good afternoon everybody

i hope you're enjoying your in frog as much as i am

today were going to talk about doa estimation with a vector sensor in the presence of noise yeah away being

direction of arrival

this work what was done together with mental have it's and strong again

so for a bit of motivation

a conventionally typically if we wanna do doa estimation or localisation

we use an array of sensors which have a certain amount of spacing between them

um and other hand uh vector sensor is entirely a compact sensor it doesn't require spacing that's it's cheap advantage

um there was some previous work done and the estimation that

we did our own research and we found out that uh we prove developed a method which turns out to

generalise

a previously proposed methods

and it also includes a perform

okay so the question is a bit of background so what it actually is a vector sensor i think the

would be to compare a vector sensor

with a typical microphone

so it table microphone has one channel

a single channel and usually doesn't have directivity may have their activity um would be amount of also of full

yeah a is actually a fear

and it doesn't make any difference of i was you can of the microphone from a C this direction

or from this direction

the signal which is received by the sensor would be identical

and other hand the vector sensor has four different channels

the first channel is a multiple

so that has um

uh that doesn't have any directionality

the other three are die people's a dipole is

um is much in of the first presentation

is of uh a three dimensional figure eight

it's sensitive at the it's is more sensitive it

two apps the ends of the uh of its directivity

and the dipole as are oriented according to the X

Y and Z axes

now um how can we construct a a vector sensor

there are two i wanna elaborate so much but they're to to uh do to uh i mean ways we

can do that

one way is with particle use answers

another way is with the difference of microphone array

and both of these options are

uh off the shelf meaning you can uh by products which do that

okay now we're going to give some notation

um so we have four different channels

the uh

um we want key is going to be used to describe the manifold channel

the X

view Y and V Z will respectively present that three different

uh they pose

where we can

uh we can uh

so is a brief is P and a vector V

uh another property which is worth mentioning and we're going to have rate that

and that further there is

that if we take a linear combination of these for different uh channels

then the then the uh effective of the beam pattern is gonna be known as a limo on which is

neither a man upon or a dipole somewhere in between

and that gives us a little bit of flexibility which were going to exploit lee

okay next we're going to discuss uh the assumptions as you made when we did our analysis

so first um are source is going to be denoted by S of a and a and being the time

index

um the source is is assumed to be in the far field and therefore the wave when it arrives at

the back sensor

can be considered to be a plane with

the doa is going to be um represented as a unit vector you that's the red vector which points from

the vector sensor chords the um there's this the sort

and our goal is going to be to try to create a you hat we're gonna try to create a

an estimator which is going to closely approximate

the true you

um if we didn't have a noise in this would be very straightforward we want have an estimate we could

actually

uh provide an accurate um

solution

but there is some noise

the noise is we're dealing with that is

uh has uncorrelated components there for channels there all uncorrelated

um such a scenario can arise um either or with device noise

coming from the sensor static

roll trendly an isotropic uh feel

okay that we're gonna give a bit of background for the at so we have a dipole and the dipole

the that three die poles are have a step fixed orientation

we might wanna die which isn't directed on the x-axis you might wanna die which is directed somewhere in between

so technically you could mechanically steer that but that's not a a uh usually a viable solution

a much better solution would be electronics year

which if you if we take a linear combination of the three dipole poles

uh then we can and the the coefficients of that um

the coefficients would be a uh a unit vector Q

then we can create and we can bind them that we create a new dipole a virtual dipole

which has an orientation so here we have a V X

and the Y

two different i poles

we weight them according to the unit vector here what and we produce a new virtual dipole able

which has an orientation and we can wear that we have any limitation we can uh

point the dipole to any direction which we

okay so the first if we have a uh the first degree of flexibility which we have is

the orientation of the dipole

but we don't have to can as of the people's we can combine the dipole with them man a pull

as we do here we have a L four times amount of multiple

one minus alpha

times the virtual dipole

and that produces what we called earlier all muscle i'm or a hybrid a a combination of a man upon

a dipole

and i give us a whole family of different beam pattern

so here for example for L pose one we have a remind able

for L C zero we have a dipole i'll equal zero point five

we have a um a card you i

and we can also have different uh uh so uh subcarrier a super cardioid uh configurations

are we won't have to see later which one is an optimal but we have this so degree of looks

ability we're going to utilise

okay no one more before we get to our method were going to have one more definition and that's

steered response power

so the to response power first we take a a a beam pattern L four

and this case we picked the uh uh to limits on

which um

which is uh super cardioid

next we steer into a a certain direction Q Q been unit vector

and after that we take over and

upper case and

uh samples we measure the power

that's uh what we have over here

uh a four times

the man poll one myself of times the virtual by paul

uh we measure we measure the but uh we measure this power hard or the average energy

and that's known as the steered response power

and we can assume that as we approach

as Q what coaches the true direction of arrival we're going to tend to have

um a how your power level and that's what we that's what we're gonna explain in our met

so now are gonna get to a are the method is self we're going to do it was we're going

to measure the S R P the steered response power

for a given out uh we have to determine which of uh is best but we can do for any

also so between zero and one

and we want to find the unit vector for which the S R P

is maximal so theoretically we can say that we're actually don't as we're going to perform a search like a

crust units are we have to

trained to to be at to have a unit norm

uh we want to perform this to perform a search we're going to find the direction of maximum power

ideally we don't wanna have to do would X uh an extensive search that would be uh a very time

consuming something or or consume resources

so we're gonna see that we can find some analytical uh solutions also

now we can take this uh

we can take this and we can uh expressed it we can do some at not but a manipulation

and represent present our estimator in terms of

are V V and R P V

are V V

is a cross covariance matrix

of the uh dipole and

R P V is uh vector

of the cross correlation between the out and dipole elements

so R V V is three by three

our P V is to be by one

this is a uh an estimate of the cross-correlation this a sample cross correlation

and we can uh and our term here we have a we have an optimization problem

here we have a a a we have two terms here the um

a linear term because hugh a clear appears only one

a second term is you transpose are E Q

and that's a um that's a quadratic term

and we have a unit constraint

and we're gonna try to solve that

and first we're gonna it's to uh extreme cases

and um

we actually we before i am for than just going to say that we we're gonna it's and spec two

cases we're going to see that

these cases can be solved analytically

and the solutions which we find a actually matched estimators which were um

previously proposed so we're we're are estimator a special case of the of two previously um proposed them

and the first case

we choose a it to be

close to one it's not quite a of close close to mine of an apple has no activity

uh if we have a a a a a half a being one minus

on we have some approach is zero there is some directivity

uh we have

the problem to the problem can be turned into T you had you arg max

of you can suppose R P V

with the you constraint

so we have a linear constraint a quadratic that's a linear uh a your term

the solution can be shown to be

uh the the vector R P V normalized

and tie surprise when we when we uh when we did the research we found out that this is a

this method

or this estimator was proposed earlier by dave is by now right party in

at seven nineteen eighty four

but they use the a different framework so our framework actually matches them for uh for one particular case

another case is where equal zero of equal zero we have a dipole directivity

we have a uh we have you constraint

we can sell this analytically in the solution as going to a

you have

is the eigenvector vector which corresponds to the largest eigenvalue of R V V that's also a solution

and this was was also propose earlier

so uh we show that our estimator is a general is to um

previously proposed estimator

but now we have a more interesting case weapons if we don't have a man pull or a dipole

we have something in between

so now we're not content also you people zero or close to one or want a also be somewhere in

between

uh the problem we have now it's you transpose R P V plus one self over to you can suppose

are V Q

this both the linear and a quadratic term

and to the best of our knowledge as there is no and local solution to this

so we tried a numerical method

and we use the method of

uh just one probably a to send take the if if we go in this in the direction of steepest

ascent as we take a step we're going to tend to

uh leave the unit sphere we're gonna reach the constraints

and we have to solve the problem with the constraint

so this is our proposed algorithm

we step in the direction of the steepest ascent

afterwards we normalize

the we normalize the this uh the vector Q

and we P this process over and over again it's so we have convergence

um it's not very tank a time consuming

and that's uh that's um but and we can use it to solve it for any uh for any L

four

so now that we have a a the we have a car we are now we see that there two

methods

we have a a more general math that would like to be able to compare the results and C

how they perform so we need some type of

um

we need some type of uh term to evaluate how well are uh

estimate or

so

we define the chair or the term as been defined earlier and error

angular error means the following and you have uh you points and one this is the true you

and we have another uh vector you had

we take a angle between those two vectors

and that's the angular are and that provides a well the information we need to know how actually one particular

estimate is

think is where had doing what on particular so we're doing with random processes

we have to

the the actual angle are for one uh predictor scenario doesn't mean that much

uh so we want to use based and anger are are we had the mean square anger are uh is

defined that was used i think was first proposed at the last of my now by nor i nineteen ninety

four

and

the mean square and there are is the angle are squared

times and as an approach isn't and

okay so now we have a a now we have a way to measure to evaluate are the different uh

estimators we're we perform the monte carlo simulations

so in the single trial first we pick a

uh you know vector you as a two D O Y

then we generate from a gaussian distribution

the signal uh this the signal components signal component the noise components for the four channels

based and uh based and uh these different uh the different symbols of we generate we can calculate

uh you have

and we don't have eight when you at we can take you had for L for ranging from zero

uh up in one

and

based on the base and the single trial we have a a a list of angular errors

and we would be this trial over and over again we did it a hundred thousand we did hundred thousand

independent trials

and then we have a we can tie really this to mangle are are is and based on the average

we can

calculate the sample mean square anger are

and we can use that to compare the different estimate

so for a particular example here are there is also which we obtain

uh that the results depend on the parameters meeting it's this depends on the signal to noise ratio depends that

there two types of noise there's um

multiple noise and not and i noise

uh this is one particular uh uh there's one particular notation

the x-axis over here

that twenty eight

uh uh out of the a parameter

the y-axis is the mean square or are

and we can see that for a equal zero

that's the dipole estimator

uh the mean square anger hours of a bit high a bit higher than ten

for the near multiple estimate

it's uh a closer to well

and

we have in between as more interesting we have as them between as we have a shape which was like

a parabola

and it tips comes lower down

and that particular value zero point four five it it's the minimum

not only does it the minimum but

if we look at the red line over here that's the cream a lower bound

um we actually

uh approach to other crime really rule lower bound so that's the best possible estimate which could be for them

based estimator which can be of

um

so uh no i as i said that this is one particular example we word uh if we were to

uh do if we did we did a numbers uh simulations

and what happens is that for different grammars is the

these two sorry

um these two edges of the graph

uh take different heights

but we're always going to have some parabola

the proposal was going to have some intermediate value L five and that value is uh a noise emissions

a he's the cramer a lower bound which shows that we have a which we that we've in the matter

of to now

um so to conclude we're going to to say we we use a vector sensor which is an really see

we're directive

we derive a method for doa estimation and we show that are D away do you estimator actually generalise is

to previous methods

and we show that if we

i have a correct um choice there selection of the parameter L for

then we we uh we have perform the previous is and we can obtain the cramer a lower bound

um i like to thank the audience and like to thank my advisers

sure and not any mental have

thank you

okay in any questions

so actually you you do have a good point and and our future research we'd like to

uh research what happens some we take

uh a number of the are sensors

but we'd also like to note that are certain situations where we have space constraints where we can't actually

um employee uh have an array of microphones with uh lemme there they're allowed for half and the spacing

and in that case super directive the a super directive beam forming is uh could be a good choice

but if we if this gives which is not constrained then we can actually use X are the microphones to

which the result

i

uh_huh

right

oh

okay

right so um

think this might uh clarify

um here here um is there two different types of uh vector sensors

the first sensor here's a sound field microphone and it uses um for different really microphones an order reduce the

the for outputs

and that could actually be viewed as a classical or there is a certain amount of spacing between them i

don't think it's desired but even so it's compact

uh the second sensor which we have here is a microphone a microphone is a relatively new device

and this is a uh uh a much mars picture it's actually about this both or the size of a

match the

this over here is the is the multiple sensor

the

a reading rena are the type of sensors

so we can pay here uh we can now a very small spacing uh that's the that's the main difference

but you could be so it could also be viewed as a

very specialised um um part or ranch of a or process

it's uh for sensor

for sense

the for uh

one okay

okay

right

okay so we uh i we could we could do this that the way the way which we did our

study with we used to a broad single uh uh signal

um we could also you we can also do very equivalent an equivalent analysis for a narrowband signal

um so and you to how we justify um so so now when would be those the spectral domain

um how do we justify this so if we if we had a four don't sensor noise

then it makes sense that the for different sensors are not correlate with it

but i said something which is a little bit more not quite intuitive but if you check the results

um um papers which are this

uh if we take uh de a diffuse noise at notes so min we have uh we have noise coming

from a unit from a sphere in all directions and it's

as a

then uh the noise than the four components are also going to be um um uncorrelated they're going to have

uh a diagonal matrix of one third of third to third

um now that that doesn't times some so to the because it's got be they're all measuring the same signal

uh it could be justified from the are you you can we justify mathematically based and the orthogonality of the

different um

spherical harmonics

and if you want i can afterwards um uh refer for you to the papers

or questions

um X the it that that's a good question it depends um

it depends very much and uh and these two parameters the parameter here this is the

man noise and this is the type noise

if the if they're going to have you noise than the than the answers going to be what you said

is going to be a

and this particular simulation we had uh the man a noise was slightly higher

and then the value that uh that i wasn't half was zero point four five

if we were to have a case where the uh um

where as the man people or the dipole poles would be have much higher noise and the other than we

have a

a parabola to be centred maybe either over here

or over here

so it depends on the scenario

a we'd also we we can and uh uh doing uh or

oh saying uh more research about that question also so

um we didn't view it we we we're i i'm i'm a researching uh um uh

a vector of vector sensor but if you were to use maybe a uh i make

so i can physically see uh uh

see that type of a a approach

um i'm just and then i'm thinking of that of "'cause" i and in do the research but instead of

having L so we have uh

since we have several harmonics we have uh uh and uh much uh which richer

uh problem

i don't know if to be able to solve analytically i don't know i know will be tracked or not

but it it is a and adjusting direction

a more class

right

okay so that that's a a good question that and um

you we could also say that how come on we have a and your mind a we have one might

uh we have um

a being one minus so one

why is there any directivity the directivity is all is really not that

um so the answer i can give was that we did and analytical evaluation

and this set near your model pull can actually be decomposed into a spherical harmonic

the dc or the uh um the zero order

is the dominant one but that one doesn't have any effect the maximisation if you have a hard to the

other very large constant

and after the can and your so something which changes

and the cast and doesn't have any uh affect so even though

the man of appears or the than your manifold to have no direct activity

when we look at the mathematics we can modify the

not directional and we're going to have uh we want to have a beam pattern which does have a

uh a considerable amount of directionality

but it's true that when you when you look at the when you look at the picture it seems it

seems that it seems that but um

the mathematical analysis the

so was that we do directivity

okay