Speech Transcript - OPTIMAL POWER ALLOCATION IN DISTRIBUTED MULTIPLE-RADAR CONFIGURATIONS

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

OPTIMAL POWER ALLOCATION IN DISTRIBUTED MULTIPLE-RADAR CONFIGURATIONS

Sensor Networks

Presented by: Hana Godrich, Author(s): Hana Godrich, Princeton University / Rutgers University, United States; Athina Petropulu, Rutgers University, United States; H. Vincent Poor, Princeton University, United States