0:00:13 | phase estimation |
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0:00:15 | in detail has many applications |

0:00:16 | such as radar are so no communications and speech and out not process |

0:00:23 | no let's begin it's example |

0:00:25 | yeah this is the example of |

0:00:27 | in on bayesian and phase estimation |

0:00:29 | in this example we demonstrate to a main problems in the general |

0:00:33 | periodic parameter estimation |

0:00:36 | so |

0:00:37 | in this example to consider the following more than |

0:00:40 | okay X and observations |

0:00:42 | a the amplitude |

0:00:43 | which is assumed to be known |

0:00:45 | data |

0:00:46 | is the unknown parameter |

0:00:48 | this is deterministic part are so we are and non based an estimation |

0:00:52 | and it is between minus and i |

0:00:55 | here we assume that we have a |

0:00:57 | so put a gaussian mean my a complex noise |

0:01:00 | with known fine |

0:01:02 | you can see that this small it is pretty a T |

0:01:04 | with this |

0:01:05 | the |

0:01:05 | that that's but to the uh problem |

0:01:08 | in this case it is a on that this is the common at on |

0:01:12 | okay it is proportional to the inverse |

0:01:14 | signal to noise ratio |

0:01:16 | and the maximum that the estimate that was given by the stuff |

0:01:20 | is |

0:01:21 | and that all of the |

0:01:22 | sample mean weight it's company |

0:01:25 | now and this data |

0:01:27 | you can see them in school uh are against the signal to noise ratio |

0:01:31 | the that is |

0:01:32 | the come but i found in and and the nine is the mse of the maximum like to estimate or |

0:01:38 | now it can defend that for and now |

0:01:41 | the common lower bound is achievable by the maximum likelihood estimate of |

0:01:45 | that is the common out about but it's of it where the performance |

0:01:48 | in the asymptotic region |

0:01:51 | i uh for less than all |

0:01:53 | you can see that the a and then estimate a close the problem |

0:01:57 | in fact and estimate or between minus a week was the bound because the body you know |

0:02:02 | it goes to infinity |

0:02:05 | a not designed for this phenomenon is that the comment is about that the performance of any unbiased |

0:02:11 | estimator |

0:02:12 | but in this case the maximum likelihood is biased estimate all |

0:02:16 | in fact there is no |

0:02:17 | a uniformly unbiased estimator for this case |

0:02:21 | so the comment on this very good asymptotic region |

0:02:25 | but this is not valid for low snrs |

0:02:28 | okay a and a |

0:02:30 | and to make a request |

0:02:34 | okay |

0:02:35 | so the conclusion the main purpose in the genital periodic parameter estimation |

0:02:40 | are different |

0:02:42 | first the conventional means got a good deal itself is inappropriate |

0:02:46 | for this estimation |

0:02:48 | a this is illustrated here |

0:02:49 | you can see that if you have to estimate and get |

0:02:52 | data |

0:02:53 | and we have a good estimate of it i |

0:02:56 | hmmm |

0:02:56 | there is this |

0:02:59 | and the mse use bits to use this uh |

0:03:02 | how important that until we discuss you know |

0:03:07 | so we should do that we should not let us instead of the mse |

0:03:12 | to make some is the prior |

0:03:15 | the second the second problem is that no uniformly unbiased estimator exists |

0:03:21 | in this case |

0:03:22 | and this is it right for the phase estimation problem |

0:03:26 | uses at and periodic parameter estimation |

0:03:30 | the periodic likelihood function |

0:03:32 | in a it's a have been proved in this too |

0:03:37 | no not at non bayesian estimation |

0:03:39 | in the minimization of the mse |

0:03:42 | another criterion |

0:03:43 | should be done under some restriction and the stuff constraint |

0:03:47 | the crime and unbiasedness |

0:03:49 | is it because no unbiased estimate like this |

0:03:52 | so we should find another constraint another station |

0:03:57 | finally |

0:03:58 | S was set to cram it may not be valued at low snrs and we want to find |

0:04:03 | to do that |

0:04:04 | which will do that at any snr |

0:04:08 | and this is exactly what we did in this right |

0:04:11 | we have a square periodic uh i'm this inequality on |

0:04:15 | instead of tennessee |

0:04:18 | a predefined periodic unbiasedness |

0:04:21 | and the constraint instead of this constraint |

0:04:25 | and the newer version of the content |

0:04:28 | which is a valid that any snr |

0:04:32 | no yeah can see the general what what in this work |

0:04:36 | yes just that our product your data |

0:04:38 | is that a many stick |

0:04:40 | and this is between minus point by |

0:04:42 | but is is on the for the sake of simplicity you can take any |

0:04:46 | time period |

0:04:48 | yes of the parabola space in which are made it's is the observation space |

0:04:51 | P |

0:04:52 | is the family of per emails |

0:04:54 | prom a tight by to by the unknown product or |

0:04:59 | a is the hundred observation vector and P that is an estimate of that |

0:05:03 | which is function from the observation space to minus by |

0:05:08 | now we not that even if the estimate all is restricted to the original of |

0:05:12 | minus by by |

0:05:14 | and the parameter at is also this a region |

0:05:17 | the is that of estimation L bit that minus the data |

0:05:21 | can be in general and in |

0:05:23 | minus two part about |

0:05:24 | so we should of the than the part of weight |

0:05:29 | and |

0:05:29 | yeah we use this quickly and the mean square to calculate you're |

0:05:33 | the S P it cost function is given here this is the square error of the preview a |

0:05:38 | estimation in or remote able to buy or |

0:05:41 | the of the estimation or |

0:05:44 | yeah the model but to by april or map |

0:05:47 | the estimation L to more by by |

0:05:50 | and you can see here |

0:05:51 | the main |

0:05:52 | scalability that the S P against |

0:05:54 | but yeah this is pretty loaded |

0:05:57 | a non-negative and and |

0:05:59 | is a better and that's a non convex |

0:06:04 | no of to define |

0:06:05 | the P the can best miss |

0:06:07 | and is the and the phonation for one best mess |

0:06:10 | and this the phonation is a a and buys this with respect to specific cost function |

0:06:16 | according to this definition and have to make but we said to be yeah have not by that |

0:06:21 | with respect to the cost function and |

0:06:24 | if is the expectation |

0:06:26 | a type like it a like to two parameter |

0:06:29 | of this cost function |

0:06:31 | if you is the true parameter but that it is there and them |

0:06:35 | and they have a parameter is that and the parameters |

0:06:39 | i the right |

0:06:40 | and estimator is on |

0:06:42 | if it was closer |

0:06:43 | to that's for parameter |

0:06:45 | but and then i mean i have a problem of in our problem space |

0:06:49 | the closeness |

0:06:50 | is a measure of using the specific cost function K |

0:06:55 | okay |

0:06:56 | the basic example sample for this a a i'm was of the best the conventional and bias net |

0:07:01 | and i'm that the mean square error cost function that unbiasedness is not you to |

0:07:06 | no no by smith |

0:07:07 | the expectation of the estimate the is equal to the to a parameter it's |

0:07:12 | so yeah no the phonation channel a i well known |

0:07:15 | min and by a |

0:07:17 | to and that's this on their |

0:07:18 | and apply to |

0:07:20 | cost function |

0:07:22 | as i said in this work we are interested in unbiased by under the S P cost function |

0:07:28 | and in this case |

0:07:29 | this is the |

0:07:30 | here and as this condition |

0:07:33 | in addition in this work we assume that we have continuous |

0:07:36 | estimator |

0:07:37 | that is estimate of |

0:07:40 | with that existing probability density function B D S |

0:07:44 | okay F |

0:07:45 | with the high |

0:07:45 | of the estimate parameter by paper |

0:07:49 | and that this assumption |

0:07:51 | that |

0:07:51 | condition |

0:07:52 | can |

0:07:53 | plus the this to conditions |

0:07:55 | the first condition is that the expectation of the pretty a K is the old |

0:08:01 | so that a and the average we have |

0:08:03 | it |

0:08:04 | the or periodic estimation all |

0:08:06 | and the second condition is that |

0:08:08 | a a in this a signal and the project of the estimate that is lower than one divided be two |

0:08:15 | a the form and them and i said that an estimate with periodic unbiased |

0:08:19 | i know that is |

0:08:20 | to conditions are satisfied |

0:08:24 | and here you can see the difference between mean and by |

0:08:27 | but you the combat |

0:08:29 | in the previous example of a phase estimation i said that no uniform an unbiased estimate legs the |

0:08:35 | but |

0:08:36 | i |

0:08:37 | yeah if the this set of estimate are that the can by |

0:08:40 | so a pretty good the mad estimator exist |

0:08:43 | and in particular the max some like to estimate of itself is periodic and by S |

0:08:48 | you can see here a bias of |

0:08:51 | the max like estimate the cans |

0:08:53 | but yeah |

0:08:54 | but more line |

0:08:55 | is the conventional and by us and |

0:08:58 | you can say that the max like to estimate of is by that |

0:09:01 | the biggest problem is that all |

0:09:02 | yeah |

0:09:04 | in big if the P L S |

0:09:06 | you can see that the maxima estimator is periodic unbiased estimator |

0:09:11 | in this case |

0:09:13 | no i want to do i knew |

0:09:16 | int |

0:09:17 | we bound the mspe mean-square politically or |

0:09:21 | of any and |

0:09:22 | by by put that to by an estimate of the that i |

0:09:25 | okay i the sound |

0:09:26 | i the probability condition |

0:09:29 | and the bound is a given here this is the preview or the calm i one |

0:09:33 | this of the crime that our bound apply |

0:09:36 | by this fact or but this nonnegative |

0:09:39 | the come our boundaries |

0:09:40 | of course that this is the best of the fisher information |

0:09:44 | okay and this fact |

0:09:46 | and have applied the common how |

0:09:49 | this is a new bound |

0:09:53 | let's see some of its but what D |

0:09:55 | but but is and the first property is |

0:09:57 | that the new about the period of and this valid that any signal-to-noise ratio |

0:10:02 | well i is three style the come at all about may not be it |

0:10:07 | the second part of the is that the you bound is always lower will it but to the problem of |

0:10:11 | our bound |

0:10:13 | for unbiased estimate |

0:10:15 | and |

0:10:16 | this can be seen here yeah |

0:10:18 | we have the kind of a applied to this fact |

0:10:21 | and this um |

0:10:23 | according to the set condition and the |

0:10:25 | but you the can but this condition |

0:10:28 | this them should be lower than one divided be two but |

0:10:31 | and of course this is a non-negative them |

0:10:33 | so all this fact there is between zero and one |

0:10:37 | so i are is that was level |

0:10:40 | however i remember that the common are bound is not to provide bad bound |

0:10:43 | for political estimation |

0:10:45 | so actually this factor keeps |

0:10:47 | are are bound to paint to permit it |

0:10:49 | permit of the region |

0:10:50 | in divided region |

0:10:54 | "'kay" that that but is that's the con that our bound so mean biased estimate of the have does it |

0:11:00 | of the common are bound to in with a bound for periodic a and by if a all |

0:11:06 | and you can see here |

0:11:08 | the by a |

0:11:09 | a of a a a a high about this specific by a |

0:11:12 | is that then they got to i'm about with the a constrained of periodic and my |

0:11:17 | this |

0:11:18 | to can so it |

0:11:20 | and this is a surprising |

0:11:22 | because our bounds a bound of the pretty good performance |

0:11:25 | on the mspe |

0:11:27 | and the kind of a bound is about bounded the non periodic performance of the M E |

0:11:31 | so this is not a trivial |

0:11:36 | finally a |

0:11:37 | in a similar manner |

0:11:39 | we can do that the bound for a vector |

0:11:41 | parameter estimation |

0:11:43 | and also for weeks |

0:11:45 | the all parameter estimation in which |

0:11:47 | part of the product or a periodic and part |

0:11:50 | of the parameters are not but |

0:11:52 | you can see have for example if we have |

0:11:54 | to parameters |

0:11:55 | one of them is |

0:11:57 | in you your that can run as well L |

0:11:59 | and the estimate are also with the same |

0:12:02 | nature |

0:12:04 | yeah the following to |

0:12:05 | constraint was the prove you the can best that's constraint |

0:12:08 | for a are one |

0:12:10 | and the main by the school constraint for pick up to |

0:12:13 | and and of this constraint |

0:12:16 | are a matrix bound |

0:12:17 | is the from nine |

0:12:18 | the covariance matrix |

0:12:20 | or of the |

0:12:21 | and a a spectral okay the but that is that a a a a big error |

0:12:25 | for the periodic part of of people one |

0:12:28 | and the |

0:12:29 | yeah irregular four |

0:12:30 | non part department |

0:12:33 | so the covariance matrix of |

0:12:35 | the aspect of is |

0:12:36 | where or equal to this data |

0:12:39 | a image which J |

0:12:40 | is the fisher information matrix |

0:12:42 | use the inverse |

0:12:44 | of this matrix |

0:12:45 | and you have a is that they have an automatic |

0:12:48 | in H |

0:12:49 | a a for a not of the parameter |

0:12:52 | we have one |

0:12:53 | and for a and the P from P test we have this data |

0:12:58 | okay so we can use |

0:12:59 | i the |

0:13:01 | for any |

0:13:02 | well i'm not a not base some parameter estimation problems to |

0:13:05 | but got the call non the parameters |

0:13:08 | and for a vector or a scalar estimation |

0:13:12 | okay okay |

0:13:14 | okay |

0:13:15 | yeah can see an example and and |

0:13:17 | this example use |

0:13:19 | example of of that to a parameter estimation |

0:13:22 | we want to estimate |

0:13:23 | i a |

0:13:25 | and |

0:13:26 | the phase fee |

0:13:28 | we have a known frequency on as they are |

0:13:31 | and |

0:13:31 | a a gaussian noise |

0:13:34 | in in this case again we have very low |

0:13:37 | i'm by april |

0:13:39 | but we don't have |

0:13:41 | conventional mean unbiased biased estimators |

0:13:44 | the crown a lower bound to metrics |

0:13:47 | is given here |

0:13:48 | and this is they have a not matrix in this case |

0:13:51 | and there are |

0:13:52 | bound is given a yeah |

0:13:54 | you can say this fact or C N |

0:13:57 | and for um is calculated using the |

0:14:00 | but it of the function of the maximum likelihood estimate them |

0:14:03 | which is that the |

0:14:05 | or are known for this |

0:14:11 | so here is a yeah can see |

0:14:14 | E |

0:14:15 | the uh is that |

0:14:17 | for |

0:14:17 | the phillies estimation |

0:14:19 | okay that really bothered |

0:14:21 | and |

0:14:22 | yeah this is the N S P in this paper you are to get against snr |

0:14:26 | the |

0:14:27 | but black time is our uh about |

0:14:29 | and the part line is the common out |

0:14:32 | this is the line |

0:14:34 | oh a |

0:14:35 | the performance of |

0:14:36 | the unbiased put the to combat the estimator |

0:14:40 | but and i mean |

0:14:41 | in the maximum an accurate estimate of |

0:14:44 | and can can that a a other bound to about it |

0:14:47 | if any snr |

0:14:49 | well i the kind of lower bound is not valid |

0:14:52 | hmmm know with an all |

0:14:53 | yeah hmmm in the so that the bound and the and the speed of the estimate of the of the |

0:14:58 | pen of the ferry |

0:15:00 | and |

0:15:00 | it a can the crime a bound and |

0:15:03 | are about |

0:15:04 | a a of a then by the that code estimate of for a high snrs |

0:15:08 | okay but |

0:15:09 | i about you was |

0:15:11 | fit of added |

0:15:12 | here |

0:15:16 | to can close in this all the concept of non bayesian periodic parameter estimation was it introduce |

0:15:23 | the periodic unbiased and that i |

0:15:25 | S P E square periodic it'll cost function |

0:15:28 | well as defined here as in the lemon definition for one best man |

0:15:32 | the project S and as of the common are bound for P you gotta parameter |

0:15:36 | for a mixed |

0:15:38 | the periodic and non |

0:15:40 | the vector parameter estimation were developed |

0:15:44 | a S we said that |

0:15:45 | put a to come as i don't provided that at lower bound in political mention |

0:15:49 | and uh a a at this so that |

0:15:52 | it but i as some periodic unbiased estimate all's and in a bound for phase estimation |

0:15:57 | a |

0:15:59 | from and different phase estimation |

0:16:01 | and a kind of yeah king |

0:16:03 | on |

0:16:04 | the relation of the but i can like |

0:16:06 | periodic bound |

0:16:07 | which supposed to be tried to of and the common are about |

0:16:11 | a a and also on a hybrid balance |

0:16:14 | based on in on a bayesian |

0:16:16 | parameter |

0:16:17 | and they finally about the periodic minimax estimation |

0:16:21 | thank you |

0:16:23 | but |

0:16:29 | we have sorry for a cushion |

0:16:43 | a |

0:16:44 | the bound is not a function of |

0:16:46 | the estimate but it is a function of its statistics |

0:16:49 | of |

0:16:50 | but you in specific point |

0:16:53 | this is a of it's a statistic |

0:16:55 | properties |

0:17:02 | uh_huh |

0:17:10 | hmmm |

0:17:24 | uh_huh |

0:17:34 | it to about it yes i |

0:17:36 | a a a a the bounds of the bounds as function of it also the colour of are bound not |

0:17:41 | only input public bounded estimation |

0:17:43 | of the parameter i yeah |

0:17:44 | cool |

0:17:46 | but uh at |

0:17:47 | it's not a function the estimate of okay you this is on the function of |

0:17:51 | it it's that is its properties |

0:17:56 | yeah |

0:18:03 | this this you know we use |

0:18:04 | but is a small or not |

0:18:06 | um |

0:18:07 | that's what is it for a new and all of a only use like on the |

0:18:12 | since you bound to used used |

0:18:16 | soon as much |

0:18:18 | but you know that you're is most useful |

0:18:25 | i say that is the L B is not used for because this is the bottle and biased estimate |

0:18:30 | but we don't have any unbiased estimate of |

0:18:33 | no not okay use is some small uh our knowledge |

0:18:36 | uh know about the the problem is that the |

0:18:39 | mse S them it's got a quite it says is inappropriate |

0:18:42 | okay if i |

0:18:46 | you are i'm pretty good |

0:18:48 | you have periodic parameter like |

0:18:50 | and again like do you way like |

0:18:52 | and this periodic parameter estimation |

0:18:55 | and we want to estimate to |

0:18:56 | okay |

0:18:57 | the ms E is "'cause" yeah could you on that |

0:19:00 | min makes you is this uh |

0:19:02 | okay |

0:19:03 | and this is not |

0:19:04 | a a a a a perfect for prosodic parameters mention |

0:19:08 | okay yeah |

0:19:09 | we really it you are |

0:19:11 | it and but with them and of and the printer to care of them are denoted by all |

0:19:16 | and also that is it the chi bounds and all the bounds our bounds on the mse |

0:19:50 | yeah |

0:19:51 | uh_huh |

0:19:53 | okay |

0:19:53 | so so the problem only not only low snrs |

0:19:57 | but that's |

0:19:58 | the problem is that the common a bound is not a and the problem is not that the bound is |

0:20:02 | more tight |

0:20:03 | okay you we can see |

0:20:06 | our example |

0:20:08 | okay that the that is not that type of the bar |

0:20:12 | okay the problem is that |

0:20:13 | and an estimate of request the bound |

0:20:16 | "'cause" so this is not a valid bound that all |

0:20:22 | okay |

0:20:25 | okay |

0:20:26 | but you again |

0:20:27 | you |