0:00:14a morning everyone so um
0:00:16yeah even here from the intelligent robotic on number to read john and
0:00:20that would present this
0:00:23first this is just was introduction so we are interested in
0:00:26separation problem like co but party problem where
0:00:30so were a speaker are talking and there's a microphone array
0:00:33so we we
0:00:35blind source separation
0:00:36in the frequency domain
0:00:38and we are in the permutation problem
0:00:41and especially in the approach based on the direction of arrival so from which direction the signal coming
0:00:47and a
0:00:48we focus on the special i'm guessing
0:00:50so what we it do we we show some model using the special a yeah thing
0:00:53that we can maybe have some sparse solution and have to relate to this
0:00:58the the permutation resolution problem
0:01:01so this is a line of the talk so first us would talk about the
0:01:04frequency domain blind signal separation
0:01:08then the permutation problem
0:01:10and after that that will go to talk about the mess with base
0:01:14do you rate estimation
0:01:19and the proposed approach so the specially in in equation of finding a solution and how to relate to the
0:01:24permutation with
0:01:28so first a domain mixtures so this is there is
0:01:31very fast to tree so we have to
0:01:33to people talking in front of the microphone array in the time domain to comparative mixture
0:01:38if we you the frequency domain we have several set of
0:01:42uh instantaneous mixture so here it was a future here we have much we see so we have a much
0:01:46we see that give us
0:01:48mixed to a mixed signal in each of the frequency bins
0:01:54but still separation in one frequency bin is just finding actually a metrics that we
0:01:59separate so weak over the signal so green and red here from the blue
0:02:03one of the problem is that actually
0:02:05when we do we
0:02:06when we this is
0:02:07using a
0:02:10statistical independence like
0:02:12independent component analysis
0:02:15we have the permutation problem meetings that
0:02:18we don't know
0:02:19the mixed are so we don't know the order of the signal when we equal
0:02:24a problem is that in uh
0:02:25a domain approach
0:02:27when we do all the processing so
0:02:29we are in the frequency domain
0:02:31we separate the mixture in each
0:02:33of the frequency band
0:02:35and at that we have to go back to time domain
0:02:37so the premise that
0:02:38usually we take the first signal is the first being as a first note is second be
0:02:42and your which is a problem
0:02:43if the order is different
0:02:46we you have mixed signal
0:02:47so meaning that
0:02:48i'm addition resolution is just actually finding in each of the frequency being
0:02:52a matrix matrix that we
0:02:54or me the signal so that we exactly have or with
0:02:58the right component to go back to time domain
0:03:03there are many review you paper so this is one of them
0:03:06that present a mini single about the convert source separation uh and also many single about
0:03:11is a permutation problem so several ms what's of some miss database was or
0:03:17checking the the signal that are separated
0:03:19and some of the on the futures so that you can be in a station of the future some smooth
0:03:24using so directivity part N
0:03:26or directly the direction of arrival so
0:03:29here we are as a interest this five
0:03:31i would also say that there is also some walk that was considering special i'm guessing
0:03:36so only walk here for example
0:03:38some of the work
0:03:39based especially on the direction of arrival
0:03:41and this walk
0:03:43we we
0:03:47here i
0:03:48give some to on so miss what was a one based on direction estimation rate especially really to this
0:03:57for simple we have a i was sick or a microphone array so the microphone a P P two P
0:04:02three and we
0:04:03i defined find some vector was actually for the position of the microphone
0:04:06and here in a far-field assumption we have a a source that is coming from far
0:04:12we have a vector of called here absolute do you way
0:04:14that is actually
0:04:16showing the direction from weight can scope
0:04:18we can have actually a a direct bus more that of the mixture
0:04:22in that
0:04:23each of the colour not the mixing matrix is actually
0:04:26the steering vector or corresponding
0:04:28to the direction
0:04:29from we just rolls is coming like here
0:04:31so we see
0:04:33uh we see a
0:04:36so this is a microphone vector or as was on this is actually just ring vectors so this scroll on
0:04:40this on the you one
0:04:42these one source of a column on the one depending on different
0:04:45during vector
0:04:46when we do the separation actually
0:04:48we recover
0:04:50the match that when we inverse it we have a and to make
0:04:53we can have an estimate of these and the problem of permutation
0:04:56is actually that
0:04:57if those are steering columns that would be steering column
0:05:00which are muted meaning that
0:05:02here for example this
0:05:04for vector
0:05:05is not the first one it can be wanting to me
0:05:08here is how the permutation appear on this and the colour of the separated met
0:05:15actually knowing this we can
0:05:17this so this is a paper
0:05:19from a
0:05:19somewhat a
0:05:21i a key mckay
0:05:22lacking of sound
0:05:23it's a
0:05:25we can see do the
0:05:26the racial of the element in the colour on
0:05:29to get
0:05:30actually is these but so we see that
0:05:32this we should know because you is
0:05:33so microphone we know the position
0:05:35and this is up to G away
0:05:37actually if we are know in a region as of argument function
0:05:41to G some meaning that Z is for example in minus spy
0:05:46we can we call work Q
0:05:48from the racial of the and so the constraint that it put on the sense or is also that
0:05:54he so this is mainly the distance between some sense as and this is absolute do you a
0:05:58and this is more or less and angle between so
0:06:01and for the actual do used like used and and go
0:06:04between this vector
0:06:06and this one
0:06:07and depending on this angle for different frequency for them for for low frequency to people are
0:06:13actually we we have a and yeah one we are over
0:06:16when we have a spacing between the microphone
0:06:19that is over the blue curve and its independent of the angle between those two
0:06:23if we have a linear rates very easy
0:06:26but for a cherry or a have for the first check or or or or even a
0:06:29different form
0:06:30is not so easy to know which pair we have
0:06:33some ideas in because we don't know usually Q
0:06:35so meaning
0:06:36and if the frequency of meant we see that
0:06:38the sense so has to be really close
0:06:41for some fixed
0:06:43send so this stance so for fixed microphones some distance between the sense or
0:06:47we can actually
0:06:48we have also limits still up to the blue curve for twenty sent to meet that will you have
0:06:52and the in so for frequency or or
0:06:55some value and depending on the angle
0:06:57so the thing here is that
0:06:59when we have a
0:07:00a linear rate
0:07:01and we can just the
0:07:04smaller a pair
0:07:05these these are
0:07:07these are constant
0:07:08but but we have we may have an array a to send you to always
0:07:12but for a a or real rate maybe we want
0:07:14we may have a
0:07:15using this power can use this all the per
0:07:18this one this one they have very different
0:07:24you there is no using a a good solution that was proposed is actually to start
0:07:28or is a value for the column
0:07:30and to stack all the position or a to make a a big uh matrix
0:07:34and we we get this kind of equations where i will
0:07:39direction of a right but we want to find should do the solution and
0:07:42we can have at least squares solution so this is still
0:07:44so walk with somewhat what was a also
0:07:48and so why do we do that is that
0:07:52blind just suppression some matrices
0:07:55from there we can get some direction of arrival
0:07:58if we see for the different frequencies the direction of a right so this is
0:08:01the permutation we see that in this
0:08:04a C is a for example the first component it's coming from
0:08:07sometimes so the blue is a first component it maybe coming from this direction
0:08:12was this direction sometimes actually if we are able to seize direction
0:08:16permit attack to that
0:08:17we will so the permutation problem
0:08:22here are in this paper we i interest in the case where they're especially i'm guessing meaning that
0:08:27this relation
0:08:28for some frequency else for some sense so pair is not
0:08:32it's not longer very fine
0:08:33so meaning that actually
0:08:35we can introduce a some values so this are in take it actually those in check your
0:08:40so that we have this relation the
0:08:43this is not true but this one is true we put some that have a you here so that this
0:08:47one is two
0:08:47so to show eight for example for frequency of
0:08:50two thousand hz and
0:08:52one sickle a rate
0:08:54this is actually
0:08:55so more less the distance between the microphone and the angle between
0:08:59so a
0:09:02between the two microphone and the absolute do way
0:09:05so in red
0:09:06we have
0:09:10okay okay
0:09:10so we see that it's over
0:09:12P and minus P and undermine the P
0:09:15so these are the
0:09:16but you in take a value
0:09:18so but you by by to by
0:09:20that we have to add
0:09:22the green curve
0:09:23and this is actually the difference of the two that is always
0:09:27in the boundary
0:09:28so this actually
0:09:29this one the press this one gives this
0:09:34we can also add this
0:09:35to the racial of the call on meaning that we have to those term as before but it also a
0:09:42difference is in figure here
0:09:44so if we stack
0:09:46the same way those resort
0:09:47we also get and
0:09:49equation that should be verified
0:09:51by a known direction of ball but also
0:09:55we have those
0:09:56in a value that appear in the second part C a and here's was value also unknown no
0:10:01a tree if if there is no single like this
0:10:04this would be this is it's simple miss we with special and this
0:10:10so here is just to show that uh
0:10:12we can transform this equation question with a Q and that of that appear we can transform it actually to
0:10:17would be to a equation that is only we i'd
0:10:19by this down that
0:10:21so yeah yeah change of bits and the patients as the first part here i've we name it G G
0:10:27it depend actually
0:10:29of this
0:10:31but use
0:10:33which we can see depend
0:10:35on the
0:10:37estimated things
0:10:38from the metrics so it you and of the
0:10:41colour on so G and of the frequency
0:10:44this part
0:10:46is just
0:10:48depending of the sense geometry and see every name E
0:10:51this one i mean
0:10:52you we see later Y
0:10:53so we have this equation
0:10:58the proposed so think is that we would like to solve as a question and
0:11:02to find actually a what delta
0:11:04the things that this equation depend we have a a actually the symmetric is not for run so that is
0:11:09an infinite number of solution
0:11:11we can
0:11:12for example simple get a solution with a minimal norm like this
0:11:16but we have a our interest in an indigo solution to the equation
0:11:20which is different from this one show
0:11:25i was talking about sparsity also in the introduction because actually
0:11:30we can know that this that that
0:11:33have a new and trees
0:11:35for the rows that correspond to
0:11:37microphone pair without and the other thing for the given frequency
0:11:41so it means that
0:11:42if we have a good initial guess but i'm pretty phi can have a initial guess like is this
0:11:47the difference between this initial guess
0:11:50and the value "'em" searching should be sparse so
0:11:52here have for an example
0:11:54that's say that
0:11:55i mean rest in
0:11:56so frequency
0:11:58two thousand and
0:11:59one hundred hz and they used to seven hz one for guess
0:12:03so these are actually here
0:12:05that that that
0:12:06in green
0:12:08and they quite similar so if we look as the difference the different is nearly always with zero
0:12:12except for some value so it's are like to be them the one
0:12:17well yeah i mean interest in Z is
0:12:19actually because
0:12:21we want to tree to like in many at the reason that solve the permutation we get we start from
0:12:26the lower what frequency where a less permutation and we
0:12:30or to higher frequency to sort of them so he the same we we
0:12:33we use the previous
0:12:34frequency be as a meaning that
0:12:37for the been G
0:12:39of for the bin F
0:12:40at the can and J we use the result we got for the previous B and we start from the
0:12:44low work
0:12:45for each case we be so for a what a question so
0:12:48with the initial guess
0:12:51and this is actually a re the solution so the the real solution of is minimal no
0:12:55then we take actually is the rounding of distribution to have
0:13:00to have an intake
0:13:01and the goal is actually we would like to have this X close
0:13:04to the indigo so that that were rounding give this it
0:13:08this is why actually
0:13:10we were trying to have this initial guess that is close
0:13:14quite spots
0:13:16this is and
0:13:17oh first approach S i guess a it's in better solution to find a sparse
0:13:22direct you sparse solution to this the question but
0:13:25i i didn't the E D
0:13:28when we have a solution that to we can define the residual which use actually how good the question was
0:13:32source to this is just actually
0:13:34the difference between
0:13:37the solution of the question
0:13:39and uh well
0:13:41what is the error
0:13:42of that
0:13:44uh a what where steep
0:13:47how do i realise is to permutation resolution
0:13:50it simply that
0:13:52when we are
0:13:53moving from the frequency being F minus one to the frequency F
0:13:58you there is a permutation say of the column K
0:14:01and G
0:14:04in the
0:14:05been F we we use this equation some innings that
0:14:08we want to solve this equation to find X
0:14:11that is solution of this which question we see that
0:14:14we are in the row K
0:14:16what the column catch
0:14:18and so we have here
0:14:20the K index but because of the permutation here
0:14:23we will be using
0:14:25the guess from that correspond to a knows or colour
0:14:30if is two solution
0:14:32for the current J and K are quite different
0:14:35we would not find a closing take a solution
0:14:37meaning that the residual so the error or on the equation
0:14:40would be large
0:14:42how large
0:14:43this we depend also of how much uh a noise there is no what estimate meaning
0:14:48uh if i where bss it's the walk where on not
0:14:52we can of the first way would be to compare as residual to a threshold
0:14:56decide if that was a permutation on not
0:14:58but this is not so easy because
0:15:00of the noise finding this threshold is not is
0:15:03and as a solution is
0:15:05we did this for the row K we can do for the road change so we we have another reason
0:15:10or with
0:15:11and X K and compares the to to decide if there was a permutation on not this will be a
0:15:15seem you know to was a ms so that compare
0:15:18uh the direction of a right
0:15:20the problem is that some
0:15:22when especially when the absolute
0:15:25direction of arrival for the colour a quite close
0:15:28so as to value may be close meetings that actually
0:15:32the the reason you're be small so we we not detect the permutation
0:15:36just make the read you are in such case
0:15:38we have actually to compute
0:15:42so absolute you way
0:15:45we we have to compute that it'd the U A and try to use a
0:15:47this up to do you a
0:15:49with the one from the previous frequency to so
0:15:51this problem
0:15:52so in this case the mess what got to bit
0:15:54close a tools or pro
0:15:59when we can see the
0:16:00uh for so this is the kind of post processing to solve this problem we first consider a all the
0:16:05frequency bins where a all the row we have small residual
0:16:10meaning that
0:16:11we we did that all the frequency be and we can see that or the one for which
0:16:15we had directly
0:16:17small residual
0:16:18for these we estimate
0:16:21the sum absolute doa is
0:16:23we have a this absolute you always along the frequency
0:16:27and for or the are as and
0:16:29we compare
0:16:30estimated you eight to this average about you and do the clustering according to
0:16:35so this is very similar ads
0:16:37a to be to the a approach
0:16:38where you find some direction of a one and you close to the
0:16:45here i i haven't for that then the lee and you now one D
0:16:48some simulation results once to make the data
0:16:52where we can see there are sixteen microphone
0:16:54uh a a race was sick or microphone so
0:16:56the drawing that was before
0:16:58it has a diameter of thirty thirty for one cent to majors
0:17:01so we can see sixteen Q has something frequency and five hundred to
0:17:06so this is how i model that of is estimated colour
0:17:10so we have
0:17:12oh this would be the problem if they don't not
0:17:14uh absolute do way
0:17:16so like these
0:17:17and i put some error or so
0:17:19on the angle
0:17:21that is
0:17:21that are uniform in gamma
0:17:23in a
0:17:24in uh
0:17:26as they are uniform on the
0:17:29interval mine got come model so meaning that a some error or on the direction of arrival
0:17:35and there is also some at know
0:17:37showing the error or the estimation of this one
0:17:42for some of the frequency
0:17:43so a random permutation of the core of the core and the percentage
0:17:48D of these frequency bins uh
0:17:51permit it
0:17:52so how about we measure the performance
0:17:56the present age of frequency been with adequate permutation after part
0:18:00after the processing
0:18:02and also as ever all
0:18:03on the absolute there or do you a
0:18:07so i that is to experiment vector in the first one we try to see in france
0:18:11of the different
0:18:12a parameter
0:18:15the additive noise
0:18:17the a row and the end goal
0:18:19and that
0:18:20is um the racial of for a permit it
0:18:23uh call
0:18:25and the second experiment
0:18:26i'm right and this is done for some fixed the absolute do you a and
0:18:30we have a rate the resort to a and some uh
0:18:32a certain number numbers
0:18:33independent run
0:18:35the second
0:18:37we want to actually
0:18:39okay so this is Q to actually
0:18:41we see we want to see the difference between
0:18:43is the angle between the absolute to you a how it's
0:18:46it seems finance
0:18:47on the result because
0:18:50this is critical especially
0:18:53this is what create this kind of problem
0:18:59so this is a result of the first experiment so
0:19:04one case we just compare the residual so meetings that there is no prob
0:19:07post processing we just compare the with you're we don't compute a
0:19:12here in this case we don't compute any direction of rival to result
0:19:16to resolve the permutation the second case
0:19:19we actually do the post processing that much
0:19:21propose that
0:19:23in this
0:19:24we make a second pass
0:19:26to get
0:19:27the direction of arrival to permit the beans that maybe had a
0:19:31it's a first call and here we see actually
0:19:34this is a in france of that the even noise
0:19:38okay okay
0:19:40on the permutation ratio and
0:19:42on the or on the deal so we see act actually
0:19:46that is an improvement when we do
0:19:48uh the post processing
0:19:50for the number of limitation that the result and
0:19:53and T it's quite constant and you
0:19:56a set and
0:19:56i'm not of uh it's quite robust yeah to the
0:19:59at no
0:20:00in the second experiment
0:20:02we we see the in france of the air or on the angle of that you way
0:20:06so no
0:20:07how there is this dispersion of that do you S so this would be the case in a room where
0:20:10there is more rubber break more less reverberation
0:20:15we see that for this one same
0:20:17and you a and there were all of around the
0:20:20fifteen degree they is not so much decrease but that does that we see a sharp you decrease meaning that
0:20:25it's quite sensitive so
0:20:26this may be a problem
0:20:28for high reverberant room
0:20:31in the third colour on we see actually
0:20:33so different uh
0:20:35amount of permit date
0:20:37a column before processing so
0:20:39how we can friends a result so we see that
0:20:42with a post processing it's quite clean
0:20:44well as
0:20:45it decrease faster are really with no post processing that meaning that
0:20:48the more corn and we have to that that the less
0:20:50good we are at
0:20:52uh something the or where as it's quite a linear so
0:20:56and same so what so ever and go to increase very fast with the post
0:21:02the second experiment is actually we have the different of us so this is the angle between
0:21:07is a two steering vector corresponding to the colour on
0:21:11and we see how this in france the mess it so the first thing is that in this case
0:21:15so two curve a very different meanings that
0:21:18the if we just use the residual
0:21:21we don't
0:21:22we don't try to use
0:21:23to get is up to you the and self
0:21:25we need actually to have absolute the you a that that separate separated to to reach
0:21:31acceptable results
0:21:32so meaning like
0:21:33there should be at least thirty forty T we between them whereas
0:21:37with a post processing
0:21:38this is not the problem we and Q fifteen degrees here it was to walking
0:21:46to compare that would say that we we consider this problem and the case of special i'm guessing
0:21:51we we introduce a kind of model for the special at using to solve the permutation so
0:21:55one thing that has to be done in sector is now my solution to the equation so to find my
0:22:00sparse solution it's a
0:22:01uh how to say
0:22:03a very easy approach that maybe some that to to do
0:22:06and of course to apply these to real data and compare it with is was of missile
0:22:19we have time for only