0:00:13phase estimation
0:00:15in detail has many applications
0:00:16such as radar are so no communications and speech and out not process
0:00:23no let's begin it's example
0:00:25yeah this is the example of
0:00:27in on bayesian and phase estimation
0:00:29in this example we demonstrate to a main problems in the general
0:00:33periodic parameter estimation
0:00:37in this example to consider the following more than
0:00:40okay X and observations
0:00:42a the amplitude
0:00:43which is assumed to be known
0:00:46is the unknown parameter
0:00:48this is deterministic part are so we are and non based an estimation
0:00:52and it is between minus and i
0:00:55here we assume that we have a
0:00:57so put a gaussian mean my a complex noise
0:01:00with known fine
0:01:02you can see that this small it is pretty a T
0:01:04with this
0:01:05that that's but to the uh problem
0:01:08in this case it is a on that this is the common at on
0:01:12okay it is proportional to the inverse
0:01:14signal to noise ratio
0:01:16and the maximum that the estimate that was given by the stuff
0:01:21and that all of the
0:01:22sample mean weight it's company
0:01:25now and this data
0:01:27you can see them in school uh are against the signal to noise ratio
0:01:31the that is
0:01:32the come but i found in and and the nine is the mse of the maximum like to estimate or
0:01:38now it can defend that for and now
0:01:41the common lower bound is achievable by the maximum likelihood estimate of
0:01:45that is the common out about but it's of it where the performance
0:01:48in the asymptotic region
0:01:51i uh for less than all
0:01:53you can see that the a and then estimate a close the problem
0:01:57in fact and estimate or between minus a week was the bound because the body you know
0:02:02it goes to infinity
0:02:05a not designed for this phenomenon is that the comment is about that the performance of any unbiased
0:02:12but in this case the maximum likelihood is biased estimate all
0:02:16in fact there is no
0:02:17a uniformly unbiased estimator for this case
0:02:21so the comment on this very good asymptotic region
0:02:25but this is not valid for low snrs
0:02:28okay a and a
0:02:30and to make a request
0:02:35so the conclusion the main purpose in the genital periodic parameter estimation
0:02:40are different
0:02:42first the conventional means got a good deal itself is inappropriate
0:02:46for this estimation
0:02:48a this is illustrated here
0:02:49you can see that if you have to estimate and get
0:02:53and we have a good estimate of it i
0:02:56there is this
0:02:59and the mse use bits to use this uh
0:03:02how important that until we discuss you know
0:03:07so we should do that we should not let us instead of the mse
0:03:12to make some is the prior
0:03:15the second the second problem is that no uniformly unbiased estimator exists
0:03:21in this case
0:03:22and this is it right for the phase estimation problem
0:03:26uses at and periodic parameter estimation
0:03:30the periodic likelihood function
0:03:32in a it's a have been proved in this too
0:03:37no not at non bayesian estimation
0:03:39in the minimization of the mse
0:03:42another criterion
0:03:43should be done under some restriction and the stuff constraint
0:03:47the crime and unbiasedness
0:03:49is it because no unbiased estimate like this
0:03:52so we should find another constraint another station
0:03:58S was set to cram it may not be valued at low snrs and we want to find
0:04:03to do that
0:04:04which will do that at any snr
0:04:08and this is exactly what we did in this right
0:04:11we have a square periodic uh i'm this inequality on
0:04:15instead of tennessee
0:04:18a predefined periodic unbiasedness
0:04:21and the constraint instead of this constraint
0:04:25and the newer version of the content
0:04:28which is a valid that any snr
0:04:32no yeah can see the general what what in this work
0:04:36yes just that our product your data
0:04:38is that a many stick
0:04:40and this is between minus point by
0:04:42but is is on the for the sake of simplicity you can take any
0:04:46time period
0:04:48yes of the parabola space in which are made it's is the observation space
0:04:52is the family of per emails
0:04:54prom a tight by to by the unknown product or
0:04:59a is the hundred observation vector and P that is an estimate of that
0:05:03which is function from the observation space to minus by
0:05:08now we not that even if the estimate all is restricted to the original of
0:05:12minus by by
0:05:14and the parameter at is also this a region
0:05:17the is that of estimation L bit that minus the data
0:05:21can be in general and in
0:05:23minus two part about
0:05:24so we should of the than the part of weight
0:05:29yeah we use this quickly and the mean square to calculate you're
0:05:33the S P it cost function is given here this is the square error of the preview a
0:05:38estimation in or remote able to buy or
0:05:41the of the estimation or
0:05:44yeah the model but to by april or map
0:05:47the estimation L to more by by
0:05:50and you can see here
0:05:51the main
0:05:52scalability that the S P against
0:05:54but yeah this is pretty loaded
0:05:57a non-negative and and
0:05:59is a better and that's a non convex
0:06:04no of to define
0:06:05the P the can best miss
0:06:07and is the and the phonation for one best mess
0:06:10and this the phonation is a a and buys this with respect to specific cost function
0:06:16according to this definition and have to make but we said to be yeah have not by that
0:06:21with respect to the cost function and
0:06:24if is the expectation
0:06:26a type like it a like to two parameter
0:06:29of this cost function
0:06:31if you is the true parameter but that it is there and them
0:06:35and they have a parameter is that and the parameters
0:06:39i the right
0:06:40and estimator is on
0:06:42if it was closer
0:06:43to that's for parameter
0:06:45but and then i mean i have a problem of in our problem space
0:06:49the closeness
0:06:50is a measure of using the specific cost function K
0:06:56the basic example sample for this a a i'm was of the best the conventional and bias net
0:07:01and i'm that the mean square error cost function that unbiasedness is not you to
0:07:06no no by smith
0:07:07the expectation of the estimate the is equal to the to a parameter it's
0:07:12so yeah no the phonation channel a i well known
0:07:15min and by a
0:07:17to and that's this on their
0:07:18and apply to
0:07:20cost function
0:07:22as i said in this work we are interested in unbiased by under the S P cost function
0:07:28and in this case
0:07:29this is the
0:07:30here and as this condition
0:07:33in addition in this work we assume that we have continuous
0:07:37that is estimate of
0:07:40with that existing probability density function B D S
0:07:44okay F
0:07:45with the high
0:07:45of the estimate parameter by paper
0:07:49and that this assumption
0:07:53plus the this to conditions
0:07:55the first condition is that the expectation of the pretty a K is the old
0:08:01so that a and the average we have
0:08:04the or periodic estimation all
0:08:06and the second condition is that
0:08:08a a in this a signal and the project of the estimate that is lower than one divided be two
0:08:15a the form and them and i said that an estimate with periodic unbiased
0:08:19i know that is
0:08:20to conditions are satisfied
0:08:24and here you can see the difference between mean and by
0:08:27but you the combat
0:08:29in the previous example of a phase estimation i said that no uniform an unbiased estimate legs the
0:08:37yeah if the this set of estimate are that the can by
0:08:40so a pretty good the mad estimator exist
0:08:43and in particular the max some like to estimate of itself is periodic and by S
0:08:48you can see here a bias of
0:08:51the max like estimate the cans
0:08:53but yeah
0:08:54but more line
0:08:55is the conventional and by us and
0:08:58you can say that the max like to estimate of is by that
0:09:01the biggest problem is that all
0:09:04in big if the P L S
0:09:06you can see that the maxima estimator is periodic unbiased estimator
0:09:11in this case
0:09:13no i want to do i knew
0:09:17we bound the mspe mean-square politically or
0:09:21of any and
0:09:22by by put that to by an estimate of the that i
0:09:25okay i the sound
0:09:26i the probability condition
0:09:29and the bound is a given here this is the preview or the calm i one
0:09:33this of the crime that our bound apply
0:09:36by this fact or but this nonnegative
0:09:39the come our boundaries
0:09:40of course that this is the best of the fisher information
0:09:44okay and this fact
0:09:46and have applied the common how
0:09:49this is a new bound
0:09:53let's see some of its but what D
0:09:55but but is and the first property is
0:09:57that the new about the period of and this valid that any signal-to-noise ratio
0:10:02well i is three style the come at all about may not be it
0:10:07the second part of the is that the you bound is always lower will it but to the problem of
0:10:11our bound
0:10:13for unbiased estimate
0:10:16this can be seen here yeah
0:10:18we have the kind of a applied to this fact
0:10:21and this um
0:10:23according to the set condition and the
0:10:25but you the can but this condition
0:10:28this them should be lower than one divided be two but
0:10:31and of course this is a non-negative them
0:10:33so all this fact there is between zero and one
0:10:37so i are is that was level
0:10:40however i remember that the common are bound is not to provide bad bound
0:10:43for political estimation
0:10:45so actually this factor keeps
0:10:47are are bound to paint to permit it
0:10:49permit of the region
0:10:50in 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
0:11:00of the common are bound to in with a bound for periodic a and by if a all
0:11:06and you can see here
0:11:08the by a
0:11:09a of a a a a high about this specific by a
0:11:12is that then they got to i'm about with the a constrained of periodic and my
0:11:18to can so it
0:11:20and this is a surprising
0:11:22because our bounds a bound of the pretty good performance
0:11:25on the mspe
0:11:27and the kind of a bound is about bounded the non periodic performance of the M E
0:11:31so this is not a trivial
0:11:36finally a
0:11:37in a similar manner
0:11:39we can do that the bound for a vector
0:11:41parameter estimation
0:11:43and also for weeks
0:11:45the all parameter estimation in which
0:11:47part of the product or a periodic and part
0:11:50of the parameters are not but
0:11:52you can see have for example if we have
0:11:54to parameters
0:11:55one of them is
0:11:57in you your that can run as well L
0:11:59and the estimate are also with the same
0:12:04yeah the following to
0:12:05constraint was the prove you the can best that's constraint
0:12:08for a are one
0:12:10and the main by the school constraint for pick up to
0:12:13and and of this constraint
0:12:16are a matrix bound
0:12:17is the from nine
0:12:18the covariance matrix
0:12:20or of the
0:12:21and a a spectral okay the but that is that a a a a big error
0:12:25for the periodic part of of people one
0:12:28and the
0:12:29yeah irregular four
0:12:30non part department
0:12:33so the covariance matrix of
0:12:35the aspect of is
0:12:36where or equal to this data
0:12:39a image which J
0:12:40is the fisher information matrix
0:12:42use the inverse
0:12:44of this matrix
0:12:45and you have a is that they have an automatic
0:12:48in H
0:12:49a a for a not of the parameter
0:12:52we have one
0:12:53and for a and the P from P test we have this data
0:12:58okay so we can use
0:12:59i the
0:13:01for any
0:13:02well i'm not a not base some parameter estimation problems to
0:13:05but got the call non the parameters
0:13:08and for a vector or a scalar estimation
0:13:12okay okay
0:13:15yeah can see an example and and
0:13:17this example use
0:13:19example of of that to a parameter estimation
0:13:22we want to estimate
0:13:23i a
0:13:26the phase fee
0:13:28we have a known frequency on as they are
0:13:31a a gaussian noise
0:13:34in in this case again we have very low
0:13:37i'm by april
0:13:39but we don't have
0:13:41conventional mean unbiased biased estimators
0:13:44the crown a lower bound to metrics
0:13:47is given here
0:13:48and this is they have a not matrix in this case
0:13:51and there are
0:13:52bound is given a yeah
0:13:54you can say this fact or C N
0:13:57and for um is calculated using the
0:14:00but it of the function of the maximum likelihood estimate them
0:14:03which is that the
0:14:05or are known for this
0:14:11so here is a yeah can see
0:14:15the uh is that
0:14:17the phillies estimation
0:14:19okay that really bothered
0:14:22yeah this is the N S P in this paper you are to get against snr
0:14:27but black time is our uh about
0:14:29and the part line is the common out
0:14:32this is the line
0:14:34oh a
0:14:35the performance of
0:14:36the unbiased put the to combat the estimator
0:14:40but and i mean
0:14:41in the maximum an accurate estimate of
0:14:44and can can that a a other bound to about it
0:14:47if any snr
0:14:49well i the kind of lower bound is not valid
0:14:52hmmm know with an all
0:14:53yeah hmmm in the so that the bound and the and the speed of the estimate of the of the
0:14:58pen of the ferry
0:15:00it a can the crime a bound and
0:15:03are about
0:15:04a a of a then by the that code estimate of for a high snrs
0:15:08okay but
0:15:09i about you was
0:15:11fit of added
0:15:16to can close in this all the concept of non bayesian periodic parameter estimation was it introduce
0:15:23the periodic unbiased and that i
0:15:25S P E square periodic it'll cost function
0:15:28well as defined here as in the lemon definition for one best man
0:15:32the project S and as of the common are bound for P you gotta parameter
0:15:36for a mixed
0:15:38the periodic and non
0:15:40the vector parameter estimation were developed
0:15:44a S we said that
0:15:45put a to come as i don't provided that at lower bound in political mention
0:15:49and uh a a at this so that
0:15:52it but i as some periodic unbiased estimate all's and in a bound for phase estimation
0:15:59from and different phase estimation
0:16:01and a kind of yeah king
0:16:04the relation of the but i can like
0:16:06periodic bound
0:16:07which supposed to be tried to of and the common are about
0:16:11a a and also on a hybrid balance
0:16:14based on in on a bayesian
0:16:17and they finally about the periodic minimax estimation
0:16:21thank you
0:16:29we have sorry for a cushion
0:16:44the bound is not a function of
0:16:46the estimate but it is a function of its statistics
0:16:50but you in specific point
0:16:53this is a of it's a statistic
0:17:34it to about it yes i
0:17:36a a a a the bounds of the bounds as function of it also the colour of are bound not
0:17:41only input public bounded estimation
0:17:43of the parameter i yeah
0:17:46but uh at
0:17:47it's not a function the estimate of okay you this is on the function of
0:17:51it it's that is its properties
0:18:03this this you know we use
0:18:04but is a small or not
0:18:07that's what is it for a new and all of a only use like on the
0:18:12since you bound to used used
0:18:16soon as much
0:18:18but you know that you're is most useful
0:18:25i say that is the L B is not used for because this is the bottle and biased estimate
0:18:30but we don't have any unbiased estimate of
0:18:33no not okay use is some small uh our knowledge
0:18:36uh know about the the problem is that the
0:18:39mse S them it's got a quite it says is inappropriate
0:18:42okay if i
0:18:46you are i'm pretty good
0:18:48you have periodic parameter like
0:18:50and again like do you way like
0:18:52and this periodic parameter estimation
0:18:55and we want to estimate to
0:18:57the ms E is "'cause" yeah could you on that
0:19:00min makes you is this uh
0:19:03and this is not
0:19:04a a a a a perfect for prosodic parameters mention
0:19:08okay yeah
0:19:09we really it you are
0:19:11it and but with them and of and the printer to care of them are denoted by all
0:19:16and also that is it the chi bounds and all the bounds our bounds on the mse
0:19:53so so the problem only not only low snrs
0:19:57but that's
0:19:58the problem is that the common a bound is not a and the problem is not that the bound is
0:20:02more tight
0:20:03okay you we can see
0:20:06our example
0:20:08okay that the that is not that type of the bar
0:20:12okay the problem is that
0:20:13and an estimate of request the bound
0:20:16"'cause" so this is not a valid bound that all
0:20:26but you again