work here
0:18:44	it's it it kind of a weird thing is that
0:18:46	it turns out that there's not really good grassmannian code-books for large codebook size is so with a vector quantisation
0:18:52	you can get
0:18:52	more or less a grassmannian cold so it's
0:18:55	you can get essentially a very good code
0:18:57	have a geometry works so this actually is a good codebook book
0:19:00	a good fixed to mention book you probably won't do much better than that
0:19:03	and in here is the standard errors for result you get a high snr your france
0:19:07	so
0:19:08	right here you can see these different uh doppler
0:19:11	a a symbol period products here so this
0:19:13	zero point zero one this is a very slow channel
0:19:16	and this is a
0:19:17	reasonably fast channel here
0:19:19	well you can see is that
0:19:21	in a of all with nine bits of feedback in a in a effectively a slow channel were able to
0:19:26	get a
0:19:26	very good
0:19:27	performance tracking that ideal sum rate
0:19:30	with a a a you know somewhat
0:19:32	fast channel
0:19:34	we still get a little bit of improvement but um the performance improvements not
0:19:38	is not
0:19:39	and so then nice thing is that you know this this means that
0:19:42	you know every user essentially K can be updating their um
0:19:46	this is for all the users have the same channel profile but the out of as flexible enough that mean
0:19:50	every user since you runs are an adaptive algorithm that predicted
0:19:53	they get the csi and if their channel happens be slow they have a better information of the channel during
0:19:57	fast to get worse
0:19:59	so it's it's um i-th that it is actually tracked quite well on the also includes
0:20:03	a five millisecond delay which is a standard assumption three D
0:20:07	so even with delay that we're not accounting for we still can track the the sum rate with uh
0:20:14	under some
0:20:15	reasonable some
0:20:18	so uh that's essentially a tears what i've done and the stalks the proposed a
0:20:22	a a many predictive coding framework
0:20:24	and this is i think um um at least from a source coding perspective
0:20:27	you know the the pictures nice the stories nice um
0:20:31	now
0:20:32	there's a lot of limitations of first formally using the previous two point
0:20:36	i like to use more than that
0:20:38	clearly if i had um i figure out how to do this with three or five previous points of would
0:20:42	just shown that to
0:20:44	it's it's hard because
0:20:46	these tangents all those concept are based on two point
0:20:48	so you have three i U you gotta do something else there
0:20:52	and so we have some ideas but that's that's proving to be very difficult
0:20:55	um my student in his dissertation has some extension of this steve manifolds to so you can do this um
0:21:01	we did some work with limited feedback mimo ofdm or you had to do a a people quantisation set of
0:21:06	grassmann quantisation
0:21:07	and this whole concept that works there as well but the feedback requirements really become high so i
0:21:13	i i i'm not sure that this is exactly the right solution for higher dimensions yet but i think it's
0:21:17	a good idea
0:21:19	and then i should you the multiuser mimo results
0:21:22	um
0:21:23	since we develop this i've another student who graduated
0:21:26	who has
0:21:27	um enhance this for multi cell my mel so where we actually predict the channels from interfering base stations using
0:21:33	a similar concept what she was actually able to come up with frankly a much better predictor
0:21:37	and that results in a at i C C
0:21:40	and then i have another result
0:21:42	where we've also come up with a better predictor for interference alignment
0:21:46	and that's gonna appear period
0:21:48	that
0:21:48	so the concept
0:21:50	the concept i think it's good and that i still think of a lot of a lot of work to
0:21:53	be done here
0:21:54	okay so that's a i pa gesture not
0:21:56	taking enough time
0:21:58	um
0:21:58	uh this is not my virus
0:22:02	oh i took exactly the right about time one
0:22:09	well
0:22:13	oh
0:22:18	uh would you know
0:22:20	i need to get back to D
0:22:23	it is not depend this this method on a a a a a a a you model list time
0:22:28	correlation
0:22:29	um it's it's not really um
0:22:32	very dependent so
0:22:34	in the in the journal version which uh i i think it
0:22:38	we actually just submitted the journal version recently and it should be an archive
0:22:41	a or tomorrow we have simulations with other temporal like a a multi tap they are models and it
0:22:47	still works well
0:22:48	so we're not really exploiting any knowledge about the channel correlation
0:22:52	you could probably come up with a correlated channel model that makes the art more for like
0:22:56	i process but
0:22:57	but we're not using
0:22:58	you know we we don't have the um
0:23:02	the the correlation function anyway
0:23:04	the there was no i'm and right i mean mean we'd like to have that or not exploiting a yeah
0:23:07	so you not exploding help
0:23:10	right right
0:23:11	i i mean it it's only coming see the this be it's really only coming when we quantise
0:23:17	the magnitude of the tangent vector
0:23:19	so the channels bearings slowly
0:23:21	what's gonna happen is um
0:23:24	we're gonna take the where the quantized nick here
0:23:26	well when we quantized this so we quantized the direction
0:23:30	and
0:23:31	um the magnitude to of tender vectors one very slowly
0:23:35	we're gonna take a small step
0:23:36	and what very quickly we're gonna take a larger step
0:23:39	so that's the only place where
0:23:41	the correlation comes in now the other paper i mentioned where we adapt the size
0:23:46	they are the statistics come in but not in uh
0:23:49	a way
0:23:57	as one the back
0:23:59	a this was not a speaker good heart
0:24:03	so yeah
0:24:04	cross
0:24:06	this
0:24:07	these correspond to "'cause" i'm an easy to
0:24:09	the best
0:24:11	presentation of the information that one can combine E
0:24:14	hmmm hmmm you do something on some in by do have to seek to unit
0:24:19	inspectors
0:24:20	um okay i can use be the last part at in
0:24:24	do you have to two
0:24:25	stick to to to to not terms
0:24:29	you could do just one uh the percent direction by by
0:24:33	some big to that's not normal estimates
0:24:37	yeah so i mean the
0:24:38	a the question is
0:24:40	yeah whether you need the
0:24:42	do you know link factors not something is that remember that this vector of lives on the grass amount for
0:24:46	model is it unit length that's also phase and vary
0:24:48	so those two properties
0:24:50	it
0:24:51	holds out degrees of freedom of the vectors we have a complex vector with and in tree you have to
0:24:56	and pram
0:24:57	you know nor we have to and mine one this phase in bearing you have to and minus two
0:25:00	so the whole reason that we use this on the grassmann manifold is so that we quantise
0:25:05	less information and you can show from a point of view of capacity you're probably of error that all you
0:25:09	need is this
0:25:11	normalized direction information
0:25:13	now if you wanna quantise the direction as well
0:25:18	with the C Q why then actually your back your back to to and minus one and so if you
0:25:22	want to combine those together you could
0:25:25	you could do that and you will end up with something different
0:25:27	but still you have the phase and variance so what the
0:25:30	it when a be like it wouldn't be like a a random vector
0:25:34	a gaussian still wanna be a gaussian problem it would be on a different fold
0:25:38	right but but the
0:25:40	doesn't really matter in this context menu you
0:25:43	one to two
0:25:46	good some some
0:25:48	and some
0:25:49	after at each step and and a doesn't matter
0:25:53	oh to the one of is
0:25:54	no number unit
0:25:56	thanks
0:25:58	and you you don't care about
0:25:59	the thanks and then
0:26:01	so anyway
0:26:02	oh okay okay so um i-th
0:26:04	i can give you another answer of this
0:26:06	this question here so one okay one thing that you could do
0:26:09	is um
0:26:11	grassmann manifold as the rim money in manifold it's local euclidean so in fact you could uh you could use
0:26:16	kind of a standard predictor
0:26:18	and then
0:26:19	you could at you can add them up and you get a point not on the manifold and then you
0:26:22	could if you need a you know norm you can project back to the manifold
0:26:26	and uh indeed indeed that works if you're operating in a local enough region
0:26:30	and so we found that um
0:26:32	you an actually the analysis in the journal version this paper
0:26:35	makes that it's like a small angle assumption but if you have
0:26:39	more variation
0:26:40	the problem is that
0:26:42	you're
0:26:42	that effectively
0:26:44	that addition in projection
0:26:46	gets you five away from this kind of addition operation so i i
0:26:50	i think there there's is actually a case for doing that to
0:26:53	it just depends and and the the one
0:26:56	the the approach that we've taken this i C C paper actually has a better predictor that that doesn't predict
0:27:02	on the grassmann manifold but we do updates on the grass amount so
0:27:06	i think it's
0:27:07	the your point is well as well taken
0:27:11	okay
0:27:12	so
0:27:13	copy break
0:27:14	right
0:27:16	oh
0:27:18	i

GRASSMANNIAN PREDICTIVE CODING FOR LIMITED FEEDBACK MULTIUSER MIMO SYSTEMS

Multiuser and Network MIMO

Přednášející: Robert W. Heath Jr., Autoři: Takao Inoue, National Instruments, United States; Robert W. Heath Jr., The University of Texas at Austin, United States