so

my name

but i

time

my

for you expansion much time

for non

system identification

and uh

this work was done in in uh what we was home

and get a and slide yeah

the

or

so uh

i go on

to the outline

notation

after representing the motivation and talking about two basis function

no

and

and because "'cause" we have more than

one type of basis function

and be talking about a piece is genetic

the of signal modeling dft domain

which you see here that you can cost on to an equivalent multichannel

i

that's bringing

in into the king

and and this is a section you see how the basis selection

from here basically C B but the fixed the multichannel system i

and you the can

so uh

uh

it's well known that you to of hammerstein fine structure is that

enables to

of a

non linearity the memory for the by a linear fire sister

so if you want to have a a a of i've

system i and

location then we need to cater

for the menu the less nonlinearity is but

which we can model by means of a power C D's or

or the norm

expansion bases

which is somewhat the more traditional and used at this one

and uh here you somewhat

propose and investigate the or for you C

which is an orthogonal are mutually orthogonal

expansion bases

and the X that investigate or in the context of what if

what it have

on the ensuing

equivalent multichannel system i

a with respect to the quality rate of convergence

and also

quality of the learning

of the underlying nonlinear a

so that just to bring uh everybody on the same page with respect to notation

as how much time structure

yeah this input signal the acceleration signal

this a nonlinear mapping it under just nonlinear transformation to get the non linearly map signal

it gets you know can vol

but in a we are in your system to get T

uh auxiliary signal D it's gets

so and post by this year in noise

finally give the observation

so these

nonlinear at

here

so uh now for the basis functions or

types

we can of the nonlinearity in the hammerstein model

by a such a a nation firefighter are basically

i i guess order

basis function of the corresponding

which

and she you over here is

the expansion or

i

if

so five i were to long to some sort of polynomial bases then simply would have a

it has the i that's power of X

and he of i would then

be the corresponding uh autonomic coefficient

and correspondingly if

this for the forty your basis

and i have would be a sinusoidal just form

the for C D

and then a a a or whatever the for your coefficient

and and or rather to L would be the fundamental period

and the selection of this fundamental period is somewhat critical

but to give a short we can do that

a assuming that uh we have

any given a nonlinear mapping F F X

and the data range

normalized minus one plus one

then if you were

to if

compute the

could which is the power C D's and you can minimize

expression of or rather than take here in the least sense

this can be any number of strategy are not focusing on that you can use a map

might have only for that

and for the for U C D's of or for you C is because you know the nonlinear that it

would be somewhat an or function

so we can use this close form expression for the computation

and again this

half of the fundamental peter comes into the play

importance of selection of L is that

this is that data range and we know that the data let's is and plus one

then we can select a one to be bit

greater than one

because of it is one and all the kind of so

near the plus minus one range would go to zero

and you wouldn't be able to model the data on the thing

so a a and an example of the manifestation of the non linearity i'd take it as a clipping function

a or whatever the linear range and then it's clipped by X max which is a threshold

and

or we set have experiment or out of experiment that other forms of clipping functions of a nonlinear functions

but this is as a very good example

have a discussion

so that the first uh a result that you you what has about a

to fitting ability of the boat C so i selected D fundamental period of a is one point five

and you can see that the clipping threshold of of of the simulated or the clipping function his point one

so we have a a minus point one and plus one one what here

and the

the expansion or or is i

so we see the start line depicts this how forty a bases is basically modeling this

non linearity

then we also see how power

polynomial a normal bases is modeling

you might

but even if i don't to you system distance but the for you based basically you has a five db

it

on the modeling

but that's must

what i'm focusing on a have because then

would have the argument let's increase the model

but in we'll see

the and so on so forth

but it is true that

forty bases is

a contend or and it comes to such model

so now the basis generic a

signal model in the dft domain because of be like to have a the system identification

frequency domain multi-channel forms so that's i we select dft to domain

and uh

because we are going to uh going to the dft domain so we try to

a find a block based definition of the input signal

no here you see or is basically the frame shift in M

a frame size

so in analogy to X

a uh i can

find the block these definition of the non linear in that

input signal

and i do that way

i in this nonlinear mapping to all these individual samples

the this vector

and this for

and uh i can replace this nonlinear mapping by

such a some nation form which i sure and one of the previous slides

i can do this

can be rearrangement spent the summation sign out here out

and then i have this

vector

compact notation X

which is basically the idea that order

off

the nonlinearly mapped input signal

that's a block this definition sort or is made from the eyes or of the bases from

which is in it's

channel from right

now we want to convert

if you do means what we do is you like for you mate

i nation

and to see what use going on we can

a place

a this definition by the summation

which will bring

coefficients

more two

a play

and then we can keep the coefficients outside side and then you would have a higher order

of the non unit the input signal the dft main

uh

if

actual

and uh uh now that we have this

a formal definition

of the input signal or the non email and signal

give you've main we can go a morning

you you but or leaner i R

system

so we basically model and minus are non-zero coefficients

basically a uh to make sure that overlaps safe

strange

remains

that it later on

so this again as forty major

is so

time domain vector

we have

yeah domain con

so uh we know that the observation can you given as a function of the convolution between B

nonlinear mapped mapped input

the equal part

oh as the observation noise

and this right S over a that is that some special

rather sensor

uh a just to linearize the convolution dft domain

the scene worse for year

as this the padding the and for your

i can

by compile this all of the form G

that i can combine G an X to get a to get a C

so C is basically a

constraint

of the non

that's

so uh this there a compact expression

for the dft domain observation

which we can

for the uh

really

to get a equivalent multichannel structure

and that the way you go about doing that is your base this a vector

to to the by the summation expression because we the to right this earlier

and instead of using the summation sign we can use such a a matrix

location

so

this basically couldn't

and i i

do so what yeah the identity matrix to make sure the dimensions are

consist

and instead of then combining a the end of it these components what i words a combined you could possibly

get

effective or virtual it couldn't multichannel four

and uh then i combined your again but this composite matrix text

the second it may trick

and this is

the observation model

a multichannel

position the most

calm

so how does it basically

look uh

dark grammatic

a you this is what

happened

we have combined he's

some

of the nonlinear

and from which has as can be any for your more power

or if anybody

good idea here

and we combine it with a the people above

to get a were true channel had

and then we C D's excitation signal

and

no

appreciate here and i see fans

of using the forty am wanting because

and this is a multichannel identification problem than all the ancient problems of multichannel channel

and if and that's a lot of correlation between these excitation signals than i would be

some works in a problem

and if i have a forty a uh basis then uh all these

a a a a a a a signals the excitation signal quite each and we usually problem

and that would be a very good thing for

convergence

you you to that

so uh uh know and we want to the results and uh what whatever

used

uh for the multichannel channel uh

evaluation

so this is a a not a very fancy other this is

block lms type

a given by

uh used to equations be

a function

and the update equation

you

the step size

and

basically the step-size size contains

a step size for

channel

and which is a function all this adaptation constant

and uh the estimate

of the power spectrum and

and the adaptation constant like in this strange

and its estimate can be achieve are obtained i've such a

a a recursive equation

okay gamma headers the forgetting factor and the range you one

so uh he can

a

for the evaluation was that were operating the multichannel frequency-domain adaptive filter

uh with a frame size of two fit

sex

a frame shift of sixty four

and the linear to nonlinear power issue of the snr and L

given by such an expression this basically differ

the input signal and the nonlinearly mouth

was five you in twenty db

if i have a time just discussed of twenty db case as well as not then

just a

a five but

the signal to observation noise or

in the you cancellation or or you code

observation noise

or a show has been kept

as for the

sixty db because

we want to concentrate on the nonlinear performance

a the robust

because it than your and observation

two types of is as we have a C D

you what for you C D's

and a performance measure would be the relative

uh error signal attenuation given by such an expression

and are also

inspect the estimated nonlinear nothing

not in ins

and the on a mapping you would

track

the nonlinear coefficients C

and that we do uh by this expression which gets just nonlinear

coefficients

and the least squares sense optimal in the least squares

there this stuff you would have a or i is the estimate of the I channel

all the eyes were true channel

so uh this is the performance comparison for uh

fight T case and that would basically mean that the threshold uh

the clipping threshold this plus as point one

so the first uh a algorithm that and using as a anchor is to a linear or uh and stuff

the single channel that stuff that any provision

for each have a or sorry nonlinear processing and easy it converges to the area of

eight db

and then we have to put normal model with how a series

it

uh

uh

a it's it's of some it'll or yeah and then it

but this slow

P convergence

street that used to go up

my believe is

keep on going up somewhere or whatever

and uh but we see that

somewhat slower or and B i

or or comment

polynomial model with gram schmidt

and uh uh we see that me to be the performance should so

and then we finally have to for you model without any additional orthogonalization we see that

somewhat matches

is

a better

then the polynomial normal plus crash

and hence the notion

you have orthogonal input

to the multichannel structure

then you would have this

same effect as the gram schmidt orthogonalization provides

so better convergence and higher

and that's have a look at the quality of the nonlinear that you have

for

so this is the ground truth plusminus point one

and we see that this is the green guy which was a polynomial night model without any gram schmidt

so it is a very of that

so there's a correspondence between the quality of the nonlinear to for a and you have a but we solve

before

and then we see that

putting on on the gram schmidt

a forty T once they are at hand and hand

two was it fringes

this is put a will that crash from goals of it you on both sides

but i'm not so what it about that

for now because

my data is more concentrated in the range plus minus point for

what of course if there's any hope live and the data would be if

in that case

would

consider

for you to be the better one

and this is a a to have time

yeah okay

so that a performance comparisons for twenty db twenty db basically means that my uh

uh

clipping threshold as plus minus

three

and this is a a a a my to the nonlinear case

this means my higher order polynomials

or not

that much and my to the coefficients

this basically means that

my polynomial a model and a point and model because the gram schmidt are having a very nice day

and this

this you see that is a

difference in the rate of convergence and this happens because

the forty a model even if a

might be be mean my

my

the the non linear any but if it is linear

to all the channels of a for a what would be active an adaptation

so that to take it

time

that would show up or take it's still wouldn't the convergence

but still a goes higher

ten the other approaches

and this uh linear model left off is

corresponding

is is

some in of the

and you range as the snr and it

the there is no direct correspondence between the nonlinear in snr

and the virtual source of noise this that is this

still

so as we seen that these two guys were also not perform that bad and

so

or not performing that that and so he's see there are also a really here are also not performing that

but the for you guy was much better than both of them so easy

follows

the ground truth somewhat better

so uh

bring

illusion

and so we uh sort of presented

a a a a four cylinder to of nonlinear hammerstein model

a a the tradition tuition C D's and or or or or on four you D's

we presented the signal model and block frequency domain

which basically uh was to ride by contain bass

channel

derivation

and was for by an efficient multichannel representation again into line on you're fifty

and uh in the results by a multichannel adaptive identification we showed to be orthogonal for us he's

lines up with a polynomial modeling of the gram schmidt orthogonalization

in the sense that it uh a uh uh that's high error signal attenuation

and effectively imitates the underlying nonlinear

oh you mister

so that from and say

Q

i

or we can thanks

is

i help these

so

oh right in your first the results on the gram schmidt response of the polynomial had a fairly high variance

uh

a a lot of fluctuation

yeah

do you have one more fluctuations

uh i and the response here

yeah

could you corpsman because when you we're going that result with the fine db in a to the twenty db

there's virtually no fluctuations at all the twenty db be a yeah i can yeah i and so but

the response is not as good so could you comment or half

somehow have me

difference is the in your response

may may be it yeah be contributing to the change

i i i i i

two my polynomial

model that's

two

this area

right right would be to stay have yeah

so if i have a let's see the twenty db case

and that would mean that

these coefficients

for a polynomial series

don't have a lot of mine to this means that these channels don't have a lot of excitation

so this basically means

a gram schmidt orthogonalization

doesn't have a lot to offer four

doesn't have a lot of influence

and that that like tuition in the fine db can use

could be because there might be some smoothing or something that

a further applied to the grams schmidt organisation

but uh i said that the by something but have been done because that does not focus of uh

what i was actually trying to do

so what i basically we is the difference in performance which does not uh come in the twenty cases because

there's not much room

of and to to the ground truth can provide

because of the depleted excitations those

i or channel

and yeah

the non but not for you

yeah

oh

i i on a say uh yeah okay yeah

i i don't uh for that directly because it this uh off a plate

that you describe like

pose for you consider it

well with all this like this