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