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 |