good afternoon everyone

yeah a hand i got each

said

uh

oh

hmmm

location

what

i

this work

uh is a joint with them

a a to patrol boat and a cup or

and was sort the uh with the short back

uh on the system that you're talking about

uh we will make should use the optimization them match to that to are going to uh exploit here which

is the cramer-rao around uh bound um metric

uh uh we're going to use this one two

uh formulate the power allocation optimization problem

where the objective is to minimize the total transmitted power in a a

my more L multiple radar uh a vector

uh

to get an efficient to

solution for that we use the the make the composition for that

and it will use a some american analysis to show how uh

uh the power location

uh is generated

and finally some concluding remarks

a a target a localisation uh yeah estimation mean-square square error is known to be lower bounded by the cramer-rao

lower bound especially if we talk about that maximum likelihood estimator

and uh based on this metric it has been shown in the past that um

a system with widely the separated

uh uh and most people uh uh or

uh

and the and systems

using a coherent or non-coherent processing

uh offers advantages in terms of the estimation is square

and

that you're see again is proportional to the product of the number of transmit and receive antenna

i i in general though if we expand this dependency the mean square error depends on

as a set and the number of transmit and receive radars

but it also depends on the geometric metric layout

of of the transmit and receive radars with respect to the target

uh it depends on the uh signal effective bandwidth

and on the signal to noise ratio and that brings us to the

transmit power which were going to focus in this store

oh

that the the of that we're looking at is new

a widely separated

multiple radar system

that is uh mobile

and we see more and more application like that

uh one example is the

ground surveillance radar

where we have a

yeah a mounted and vehicles

that are spread the along borders we can see them mean um yeah pro controls and things like that

a in this type of

uh application

yeah it makes sense to be more conscious

about are the use of resources

and uh uh what you cover in uh

this work is

uh resource uh awareness in terms of the transmitted power

so our objective here

as we can see here used to minimize

the total transmitted power such that the predetermined estimation mean-square square error is a thing

uh uh white and the transmit power

at each a station within an acceptable range

so well we not that yeah

extending the

the number of of

uh rate hours

provides a higher accuracy in practical we need some level of your see which can can serve as a trash

threshold

and the question is how do we minimize the powers

the system before form a a at the threshold that we one

uh and this is what we going to do

so this is the system that we talking in in this figure

uh

an example of such a system and we keep it very general

in terms that

our readers can be

you or transmit or or cu

a or or can be vote i mean each one of this point can be a transmit receive a radar

and assumption something that the uh

are the information jointly so

from all of this element

and you have a target uh

here that we want to estimate its location

specifically a of before

with assume we have a a a as a C M transmit radars and receive radar

the target is modelled as an extended target

with the center mass located uh

position

S

we use of and all signal and assume we have

M and and the had propagation path

uh the transmitted power vector is given here as P of T uh

for each one of the transmitting ten

uh as we know the

time delay of propagation of each one of

pat

uh a time is a function of the range from to transmitting uh uh radar

uh to the target

and from the

target to the receive radar also

tao

and and basically a a measure those time delays for example if we use this one

as a transmit

to the target

and received here this would be

uh

this propagation that would be proportional to the

range sample

uh this

brings us to the received signal

on the specific path and pat to "'em" and

and we see that i went to that you take into account uh a in our model which we have

here off i of and basically is proportional to

uh the path so it was that the path loss

uh P of T in P of M T X is the transmitted power

a a or friends then it's is a complex coefficient

basically takes into account the

uh rate cross section on to M and

a plus and any phase offsets and this path

uh we have your uh uh uh delayed

time delayed version of the transmitted power

that's transmitted signals or and

oh a white gaussian the

voice

um

we actually a defined all of this

so we can find some metric you said that

the constrained our system are giving in terms of

square

so we need to find a a a a metric that labour enable us to

uh represent this man

for this were using the cramer-rao bound

i where we using the trace of the cramer-rao bound metrics

uh two

provide the bound on the mean square on the X action

and and the white direction one

uh a the previous work

trade between the two so

uh

optimising one of them

we just uh maximise the that

um

the the around on that because it was developed in previous studies it's not and you result

uh what we have to do your do is uh

re

state it

so it can be used to optimize power

so what we did here is we talk

the original expression and defined it as a sum

of some

elements here multiplied by the pope power transmitted by transmitter am and have end of the

i if we go one step forward and you can see your by the way

that the elements of this matrix

are dependent on off a age

which were uh what the code channel correct for state on path and man

and um

we have you the location of the transmit and receive radars with respect to the target

uh uh incorporated through this expression which are basically cosine and sign

of the angles between the transmitter and receiver to the target

uh the vector of a in this case

you use the target location X Y

and the channel

vector eight

uh using this type of uh

expression and us was us to

expressed the trace of the cramer-rao bound

in the form that you can see here

well basically we have some vector B

multiplying the vector of power

and in the denominator we have

a a uh metrics eight

that

second second order

expression for the the same power

and it you can see that basically be and any incorporate

all the existing system

a just the geometric spread the channel

that fading and so forth so these are

oh coming in to play through uh metrics as

metrics a and vector B

now that we have an expression for the cramer-rao bound we can formulate

position

uh a and as we said our objective is

uh uh that given a a predetermined threshold

but is if a mean square error or

uh we would like to optimize

uh

uh the the pa out basically um

minimize the total contrast

and this is the mathematical formulation for that

so we we minimize the total transmitted power

a a given a specific threshold are the cramer-rao bound where

uh we use of previous uh

estimate of the target location in age

to calculate

C

and also need for some limitation under transmitted power we we assume as you we transmit the minimal power

uh P T X minimum and the maximum power uh uh uh

P M T X max

ah

taking

just go back to second this this is obviously an an uh nonlinear optimization problem

uh

due to the structure of C

the trace of C

and what we're doing the is basically um relaxation of the region problem and we using the expression that we

just developed previously at these solo using a vector B in metrics say

and you get this type of um

expression for the optimization problem

ah

no for this problem since is a non-convex

problem

uh we decided to go um using uh the like you on and uh the K can take a kick

it T conditions to find a us uh

until a solution

so next uh in the bottom here you see the lagrangian and uh a function

for this optimization problem

where we incorporate the objective

the first equality constraint multiplied by

no no and we have the two

yeah uh sets of uh inequality constraint multi by by you and you

uh the cake the condition formulated here

uh uh uh where you see that basically this expression is by just by long down "'cause" our

um

a train here are equally equal to constraint was uh uh metric uh a one parameter

um

to solve that

we take one step

i had and we basically or

and the constraint on P max

a mean by choosing me you when you E close to the zero

we in Z want all those two um

equation

and we

uh

get from this set we get

the three questions that we have here

and this has a have an analytical solution a very simple analytical so

then it could can solution is given here

and

what you see by ignoring uh

for for temporal ignoring down

the restriction and the power

is is that the optimal power allocation

has uh uh basically a um a levelling mechanism here

one of "'em" though

and one are all of them and the uh what it does has to be E and by the the

the

um

by the way B E and eight E represent be in a good we had previously we just

use

a uh the uh last

estimate to make we have the location but the channel to actually calculate the of so it's an actually a

value based on estimate

and you can see that basically what it does

it moves

it we levels

the elements of B

B

uh we uh we do a value inversely proportional to one that's quite at on that

start here

and one of the star you can see that

this levelling mechanism

incorporates it's a

mixture

of what element that naturally uh

um

i think the system such as the location the channel

the uh propagation loss and so forth

uh and an important thing that you know this different is different for communication

uh a system or as passive sensor system in this case it we have a transmitter

that ready it's energy

the he's reflected back to the targets so there is a cross dependency between the

a selected

power level at the specific the transmitter

and uh a signal that we get at all

and receiver

so when we that

specific transmit it fact

and propagation path

and the are this is why we get you few more complex value for long

um

we can see here data

uh

uh a fact of the two track actual uh that to be

introduced

i do think about this solution is as i just a

we can or the constraint

part

right so we can get an analytical solution here but we can be

outside

the ability of our sets them in terms of transmit and receive a a transmit power mean and max

so

well this gives a something inside of how

the power is distributed between the the different transmitters

uh oh

we were looking for something for uh and then the could way to get

more solution feasible solution

so we

oh when

and

yeah used uh the composition that that's and basically what we did in this uh approach

since were looking for mean

transmitted power

we can use a boundary

a points

to fine

it's solutions

so for example if we looking into the minimal value that each transmitter can you

we can

take a scenario where we take for example one transmitter

make these transmitter transmit the minimum value

and then calculate all the other

uh in my one analytically

but for this we need to mathematically formulate it

such that we can separate between

a group of transmitter

where actually enforce either a minimal maximum value on them

and the set of transmit that we analytically calc

and basically the uh structures that you see here

be one one B to one

do you and B two

are we we organise are vector are transmitted vector so the first portion

of this vector or uh you see here as the P T X one

represents a uh the one to our car

and P P X two what are the one that we enforce and we enforce force K elements

to be on the bound

now for the boundaries we

are are a select think in the minimum

what we can select

maximum a minimal points

to not to know uh used to much

um

yeah i uh search unnecessary search

we evaluate what would be the power in case of uniform and any

uh some of uh the powers that is beyond this a uniform power allocation we are not even investigating that

uh doing so and you have the details in the paper or of how this is the derive but we

basically the a a a a an analytical form

to calculate the remaining vector that we did not enforce any boundary point on

and you see the same

uh a structure all

a levelling

um

mechanism

uh that works again on

the the lot of the uh

uh metrics is uh B and that uh vectors

a a a a a capital B and a vector B

that

that to represent and the system structure

and uh we have a more complex um

a calculation for on the squared but uh again this is simple a uh and a solution once we have

the form a let's take that takes a little bit longer

that the uh uh resulting in question a very simple to you

uh and basically what what it gives that sees the set all

optimization problem that can be used to be either or or you can use this to the processing to get

uh

the solutions here or you can um

send a have them

calculated that the different receivers where the information is uh available

uh the only information each one of these sound problems needs

to calculate is the uh a a a a a a a estimated location of the target X line

and that's to channel age and all the rest of the

data are is uh existing data are related to the structure of the system

uh so each one of this problem basically K means that we take

K elements and put them on the boundaries

uh K going from one to in minus one

and each one of these we get

an optimal set of solution

which the minimum one is transmitted to the fusion center

which are select the mean one

one out

a a to see how this um

i'll go with them

how this uh method work we we use a few scenarios here

oh okay so as want to four for and the left side are cases where we assume the distances

from the uh um elements to the target or equal i would basically a human a the effect of about

five

a a a a a on the left side and right sides or a case five to eight

um generate different

this and if i would use that the the right hand side the the right hand side it

uh

all channels like equal to one we can actually

uh

have an option of seeing what

how that the geometric effect

uh uh what do the do you could uh they'll pay yeah affect how it fact the power distribution between

the transmitter still

um the right hand side will help us um

understand that

which chose a a a few are possible for the channel or as i said uh one of them is

all the trend channels are perfect in terms of um

uh a target rcs

uh the second one has to

a good transmitters this other one has to be good transmitter

and

the question

before we you know

before we we we we we go forward is

you know of a valid question would be why not just take the expression we had previously

find a uniform power allocation and use it i mean we have an expression we can easily calculate what would

be the total power for uniform case then you have it here

and what we don't axis compare

optimally uh i don't think that the power to the to the scenario or just using uniform

and you see the results for case one case for using H two which H two means that

one and two are a one and for uh one and uh

transmitter one and five are the bad

you can see here that the total you the four would be one sixty two

people were compared to nineteen which has a fifty six percent saving power

so when compared to uniform power allocation be the same mean square error we save here about around fifty percent

by adapting the power

and not using uniform allocation

i this an i where these to we are the best we can see that

uh basically doesn't need to be transmitted together a performance

so you can see that

it which was different transmit based on geometry

well that they are uh a um the it it looks to uh white and uh that i aperture of

the um

a a a a a a the of the set of

transmitted that it uses

again you see the saving compared to a uniform a case

uh this is case as five to eight where we don't have the

lost or channel was but the only think that these a fact and you see that even when only in

terms of distance

there is a point in in using power location

it's still same some power

uh so two

and the summarise everything i well we look

into a resource away way operation of this to put multiple radar system

a by minimizing the total radiating eighteen power a a uh uh a a for a given in score trash

well

the optimization problem was solved to domain the composition at all do which basically generated

probably set of optimization problem that can be distributed

and in terms of processing

uh the power allocation expression we've level levelling

uh mechanism a a which gives the since like to how the system actually

um um i look at the power and we also showed that you for power allocation is not necessary or

optimal and that a

adapting the power

i in the way we suggested is uh offering saving in terms of power

i

i

hmmm

yeah

a

okay well

so i so was really you

sounds to switch are you just surface i'm to from the one into a is that correct

so

actually some of the points on the boundaries

yeah yeah but so to have to constrain so you are for my like to transform or and that you

want to a problems also four

you how missions you really have

do with

or

to

oh

something like that right right and the you from the which are on the bottom were

and also so of the gene that you wait until you

so to that to just low red

oh okay just one subject

thank you

yes

oh

oh

yeah

a a well we assume in this case we using the cramer-rao bound when we tracking a target and you

assume you have a uh uh uh a um you track the target to file the target then you have

some estimate on it and you keep and tracking it and you want to keep a tracking it in a

resource away way man so you use every time the previous estimate

to adapt the power

i