CAPACITY OF MULTIPLE ACCESS CHANNELS

WITH CORRELATED JAMMING∗

Shabnam Shafiee and Sennur Ulukus

Department of Electrical and Computer Engineering

University of Maryland

College Park, MD

ABSTRACT

We investigate the behavior of two users and one jammerin an AWGN channel with and without fading when theyparticipate in a non-cooperative zero-sum game, with thechannel’s input/output mutual information as the objec-tive function. We assume that the jammer can eaves-drop the channel and can use the information obtainedto perform correlated jamming. Under various assump-tions on the channel characteristics, and the extent ofinformation available at the users and the jammer, weshow the existence, or otherwise nonexistence of a simul-taneously optimal set of strategies for the users and thejammer. Whenever the game has a solution, we find thecorresponding user and jammer strategies, and wheneverthe game does not admit a solution, we find the max-minuser strategies and the corresponding jammer strategy.

INTRODUCTION

Correlated jamming, the situation where the jammerhas full or partial knowledge about the user signals hasbeen studied in the information-theoretic context un-der various assumptions [1–3]. In [1] the best transmit-ter/jammer strategies are found for a single user AWGNchannel with a jammer who has full or partial knowledgeof the transmitted signal. In [2], the problem is extendedto a single user MIMO fading channel. This model hasbeen further extended in [3], to consider fading in thechannel between the jammer and the receiver. In [3]various assumptions are made on the availability of theuser channel state at the user, and the jammer channelstate at the jammer.

In this paper, we study a multi-user system under cor-related jamming. Partial results of this study have beenreported in [4]. We consider a system of two users and

∗This work was supported by NSF Grants ANI 02-05330 andCCR 03-11311; and ARL/CTA Grant DAAD 19-01-2-0011.

one jammer who has full or partial knowledge of theuser signals through eavesdropping. In the non-fadingtwo user channel, we show that the game has a solutionwhich is Gaussian signalling for the users, and linearjamming for the jammer. Here we define linear jammingas employing a linear combination of the available infor-mation about the user signals plus Gaussian noise. Wethen consider fading in the user channels. We assumethat the jammer may have the user channel states byeavesdropping on the channel state feedback and showthat if the jammer is not aware of the user channel states,it would disregard its eavesdropping information andonly transmit Gaussian noise. If the jammer knows theuser channel states but not the user signals, the gamehas a solution which is composed of the optimal userand jammer power allocation strategies over the chan-nel states, together with Gaussian signalling and linearjamming at each channel state. If the jammer knows theuser channel states and the user signals, the game doesnot always have a Nash equilibrium solution, in whichcase, we characterize the max-min user strategies, andthe corresponding jammer best response.

The term capacity will hereafter always refer to thechannel’s information capacity, defined as the channel’smaximum input/output mutual information [5].

SYSTEM MODEL

Figure 1 shows a communication system with two usersand one jammer. In the absence of fading, the attenu-ations of the user channels are known to everyone. TheAWGN channel with two users and one jammer is

Y =√

h1X1 +√

h2X2 +√

γJ + N (1)

where Xi is the the ith user’s signal, hi is the attenuationof the ith user’s channel, J is the jammer’s signal, γ isthe attenuation of the jammer’s channel and N is a zero-

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USER 2

JAMMER

USER 1 RECEIVER

Figure 1: A communication system with two users andone jammer.

mean Gaussian random variable with variance σ2N . To

model fading in the received powers, we consider hi asfading random variables, and to further model the phaseof the user channel coefficients, we substitute the scalarattenuations

√hi with complex fading random variables

Hi, i = 1, 2. The power constraints are

E[X2i ] ≤ Pi, i = 1, 2 (2)

E[J2] ≤ PJ (3)

We analyze both cases when the jammer has perfect in-formation about the user signals, and imperfect informa-tion gained through eavesdropping, where the jammer’seavesdropping link is also a multi-user AWGN channel

Ye =√

g1X1 +√

g2X2 + Ne (4)

where gi is the attenuation of the ith user’s eavesdrop-ping channel, i = 1, 2, and Ne is a zero-mean Gaussianrandom variable with variance σ2

e .

JAMMING WITH COMPLETE

INFORMATION

In this section, we consider the two user non-fading chan-nel in (1), and assume that the jammer knows the usersignals. The jammer and the two users are involved in azero-sum game with the input/output mutual informa-tion as the objective function. We investigate the exis-tence and uniqueness of a Nash equilibrium solution forthis game [6]. A Nash equilibrium is a set of strategies,one for each player, such that no player has an incentivefor unilaterally changing its own strategy.

For (X1, X2, Y ) ∼ f(x1)f(x2)f(y|x1, x2), the in-put/output mutual information I(X1, X2; Y ) is a con-cave function of f(x1) for fixed f(x2) and f(y|x1, x2), aconcave function of f(x2) for fixed f(x1) and f(y|x1, x2),

and a convex function of f(y|x1, x2) for fixed f(x1)and f(x2), therefore, I(X1, X2; Y ) has a saddle point,which is the Nash equilibrium solution of the game [4].In the sequel, we show that when the users employGaussian signalling, the best jamming strategy is lin-ear jamming, and when the jammer employs linear jam-ming, the best strategy for the users is Gaussian sig-nalling, which proves that this set of strategies is a Nashequilibrium solution for the game.

First assume that the jammer employs linear jamming

J = ρ1X1 + ρ2X2 + NJ (5)

Using (1), the output of the channel will be

Y =(√

h1 +√

γρ1)X1 + (√

h2 +√

γρ2)X2

+√

γNJ + N (6)

which describes an AWGN multiple access channel, forwhich the best signalling for the users is Gaussian [5].

Next, we should show that if the users performGaussian signalling, then the best jamming strategy islinear jamming. The channel output is as in (1) withthe input/output mutual information

I(Y ; X1, X2) = h(X1, X2) − h(X1, X2|Y ) (7)

The jammer can only affect h(X1, X2|Y )

h(X1, X2|Y ) = h(X1 − a1Y, X2 − a2Y |Y ) (8)

≤ h(X1 − a1Y, X2 − a2Y ) (9)

≤ 1

2log(

(2πe)2|Λ|)

(10)

where Λ is the covariance matrix of (X1 − a1Y, X2 −a2Y ). We choose a1 = E[X1Y ]/E[Y 2] and a2 =E[X2Y ]/E[Y 2] and prove the optimality of linear jam-ming in two steps. First, we consider the set of all jam-ming signals which result in the same Λ value, and showthat if this set includes a linear jammer, then that linearjammer is optimal over this set. Then, we consider theset of all feasible jamming signals and show that for anyjamming signal in this set, there exists a linear jammerin this set resulting in the same Λ value. The feasibilityis in the sense of the jammer’s available power.

Consider the set of all jamming signals which resultin the same Λ value in (10). Assume that there is alinear jamming signal in this set. For this jamming sig-nal, X1 − a1Y, X2 − a2Y and Y are jointly Gaussian,and X1 − a1Y and X2 − a2Y are uncorrelated with andconsequently, independent of Y. We conclude that this

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jamming signal achieves both (9) and (10) with equalityand is the optimal jamming strategy over this set.

Now we show that any Λ achievable by any feasiblejammer, is also achievable by a feasible linear jammer.For the chosen values of a1 and a2, Λ can be expressedas a function of E[X1J ] and E[X2J ] [4]. Consider anyjamming signal J . Define R as

R = J − X1E[X1J ]

P1− X2

E[X2J ]

P2(11)

Note that R is uncorrelated with X1 and X2. The powerof this jamming signal is

E[J2] =E[X1J ]2

P1+

E[X2J ]2

P2+ E[R2] (12)

For this jamming signal to be feasible, we should have

E[X1J ]2

P1+

E[X2J ]2

P2≤ PJ (13)

Now define a linear jamming signal as in (5), whereρi = E[XiJ ]/Pi, i = 1, 2, and NJ is an independentGaussian random variable with power E[R2]. This lin-ear jammer has the same power as J and therefore isfeasible. Moreover, it results in the same |Λ| value as J .This means that, there exists a feasible linear jammingsignal which is as effective as any other feasible jammingsignal, and this concludes the proof.

The next step is to find the jamming coefficients ρ1

and ρ2 that minimize the mutual information in (7).Minimizing the mutual information here is equivalentto minimizing the SNR [4]

SNR =(√

h1 +√

γρ1)2P1 + (

√h2 +

√γρ2)

2P2

γσ2NJ

+ σ2N

(14)

The Karush-Kuhn-Tucker (KKT) necessary conditionsresult in [4]

(ρ1, ρ2) =

{

(−√

h1√γ

,−√

h2√γ

) if γPJ ≥ h1P1 + h2P2

(−ρ√

h1,−ρ√

h2) if γPJ < h1P1 + h2P2

(15)

where

ρ = min

{

√

PJ

h1P1 + h2P2,

γPJ + σ2N√

γ(h1P1 + h2P2)

}

(16)

and the jammer transmits as in (5). We observe thatthe amount of power the jammer allocates for jamming

each user is proportional to that user’s effective receivedpower which is hiPi for user i, i = 1, 2.

JAMMING WITH EAVESDROPPING

INFORMATION

Now suppose that the jammer gains information aboutthe user signals through an eavesdropping channel,

Ye =√

g1X1 +√

g2X2 + Ne (17)

We define linear jamming as

J = ρYe + NJ (18)

Here, we will prove that in the eavesdropping case aswell, linear jamming and Gaussian signalling is a gamesolution. The proof of the optimality of Gaussian sig-nalling, when the jammer is linear is similar to the pre-vious section [4]. However, when it comes to show-ing the optimality of linear jamming when the usersemploy Gaussian signalling, the method of the previ-ous section cannot be used, since from (17) and (18),the values of E[X1J ] and E[X2J ] that are achiev-able through linear jamming, should further satisfyE[X1J ]/

√g1 = E[X2J ]/

√g2. Therefore, linear jamming

may not achieve all |Λ| values in (10) that are allowedunder the power constraints. Here, we show the opti-mality of linear jamming, by setting up an equivalentchannel. Define random variables Z1 and Z2 in terms ofX1 and X2 as

Z1 = X1 +

√g2√g1

X2 (19)

Z2 = −√

g1g2P2

g1P1 + g2P2X1 +

g1P1

g1P1 + g2P2X2 (20)

It is straightforward to verify that Z1 and Z2 are un-correlated, and hence, independent Gaussian randomvariables. Also I(X1, X2; Y ) = I(Z1, Z2; Y ). There-fore, the game’s objective function can be replaced withI(Z1, Z2; Y ). Now, using (1), (19) and (20),

Y = u1Z1 + u2Z2 +√

γJ + N (21)

where ui can be expressed in terms of hi, gi and Pi, i =1, 2. Using (17), (19) and (20),

Ye =√

g1Z1 + Ne (22)

Note that Ye is independent of Z2. Equations (21) and(22) define a two user, one jammer system, depicted in

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X2

X1

Z2

Z1

JAMMER

RECEIVER

Figure 2: An interpretation of a communication systemwith two users and one jammer with eavesdropping in-formation.

Figure 2, where the jammer has eavesdropping informa-tion only about one of the users. Now, the equivalentinput/output mutual information is

I(Z1, Z2; Y ) = h(Z1, Z2) − h(Z1, Z2|Y ) (23)

The jammer can only affect the second term above,

h(Z1, Z2|Y ) = h(Z1 − a1Y, Z2 − a2Y |Y ) (24)

≤ h(Z1 − a1Y, Z2 − a2Y ) (25)

≤ 1

2log(1 + |Σ|) (26)

where Σ is the covariance matrix of Z1 − a1Y andZ2 − a2Y. Following steps similar to those in the pre-vious section, when the users are Gaussian, employinglinear jamming together with a good choice of a1 anda2 can make both inequalities hold with equality, and|Σ| will only be a function of E[Z1J ] and E[Z2J ]. How-ever, E[Z2J ] = 0 and therefore, |Σ| is only a function ofE[Z1J ]. In the sequel, we show that all E[Z1J ] valuesthat are achievable by all feasible jamming strategies,are also achievable by some feasible linear jamming, andtherefore, linear jamming achieves the largest possibleupper bound in (26) and is optimal.

Using (22), the linear least squared error (LLSE) es-timate of Z1 from Ye is [7]

Z1(Ye) =

√g1E[Z2

1 ]

σ2Ne

+ g1E[Z21 ]

Ye (27)

Since Z1 and Ne are Gaussian, this estimate is also theminimum mean squared error (MMSE) estimate of Z1,therefore, any other estimate of Z1 results in a highermean squared error. Consider any jamming signal J .

The LLSE estimate of Z1 from J is

Z1 =E[Z1J ]

E[J2]J (28)

This is also another estimator of Z1 from Ye, hence themean squared error for Z1 is greater than or equal to themean squared error for Z1, resulting in

E2[Z1J ] ≤ g1E2[Z2

1 ]PJ

σ2Ne

+ g1E[Z21 ]

(29)

Meanwhile, using (22), all E2[Z1J ] values smaller thanthe right hand side of (29) are achievable by linear jam-ming, which means that linear jamming can achieve allfeasible |Σ| values in (26), and this concludes the proof.

To find the optimal jamming coefficient, using (17)and (18), we write the jamming signal as

J = ρ(√

g1X1 +√

g2X2 + Ne) + NJ (30)

and the jammer’s optimization problem is

min{ρ,σ2

NJ}

(√

h1 + ρ√

γg1)2P1 + (

√h2 + ρ

√γg2)

2P2

γρ2σ2Ne

+ γσ2NJ

+ σ2N

s.t. ρ2(g1P1 + g2P2 + σ2Ne

) + σ2NJ

≤ PJ (31)

The KKTs for this problem can be solved using numer-ical optimization [4].

JAMMING IN FADING MULTI-USER AWGN

CHANNELS

In the rest of the paper, we investigate the optimumuser/jammer strategies when the channels are fading.We use the term CSI, for the channel state informationon the links from the users to the receiver, and assumethat the link between the jammer and the receiver isnon-fading. The receiver is assumed to know the CSI,while various assumptions are made on the availabilityof the CSI at the jammer.

NO CSI AT THE TRANSMITTERS

When the transmitters do not have the CSI, it is reason-able to assume that the jammer does not have the CSIeither. In the sequel, we show that the jammer’s infor-mation about the transmitted signals will be irrelevantand therefore, it will not make any difference whether ithas perfect or noisy information about the transmittedsignals. This is a multi-user generalization of the resultsof [2] in a SISO system.

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Assuming that the user links are fading and the jam-mer link is non-fading, the received signal is

Y = H1X1 + H2X2 +√

γJ + N (32)

Similar arguments as in the case of correlated jammingin non-fading channels can be used to show that thestrategies corresponding to the game solution will beGaussian signalling and linear jamming [4]. The jam-ming signal is as in (5) and the jammer minimizes

E

[

log

(

1 +|H1 +

√γρ1|2P1 + |H2 +

√γρ2|2P2

γσ2NJ

+ σ2N

)]

subject to ρ21P1 + ρ2

2P2 + σ2NJ

≤ PJ . The randomvariables H1 and H2 are circularly symmetric complexGaussian with zero mean, therefore, the optimal jam-ming coefficients are ρ1 = ρ2 = 0 and the jammer disre-gards its information about the user signals [4].

UNCORRELATED JAMMING WITH CSI AT

THE TRANSMITTERS

We now consider a two user fading channel with a jam-mer who does not have any information about the usersignals. We assume that the state of the user links areknown to the users. Capacity of fading channels withCSI both at the transmitter and the receiver when thereis no jammer, has been investigated in [8] and [9], andoptimum user strategies have been derived. We considerthe same problem when there is a jammer in the system.First consider the single-user case

Y = HX +√

γJ + N (33)

When the CSI is available both at the transmitter andthe receiver, the input/output mutual information is

C = I(X; Y |H) (34)

For (X1, X2, Y |H) ∼ f(x1|h)f(x2|h)f(y|x1, x2, h), C isa convex function of f(y|x, h) for any f(x|h), and a con-cave function of f(x|h) for any fixed f(y|x, h). The mu-tual information sub-game for any given channel statehas a Nash equilibrium solution which is Gaussian sig-nalling for the user and linear jamming for the jammer.If in addition, a Nash equilibrium solution exists overall feasible power allocation strategies of the user andthe jammer, that solution along with the signalling andjamming strategies specified as the solution of the sub-games at each channel state, will give the overall game

solution. We proceed with first assuming that while theuser has the CSI, the jammer does not have the CSI,and then assuming that the CSI is available both at theuser and at the jammer.

If the jammer has no information about the fadingchannel state, the best strategy for the jammer is totransmit Gaussian noise and the capacity is

C =1

2E

[

log

(

1 +hP (h)

σ2N + γPJ

)]

(35)

where P (h) is the user power at fading level h whichshould satisfy

E [P (h)] ≤ P (36)

The best user power allocation is waterfilling over theequivalent parallel AWGN channels [8], i.e.,

P (h) =

(

1

λ− σ2

N + γPJ

h

)+

(37)

where (x)+ = x if x ≥ 0, and 0 otherwise, and λ isa constant chosen to enforce the user power constraint.The corresponding two user system, where the jammeris not aware of the user channel coefficients, is a straight-forward extension of the results in [10] where only oneuser transmits at a time. The jammer will again use allits power to add Gaussian noise.

Next, we assume that the uncorrelated jammer hasfull CSI. At each channel state, the jammer transmitsGaussian noise at the power level allocated to that stateand the capacity is

C =1

2E

[

log

(

1 +hP (h)

σ2N + γJ(h)

)]

(38)

where J(h) is the jammer power at fading level h. Theuser power constraint is the same as (36), and the jam-mer power constraint is

E [J(h)] ≤ PJ (39)

The capacity is a concave function of P for fixed J anda convex function of J for fixed P , therefore, the mutualinformation game has a solution [4]. The KKTs for eachstate-allocated user and jammer power result in [4]

P (h) =

( 1λ− γσ2

N

h)+ if h <

σ2

Nγλ

γ−σ2

Nµ

h

λ(h+γ λµ

)if h ≥ σ2

Nγλ

γ−σ2

Nµ

(40)

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and

J(h) =

0 if h <σ2

Nγλ

γ−σ2

Nµ

h

µ(h+γ λµ

)− σ2

N

γif h ≥ σ2

Nγλ

γ−σ2

Nµ

(41)

where λ and µ are found using the user and jammerpower constraints.

Following similar arguments as in the two user systemwithout a jammer in [10], the results can be extended toa two user system where the jammer is uncorrelated butit has access to the CSI. The power allocation strategieswill be functions of both channel states h1 and h2, andat any given pair of h1 and h2, only the user with arelatively better channel state transmits [4].

CORRELATED JAMMING WITH CSI AT

THE TRANSMITTERS

In this section, we consider a two user fading channelwith a jammer who knows the user signals. We assumethat the user links are fading and the state of the userlinks are known to the users and the correlated jammer.We first show that this game does not always have aNash equilibrium solution, and then, find the max-minuser strategies and the corresponding jamming strat-egy. The input/output mutual information is as in (34).The user and jammer power constraints can be writ-ten as E

[

E[X2|H]]

≤ P and E[

E[J2|H]]

≤ PJ whereE[X2|H = h] and E[J2|H = h] are the user and jammerpowers allocated to the fading level H = h. Any pair ofuser and jammer strategies, results in a pair of user andjammer power allocations. Therefore, the game’s solu-tion can be described as a pair of user and jammer powerallocations, along with the user and jammer signallingfunctions at each channel state. Using our results forthe non-fading channels, we have that irrespective of theexistence or non-existence of optimal power allocationfunctions for the user and the jammer, the sub-gamesat each channel state, always have a solution which isGaussian signalling for the users and linear jamming forthe jammer.

Since the jammer knows the channel state and thetransmitted signal, the received signal is

Y =(√

h + ρ(h))

X + NJ + N (42)

where the variance of NJ is also a function of h, σ2NJ

(h).Given a pair of power allocation functions P (h) and

J(h), the capacity is

C =1

2E

log

1 +

(√h + ρ(h)

)2P (h)

σ2N + σ2

NJ(h)

(43)

where ρ(h) and σ2NJ

(h) describe the optimal linear jam-ming for an equivalent non-fading channel with atten-uation h, and user and jammer powers P (h) and J(h).The power constraints of the user and the jammer areas in (36) and (39). The capacity here does not havethe convexity/concavity properties. In the sequel, weshow that a pair of strategies, which is simultaneouslyoptimal for the user and the jammer, does not alwaysexist. We first assume that the user chooses its strategyonce at the beginning of the communication, knowingthat the jammer will employ the corresponding optimaljamming strategy. We then characterize the user andjammer strategies in this scenario. If the game had a so-lution, it would have been this pair of user and jammerstrategies, however we prove the converse.

Given any user power allocation function P (h), wefind the jammer’s best response, where a best responsedescribes what a player does, given the other player’smove [6]. In this case, a best response is the jammer’sbest power allocation, given a fixed user power alloca-tion. The jammer’s best response can also be thought ofas a pair of functions ρ(h) and σ2

NJ(h) which minimizes

C =1

2E

log

1 +

(√h + ρ(h)

)2P (h)

σ2N + σ2

NJ(h)

(44)

and ρ(h) and σ2NJ

(h) are constrained such that

E[

ρ2(h)P (h) + σ2NJ

(h)]

≤ PJ . Using the first order

KKT conditions for the jammer, the optimal jammingstrategy is

(

ρ(h), σ2NJ

(h))

={

(−√

h, 0) if hP (h) ≤ 1λ

(

− 1λ√

hP (h), 1

λ− 1

λ2hP (h)

)

if hP (h) > 1λ

(45)

where λ is chosen to satisfy the jammer’s power con-straint. The total power that the jammer allocates toeach channel state is found as

J(h) =

{

hP (h) if hP (h) ≤ 1λ

1λ

if hP (h) > 1λ

(46)

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h

1/λ

1/µP(h)

J(h)

µ/λ

Figure 3: Max-min user power allocation and the corre-sponding jammer best response power allocation.

which describes the best response of the jammer, to theuser power allocation P (h).

We now derive the max-min user power allocation.First note that the function inside the expectation in(44) is zero for hP (h) ≤ 1/λ, therefore, in the opti-mal user power allocation, P (h) is either zero, or suchthat hP (h) > 1/λ. We now show that the optimal userpower allocation is also such that σ2

NJ(h) is nonzero in

all fading levels where both the user and jammer trans-mit. Assume that at some fading level h, σ2

NJ(h) is zero.

Define Ch as

Ch =1

2log

1 +

(√h − 1

λ√

hP (h)

)2P (h)

σ2N

(47)

Ch and consequently C are convex in P (h), therefore anyP (h) value in this region is not optimal and the optimaluser power allocation should not have any point resultingin σ2

NJ(h) = 0. The capacity can now be written as

C =1

2E[

log(

λ√

hP (h))]

(48)

The KKT condition for the user power, whenever theuser transmits, results in P (h) = 1

µ, for which

J(h) =

{

0 if h ≤ µλ

1λ

if h > µλ

(49)

These user and jammer power allocations are illustratedin Figure 3 where µ and λ are chosen to satisfy the userand jammer power constraints.

We now show that when the jammer employs (49),

the current user strategy is not optimal. Consider twofading levels u+ > µ/λ and u− < µ/λ close enough toh = µ/λ such that

u− ≃ u+ ≃ µ/λ (50)

We have J(u−) = 0 and J(u+) = 1/λ. Obviously, itis not optimal for the user to transmit at u+ while nottransmitting at u−, therefore, the pair of the user andjammer power allocations derived (which corresponds tothe user max-min solution), is not a game solution, andthe game does not admit a solution.

Even though the max-min solution derived above isnot the game Nash equilibrium solution, it is the optimalpair of user and jammer strategies in a system wherea conservative user would like to guarantee itself withsome capacity value. It also describes the best strategyfor a user which is less dynamic than the jammer interms of changing the transmission strategy, and canchoose its strategy only once.

References

[1] M. Medard, “Capacity of Correlated Jamming Chan-nels,” Allerton, 1997.

[2] A. Kashyap, T. Basar and R. Srikant, “Correlated jam-ming on MIMO Gaussian fading channels,” IEEE Tran.on Inf. The., Sep. 2004.

[3] A. Bayesteh, M. Ansari and A. K. Khandani, “Effect ofJamming on the Capacity of MIMO Channels,” Allerton,2004.

[4] S. Shafiee and S. Ulukus, “Correlated Jamming in Mul-tiple Access Channels,” CISS, 2005.

[5] T. M. Cover and J. A. Thomas, Elements of InformationTheory, John Wiely & Sons, 1991.

[6] D. Fudenberg and J. Tirole, Game Theory, MIT Press,1991.

[7] H. V. Poor, An Introduction to Signal Detection and Es-timation, 2nd Ed., Springer-Verlag, 1994.

[8] A. Goldsmith and P. Varaiya, “Capacity of fading chan-nels with channel side information,” IEEE Tran. on Inf.The., Nov. 1997.

[9] G. Caire and S. Shamai, “On the Capacity of SomeChannels with Channel State Information,” IEEE Tran.on Inf. The., Sep. 1999.

[10] R. Knopp and P. A. Humblet, “Information capacity andpower control in single-cell multiuser communications,”IEEE ICC, Jun. 1995.

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