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