On the Dirty Paper Channel with Fast Fading Dirt

Transcription

1 On the Dirty Paper Channel with Fast Fading Dirt Stefano Rini and Shlomo Shamai Shitz) National Chiao-Tung University, Hsinchu, Taiwan, Technion-Israel Institute of Technology, Haifa, Israel, arxiv: v [cs.it] 4 Jun 05 Abstract Costa s writing on dirty paper result establishes that full state pre-cancellation can be attained in the Gel fand-pinsker problem with additive state and additive white Gaussian noise. This result holds under the assumptions that full channel knowledge is available at both the transmitter and the receiver. In this work we consider the scenario in which the state is multiplied by an ergodic fading process which is not known at the encoder. We study both the case in which the receiver has knowledge of the fading and the case in which it does not: for both models we derive inner and outer bounds to capacity and determine the distance between the two bounds when possible. For the channel without fading knowledge at either the transmitter or the receiver, the gap between inner and outer bounds is finite for a class of fading distributions which includes a number of canonical fading models. In the capacity approaching strategy for this class, the transmitter performs Costa s pre-coding against the mean value of the fading times the state while the receiver treats the remaining signal as noise. For the case in which only the receiver has knowledge of the fading, we determine a finite gap between inner and outer bounds for two classes of discrete fading distribution. The first class of distributions is the one in which there exists a probability mass larger than one half while the second class is the one in which the fading is uniformly distributed over values that are exponentially spaced apart. Unfortunately, the capacity in the case of a continuous fading distribution remains very hard to characterize. Index Terms Gel fand-pinsker Problem; Writing on Fading Dirt; Ergodic Fading; Imperfect Channel Side Information; I. INTRODUCTION In the Gel fand-pinsker GP) model [] the output of a point-to-point memoryless channel is obtained as a function of the channel input, a noise term and a state variable which is non-causally provided to the transmitter but is unknown at the receiver. In this channel the state may represent the interference caused by another user in a wireless network which is also communicated to the transmitter by the network infrastructure. In the original setup, both transmitter and receiver are assumed to have perfect channel knowledge: while it is reasonable to assume that a transmitter knows the channel toward its intended receiver and vice-versa, it is not always realistic to suppose that a transmitter knows the channel between an interfering user and the receiver. This is especially true in wireless network, since here channel conditions vary continuously over time and reliable channel estimates are hard to obtain. The work of S. Rini was partially funded by the Ministry Of Science and Technology MOST) under grant 03-8-E MY. The work of S. Shamai was supported by the Israel Science Foundation ISF) and by the European FP7 NEWCOM#.

2 Fig.. The Dirty Paper Channel with Fast Fading Dirty DPC-FFD). The dotted line represent the state information provided at the transmitter. The writing on dirty paper result from Costa [] establishes a closed-form characterization of the capacity of the GP problem in the additive state and additive white Gaussian noise setting. Perhaps surprisingly, the presence of the state does not reduce the capacity of this model, regardless of the distribution or power of this sequence. In this work we are interested in characterizing the effect of fading on the capacity of this model and determine the optimal transmission strategies in this scenario. In the literature, different variations of Costa s setup which also include fading have been considered. The writing on fading dirt channel in [3] is a variation of the channel of [] in which both the channel input and the state sequence are multiplied by a fading value known at the receiver but not at the transmitter. The authors of [3] evaluate the achievable region with Costa s assignment and show that the rate loss from full state pre-cancellation is vanishing in both the ergodic and quasi-static fading case. In the compound dirty-paper channel of [4] only the state is multiplied by a quasi-static fading coefficient know at the receiver but unknown at the transmitter. For this model, an inner bound based on lattice strategies is derived to compensate for the channel uncertainty at the transmitter. Achievable rates under Gaussian signaling and lattice strategies for this channel are derived in [5] while outer and inner bounds to the capacity of the writing on fading dirt channel with phase fading are derived in [6]. The approximate capacity of this channel is obtained in [7] for the case of binomial and uniform phase fading case. In this paper we study the writing on fading dirt model, a variation of the classic model in which the state sequence is multiplied by an ergodic fading coefficient which is not known at the transmitter. We derive inner and outer bounds to capacity for both the case in which the fading is known at the receiver and for the case in which it is not. When neither the transmitter nor receiver have fading knowledge, we show that the outer bound can be attained to within a finite gap for a class of fading distribution which includes the Gaussian, the uniform and the Rayleigh distribution but does not include the log-normal distribution. For the case in which only the receiver has fading knowledge, we show a finite gap between inner and outer bound for two classes of discrete distributions: when the fading distribution has a mass function greater than a half and when it is uniformly distributed over a set of points that are exponentially spaced apart. The remainder of the paper is organized as follows: Sec. II introduces the channel model and the some related results. Sec. III investigates the capacity for the case in which neither the transmitter nor the receiver have fading knowledge while Sec. IV focuses on the case in which only the receiver has fading knowledge. Finally, Sec. V concludes the paper.

3 Fig.. The Dirty Paper Channel with Fast Fading Dirty and Receiver Channel Side Information DPC-FFD-RCSI). II. DIRTY PAPER CHANNEL WITH FADING DIRT In Dirty Paper Channel with Fast Fading Dirt DPC-FFD), also depicted in Fig., the channel output is obtained as Y i = X i + ca i S i + Z i, i [...N], ) for c R and where X i is the channel input, S i the state, A i the fading realization and Z i the additive noise. The channel input X i is subject to a second moment constraint E [ X j ] P while the state S i and the noise term Z i are distributed as S i N µ S,), Z i N 0,), i.i.d. ) where N µ,σ ) indicates the Gaussian Random Variable RV) with mean µ and variance σ. The fading RV A i is drawn from a distribution p A which has variance one and mean µ A. The state sequence S N is assumed to be non-causally available at the transmitter while fading sequence A N is unknown at both the transmitter or the receiver. A related model to the DPC-FFD in Fig. is the model in which the fading sequence is provided to the receiver. We refer to this model as the Dirty Paper Channel with Fast Fading Dirty and Receiver Channel Side Information DPC-FFD-RCSI), also depicted in Fig.. For the DPC-FFD-RCSI the receiver side information can be seen as an additional channel output, that is, the channel output is the vector [Y i A i ] for Y i in 5). Remark II.. Mean of the state and the fading. The channel output in ) can be rewritten as Y = X+ ca 0 µ A )S 0 µ S )+Z = X+ ca 0 S 0 µ A S 0 µ S A 0 + µ A µ S )+Z, 3) where A 0 = A µ A and S 0 = S µ S. that Each of the term in 3) can be seen as follows cµ S A 0 can be cancelled at the receiver when it posses fading knowledge. Without receiver fading knowledge, this term is unknown at both the receiver and the transmitter and is equivalent to additive noise. cµ A S 0 can be pre-cancelled with Costa coding by the transmitter as in [] Costa pre-coding in the following). ca 0 S 0 requires the cooperation of both transmitter and receiver, since they each have a knowledge of one of the terms in the multiplication. The DPC-FFD and the DPC can be used to model the downlink scenario in which a base station is aware of the signal transmitted by a neighbouring base station but has only partial or no knowledge on the channel between the interference and the intended receiver. In this scenario it is not clear whether the knowledge of the interfering message is at all useful at the

4 base station since the pre-coding operations heavily rely on the knowledge of the channel gains. A. Related Results Gelfand-Pinsker GP) channel. The DPC-FFD and the DPC-FFD-RCSI are a special case of the GP problem for which capacity is obtained in []. Theorem II.. Capacity of the DPC-FFD/-RCSI []. The capacity C of the DPC-FFD in 5) is C = max P U,X S IY ;U) IU;S), 4) while the capacity of the DPC-FFD-RCSI is obtained from 4) by considering the channel output [Y A]. The expression in 4) contains an auxiliary RV U and entails the maximization over the distribution P U,X S. For this reason a closed-form expression cannot be evaluated easily, either analytically or numerically. Dirty paper channel with receiver side information and phase fading. In [7], we have derived the approximate capacity of the DPC-FFD-RCSI for the case in which p A is a circularly binomial distribution. Theorem II.3. Capacity of the DPC-FFD-RCSI with circularly binomial fading [7, Th. IV.5]. Consider the DPC-FFD-RCSI Y i = X i + e jθ i S R,i + Z i, i [...N], 5) where the state S R,i is a Gaussian RV with zero mean and variance Q and while the fading is A=exp{θ} for P θ t)= {t=+ } t)+ {t= } t) ), [0,π/], 6) then, if π/4 π/, the capacity lies to within constant gap of 3 bits per channel use from the outer bound logp+ )+ c 3 R OUT 4 = logp+ )+ c P+ logp+ )+ log + P+ c) ) 4 logc )+ <c < P+ where c=sin ) Q. Carbon copying onto dirty paper. A model related to the DPC-FFD is the carbon copying onto dirty paper of [8]: in this channel model there are M possible state sequences S j that can possibly affect in the channel output. The transmitter has knowledge of each sequence but does not know which one will appear. Correct decoding must be granted regardless of the state realization and for each of the possible channel output. Y N j = X N + cs N j + Z N j, j [...M], 7)

5 where S N j is an i.i.d. Gaussian sequence for each j [...M]. In [8] inner and outer bound to the capacity region are derived but capacity has yet to been determined. III. THE DIRTY PAPER CHANNEL WITH FAST FADING DIRT We begin by investigating the capacity of DPC-FFD in Fig : since no closed-form expression for the optimization in 4) is available, we derive a novel outer bound that is expressed solely as a function of the channel parameters. This outer bound can be approached, for some models, by a simple achievable strategy in which the transmitter to performs Costa pre-coding against the term cµ A S, the average realization of the fading times state. For the DPC-FFD the term cµ S A acts as additional noise, since it is unknown at both the transmitter and the receiver: for this reason in the following we assume that µ S = 0. Theorem III.. Outer bound and partial approximate capacity for DPC-FFD. Consider the DPC-FFD in Fig. and let ha)= logπeα) for some α [0,], then the capacity C is upper bounded as C R OUT = P+ log c α + ) + α, 8) and the capacity is to within a gap G bits/channel-use from R OUT where Proof: The proof can be found in App. A. G= logα) +. 9) The gap from capacity in Th. III. can be easily evaluated for some canonical fading distributions. Lemma III.. Gap from for some fading distributions. When A is Gaussian distributed with mean µ A and unitary variance, the capacity is known to within a gap G N G N =. When A is uniformly distributed between [µ A, µ A+ ], the capacity can be attained to within a gap G U G U = ) πe log +. When A is Rayleigh distributed, i.e. A= U +V for U,V N 0,/4 π)) and independent, capacity can be attained to within a gap G R defined as G R = log)+γ+ +.08, where γ is the Euler-Mascheroni constant. When A is log-normal distributed, i.e. A = e Z Ae µ σ e σ ) for Z A µ,σ ), capacity can be attained to within a gap G log defined as ) G log = log e σ + µ+ σ + µ+ σ,

6 which is not a finite value for all values of µ and σ. The result in Th. III. is substantially a negative result since in establishes that, for a number of fading distributions for which α is close to one, the best strategy is to Costa pre-code against the mean value of the fading times the state and treat the term A 0 S 0 as additional noise. This strategy performs very poorly when compared to the full state pre-cancellation and indeed, for any choice of the power P, capacity tends to a small constant as the term c increases. Note that the gap G in 9) for the log-normal distribution is not bounded: the variance of this distribution grows exponentially with σ while the entropy grows logarithmically with σ, therefore α can be made arbitrarily small and G arbitrarily large. In actuality, we expect the outer bound in 8) to be close to capacity for a larger set of distributions than that for which α is close to one. The difficulty in developing a more general result lies in the lack of tighter outer bound. Note also that this result does not hold for discrete fading distributions and thus does not include extensions of the result in Th. II.3 for the case with no RCSI. IV. DIRTY PAPER CHANNEL WITH FAST FADING DIRT AND RECEIVER SIDE INFORMATION We now turn our attention to the DCP-FFD-RCSI: also for this channel capacity can be obtained from Th. II. but the optimization is extremely hard to express in closed-form. This case is significantly harder to study than the case with no receiver fading information because of the distributed way in which transmitter and receiver can cooperate in dealing with the term cas. As an illustrative example, consider the DPC-FFD with no additive noise and in which the state and the input are restricted to take value ±, that is Y = X+ AS, X,S {,}, 0) while A has any distribution. Given the cardinality of the input, the capacity of this channel is at most bit/channel-use. This rate can be attained by setting X )=X+)=/, independent from S and by setting U = XS and independent from S in 4). With this assignment, U can be recovered from the channel output by considering the squared channel output, in fact: Y A=a)=X + a S + axs= +a + au, ) so that U = Y A )/A, regardless of the distribution of A. This simple example shows that the maximization in 4) might yields some unexpected results. Given the difficulty of the problem at hand, we are able to make only partial progress in characterizing the capacity of the DPC-FFD-RCSI. In the following we provide two approximate capacity results for two classes of discrete distributions of A: i) for the class of discrete distributions in which one of the probability masses is larger or equal to one half and ii) for the class of uniform distributions over the discrete set in which points are incrementally spaced apart. Both results are a generalization of our previous result in Th. II.3 and employ a similar inner bound in which the transmitter simply performs Costa pre-coding against one realization of the fading times the state. Our contributions is, therefore, to identify a set of channels in which Costa pre-coding is optimal, although it is clear that this coding strategy is not be capacity achieving in general.

7 Note that, for the DPC-FFD-RCSI, we again consider the case in which µ S is equal to zero: since the receiver has knowledge of A, it can subtract cµ S A from the channel output. We also let µ A = 0 for simplicity: the general case is considered in the journal version of this work. Let s consider first the class of distribution in which there exists an outcome A = a with P A a ) /: this class of distributions generalizes the distribution considered in our result in Th. II.3. For this fading model the transmitter can Costa pre-code against the realization ca S and obtain full state cancellation for approximatively a portion P A a ) of the time. The performance of this strategy can be improved upon letting the channel input be composed of two codewords: one treating the state times fading as noise and one that Costa pre-codes against ca S. By optimizing over the power allocated to each codeword, one obtains a larger inner bound. Theorem IV.. Approximate capacity for a discrete distribution with a mass larger than half. Consider a DPC-FFD-RCSI in Fig. and let A have a discrete distribution P A a ) with support A where there exists A=a such that P A a ) /. Define moreover P A = P A a ), P A = P A a ) G=P AE[loga a ) a a ] [ G = P a a ) ] AE log a + ) a a, then the capacity C is upper bounded as C R OUT = log+p)+ P A P A c P A log+p) P A c P A P+) + P A log Pc ) + G/ P A log+p)+ 3 G/ P A c > P A P+) an the capacity lies to within G G+3 bits per channel use from R OUT. Proof: The proof can be found in App. B. The result of Th. IV. can be evaluated for some discrete fading distributions. Lemma IV.. Gap from for some discrete distributions. When A is distributed according to a geometric distribution, i.e. P A k a + n )= p) n p, n N, ) for some p [0,], >0 and p = p to obtain a unitary variance) ani k a = p)/p to obtain zero mean), Th. IV. can be applied for p /. For this choice of p, A = k a has probability larger than a half and the best strategy for the transmitter is to Costa pre-code against the sequence ck a S or otherwise treat the fading times state as noise. The value of the

8 outer bound in ) depends on the value G, while the gap from capacity on G which are obtained as for which G= G = n= n= logn ) p p) n p)log log n ) k a + n) + p p) n ka p), 3) G G p)k a + log ). 4) The gap between inner and outer bound goes to infinite as goes to zero: in this regime the channel reduces to the classic DPC with no fading for which the bounding techniques in Th. IV. are no longer tight. Note that 4) goes to infinity as k a goes to zero, but this is only a consequence of the bounding in 3). Binomial Distribution. Consider now the case in which A has a binomial distribution of the form p A k a + n,n)= N n p) n p N n, n [ N...+N], and N p p)= to maintain the variance unitary and k a = N p to have zero mean. By simple enumeration we see that for N > no assignment of p gives a probability mass larger than a half. For N = we have only one p which makes the theorem applicable: p = / which corresponds to the probability vector [/4////4]. This result extends the case where the probability vector is [///] which corresponds to the case it Th. II.. Another possible extension of the result in Th. II.3 is the case in which A is uniformly distributed over a set with more than two elements. In the following we indeed show such a generalization: the caveat is that the points in the support of the distribution must be increasingly spaced apart points. This result is similar in spirit to our result in [9] for the DPC with slow fading, that is, for the channel in which a fading coefficient is randomly drawn from a set of possible values before transmission and is kept constant through the channel transmission. The intuitive interpretation of this result is as follows: when two fading value are sufficiently spaced apart, the transmitter cannot exploit the correlation between the two different channel outputs corresponding to the two different fading realizations. For this reason the best choice for the transmitter is to Costa pre-code against one realization of the fading times state. Theorem IV.3. Approximate capacity in the strong fading regime. Consider the case in which A is uniformly distributed over the set AM)={a 0,a...a M, a i R}, 5) with VarA)= and let i be the distance between two consecutive points in A, that is i+ = a i+ a i, i [0...M ] 6)

9 and > α, then, if i+ αc i ) i++, i> 7) for some α 0 then an outer bound to capacity is ) log + P M + c + M c M R OUT M = log+p)+ c M M M P+) + M M log c ) + +logα/ M log+p)++logα/ c M > M M P+) and the exact capacity lies to within a gap of max{logα)/ G+ 3,} where Proof: The proof is provided in App. C. j= [ a a G=M ) ] )E log a + ) a a, 8) As an example of Th. IV.3 consider the case in which α = c /c + ): in this case the condition in 7) translates to the set AM) defined as where 0 is determined so that the variance is equal to one, that is AM)= { 0,,c,c...c M } c 0, 9) c c M M c M which follows from the properties of the geometric series. c M ) =, 0) c Note that Th. IV.3 implies that, when c is much larger than P, then the capacity of the DPC-FFD-RCSI as /M times the capacity of the channel without state. We conclude by providing an outer bound for the case of a continuous fading distribution. Unfortunately this bound is not tight in general: this reflect the fact that the outer bounding techniques employed so far are too crude to address this general case. Theorem IV.4. Outer Bound for continuous fading distributions. Consider the case in which A has a continuous distribution with such that there exists a an interval I = [a,b] R with P A I) /, let moreover a [a,b] s.t. Pa )b a)= PI) G= log a a ) ) dp a, ) R\I

10 then the capacity C is upper bounded as C R OUT = log+p)+ P AI) P A I)c P A log+p)+ P A log Pc ) + G/ P A log+p)+ G/ P A I)c P A I)P+) P A I)c > P A I)P+) It is straightforward to verify that the above bound cannot be attained by simply performing Costa pre-coding against a value of ca S for some a of choice: in fact this strategy achieves R IN = log+p) [ E Pc )] A log P+ c a + a a ) + log+p) [ E A log min{a c,p} a a ) )] a +, which goes to zero as P or c grows, unless A is mostly concentrated around a ± /c. V. CONCLUSION In this paper we studied a variation of the classic dirty paper channel in which the channel state is multiplied by a fast fading process which is unknown at the transmitter. We consider both the case in which the decoder has knowledge of the fading and the case in which it does not. For this model we derive inner and outer bounds to capacity and bound the difference between the two when possible. When fading knowledge in not available at the receiver, the gap between inner and outer bounds is small for a number of classic fading distributions but it is not bounded for others. When fading knowledge is available at the receiver we can characterize capacity for some specific discrete distributions of the fading. REFERENCES [] S. Gel fand and M. Pinsker, Coding for channel with random parameters, Problems of control and information theory, vol. 9, no., pp. 9 3, 980. [] M. Costa, Writing on dirty paper. IEEE Trans. Inf. Theory, vol. 9, no. 3, pp , 983. [3] W. Zhang, S. Kotagiri, and J. N. Laneman, Writing on dirty paper with resizing and its application to quasi-static fading broadcast channels, in Information Theory, 007. ISIT 007. IEEE International Symposium on. IEEE, 007, pp [4] A. Khina and U. Erez, On the robustness of dirty paper coding, Communications, IEEE Transactions on, vol. 58, no. 5, pp , 00. [5] Y. Avner, B. M. Zaidel, S. Shamai, and U. Erez, On the dirty paper channel with fading dirt, in Electrical and Electronics Engineers in Israel IEEEI), 00 IEEE 6th Convention of. IEEE, 00, pp [6] P. Grover and A. Sahai, On the need for knowledge of the phase in exploiting known primary transmissions, in New Frontiers in Dynamic Spectrum Access Networks, 007. DySPAN 007. nd IEEE International Symposium on. IEEE, 007, pp [7] S. Rini and S. Shamai, The impact of phase fading on the dirty paper coding channel, in Information Theory ISIT), 04 IEEE International Symposium on. IEEE, 04, pp [8] A. Khisti, U. Erez, A. Lapidoth, and G. Wornell, Carbon copying onto dirty paper, IEEE Trans. Inf. Theory, vol. 53, no. 5, pp , May 007. [9] S. Rini and S. Shamai, On capacity of the dirty paper channel with fading dirt in the strong fading regime, in Information Theory Workshop ITW), 04 IEEE. IEEE, 04, pp

11 APPENDIX A. Proof of Th. III. Capacity outer bound Consider the following series of inequalities developed from Fano s inequality NR ε N ) IY N ;W) IY N ;W S N ) a) b) c) = hy N S N ) hy N W,S N,X N ) d) = N max j hy j S j ) hy N W,S N,X N ) e) = N max j E S j [hx j + csa j + Z j )] hy N W,S N,X N ) f) N E S[ logπe P+ c s + )] hy N W,S N,X N ) g) N logπe P+ c + ) hy N W,S N,X N ), h) where g) follows from the GME property given that A X and Var[A] = by definition while follows from Jensen s inequality and from the fact that E[S]=0. Note that the mean of A does not influence this bound. For the term hy N W,S N,X N ) we have: hy N W,S N,X N ) = hcs N A N + Z N W,S N,X N ) = hcs N A N + Z N S N ) = NhcS j A j + Z j S j ) NHS j A j S j ) Nlog c, 3a) 3b) 3c) 3d) 3e) where 3c) follows from the Markov Chain cs N A N + Z N S N W,X N, 3c) from the fact that A i,s i and Z i are iid RV. The term hs j,a j S j ) can be rewritten as [ ] hs j A j S j )= ha j ) E S logs ) = ha j )+ γ, where γ is the Euler s constant γ Note that the derivation holds for A both continuous or discrete. Combining the bounds in ) and 3) we obtain the expression in 8). Capacity inner bound For the inner bound, we consider Costa s dirty paper coding strategy to pre-cancel µ A S while disregarding the remaining

12 randomness in the fading. This strategy attains R IN = IY ;U A) IU;S) = HU S) HU Y). Considering now the assignment in which X and U X N 0,P), U = X+ ks, which attains R IN log P 4) P+ k P+kcµ A) P+c +µ A )+ by upper bounding hu Y) using the GME property. The optimal choice of k is k = P P++c cµ A 5) which achieves R IN log + P ) c, 6) + as expected. Gap between inner and outer bound By comparing the outer bound expression in 8)and the inner bound expression in 6) have that the difference in the two expressions is G=R OUT R IN 7a) = logπep+ +c ) logπec α)+ γ logπep+ +c ) ) logπec + ) 7b) = c ) log + αc + γ 7c) ) 4 log 3αc + γ 7d) ) log + α, 7e) where 7d) follows from the fact that capacity is known to within bit for c 3. Equation 7) concludes the proof.

13 B. Proof of Th. IV. Capacity outer bound Using Fano s inequality we write NR ε) IY N ;W A N ) N [ E A logπep+ a c + c a ] P+ ) HY N W,A N )+ N E [ A logπep+ a c + ) ] HY N W,A N )+ N E A[ logπep+ c + ) ] HY N W,A N = a N )+, a N A N 8a) 8b) 8c) 8d) where 8d) follows from Jensen s inequality. Next we derive a bound on HY N W,A N ) based on the letter-typicality of the sequence a n, defined as n Nk an ) P A k) εp Ak) k A, 9) where Nk a N ) is the number of symbols is the sequence a N which are equal to k, i.e. Nk a N )= N j= Accordingly, the ε-typical set T N ε P A ) is defined as the set of a N which satisfy 9): { Tε N P A)= a N, Using the letter-typicality in 9), we write: {k=a j }. 30) n Nk an ) P A k) εp Ak), k A }. 3) a N A N Pa N )HY N W,A N = a N ) a N T N ε P A ) Pa N )HY N W,A N = a N ). 3a) 3b) Let now ε P A P A sequence a N where so that Na x N ) > /. With this provision, we can define the sequence a N as a permutation of the if a i a, then a i = a, if a i a, then a i = a. This permutation is also depicted in Fig. 3: the sequence a N is obtained by permuting the positions i for which a i a with some of the positions j for which a j = a : since Na x N )>/, this can always be done. Note that N N Na a n ))= Na a n ) N positions are such that a i = a i = a. With this definition of a N we next define the equivalent channel output Y = X N + ca N S N + Z N, 33)

14 Fig. 3. The permutation that generates a N from a N in the proof of Th. IV. in App. B. where Z N has the same marginal distribution of Z N and any chosen joint distribution with this term. With these definitions in place, we write: a N T N ε P A ) = Pa N ) a N Tε N P A ) a N T N ε P A ) Pa N )HY N W,A N = a N ) = Pa N )H a N Tε N P A ) = Pa N ) a N Tε N P A ) a N T N ε P A ) = Pa N ) a N Tε N P A ) ) HY N W,A N = a N )+HY N W,A N = a N ) ) Pa N ) HX N + ca N S N + Z N,X N + ca N S N + Z N W) ) ca N a N )S N + Z N Z N,X N + ca N S N + Z N W H ca N a N )S N + Z N Z N) ) + HY Y Y,W,S N,X N ) Pa N ) H ca N a N )S N + Z N Z N) ) + HZ N ) H ca N a N )S N + Z N Z N) + N ) logπe) 34a) 34b) 34c) 34d) 34e) 34f) 34g) where 34e) follows from the fact that S N and the additive noises are independent from W. Let us now focus solely on the term / a N Tε N P A ) PaN )H ca N a N )S N + Z N Z N) : we can make use of the following properties of the typical sets: Pa N ) n+ε)ha), an Tε N T N ε P A ) δε ) n ε)ha) Nk a N ) NP A k)a) ε), 35a) 35b) 35c)

15 for δ ε = A e nmin k P A k). 36) Using the properties in 35) we now write: a N T N ε P A ) Pa N )H ca N a N )S N + Z N Z N) n+ε)ha) a N Tε N P A ) n+ε)ha) a N T N ε P A ) i= H ca N a N )S N + Z N Z N) N 37a) 37b) H cai a i )S i + Z i Z i )). 37c) We would now wish to change the summation in the right hand side of 37c) from i [...N] to k A. To do so we need to remember how a N was defined: a i a i can take values: a a, a a and 0. Since the entropy term H ca i a i )S i + Z i Z i ) is not affected by the sign of a i a i, we conclude that there are N Na a N )) times in which we have H ca k)s i + Z i Z i ) for some k a and Na a N ) N terms with value H Z i Z i ). Additionally, for a given k, H ca k)s i + Z i Z i ) appears Nk a N ) times. With these observations we now write n+ε)ha) a N T N = N ε P A ) i= n+ε)ha) a N T N ε P A ) Na a N ) N)H Z i Z i ). H cai a i )S i + Z i Z i )) Nk a N )H ca ) k)s i + Z i Z i k A\a 38a) 38b) We can now choose the joint distribution between Z i ani Z i to simplify the bound above: for simplicity we choose Z N = Z N. With this choice, we can write n+ε)ha) a N T N ε P A ) Na a N ) N)H Z i Z i ) Nk a N )H ca ) k)s i + Z i Z i k A\a = n+ε)ha) log4πe) 39b) a N Tε N P A ) = n+ε)ha) Nk a N ) a N Tε N P A ) k A\a logπec a k) ) N 4 log4πe) 39c) = n+ε)ha) δ ε) n ε)ha) Nk a N ) k A\a logπec a k) ) Nk a N )H ca ) N k)s i k A\a 4 39a) N 4 log4πe) = n+ε)ha) δ ε) n ε)ha) ε)n P A k)a) k A\a logπec a k) ) N 4 log4πe). 39d) 39e)

16 When N is sufficiently large and ε sufficiently small, we then have that HY N W,A N ) 40a) for some ε all that goes to zero as N. k A\a P A k)a) logπec a k) ) N 4 log4πe) ε all 40b) NP A logc NG N log4πe) ε all, 40c) Using the bound in 40) in 8d) and for some ε all sufficiently small, we obtain R OUT = log πep+ c + ) ) P A logc G 4 logπe)+ 4a) log P+ c + ) P A logc ) G +, 4b) We next optimize the above expression over the parameter c over the set [0,c ] since capacity must be decreasing in c. The optimal value of c in 4) is c ) {P = min A P A +P),c }. 4) When P A c P A +P) this optimization yield the tighter outer bound than the original outer bound in 4) R OUT P A c P A+P) 43) = P A log+p)+ h P A ) G + 44) P A log+p) G + 3, 45) where h x) indicates the binary entropy. so that the overall outer bound can be further simplified as log P+c + ) R OUT = P A logc ) G + P A c P AP+ ) P A log+p) G + 3 P A c > P AP+ ). 46) Capacity inner bound For the inner bound consider the simple scenario in which the transmitter Costa pre-codes against the realization ca S, which occurs more than half of the time. That is, consider the assignment X N 0,P) U = X+ P P+ a cs, U X.

17 The attainable rate of this scheme is R IN [ =E A [IY ;U A) IU;S)] + ] 47a) P A log+p)+ P A a) +c a ) + P)+P) log A,a a P A c a a ) + P+ c a, + 47b) the latter term is bounded as P A a) A,a a +c a + P)+P) log Pc a a ) + P+ c a + P A a) = log+p) P Aa) Pa c log A,a a P+ c a + P A a) logp) P Aa) min{p,a c }a a ) ) log A,a a a + P Aa) log min {, a c } a a ) A,a a P a + ) P P Aa) log min {, a c } a a ) ) A,a a P a + P Aa) a a ) ) log A,a a a + = G. ) a a ) ) a + 48a) 48b) 48c) 48d) 48e) 48f) This attainable rate can be improved upon by using two codewords: one that treats the interference as noise. We can assign power α to one codeword and power α = α to the other and successively optimize over the power assigned to each codeword. This yield the achievable rate [ ) R IN = max E A α [0,] log αp + +c a + αp + P A log+αp)+ P A a) +c a ) ] + αp)+αp) log A,a a P A c a a ) + αp+ c a 49a) + [ ) max E A α [0,] log αp + +c a + P A ] + αp log+αp) G 49b) ) max α [0,] log αp + +c + P A G log+αp) + αp, 49c) where 49b) follows from the fact that the bound in 48) holds for any P. the optimal value of αp is then { α P=max min { P A P A c,p } },0, 50) so that, when P A P A c P A P+) we have R IN = logp+ c + ) P A log c ) h P A) 5a) logp+ c + ) P A log c ) G. 5b)

18 Finally, we have shown the achievability of the outer bound ) log + P +c R IN = P A P A c logp+ c + ) P A P A c P A P+ ) P A log c ) G P A log+p) G P A c > P A P+ ) 5) Gap between inner and outer bound A gap between inner and outer bound of 3 bits in the interval P A c > P A can be obtained by comparing the two expressions in 5) and 46) in the cases i) P A P A c, ii) P A P A c P A P+) and iii) P A c > P A P+ ). For the case in which P A P A c we have that c so that the capacity can be approached to within bit by treating the interference as noise with a variance partially known at the receiver. In the other two cases the gap is at most G + G + 3. C. Proof of Th. IV.3 Capacity outer bound We proceed in the bounding from Fano s inequality up to 8) in App. B. We next wish to construct now a sequence a N from a N as done it the proof of Th. IV.: for this proof we actually need to construct M auxiliary sequences, a N, obtained as k) a N k) ={ a i = α j = a k),i = α modk+ j,m), j [...]A } k [0...M ]. 53) Accordingly we define Yk) N as the channel output obtained when the fading sequence is an as in 33), k) Y N k) = X+ an k) SN + Z N k). 54) Note that, as in the proof of Th. IV., we can associate a different noise to each Yk) N in 54) and later choose the joint distribution among these noise terms. Since the symbols are equiprobable, we have that PY N W,A N = a N 0) )=PY N W,A N = a N ) for all k) k. Additionally, given the definition of typicality in 9), if a N T N ε, we have that also a N k) T N ε. As a last definition, let Yk) N j) be the subset of position of Y N k) j) in which a k), j = α j for j [...M] in the chosen ordering of A, that is Y N k) j)={ Y k),i j), s.t.,a k),i = α j, i [...N] }, j [...M], 55) Accordingly Y 0) m) is the subsets of channel outputs in which a j = x m and Y k) modm+k)) are the same subsets of outputs but in which a j = x modm+k). A first part of the proof involves extending the bounding in 34) to the case of any number of passible fading realization M = A. This derivation involves a recursion which we illustrate this using the case M = 3: the general case is inferred from

19 Fig. 4. An illustration of the The sequences Y N k) and the subsequences Y N j) for k, j {,,3} in App. C k) this derivation. We shall continue the derivation of the outer bound from 8d) and focusing on the bounding of the term HY N W,A N ). Case for M = 3 Consider M = 3 and A ={a,a,a 3 } for some ordering of the elements in A and, as in 38) note that HY N W,A N ) a N T N ε P A ) = a N T N ε P A ) Pa N )HY N W,a N ) 56a) 3 N HY N W,a N )+ε all 56b) The sequences Yk) N and the subsequences Y N k) j) for k, j {,,3} are illustrated it Fig. 4 from which we see that Y 0)), Y ) ) and Y ) 3) are obtained from the same set of Xs, Ss and Zs but different fading value. For this reason we can write HY N A N = a N ) 57a) = ) HY N W,A N = a N 0) 3 )+HY N W,A N = a N ) )+HY N W,A N = a N ) ) 57b) ) HY0) N 3,Y ) N,Y ) N W) 57c) = HY N 3 0) ),Y 0) N ),Y 0) N 3),Y ) N ),Y ) N ),Y ) N 3),Y ) N ),Y ) N ),Y ) ), N 3) W) 57d) where 57d) follows form the fact that the transformation of variables has Jacobian one. Using the definition of Yk) N j) in 55) we conclude that the vector [ ] Y ) ) Y0) N ), Y 0) N ) Y )), Y ) ) Y ) ), 58)

20 is a permutation of the vector ca a )S N + Z N, 59) where Z N is a permutation of the terms [ Z0) ) Z ) ), Z ) ) Z 0) ), Z ) ) Z ) ) ]. 60) We then have 3HY N W,A N = a N ) 6a) HY N 0) ),Y N 0) 3),Y N ) ),Y N ) 3),Y N 3) ),Y N 3) 3) W,ca a )S N )+ Z N W) Hca a )S N + Z N W), 6b) where 6b) follows from the fact that this transformation has unitary Jacobian. Consider now the vector [ ] Y) N 3) Y )), Y ) 3) Y0) N ), Y 0)3) Y ) ), 6) which is again a permutation of the vector ca 3 a )S N + Z N 3, 63) where Z 3 N is a permutation of the noise vector [ ] Z) N 3) Z )), Z ) 3) Z0) N ), Z 0)3) Z ) ). 64) With this definition we can write 3HY N W,A N = a N ) 65a) Hca 3 a )S N + Z N 3 ca a )S N + Z N ) Hca a )S N + Z N ) HY 0) 3),Y ) 3),Y ) 3) ca a )S N + Z N,ca 3 a )S N + Z N 3,W) Hca 3 a )S N + Z N 3 ca a )S N + Z N ) Hca a )S N + Z N ) H Z N 3 ), 65b) 65c) where Z N 3 is a permutation of the noise terms [ Z0) 3), Y ) 3), Y ) 3) ]. 66) The expression in 65c) is composed of vectors of independent terms, but the distribution of Z N and Z N 3 might not be identical, since we haven t chosen a joint distribution between the noise terms. At this point in the proof we can sen the noises

21 to be independent so that Z,i, Z 3,i N 0,), 67) and iid for all i [...N]. We can now evaluate the terms in 65c) for this assignment as Hca a )S N + Z N )= N logπe c + ), 68) and Hca 3 a )S N + Z 3 N ca a )S N + Z N ) = NH c a 3 a )a a ) ca a ) c a a ) + S ) 69a) + Z 3 c a 3 a )a a ) Z c a a ) ), 69b) + where we have Z 3 and Z are zero mean Gaussian with variance two. which can be further simplified as Hca 3 a )S+ Z 3 ca a )S+ Z ) 70a) = Hc S+ Z c S+ Z ) 70b) = c logπe + c4 ) c + 70c) = c logπe + )+4 ) c + 70d) = c log + )+ ) c + + log πe 70e) The conditions in 7) for M = 3 become α, 7a) αc ) = + αc, 7b) for some α > 0 so that we can write 3HY N A N = a N ) N logπec ) N ) logα N logπe. 7) Note that when 7b) holds, then the entropy term HY a N,W) no longer depends on a N and thus we have that Pa N )HY N A N = a N ) M logπec ) M logα ε all, 73) a N A N for some ε all which goes to zero as N goes to infinity. Equation 73) follows, similarly to 40), from the fact that the typical set T N ε P A) contains most of the probability and that the sequences in the typical set have a sample probability close to the P A. Case for general M The derivation for the case M = 3 can be extended to the general case by generalizing the bounding in 57), 6) and 70)

22 to any M. Typicality, as in 73), can be invoked to obtain a bound on the term HY N W,A N ). The bound in 57) produce M sequence a N k) and the corresponding sequences Y N k) so that HY N W,a N )= M N k= HY N k) W) M HY N 0),...,Y N M ) W). 74a) 74b) This expands on the bounding in 57) Obtain the term a a )S N + Z as combination of the terms Y k) ) Y modk+m,m)) ) from the entropy term in 74b): this transformation is composed of a circular matrix and an identity matrix which can be shown to have unitary determinant. This term can be removed it from the term in using the definition of conditional entropy and bounded as N/logπec )+/logα) This generalizes the passage in 6). Successively remove the terms i S N + Z ii ) so that HY N A n = a n ) M where Z i,i+ is defined analogously to Z in 60). M NH i S+ Z i S+ Z... i S+ Z i ) H Z M ), 75) i= Each term H i S+ Z i S+ Z... i S+ Z i ) in 75) can be evaluated as H i S+ Z i S+ Z... i S+ Z i )= log c i j= j )+ ). 76) c + i j= j This term, under the condition in 7) can be bounded as /logπec + /logα. This generalizes the bounding in 70). With the above recursion we come to the outer bound R OUT = log++ µ A )c + P) M M log+ µ A )c )+ M logα) M + 77) This expression correspond to the expression in 4) in the proof of Th. IV., consequently in can be optimized over c as such said expression. This results in the outer bound R OUT = log P+c + µ A )+) M M logc + µ A )) M M logα)+ M M log+p) M logα)+ 3 M c + µ A M c + µ A M ) M M )> M P+ ) P+ ). 78) Capacity inner bound For the inner bound, consider the case in which the transmitter pre-codes against one of the realizations of the state times

23 the fading. Let such realization be a S N so that we attain the rate as in 48). Using the definition of G in R IN M log+p) M Pc a a ) ) log A,a a P+ c a ) R IN log+p). 80) M By combining the scheme in 79) with the scheme that treats the fading-times-state as noise we attain the bound ) R IN = max δ [0,] log δp + +c + µ A )+δp + ) M log +δp, 8) and the optimization over δ yields R IN = ) log P + +c +µ A ) M M > M c + µ A ) logp+ c + µ A )+) M M M c + µ A M M log c + µ A )) G M log+p) G M c + µ A M )> M M ) M P+) P+ ) 8) Gap between inner and outer bound The gap between inner and outer bound is obtained by comparing the expressions in 78) and the expression in 8). D. Proof of Th. IV.4 Similarly to the proof of Th. IV. in App. B when deriving an outer bound to capacity. NR ε) IY N ;W A N ) 83a) N E A[ logπep+ a c + ) ] N HY N W,A N ) 83b) N logπep++ µ A)c + ) N PaN )HY N W,A N = a N )da N, 83c) a N A N where Given the condition in ) and since HY N A N,W)= PaN )HX N + cas N + Z N W)da N I N + PaN )HX N + cas N + Z N W)da N. R N \I N N logπe) HX N + cas N + Z N W) N logp+c + )+, 84) and I N is a closed interval, we can apply the mean value theorem and conclude that I N PaN )HX N + ca N S N + Z N W,A N = a N )da N = P A I N )HX N + ca N S N + Z N W), 85)

24 for some a N I N. Note that this holds even if the distribution P X N,SN has some discrete points because of the convolution with the distribution of Z N. We can now write PaN )HX N + cas N + Z N W)da N I N + PaN )HX N + cas N + Z N W)da N R N \I N =P A I) P A I))) N HX N + ca N S N + Z N W) + P A a) HX N + cas N + Z N W)+HX N + ca N S N + Z N W) ) da N R\I P A I) P A I))) N HX N + ca N S N + Z N X N,S N ) + PaN ) HX N + ca N S N + Z N,X N + ca N S N + Z N W) ) da N R N \I N N P AI) P A I)) N logπe) + PaN ) Hca N a N )S N + Z N,X N + ca N S N + Z N W) ) da N R N \I N = N P AI) P A I)) logπe) + Pa N ) logπe c a N a N ) + ) R\I +HZ N ca N a N )S N + Z N,S N,X N ) ) da N N P AI) logπe)+ P AR\I) logπe+ µ A )c ) P A a) a a) + N log R\I + µ A da N P AI) logπe)+ P AR\I) logπe+ µ A)c + Ĝ. This yields the same outer bound as 4) but with an updated expression for G. As for Thm. IV. we can optimize the expression in c and obtain the same outer bound.