0:00:14good afternoon everyone
0:00:15yeah a hand i got each
0:00:25this work
0:00:26uh is a joint with them
0:00:29a a to patrol boat and a cup or
0:00:36and was sort the uh with the short back
0:00:39uh on the system that you're talking about
0:00:41uh we will make should use the optimization them match to that to are going to uh exploit here which
0:00:47is the cramer-rao around uh bound um metric
0:00:51uh uh we're going to use this one two
0:00:54uh formulate the power allocation optimization problem
0:00:57where the objective is to minimize the total transmitted power in a a
0:01:01my more L multiple radar uh a vector
0:01:06to get an efficient to
0:01:07solution for that we use the the make the composition for that
0:01:11and it will use a some american analysis to show how uh
0:01:16uh the power location
0:01:17uh is generated
0:01:19and finally some concluding remarks
0:01:23a a target a localisation uh yeah estimation mean-square square error is known to be lower bounded by the cramer-rao
0:01:29lower bound especially if we talk about that maximum likelihood estimator
0:01:33and uh based on this metric it has been shown in the past that um
0:01:37a system with widely the separated
0:01:40uh uh and most people uh uh or
0:01:43and the and systems
0:01:45using a coherent or non-coherent processing
0:01:48uh offers advantages in terms of the estimation is square
0:01:53that you're see again is proportional to the product of the number of transmit and receive antenna
0:01:59i i in general though if we expand this dependency the mean square error depends on
0:02:04as a set and the number of transmit and receive radars
0:02:07but it also depends on the geometric metric layout
0:02:10of of the transmit and receive radars with respect to the target
0:02:13uh it depends on the uh signal effective bandwidth
0:02:17and on the signal to noise ratio and that brings us to the
0:02:20transmit power which were going to focus in this store
0:02:25that the the of that we're looking at is new
0:02:28a widely separated
0:02:30multiple radar system
0:02:32that is uh mobile
0:02:34and we see more and more application like that
0:02:37uh one example is the
0:02:39ground surveillance radar
0:02:41where we have a
0:02:42yeah a mounted and vehicles
0:02:45that are spread the along borders we can see them mean um yeah pro controls and things like that
0:02:52a in this type of
0:02:53uh application
0:02:55yeah it makes sense to be more conscious
0:02:57about are the use of resources
0:03:00and uh uh what you cover in uh
0:03:03this work is
0:03:05uh resource uh awareness in terms of the transmitted power
0:03:11so our objective here
0:03:12as we can see here used to minimize
0:03:15the total transmitted power such that the predetermined estimation mean-square square error is a thing
0:03:20uh uh white and the transmit power
0:03:22at each a station within an acceptable range
0:03:25so well we not that yeah
0:03:27extending the
0:03:29the number of of
0:03:31uh rate hours
0:03:32provides a higher accuracy in practical we need some level of your see which can can serve as a trash
0:03:38and the question is how do we minimize the powers
0:03:41the system before form a a at the threshold that we one
0:03:45uh and this is what we going to do
0:03:49so this is the system that we talking in in this figure
0:03:53an example of such a system and we keep it very general
0:03:57in terms that
0:03:58our readers can be
0:04:00you or transmit or or cu
0:04:02a or or can be vote i mean each one of this point can be a transmit receive a radar
0:04:07and assumption something that the uh
0:04:08are the information jointly so
0:04:11from all of this element
0:04:13and you have a target uh
0:04:15here that we want to estimate its location
0:04:17specifically a of before
0:04:20with assume we have a a a as a C M transmit radars and receive radar
0:04:24the target is modelled as an extended target
0:04:27with the center mass located uh
0:04:31we use of and all signal and assume we have
0:04:34M and and the had propagation path
0:04:37uh the transmitted power vector is given here as P of T uh
0:04:41for each one of the transmitting ten
0:04:45uh as we know the
0:04:46time delay of propagation of each one of
0:04:50uh a time is a function of the range from to transmitting uh uh radar
0:04:55uh to the target
0:04:56and from the
0:04:58target to the receive radar also
0:05:01and and basically a a measure those time delays for example if we use this one
0:05:05as a transmit
0:05:06to the target
0:05:07and received here this would be
0:05:11this propagation that would be proportional to the
0:05:13range sample
0:05:16uh this
0:05:17brings us to the received signal
0:05:19on the specific path and pat to "'em" and
0:05:22and we see that i went to that you take into account uh a in our model which we have
0:05:27here off i of and basically is proportional to
0:05:30uh the path so it was that the path loss
0:05:33uh P of T in P of M T X is the transmitted power
0:05:37a a or friends then it's is a complex coefficient
0:05:39basically takes into account the
0:05:41uh rate cross section on to M and
0:05:44a plus and any phase offsets and this path
0:05:48uh we have your uh uh uh delayed
0:05:50time delayed version of the transmitted power
0:05:53that's transmitted signals or and
0:05:55oh a white gaussian the
0:06:00we actually a defined all of this
0:06:02so we can find some metric you said that
0:06:04the constrained our system are giving in terms of
0:06:07so we need to find a a a a metric that labour enable us to
0:06:11uh represent this man
0:06:14for this were using the cramer-rao bound
0:06:16i where we using the trace of the cramer-rao bound metrics
0:06:20uh two
0:06:21provide the bound on the mean square on the X action
0:06:25and and the white direction one
0:06:27uh a the previous work
0:06:29trade between the two so
0:06:31optimising one of them
0:06:33we just uh maximise the that
0:06:38the the around on that because it was developed in previous studies it's not and you result
0:06:42uh what we have to do your do is uh
0:06:46state it
0:06:47so it can be used to optimize power
0:06:50so what we did here is we talk
0:06:53the original expression and defined it as a sum
0:06:56of some
0:06:57elements here multiplied by the pope power transmitted by transmitter am and have end of the
0:07:03i if we go one step forward and you can see your by the way
0:07:06that the elements of this matrix
0:07:08are dependent on off a age
0:07:11which were uh what the code channel correct for state on path and man
0:07:15and um
0:07:16we have you the location of the transmit and receive radars with respect to the target
0:07:21uh uh incorporated through this expression which are basically cosine and sign
0:07:26of the angles between the transmitter and receiver to the target
0:07:30uh the vector of a in this case
0:07:33you use the target location X Y
0:07:35and the channel
0:07:36vector eight
0:07:39uh using this type of uh
0:07:41expression and us was us to
0:07:43expressed the trace of the cramer-rao bound
0:07:46in the form that you can see here
0:07:48well basically we have some vector B
0:07:50multiplying the vector of power
0:07:52and in the denominator we have
0:07:55a a uh metrics eight
0:07:58second second order
0:07:59expression for the the same power
0:08:01and it you can see that basically be and any incorporate
0:08:05all the existing system
0:08:07a just the geometric spread the channel
0:08:10that fading and so forth so these are
0:08:12oh coming in to play through uh metrics as
0:08:15metrics a and vector B
0:08:18now that we have an expression for the cramer-rao bound we can formulate
0:08:24uh a and as we said our objective is
0:08:27uh uh that given a a predetermined threshold
0:08:30but is if a mean square error or
0:08:32uh we would like to optimize
0:08:35uh the the pa out basically um
0:08:38minimize the total contrast
0:08:40and this is the mathematical formulation for that
0:08:43so we we minimize the total transmitted power
0:08:46a a given a specific threshold are the cramer-rao bound where
0:08:50uh we use of previous uh
0:08:52estimate of the target location in age
0:08:54to calculate
0:08:57and also need for some limitation under transmitted power we we assume as you we transmit the minimal power
0:09:03uh P T X minimum and the maximum power uh uh uh
0:09:08P M T X max
0:09:14just go back to second this this is obviously an an uh nonlinear optimization problem
0:09:20due to the structure of C
0:09:23the trace of C
0:09:24and what we're doing the is basically um relaxation of the region problem and we using the expression that we
0:09:29just developed previously at these solo using a vector B in metrics say
0:09:33and you get this type of um
0:09:35expression for the optimization problem
0:09:39no for this problem since is a non-convex
0:09:44uh we decided to go um using uh the like you on and uh the K can take a kick
0:09:49it T conditions to find a us uh
0:09:52until a solution
0:09:54so next uh in the bottom here you see the lagrangian and uh a function
0:09:59for this optimization problem
0:10:01where we incorporate the objective
0:10:03the first equality constraint multiplied by
0:10:05no no and we have the two
0:10:08yeah uh sets of uh inequality constraint multi by by you and you
0:10:13uh the cake the condition formulated here
0:10:17uh uh uh where you see that basically this expression is by just by long down "'cause" our
0:10:25a train here are equally equal to constraint was uh uh metric uh a one parameter
0:10:32to solve that
0:10:34we take one step
0:10:36i had and we basically or
0:10:39and the constraint on P max
0:10:41a mean by choosing me you when you E close to the zero
0:10:45we in Z want all those two um
0:10:49and we
0:10:51get from this set we get
0:10:53the three questions that we have here
0:10:55and this has a have an analytical solution a very simple analytical so
0:11:01then it could can solution is given here
0:11:05what you see by ignoring uh
0:11:07for for temporal ignoring down
0:11:11the restriction and the power
0:11:13is is that the optimal power allocation
0:11:16has uh uh basically a um a levelling mechanism here
0:11:20one of "'em" though
0:11:22and one are all of them and the uh what it does has to be E and by the the
0:11:28by the way B E and eight E represent be in a good we had previously we just
0:11:33a uh the uh last
0:11:35estimate to make we have the location but the channel to actually calculate the of so it's an actually a
0:11:39value based on estimate
0:11:42and you can see that basically what it does
0:11:45it moves
0:11:46it we levels
0:11:47the elements of B
0:11:50uh we uh we do a value inversely proportional to one that's quite at on that
0:11:55start here
0:11:56and one of the star you can see that
0:11:58this levelling mechanism
0:12:01incorporates it's a
0:12:04of what element that naturally uh
0:12:08i think the system such as the location the channel
0:12:11the uh propagation loss and so forth
0:12:15uh and an important thing that you know this different is different for communication
0:12:20uh a system or as passive sensor system in this case it we have a transmitter
0:12:25that ready it's energy
0:12:27the he's reflected back to the targets so there is a cross dependency between the
0:12:31a selected
0:12:33power level at the specific the transmitter
0:12:36and uh a signal that we get at all
0:12:39and receiver
0:12:40so when we that
0:12:42specific transmit it fact
0:12:45and propagation path
0:12:46and the are this is why we get you few more complex value for long
0:12:53we can see here data
0:12:56uh a fact of the two track actual uh that to be
0:13:00i do think about this solution is as i just a
0:13:03we can or the constraint
0:13:05right so we can get an analytical solution here but we can be
0:13:11the ability of our sets them in terms of transmit and receive a a transmit power mean and max
0:13:18well this gives a something inside of how
0:13:21the power is distributed between the the different transmitters
0:13:24uh oh
0:13:25we were looking for something for uh and then the could way to get
0:13:29more solution feasible solution
0:13:32so we
0:13:33oh when
0:13:35yeah used uh the composition that that's and basically what we did in this uh approach
0:13:40since were looking for mean
0:13:42transmitted power
0:13:43we can use a boundary
0:13:45a points
0:13:46to fine
0:13:49it's solutions
0:13:50so for example if we looking into the minimal value that each transmitter can you
0:13:56we can
0:13:58take a scenario where we take for example one transmitter
0:14:01make these transmitter transmit the minimum value
0:14:04and then calculate all the other
0:14:07uh in my one analytically
0:14:10but for this we need to mathematically formulate it
0:14:13such that we can separate between
0:14:15a group of transmitter
0:14:17where actually enforce either a minimal maximum value on them
0:14:21and the set of transmit that we analytically calc
0:14:26and basically the uh structures that you see here
0:14:29be one one B to one
0:14:30do you and B two
0:14:32are we we organise are vector are transmitted vector so the first portion
0:14:37of this vector or uh you see here as the P T X one
0:14:41represents a uh the one to our car
0:14:44and P P X two what are the one that we enforce and we enforce force K elements
0:14:49to be on the bound
0:14:52now for the boundaries we
0:14:55are are a select think in the minimum
0:14:57what we can select
0:14:58maximum a minimal points
0:15:00to not to know uh used to much
0:15:04yeah i uh search unnecessary search
0:15:06we evaluate what would be the power in case of uniform and any
0:15:10uh some of uh the powers that is beyond this a uniform power allocation we are not even investigating that
0:15:17uh doing so and you have the details in the paper or of how this is the derive but we
0:15:21basically the a a a a an analytical form
0:15:23to calculate the remaining vector that we did not enforce any boundary point on
0:15:30and you see the same
0:15:31uh a structure all
0:15:34a levelling
0:15:37uh that works again on
0:15:40the the lot of the uh
0:15:42uh metrics is uh B and that uh vectors
0:15:45a a a a a capital B and a vector B
0:15:48that to represent and the system structure
0:15:52and uh we have a more complex um
0:15:56a calculation for on the squared but uh again this is simple a uh and a solution once we have
0:16:02the form a let's take that takes a little bit longer
0:16:04that the uh uh resulting in question a very simple to you
0:16:10uh and basically what what it gives that sees the set all
0:16:14optimization problem that can be used to be either or or you can use this to the processing to get
0:16:20the solutions here or you can um
0:16:22send a have them
0:16:24calculated that the different receivers where the information is uh available
0:16:29uh the only information each one of these sound problems needs
0:16:32to calculate is the uh a a a a a a a estimated location of the target X line
0:16:37and that's to channel age and all the rest of the
0:16:41data are is uh existing data are related to the structure of the system
0:16:46uh so each one of this problem basically K means that we take
0:16:50K elements and put them on the boundaries
0:16:53uh K going from one to in minus one
0:16:56and each one of these we get
0:16:58an optimal set of solution
0:17:00which the minimum one is transmitted to the fusion center
0:17:04which are select the mean one
0:17:05one out
0:17:10a a to see how this um
0:17:12i'll go with them
0:17:13how this uh method work we we use a few scenarios here
0:17:18oh okay so as want to four for and the left side are cases where we assume the distances
0:17:24from the uh um elements to the target or equal i would basically a human a the effect of about
0:17:31a a a a a on the left side and right sides or a case five to eight
0:17:35um generate different
0:17:37this and if i would use that the the right hand side the the right hand side it
0:17:43all channels like equal to one we can actually
0:17:47have an option of seeing what
0:17:49how that the geometric effect
0:17:51uh uh what do the do you could uh they'll pay yeah affect how it fact the power distribution between
0:17:57the transmitter still
0:17:58um the right hand side will help us um
0:18:01understand that
0:18:03which chose a a a few are possible for the channel or as i said uh one of them is
0:18:08all the trend channels are perfect in terms of um
0:18:12uh a target rcs
0:18:14uh the second one has to
0:18:17a good transmitters this other one has to be good transmitter
0:18:22the question
0:18:23before we you know
0:18:25before we we we we we go forward is
0:18:28you know of a valid question would be why not just take the expression we had previously
0:18:33find a uniform power allocation and use it i mean we have an expression we can easily calculate what would
0:18:39be the total power for uniform case then you have it here
0:18:42and what we don't axis compare
0:18:45optimally uh i don't think that the power to the to the scenario or just using uniform
0:18:50and you see the results for case one case for using H two which H two means that
0:18:55one and two are a one and for uh one and uh
0:18:59transmitter one and five are the bad
0:19:02you can see here that the total you the four would be one sixty two
0:19:06people were compared to nineteen which has a fifty six percent saving power
0:19:11so when compared to uniform power allocation be the same mean square error we save here about around fifty percent
0:19:17by adapting the power
0:19:19and not using uniform allocation
0:19:22i this an i where these to we are the best we can see that
0:19:25uh basically doesn't need to be transmitted together a performance
0:19:28so you can see that
0:19:30it which was different transmit based on geometry
0:19:33well that they are uh a um the it it looks to uh white and uh that i aperture of
0:19:38the um
0:19:39a a a a a a the of the set of
0:19:41transmitted that it uses
0:19:43again you see the saving compared to a uniform a case
0:19:46uh this is case as five to eight where we don't have the
0:19:51lost or channel was but the only think that these a fact and you see that even when only in
0:19:55terms of distance
0:19:57there is a point in in using power location
0:20:00it's still same some power
0:20:03uh so two
0:20:06and the summarise everything i well we look
0:20:09into a resource away way operation of this to put multiple radar system
0:20:14a by minimizing the total radiating eighteen power a a uh uh a a for a given in score trash
0:20:19the optimization problem was solved to domain the composition at all do which basically generated
0:20:25probably set of optimization problem that can be distributed
0:20:29and in terms of processing
0:20:30uh the power allocation expression we've level levelling
0:20:33uh mechanism a a which gives the since like to how the system actually
0:20:37um um i look at the power and we also showed that you for power allocation is not necessary or
0:20:43optimal and that a
0:20:45adapting the power
0:20:46i in the way we suggested is uh offering saving in terms of power
0:21:21okay well
0:21:22so i so was really you
0:21:24sounds to switch are you just surface i'm to from the one into a is that correct
0:21:30actually some of the points on the boundaries
0:21:32yeah yeah but so to have to constrain so you are for my like to transform or and that you
0:21:37want to a problems also four
0:21:39you how missions you really have
0:21:41do with
0:21:44something like that right right and the you from the which are on the bottom were
0:21:47and also so of the gene that you wait until you
0:21:51so to that to just low red
0:21:52oh okay just one subject
0:21:55thank you
0:22:02a a well we assume in this case we using the cramer-rao bound when we tracking a target and you
0:22:08assume you have a uh uh uh a um you track the target to file the target then you have
0:22:11some estimate on it and you keep and tracking it and you want to keep a tracking it in a
0:22:15resource away way man so you use every time the previous estimate
0:22:18to adapt the power