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