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