0:00:13good afternoon everybody
0:00:15i hope you're enjoying your in frog as much as i am
0:00:18today were going to talk about doa estimation with a vector sensor in the presence of noise yeah away being
0:00:24direction of arrival
0:00:25this work what was done together with mental have it's and strong again
0:00:31so for a bit of motivation
0:00:32a conventionally typically if we wanna do doa estimation or localisation
0:00:37we use an array of sensors which have a certain amount of spacing between them
0:00:41um and other hand uh vector sensor is entirely a compact sensor it doesn't require spacing that's it's cheap advantage
0:00:49um there was some previous work done and the estimation that
0:00:52we did our own research and we found out that uh we prove developed a method which turns out to
0:00:57a previously proposed methods
0:00:59and it also includes a perform
0:01:03okay so the question is a bit of background so what it actually is a vector sensor i think the
0:01:08best way to answer that
0:01:09would be to compare a vector sensor
0:01:11with a typical microphone
0:01:13so it table microphone has one channel
0:01:16a single channel and usually doesn't have directivity may have their activity um would be amount of also of full
0:01:23yeah a is actually a fear
0:01:25and it doesn't make any difference of i was you can of the microphone from a C this direction
0:01:30or from this direction
0:01:31the signal which is received by the sensor would be identical
0:01:35and other hand the vector sensor has four different channels
0:01:38the first channel is a multiple
0:01:40so that has um
0:01:43uh that doesn't have any directionality
0:01:45the other three are die people's a dipole is
0:01:48um is much in of the first presentation
0:01:50is of uh a three dimensional figure eight
0:01:53it's sensitive at the it's is more sensitive it
0:01:56two apps the ends of the uh of its directivity
0:01:59and the dipole as are oriented according to the X
0:02:02Y and Z axes
0:02:06now um how can we construct a a vector sensor
0:02:10there are two i wanna elaborate so much but they're to to uh do to uh i mean ways we
0:02:14can do that
0:02:15one way is with particle use answers
0:02:18another way is with the difference of microphone array
0:02:20and both of these options are
0:02:23uh off the shelf meaning you can uh by products which do that
0:02:28okay now we're going to give some notation
0:02:30um so we have four different channels
0:02:33the uh
0:02:35um we want key is going to be used to describe the manifold channel
0:02:39the X
0:02:40view Y and V Z will respectively present that three different
0:02:44uh they pose
0:02:45where we can
0:02:46uh we can uh
0:02:48so is a brief is P and a vector V
0:02:50uh another property which is worth mentioning and we're going to have rate that
0:02:54and that further there is
0:02:55that if we take a linear combination of these for different uh channels
0:02:59then the then the uh effective of the beam pattern is gonna be known as a limo on which is
0:03:04neither a man upon or a dipole somewhere in between
0:03:07and that gives us a little bit of flexibility which were going to exploit lee
0:03:13okay next we're going to discuss uh the assumptions as you made when we did our analysis
0:03:20so first um are source is going to be denoted by S of a and a and being the time
0:03:26um the source is is assumed to be in the far field and therefore the wave when it arrives at
0:03:31the back sensor
0:03:33can be considered to be a plane with
0:03:35the doa is going to be um represented as a unit vector you that's the red vector which points from
0:03:41the vector sensor chords the um there's this the sort
0:03:45and our goal is going to be to try to create a you hat we're gonna try to create a
0:03:49an estimator which is going to closely approximate
0:03:51the true you
0:03:53um if we didn't have a noise in this would be very straightforward we want have an estimate we could
0:03:58uh provide an accurate um
0:04:01but there is some noise
0:04:02the noise is we're dealing with that is
0:04:05uh has uncorrelated components there for channels there all uncorrelated
0:04:08um such a scenario can arise um either or with device noise
0:04:14coming from the sensor static
0:04:15roll trendly an isotropic uh feel
0:04:21okay that we're gonna give a bit of background for the at so we have a dipole and the dipole
0:04:26the that three die poles are have a step fixed orientation
0:04:29we might wanna die which isn't directed on the x-axis you might wanna die which is directed somewhere in between
0:04:35so technically you could mechanically steer that but that's not a a uh usually a viable solution
0:04:40a much better solution would be electronics year
0:04:43which if you if we take a linear combination of the three dipole poles
0:04:47uh then we can and the the coefficients of that um
0:04:51the coefficients would be a uh a unit vector Q
0:04:55then we can create and we can bind them that we create a new dipole a virtual dipole
0:04:59which has an orientation so here we have a V X
0:05:01and the Y
0:05:02two different i poles
0:05:04we weight them according to the unit vector here what and we produce a new virtual dipole able
0:05:08which has an orientation and we can wear that we have any limitation we can uh
0:05:12point the dipole to any direction which we
0:05:17okay so the first if we have a uh the first degree of flexibility which we have is
0:05:21the orientation of the dipole
0:05:23but we don't have to can as of the people's we can combine the dipole with them man a pull
0:05:27as we do here we have a L four times amount of multiple
0:05:31one minus alpha
0:05:33times the virtual dipole
0:05:34and that produces what we called earlier all muscle i'm or a hybrid a a combination of a man upon
0:05:39a dipole
0:05:40and i give us a whole family of different beam pattern
0:05:42so here for example for L pose one we have a remind able
0:05:46for L C zero we have a dipole i'll equal zero point five
0:05:50we have a um a card you i
0:05:52and we can also have different uh uh so uh subcarrier a super cardioid uh configurations
0:05:57are we won't have to see later which one is an optimal but we have this so degree of looks
0:06:01ability we're going to utilise
0:06:04okay no one more before we get to our method were going to have one more definition and that's
0:06:08steered response power
0:06:10so the to response power first we take a a a beam pattern L four
0:06:15and this case we picked the uh uh to limits on
0:06:18which um
0:06:20which is uh super cardioid
0:06:23next we steer into a a certain direction Q Q been unit vector
0:06:27and after that we take over and
0:06:29upper case and
0:06:30uh samples we measure the power
0:06:33that's uh what we have over here
0:06:35uh a four times
0:06:37the man poll one myself of times the virtual by paul
0:06:40uh we measure we measure the but uh we measure this power hard or the average energy
0:06:44and that's known as the steered response power
0:06:47and we can assume that as we approach
0:06:51as Q what coaches the true direction of arrival we're going to tend to have
0:06:54um a how your power level and that's what we that's what we're gonna explain in our met
0:07:00so now are gonna get to a are the method is self we're going to do it was we're going
0:07:03to measure the S R P the steered response power
0:07:06for a given out uh we have to determine which of uh is best but we can do for any
0:07:10also so between zero and one
0:07:13and we want to find the unit vector for which the S R P
0:07:17is maximal so theoretically we can say that we're actually don't as we're going to perform a search like a
0:07:21crust units are we have to
0:07:24trained to to be at to have a unit norm
0:07:27uh we want to perform this to perform a search we're going to find the direction of maximum power
0:07:32ideally we don't wanna have to do would X uh an extensive search that would be uh a very time
0:07:36consuming something or or consume resources
0:07:39so we're gonna see that we can find some analytical uh solutions also
0:07:45now we can take this uh
0:07:47we can take this and we can uh expressed it we can do some at not but a manipulation
0:07:52and represent present our estimator in terms of
0:07:55are V V and R P V
0:07:57are V V
0:07:58is a cross covariance matrix
0:08:00of the uh dipole and
0:08:03R P V is uh vector
0:08:06of the cross correlation between the out and dipole elements
0:08:09so R V V is three by three
0:08:11our P V is to be by one
0:08:13this is a uh an estimate of the cross-correlation this a sample cross correlation
0:08:18and we can uh and our term here we have a we have an optimization problem
0:08:22here we have a a a we have two terms here the um
0:08:26a linear term because hugh a clear appears only one
0:08:29a second term is you transpose are E Q
0:08:32and that's a um that's a quadratic term
0:08:36and we have a unit constraint
0:08:37and we're gonna try to solve that
0:08:39and first we're gonna it's to uh extreme cases
0:08:43and um
0:08:46we actually we before i am for than just going to say that we we're gonna it's and spec two
0:08:49cases we're going to see that
0:08:51these cases can be solved analytically
0:08:53and the solutions which we find a actually matched estimators which were um
0:08:58previously proposed so we're we're are estimator a special case of the of two previously um proposed them
0:09:05and the first case
0:09:07we choose a it to be
0:09:08close to one it's not quite a of close close to mine of an apple has no activity
0:09:13uh if we have a a a a a half a being one minus
0:09:16on we have some approach is zero there is some directivity
0:09:20uh we have
0:09:21the problem to the problem can be turned into T you had you arg max
0:09:26of you can suppose R P V
0:09:28with the you constraint
0:09:29so we have a linear constraint a quadratic that's a linear uh a your term
0:09:33quadratic constraint
0:09:35the solution can be shown to be
0:09:37uh the the vector R P V normalized
0:09:40and tie surprise when we when we uh when we did the research we found out that this is a
0:09:44this method
0:09:45or this estimator was proposed earlier by dave is by now right party in
0:09:50at seven nineteen eighty four
0:09:51but they use the a different framework so our framework actually matches them for uh for one particular case
0:09:57another case is where equal zero of equal zero we have a dipole directivity
0:10:02we have one quadratic from
0:10:04we have a uh we have you constraint
0:10:06we can sell this analytically in the solution as going to a
0:10:09you have
0:10:10is the eigenvector vector which corresponds to the largest eigenvalue of R V V that's also a solution
0:10:16and this was was also propose earlier
0:10:17so uh we show that our estimator is a general is to um
0:10:22previously proposed estimator
0:10:25but now we have a more interesting case weapons if we don't have a man pull or a dipole
0:10:28we have something in between
0:10:30so now we're not content also you people zero or close to one or want a also be somewhere in
0:10:36uh the problem we have now it's you transpose R P V plus one self over to you can suppose
0:10:41are V Q
0:10:42this both the linear and a quadratic term
0:10:44a quadratic constraint
0:10:46and to the best of our knowledge as there is no and local solution to this
0:10:50so we tried a numerical method
0:10:52and we use the method of
0:10:54um gradient to
0:10:56the gradient is computed below
0:10:59uh just one probably a to send take the if if we go in this in the direction of steepest
0:11:03ascent as we take a step we're going to tend to
0:11:06uh leave the unit sphere we're gonna reach the constraints
0:11:09and we have to solve the problem with the constraint
0:11:13so this is our proposed algorithm
0:11:16uh we start with that we see start with initial guess
0:11:19we step in the direction of the steepest ascent
0:11:22afterwards we normalize
0:11:24the we normalize the this uh the vector Q
0:11:29and we P this process over and over again it's so we have convergence
0:11:33um it's not very tank a time consuming
0:11:36and that's uh that's um but and we can use it to solve it for any uh for any L
0:11:42so now that we have a a the we have a car we are now we see that there two
0:11:45we have a a more general math that would like to be able to compare the results and C
0:11:49how they perform so we need some type of
0:11:53we need some type of uh term to evaluate how well are uh
0:11:58estimate or
0:12:01we define the chair or the term as been defined earlier and error
0:12:05angular error means the following and you have uh you points and one this is the true you
0:12:10and we have another uh vector you had
0:12:13we take a angle between those two vectors
0:12:15and that's the angular are and that provides a well the information we need to know how actually one particular
0:12:21estimate is
0:12:22think is where had doing what on particular so we're doing with random processes
0:12:26we have to
0:12:26the the actual angle are for one uh predictor scenario doesn't mean that much
0:12:31uh so we want to use based and anger are are we had the mean square anger are uh is
0:12:36defined that was used i think was first proposed at the last of my now by nor i nineteen ninety
0:12:43the mean square and there are is the angle are squared
0:12:46times and as an approach isn't and
0:12:50okay so now we have a a now we have a way to measure to evaluate are the different uh
0:12:54estimators we're we perform the monte carlo simulations
0:12:58so in the single trial first we pick a
0:13:01uh you know vector you as a two D O Y
0:13:03then we generate from a gaussian distribution
0:13:06the signal uh this the signal components signal component the noise components for the four channels
0:13:12based and uh based and uh these different uh the different symbols of we generate we can calculate
0:13:18uh you have
0:13:19and we don't have eight when you at we can take you had for L for ranging from zero
0:13:23uh up in one
0:13:27based on the base and the single trial we have a a a list of angular errors
0:13:30and we would be this trial over and over again we did it a hundred thousand we did hundred thousand
0:13:34independent trials
0:13:35and then we have a we can tie really this to mangle are are is and based on the average
0:13:39we can
0:13:40calculate the sample mean square anger are
0:13:43and we can use that to compare the different estimate
0:13:48so for a particular example here are there is also which we obtain
0:13:52uh that the results depend on the parameters meeting it's this depends on the signal to noise ratio depends that
0:13:58there two types of noise there's um
0:14:00multiple noise and not and i noise
0:14:03uh this is one particular uh uh there's one particular notation
0:14:07the x-axis over here
0:14:09that twenty eight
0:14:10uh uh out of the a parameter
0:14:12the y-axis is the mean square or are
0:14:15and we can see that for a equal zero
0:14:18that's the dipole estimator
0:14:20uh the mean square anger hours of a bit high a bit higher than ten
0:14:24for the near multiple estimate
0:14:26it's uh a closer to well
0:14:29we have in between as more interesting we have as them between as we have a shape which was like
0:14:33a parabola
0:14:34and it tips comes lower down
0:14:37and that particular value zero point four five it it's the minimum
0:14:40not only does it the minimum but
0:14:42if we look at the red line over here that's the cream a lower bound
0:14:45um we actually
0:14:47uh approach to other crime really rule lower bound so that's the best possible estimate which could be for them
0:14:53based estimator which can be of
0:14:57so uh no i as i said that this is one particular example we word uh if we were to
0:15:01uh do if we did we did a numbers uh simulations
0:15:05and what happens is that for different grammars is the
0:15:08these two sorry
0:15:10um these two edges of the graph
0:15:12uh take different heights
0:15:13but we're always going to have some parabola
0:15:16the proposal was going to have some intermediate value L five and that value is uh a noise emissions
0:15:21a he's the cramer a lower bound which shows that we have a which we that we've in the matter
0:15:26of to now
0:15:28um so to conclude we're going to to say we we use a vector sensor which is an really see
0:15:32we're directive
0:15:34we derive a method for doa estimation and we show that are D away do you estimator actually generalise is
0:15:39to previous methods
0:15:41and we show that if we
0:15:42i have a correct um choice there selection of the parameter L for
0:15:47then we we uh we have perform the previous is and we can obtain the cramer a lower bound
0:15:53um i like to thank the audience and like to thank my advisers
0:15:56sure and not any mental have
0:15:58thank you
0:16:04okay in any questions
0:16:29so actually you you do have a good point and and our future research we'd like to
0:16:34uh research what happens some we take
0:16:36uh a number of the are sensors
0:16:38but we'd also like to note that are certain situations where we have space constraints where we can't actually
0:16:44um employee uh have an array of microphones with uh lemme there they're allowed for half and the spacing
0:16:49and in that case super directive the a super directive beam forming is uh could be a good choice
0:16:55but if we if this gives which is not constrained then we can actually use X are the microphones to
0:17:00which the result
0:17:39right so um
0:17:40think this might uh clarify
0:17:44um here here um is there two different types of uh vector sensors
0:17:49the first sensor here's a sound field microphone and it uses um for different really microphones an order reduce the
0:17:55the for outputs
0:17:56and that could actually be viewed as a classical or there is a certain amount of spacing between them i
0:18:01don't think it's desired but even so it's compact
0:18:04uh the second sensor which we have here is a microphone a microphone is a relatively new device
0:18:10and this is a uh uh a much mars picture it's actually about this both or the size of a
0:18:15match the
0:18:17this over here is the is the multiple sensor
0:18:20a reading rena are the type of sensors
0:18:23so we can pay here uh we can now a very small spacing uh that's the that's the main difference
0:18:29but you could be so it could also be viewed as a
0:18:32very specialised um um part or ranch of a or process
0:18:38it's uh for sensor
0:18:42for sense
0:18:43the for uh
0:18:45one okay
0:19:07okay so we uh i we could we could do this that the way the way which we did our
0:19:11study with we used to a broad single uh uh signal
0:19:16um we could also you we can also do very equivalent an equivalent analysis for a narrowband signal
0:19:21um so and you to how we justify um so so now when would be those the spectral domain
0:19:27um how do we justify this so if we if we had a four don't sensor noise
0:19:32then it makes sense that the for different sensors are not correlate with it
0:19:36but i said something which is a little bit more not quite intuitive but if you check the results
0:19:40um um papers which are this
0:19:43uh if we take uh de a diffuse noise at notes so min we have uh we have noise coming
0:19:47from a unit from a sphere in all directions and it's
0:19:50as a
0:19:52then uh the noise than the four components are also going to be um um uncorrelated they're going to have
0:19:56uh a diagonal matrix of one third of third to third
0:20:00um now that that doesn't times some so to the because it's got be they're all measuring the same signal
0:20:05uh it could be justified from the are you you can we justify mathematically based and the orthogonality of the
0:20:10different um
0:20:11spherical harmonics
0:20:13and if you want i can afterwards um uh refer for you to the papers
0:20:20or questions
0:20:33um X the it that that's a good question it depends um
0:20:41it depends very much and uh and these two parameters the parameter here this is the
0:20:45man noise and this is the type noise
0:20:48if the if they're going to have you noise than the than the answers going to be what you said
0:20:52is going to be a
0:20:53and this particular simulation we had uh the man a noise was slightly higher
0:20:58and then the value that uh that i wasn't half was zero point four five
0:21:03if we were to have a case where the uh um
0:21:06where as the man people or the dipole poles would be have much higher noise and the other than we
0:21:10have a
0:21:11a parabola to be centred maybe either over here
0:21:14or over here
0:21:15so it depends on the scenario
0:21:17a we'd also we we can and uh uh doing uh or
0:21:21oh saying uh more research about that question also so
0:21:33um we didn't view it we we we're i i'm i'm a researching uh um uh
0:21:39a vector of vector sensor but if you were to use maybe a uh i make
0:21:44so i can physically see uh uh
0:21:46see that type of a a approach
0:21:48um i'm just and then i'm thinking of that of "'cause" i and in do the research but instead of
0:21:52having L so we have uh
0:21:54since we have several harmonics we have uh uh and uh much uh which richer
0:21:58uh problem
0:22:00i don't know if to be able to solve analytically i don't know i know will be tracked or not
0:22:04but it it is a and adjusting direction
0:22:08a more class
0:22:31okay so that that's a a good question that and um
0:22:36you we could also say that how come on we have a and your mind a we have one might
0:22:39uh we have um
0:22:40a being one minus so one
0:22:42why is there any directivity the directivity is all is really not that
0:22:46um so the answer i can give was that we did and analytical evaluation
0:22:51and this set near your model pull can actually be decomposed into a spherical harmonic
0:22:57the dc or the uh um the zero order
0:23:00is the dominant one but that one doesn't have any effect the maximisation if you have a hard to the
0:23:04other very large constant
0:23:06and after the can and your so something which changes
0:23:09and the cast and doesn't have any uh affect so even though
0:23:12the man of appears or the than your manifold to have no direct activity
0:23:16when we look at the mathematics we can modify the
0:23:19not directional and we're going to have uh we want to have a beam pattern which does have a
0:23:24uh a considerable amount of directionality
0:23:26but it's true that when you when you look at the when you look at the picture it seems it
0:23:30seems that it seems that but um
0:23:32the mathematical analysis the
0:23:35so was that we do directivity