Multiband Jamming Strategies with Minimum Rate Constraints

Transcription

1 Multband Jammng Strateges wth Mnmum Rate Constrants Karm Banawan, Sennur Ulukus, Peng Wang, and Bran Henz Department of Electrcal and Computer Engneerng, Unversty of Maryland, College Park, MD 7 US Army Research Laboratory, Aberdeen Provng Ground, MD 5 Abstract We consder a channel wth N parallel sub-bands. There s a sngle user that can access exactly k channels, whle mantanng some mnmum rate at each accessed channel. The transmsson takes place n the presence of a jammer whch can access at most m channels. We cast the problem as an extensveform game and derve the optmal power allocaton strateges for both the user and the jammer. We present extensve smulaton results regardng convergence of rates, effect of changng the number of accessed bands for the user and the jammer, and the mnmum rate constrant. I. INTRODUCTION Relable communcaton n the presence of a malcous jammer has attracted consderable amount of research. ] fnds the worst addtve nose for a communcaton channel satsfyng a covarance constrant and derves the saddle ponts correspondng to equlbrum dstrbuton of nose and transmtted sgnals. For the specal case of memoryless channels, ] shows that Gaussan codebooks for both jammer and user satsfy the equlbrum condtons based on mn-max problem. ] consders the case where the jammer can eavesdrop on the channel and use the nformaton obtaned to perform correlated jammng. Consequently, ] examnes the exstence of a smultaneously optmal set of strateges for the users and the jammer. A multuser, mult-tone verson of ] s consdered n ], where a generalzed water-fllng algorthm s proposed for the user and the jammer power allocaton. In 5], the authors consder a jammed sngle-hop wreless network wth N ndependent channels, n non-cooperatng users and m non-colludng jammers. The transmsson s assumed to be noseless and non-faded. The model assumes that whenever a jammer hts an occuped channel, the rate of ths channel drops drectly to zero. Dependng on whether each occuped channel s jammed or not, the jammers and the users change ther frequency bands accordng to a fxed transmsson strategy. 5] calculates the steady state normalzed rate by formulatng the system model as a Markov chan, where the throughput can be obtaned from the statonary dstrbuton. In ths paper, we consder an extenson of the work n 5]. We consder a nosy, fadng channel model of N parallel channels. Our system has one user that can access exactly k channels subject to the constrant that the mnmum rate at each accessed channel should at least be θ. The motvaton of Ths work was supported by ARL Cooperatve Agreement W9NF the mnmum rate constrant s that n some communcaton systems lke broadcastng systems, the system may be forced to convey nformaton for specfc number of channels and the rate of transmsson may be lower bounded by servce requrements. Consequently, the communcaton system may not be able to smply swtch off some channels n order to maxmze ts rate, but rather t may be forced to use exactly k channels wth ndvdual rates of at least θ n each channel. The transmsson s dsrupted by a jammer who s able to access at most m channels. Although our model deals wth a sngle user and a sngle jammer, t can be thought of as a generalzaton of the work n 5], snce t permts cooperaton n the transmtter and jammer sdes. Instead of fxng the strateges of the jammer and the user and analyzng the correspondng rate as n 5], we derve the optmal power allocaton polces for the jammer and the user under transmsson and jammng power constrants and a mnmum rate constrant for each used channel. We cast the problem as an extensve-form game, where the jammer and the user take turns to respond to each other s strategy. Our model admts a softer verson of the jammng effect, where the jammer decreases the rate of a channel down to a level θ. Once the rate decreases to ths level, ths sub-channel contrbutes zero rate to the throughput, and therefore, there s no need for the jammer to expend any more jammng power to decrease the rate to zero. In ths paper, we frst show that the problem under the mnmum rate constrants s concave n the user power allocaton polcy and convex n the jammer power allocaton polcy. Next, we determne the optmal channel selecton strategy for the transmtter, and derve the correspondng optmal user power allocaton strategy over the selected set of channels. The optmal allocaton strategy s a modfed water-fllng algorthm where weaker channels are provded wth suffcent power to mantan the mnmum rate constrant. We show that the optmal power allocaton strategy for the jammer s a generalzed water-fllng algorthm. We observe that the jammer does not target channels that barely satsfy the mnmum rate constrant. We provde the condtons under whch the jammer chooses not to jam a specfc channel. We verfy our theoretc fndngs wth extensve smulaton results. We observe that an equlbrum may not be obtaned n case of partal band utlzaton. We dscuss also the effects of changng the number of accessed bands for the user and the jammer, and the mnmum rate constrant, on the system performance.

2 II. SYSTEM MODEL Consder a system wth N parallel channels. Assume that there s a user who can access exactly k of these channels to send ts message to the recever n the presence of a malcous jammer who can access at most m channels to nflct the maxmum hurt on ths transmsson. We consder the case where the user and the jammer encode ther sgnals n response to each other,.e., they are nvolved n a perfect nformaton extensve-form game ]. The user begns ts transmsson by choosng the best possble k channels to send ts message wth the hghest possble rate. The user has a mnmum rate constrant θ on each channel t uses. The user performs power allocaton along the set of channels S u that t chooses for transmsson. The jammer chooses a jammng power allocaton strategy j such that the jammer pulls the rate of the k channels below θ, and hence these channels are no longer actve and the user s forced to leave these channels for worse channels. Consequently, the jammer performs the followng optmzaton problem subject to the total jammng power constrant J mn j,s j s.t. S u log S j j J + h p + g j θ ] + S j m where h, g are the channel gans from the user and the jammer, respectvely, to the recever over the th channel, p s the power of the user n the th channel, and S u, S j are the szes of the transmsson set S u and the jammng set S j, respectvely. Whenever the rate of any channel s below θ, the user chooses another channel to replace the faled channel. That means that the user consders channel completely jammed whenever R < θ, where R s the rate of the th channel. Thus the set S u should be updated by replacng channel S u by a new channel from Su. c The user performs the followng power allocaton strategy over the updated S u set of channels n response to the jammng strategy {j } Sj max p,s u log + h p + g S u j s.t. p P S u log + h p + g j θ, S u S u = k where P s the total power of the user. III. OPTIMALITY CONDITIONS In ths secton, we derve the optmalty condtons for the transmtter and jammer optmzaton problems. Consequently, we provde some structural propertes of the optmal soluton. A. Convexty-Concavty Property We start our dscusson by consderng the objectve payoff functons of both the transmtter and the jammer. Although the objectve functons of the two problems are dfferent, we can cast the transmtter s payoff functon to have the same optmal soluton as the jammer s payoff functon under the mnmum rate constrant. Defne the followng objectve Rp, j Rp, j = S u log + h p + g j θ] + where p = p,..., p N, and j = j,..., j N are the power allocaton strateges for the transmtter and jammer, respectvely. Lemma Maxmzaton of Rp, j s equvalent to maxmzaton n under the mnmum rate constrants. Proof: Snce subtractng a constant term S u θ = S u θ does not change the optmal soluton of the problem, the optmal power allocaton strategy of the problem n s the same as the optmal power allocaton of the objectve functon ] S u log + h p θ. From the mnmum rate constrants log + h p θ, +g j +g j + S u. Consequently, S u log + h p θ] = ] +g S j u log + h p θ, and the two objectve functons are equvalent. j +g The followng lemma states the convexty-concavty property of the payoff functon Rp, j. Lemma Rp, j s concave n p and convex n j under the mnmum rate constrants. Proof: For the convexty n j, we do not need the mnmum rate constrant. Consder the followng functon fj fj = log + h p + g θ j The functon fj s convex n j for j ]. Defne gj = max{fj, }. Snce, the maxmum of two convex functons s convex 7], g s convex n j. Snce the sum of convex functons s convex, Rp, j s convex n j. In addton, + snce log + h p +g j θ] = log + h p +g j θ under the mnmum rate constrant as n Lemma, t s concave n p. Thus, Rp, j s concave n p. B. Transmtter Sde Problem In ths secton, we consder the soluton of the transmtter s optmzaton problem n. We begn by dentfyng S u n the next lemma. Lemma The transmtter chooses S u such that t ncludes the hghest k channels n the normalzed sgnal to jammng and nose rato SJNR sense.

3 Proof: We defne the normalzed SJNR at the th channel as h q = + g j Now, wthout loss of generalty assume that the channels are ordered n the sense of normalzed SJNR. Assume for the sake of contradcton that Su = {,,..., k, k + },.e., we choose the k + th nstead of the kth largest SJNR, wth optmal power dstrbuton p. Snce log + x s monotone n x, t s clear that wth the same power p k, we have log + h k p h k k+ + gk j > log + p k k + gk+ j k+ If p k s feasble when usng the k + th channel, t satsfes the mnmum rate constrant when usng the kth channel. Hence, the total rate can be ncreased wth the same optmal power allocaton and ths contradcts the optmalty of Su. The next theorem characterzes the optmal strategy of the transmtter n response to the jammer s strategy. Theorem The optmal power allocaton strategy p of the transmtter n response to the jammer s strategy j under the mnmum rate constrant θ s gven by p q e θ, S u, q λe θ = λ q, S u, q > λe θ 7, Su c where q = h s the normalzed SJNR at the th channel, S j u s the set of channels correspondng to the hghest +g normalzed SJNR, and λ s chosen such that S u p = P. Proof: The optmal S u s obtaned by Lemma. The Lagrangan of the optmzaton problem n s gven by L = log + h p + g S u j + λ p P S u + µ θ log + h p + g S u j 8 = log + q p + λ p P S u S u + θ log + q p 9 S u µ The optmalty condtons are gven by q + q p + λ µ q + q p 5 = If on the th channel the mnmum rate constrant s satsfed wth equalty,.e., log + q p = θ and p = q e θ. Snce µ from, we have q + q p + λ = µ q + q p Hence, the condton of satsfyng constrant wth equalty s q + q p + λ whch further mples q λe θ. On the other hand, f the mnmum rate constrant s a strct nequalty, then µ = n, we have whch mples q + q p = λ p = λ q and ths occurs f q > λe θ. We note that for the specal case of k = N, θ =, 7 reduces to the classcal water-fllng strategy n ]. C. Jammer Sde Problem In ths secton, we consder the jammer sde optmzaton problem n response to the transmtter power allocaton strategy. The epgraph form of the jammer s problem s mn j,s j,t S u t s.t. t j log + h p + g j S j j J θ t S j m 5 For a fxed S j, 5 becomes a convex optmzaton problem. The followng theorem derves the optmal power allocaton strategy for the jammer n response to the transmtter strategy. Theorem The optmal power allocaton strategy j of the jammer n response to the transmtter s strategy p under the mnmum rate constrant θ s gven by j = { + w + µ λw, S j, µ g r λ, otherwse s the to sgnal to jammng rato SJR of g where w = h p g the th channel, r = h p s the useful sgnal rato, µ +h s p the Lagrange multpler correspondng to the rate constrant and λ s chosen such that S j j = J. Proof: The Lagrangan of the optmzaton problem s L = t + µ log + h p + g S u S u j θ t + λ j J ν t η j 7 S u S u S u

4 The optmalty condtons are µ µ + ν = 8 g g ] + h p + g j + g j + λ η = 9 When the jammer does not jam a channel,.e., j =, then η, and from 9 we have µ g ] + h p g + λ = η whch mples whch further mples µ g h p + h p λ µ g r λ On the other hand, f the jammer jams the th channel, then η = and hence 9 becomes µ g h p ] + h p + g j + g j + λ = whch s equvalent to λ + h p + g j + g j = µ g h p whch s quadratc n j. By expressng the roots of ths quadratc equaton n explct form, we obtan j = g + h p h g + p g + µ h p λg 5 = g + w + µ λw wth w = h p. g In the followng lemmas, we nvestgate some propertes of the jammer s constrants. The frst lemma deals wth the constrant of log + h p θ t +g. j Lemma The constrant log + h p θ t +g should be satsfed wth equalty and hence µ j. Proof: There are two cases to be consdered. The frst case s t >. Assume for sake of contradcton that the constrant s strct for the optmal j, t,.e., log + h p θ < t +g j and t >. In ths case, we can decrease the value of t untl the equalty holds. Ths s feasble and decreases the objectve functon and hence we have contradcton that t s optmal. On the other hand, f t =, then f the constrant s also strct, then one can decrease j such that log + h p +g j θ =. Ths s feasble under the total jammng power constrant, whle the objectve functon wll not ncrease. Hence, the constrant s satsfed by equalty n all cases. The followng lemma concerns about the total jammng power constrant. user s channel qualty h jammer s channel qualty g ndex of user s channel ndex of jammer s channels Fg.. Channel gans. Lemma 5 The total jammng power constrant should be satsfed wth equalty. Proof: Frst, f J S j j = > and there exsts t l for some l S j, we let j l = jl +, whch s a feasble power allocaton strategy. Then, by the monotoncty of log, we can have t l < t l. Moreover, f t =, S j, then any power allocaton strategy s optmal and hence we restrct ourselves to satsfy the jammng power constrant wth equalty. The followng lemma states the condtons under whch the jammer does not jam the th channel. Lemma The jammer chooses not to jam the th channel f the SNR of the channel s low or the jammer s channel gan s low. More specfcally, j = f µ g r λ. Proof: The proof follows from the optmalty condtons derved n Theorem. r = h p =, whch s a +h p + h p monotone functon n h p,.e., the SNR of channel. The SNR also controls µ, snce from 8 we have µ + ν =. If t = whch corresponds to the case where the channel barely satsfes the θ constrant, then ν, whch means that µ n contrary to µ = for the channels that exceed rate θ. Hence, f channel SNR or jammer s channel gan decrease, the product µ g r also decreases and the jammer chooses not to jam ths channel, snce t carres lttle rate or does not hurt the man lnk as much. IV. SIMULATION RESULTS In ths secton, we present some smulaton results for the presented system model. In all smulatons, we fx N =. The user and the jammer repeat ther encodng over transmsson blocks each,.e., encodng frames. We use fxed channel gans, whch are shown n Fg.. A. Power Allocaton Results We choose m = k = N =, and θ =.. In ths case, we have an equlbrum n the sense that nether the user nor the jammer changes ts power allocaton, snce the

5 achevable rate bts/channel use User s strategy Jammer s strategy user s power unt jammer s power unt user s channel ndex encodng frame ndex jammer s channel ndex Fg.. Equlbrum of achevable rates under P = J =, θ =., and m = k = N = at each encodng frame for the user and the jammer. strateges acheve ther optmal payoff functons as shown n Fg.. In Fg., we show the power strateges of the user and the jammer at each channel. The colored bars represent the encodng frame power. We note that for the channels {,, 9, }, the user apples ordnary water-fllng n the sense that the hgher the channel gan, the hgher the transmtted power. We see slght varatons n power dstrbuton along these channels over tme, because the nose levels change due to the jammng power. For the rest of the channels,.e., channels {,,, 5, 7, 8}, we see an nverse behavor, where the worse the channel gan, the hgher the power njected by the user to mantan the requred mnmum rate θ. Hence, we have fxed power dstrbuton along these channels. We note that the jammer does not waste ts power on the channels that barely acheve θ, snce any power ɛ > drves the rate on these channels to zero. Consequently, the jammer concentrates on good channels,.e., the set {,, 9, }. However, for channel, we note that the channel qualty from jammer to recever s bad. Hence, jammer uses channel nstead whch has the maxmum channel gan and n the meanwhle carres rate larger than θ for the frst two encodng frames. The correspondng rates of every channel s gven n Fg.. We note that not all model settngs lead to equlbrum. More specfcally, when m, k < N, the user can possbly move from the jammed band to other channel whch was ntally worse n order to ncrease ts rate. Ths potentally leads to an oscllatons between multple sets of channels wth dfferent payoffs and hence no equlbrum can be acheved. Fgs. 5, and show an example of ths non-equlbrum case wth N =, m = k = wth same channel gans. The fgures show that the user and jammer jump between two set of channels S = {, } and S = {9, }. Hence, we have mssng bars n the power allocaton and the achevable rates oscllate. B. Effect of the Mnmum Rate Constrant In Fg. 7, we nvestgate changng the mnmum rate constrant θ. We consder the achevable rate after jammer determnes ts encodng strategy. We choose that m = k = N = for θ = {.5,.,.}. We note at frst that f we ncrease Fg.. Equlbrum power allocaton for the user and the jammer under P = J =, θ =., and m = k = N =. achevable rate bts/channel use user s channel ndex Fg.. Equlbrum achevable rates for P = J =, θ =., and m = k = N =. θ >., the problem becomes nfeasble for the channel gans under dscusson. We also note that as θ ncreases, the achevable rate decreases. Ths s because the user must provde excessve power n the bad channels to mantan rate θ at each bad channel. Ths decreases the avalable power for other channels. C. Effect of the Number of Channels In Fg. 8, we nvestgate the effects of changng the number of accessed channels by the user and the jammer. We consder the case where m = k. In another words, we consder the specal case where the jammer has the ablty to jam all the channels of the user. Snce equlbrum may not be obtaned, we use the average rate over all encodng ntervals,.e., R = T T t= Rt where T s the total encodng ntervals for the user and the jammer, we take T =, and Rt s the achevable rate n the tth encodng nterval. From Fg. 8, we note that the average rate ncreases untl k = 5, because the domnant effect untl that pont s that we are addng channels to S u that can acheve rates larger than θ. However, after we reach k = 5, the problem s more confned snce we are forcng the user to nject power n bad channels to barely acheve rate θ and ths sacrfces from the maxmum rate that

6 achevable rate bts/channel use User s strategy jammer s strategy encodng frame ndex achevable rate bts/channel use θ =.5 θ =. θ =. Fg. 5. Achevable rates at every encodng frame for the non-equlbrum nstance m = k =, N = and θ =.. user s power unt jammer s power user s channel ndex jammer s channel ndex Fg.. Non-equlbrum power allocaton for the case N =, m = k = and θ =.. the user can acheve f t follows the ordnary water-fllng and swtches off these bad channels. In Fg 9, we fx k = and we nvestgate the effect of changng the number of accessed channels for the jammer. We note that the as m ncreases, we have monotone non-ncreasng graph whch shows the fact that as m ncrease, the degrees of freedom of the jammer to hurt the user ncreases. REFERENCES ] S. Dggav and T. Cover, The worst addtve nose under a covarance constrant, IEEE Trans. on Info. Theory,, vol. 7, no. 7, pp. 7 8, Nov. ] T. M. Cover and J. A. Thomas, Elements of Informaton Theory. John Wley & Sons,. ] S. Shafee and S. Ulukus, Mutual nformaton games n multuser channels wth correlated jammng, IEEE Trans. on Info. Theory,, vol. 55, no., pp , Oct 9. ] R. Gohary, Y. Huang, Z.-Q. Luo, and J.-S. Pang, A generalzed teratve water-fllng algorthm for dstrbuted power control n the presence of a jammer, IEEE Trans. on Sgnal Processng,, vol. 57, no. 7, pp. 7, July 9. 5] P. Wang and B. Henz, Performance analyss of jammed sngle-hop wreless networks, n IEEE MILCOM, November. ] S. Tadels, Game Theory: An Introducton. Prnceton Unv. Press,. 7] S. Boyd and L. Vandenberghe, Convex Optmzaton. Cambrdge Unv. press, encodng frame ndex for the jammer Fg. 7. Effect of changng θ for m = k = N =. average achevable rate bts/channel use number of accessed channels m = k Fg. 8. Effect of changng m, k such that m = k, N = and θ =.. average rate bts/channel use maxmum possble number of jammed channels m Fg. 9. Effect of changng m wth fxed k =.