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