CAPACITY OF MULTIPLE ACCESS CHANNELS WITH CORRELATED JAMMING

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1 CAPACITY OF MULTIPLE ACCESS CHANNELS WITH CORRELATED JAMMING Sabnam Safiee and Sennur Ulukus Department of Electrical and Computer Engineering University of Maryland College Park, MD ABSTRACT We investigate te beavior of two users and one jammer in an AWGN cannel wit and witout fading wen tey participate in a non-cooperative zero-sum game, wit te cannel s input/output mutual information as te objective function. We assume tat te jammer can eavesdrop te cannel and can use te information obtained to perform correlated jamming. Under various assumptions on te cannel caracteristics, and te extent of information available at te users and te jammer, we sow te existence, or oterwise nonexistence of a simultaneously optimal set of strategies for te users and te jammer. Wenever te game as a solution, we find te corresponding user and jammer strategies, and wenever te game does not admit a solution, we find te max-min user strategies and te corresponding jammer strategy. INTRODUCTION Correlated jamming, te situation were te jammer as full or partial knowledge about te user signals as been studied in te information-teoretic context under various assumptions [ 3]. In [] te best transmitter/jammer strategies are found for a single user AWGN cannel wit a jammer wo as full or partial knowledge of te transmitted signal. In [2], te problem is extended to a single user MIMO fading cannel. Tis model as been furter extended in [3], to consider fading in te cannel between te jammer and te receiver. In [3] various assumptions are made on te availability of te user cannel state at te user, and te jammer cannel state at te jammer. In tis paper, we study a multi-user system under correlated jamming. Partial results of tis study ave been reported in [4]. We consider a system of two users Tis work was supported by NSF Grants ANI and CCR 03-3; and ARL/CTA Grant DAAD and one jammer wo as full or partial knowledge of te user signals troug eavesdropping. In te nonfading two user cannel, we sow tat te game as a solution wic is Gaussian signalling for te users, and linear jamming for te jammer. Here we define linear jamming as employing a linear combination of te available information about te user signals plus Gaussian noise. We ten consider fading in te user cannels. As opposed to [3], were te user cannel states could only be known at te users, we assume te possibility of te jammer gaining information about te user cannel states by eavesdropping te feedback cannel from te receiver to te users and sow tat if te jammer is not aware of te user cannel states, it would disregard its eavesdropping information and only transmit Gaussian noise. If te jammer knows te user cannel states but not te user signals, te game as a solution wic is composed of te optimal user and jammer power allocation strategies over te cannel states, togeter wit Gaussian signalling and linear jamming at eac cannel state. If te jammer knows te user cannel states and te user signals, te game does not always ave a Nas equilibrium solution, in wic case, we caracterize te max-min user strategies, and te corresponding jammer best response. Te term capacity will ereafter always refer to te cannel s information capacity, defined as te cannel s maximum input/output mutual information [5]. SYSTEM MODEL Figure sows a communication system wit two users and one jammer. In te absence of fading, te attenuations of te user cannels are known to everyone. Te AWGN cannel wit two users and one jammer is Y = X + 2 X 2 + γj + N () of 7

2 USER USER 2 JAMMER RECEIVER Figure : A communication system wit two users and one jammer. were X i is te te i t user s signal, i is te attenuation of te i t user s cannel, J is te jammer s signal, γ is te attenuation of te jammer s cannel and N is a zeromean Gaussian random variable wit variance σ 2 N. To model fading in te received powers, we consider i as fading random variables, and to furter model te pase of te user cannel coefficients, we substitute te scalar attenuations i wit complex fading random variables H i, i =, 2. Te power constraints are E[X 2 i ] P i, i =, 2 (2) E[J 2 ] P J (3) We analyze bot cases wen te jammer as perfect information about te user signals, and imperfect information gained troug eavesdropping, were te jammer s eavesdropping link is also a multi-user AWGN cannel Y e = g X + g 2 X 2 + N e (4) were g i is te attenuation of te i t user s eavesdropping cannel, i =, 2, and N e is a zero-mean Gaussian random variable wit variance σ 2 e. JAMMING WITH COMPLETE INFORMATION In tis section, we consider te two user non-fading cannel in (), and assume tat te jammer knows te user signals. Te jammer and te two users are involved in a zero-sum game wit te input/output mutual information as te objective function. We investigate te existence and uniqueness of a Nas equilibrium solution for tis game [6]. A Nas equilibrium is a set of strategies, one for eac player, suc tat no player as an incentive for unilaterally canging its own strategy. For (X, X 2, Y ) f(x )f(x 2 )f(y x, x 2 ), te input/output mutual information I(X, X 2 ; Y ) is a concave function of f(x ) for fixed f(x 2 ) and f(y x, x 2 ), a concave function of f(x 2 ) for fixed f(x ) and f(y x, x 2 ), and a convex function of f(y x, x 2 ) for fixed f(x ) and f(x 2 ), terefore, I(X, X 2 ; Y ) as a saddle point, wic is te Nas equilibrium solution of te game [4]. In te sequel, we sow tat wen te users employ Gaussian signalling, te best jamming strategy is linear jamming, and wen te jammer employs linear jamming, te best strategy for te users is Gaussian signalling, wic proves tat tis set of strategies is a Nas equilibrium solution for te game. First assume tat te jammer employs linear jamming J = ρ X + ρ 2 X 2 + N J (5) Using (), te output of te cannel will be Y =( + γρ )X + ( 2 + γρ 2 )X 2 + γn J + N (6) wic describes an AWGN multiple access cannel, for wic te best signalling for te users is Gaussian [5]. Next, we sould sow tat if te users perform Gaussian signalling, ten te best jamming strategy is linear jamming. Te cannel output is as in () wit te input/output mutual information I(Y ; X, X 2 ) = (X, X 2 ) (X, X 2 Y ) (7) Te jammer can only affect (X, X 2 Y ) (X, X 2 Y ) = (X a Y, X 2 a 2 Y Y ) (8) (X a Y, X 2 a 2 Y ) (9) 2 log ( (2πe) 2 Λ ) (0) were Λ is te covariance matrix of (X a Y, X 2 a 2 Y ). We coose a = E[X Y ]/E[Y 2 ] and a 2 = E[X 2 Y ]/E[Y 2 ] and prove te optimality of linear jamming in two steps. First, we consider te set of all jamming signals wic result in te same Λ value, and sow tat if tis set includes a linear jammer, ten tat linear jammer is optimal over tis set. Ten, we consider te set of all feasible jamming signals and sow tat for any jamming signal in tis set, tere exists a linear jammer in tis set resulting in te same Λ value. Te feasibility is in te sense of te jammer s available power. Consider te set of all jamming signals wic result in te same Λ value in (0). Assume tat tere is a linear jamming signal in tis set. For tis jamming sig- 2 of 7

3 nal, X a Y, X 2 a 2 Y and Y are jointly Gaussian, and X a Y and X 2 a 2 Y are uncorrelated wit and consequently, independent of Y. We conclude tat tis jamming signal acieves bot (9) and (0) wit equality and is te optimal jamming strategy over tis set. Now we sow tat any Λ acievable by any feasible jammer, is also acievable by a feasible linear jammer. For te cosen values of a and a 2, Λ can be expressed as a function of E[X J] and E[X 2 J] [4]. Consider any jamming signal J. Define R as R = J X E[X J] P X 2 E[X 2 J] P 2 () Note tat R is uncorrelated wit X and X 2. Te power of tis jamming signal is E[J 2 ] = E[X J] 2 P + E[X 2J] 2 P 2 + E[R 2 ] (2) For tis jamming signal to be feasible, we sould ave E[X J] 2 P + E[X 2J] 2 P 2 P J (3) Now define a linear jamming signal as in (5), were ρ i = E[X ij] P i, i =, 2 (4) and N J is an independent Gaussian random variable wit power E[R 2 ]. Tis linear jammer as te same power as J and terefore is feasible. Moreover, it results in te same Λ value as J. Tis means tat, tere exists a feasible linear jamming signal wic is as effective as any oter feasible jamming signal, and tis concludes te proof. Te next step is to find te jamming coefficients ρ and ρ 2 tat minimize te mutual information in (7). Minimizing te mutual information ere is equivalent to minimizing te SNR [4] SNR = ( + γρ ) 2 P + ( 2 + γρ 2 ) 2 P 2 γσ 2 N J + σ 2 N (5) Te Karus-Kun-Tucker (KKT) necessary conditions result in [4] { ( γ, γ 2 ) if γp (ρ, ρ 2 ) = J P + 2 P 2 ( ρ, ρ 2 ) if γp J < P + 2 P 2 (6) were ρ = min { } P J γp J + σn 2, P + 2 P 2 γ( P + 2 P 2 ) (7) and te jammer transmits as in (5). We observe tat te amount of power te jammer allocates for jamming eac user is proportional to tat user s effective received power wic is i P i for user i, i =, 2. JAMMING WITH EAVESDROPPING INFORMATION Now suppose tat te jammer gains information about te user signals troug an eavesdropping cannel, Y e = g X + g 2 X 2 + N e (8) We define linear jamming as J = ρy e + N J (9) Here, we will prove tat in te eavesdropping case as well, linear jamming and Gaussian signalling is a game solution. Te proof of te optimality of Gaussian signalling, wen te jammer is linear is similar to te previous section [4]. However, wen it comes to sowing te optimality of linear jamming wen te users employ Gaussian signalling, te metod of te previous section cannot be used, since from (8) and (9), te values of E[X J] and E[X 2 J] tat are acievable troug linear jamming, sould furter satisfy E[X J] g = E[X 2J] g2 (20) Terefore, linear jamming may not acieve all Λ values in (0) tat are allowed under te power constraints. Here, we sow te optimality of linear jamming, by setting up an equivalent cannel. Define random variables Z and Z 2 in terms of X and X 2 as g2 Z = X + X 2 (2) g g g 2 P 2 g P Z 2 = X + X 2 (22) g P + g 2 P 2 g P + g 2 P 2 It is straigtforward to verify tat Z and Z 2 are uncorrelated, and ence, independent Gaussian random variables. Also I(X, X 2 ; Y ) = I(Z, Z 2 ; Y ) (23) 3 of 7

4 Terefore, te game s objective function can be replaced wit I(Z, Z 2 ; Y ). Now, using (), (2) and (22), JAMMER Y = u Z + u 2 Z 2 + γj + N (24) X Z RECEIVER were u i can be expressed in terms of i, g i and P i, i =, 2. Using (8), (2) and (22), Y e = g Z + N e (25) Note tat Y e is independent of Z 2. Equations (24) and (25) define a two user, one jammer system, depicted in Figure 2, were te jammer as eavesdropping information only about one of te users. Now, te equivalent input/output mutual information is I(Z, Z 2 ; Y ) = (Z, Z 2 ) (Z, Z 2 Y ) (26) Te jammer can only affect te second term above, (Z, Z 2 Y ) = (Z a Y, Z 2 a 2 Y Y ) (27) (Z a Y, Z 2 a 2 Y ) (28) log( + Σ ) (29) 2 were Σ is te covariance matrix of Z a Y and Z 2 a 2 Y. Following steps similar to tose in te previous section, wen te users are Gaussian, employing linear jamming togeter wit a good coice of a and a 2 can make bot inequalities old wit equality, and Σ will only be a function of E[Z J] and E[Z 2 J]. However, E[Z 2 J] = 0 and terefore, Σ is only a function of E[Z J]. In te sequel, we sow tat all E[Z J] values tat are acievable by all feasible jamming strategies, are also acievable by some feasible linear jamming, and terefore, linear jamming acieves te largest possible upper bound in (29) and is optimal. Using (25), te linear least squared error (LLSE) estimate of Z from Y e is [7] Z (Y e ) = g E[Z 2 ] σ 2 N e + g E[Z 2 ]Y e (30) Since Z and N e are Gaussian, tis estimate is also te minimum mean squared error (MMSE) estimate of Z, terefore, any oter estimate of Z results in a iger mean squared error. Consider any jamming signal J. Te LLSE estimate of Z from J is Ẑ = E[Z J] E[J 2 ] J (3) X2 Z2 Figure 2: An interpretation of a communication system wit two users and one jammer wit eavesdropping information. Tis is also anoter estimator of Z from Y e, ence te mean squared error for Ẑ is greater tan or equal to te mean squared error for Z, resulting in E 2 [Z J] g E 2 [Z 2 ]P J σ 2 N e + g E[Z 2 ] (32) Meanwile, using (25), all E 2 [Z J] values smaller tan te rigt and side of (32) are acievable by linear jamming, wic means tat linear jamming can acieve all feasible Σ values in (29), and tis concludes te proof. To find te optimal jamming coefficient, using (8) and (9), we write te jamming signal as J = ρ( g X + g 2 X 2 + N e ) + N J (33) and te jammer s optimization problem is min {ρ,σ 2 N J } ( + ρ γg ) 2 P + ( 2 + ρ γg 2 ) 2 P 2 γρ 2 σ 2 N e + γσ 2 N J + σ 2 N s.t. ρ 2 (g P + g 2 P 2 + σ 2 N e ) + σ 2 N J P J (34) Te KKTs for tis problem can be solved using numerical optimization [4]. JAMMING IN FADING MULTI-USER AWGN CHANNELS In te rest of te paper, we investigate te optimum user/jammer strategies wen te cannels are fading. We use te term CSI, for te cannel state information on te links from te users to te receiver, and assume tat te link between te jammer and te receiver is non-fading. Te receiver is assumed to know te CSI, wile various assumptions are made on te availability of te CSI at te jammer. 4 of 7

5 NO CSI AT THE TRANSMITTERS Wen te transmitters do not ave te CSI, it is reasonable to assume tat te jammer does not ave te CSI eiter. In te sequel, we sow tat te jammer s information about te transmitted signals will be irrelevant and terefore, it will not make any difference weter it as perfect or noisy information about te transmitted signals. Tis is a multi-user generalization of te results of [2] in a SISO system. Assuming tat te user links are fading and te jammer link is non-fading, te received signal is Y = H X + H 2 X 2 + γj + N (35) Similar arguments as in te case of correlated jamming in non-fading cannels can be used to sow tat te strategies corresponding to te game solution will be Gaussian signalling and linear jamming [4]. Te jamming signal is as in (5) and te jammer minimizes [ ( )] E log + H + γρ 2 P + H 2 + γρ 2 2 P 2 γσ 2 N J + σ 2 N subject to ρ 2 P + ρ 2 2P 2 + σ 2 N J P J. Te random variables H and H 2 are circularly symmetric complex Gaussian wit zero mean, terefore, te optimal jamming coefficients are ρ = ρ 2 = 0 and te jammer disregards its information about te user signals [4]. UNCORRELATED JAMMING WITH CSI AT THE TRANSMITTERS We now consider a two user fading cannel wit a jammer wo does not ave any information about te user signals. We assume tat te state of te user links are known to te users. Capacity of fading cannels wit CSI bot at te transmitter and te receiver wen tere is no jammer, as been investigated in [8] and [9], and optimum user strategies ave been derived. We consider te same problem wen tere is a jammer in te system. First consider te single-user case Y = HX + γj + N (36) Wen te CSI is available bot at te transmitter and te receiver, te input/output mutual information is C = I(X; Y H) (37) For (X, X 2, Y H) f(x )f(x 2 )f(y x, x 2, ), C is a convex function of f(y x, ) for any f(x ), and a concave function of f(x ) for any fixed f(y x, ). Te mutual information sub-game for any given cannel state as a Nas equilibrium solution wic is Gaussian signalling for te user and linear jamming for te jammer. If in addition, a Nas equilibrium solution exists over all feasible power allocation strategies of te user and te jammer, tat solution along wit te signalling and jamming strategies specified as te solution of te subgames at eac cannel state, will give te overall game solution. We proceed wit first assuming tat wile te user as te CSI, te jammer does not ave te CSI, and ten assuming tat te CSI is available bot at te user and at te jammer. If te jammer as no information about te fading cannel state, te best strategy for te jammer is to transmit Gaussian noise and te capacity is C = ( [log 2 E + P() )] σn 2 + γp (38) J were P() is te user power at fading level wic sould satisfy E [P()] P (39) Te best user power allocation is waterfilling over te equivalent parallel AWGN cannels [8], i.e., P() = ( σ2 N + γp ) + J (40) were (x) + = x if x 0, and 0 oterwise, and is a constant cosen to enforce te user power constraint. Te corresponding two user system, were te jammer is not aware of te user cannel coefficients, is a straigtforward extension of te results in [0] were only one user transmits at a time. Te jammer will again use all its power to add Gaussian noise. Next, we assume tat te uncorrelated jammer as full CSI. At eac cannel state, te jammer transmits Gaussian noise at te power level allocated to tat state and te capacity is C = ( [log 2 E + P() )] σn 2 + γj() (4) were J() is te jammer power at fading level. Te user power constraint is te same as (39), and te jammer power constraint is E [J()] P J (42) 5 of 7

6 Te capacity is a concave function of P for fixed J and a convex function of J for fixed P, terefore, te mutual information game as a solution [4]. Te KKTs for eac state-allocated user and jammer power result in [4] and P() = J() = ( γσ2 N ) + (+γ µ ) if < σ2 N γ γ σn 2 µ if σ2 N γ (43) γ σn 2 µ 0 if < σ2 N γ γ σ 2 N µ µ(+γ µ ) σ2 N γ if σ2 N γ γ σ 2 N µ (44) were and µ are found using te user and jammer power constraints. Following similar arguments as in te two user system witout a jammer in [0], te results can be extended to a two user system were te jammer is uncorrelated but it as access to te CSI. Te power allocation strategies will be functions of bot cannel states and 2, and at any given pair of and 2, only te user wit a relatively better cannel state transmits [4]. CORRELATED JAMMING WITH CSI AT THE TRANSMITTERS In tis section, we consider a two user fading cannel wit a jammer wo knows te user signals. We assume tat te user links are fading and te state of te user links are known to te users and te correlated jammer. We first sow tat tis game does not always ave a Nas equilibrium solution, and ten, find te max-min user strategies and te corresponding jamming strategy. Te input/output mutual information is as in (37). Te user and jammer power constraints can be written as E [ E[X 2 H] ] P (45) E [ E[J 2 H] ] P J (46) were E[X 2 H = ] and E[J 2 H = ] are te user and jammer powers allocated to te fading level H =. Any pair of user and jammer strategies, results in a pair of user and jammer power allocations. Terefore, te game s solution can be described as a pair of user and jammer power allocations, along wit te user and jammer signalling functions at eac cannel state. Using our results for te non-fading cannels, we ave tat irrespective of te existence or non-existence of optimal power allocation functions for te user and te jammer, te sub-games at eac cannel state, always ave a solution wic is Gaussian signalling for te users and linear jamming for te jammer. Since te jammer knows te cannel state and te transmitted signal, te received signal is ( ) Y = + ρ() X + N J + N (47) were te variance of N J is also a function of, σn 2 J (). Given a pair of power allocation functions P() and J(), te capacity is ( ) 2 C = 2 E + ρ() P() log + σn 2 + (48) σ2 N J () were ρ() and σ 2 N J () describe te optimal linear jamming for an equivalent non-fading cannel wit attenuation, and user and jammer powers P() and J(). Te power constraints of te user and te jammer are as in (39) and (42). Te capacity ere does not ave te convexity/concavity properties. In te sequel, we sow tat a pair of strategies, wic is simultaneously optimal for te user and te jammer, does not always exist. We first assume tat te user cooses its strategy once at te beginning of te communication, knowing tat te jammer will employ te corresponding optimal jamming strategy. We ten caracterize te user and jammer strategies in tis scenario. If te game ad a solution, it would ave been tis pair of user and jammer strategies, owever we prove te converse. Given any user power allocation function P(), we find te jammer s best response, were a best response describes wat a player does, given te oter player s move [6]. In tis case, a best response is te jammer s best power allocation, given a fixed user power allocation. Te jammer s best response can also be tougt of as a pair of functions ρ() and σ 2 N J () wic minimizes C = 2 E log + ( ) 2 + ρ() P() σn 2 + (49) σ2 N J () and ρ() and σ 2 N J () are constrained suc tat E [ ρ 2 ()P() + σ 2 N J () ] P J (50) Assuming tat σn 2 = 0, te jammer s strategy sould always be suc tat wenever te user transmits, σn 2 J () > 0, since oterwise, te capacity would be in- 6 of 7

7 finite. Te first order KKT conditions for te optimal jamming strategy result in ( ρ(), σ 2 NJ () ) = { (, 0) if P() ( ), P() if P() > 2 P() (5) /µ / P() J() were is cosen to satisfy te jammer s power constraint. Te total power tat te jammer allocates to eac cannel state is found as { P() if P() J() = if P() > (52) wic describes te best response of te jammer, to te user power allocation P(). We now derive te max-min user power allocation. First note tat te function inside te expectation in (49) is zero for P() /, terefore, in te optimal user power allocation, P() is eiter zero, or suc tat P() > /. Te capacity can now be written as C = ( [log 2 E )] P() (53) Te KKT condition for te user power, wenever te user transmits, results in P() = µ, for wic { 0 if µ J() = if > µ (54) Tese user and jammer power allocations are illustrated in Figure 3 were µ and are cosen to satisfy te user and jammer power constraints. We now sow tat wen te jammer employs (54), te current user strategy is not optimal. Consider two fading levels u + > µ/ and u < µ/ close enoug to = µ/ suc tat u u + µ/ (55) We ave J(u ) = 0 and J(u + ) = /. Obviously, it is not optimal for te user to transmit at u + wile not transmitting at u, terefore, te pair of te user and jammer power allocations derived (wic corresponds to te user max-min solution), is not a game solution, and te game does not admit a solution. Even toug te max-min solution derived above is not te game Nas equilibrium solution, it is te optimal pair of user and jammer strategies in a system were a conservative user would like to guarantee itself wit µ/ Figure 3: Max-min user power allocation and te corresponding jammer best response power allocation. some capacity value. It also describes te best strategy for a user wic is less dynamic tan te jammer in terms of canging te transmission strategy, and can coose its strategy only once. References [] M. Médard, Capacity of Correlated Jamming Cannels, Allerton, 997. [2] A. Kasyap, T. Basar and R. Srikant, Correlated jamming on MIMO Gaussian fading cannels, IEEE Tran. on Inf. Te., Sep [3] A. Bayeste, M. Ansari and A. K. Kandani, Effect of Jamming on te Capacity of MIMO Cannels, Allerton, [4] S. Safiee and S. Ulukus, Correlated Jamming in Multiple Access Cannels, CISS, [5] T. M. Cover and J. A. Tomas, Elements of Information Teory, Jon Wiely & Sons, 99. [6] D. Fudenberg and J. Tirole, Game Teory, MIT Press, 99. [7] H. V. Poor, An Introduction to Signal Detection and Estimation, 2nd Ed., Springer-Verlag, 994. [8] A. Goldsmit and P. Varaiya, Capacity of fading cannels wit cannel side information, IEEE Tran. on Inf. Te., Nov [9] G. Caire and S. Samai, On te Capacity of Some Cannels wit Cannel State Information, IEEE Tran. on Inf. Te., Sep [0] R. Knopp and P. A. Humblet, Information capacity and power control in single-cell multiuser communications, IEEE ICC, Jun of 7