and
and i'm the last one i'll try to make it a fish and
a is but yeah yeah that time and now we can spend a whole leaving here no i don't think
um
this article of a
this paper or uh is a knapsack problem uh formulation for really selection in secure corporate people that wireless was
communication
i i this work yeah yeah um is a collaboration with a one Q will uh
a professor for trouble profess of a and myself
and will start by a short introduction talking about the uh um
uh a cooperative jamming uh the system will introduce the system the model that we are considering for this
um and then we will uh formulate their uh an set
uh
problem uh for the selection of um minimal set
of really is uh to cooperate um
a a in the jamming
uh as uh this result that you know uh optimization problem that requires exponential complexity we will um
all uh all four three uh alternative a solution that um
oh four as um significant the
saving in complexity
and we will introduce a it would show some assimilation analysis of but these algorithms
and finally some concluding remarks
a so we know
a a cooperative the wireless communication system
the additional uh um
and degrees of freedom that are offered by those cooperating uh really is uh maybe use them to degrade great
uh channel
to uh potentially drop or
uh and they are uh
they can use the uh uh jointly
some way a
B to send a a collaborative them
a jamming signal
uh
to that you've drop
uh this type of uh a scheme was uh considered uh a would for protection meaning that if we have
a a an available really we use all in available relays
a for the purpose of jamming
uh and they a send a way uh version of the same jamming signal uh
for the two D pro
and this work was been done in two thousand nine by
don't hand problem
uh since no
it has been shown that
at some point
and as the that lot number of uh
grows
at some point
the that we get a introducing the relay for the jamming
is is small
uh and this has been a shown by um
one and uh
swindle to her in two thousand nine
uh considering that that and the fact that
the larger the system is
the larger uh the communication that a it is and uh
the synchronization requirement
that bring just to to the full question uh when you have a and collaborating really is and we previously
used all of them
uh can now the sec you rate requirement be a cheap tool close to to the maximum be achieved
we do a smaller or you're
uh uh uh activity
and the objective that
we we put forth and this is to identify i mean a more set of of uh relays
such that the predetermined secrecy receive rate the objective is a G
uh
first let's in the system that are considering
assuming we have a a a a a source transmitting a signal to the destination and
that you drop or is hearing the same message
oh we know the channel a between this uh a source and the destination by a just zero and between
the source and the proper or by G zero
also seem we have and
uh cooperating relays
and i and for each one of the really a uh with assume we have
we know the channel uh from
for example a really really high
to the destination marked by a giant again from the really
uh to that you drop or might but by G I and you seem we know we have knowledge of
the channel and bob
uh we assume um white gaussian noise of be zero mean and variance sigma squared
assume the energy for the
transmitted symbol X is uh one
and at the sort transmission power for the
the
so the symbol is P
uh all S
uh the really transmit meet a common jamming signal uh Z and we assume that each one transmit a weighted
version of signal
a for this uh scenario we basically have here
for the the signal received at the destination
is composed of the first element that is the message that was sending
uh in the second part is uh the version of the jamming signal that we have transmitted
multiply force by the channel to the destination
um
the signal received at the E dropper again has the same structure where we have
a version of the
symbol is received at the dropper
uh once the jamming signal that we have
france
uh we have here the notation for the various vectors
and basically in this case the secrecy rate may be written in this form where the first
element here represents the rate
yeah to the destination
and the second to rake at T
um
as i said the the first uh
what that has been done on um
and this uh uh and is assumed that
or and
a really are called right
the G
and for this and really is
uh a are the power
uh the power P S and the vector or all the weights vector
uh only got
maybe be uh such that
yeah we get a maximum
sec receive and this is basically the optimization problem that
design
a P S O
oh make the
uh oh yeah O optimal
um
such that
that set was rate is
max
as we said were looking for more
uh energy efficient the uh weighting and we trying to minimize the number of really at
so we are basically um defining a racial and the threshold these is with here is the are sub S
a requirement
and we we uh define it as a fraction of the maximum a C rate that
so
if we are all K be eighty percent of a
that would be a zero point eight as we set this threshold
uh and therefore we
next we want to define an optimization problem that will help us identify
the best realise to use to get this accuracy rate which is the minimal one as well
uh to do so we first started um
um by introducing
a binary variable
and we just a vector Q where each element of this queue Q why
can be wanting to really
i exactly or zero if it's not
uh our objective eventually is to write it and that combinatorial uh a framework
and uh
uh yeah
and specifically in a knapsack problem yeah therefore we take
the original a sec receiver rate expression that we had before and we rewrite it so we can use it
in this frame
and we define um
are uh uh that the sec C rate
uh we
and E here
a out which is a uh a and X exponent um
function of the do what we can secrecy rate
and it a to be an expression and basically is a sum and high and J work Q one Q
J
uh
as you you know you recall from here basically represent present a really is T one and
uh in the um
uh jamming
um multiplies applies
a a function here are are i J
but are i J is you can see is a function of Q as well
uh so this uh F F of to here is is given here
and you can see to F age of Q and F G of Q
where the dependencies still and Q one Q J
which is given a general form here
uh so it's a nonlinear your function basically of Q i and Q
but this type of formulation now enables us to right
uh the search
uh
problem for the minimal set E as an that
and that six probably it means that we are trying to minimize
the number of element
or the cost of that element there are putting in and a
well at this is a are capacity this is what we want to go
and a and as i said it's a combinatorial optimisation problem Q why can be zero
um
as you can see when we use in here that can a sec receive a to are using it read
or my got a star and P star
to mean that we are using it with a
vector or a mean god that was to my
for the sec rate so
you can see here
but to basically uh do for a given set of Q let's say that we follow that we we
we are looking for Q when where
oh forwarding a possible solution for Q
for this Q we are going and optimising in and then P S
such that it maximises the set or C
uh so this is this is the do know um
and knapsack optimization problem
and
one of the well known problems all of this is that a complexity it it has an exponential complexity
uh it's uh
you for larger number of of of and it's
impossible possible to solve
so of course
to make it feasible and and usable they are looking for
a a fast approximation algorithm that was still be a fish
and um
in this work we have proposed a three different
uh
a approximation algorithms
each one has its advantages or disadvantages as we will discuss later
and each one has different the a level of complexity of course
uh the first one is uh
yeah individual sectors you rate than a base
really selection
uh
the second is a weighted norm really selection and the last one
is a successive a a really uh selection algorithm that is based on
what power uh start local search uh approach
um we start for the first one
the first one is is the
think five one
and what it it does it it relaxes the original position problem
and says we not going to optimize it's for each vector
what what are going to do we are going to
first take each individual really
are going to find a a
the optimal value for a i and P S time
that maximise the sectors C
for this uh a specific one
and
once we generate a set of uh of value
we are so like think the the we are successively selecting them by
they're value so we're selecting the highest one first and keep on adding
uh by there
and feel we we basically have some
uh um threshold uh pass
um
i don't note and how what what what is basically down we use this a figure to explain how
we basically find uh the the palm maximum um
oh values for P S and all my got to this process
and you have your plot for to to really
call really and really be
and you can see them
sec to see wait for each one of them as a function of the
of P S
and basically it it the the algorithm that to show identifies stills maximal point and the appropriate P S for
them
which are then used
a selection out
uh the second approach
yeah
is going if
first we went for the individual one here we going on the full system
so we assume we take all and really
and we doing what has been that in previous work which is optimising
the weight the vector or my guy and P S
such that the total
step C rate is all is met
no when not
repeating that that actually taking the values that we got
are this optimal value
and we are using them
in our uh expression for the set to see so yeah we not
calculate that again and again
so we're saving uh on this process
and uh we are selecting uh
uh the the the larger one of course which use a basically equivalent to
selecting the
uh i
a really is with their a larger storm and this is like like weighted norm uh
selection
the third one
does not assume any uh
relaxation on the original problem
that's on basically takes the we problem
and
is um
in a sense
in a greedy way
ah
and it's a a really is to the group
so we have a starting point
we we we start the let's say we do really one
and we keep and adding really
and each time we adding relays
we are we
going and maximizing uh the vectors
uh W and P again
and we are choosing the relay that
maximise the sec receiver in this case
now once we add a a a a a a ad
the the one that maximises with keep on adding and till we cross trash
uh
for a
a better performance in terms of of uh reading the optimum we we be
this search for every possible forced really
and this is one call
a most people start
yeah i am with people start a local search
uh
we eventually end up with them
and optimal solutions
and we choose the the one with the least number or
um
sure an example of that
uh
by the way the the here though optimization used done using um
an algorithm that was suppose to the proposed by um the at a bit trouble in web
and two thousand ten at each that
um um
to show uh an example of that
a you see C or scenario all of a source destination
and and you drop or or and we assume we have a set of relays
a a spreading a given area
ah
we assume fifteen uh really is are a given here
um and
uh we we normalize the sec to C rate compare sent to the maximum achievable one
and we we used to threshold one of them is ninety percent of zero point nine five and seventy five
percent
zero point somebody five
i you see for an example
of of the results that we get for this scenario
and
in in in black though it's not seen because the green is over that
and i and like to use a a is a uh the results of an exhaustive search
so basically if i would take for example for
really is
this points says
if i would do an exhaustive search for the best for really is this would be the the value the
to see way that that would get and so
and
the third on go with him a very nice we'd we'd the optimal one
uh uh we it for different scenario in in general it fold was very close to the optimal
uh it also more we'll bossed uh there there was information in paper it also more boss to the location
of the the sensors
respect to
if dropped or and
nation
uh in terms that it falls again candle
uh a the on a two
uh uh um
there are behaviour is more uh reliant on location
uh in many cases uh
as expected algorithm uh be performs better than algorithm a a a board to many have the stalls
slow reaction as you can see from here so
um uh for for low uh
low threshold if forms a we well but when the threshold is high as you can see here
it will give fifteen sensors or something like that
C right there
uh it would give sort twelve
a a sensor as the number of sensor that that we need for that
um
we also checked or robust most of that too small error in the estimation of the channel
uh between the source and you drop or
and uh again the the algorithm C is falling that before forming quite well in in uh in this sense
as well and most of them
a stay within a reasonable before
so to wrap things up uh
i the problem of selecting mean set of really seeing um
well wire score a cooperative system uh for the purpose of jamming
uh has been formulated as a knapsack problem
uh the first two algorithm of four or complexity all and uh
order of L
a while the um
the third one is
it's it's here
it sort of a a L
and time L and L is the number of a is we eventually end up
having in the subset
um
um
uh in terms of complexity what that explains is a little what this so uh in the sense that in
small or for smaller or uh on
a threshold
uh i'll go "'em" one free form relatively good but in large a threshold it you can perform as well
well do for like special algorithm be free phone
or not converge conversion to to the optimal one and you know the case as well
so there is a trade of your between performance of course and um
and complexity not to large trade off but there is a tradeoff
and
gives a few option of uh
uh
implementing that
in real
thank you