Speech Transcript - RECURSIVE ESTIMATION OF ROOM IMPULSE RESPONSES WITH ENERGY CONSERVATION CONSTRAINTS

0:00:15	okay thank as the german um but can i as my presentation which is about
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

RECURSIVE ESTIMATION OF ROOM IMPULSE RESPONSES WITH ENERGY CONSERVATION CONSTRAINTS

Target Detection and Localisation

Presented by: Stefan Uhlich, Author(s): Stefan Uhlich, Bin Yang, Universitaet Stuttgart, Germany