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 best way to answer that 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 quadratic constraint 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 one quadratic from 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 a quadratic constraint 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 um gradient to the gradient is computed below 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 uh we start with that we see start with initial guess 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