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