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