okay thank as the german um but can i as my presentation which is about because of estimation of room impulse responses and this is a joint back to go of my peachy chi supervisor to be known so that's the goal of this work we want to track a time varying room impulse response so we have the following scenario we have a so as of "'em" which could be moving in time and we have uh the microphone here at that position and you want to estimate the um room impulse response which is in between the source and the microphone and as we have a time varying scenario we want to to track to this room impulse response so down here we have the simple model we have X X of and the convolution of T of end time um can have that as of and as of and we assumed to be known this is to use source signal and X of this microphone signal and you have to estimate this can entire fan uh given some additive caution noise that we have for convenience be we the in a matrix notation a given here um so if it introduce some type it's mattresses as you the feature of and um which present a convolution or equivalently some matrix as of and which just place then we can uh we present this uh a convolution as a metric vector multiplication and and end up that's a signal model that we have a pure and i O problem has solution approach to this um problem is the weighted at least squares estimator i think most a few know um and you do is to minimize to weighted likely an um i introducing some uh forgetting factor which just says that past measurements are not so important and then you observations that i have and this forgetting factor has to lie between zero and one so that's not the idea of this presentation yeah does not improve this way at least squares estimate by incorporating additional information and the additional information at you want to uh in corporate um this will be an energy conservation constraint so basically we have the two questions one i or where the answer this is what is the a prior information that be normally have so we will um assume that the energy that i was see at the microphone has to be less less equal to the energy that i image that the source and um that this is the a a not knowledge that we have a and you want to model this as a constraint into our because if estimation and the second question is how can be efficiently come up with a estimators that a really include incorporate this additional knowledge that we have and and will propose a we look at for different methods to do with this okay so this brings me to the contents well of my talk um it basically has two parts one is the first part is answering the first question um what is the constraint how does to constraint look like and that we can exploit and a second part is about the uh estimate was that i can use to estimate because of three D room impulse response and then i will show you some simulation results and and some up the presentation okay so first of all um how can express this energy conservation constraint and that's has set what you want to exploit is that these signal energy at a microphone has to be less or equal lead a signal energy at the source just written down here and um if i just but this in a mathematical um from i what i need to ensure is that you are clean distance of it and time times as of then to this is the the signal that i have at the microphone has to be less or equal to the i clean distance of my emitted signal and if are just multiply this out and keep in mind that this has to hold for all signals as of and then i see that this metric heat of and france post times you'd of and that it of and was this step it's metrics we had to you room impulse responses um on the diagonal um this has to be nonnegative and negative semidefinite uh sorry this one here and um this condition has as for all signal length as and um if be be um i know that sick conditions for for for particular signal length as zero then we know um from the fact that um the upper left matrix of the negative semidefinite matrix is also again negative semidefinite a note that this condition also holds for all signal can that are smaller than assume so it's for us it's just important to look at the case that the signal length S goes to infinity uh and this is what you do in the following so that this is the constraint that we can exploit and on the next slide i will show you two equivalent representations um of this set so the set constraints all room impulse responses which fulfil this energy conservation constraint and the first representation presentation that we can use this T is an L i representation a presentation if i just use shows slim a um than i see that i can work express this constraint here by this um plot metrics which has to be positive semidefinite and um as it this linear and detail and you each element of this um matrix elements i know that this is not an my and i immediately see that the set um um as a convex set but this presentation is not so convenient for us to include um it did the you because if estimation later that as we will see and the five will use the following frequency domain representation and using the equivalence of the eigenvalues of a band limited toeplitz matrix and the corresponding circulant metrics for the case that my second next goes to infinity um i can come up with the following constraint and i the details um we can find in the people are at paper or or to put them and and backup slide but what you will see and this is also somehow into um that the room frequency response so this is nothing else than just a four you transform of my room impulse response um if i take the magnitude skirt of this this has to be less or equal to one for all frequencies so this basically says is there was not of single frequency or make them um but the remember a room frequency response this larger than one so no single frequency um is um a increased E D is not there's no i'm additional gain for each frequency a and we will see that all or uh three recursive estimators are based on this frequency domain representation and um to really to come up with some can be and um computational from less we will approximate um the question at we had before by introducing a if T and now we just um be you room frequency response at discrete frequencies so we now have on the guy was to pi are divided by have this i was see if that you crawl to M or logic to M which just corresponds to a case of zero padding and how um if and should use some selection matrix P what i basically see um here is that this what i have is that my uh room frequency response at this all my get our now and time instants and has to be less or equal to one as just the constraint that we have that we have to ensure for all L from zero up to basically L over two um and this what what's then he this is basically just a quadratic form that i have so my constraint um so this family of room impulse response um or or a room impulse responses which i and you can seven um i can rupture sent this constraint just by a set of uh a quadratic forms that i have to so okay this was the energy conservation constraint and a the question is how can be incorporate this knowledge into the recursive estimation and therefore i will a proof talk about the channel set up that we have and then um come up with these specific estimators for our we're impulse response estimation problem so channel we have to following problem we have X of and as as of and times it it zero of and so this is the time varying power me that we want to estimate from our observations X of N um given some additive cost noise and B no a priori that might power me that i want to estimate this lies in the subset it it which which is a subset of the um and dimensional um space and smiling motivated it by the at least squares estimator we can we formulate a uh are we can um come up with the signal model is given here if i introduce a the observation vector X of and which just contains all observations that i have if i introduce introduces stacked um model metrics as of an which contains all model match races for all time instants and also introduce um and noise vector set of and which is not not merely the only these stacking of the um the show more terms but also incorporates um this don waiting that i have which just says that past measurements are not so important then um you observations i can come up with the signal model and um if i just for not um don't consider this constraint here then it's well known that the maximum likelihood estimate of this one is just a at least squares estimator and that for a um this is was the motivation for us to consider the stick model and now with the additional constraint that's we know that this it of and has to lie in the subset it okay okay and now the question is how can we applied estimators with the crow computational complexity so if i go back if you just look here to sex of and and also the other terms there are um and growing with time because i just stack all observations that i have into this um large vector X of and and that for the questions how can be avoid estimators with this growing computational complexity and um what we use is the concept of sufficient statistics and a sufficient statistic of the signal model that up we had before so of this linear cost model is um given by the following um and you just look closely at this time that we have few this is nothing else than the maximum likelihood estimator of the equation that we had before and this is just a weighted at least squares estimator or S that's so was sufficient statistic that i can use is um the rate feast squares estimator of the plot problem that i had before and be um um or perhaps first of this um this sufficient statistic um it's about on that this can be efficiently computed by recursive we discuss a estimator and this um because if at feast was um i'm with them or um i dates my sufficient statistic um because it's solves um for the um we discuss estimate in each iteration and also gives me the inverse correlation metrics and these both quantities are will need um for the estimators and that will follow so the idea of now um what i can use as the sufficient statistics in a first step so use the way at least squares estimator and now i'm thinking of different estimators in a second step or different ways to incorporate that knowledge that i have this energy conservation constraint to get a best but estimate then the you wait at least squares estimate of look give me okay and now this is the first estimate at we can use this is um um quite simple we use um the maximal like a estimator of our signal model but this time with the knowledge that might it has to lie in the subset term so and the set of all the room impulse responses which are and cheek can serving um and if you just had the this out and keeping in mind that um we know the sufficient statistics then it's then we come up to the following and quadratic form um we have do the is um um the you body discuss estimate times the correlation matrix and times this button we have to minimize this quadratic roddick from a subject to our constraint so at each time step um we have to check either is the least squares estimate inside my um constraint if yes then this is also my because of um can makes "'em" like that to estimate or if not we have to find the minimum of this quadratic form on the boundary of this um constraints at that i have so now for our a room impulse response tracking problem we know that the constraint is just given by these uh a quadratic form as and that all i have to do is um and the case that my um way discuss estimate is not inside this constraint what i have to do is stand um i have to solve well have to minimize a quadratic form or what quadratic constraints which is a quadratically constrained quadratic program which we can efficiently solve to this is the first um estimate that we can use a second estimate is the recursive if you'd minimax estimator um and it was shown by L or that um this a few minimax estimator has the following form um i know the efficient estimator form Y um supermodel model um then i have to of um the than of fine from you same this metric i'm of N oh sorry and this um that so you of and and M of and and you of and uh fine a i can be found by solving min next problem given here and if you look close that this one um this just quote um depends on the inverse correlation matrix which is also computed by they were "'cause" if at least squares estimator so i can we have here um um it depends on the um sufficient statistics and you the inverse correlation matrix and both are i did by do because if a this squares estimator so i'm still a um i also have few at the problem of the the the the um nice fact the like and relied on this way least squares estimator and the for so now for our a room impulse response tracking problem we do the following simplification to reduce used the computational complexity we just that you of to zero and assume that this i'm of and it's just a bike mot metrics um which just have one meet the all from that we tried to to mess so we saw of those not overall i meant you but just a with this i'll and um to corporate this um set of quadratic ready constraints so i have um it's um one has to first um transform the problem into a the graphic form and then use the as procedure as as written down here and then i can we formulate this optimization problem into a semidefinite program and this is um what you will do um in the simulation results that this was to a because of affine minimax estimator and the slide your um to introduce or to incorporate this um now which show this a or no that i have this constraint is to use the minimum mean squared error estimator where and say um that i have a uniform prior on this constraint set T time and this can be motivated just by D makes entropy principle which just says out of of family of prior densities i should choose that prior or which has a maximum entropy and this is the uniform prior on this set you time um so you using the concept of by asian um sufficient statistics we can come up with the following because as minimum mean squared error estimator which is just written here and again be C it's also just depends on the sufficient statistics and on the inverse correlation matrix so we also have not the problem of and enough and crawling that much or um vector X of and and to have to close um we use rejection sampling um so we use basically much colour integration and the samples from the posterior a formed by um something thing um using rejection sample and the posterior i you can see year it's just a caution densities as a quadratic form and this exponential which is a truncated to the uh features as this part of this at is given by this um constraints at you time and for our room impulse response tracking problem um we sample from a caution and then we have just a check um is this constraint for do not for that sample and if it's not fulfilled food than we just um or we got um this or reject this sample okay this point not to the simulation results and the simulation was that simulations to be have the following problem we have a we want to estimate a room impulse response from a moving source to six microphone so the setup is as following um the source moves along a straight line and we have ten centimetres speech uh between neighbouring position or all we have a let L eleven positions so the source moves in total of one are and the source signal is assumed to be caution um just caution lies of length one hundred example and the room has a size of three by three meters and a i hate of two point three meters and the reverberation time is a a one hundred and twenty miliseconds and use a image source model to obtain the room impulse responses uh with the sampling frequency of to both two has which um if us a room impulse responses of length of one thousand two hundred forty one taps that we have to estimate in each time step um and the all three estimators um as a set use this give the approximation and V use or are sampling uh by roughly a factor of ten so we use um L is you put to to to the power fourteen and for this um because of minimum mean square error estimator use three thousand samples to um approximate these integrals by monte colour integration that just preview T um definitions of signal to noise ratio is just the to noise ratio at the microphone and be use a normalized are emission to show the results here which just just um could difference between my estimate and my to room impulse response divided by the um room impulse response and energy okay first of all um Q the results for the instantaneous estimators which corresponds to a case to better that you zero um and that's a do not have much time left um what you can see all three estimators um are better than you but if he squares estimator and especially the um minimum is could have estimator this small here and if i allow for better on equal to zero um you see that all values gets smaller from um up you to down here and again this is because of minimum is mean squared error estimate which has this uniform prior on this constraint it and gives the best results so um but me quickly some uh up to my presentation um we talked about the because of room impulse response estimation image with with an energy conservation constraint and as you could see in the simulation results this constraint will helps to improve the performance and the because of minimum means good it um i estimator post estimator with the um best to form an and now to um come up with better estimate is a with a um to improve the performance you more you can't with we could now in corporate additional information that you could have um so this would mean additional uh constraints and this is some future work are thinking about thank you very much for a attention you in like i was wondering how what is the physical meaning of this constraints a yeah i mean you talk about a discrete digital signals and uh the the quantity it you call in G well it's somehow how related to real world for you know a G but you know they are apply five say and and don't device in between and a well that that that response that you may i can mess are used using K is and an estimator could have easily like a twenty db gain i have for a particular frequency um so um in the simulation results we see that um this is and as you conservation constraint just gives us um room impulse response estimates which are more smooth than if you just compared to the ordinary weighted feast squares estimates of this this is basically what this constraint performs but now the question as you're right um if there devices in between some amplifiers um you need to have one calibration step before and in a real application to really use this set up because what you really want to have is that this energy at T um a microphone is less or equal to the energy of at the source um so if that would be an empty in between um we we need one calibration step um to really to use this question okay uh one i you what for example if i know that my source will not be close to to the microphone than for example two meters or something like this but know from just um the error propagation what what could be a what what is the minimum um um iteration that double have and and stuff like this could could be incorporate or also there are uh some work um publish already which include sparsity prior so if i know that my um room impulse response just as as some strong pad this is something additional that i can put on top to come up with a better estimate of this room impulse response a how likely would the then mse the normal the and see solution be a a a a of than one because here you have this constraint on the norm of of of the channel yeah a a how like if you have been mention and the receive the power is always yes smaller than the transmit by so how likely would the now of the and then C solution i can would be larger than one in which case you would need your okay even mean how many for example for yeah "'cause" minimum mean square error estimate to i have to do a rich action or a um this depends strongly on the signal to noise ratio and um so for a signal to noise ratio of for example five to be um rough if if if you sample from this posterior density i'm you have to um exclude every second from sample that it it's a draw from this your density for some um at a or a if for the three "'cause" of constrained maximum likelihood estimator i do it know the ratio ratios at all how many times exactly with at least squares estimator and how many times you have to solve the optimization problem in um one E i'm i'm i'm not sure how many times this will happen for this or C M L oh check this yeah left things you can again