ISIT2007, Nice, France, June 24 - June 29, 200O

Gaussian Jamming in Block-Fading Channels under

Long Term Power ConstraintsGeorge T. Amariucai and Shuangqing Wei and Rajgopal Kannan

Abstract- We formulate a Gaussian uncorrelated jammingproblem in block fading channels under long term powerconstraints. Source aims at minimizing the outage probabilityof its transmission under the presence of a malicious jammer,while the jammer attempts to maximize the corresponding outageprobability under its average power constraint. Optimal powercontrol strategies for both source and jammer are obtained forminimax and maxmin problems, respectively, for any arbitraryfinite number of blocks in block fading channels. Our resultsdemonstrate the non-existence of Nash-equilibria of this two-person zero-sum game.

I. INTRODUCTION

The problem of jamming in wireless networks started toattract interest in the '80s [1] and was focused on simple,point-to point communication systems affected by intelligentjammer.

The jammer was assumed to have access to a noise-distortedversion of the transmitter's output [1]. The mean-squared errorwas considered as a performance indicator. A saddlepoint forthe jamming game of [1] consists of an amplifying transmitterand a jammer that performs a linear transformation of its avail-able version of the transmitter's output signal. Similar resultswere obtained in [2] for correlated jammers suffering fromphase/time jitters at acquisition or at transmission. Channelcapacity was used as a performance indicator. Extensions tomore complex, multi-user channels with fading, were derivedin [3], [4] and [5].The general tendency seems to be in favor of an assumption

that jammer has access to either the transmitter's output orinput and consequently is able to produce correlated jam-ming signals. Uncorrelated jammers are often studied onlyas a particular case. We, however, argue that the correlationassumption is sometimes inappropriate because of the effectof causality. In addition, most recent works adopt ergodic ca-pacity as a common objective function over which transmitterand jammer fight against each other [5], [3], which is not asuitable metric when a delay constraint is imposed.

In this paper, we take a look at a constant-rate wirelesssystem with power and delay constraints, which is depictedin Figure 1. This system model is similar to the one usedin [5]. However, the major difference is that we investigatejamming in delay constrained block fading channels, andtherefore adopt outage probability [6] as an objective overwhich jammer and transmitter fight against each other. In

'G. Amafiucai and S. Wei are with the Department of ECE, Louisiana StateUniversity

2R. Kannan is with the Department of CS, Louisiana State University

addition, in our formulation the jammer is assumed to possess

no knowledge about the output of or the codebook employedby the transmitter.

Our problem is formulated as a two-player, zero-sum game,

where only pure strategies (no randomized strategies) are

considered. The power constraint in a block fading channelcan be manifested in terms of either short term or long term[6]. In this paper, we only present the long-term case.

We have already solved theM = 1 case in [7]. In this paper,

we provide an extension to the more general case of any finiteM > 1. The extension is far from trivial. Not only are theregular convex optimization techniques (e.g.KKT conditions)unable to yield a closed form solution, but the methods usedin [7] are no longer applicable. We therefore develop a novelapproach to solving both the intra-frame and inter-frame powerallocation problems under the minimax and maxmin criteria,respectively.

II. CHANNEL MODEL AND NOTATIONS

The channel model is depicted in Figure 1. Each codewordspans a concatenation of M blocks, each of which has Nchannel uses. As assumed in [6], we let N -> Xo in order toaverage out the impact of Gaussian noise.

Over a given frame, the transmitter (Tx) allocates power Pmto block m, 0 < m < M -1, while the jammer (Jx) investspower Jm in jamming the same block with the worst possiblejamming signal which is uncorrelated with the transmitter'soutput, white and Gaussian distributed [8].The channel squared fading coefficient hm is constant over

the length of one block. The vector h = [ho,... , hm- ] ofchannel coefficients over a whole frame is assumed to beperfectly known to transmitter, receiver and jammer beforetransmission begins. This condition does not imply non-

causality if we think of modeling a multicarrier system [6].The mutual information over a subchannel m when trans-

mitter uses a Gaussian codebook is given by I(hm, Pm, Jm) =

Fig. 1. Channel model

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ISIT2007, Nice, France, June 24 - June 29, 200O

1/2 log(l + Ch_P ), where 4- is the variance of the AWGN.In the sequel, the following denotations will be repeatedly

used: (1) Power allocated by transmitter to a frame: PM =

M m-1 Pm; (2) Power allocated by jammer to a frame:JM = jM-1 Jm; (3) Instantaneous mutual informationfor a frame:IM = M m-1 I(h, Pm, Jm).

Also note that PM is a function of the channel realizationh, so we often write PM (h) when this relation needs to bepointed out. This can be interpreted as the function giving thepower distribution between frames. The notation P(h) andJ(h) will denote the functions giving the power distributionwithin a frame (between blocks).The probability of outage will be the cost/reward function.

It is defined as [6] P0ot = Pr(IM < R), with R denoting thefixed rate of the system.The power required for a player to achieve its objective (re-

liable communication for transmitter, and outage for jammer)over some frame, given a fixed opponent's behavior, will bedenoted as required power. Depending on its optimal strategy,this power may or may not be actually matched by the player.The long-term power constrained jamming game can be

described as:

TxMinimize

T Subject toPr(IM(h, P(h), J(h)) < R)

E[PM(h)] <P

J{ Maximize Pr(IM(h,P(h),J(h)) <R)J Subject to E[JM(h)] <J

and has the probability of outage as cost/reward function. Thecase of M = 1 is only concerned with this level. The second(finer) level is that of power allocation between blocks withina frame.The probability of outage is determined by the m-measure of

the set over which the transmitter is not present or transmissionis jammed. This set is established in the first level of powercontrol. In this subsection, we study the second level strategies.

In the maximin case (when jammer plays first), assumethat the jammer has already allocated power JM to a givenframe. Depending on the channel realization, the value ofJM, and its power constraints, the transmitter decides whetherit wants to achieve reliable communication over that frame.If it decides to transmit, it needs to spend as little poweras possible (transmitter will be able to use the saved powerfor achieving reliable communication over another set ofpositive m-measure, and thus to decrease the probability ofoutage). Therefore, the transmitter's objective is to minimizethe power PM spent for achieving reliable communication.The transmitter will adopt this strategy whether the jammeris present over the frame, or not. The jammer's objective isthen to allocate JM between the blocks such that the requiredPM is maximized. Similar arguments apply for the minimaxscenario.The two problems can be formulated as:Problem ] (for the maximin solution)

(2)

where expectations are taken with respect to the vector ofchannel coefficients h = (ho, h1,..., hM-1) C RM, and Pand J denote the upper-bounds on the average transmissionpower of the source and jammer, respectively.

Each player needs to make a decision on whether or notto transmit over a frame, given its opponent's strategy. In thesequel, we look at both maximin and minimax solutions tothe above game and determine the associated optimal powercontrol functions PM(h) and JM(h).The maximin solution is defined as the set of optimal

strategies when jammer plays first. No matter what strategyjammer uses, transmitter will take advantage of its weaknesses,and aim at a minimum of the outage probability. The bestoption for jammer is to maximize the minimum achievable bythe transmitter. On the other hand, the minimax solution isdefined as the set of optimal strategies when the transmitterplays first. Jammer aims at a maximum outage probability.The best option for transmitter is to minimize the maximumachievable by jammer. In [7] we found these solutions for thecase M= 1.

Let m denote the probability measure introduced by theprobability density function (p.d.f.) f (h)d of h, i.e., for a setc/ CR', we have m(S) = f (h)dh.

III. POWER ALLOCATION BETWEEN BLOCKS WITHIN AFRAME FOR M > 1

If the number of blocks M in each frame is larger than 1,the game between transmitter and jammer has two levels. Thefirst (coarser) level is about power allocation between frames,

max [min PM{Jm} {Pm}

1M-1M E: Pm, s.t. IM({Pm}, {Jm}) > R]

m=O

1M-1

s.t.yM S Jm < JM;m=O

Problem 2 (for the minimax solution)

max [min JM{Pm} {Jm}

(3)

1M-1M , Jm, s.t. IM({Pm}, {Jm}) < R]

m=O

1 M-1

s.t. M: Pm <-PM.mO

(4)

Note that the first level power allocation strategies cannotbe derived before the second level strategies are available.Due to the linearity of the cost function and convexity of theconstraints, the solutions of the above optimization problemscan be sought by solving the KKT conditions.The following proposition provides a result that we shall

use in the sequel. Due to the intuitive nature of the proof, aswell as space limitations, we put the proof in [9].

Proposition 1: The optimal solution of either of the twoproblems above satisfies both constraints with equality.

Denote xCm = Jm + oN. Without loss of generality,throughout the sequel we assume that the ratios x,mjhm, withm = 0, 1, 2, ... , M -1 are always ordered increasingly, i.e.xo/ho < xilhi < ... < xM-llhM-, [6]. Note that thejammer's optimal strategy should imply 1/ho < l/hi < ... <

llhM-1.Solution of Problem 1

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The transmitter's problem can be written as:

nin + E1 Pm, S. t E log h1+ P > 2RM.=0 .'=io (i N

(5)With the notation c exp(2RM), the KKT conditions

yield:

Pn= A-

where A = (i m=)(1 JM') , and M' is the least

integer in {1, 2,...,M} such that A < xz,/hn for n > M'[6], and [z]±+ = max{z, 0}. We say the transmitter is "non-absent" over blocks {0, 1, M'- 1}. We can now show thatthe solution is unique.

Proposition 2: Problem ] has a unique solution.Proof: PM is a strictly concave function of the vector

X = (xo,1,.. ,xM-) (for a detailed proof see [9]). Thesecond part of Problem ] can be written as

M-1

min(- PM(3X)) subject to XTm = M(JM + c2), (7)mNm=0

which is a nonlinear minimization problem, with convex costfunction and linear equality constraints, thus has a uniquesolution. 0Jammer should only transmit on blocks where transmitter

is "non-absent". Otherwise, if Jn > 0 and A < Xn/hn in (6),the jammer can decrease Jn, n > M' until A = Xn/hn in (6)and maintain the same outcome without affecting the originalA. Hence, jammer's problem can be written as:

Find m/ar M ((HlM1 h ))) ' (1II1 xm)

1ME1z (8)

Ihm

m=O

X,nM MI-1h1

subject to WIZX, = (JM+4) (9)m=0

and xm > o-2 form = 1,2,...,M -1. (10)

The (unique) solution can be found by the following algo-rithm:

1) Fix M' in the set {1, 2,...., M};2) solve the new KKT conditions given by

I-Ml

II xm |A I| xm|km=O,mzhn J m=o

1hn1

along with (9) and (10) (the notation AX (1/M')

K (H/= hm)) is used for simplicity);

3) check (10) while assuming the jammer is present on anumber M" of blocks which decreases from M' to 1;

4) for each feasible vector (Xo, ... XM-1) verify that theresulting powers Pm, m = 0,... M -1 are non-negative;

5) pick the feasible solution that maximizes PM.An interesting situation occurs when both transmitter and

jammer are present over the same M' blocks. In this case, thefunction PM(JM) is linear (see [7] for a proof).

However, if jammer and transmitter are present over differ-ent blocks linearity of PM(JM) function no longer holds.

Solution of Problem 2The minimax intra-frame power allocation problem can also

be solved by writing the KKT conditions. However, we werenot able to reach satisfying results by this method. Instead weuse the above solution of Problem] and show that for bothproblems, the power allocation should be the same.

Theorem 1: If JM,1 is the value used for the second con-straint in Problem ] above, and PM,1 is the resulting solution,then solving Problem 2 with PM = PM,1 yields the solutionJM = JM,1. Moreover, the power distributions should be thesame, in both problems.

Proof: Assume that the vectors * =(P&,P*1* . PMj-1) and QY* = (JO*: J1*. ., JM-_1) arethe solution of Problem 1. Then M E M-1 J- = JM,1 and

ZMj,M Pm = PM,1.Since q* and QJ* form a solution, by Proposition 1, they

satisfy the first constraint in Problem] with equality, and sothey also satisfy the first constraint in Problem 2. Furthermore,setting the second constraint of Problem 2 as PM = PM,1, wenote that q* and QJ* are in the feasible set of this problem. Ifwe evaluate the cost function at this point, we get JM = JM,l.

Thus, keeping the power distribution given by q* in thesecond problem, we can only obtain JM,2 < JM,l, byminimizing the cost function over (JO, J1... , JM-1).Now take any different power distribution q3'

(PO: PPM-) 7& 9, satisfying MO Pm PM,1.The pair of vectors (Op', Q*) cannot satisfy the first constraintin Problem 1, because if it did, it would make a second solutionof this problem, and the solution of Problem] is unique, asspecified by Proposition 2 above.

Thus, this pair has to satisfy the first constraint in Problem2 (with strict inequality). We know this could not possibly bea solution of the second problem, since the first constraint isnot tight, but it is a feasible point and, by evaluating the costfunction at this point, we get JM = JM,1.

Thus, any power distribution of PM,1 we pick, we shouldalways obtain JM < JM,1 in Problem 2. But any solution ofProblem 2, with PM = PM,1 is also a solution of Problem 1,and so we cannot have JM < JM,.

Therefore, T* and Q* are a solution of Problem 2. UWe have shown that the second level optimal power alloca-

tion strategies for the maximin and minimax problems coin-cide. We need to characterize a particular channel realizationin terms of this power allocation technique. Considering themaximin problem, we can map each channel vector h to aunique curve in the plane PM(JM). That is, for fixed h, we

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ISIT2007, Nice, France, June 24 - June 29, 2007

increase the jammer's power over the frame from 0 to oo,and compute the transmitter power PM (JM, h) required forachieving reliable communication. We have already mentionedthat PM(JM) is a continuous, concave function.

In the remainder of this section we present the particularcase of M = 2 as an example of intra-frame power allocation.

Particular case: M=2The case of M = 2 is the simplest and most intuitive

illustration of the second-level power control strategy. Sincewe already showed that the minimax and maximin solutionscoincide, the following considerations refer to the maximinscenario only.

Particularizing (6) to M = 2, for n C {1, 2} and taking theaverage of the two quantities we get:

PM = h1h,22 hl+h2 if CXI > X2(c-1) xI if CXL < X2

This is the minimum required transmitter power PM, obtainedby optimally distributing transmitter power between the twoblocks.

If the transmitter is only present on the first block, i.e. c XI <

T2, then PM only depends on x1 /hi . In order to maximize thisquantity, the jammer should allocate all its power to the firstblock. Therefore, this situation is only possible if X2 = CN.The jammer's strategy in this case is to decrease the ratior = (X2/h2)j(Xljhl) by increasing the denominator.Next assume that the transmitter is present over both blocks

(as a result of either channel conditions or ofjammer allocatingenough power over the first block). Using the ratio r definedas above, and the fact that X1 + X2 = 2(JM + s)2 we obtain:

(JM -2-o)(2 VIcrT r -21 x1 x2PM -N if c > (12)h2r + hi hi h2

The ratio r that maximizes PM is found by setting thederivative equal to zero, which yields

(-K(hih2)2-[4hih2C1opt- 2h2C

2(hi -h2)

and is between 1 (for hi = h2) and c (for h2 = 0)Furthermore, PM(r) is strictly increasing for r C [1, r,pt) andstrictly decreasing for r C (ropt, c] .

This implies that the optimal jammer strategy is to allocateits power such that the ratio r = (X2jh2)j(Xlihl) approachesthe optimal ratio ropt. If the optimal ratio is attained, jammershould further maintain the ratio.

If JM increases from 0 to oo, we can define the character-istic curve PM(JM). For instance, if (0n2 /h2) /(-2 /lhl) > cthe transmitter transmits on the first block only. As jammerstarts transmitting, it will concentrate its power over the firstblock, until the ratio (ofj/h2)/[(2JM+- 2)jhl] reaches ropt.Note that in doing so, the ratio passes through c, and thatis when the transmitter starts transmitting on both blocks.After optimal ratio is attained, as JM increases the jammerkeeps allocating power over both blocks, while keeping theratio r constant and equal to ropt. Similar strategies apply if(u-2/h2) /j(ufj/h1 ) e [ropt, c]. Note that hi /h2 > ropt for anychannel realization (hi, h2), and so the two scenarios abovecover all possible situations.

Fig. 2. First level power allocation techniques for the maximin (left) andminimax (right) solutions

IV. INTER-FRAME POWER ALLOCATION FOR M > 1:MAXIMIN SOLUTION

In this subsection we present the first level optimal powerallocation strategies for the maximin problem, in the generalcase M > 1. The jammer needs to find the best choice of theset X c RLM of channel realizations over which it should bepresent, and the optimal way JM(h) to distribute its powerover 9, such that when the transmitter employs its optimalstrategy, the probability of outage is maximized.The solution of this problem is presented in the following

theorem.Theorem 2: It is optimal for jammer to make JM(h) sat-

isfy the power constraint with equality. The optimal jammerstrategy for allocating power across frames is to increase therequired transmission power, starting with those frames whosechannel realizations exhibit the steepest instantaneous slopeof the characteristic PM(JM) curve. This increase shouldbe done such that the required transmitter power over eachchannel realization where the jammer is present does notexceed a pre-defined level K.A description of the technique is given in Figure 2.The optimal value for K that maximizes the outage proba-

bility can be found numerically.Proof: Let 5/, X C R"" denote the sets of channel

realizations over which the transmitter and the jammer arepresent, respectively.

Let jammer pick a certain strategy JM (h). Since the trans-mitter's strategy is predictable, the jammer knows the set 5as well as the maximum level of required power that willbe matched by the transmitter. Denote this level by K. Therequired transmitter power should be equal to K over 9 \5,since otherwise either the jammer (if larger) or the transmitter(if smaller) would be wasting power.Assume there exist two sets X, X c 5 n x of non-zero

m-measure such that dPm(hl) > dP(h2) V hl C v and h2 Cdim dJm V1~ad2X, and such that the required PM is less than K on c andJM > 0 on 6.

Consider a small enough amount of jamming power 6JM,such that for any channel realization h C c U X, we canmodify the jamming power by 6JM without changing the slopeof the PM (JM) curve. Subtracting 6JM from all frames in X,the jammer obtains the excess power 6JMm(_'), which it canallocate uniformly over S/. This way, the jammer improves itsstrategy by forcing the transmitter to allocate more power tothe set c U X, and hence increases the probability of outage.

Note that the optimal pre-defined constant K should be thelimit of at lest one sequence of power levels PM(h) matchedby the transmitter. M

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ISIT2007, Nice, France, June 24 - June 29, 200O

V. INTER-FRAME POWER ALLOCATION FOR M > 1:MINIMAX SOLUTION

In Theorem 1, we showed that for the minimax problem,the power allocation within a frame, as well as the relationshipbetween the total powers used by transmitter and receiverover a particular frame, are identical to the maximin problemunder a same pair of JM and PM. Hence, by rotating thePM (JM) plane, we get the characteristic JM (PM) curves forthe minimax problem.The main result of this section is presented in the following

theorem.Theorem 3: It is optimal for transmitter to make PM(h)

satisfy the long-term power constraint with equality. The op-timal transmitter power allocation across frames is to increasethe required jamming power up to some pre-defined level K,starting with those frames on which the required transmitterpower to achieve this goal is least.A description of the technique is given in Figure 2.The optimal value for K that minimizes the outage proba-

bility can be found numerically.Proof: Let 5 and Y denote the sets over which

the transmitter and the jammer are present, respectively. Lettransmitter pick a certain strategy PM(h). Since jammer'sstrategy is predictable, the transmitter knows the maximumlevel of required power that will be matched by jammer.Denote this level by K and note that the required jammingpower over 5 \ X should be equal to K (otherwise eitherthe jammer - if smaller than K - or the transmitter - if larger- would be wasting power).Assume there exist two sets c, _ C5c n x of non-

zero m-measure such that PMi(hi,K) > K( Vh,v and h2 C X, and such that the required JM is less thanK on c and JM > 0 on X~. Denote the original transmitterpower allocations by PLM0 (h) and PM 0 (h) respectively.We know that JM(PM) is convex, and hence

KPM(hl, K)

JM,iPM(h1, JM,1)

KPM(hl, K) >

> > J__MPM (h2, K) PM (h2, JM,2)

V h, C X, h2 C 6 and JM,, JM,2 < K. (14)

If the transmitter cuts off transmission over a subset 6' cX, it obtains the excess power fe, PM(h)dm(h), which itcan allocate to a subset cY' c W such that the required JMis equal to K over Y', i.e.

Ii, PM o(h)dm(h) J [PM (h, K) -PM, 0(h)] dm(h)(15)

Replacing PM(hl, JM,1) by PM 0(h) and PM(h2, JM,2) by

PM, o(h) in (14), we see the transmitter improves its strategyby forcing the jammer to allocate more power to the setv U X, and hence decreases the probability of outage.Note that since W' C Son x, the set ' is in outage,

regardless of whether the transmitter is present or not. Thus,transmitter does not increase Po,t by cutting off transmissionon $'. X

P_,,. vs. P for R=1, N=5, J=10, i.i.d exponential hm with parameter X=1/6

Fig. 3. Outage probability vs. P for M = 1 and Mminimax and maximin cases - and when J = 0

= 2 when j = 10

VI. NUMERICAL RESULTS

We have computed the outage probabilities for both min-imax and maximin problems when M = 2. The channelcoefficients are assumed i.i.d. exponentially distributed withparameter A = 1/6. Figure 3 shows the outage probabilityvs. the maximum allowable average transmitter power P forfixed J = 10 when R = 1. For comparison purposes, wealso provide results for the cases when M = 1 and when thejammer is not present (J = 0).

The numerical results demonstrate a sharp difference be-tween minimax solutions and maxmin solutions, which impliesthe non-existence of Nash-equilibria of this two-person zero-sum game.

In addition, note that increasing M from 1 to 2 producesan increase in the outage probability for the minimax, anda decrease for the maximin. This can be explained by thefact that the first player is always at a disadvantage, and thisdisadvantage increases as the second player gains degrees offreedom.

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[2] M. Medard, "Capacity of correlated jamming channels," Allerton AnnualConf on Comm., Control and Computing, 1997.

[3] A. Kashyap, T. Basar, and R. Srikant, "Correlated jamming on mimogaussian fading channels," IEEE Trans. Inform. Theory, vol. 50, pp. 2119-2123, Sept. 2004.

[4] M. H. Brady, M. Mohseni, and J. M. Cioffi, "Spatially-correlated jammingin gaussian multiple access and broadcast channels," Proc. Confe. onInform., Science, and Systems, Princeton, March 2006.

[5] S. Shafiee and S. Ulukus, "Correlated jamming in multiple accesschannels," Conference on Information Sciences and Systems, March 2005.

[6] G. Caire, G. Taricco, and E. Biglieri, "Optimum power control over fadingchannels," IEEE Trans. Inform. Theory, vol. 45, pp. 1468-1489, July1999.

[7] G. T. Amariucai, S. Wei, and R. Kannan, "Minimax and maxmin solutionsto gaussian jamming in block-fading channels under long term powerconstraints," CISS, 2007.

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