0:00:13 | good afternoon everybody |
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0:00:15 | i hope you're enjoying your in frog as much as i am |

0:00:18 | today were going to talk about doa estimation with a vector sensor in the presence of noise yeah away being |

0:00:24 | direction of arrival |

0:00:25 | this work what was done together with mental have it's and strong again |

0:00:31 | so for a bit of motivation |

0:00:32 | a conventionally typically if we wanna do doa estimation or localisation |

0:00:37 | we use an array of sensors which have a certain amount of spacing between them |

0:00:41 | um and other hand uh vector sensor is entirely a compact sensor it doesn't require spacing that's it's cheap advantage |

0:00:49 | um there was some previous work done and the estimation that |

0:00:52 | we did our own research and we found out that uh we prove developed a method which turns out to |

0:00:56 | generalise |

0:00:57 | a previously proposed methods |

0:00:59 | and it also includes a perform |

0:01:03 | okay so the question is a bit of background so what it actually is a vector sensor i think the |

0:01:08 | best way to answer that |

0:01:09 | would be to compare a vector sensor |

0:01:11 | with a typical microphone |

0:01:13 | so it table microphone has one channel |

0:01:16 | a single channel and usually doesn't have directivity may have their activity um would be amount of also of full |

0:01:23 | yeah a is actually a fear |

0:01:25 | and it doesn't make any difference of i was you can of the microphone from a C this direction |

0:01:30 | or from this direction |

0:01:31 | the signal which is received by the sensor would be identical |

0:01:35 | and other hand the vector sensor has four different channels |

0:01:38 | the first channel is a multiple |

0:01:40 | so that has um |

0:01:43 | uh that doesn't have any directionality |

0:01:45 | the other three are die people's a dipole is |

0:01:48 | um is much in of the first presentation |

0:01:50 | is of uh a three dimensional figure eight |

0:01:53 | it's sensitive at the it's is more sensitive it |

0:01:56 | two apps the ends of the uh of its directivity |

0:01:59 | and the dipole as are oriented according to the X |

0:02:02 | Y and Z axes |

0:02:06 | now um how can we construct a a vector sensor |

0:02:10 | there are two i wanna elaborate so much but they're to to uh do to uh i mean ways we |

0:02:14 | can do that |

0:02:15 | one way is with particle use answers |

0:02:18 | another way is with the difference of microphone array |

0:02:20 | and both of these options are |

0:02:23 | uh off the shelf meaning you can uh by products which do that |

0:02:28 | okay now we're going to give some notation |

0:02:30 | um so we have four different channels |

0:02:33 | the uh |

0:02:35 | um we want key is going to be used to describe the manifold channel |

0:02:39 | the X |

0:02:40 | view Y and V Z will respectively present that three different |

0:02:44 | uh they pose |

0:02:45 | where we can |

0:02:46 | uh we can uh |

0:02:48 | so is a brief is P and a vector V |

0:02:50 | uh another property which is worth mentioning and we're going to have rate that |

0:02:54 | and that further there is |

0:02:55 | that if we take a linear combination of these for different uh channels |

0:02:59 | then the then the uh effective of the beam pattern is gonna be known as a limo on which is |

0:03:04 | neither a man upon or a dipole somewhere in between |

0:03:07 | and that gives us a little bit of flexibility which were going to exploit lee |

0:03:13 | okay next we're going to discuss uh the assumptions as you made when we did our analysis |

0:03:20 | so first um are source is going to be denoted by S of a and a and being the time |

0:03:24 | index |

0:03:26 | um the source is is assumed to be in the far field and therefore the wave when it arrives at |

0:03:31 | the back sensor |

0:03:33 | can be considered to be a plane with |

0:03:35 | the doa is going to be um represented as a unit vector you that's the red vector which points from |

0:03:41 | the vector sensor chords the um there's this the sort |

0:03:45 | and our goal is going to be to try to create a you hat we're gonna try to create a |

0:03:49 | an estimator which is going to closely approximate |

0:03:51 | the true you |

0:03:53 | um if we didn't have a noise in this would be very straightforward we want have an estimate we could |

0:03:57 | actually |

0:03:58 | uh provide an accurate um |

0:04:00 | solution |

0:04:01 | but there is some noise |

0:04:02 | the noise is we're dealing with that is |

0:04:05 | uh has uncorrelated components there for channels there all uncorrelated |

0:04:08 | um such a scenario can arise um either or with device noise |

0:04:14 | coming from the sensor static |

0:04:15 | roll trendly an isotropic uh feel |

0:04:21 | okay that we're gonna give a bit of background for the at so we have a dipole and the dipole |

0:04:26 | the that three die poles are have a step fixed orientation |

0:04:29 | we might wanna die which isn't directed on the x-axis you might wanna die which is directed somewhere in between |

0:04:35 | so technically you could mechanically steer that but that's not a a uh usually a viable solution |

0:04:40 | a much better solution would be electronics year |

0:04:43 | which if you if we take a linear combination of the three dipole poles |

0:04:47 | uh then we can and the the coefficients of that um |

0:04:51 | the coefficients would be a uh a unit vector Q |

0:04:55 | then we can create and we can bind them that we create a new dipole a virtual dipole |

0:04:59 | which has an orientation so here we have a V X |

0:05:01 | and the Y |

0:05:02 | two different i poles |

0:05:04 | we weight them according to the unit vector here what and we produce a new virtual dipole able |

0:05:08 | which has an orientation and we can wear that we have any limitation we can uh |

0:05:12 | point the dipole to any direction which we |

0:05:17 | okay so the first if we have a uh the first degree of flexibility which we have is |

0:05:21 | the orientation of the dipole |

0:05:23 | but we don't have to can as of the people's we can combine the dipole with them man a pull |

0:05:27 | as we do here we have a L four times amount of multiple |

0:05:31 | one minus alpha |

0:05:33 | times the virtual dipole |

0:05:34 | and that produces what we called earlier all muscle i'm or a hybrid a a combination of a man upon |

0:05:39 | a dipole |

0:05:40 | and i give us a whole family of different beam pattern |

0:05:42 | so here for example for L pose one we have a remind able |

0:05:46 | for L C zero we have a dipole i'll equal zero point five |

0:05:50 | we have a um a card you i |

0:05:52 | and we can also have different uh uh so uh subcarrier a super cardioid uh configurations |

0:05:57 | are we won't have to see later which one is an optimal but we have this so degree of looks |

0:06:01 | ability we're going to utilise |

0:06:04 | okay no one more before we get to our method were going to have one more definition and that's |

0:06:08 | steered response power |

0:06:10 | so the to response power first we take a a a beam pattern L four |

0:06:15 | and this case we picked the uh uh to limits on |

0:06:18 | which um |

0:06:20 | which is uh super cardioid |

0:06:23 | next we steer into a a certain direction Q Q been unit vector |

0:06:27 | and after that we take over and |

0:06:29 | upper case and |

0:06:30 | uh samples we measure the power |

0:06:33 | that's uh what we have over here |

0:06:35 | uh a four times |

0:06:37 | the man poll one myself of times the virtual by paul |

0:06:40 | uh we measure we measure the but uh we measure this power hard or the average energy |

0:06:44 | and that's known as the steered response power |

0:06:47 | and we can assume that as we approach |

0:06:51 | as Q what coaches the true direction of arrival we're going to tend to have |

0:06:54 | um a how your power level and that's what we that's what we're gonna explain in our met |

0:07:00 | so now are gonna get to a are the method is self we're going to do it was we're going |

0:07:03 | to measure the S R P the steered response power |

0:07:06 | for a given out uh we have to determine which of uh is best but we can do for any |

0:07:10 | also so between zero and one |

0:07:13 | and we want to find the unit vector for which the S R P |

0:07:17 | is maximal so theoretically we can say that we're actually don't as we're going to perform a search like a |

0:07:21 | crust units are we have to |

0:07:24 | trained to to be at to have a unit norm |

0:07:27 | uh we want to perform this to perform a search we're going to find the direction of maximum power |

0:07:32 | ideally we don't wanna have to do would X uh an extensive search that would be uh a very time |

0:07:36 | consuming something or or consume resources |

0:07:39 | so we're gonna see that we can find some analytical uh solutions also |

0:07:45 | now we can take this uh |

0:07:47 | we can take this and we can uh expressed it we can do some at not but a manipulation |

0:07:52 | and represent present our estimator in terms of |

0:07:55 | are V V and R P V |

0:07:57 | are V V |

0:07:58 | is a cross covariance matrix |

0:08:00 | of the uh dipole and |

0:08:03 | R P V is uh vector |

0:08:06 | of the cross correlation between the out and dipole elements |

0:08:09 | so R V V is three by three |

0:08:11 | our P V is to be by one |

0:08:13 | this is a uh an estimate of the cross-correlation this a sample cross correlation |

0:08:18 | and we can uh and our term here we have a we have an optimization problem |

0:08:22 | here we have a a a we have two terms here the um |

0:08:26 | a linear term because hugh a clear appears only one |

0:08:29 | a second term is you transpose are E Q |

0:08:32 | and that's a um that's a quadratic term |

0:08:36 | and we have a unit constraint |

0:08:37 | and we're gonna try to solve that |

0:08:39 | and first we're gonna it's to uh extreme cases |

0:08:43 | and um |

0:08:46 | we actually we before i am for than just going to say that we we're gonna it's and spec two |

0:08:49 | cases we're going to see that |

0:08:51 | these cases can be solved analytically |

0:08:53 | and the solutions which we find a actually matched estimators which were um |

0:08:58 | previously proposed so we're we're are estimator a special case of the of two previously um proposed them |

0:09:05 | and the first case |

0:09:07 | we choose a it to be |

0:09:08 | close to one it's not quite a of close close to mine of an apple has no activity |

0:09:13 | uh if we have a a a a a half a being one minus |

0:09:16 | on we have some approach is zero there is some directivity |

0:09:20 | uh we have |

0:09:21 | the problem to the problem can be turned into T you had you arg max |

0:09:26 | of you can suppose R P V |

0:09:28 | with the you constraint |

0:09:29 | so we have a linear constraint a quadratic that's a linear uh a your term |

0:09:33 | quadratic constraint |

0:09:35 | the solution can be shown to be |

0:09:37 | uh the the vector R P V normalized |

0:09:40 | and tie surprise when we when we uh when we did the research we found out that this is a |

0:09:44 | this method |

0:09:45 | or this estimator was proposed earlier by dave is by now right party in |

0:09:50 | at seven nineteen eighty four |

0:09:51 | but they use the a different framework so our framework actually matches them for uh for one particular case |

0:09:57 | another case is where equal zero of equal zero we have a dipole directivity |

0:10:02 | we have one quadratic from |

0:10:04 | we have a uh we have you constraint |

0:10:06 | we can sell this analytically in the solution as going to a |

0:10:09 | you have |

0:10:10 | is the eigenvector vector which corresponds to the largest eigenvalue of R V V that's also a solution |

0:10:16 | and this was was also propose earlier |

0:10:17 | so uh we show that our estimator is a general is to um |

0:10:22 | previously proposed estimator |

0:10:25 | but now we have a more interesting case weapons if we don't have a man pull or a dipole |

0:10:28 | we have something in between |

0:10:30 | so now we're not content also you people zero or close to one or want a also be somewhere in |

0:10:34 | between |

0:10:36 | uh the problem we have now it's you transpose R P V plus one self over to you can suppose |

0:10:41 | are V Q |

0:10:42 | this both the linear and a quadratic term |

0:10:44 | a quadratic constraint |

0:10:46 | and to the best of our knowledge as there is no and local solution to this |

0:10:50 | so we tried a numerical method |

0:10:52 | and we use the method of |

0:10:54 | um gradient to |

0:10:56 | the gradient is computed below |

0:10:59 | uh just one probably a to send take the if if we go in this in the direction of steepest |

0:11:03 | ascent as we take a step we're going to tend to |

0:11:06 | uh leave the unit sphere we're gonna reach the constraints |

0:11:09 | and we have to solve the problem with the constraint |

0:11:13 | so this is our proposed algorithm |

0:11:16 | uh we start with that we see start with initial guess |

0:11:19 | we step in the direction of the steepest ascent |

0:11:22 | afterwards we normalize |

0:11:24 | the we normalize the this uh the vector Q |

0:11:29 | and we P this process over and over again it's so we have convergence |

0:11:33 | um it's not very tank a time consuming |

0:11:36 | and that's uh that's um but and we can use it to solve it for any uh for any L |

0:11:40 | four |

0:11:42 | so now that we have a a the we have a car we are now we see that there two |

0:11:45 | methods |

0:11:45 | we have a a more general math that would like to be able to compare the results and C |

0:11:49 | how they perform so we need some type of |

0:11:52 | um |

0:11:53 | we need some type of uh term to evaluate how well are uh |

0:11:58 | estimate or |

0:12:00 | so |

0:12:01 | we define the chair or the term as been defined earlier and error |

0:12:05 | angular error means the following and you have uh you points and one this is the true you |

0:12:10 | and we have another uh vector you had |

0:12:13 | we take a angle between those two vectors |

0:12:15 | and that's the angular are and that provides a well the information we need to know how actually one particular |

0:12:21 | estimate is |

0:12:22 | think is where had doing what on particular so we're doing with random processes |

0:12:26 | we have to |

0:12:26 | the the actual angle are for one uh predictor scenario doesn't mean that much |

0:12:31 | uh so we want to use based and anger are are we had the mean square anger are uh is |

0:12:36 | defined that was used i think was first proposed at the last of my now by nor i nineteen ninety |

0:12:40 | four |

0:12:42 | and |

0:12:43 | the mean square and there are is the angle are squared |

0:12:46 | times and as an approach isn't and |

0:12:50 | okay so now we have a a now we have a way to measure to evaluate are the different uh |

0:12:54 | estimators we're we perform the monte carlo simulations |

0:12:58 | so in the single trial first we pick a |

0:13:01 | uh you know vector you as a two D O Y |

0:13:03 | then we generate from a gaussian distribution |

0:13:06 | the signal uh this the signal components signal component the noise components for the four channels |

0:13:12 | based and uh based and uh these different uh the different symbols of we generate we can calculate |

0:13:18 | uh you have |

0:13:19 | and we don't have eight when you at we can take you had for L for ranging from zero |

0:13:23 | uh up in one |

0:13:25 | and |

0:13:27 | based on the base and the single trial we have a a a list of angular errors |

0:13:30 | and we would be this trial over and over again we did it a hundred thousand we did hundred thousand |

0:13:34 | independent trials |

0:13:35 | and then we have a we can tie really this to mangle are are is and based on the average |

0:13:39 | we can |

0:13:40 | calculate the sample mean square anger are |

0:13:43 | and we can use that to compare the different estimate |

0:13:48 | so for a particular example here are there is also which we obtain |

0:13:52 | uh that the results depend on the parameters meeting it's this depends on the signal to noise ratio depends that |

0:13:58 | there two types of noise there's um |

0:14:00 | multiple noise and not and i noise |

0:14:03 | uh this is one particular uh uh there's one particular notation |

0:14:07 | the x-axis over here |

0:14:09 | that twenty eight |

0:14:10 | uh uh out of the a parameter |

0:14:12 | the y-axis is the mean square or are |

0:14:15 | and we can see that for a equal zero |

0:14:18 | that's the dipole estimator |

0:14:20 | uh the mean square anger hours of a bit high a bit higher than ten |

0:14:24 | for the near multiple estimate |

0:14:26 | it's uh a closer to well |

0:14:28 | and |

0:14:29 | we have in between as more interesting we have as them between as we have a shape which was like |

0:14:33 | a parabola |

0:14:34 | and it tips comes lower down |

0:14:37 | and that particular value zero point four five it it's the minimum |

0:14:40 | not only does it the minimum but |

0:14:42 | if we look at the red line over here that's the cream a lower bound |

0:14:45 | um we actually |

0:14:47 | uh approach to other crime really rule lower bound so that's the best possible estimate which could be for them |

0:14:53 | based estimator which can be of |

0:14:55 | um |

0:14:57 | so uh no i as i said that this is one particular example we word uh if we were to |

0:15:01 | uh do if we did we did a numbers uh simulations |

0:15:05 | and what happens is that for different grammars is the |

0:15:08 | these two sorry |

0:15:10 | um these two edges of the graph |

0:15:12 | uh take different heights |

0:15:13 | but we're always going to have some parabola |

0:15:16 | the proposal was going to have some intermediate value L five and that value is uh a noise emissions |

0:15:21 | a he's the cramer a lower bound which shows that we have a which we that we've in the matter |

0:15:26 | of to now |

0:15:28 | um so to conclude we're going to to say we we use a vector sensor which is an really see |

0:15:32 | we're directive |

0:15:34 | we derive a method for doa estimation and we show that are D away do you estimator actually generalise is |

0:15:39 | to previous methods |

0:15:41 | and we show that if we |

0:15:42 | i have a correct um choice there selection of the parameter L for |

0:15:47 | then we we uh we have perform the previous is and we can obtain the cramer a lower bound |

0:15:53 | um i like to thank the audience and like to thank my advisers |

0:15:56 | sure and not any mental have |

0:15:58 | thank you |

0:16:04 | okay in any questions |

0:16:29 | so actually you you do have a good point and and our future research we'd like to |

0:16:34 | uh research what happens some we take |

0:16:36 | uh a number of the are sensors |

0:16:38 | but we'd also like to note that are certain situations where we have space constraints where we can't actually |

0:16:44 | um employee uh have an array of microphones with uh lemme there they're allowed for half and the spacing |

0:16:49 | and in that case super directive the a super directive beam forming is uh could be a good choice |

0:16:55 | but if we if this gives which is not constrained then we can actually use X are the microphones to |

0:17:00 | which the result |

0:17:04 | i |

0:17:07 | uh_huh |

0:17:14 | right |

0:17:27 | oh |

0:17:30 | okay |

0:17:39 | right so um |

0:17:40 | think this might uh clarify |

0:17:44 | um here here um is there two different types of uh vector sensors |

0:17:49 | the first sensor here's a sound field microphone and it uses um for different really microphones an order reduce the |

0:17:55 | the for outputs |

0:17:56 | and that could actually be viewed as a classical or there is a certain amount of spacing between them i |

0:18:01 | don't think it's desired but even so it's compact |

0:18:04 | uh the second sensor which we have here is a microphone a microphone is a relatively new device |

0:18:10 | and this is a uh uh a much mars picture it's actually about this both or the size of a |

0:18:15 | match the |

0:18:17 | this over here is the is the multiple sensor |

0:18:20 | the |

0:18:20 | a reading rena are the type of sensors |

0:18:23 | so we can pay here uh we can now a very small spacing uh that's the that's the main difference |

0:18:29 | but you could be so it could also be viewed as a |

0:18:32 | very specialised um um part or ranch of a or process |

0:18:38 | it's uh for sensor |

0:18:42 | for sense |

0:18:43 | the for uh |

0:18:45 | one okay |

0:18:47 | okay |

0:18:59 | right |

0:19:07 | okay so we uh i we could we could do this that the way the way which we did our |

0:19:11 | study with we used to a broad single uh uh signal |

0:19:16 | um we could also you we can also do very equivalent an equivalent analysis for a narrowband signal |

0:19:21 | um so and you to how we justify um so so now when would be those the spectral domain |

0:19:27 | um how do we justify this so if we if we had a four don't sensor noise |

0:19:32 | then it makes sense that the for different sensors are not correlate with it |

0:19:36 | but i said something which is a little bit more not quite intuitive but if you check the results |

0:19:40 | um um papers which are this |

0:19:43 | uh if we take uh de a diffuse noise at notes so min we have uh we have noise coming |

0:19:47 | from a unit from a sphere in all directions and it's |

0:19:50 | as a |

0:19:52 | then uh the noise than the four components are also going to be um um uncorrelated they're going to have |

0:19:56 | uh a diagonal matrix of one third of third to third |

0:20:00 | um now that that doesn't times some so to the because it's got be they're all measuring the same signal |

0:20:05 | uh it could be justified from the are you you can we justify mathematically based and the orthogonality of the |

0:20:10 | different um |

0:20:11 | spherical harmonics |

0:20:13 | and if you want i can afterwards um uh refer for you to the papers |

0:20:20 | or questions |

0:20:33 | um X the it that that's a good question it depends um |

0:20:41 | it depends very much and uh and these two parameters the parameter here this is the |

0:20:45 | man noise and this is the type noise |

0:20:48 | if the if they're going to have you noise than the than the answers going to be what you said |

0:20:52 | is going to be a |

0:20:53 | and this particular simulation we had uh the man a noise was slightly higher |

0:20:58 | and then the value that uh that i wasn't half was zero point four five |

0:21:03 | if we were to have a case where the uh um |

0:21:06 | where as the man people or the dipole poles would be have much higher noise and the other than we |

0:21:10 | have a |

0:21:11 | a parabola to be centred maybe either over here |

0:21:14 | or over here |

0:21:15 | so it depends on the scenario |

0:21:17 | a we'd also we we can and uh uh doing uh or |

0:21:21 | oh saying uh more research about that question also so |

0:21:33 | um we didn't view it we we we're i i'm i'm a researching uh um uh |

0:21:39 | a vector of vector sensor but if you were to use maybe a uh i make |

0:21:44 | so i can physically see uh uh |

0:21:46 | see that type of a a approach |

0:21:48 | um i'm just and then i'm thinking of that of "'cause" i and in do the research but instead of |

0:21:52 | having L so we have uh |

0:21:54 | since we have several harmonics we have uh uh and uh much uh which richer |

0:21:58 | uh problem |

0:22:00 | i don't know if to be able to solve analytically i don't know i know will be tracked or not |

0:22:04 | but it it is a and adjusting direction |

0:22:08 | a more class |

0:22:24 | right |

0:22:31 | okay so that that's a a good question that and um |

0:22:36 | you we could also say that how come on we have a and your mind a we have one might |

0:22:39 | uh we have um |

0:22:40 | a being one minus so one |

0:22:42 | why is there any directivity the directivity is all is really not that |

0:22:46 | um so the answer i can give was that we did and analytical evaluation |

0:22:51 | and this set near your model pull can actually be decomposed into a spherical harmonic |

0:22:57 | the dc or the uh um the zero order |

0:23:00 | is the dominant one but that one doesn't have any effect the maximisation if you have a hard to the |

0:23:04 | other very large constant |

0:23:06 | and after the can and your so something which changes |

0:23:09 | and the cast and doesn't have any uh affect so even though |

0:23:12 | the man of appears or the than your manifold to have no direct activity |

0:23:16 | when we look at the mathematics we can modify the |

0:23:19 | not directional and we're going to have uh we want to have a beam pattern which does have a |

0:23:24 | uh a considerable amount of directionality |

0:23:26 | but it's true that when you when you look at the when you look at the picture it seems it |

0:23:30 | seems that it seems that but um |

0:23:32 | the mathematical analysis the |

0:23:35 | so was that we do directivity |

0:23:43 | okay |