Přepis řeči - FOURIER EXPANSION OF HAMMERSTEIN MODELS FOR NONLINEAR ACOUSTIC SYSTEM IDENTIFICATION

0:00:13	so
0:00:14	my name
0:00:15	but i
0:00:17	time
0:00:18	my
0:00:19	for you expansion much time
0:00:22	for non
0:00:23	system identification
0:00:25	and uh
0:00:26	this work was done in in uh what we was home
0:00:30	and get a and slide yeah
0:00:32	the
0:00:32	or
0:00:35	so uh
0:00:36	i go on
0:00:37	to the outline
0:00:39	notation
0:00:40	after representing the motivation and talking about two basis function
0:00:44	no
0:00:45	and
0:00:46	and because "'cause" we have more than
0:00:47	one type of basis function
0:00:50	and be talking about a piece is genetic
0:00:53	the of signal modeling dft domain
0:00:56	which you see here that you can cost on to an equivalent multichannel
0:01:00	i
0:01:01	that's bringing
0:01:02	multichannel adaptive
0:01:04	in into the king
0:01:05	and and this is a section you see how the basis selection
0:01:09	from here basically C B but the fixed the multichannel system i
0:01:13	and you the can
0:01:18	so uh
0:01:19	uh
0:01:20	it's well known that you to of hammerstein fine structure is that
0:01:23	enables to
0:01:25	of a
0:01:26	non linearity the memory for the by a linear fire sister
0:01:30	so if you want to have a a a of i've
0:01:32	system i and
0:01:33	location then we need to cater
0:01:35	for the menu the less nonlinearity is but
0:01:38	which we can model by means of a power C D's or
0:01:41	or the norm
0:01:42	expansion bases
0:01:44	which is somewhat the more traditional and used at this one
0:01:48	and uh here you somewhat
0:01:50	propose and investigate the or for you C
0:01:53	which is an orthogonal are mutually orthogonal
0:01:55	expansion bases
0:01:57	and the X that investigate or in the context of what if
0:02:00	what it have
0:02:01	on the ensuing
0:02:02	equivalent multichannel system i
0:02:04	a with respect to the quality rate of convergence
0:02:07	and also
0:02:09	quality of the learning
0:02:10	of the underlying nonlinear a
0:02:14	so that just to bring uh everybody on the same page with respect to notation
0:02:19	as how much time structure
0:02:21	yeah this input signal the acceleration signal
0:02:23	this a nonlinear mapping it under just nonlinear transformation to get the non linearly map signal
0:02:29	it gets you know can vol
0:02:32	but in a we are in your system to get T
0:02:34	uh auxiliary signal D it's gets
0:02:36	so and post by this year in noise
0:02:39	finally give the observation
0:02:40	so these
0:02:41	this modeling that i'm been talking about is about
0:02:44	nonlinear at
0:02:45	here
0:02:49	so uh now for the basis functions or
0:02:51	types
0:02:52	we can of the nonlinearity in the hammerstein model
0:02:55	by a such a a nation firefighter are basically
0:03:00	i i guess order
0:03:01	basis function of the corresponding
0:03:04	which
0:03:04	and she you over here is
0:03:06	the expansion or
0:03:08	i
0:03:08	if
0:03:09	so five i were to long to some sort of polynomial bases then simply would have a
0:03:15	it has the i that's power of X
0:03:17	and he of i would then
0:03:19	be the corresponding uh autonomic coefficient
0:03:22	and correspondingly if
0:03:24	this for the forty your basis
0:03:26	and i have would be a sinusoidal just form
0:03:28	the for C D
0:03:30	and then a a a or whatever the for your coefficient
0:03:33	and and or rather to L would be the fundamental period
0:03:36	and the selection of this fundamental period is somewhat critical
0:03:39	but to give a short we can do that
0:03:43	a assuming that uh we have
0:03:45	any given a nonlinear mapping F F X
0:03:48	and the data range
0:03:49	normalized minus one plus one
0:03:51	then if you were
0:03:52	to if
0:03:53	compute the
0:03:54	could which is the power C D's and you can minimize
0:03:57	expression of or rather than take here in the least sense
0:04:01	this can be any number of strategy are not focusing on that you can use a map
0:04:04	might have only for that
0:04:07	and for the for U C D's of or for you C is because you know the nonlinear that it
0:04:11	would be somewhat an or function
0:04:12	so we can use this close form expression for the computation
0:04:16	and again this
0:04:17	half of the fundamental peter comes into the play
0:04:19	importance of selection of L is that
0:04:22	this is that data range and we know that the data let's is and plus one
0:04:26	then we can select a one to be bit
0:04:28	greater than one
0:04:29	because of it is one and all the kind of so
0:04:32	near the plus minus one range would go to zero
0:04:35	and you wouldn't be able to model the data on the thing
0:04:39	so a a and an example of the manifestation of the non linearity i'd take it as a clipping function
0:04:43	a or whatever the linear range and then it's clipped by X max which is a threshold
0:04:48	and
0:04:49	or we set have experiment or out of experiment that other forms of clipping functions of a nonlinear functions
0:04:54	but this is as a very good example
0:04:56	have a discussion
0:04:59	so that the first uh a result that you you what has about a
0:05:03	to fitting ability of the boat C so i selected D fundamental period of a is one point five
0:05:09	and you can see that the clipping threshold of of of the simulated or the clipping function his point one
0:05:15	so we have a a minus point one and plus one one what here
0:05:18	and the
0:05:19	the expansion or or is i
0:05:21	so we see the start line depicts this how forty a bases is basically modeling this
0:05:25	non linearity
0:05:26	then we also see how power
0:05:28	polynomial a normal bases is modeling
0:05:30	you might
0:05:31	but even if i don't to you system distance but the for you based basically you has a five db
0:05:35	it
0:05:36	on the modeling
0:05:37	but that's must
0:05:38	what i'm focusing on a have because then
0:05:40	would have the argument let's increase the model
0:05:43	but in we'll see
0:05:44	the and so on so forth
0:05:45	but it is true that
0:05:47	forty bases is
0:05:48	a contend or and it comes to such model
0:05:52	so now the basis generic a
0:05:55	signal model in the dft domain because of be like to have a the system identification
0:05:59	frequency domain multi-channel forms so that's i we select dft to domain
0:06:04	and uh
0:06:05	because we are going to uh going to the dft domain so we try to
0:06:08	a find a block based definition of the input signal
0:06:12	no here you see or is basically the frame shift in M
0:06:14	a frame size
0:06:16	so in analogy to X
0:06:17	a uh i can
0:06:18	find the block these definition of the non linear in that
0:06:21	input signal
0:06:22	and i do that way
0:06:23	i in this nonlinear mapping to all these individual samples
0:06:27	the this vector
0:06:27	and this for
0:06:29	and uh i can replace this nonlinear mapping by
0:06:33	such a some nation form which i sure and one of the previous slides
0:06:37	i can do this
0:06:37	can be rearrangement spent the summation sign out here out
0:06:41	and then i have this
0:06:42	vector
0:06:43	compact notation X
0:06:45	which is basically the idea that order
0:06:47	off
0:06:47	the nonlinearly mapped input signal
0:06:50	that's a block this definition sort or is made from the eyes or of the bases from
0:06:55	which is in it's
0:06:56	channel from right
0:06:59	now we want to convert
0:07:01	if you do means what we do is you like for you mate
0:07:03	i nation
0:07:05	and to see what use going on we can
0:07:07	a place
0:07:08	a this definition by the summation
0:07:10	which will bring
0:07:11	coefficients
0:07:12	more two
0:07:13	a play
0:07:14	and then we can keep the coefficients outside side and then you would have a higher order
0:07:18	of the non unit the input signal the dft main
0:07:21	uh
0:07:22	if
0:07:23	actual
0:07:24	and uh uh now that we have this
0:07:25	a formal definition
0:07:27	of the input signal or the non email and signal
0:07:29	give you've main we can go a morning
0:07:31	you you but or leaner i R
0:07:33	system
0:07:34	so we basically model and minus are non-zero coefficients
0:07:37	basically a uh to make sure that overlaps safe
0:07:40	strange
0:07:40	remains
0:07:41	that it later on
0:07:42	so this again as forty major
0:07:44	is so
0:07:45	time domain vector
0:07:46	we have
0:07:46	yeah domain con
0:07:49	so uh we know that the observation can you given as a function of the convolution between B
0:07:54	nonlinear mapped mapped input
0:07:55	the equal part
0:07:57	oh as the observation noise
0:07:58	and this right S over a that is that some special
0:08:01	rather sensor
0:08:03	uh a just to linearize the convolution dft domain
0:08:06	the scene worse for year
0:08:07	as this the padding the and for your
0:08:11	i can
0:08:13	by compile this all of the form G
0:08:15	that i can combine G an X to get a to get a C
0:08:18	so C is basically a
0:08:20	constraint
0:08:21	of the non
0:08:21	that's
0:08:23	so uh this there a compact expression
0:08:26	for the dft domain observation
0:08:28	which we can
0:08:29	for the uh
0:08:31	really
0:08:33	to get a equivalent multichannel structure
0:08:36	and that the way you go about doing that is your base this a vector
0:08:41	to to the by the summation expression because we the to right this earlier
0:08:45	and instead of using the summation sign we can use such a a matrix
0:08:48	location
0:08:49	so
0:08:50	this basically couldn't
0:08:51	and i i
0:08:52	do so what yeah the identity matrix to make sure the dimensions are
0:08:56	consist
0:08:57	and instead of then combining a the end of it these components what i words a combined you could possibly
0:09:02	get
0:09:03	effective or virtual it couldn't multichannel four
0:09:07	and uh then i combined your again but this composite matrix text
0:09:10	the second it may trick
0:09:12	and this is
0:09:13	the observation model
0:09:14	a multichannel
0:09:15	position the most
0:09:16	calm
0:09:18	so how does it basically
0:09:19	look uh
0:09:21	dark grammatic
0:09:21	a you this is what
0:09:23	happened
0:09:24	we have combined he's
0:09:25	some
0:09:25	of the nonlinear
0:09:26	and from which has as can be any for your more power
0:09:29	or if anybody
0:09:31	good idea here
0:09:32	and we combine it with a the people above
0:09:34	to get a were true channel had
0:09:36	and then we C D's excitation signal
0:09:39	and
0:09:40	no
0:09:40	appreciate here and i see fans
0:09:42	of using the forty am wanting because
0:09:45	and this is a multichannel identification problem than all the ancient problems of multichannel channel
0:09:50	adaptive filtering with resurface
0:09:52	and if and that's a lot of correlation between these excitation signals than i would be
0:09:56	some works in a problem
0:09:58	and if i have a forty a uh basis then uh all these
0:10:02	a a a a a a a signals the excitation signal quite each and we usually problem
0:10:06	and that would be a very good thing for
0:10:08	convergence
0:10:09	you you to that
0:10:12	so uh uh know and we want to the results and uh what whatever
0:10:15	used
0:10:17	uh for the multichannel channel uh
0:10:19	evaluation
0:10:20	so this is a a not a very fancy other this is
0:10:23	block lms type
0:10:25	a a multichannel frequency-domain adaptive
0:10:27	a given by
0:10:28	uh used to equations be
0:10:31	a function
0:10:31	and the update equation
0:10:33	you
0:10:34	the step size
0:10:35	and
0:10:36	basically the step-size size contains
0:10:38	a step size for
0:10:39	channel
0:10:40	and which is a function all this adaptation constant
0:10:43	and uh the estimate
0:10:45	of the power spectrum and
0:10:47	and the adaptation constant like in this strange
0:10:50	and its estimate can be achieve are obtained i've such a
0:10:54	a a recursive equation
0:10:55	okay gamma headers the forgetting factor and the range you one
0:11:00	so uh he can
0:11:01	a
0:11:01	for the evaluation was that were operating the multichannel frequency-domain adaptive filter
0:11:07	uh with a frame size of two fit
0:11:08	sex
0:11:09	a frame shift of sixty four
0:11:11	and the linear to nonlinear power issue of the snr and L
0:11:14	given by such an expression this basically differ
0:11:17	the input signal and the nonlinearly mouth
0:11:19	was five you in twenty db
0:11:21	if i have a time just discussed of twenty db case as well as not then
0:11:25	just a
0:11:25	a five but
0:11:26	the signal to observation noise or
0:11:29	in the you cancellation or or you code
0:11:31	observation noise
0:11:33	or a show has been kept
0:11:34	as for the
0:11:35	sixty db because
0:11:36	we want to concentrate on the nonlinear performance
0:11:39	a the robust
0:11:40	because it than your and observation
0:11:42	two types of is as we have a C D
0:11:45	you what for you C D's
0:11:47	and a performance measure would be the relative
0:11:49	uh error signal attenuation given by such an expression
0:11:52	and are also
0:11:53	inspect the estimated nonlinear nothing
0:11:56	not in ins
0:11:57	and the on a mapping you would
0:11:59	track
0:12:00	the nonlinear coefficients C
0:12:02	and that we do uh by this expression which gets just nonlinear
0:12:06	coefficients
0:12:07	and the least squares sense optimal in the least squares
0:12:10	there this stuff you would have a or i is the estimate of the I channel
0:12:14	all the eyes were true channel
0:12:18	so uh this is the performance comparison for uh
0:12:21	fight T case and that would basically mean that the threshold uh
0:12:24	the clipping threshold this plus as point one
0:12:27	so the first uh a algorithm that and using as a anchor is to a linear or uh and stuff
0:12:32	the single channel that stuff that any provision
0:12:34	for each have a or sorry nonlinear processing and easy it converges to the area of
0:12:39	eight db
0:12:40	and then we have to put normal model with how a series
0:12:43	it
0:12:44	uh
0:12:44	uh
0:12:45	a it's it's of some it'll or yeah and then it
0:12:48	but this slow
0:12:49	P convergence
0:12:49	street that used to go up
0:12:51	my believe is
0:12:52	keep on going up somewhere or whatever
0:12:54	and uh but we see that
0:12:56	somewhat slower or and B i
0:12:58	or or comment
0:12:59	polynomial model with gram schmidt
0:13:01	uh uh data adaptive orthogonalization
0:13:05	and uh uh we see that me to be the performance should so
0:13:09	and then we finally have to for you model without any additional orthogonalization we see that
0:13:15	somewhat matches
0:13:16	is
0:13:16	a better
0:13:17	then the polynomial normal plus crash
0:13:19	and hence the notion
0:13:20	you have orthogonal input
0:13:22	to the multichannel structure
0:13:23	then you would have this
0:13:25	same effect as the gram schmidt orthogonalization provides
0:13:28	so better convergence and higher
0:13:30	and that's have a look at the quality of the nonlinear that you have
0:13:33	for
0:13:34	so this is the ground truth plusminus point one
0:13:38	and we see that this is the green guy which was a polynomial night model without any gram schmidt
0:13:43	so it is a very of that
0:13:44	so there's a correspondence between the quality of the nonlinear to for a and you have a but we solve
0:13:49	before
0:13:51	and then we see that
0:13:52	putting on on the gram schmidt
0:13:53	a forty T once they are at hand and hand
0:13:56	two was it fringes
0:13:57	this is put a will that crash from goals of it you on both sides
0:14:01	but i'm not so what it about that
0:14:03	for now because
0:14:05	my data is more concentrated in the range plus minus point for
0:14:09	what of course if there's any hope live and the data would be if
0:14:11	in that case
0:14:12	would
0:14:13	consider
0:14:14	for you to be the better one
0:14:17	and this is a a to have time
0:14:18	yeah okay
0:14:19	so that a performance comparisons for twenty db twenty db basically means that my uh
0:14:25	uh
0:14:26	clipping threshold as plus minus
0:14:28	three
0:14:29	and this is a a a a my to the nonlinear case
0:14:33	this means my higher order polynomials
0:14:35	or not
0:14:36	that much and my to the coefficients
0:14:39	this basically means that
0:14:40	my polynomial a model and a point and model because the gram schmidt are having a very nice day
0:14:46	and this
0:14:47	this you see that is a
0:14:49	difference in the rate of convergence and this happens because
0:14:52	the forty a model even if a
0:14:54	might be be mean my
0:14:56	my
0:14:57	the the non linear any but if it is linear
0:14:59	to all the channels of a for a what would be active an adaptation
0:15:03	so that to take it
0:15:04	time
0:15:06	that would show up or take it's still wouldn't the convergence
0:15:08	but still a goes higher
0:15:10	ten the other approaches
0:15:12	and this uh linear model left off is
0:15:14	corresponding
0:15:15	is is
0:15:15	some in of the
0:15:17	and you range as the snr and it
0:15:19	the there is no direct correspondence between the nonlinear in snr
0:15:23	and the virtual source of noise this that is this
0:15:26	still
0:15:28	so as we seen that these two guys were also not perform that bad and
0:15:33	so
0:15:34	or not performing that that and so he's see there are also a really here are also not performing that
0:15:38	bad
0:15:39	but the for you guy was much better than both of them so easy
0:15:43	follows
0:15:43	the ground truth somewhat better
0:15:47	so uh
0:15:48	bring
0:15:50	illusion
0:15:51	and so we uh sort of presented
0:15:54	a a a a four cylinder to of nonlinear hammerstein model
0:15:58	a a the tradition tuition C D's and or or or or on four you D's
0:16:02	we presented the signal model and block frequency domain
0:16:05	which basically uh was to ride by contain bass
0:16:09	channel
0:16:10	derivation
0:16:11	and was for by an efficient multichannel representation again into line on you're fifty
0:16:16	and uh in the results by a multichannel adaptive identification we showed to be orthogonal for us he's
0:16:23	lines up with a polynomial modeling of the gram schmidt orthogonalization
0:16:28	in the sense that it uh a uh uh that's high error signal attenuation
0:16:32	and effectively imitates the underlying nonlinear
0:16:36	oh you mister
0:16:37	so that from and say
0:16:40	Q
0:16:43	i
0:16:47	or we can thanks
0:16:48	is
0:16:52	i help these
0:16:58	so
0:16:59	oh right in your first the results on the gram schmidt response of the polynomial had a fairly high variance
0:17:06	uh
0:17:07	a a lot of fluctuation
0:17:09	yeah
0:17:10	do you have one more fluctuations
0:17:12	uh i and the response here
0:17:13	yeah
0:17:14	could you corpsman because when you we're going that result with the fine db in a to the twenty db
0:17:19	there's virtually no fluctuations at all the twenty db be a yeah i can yeah i and so but
0:17:24	the response is not as good so could you comment or half
0:17:28	somehow have me
0:17:29	difference is the in your response
0:17:31	may may be it yeah be contributing to the change
0:17:36	i i i i i
0:17:38	two my polynomial
0:17:39	model that's
0:17:41	two
0:17:42	this area
0:17:44	right right would be to stay have yeah
0:17:47	so if i have a let's see the twenty db case
0:17:50	and that would mean that
0:17:51	these coefficients
0:17:52	for a polynomial series
0:17:54	don't have a lot of mine to this means that these channels don't have a lot of excitation
0:17:58	so this basically means
0:18:00	a gram schmidt orthogonalization
0:18:02	doesn't have a lot to offer four
0:18:04	doesn't have a lot of influence
0:18:06	and that that like tuition in the fine db can use
0:18:09	could be because there might be some smoothing or something that
0:18:12	a further applied to the grams schmidt organisation
0:18:15	but uh i said that the by something but have been done because that does not focus of uh
0:18:20	what i was actually trying to do
0:18:22	so what i basically we is the difference in performance which does not uh come in the twenty cases because
0:18:28	there's not much room
0:18:30	of and to to the ground truth can provide
0:18:33	because of the depleted excitations those
0:18:35	i or channel
0:18:41	and yeah
0:19:04	but he was asking
0:19:05	the non but not for you
0:19:19	yeah
0:19:26	oh
0:19:31	i i on a say uh yeah okay yeah
0:19:33	i i don't uh for that directly because it this uh off a plate
0:19:37	that you describe like
0:19:39	pose for you consider it
0:19:44	well with all this like this

FOURIER EXPANSION OF HAMMERSTEIN MODELS FOR NONLINEAR ACOUSTIC SYSTEM IDENTIFICATION

Echo Cancellation

Přednášející: Sarmad Malik, Autoři: Sarmad Malik, Gerald Enzner, Ruhr-Universität Bochum, Germany