0:00:15 | okay thank as the german um but can i as my presentation which is about |
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0:00:19 | because of estimation of room impulse responses and this is a joint back to go of my |

0:00:24 | peachy chi supervisor |

0:00:25 | to be known |

0:00:27 | so that's the goal of this work |

0:00:29 | we want to track a time varying room impulse response |

0:00:32 | so we have the following scenario we have a so as of "'em" which could be |

0:00:37 | moving in time and we have uh the microphone |

0:00:40 | here at that position and you want to estimate the um room impulse response which is in between |

0:00:45 | the source and the microphone |

0:00:47 | and as we have a time varying scenario we want to to track to this room impulse response |

0:00:52 | so down here we have the simple model |

0:00:54 | we have X X of and the convolution of T of end time um can have that as of and |

0:00:59 | as of and we assumed to be known this is to use source signal and X of this microphone signal |

0:01:04 | and you have to estimate this can entire fan |

0:01:06 | uh given some additive caution noise that we have |

0:01:11 | for convenience be |

0:01:12 | we the in a matrix notation a given here |

0:01:15 | um so if it introduce some type it's mattresses as you the feature of and |

0:01:20 | um which present a convolution or equivalently |

0:01:23 | some matrix as of and which just place |

0:01:26 | then we can uh we present this uh a convolution as a metric vector multiplication |

0:01:32 | and and end up that's a signal model that we have a pure and i O problem has solution approach |

0:01:36 | to this um problem is the weighted at least squares estimator i think |

0:01:40 | most a few know |

0:01:41 | um and you do is to minimize to weighted likely an um |

0:01:45 | i introducing some uh forgetting factor which just says that past measurements are not so important |

0:01:50 | and then you observations that i have |

0:01:52 | and this forgetting factor has to lie between zero and one |

0:01:57 | so that's not the idea of |

0:01:58 | this presentation yeah does not improve |

0:02:01 | this way at least squares estimate by incorporating additional information and the additional information at you want to |

0:02:07 | uh in corporate um this will be an energy conservation constraint |

0:02:11 | so basically we have the two questions one i or where the answer this is |

0:02:15 | what is the a prior information that be normally have |

0:02:18 | so we will um assume that the energy that i was see at the microphone has to be less less |

0:02:23 | equal to the energy that i image that the source |

0:02:26 | and um that this is the a a not knowledge that we have a and you want to model this |

0:02:31 | as a constraint |

0:02:32 | into our because if estimation |

0:02:34 | and the second question is how can be efficiently come up with a estimators that a really include incorporate this |

0:02:41 | additional knowledge that we have and and will propose a we look at for different methods to do with this |

0:02:48 | okay |

0:02:48 | so this brings me to the contents |

0:02:50 | well of my talk um |

0:02:52 | it basically has two parts one is the first part is answering the first question |

0:02:56 | um what is the constraint how does to constraint look like and that we can exploit |

0:03:00 | and a second part is about the |

0:03:02 | uh estimate was that i can use to estimate because of three D room impulse response |

0:03:07 | and then i will show you some simulation results and |

0:03:10 | and some up the presentation |

0:03:12 | okay so first of all |

0:03:14 | um |

0:03:15 | how can express this energy conservation constraint |

0:03:18 | and that's has set what you want to exploit is that these signal energy at a microphone has to be |

0:03:22 | less or equal lead a signal energy at the source |

0:03:24 | just written down here |

0:03:25 | and um if i just |

0:03:27 | but this in a mathematical um from i what i need to ensure is that you are clean distance of |

0:03:32 | it and time times as of then |

0:03:34 | to this is the the signal that i have at the microphone |

0:03:37 | has to be less or equal to the i clean distance of my emitted signal |

0:03:42 | and if are just multiply this out |

0:03:43 | and keep in mind that this has to hold for all signals as of and |

0:03:47 | then i see that this metric heat of and france post times you'd of and that it of and was |

0:03:51 | this step it's metrics |

0:03:53 | we had to you room impulse responses |

0:03:55 | um |

0:03:56 | on the diagonal |

0:03:57 | um this has to be nonnegative and negative semidefinite |

0:04:03 | uh sorry |

0:04:04 | this one here |

0:04:05 | and um |

0:04:06 | this condition has as for all signal length as |

0:04:09 | and um if |

0:04:10 | be be um i know that sick |

0:04:12 | conditions for for for particular signal length as zero |

0:04:15 | then we know um from the fact that um the upper left matrix of the negative semidefinite matrix is also |

0:04:21 | again negative semidefinite |

0:04:22 | a note that this condition also holds for all |

0:04:25 | signal can that are smaller than assume |

0:04:27 | so it's for us it's just important to look at the case that the signal length S goes to infinity |

0:04:32 | uh and this is what you do in the following |

0:04:35 | so that this is the constraint that we can exploit |

0:04:38 | and on the next slide i will show you two |

0:04:40 | equivalent representations um of this set |

0:04:43 | so the set constraints all room impulse responses |

0:04:46 | which fulfil this energy conservation constraint |

0:04:49 | and the first representation presentation that we can use this T is an L i representation a presentation if i |

0:04:53 | just use shows slim a um than i see that i can work express this constraint here by this um |

0:04:59 | plot metrics |

0:05:00 | which has to be positive semidefinite |

0:05:02 | and um as it this linear and detail and you each element of this um |

0:05:06 | matrix elements i know that this is not an my and i immediately see that the set um um as |

0:05:11 | a convex set |

0:05:12 | but this presentation is not so convenient for us to include um |

0:05:16 | it did the you because if estimation later that as we will see |

0:05:19 | and the five will use the following frequency domain representation |

0:05:23 | and |

0:05:24 | using the equivalence |

0:05:26 | of the eigenvalues of a band limited toeplitz matrix and the corresponding circulant metrics |

0:05:30 | for the case that my second next goes to infinity |

0:05:33 | um i can come up with the following constraint and i the details um we can find in the people |

0:05:38 | are at paper |

0:05:39 | or or to put them and and backup slide |

0:05:41 | but what you will see and this is also somehow into |

0:05:45 | um that the room frequency response so this is nothing else than just a four you transform of my room |

0:05:50 | impulse response |

0:05:52 | um if i take the magnitude skirt of this this has to be less or equal to one for all |

0:05:56 | frequencies so this basically says is there was not of single frequency or make them |

0:06:01 | um |

0:06:02 | but the remember a room frequency response this larger than one so no single frequency |

0:06:07 | um |

0:06:08 | is um |

0:06:10 | a increased |

0:06:11 | E D is not there's no i'm additional gain for each frequency |

0:06:15 | a and we will see that all or uh |

0:06:17 | three recursive estimators are based on this frequency domain representation |

0:06:21 | and um to really to come up with some can be and um computational from less |

0:06:26 | we will approximate |

0:06:27 | um the question at we had before |

0:06:29 | by introducing a if T and now we just um be you room frequency response |

0:06:35 | at discrete frequencies so we now have on the guy was to pi are divided by have this i was |

0:06:40 | see if that you crawl to M or logic to M which just corresponds to a case of zero padding |

0:06:46 | and how um |

0:06:47 | if and should use some |

0:06:48 | selection matrix P what i basically see um here is that this |

0:06:52 | what i have is that my |

0:06:54 | uh room frequency response |

0:06:55 | at this all my get our now and time instants and |

0:06:58 | has to be less or equal to one |

0:06:59 | as just the constraint that we have that we have to ensure for all L from zero |

0:07:03 | up to basically L over two |

0:07:05 | um and this what what's then he this is basically just |

0:07:09 | a quadratic form that i have so my constraint |

0:07:11 | um so this |

0:07:13 | family of room impulse response um |

0:07:15 | or or a room impulse responses |

0:07:17 | which i and you can seven |

0:07:19 | um i can rupture sent this constraint just by a set of |

0:07:22 | uh a quadratic forms that i have to so |

0:07:26 | okay this was the energy conservation constraint and a the question is how can be |

0:07:29 | incorporate this knowledge into |

0:07:31 | the recursive estimation |

0:07:33 | and therefore i will |

0:07:35 | a proof talk about the channel set up that we have and then um come up with these specific estimators |

0:07:40 | for our |

0:07:41 | we're impulse response estimation problem |

0:07:43 | so channel we have to following problem we have X of and as as of and times |

0:07:47 | it it zero of and so this is the time varying power me that we want to estimate from our |

0:07:52 | observations X of N |

0:07:53 | um given some additive cost noise and B no a priori that might power me that i want to estimate |

0:07:59 | this lies in the subset it it which which is a subset of |

0:08:03 | the um and dimensional |

0:08:04 | um space |

0:08:06 | and smiling motivated it by the at least squares estimator we can we formulate a uh are we can |

0:08:11 | um come up with the signal model is given here |

0:08:13 | if i introduce |

0:08:15 | a the observation vector X of and which just contains all observations that i have |

0:08:19 | if i introduce introduces stacked um model metrics as of an which contains all model match races for all time |

0:08:25 | instants |

0:08:26 | and also introduce |

0:08:27 | um and noise vector set of and |

0:08:29 | which is not not merely the only these stacking of the um |

0:08:33 | the show more terms but also incorporates |

0:08:36 | um this |

0:08:37 | don waiting that i have |

0:08:39 | which just says that past measurements are not so important then um you observations |

0:08:44 | i can come up with the signal model and um if i just for not um don't consider this constraint |

0:08:49 | here |

0:08:50 | then it's well known that the maximum likelihood estimate of this one is just a at least squares estimator |

0:08:54 | and that for a um this is was the motivation for us to consider the stick model |

0:08:59 | and now with the additional constraint |

0:09:01 | that's we know that this it of and has to lie in the subset it |

0:09:07 | okay okay and now the question is how can we applied estimators with the crow computational complexity |

0:09:12 | so if i go back if you just look here to sex of and |

0:09:15 | and also the other terms there are um |

0:09:17 | and growing with time |

0:09:18 | because i just stack all observations that i have into this um large vector X of and |

0:09:24 | and that for the questions how can be avoid estimators with this growing computational complexity |

0:09:29 | and um what we use is the concept of sufficient statistics |

0:09:33 | and a sufficient statistic of the signal model that up we had before so of this linear cost model |

0:09:37 | is um given by the following |

0:09:39 | um and |

0:09:40 | you just look closely at this time that we have few this is nothing else than the maximum likelihood estimator |

0:09:46 | of the equation that we had before |

0:09:48 | and this is just a weighted at least squares estimator or S that's so was sufficient statistic that i can |

0:09:52 | use |

0:09:52 | is um |

0:09:54 | the rate feast squares estimator of the plot problem that i had before |

0:09:57 | and be um |

0:10:00 | um or |

0:10:01 | perhaps |

0:10:01 | first of this um |

0:10:03 | this sufficient statistic um it's about on that this can be efficiently computed by recursive we discuss a estimator and |

0:10:09 | this |

0:10:09 | um because if at feast was um i'm with them |

0:10:12 | or um i dates |

0:10:14 | my sufficient statistic |

0:10:15 | um because it's solves um for the um we discuss estimate in each iteration and also gives me the inverse |

0:10:21 | correlation metrics and these both quantities are will need um for the estimators |

0:10:26 | and that will follow |

0:10:28 | so the idea of now um |

0:10:30 | what i can use as the sufficient statistics in a first step |

0:10:32 | so use the way at least squares estimator |

0:10:35 | and now i'm thinking of |

0:10:36 | different estimators in a second step or different ways to incorporate that knowledge that i have this energy conservation constraint |

0:10:42 | to get a best but estimate then the you wait at least squares estimate of look give me |

0:10:48 | okay and now this is the first estimate at we can use this is |

0:10:52 | um |

0:10:53 | um quite simple we use um the maximal like a estimator of our signal model but this time with the |

0:10:58 | knowledge that might it has to lie in the subset term so and the set of all the room impulse |

0:11:03 | responses |

0:11:03 | which are and cheek can serving |

0:11:05 | um and if you just had the this out |

0:11:08 | and keeping in mind that um we know the sufficient statistics then it's |

0:11:11 | then we come up to the following and quadratic form um we have |

0:11:14 | do the is um |

0:11:16 | um the you body discuss estimate |

0:11:18 | times the correlation matrix |

0:11:21 | and times this button |

0:11:22 | we have to minimize this quadratic roddick from a subject to our constraint |

0:11:26 | so at each time step |

0:11:27 | um we have to check either is the least squares estimate inside my um constraint if yes then this is |

0:11:33 | also my |

0:11:34 | because of um |

0:11:35 | can makes "'em" like that to estimate or if not we have to find the minimum of this quadratic form |

0:11:39 | on the boundary of this um constraints at that i have |

0:11:43 | so now for our a room impulse response tracking problem |

0:11:46 | we know that the constraint is just given by these uh a quadratic form as |

0:11:50 | and that all i have to do is |

0:11:52 | um and the case that my um way discuss estimate is not inside this constraint |

0:11:57 | what i have to do is stand um i have to solve |

0:11:59 | well have to minimize a quadratic form or what quadratic constraints which is a quadratically constrained quadratic program |

0:12:05 | which we can efficiently solve |

0:12:07 | to this is the first um estimate that we can use |

0:12:10 | a second estimate is the recursive if you'd minimax estimator |

0:12:14 | um and it was shown by L or that um |

0:12:16 | this a few minimax estimator has the following form |

0:12:19 | um i know the efficient estimator form Y um |

0:12:23 | supermodel model |

0:12:24 | um then i have to of um the than of fine from you same |

0:12:28 | this metric i'm of N |

0:12:30 | oh sorry |

0:12:31 | and this |

0:12:32 | um that so you of and |

0:12:34 | and M of and and you of and uh fine a i can be found by solving min next problem |

0:12:38 | given here |

0:12:39 | and if you look close that this one |

0:12:41 | um this just quote um depends on the |

0:12:44 | inverse correlation matrix which is also computed by they were "'cause" if at least squares estimator so i can |

0:12:49 | we have here |

0:12:50 | um um |

0:12:51 | it depends on the um sufficient statistics and you the inverse correlation matrix |

0:12:55 | and both are |

0:12:56 | i did by do because if a this squares estimator so i'm still a um |

0:13:00 | i also have few at the problem of the the the the um nice fact the like and |

0:13:04 | relied on this way least squares estimator and the for |

0:13:08 | so now for our a room impulse response tracking problem we do the following simplification |

0:13:13 | to reduce used the computational complexity we just that you of to zero |

0:13:17 | and assume that this i'm of and it's just a bike mot metrics um which just have one |

0:13:22 | meet the all from that we tried to to mess |

0:13:24 | so we saw of those not overall i meant you but just a with this i'll |

0:13:29 | and um to corporate this um set of quadratic ready constraints so i have um |

0:13:35 | it's um |

0:13:36 | one has to first |

0:13:37 | um transform the problem into a the graphic form and then use the as procedure as as written down here |

0:13:41 | and then i can we formulate this optimization problem into a semidefinite program |

0:13:45 | and this is um what you will do |

0:13:48 | um in the simulation results |

0:13:49 | that this was to a because of affine minimax estimator |

0:13:52 | and the slide your |

0:13:53 | um to introduce or to incorporate this um now which show this a or no that i have this constraint |

0:13:59 | is |

0:13:59 | to use the minimum mean squared error estimator |

0:14:01 | where and say um |

0:14:03 | that i have a uniform prior on this constraint set T time |

0:14:07 | and |

0:14:07 | this can be motivated just by D makes entropy principle which just says out of of family of prior densities |

0:14:13 | i should choose that prior or which has a maximum entropy |

0:14:17 | and this is the uniform prior on this set you time |

0:14:20 | um so you using the concept of by asian um sufficient statistics we can |

0:14:25 | come up with the following because as minimum mean squared error estimator |

0:14:28 | which is just written here and again be C it's also just depends on the sufficient statistics and on the |

0:14:33 | inverse correlation matrix so we also have not the problem of and enough and crawling |

0:14:38 | that much or um vector X of and |

0:14:41 | and to have to close um we use rejection sampling um so we use basically much colour integration |

0:14:48 | and the samples from the posterior a formed by um something thing |

0:14:52 | um using rejection sample |

0:14:53 | and the posterior |

0:14:55 | i you can see year |

0:14:56 | it's just a caution densities as a quadratic form and this exponential which is a truncated to the |

0:15:01 | uh features as this part of this at is given by this um constraints at you time |

0:15:06 | and for our room impulse response tracking problem |

0:15:09 | um we sample from a caution and then we have just a check |

0:15:12 | um is this constraint for do not for that sample and if it's not fulfilled food than we just um |

0:15:17 | or we got um this |

0:15:19 | or reject this sample |

0:15:21 | okay this point not to the simulation results |

0:15:24 | and the simulation was that simulations to be |

0:15:27 | have the following problem we have a we want to estimate a room impulse response from a moving source to |

0:15:31 | six microphone |

0:15:33 | so the setup is as following um |

0:15:35 | the source moves along a straight line |

0:15:37 | and we have ten centimetres speech uh between neighbouring position |

0:15:41 | or all we have a let L eleven positions |

0:15:44 | so the source moves in total of one are |

0:15:47 | and the source signal is assumed to be caution |

0:15:49 | um just caution lies of length one hundred example |

0:15:52 | and the room has a size of three by three meters and a i hate of two point three meters |

0:15:57 | and the reverberation time is a a one hundred and twenty miliseconds |

0:16:00 | and use a image source model to obtain the room impulse responses |

0:16:04 | uh with the sampling frequency of to both two has |

0:16:07 | which |

0:16:07 | um |

0:16:08 | if us |

0:16:09 | a room impulse responses of length of one thousand two hundred forty one taps that we have to estimate |

0:16:13 | in each time step |

0:16:15 | um |

0:16:16 | and the all three estimators um |

0:16:18 | as a set use this give the approximation and V use |

0:16:21 | or are sampling uh by roughly a factor of ten so we use um L is |

0:16:26 | you put to to to the power fourteen |

0:16:28 | and for this um because of minimum mean square error estimator |

0:16:31 | use three thousand samples to um approximate these integrals by monte colour integration |

0:16:39 | that just preview T um definitions of signal to noise ratio is just the to noise ratio at the microphone |

0:16:44 | and be use a normalized are emission |

0:16:46 | to show the results here |

0:16:47 | which just just um could difference between my estimate and my to room impulse response |

0:16:52 | divided by the um room impulse response and energy |

0:16:56 | okay |

0:16:56 | first of all um Q the results for the instantaneous estimators which corresponds to a case |

0:17:01 | to better that you zero |

0:17:03 | um |

0:17:04 | and that's a do not have much time left um what you can see |

0:17:06 | all three estimators um are better than you but if he squares estimator and especially the |

0:17:11 | um minimum is could have estimator this small here and if i allow for better on equal to zero |

0:17:17 | um you see that all values gets smaller from |

0:17:20 | um up you to down here |

0:17:22 | and |

0:17:22 | again this is because of minimum is mean squared error estimate which has this uniform prior on this constraint it |

0:17:27 | and gives the best results |

0:17:30 | so |

0:17:30 | um but me quickly some uh up to my presentation um |

0:17:34 | we talked about the because of room impulse response estimation image with with an energy conservation constraint |

0:17:39 | and as you could see in the simulation results this constraint will helps to improve the performance |

0:17:44 | and the because of minimum means good it um |

0:17:47 | i estimator |

0:17:48 | post estimator with the um |

0:17:51 | best to form an |

0:17:52 | and now to |

0:17:53 | um come up with better estimate is a with a um to improve the performance you more |

0:17:58 | you can't with we could now in corporate additional information that you could have um so this would mean additional |

0:18:04 | uh constraints |

0:18:05 | and this is some future work are thinking |

0:18:07 | about |

0:18:08 | thank you very much for a attention |

0:18:26 | you in like |

0:18:27 | i was wondering how |

0:18:29 | what is the physical meaning of this constraints |

0:18:32 | a yeah i mean |

0:18:34 | you talk about a discrete |

0:18:36 | digital signals |

0:18:37 | and uh |

0:18:39 | the the quantity it you call in G |

0:18:42 | well it's somehow how related to real world for you know a G but you know they are apply five |

0:18:46 | say and |

0:18:47 | and don't device |

0:18:48 | in between |

0:18:49 | and a |

0:18:51 | well that |

0:18:52 | that that response that you may i can mess are used using K is and an estimator |

0:18:58 | could have easily |

0:19:00 | like a twenty db gain |

0:19:02 | i have for a particular frequency |

0:19:05 | um |

0:19:06 | so um in the simulation results we see that um this is |

0:19:09 | and as you conservation constraint just gives us um |

0:19:12 | room impulse response estimates which are more smooth than if you just compared to the ordinary weighted feast squares estimates |

0:19:18 | of this this is basically what this constraint performs but now the question as you're right um if there devices |

0:19:23 | in between some amplifiers um |

0:19:25 | you need to have one calibration step before and in a real application to really use this set up because |

0:19:30 | what you really want to have is that this energy at T |

0:19:33 | um |

0:19:35 | a microphone is less or equal to the energy of |

0:19:38 | at the source |

0:19:39 | um so if that would be an empty in between um we we need one calibration step |

0:19:44 | um |

0:19:45 | to really to use this |

0:19:49 | question |

0:19:54 | okay |

0:19:55 | uh one i you what for example if i know that my source will not be close to to the |

0:19:58 | microphone than for example two meters or something like this but know from |

0:20:02 | just um |

0:20:04 | the error propagation what what could be a what what is the minimum um um iteration that double have |

0:20:09 | and and stuff like this could could be incorporate |

0:20:12 | or also there are uh some work um |

0:20:14 | publish already |

0:20:15 | which include sparsity prior so if i know that my um room impulse response |

0:20:19 | just as as some strong pad |

0:20:21 | this is something additional that i can put on top |

0:20:23 | to come up with a better estimate of this room impulse response |

0:20:27 | a |

0:20:29 | how likely would the then mse the normal the and see solution be a a a a of than one |

0:20:37 | because here you have this constraint on the norm of of of the channel yeah |

0:20:41 | a a how like if you have been mention and the receive the power is always |

0:20:45 | yes smaller than the transmit by |

0:20:47 | so how likely would the now of the and then C solution i can would be larger than one in |

0:20:52 | which case you would need your |

0:20:53 | okay even mean how many for example for yeah "'cause" minimum mean square error estimate to i have to do |

0:20:57 | a rich action or a um |

0:20:59 | this depends strongly on the signal to noise ratio |

0:21:01 | and |

0:21:02 | um so for a signal to noise ratio of for example five to be |

0:21:06 | um rough if if if you sample from this posterior density |

0:21:09 | i'm you have to um exclude every second from sample that it it's a draw from this your density for |

0:21:14 | some |

0:21:15 | um |

0:21:16 | at a or a if for the three "'cause" of constrained maximum likelihood estimator |

0:21:19 | i do it know the ratio ratios at all how many times exactly with at least squares estimator and how |

0:21:24 | many times you have to solve the optimization problem in um one E |

0:21:28 | i'm i'm i'm not sure how many times this will happen for this |

0:21:31 | or C M L |

0:21:32 | oh check this yeah |

0:21:34 | left things you can again |