a Q mister chain um a break go up up the ladies and gentlemen my name is challenging light and it's my honour to present to you the design of robust the of a broadband beam formers right the microphone gain in base error of right there is day now during the course of my presentation of first find the problem that we want to solve and then now move to discuss about the array geometry that we use for all beamformer design after that out top some up just the of a broadband beamformer structure use and robust design formulation then out a provide us sign example and finally out close my presentation now in speech acquisition as applications such a teleconferencing and audio surveillance it is likely that the speaker a time was to move around the room so you be best if we have a beamformer form that can be steel to the directions of the speaker in order to acquire the speech signal and because we are dealing with a speech signal here which use of rock band we want our be former to have a a frequency invariant property and lastly we want our beamformer to robot um to be robust to microphones errors and i the deviation so in other what's these three card to stick is what we are trying to achieve a our design and in these presentations only meet the discussion in the five few more that you as you move and only but it should be not that it can be these design can be easily tend to the knee a few more than not the original tree that we propose here is what we call the spider a i'm alright right basically is the mouth you reading can are right where if we can see the that for example um uh michael here for the first all of the zero already and then the microphone at the first three and the next ring we can see that you've form the spiral um and the next one is well so that the reason by would call it the spiral i'm a now one of the good properties of this spire i'm i race that's it has uh circular symmetric properties which we we can exploit in order to have a three hundred and sixty degrees during compatibility not the it response of these microphone are rate is given by these equations where the in that C and K we was to the K microphone in the P three and the only got are represent a frequency five represent a as he move and a our P P we present the radius of the P reading a C is the speech of the propagating eighteen where now as for the steerable a broadband beamformer structures i'm we are using the farrow structures at each of the microphone so here we can use the for you by bill from the far all structures do not that has the thought here the all main B and the patient is given by these so basically the side here represent the steering and a and a sign met here is the maximum steering range so basically these inspirations is just to scale the theory and a to be we in the range of plus minus the a five for the five farrow structure no do not mean a beam pattern of these beamformer structure is given by these inspiration which can be written compactly in terms of but the form where the back the a a and a ah C K and and times one long vector wait P C is the number of reading K is the number of microphone re and mine one he's the order of farrow structure and and is the number of tech for the if a a few of now in print um in print car environment we always has some deviations from the idea more and this deviations can come from a so such as mismatches between microphone elements not i knew characteristic of the microphones positions a those in the microphones elements and also low cost get bring effect so if we want to um have uh a robust side of being form of then we need to include some sort of error of modeling into our design so if we include the error modelling into our design then we have oh but to alima response people by D K where cut a here we present the gain deviations for each microphone and gamma we present the face deviations for each microphone and if we use these but to my element rest balls then will have the beam paid them with but the element response given by D which we can use then use this in our design now for the robust design we and to optimize the design based on the mean of the deviations which is basically the S but the value you of how a deviation not if we formulate our design in least ways formulation as we have the cost functions people by these what the compute the side we present the desired steering range the tape make a we present the frequency range of interest and the to Y represent present a as a move in of interest and it is well known that these these where cost function can be written in a compact matrix form given by these where the element is uh the matrix chi bet the P but the B and a D is given by these now the mean error terms is given by the matrix you bob L talk and B but the tar and there are and a into the metric Q you and that the B as you can see here you but with the robust sorry if but with the robust these i'm formulations with do you have this and that form of these ways formulations this mean that this means that to solve this formulations we can still use this then that these squares this i'm at not as and design example suppose that we want to design a being and that can be as a from minus the T six degree tools the T six degree and you has a spectral pass band from two hundred to three thousand and eight hundred uh and you has a fall reading we've the ring id given by these and for each reading we have four five microphone and the order of fire structure is four and the number of you that the is study two now or the microphone gain deviations we use the rally our rally distribution we've sick my was to one and for the slice deviations we use a uniform K a we've the actual minus pile but to to pile over two in order to illustrate last the robustness of our design we introduced two types of perturbation to our um beam patterns the first but the nations we want to model the bees mismatch between michael so here each of the microphone is more than the as the fifty texts band pass if i a people talk and then the coefficients of this field those is perturbed by uniform random form random was given by D so as can be seen here the two graph here we represent a frequency response of the microphones where the first the left hand graph represent the microphone case and uh right graph we present the microphone groups delay now each line here give the response of each of the microphone elements used as can be seen from these two graph we can see that for all the michael form in them once we do not have i you microphone characteristic and a microphone elements ah not match with each other not on top of that we introduced in either the perturbations which is to model the error in the microphone a positions so here the X and Y coding net of all microphone uses is put up with zero-mean gaussian pdf with standard deviation of once same thing we do now this for graph shows the beam pattern all sparse are and spectral response of a beamformer the left column here shows the beam pattern them for the non robust design yeah et minus twenty degree and thirty five degree the right hence that here shows the beam pattern for the robust design the a minus twenty green and the five degree not a few comments here for the robust design we can see that the beam pattern you meant tend the properties that we this or we and to design where um why we have a frequency invariance property and just do you it is you clear that the as a man being at the directions of the steering and a we she's minus twenty degree his case and the D five degree these case now for the non robust design or the only case do see some beam but then a sum men be at the higher frequency and but and the low frequency and the beam but than just blows out in the presence of perturbation so this clearly shows that i'm the is improvement in the robust design no these for beam pattern shows the beam patterns without any perturbation now if the is not that the missions then the non robust design even a a a at the left two graphs in the perform better as shown by the more was quite low where for the robust designs we have a be high side not so this is the tradeoff off between have been a lot was a low side and the robust design so in order to achieve robust design we need to trail the side lot level i as compared to the previous slide we can see that for the robust design the beam pattern is maintain even in the presence of perturbation now to conclude we have proposed a robust steerable broadband beamformer design and the steering capability is achieved by using the pharaohs if few those structure not the robust formulations is more though using stochastic model and it is optimized for the mean performance and from the design example us it is clearly show that um the robust design achieve the tree i to the that initially we set up to solve namely it has a steering actually T and it has a frequency variance properties and lastly it is robust against the perturbations not with these i and my presentation and thank you for your attention questions use use the mac a case you for in simply we don't and some a question is that the is so i look but you corporate thing to be to to control to the microphone again and and arrow can be a a a a uh a peak move for the the raise for instance here you you use a like a spider of um race where yes but yes S is a a a few cable for linear or no the right yes um this that can be applicable to any other the um array geometry that in that we need to modify is the a response functions a on the uh original array three that we you uh we design or i guess okay thank you um as you know we the more the average response it could still happen that for some special division of giving and uh a actually there is a a large deviation for the we won you can comment on that and do you also look at worst case optimization performance okay um if if the error a D too much from the nominal mean a value of than definitely the you know what think for or if you trim a the mean the average performing sparse specific or additional doing and the is it could go can for the specific variation yeah that a diffusion is very large but still on average is good performance but did you also look at its worst case optimization um we have a look at the worst case a P my stations yet currently we only look look at the mean performance thank you and more questions thank you