Binary Dirty MAC with Common State Information Anatoly Khina Email: anatolyk@eng.tau.ac.il Tal Philosof Email: talp@eng.tau.ac.il Ram Zamir Email: zamir@eng.tau.ac.il Uri Erez Email: uri@eng.tau.ac.il Astract The general two-user memoryless multiple-access channel, with common channel state information known to the encoders, has no single-letter solution which explicitly characterizes its capacity region. In this paper a inary dirty multipleaccess channel MAC with common interference, when the interference sequence is known to oth encoders, is considered. We determine its sum-capacity, which equals to the capacity when full-cooperation etween transmitters is allowed, contrary to the Gaussian case. We further derive an achievale rate region for this channel, y adopting the onion-peeling strategies which achieve the capacity region of the clean inary MAC. We show that the gap etween the capacity region of the clean MAC and the achievale rate region of dirty MAC stems from the loss of the point-to-point inary dirty channel relative to the corresponding clean channel. I. INTRODUCTION Consider the two-user memoryless state-dependent multiple-access channel MAC with transition and state proaility distriutions py x,x,s and ps, where s Sis known non-causally at oth encoders, ut not to the decoder. The channel inputs are x X and x X, and the channel output is y Y. The memoryless property of the channel implies that n py x, x, s = py i x i,x i,s i. i= The capacity region of this channel is still not known in general, and remains an open prolem. See, e.g., []. Interestingly, this model appears to e a ottleneck in many wireless networks, ad hoc networks and relay prolems. The MAC model in generalizes the point-to-point channel with side information SI at the transmitter considered y Gel fand and Pinsker []. Gel fand and Pinsker proved a direct coding theorem using a random inning technique, an approach widely used in the analysis of multi-terminal source and channel coding prolems [3]. They otained a general capacity expression which is given in terms of an auxiliary random variale U: C = max HU S HU Y, 3 pu,x s This work was supported in part y the U.S. - Israel Binational Science Foundation under grant 008/455. where the maximization is over all joint distriutions of the form pu, s, y, x =pspu, x spy x, s. Using this result, Costa [4] showed that in the Gaussian additive channel with known interference, the capacity is equal to that of the AWGN channel, i.e., as if the interference S were not present. Nevertheless, this does not carry on to the inary modulo-additive case inary dirty-paper channel : Y = X S Z, where X, S, Z Z and denotes addition mod XOR. The input constraint is n w Hx q, where 0 q / We shall refer to this constraint as power constraint, w H denotes Hamming weight, and n is the length of the codeword. The noise Z Bernoulliε is independent of S, X w.l.o.g. we assume ε / ; the state information interference S Bernoulli / is known non-causally to the encoder. The capacity of this dirty channel is equal to CDirty PP =uce max H q H ε, 0, 4 where H denotes the inary entropy [3] and ucre is the upper convex envelope operation with respect to q ε is held constant. The capacity of the interference-free clean channel the inary symmetric channel with a Hamming input constraint, given y CClean PP = H q ε H ε, 5 is higher than that of the dirty inary channel 4, since H q ε H q, where denotes inary convolution, defined as p p p p + p p. See [5], [6]. One approach to finding achievale rates for the MAC with common interference, is to extend the Gel fand and Pinsker result [] to the two-user case []. This extension leads to the following inner ound for the capacity region of see []: R cl conv R,R :R IU; Y V IU; S V R IV ; Y U IV ; S U R + R IU, V ; Y IU, V ; S, 6 where cl and conv are the closure and convex hull, respectively, taken over all admissile auxiliary pairs U, V satisfying:
W W S Enc. Enc. X X Z Y Ŵ Dec Ŵ Fig.. Dirty MAC with common state information U, X S V,X U, V X,X,S Y. Philosof et al. [7], [8] considered the douly-dirty MAC : a Gaussian additive MAC, with an additive interference which is composed of a sum of two independent Gaussian interferences, where each interference is known non-causally only to one of the encoders. The capacity region of the Gaussian dirty MAC, where the interference is known non-causally to oth transmitters inary MAC with common interference, was found y Gel fand and Pinsker [9] and rediscovered y Kim, Sutivong and Sigurjónsson [0], to e equal to the interference-free MAC channel, y applying dirty paper coding y oth users. Philosof, Zamir and Erez [] considered a inary moduloadditive version of this channel inary DMAC, depicted also in Figure : Y = X X S Z, 7 where X,X,S,Z Z. The input power constraints are n w Hx i q i, i =,, 8 where 0 q,q /. The noise Z has Bernoulliε distriution and is independent of S, X,X ; the state information S Bernoulli / is referred to as interference. The rates, R and R, of the two users are given y R i = n log W i, where W i is the set of messages of user i. In [], the capacity for two different scenarios is derived: The inary douly-dirty MAC: in this scenario S = S S, where S,S Bernoulli/ are independent and known non-causally to encoders and, respectively. The capacity region of this channel is given y the set of all rate pairs R,R satisfying: Cq,q R,R : R + R uce max [H q min H ε], where q min minq,q and the upper convex envelope operation is w.r.t. q and q. The inary MAC with a single informed user: in this scenario S is known only to user. The capacity region of this channel is given y the set of all rate pairs R,R satisfying: Cq,q cl conv R,R : R H q ε H ε R + R H q H ε. 9 Unlike in the Gaussian case, in which the common interference capacity region is the same as the interference-free region, and is achieved using stationary inputs, in the inary DMAC, we shall see that there is a loss. In this work we consider the inary MAC with common interference, depicted in Figure, where the interference S is known non-causally at oth encoders. We assume that the interference is strong S Bernoulli /. This is the worst-case interference, as any other distriution of the interference can e transformed into a uniform one y incorporating dithering at the receiver s end. We show that using the dirty-paper strategies that achieve 5 in 6, along with successive decoding of the messages onion peeling allows to achieve a rate region of the inary DMAC 7, that is equal to the inary clean MAC capacity region up to a loss which stems from the loss seen in the point-to-point case 4,5. Moreover, we show that these strategies achieve the sum-capacities of the inary clean MAC and dirty MAC with common interference, which are equal to the sum-capacities of these channels when full cooperation etween the encoders is allowed, unlike in the Gaussian MAC with common interference. To simplify the treatment we concentrate on the noiseless case, i.e., Z =0. The paper is organized as follows: We first consider the inary clean MAC in Section II and then turn to treating the inary MAC with common interference in Section III. II. CLEAN MAC In this section we consider the clean inary moduloadditive channel: Y = X X Z 0 with input constraints q,q. The capacity region of this channel contains the capacity region of the inary DMAC 7, and therefore serves as an outer ound. Furthermore, the capacity achieving strategies are useful also for the DMAC case, as is discussed in Section III. As mentioned earlier, we concentrate on the noiseless case Z =0. The inary additive model 0 is a special case of the general clean MAC channel, the capacity region of which is known to e [], [3]: C cl conv R,R : R IX ; Y X R IX ; Y X R + R IX,X ; Y, where the closure and convex hull operations are taken over all product distriutions p x p x on X X. In the Gaussian additive MAC, any point within its capacity region can e achieved using Gaussian stationary inputs. Hence the convex hull operation is superfluous see, e.g., [3]. In the inary case, however, the use of stationary inputs is not optimal and convex hull is necessary to achieve the capacity
region envelope. To see this, we rewrite explicitly for the inary case: C cl conv R,R :R i H X i, i =, R +R H X X, where the closure and the convex hull are taken over all admissile distriutions of the form p x p x on 0, 0,, such that the input constraints 8 are satisfied. One easily verifies that, y allowing only stationary inputs in,.i.e., relinquishing the convex hull, the sum-rate R + R cannot exceed R + R H q q, 3 which is suoptimal, as indicated y the following theorem. Theorem Sum-Rate Capacity of the Binary Clean MAC: The sum-capacity of the inary noiseless modulo-additive MAC 0 with input constraints n w Hx i q i, i =,, is: Cclean sum = H + q + q, 4 where H + q H min q, /. Proof: Direct: Using time-sharing one can divide each lock into two parts: in the first n lock samples user spends all of its power to convey his private message, while user is silent transmits zeros, whereas in the remaining n lock samples user spends all of its transmission power to convey his message, while user is silent. This leads to the sum-rate R + R = H + q + H + q, which is equal to H + q + q for = q q +q. Converse: Full cooperation etween the transmitters can only increase the sum-capacity. Full cooperation transforms the prolem into a point-to-point prolem of transmitting over a inary clean channel with power constraint n w Hx q + q, the capacity of which is H + q + q. Thus, the sum-capacity of the inary clean MAC 4 is strictly greater than the est achievale rate using only stationary inputs 3. Remark : The sum-capacity of 0 can e shown, using the same methods, to e: C sum clean = H+ q + q ε H ε. If we allow full cooperation etween the transmitters, the capacity of the channel does not outperform 4, as pointed out in the converse part of the proof. In the Gaussian case, on the other hand, the sum-capacity of the MAC channel is equal to log + SNR + SNR, which is strictly smaller than the full-cooperation capacity, log + SNR + SNR + SNR SNR. This R [its] 0.5 0.45 0.4 0.35 0.3 0.5 0. 0.5 0. 0.05 Capacity Region Sum Capacity Improved Onion Peeling Onion Peeling Time Sharing one Tx at a time h q*q h q h q*q h q 0 0 0. 0. 0.3 0.4 0.5 0.6 0.7 R [its] Fig.. Rate Regions for the inary clean MAC and input constraints: q =/6, q =/0. dissimilarity stems from the difference of the alphaets that we work with in oth prolems and the nature of the addition. In the inary case, no coherence can e attained y transmitting the same message, and additional power can only assist in exploiting more time slots within a lock. In the Gaussian case, on the other hand, cooperation allows additional coherence gain, which cannot e achieved in the inary case. To find the capacity region of explicitly, we replace the convex hull with a time-sharing variale Q, with alphaet of size Q =see, e.g., [4]. C R,R :R H X Q R H X Q R +R H X X Q, 5 where the union is over all admissile Markov chains X Q X, satisfying the input constraints EX i q i, i =,. Remark : Note that X and X are not independent in 5, ut rather independent given the time-sharing parameter Q. Deriving an explicit analytical expression for 5 is hard, and numerical solutions are needed instead. Using the scheme proposed in the proof of Theorem, in which only one user transmits at each time instance, whereas the other user remains silent, is suoptimal in general, as is illustrated in Figure. Exploring the capacity region in 5, we note that the corner points of the pentagons, which constitute the capacity region, i.e., points that satisfy one of the first two inequalities and the third one with equality in 5, can e achieved y incorporating the successive cancellation or onion peeling method, in which the decoder treats the message of one of the users as noise, recovers the message of the other user, and sutracts it to recover the remaining message. Due to the time-sharing variale Q of cardinality, two such strategies need to e considered, to achieve a general point in the capacity region 5, such that the power constraints
are satisfied on the average. Nevertheless, we examine these rates for stationary points viz. P Q =0=, to otain etter understanding. Thus, oth users transmit simultaneously at all times, such that user uses all of its availale power EX = q, whereas user uses only some portion of its power EX = q 0 q q. User treats q as noise and can achieve a rate of R = H q q H q. After recovering the message of user, it can e sutracted, such that user sees a clean point-to-point channel and hence can achieve a rate of R = H q. Note that even though using this strategy alone the capacity region cannot e achieved, it does achieve certain rate pairs which cannot e achieved y simple time-sharing, like the one used in Lemma and depicted in Figure. Remark 3: When using this onion peeling strategy, user does not exploit all of its power, ut only a portion 0 q q. Hence a residual power of q q is left unexploited. This implies that this strategy is not optimal except when q = q as is, and a way to exploit this residual power needs to e constructed. As mentioned earlier, time-sharing etween such onion peeling strategies allows to achieve capacity. However, numerical evidence suggest that a simpler scheme suffices to achieve the capacity region, as is depicted in Figure. In this scheme we divide the transmission lock into two parts, where in the first we use onion peeling, such that the user eing pilled first uses all of its power, whereas the other user uses a portion of its power in the first su-lock, and transmits with its remaining power in the second-lock whereas the other user is silent. If we denote y the lock portion allotted to onion peeling and y q 0 q q the power of user, used during this period, this scheme supplies us with the following achievale rates: R = H + R = H + q q q H + + H + q, q q, 6 Or in the noise case: R = H + q q q ε H + ε, q R = H + ε + H + q q ε H ε, in a similar manner. The roles of user and user are not symmetric: the achievale rate pairs, using onion peeling, when user is peeled, differ from the rate pairs that are achieved when user is peeled. Hence, y switching roles etween the two users, one may achieve additional rate points. III. DIRTY MAC WITH COMMON INTERFERENCE We adopt the strategies introduced in Section II to the dirty case 7 depicted also in Figure, and derive an achievale rate region. Similarly to the clean MAC case, the sum-capacity of the inary DMAC with common interference is equal to the capacity of this channel when oth encoders can fully cooperate, as indicated y the following theorem. Theorem Sum-Rate Capacity of DMAC with Common SI: The sum-capacity of the inary noiseless modulo-additive MAC with common interference 7 and input constraints n w Hx i q i, i =,, is: Cdirty sum = H + q + q. 7 Proof: Direct: We repeat the proof of Lemma, only now the point-to-point BSC capacity 4 should e replaced y the inary dirty paper channel capacity 5. Nevertheless, in the noiseless case Z =0, there is no difference etween the two expressions, and thus R + R = H + q + q. Converse: Again, like in the proof of Lemma, we allow full cooperation etween the transmitters, which in turn transforms the prolem into a point-to-point channel, the capacity of which is H + q + q. Remark 4: In the presence of noise, the sum-capacity of this channel is C sum Dirty =uchmax H + q + q H ε, 0. In the noiseless case, the sum-capacities of the inary clean and dirty MACs are equal. However, in the presence of noise Z, the sum-capacity of the dirty MAC channel is strictly smaller than that of the clean MAC channel for q + q <. This difference stems from the capacity loss, due to the presence of interference, of the point-topoint setting 4,5 As was mentioned in Remark, if we allow full cooperation etween the transmitters, the capacity of the channel cannot exceed 7, in contrast to the Gaussian case, in which additional coherence gain can e achieved. The capacity region of the single informed user 9 serves as an inner ound for the capacity region of the common interference dirty MAC. To improve the achievale region of our channel of interest, we allow time-sharing etween single informed user strategies, where the informed user is alternately user or user. Note that y this only the user that is pilled first needs to know the interference. This is also true for the sum-capacity achieving strategy presented in the proof of Theorem. Remark 5: The strategies used in [], to achieve the capacity region of the single informed user 9, can e viewed as onion
peeling, where user assumes a point-to-point dirty paper channel and input constraint 0 q q ; and user treats the signal of user, X, as noise, and uses dirty paper coding of the form 5. The achievale rates, using this strategy, are of the form: R = H q H q, R = H q, where since q can take any value in the interval [0,q ], the single informed user capacity 9 is achieved y timesharing etween such strategies where user is always pilled first, since user is ignorant of the interference sequence. Using such stationary strategies alone with no timesharing, one cannot hope to achieve the sum-capacity of Lemma or the whole capacity region of the singleinformed user prolem 9, since there is an average residual power of q q, for each sample, left unexploited. As in the clean MAC case Section II, we conjecture that rather than using time-sharing etween two onion peeling strategies, a simplified scheme that divides the transmission locks into two parts, where in the first su-lock onion peeling is performed, where the user that is pilled first exploits all of power, and in the second su-lock the other user transmits with all of its remaining power. This allows to achieve rate pairs of the form: R = H + R = H + q H + q q + H +, q q, 8 where q [0,q ]. See Figure 3. In the noisy case, this scheme achieves the following rates: R = H + q q H + ε, q R = H + ε + H + q q ε H ε. Even in the noiseless case Z =0 the achievale rate region of the dirty channel 8 is properly contained in its corresponding clean counterpart 6 as depicted in Figure 3, in contrast to the point-to-point setting, in which the capacities are equal in the asence of noise 4, 5. The gap etween the two regions stems from the fact that in the first su-lock, the user eing peeled first, treats the signal of the other user as noise, in the presence of interference. Hence the achievale rate during this stage is strictly smaller, due to the point-to-point loss of inary dirty paper coding 4, 5. This strategy is asymmetric in user and user, as was explained in Remark 3. R [its] 0.5 0.45 0.4 0.35 0.3 0.5 0. 0.5 0. 0.05 Clean Sum Capacity UB Clean MAC Capacity UB TS etween OP Improved Onion Peeling Time Sharing one Tx at a time Onion Peeling h q*q h q h q*q h q 0 0 0. 0. 0.3 0.4 0.5 0.6 0.7 R [its] Fig. 3. Rate regions for the inary DMAC with common interference and input constraints q =/6, q =/0. [] S. I. Gel fand and M. S. Pinsker, Coding for channel with random parameters, Prolemy Pered. Inform. Prolems of Inform. Trans., vol. 9, No., pp. 9 3, 980. [3] T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: Wiley, 99. [4] M. H. M. Costa, Writing on dirty paper, IEEE Trans. Information Theory, vol. IT-9, pp. 439 44, May 983. [5] R. J. Barron, B. Chen, and G. W. Wornell, The duality etween information emedding and source coding with side information and some applications, IEEE Trans. Information Theory, vol. 49, pp. 59 80, May 003. [6] R. Zamir, S. S. Shitz, and U. Erez, Nested linear/lattice codes for structured multiterminal inning, IEEE Trans. Information Theory, vol. 48, pp. 50 76, June 00. [7] T. Philosof, R. Zamir, U. Erez, and A. Khisti, Lattice strategies for the dirty multiple access channel, arxiv:0904.89v, also sumitted to Transaction on Information Theory, 009. [8] T. Philosof and R. Zamir, On the loss of single-letter characterization: The dirty multiple access channel, IEEE Trans. Information Theory, vol. IT-55, pp. 44 454, June 009. [9] S. I. Gel fand and M. S. Pinsker, On Gaussian channels with random parameters, in Astracts of Sixth International Symposium on Information Theory, Tashkent, U.S.S.R, Sep. 984, pp. 47 50. [0] Y. H. Kim, A. Sutivong, and S. Sigurjónsson, Multiple user writing on dirty paper, in Proceedings of IEEE International Symposium on Information Theory, Chicago, USA, June 004. [] T. Philosof, R. Zamir, and U. Erez, The capacity region of the inary dirty MAC, in Proc. of Info. Th. Workshop, Sicily, Italy, Oct. 009, pp. 73 77. [] R. Ahlswede, Multi-way communication channels, in Proceedings of nd International Symposium on Information Theory, Thakadsor, Armenian S.S.R., Sept. 97. Akadémiai Kiadó, Budapest, 973, pp. 3 5. [3] H. Liao, Multiple access channels, Ph.D. dissertation, Department of Electrical Engineering, University of Hawaii, Honolulu, 97. [4] I. Csiszar and J. Korner, Information Theory - Coding Theorems for Discrete Memoryless Systems. New York: Academic Press, 98. REFERENCES [] T. Philosof, R. Zamir, and U. Erez, Technical report: Achievale rates for the MAC with correlated channel-state information, in arxiv:08.4803, 008.