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 multichannel adaptive 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 this modeling that i'm been talking about is about 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 adaptive filtering with resurface 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 a multichannel frequency-domain adaptive 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 uh uh data adaptive orthogonalization 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 bad 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 but he was asking 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