On Capacity of. the Writing onto Fast Fading Dirt Channel

Transcription

1 On Capacity of the Writing onto Fast Fading Dirt Channel Stefano Rini and Shlomo Shamai (Shitz) National Chiao-Tung University, Hsinchu, Taiwan arxiv: v [cs.it] 0 Jun 06 Technion-Israel Institute of Technology, Haifa, Israel sshlomo@ee.technion.ac.il Abstract The classic Costa s writing on dirty paper capacity result establishes that full state pre-cancellation can be attained in the Gel fand-pinsker channel with additive state and additive Gaussian noise. This result holds under the assumption that perfect channel knowledge is available at both transmitter and receiver: such an assumption is not valid in wireless scenarios affected by fading and with limited feedback rates. For this reason, we investigate the capacity of the writing on fast fading dirt channel, a variation of the writing on dirty paper channel in which the state sequence is multiplied by an ergodic fading process unknown at the transmitter. We consider two scenarios: when the fading process in not known at either the transmitter or the receiver and when it is known only at the receiver. For both, cases we derive novel assessments of capacity which clearly indicate the limits of state pre-cancellation in the presence of fast fading. Index Terms Gel fand-pinsker Problem; Writing on Fading Dirt; Carbon Copying onto Dirty Paper; Ergodic Fading; Costa pre-coding; INTRODUCTION Costa [] showed that the capacity of the writing on dirty paper channel, the Gaussian version of the Gel fand-pinsker [] channel, is equal to the capacity of the Gaussian point-to-point channel without state, regardless of the power or the distribution of the state sequence. Albeit very promising, this result assumes that perfect channel knowledge is available at the users and the attainable rate quickly degrade with any uncertainty in the channel knowledge. The assumption of perfect channel knowledge does not hold in communication scenarios where channel conditions vary over time and with limited feedback between the receiver and the transmitter. This paper investigates the effects of partial channel knowledge on the performance of state precancellation and the topic of robust pre-coding. More specifically, we study the dirty paper channel with fast fading dirt, a variation of Costa s writing on dirty paper channel in which the state sequence is multiplied by an ergodic fading process. We consider both the case in which neither the transmitter nor the receiver have knowledge of the fading values and the case The work of S. Rini was funded by the Ministry Of Science and Technology (MOST) under the 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 in which only the receiver has knowledge of the fading realization. For both models, we derive the approximate capacity for some classes of fading distributions, also including classic distributions such as Rayleigh and Gaussian fading. These results are the first closed-form characterizations of capacity to be derived for models combining state pre-cancellation and ergodic fading. Related Results: The Gel fand-pinsker (GP) channel is the point-to-point channel in which the channel output is obtained as a random function of the input and a state sequence which is provided non-causally at the transmitter. The capacity of this model is determined in [] and is expressed as the maximization of a non-convex function for which the optimal solution is not easily determined, either explicitly or through numerical optimization. For this reason, very few closed-form expressions of the capacity result in [] are available in the literature. In [], Costa determines an explicit expression of capacity for the Gaussian version of the GP channel, termed the Writing on Dirty Paper (WDP) channel. Perhaps surprisingly, the capacity of the WDP channel coincides with the capacity of the Gaussian point-to-point channel, thus implying that it is possible for the transmitter to fully pre-code its transmission against the channel state sequence. In the literature, few authors have investigated extensions of the WDP channel to include fading and partial side information. The first channel model to include state pre-cancellation and slow fading is the carbon copying onto dirty paper (CCDP) channel of [3]: the CCDP channel is the M-user compound channel in which the output at each receiver is obtained as a linear combination of the input, Gaussian noise and one of M possible state sequences, all non-causally known at the transmitter. The CCDP channel in which the M state sequences are scaled versions of an identical sequence, reduces to the WDP channel with slow fading and each receiver corresponds to one of M possible fading realizations. The authors of [4] consider the case in which both the input and the state sequence are multiplied by the same fading value which is known at the decoder but not at the encoder. They consider both the case of fast and slow fading and evaluate the achievable rates using Costa pre-coding for these scenarios. It is show, in particular, that the rate loss from full state pre-cancellation is vanishing in both the fast and slow fading case as the transmit power grows. This result holds because the fading identically affects the channel input and the state: for this reason Costa pre-coding is still as effective as in the WDP channel. The WDP channel with fast fading in which fading only affects the state sequence is first studied in [5] which focuses, in particular, on the case of phase fading. In [6], the same authors study this model from an outage analysis perspective and derive inner and outer bounds for the outage probability. Achievable rates under Gaussian signaling are derived in [7] for the WDP channel in which the state sequence is multiplied by an iid Gaussian fast fading process. These attainable rates are also compared to lattice coding strategies and some numerical observation are provided on the performance of various coding choices. In [8] we derive the first approximate capacity result for a class of WDP channel with slow fading in which a Gaussian state sequence is multiplied by one of M possible fading realizations. In this class of channels, which we term strong fading regime, the fading coefficients are increasingly spaced apart which makes it is impossible for the transmitter to simultaneously code for multiple fading realizations. Consequently, the best transmission strategy is to pre-code against the fading-time-state realization at each receiver for a portion /M of the time. Accordingly, capacity scales as /M times the capacity of the

3 3 Gaussian point-to-point channel: this shows that the presence of fading can drastically affect the ability of the transmitter to exploit the channel state knowledge. Contributions: In this correspondence, we investigate the capacity of the Writing onto Fast Fading Dirt (WFFD) channel, a variation of the WDP channel in which the channel state is multiplied by an ergodic fading sequence. We focus on the case in which the state sequences is a zero-mean iid Gaussian process so that the channel is parameterized by the transmit power, the state variance and the fading distribution. We study (i) the case of No Channel Side Information (WFFD-NCSI), when the fading sequence is unknown at both the transmitter and the receiver and (ii) the case of Receiver Channel Side Information (WFFD-RCSI), when the fading sequence is known only at the receiver. In both scenarios, we derive the first approximate characterizations of capacity available for this class of models which include both channel state and fast fading. A summary of the contributions for these two models is as follows: Sec. III, Writing onto Fast Fading Dirt with No Channel Side Information (WFFD-NCSI) channel: We show that, when the fading entropy is sufficiently high, capacity can be approached by pre-coding the input against the state times the mean of the fading. Although conceptually simple, this result clearly shows that channel state knowledge can have very little value in the presence of even a small amount of channel uncertainty. Sec. IV, Writing onto Fast Fading Dirt Channel with Receiver Channel Side Information (WFFD-RCSI) channel: The availability of receiver channel side information makes it possible the transmitter and the receiver to develop distributed pre-coding strategies which take advantage of the partial knowledge available at each terminal. The analysis is divided into the case of discrete and continuous fading distributions: Sec. IV-A: Discrete fading distribution: We begin by determining capacity to within a constant gap for the case of uniform, antipodal fading: for this simple fading distribution capacity can be approached by transmitting the superposition of two codewords: one codeword treats the fading-times-state as noise while another codeword is pre-coded against one of the fading realization times the channel state. This result is extended to two classes of fading distributions: the class of distributions with a mode larger than a half and the class of uniform distributions with exponentially spaced points in the support. Sec. IV-B: Continuous fading distribution: We investigate the case of continuous fading by first discretizing the continuous fading values and then applying the approximate capacity results in Sec. IV-A. We numerically evaluate the gap between inner and outer bound for some typical fading distributions such Gaussian and Rayleigh fading. Unfortunately, the case of continuous fading is fundamentally harder to characterize and we are able to derive only partial results for this scenario. The main theoretical contributions of the paper consist in the development of new outer bounding techniques to characterize the capacity of a model comprising both channel states and fading. In particular, we adapt the outer bounding techniques of [3] for the CCDP channel to the case of fast fading and further improve on the results in [3] by showing a tightening of the outer bounds for both the -user CCDP channel as well as the M-user CCDP channel. On the other hand, the inner bound This channel, in the literature, has also been referred to as dirty paper channel with fading dirt, writing on fast faded dirt and dirty paper coding channel with fast fading, we prefer the term writing on fast fading dirt for both brevity and clarity.

4 4 used throughout the paper are rather straightforward combinations of time-sharing, Costa pre-coding and superposition coding. This is somewhat disappointing as it Paper Organization: The remainder of the paper is organized as follows: in Sec. I we introduce the channel models of the WFFD-NCSI channel and WFFD-RCSI channel. Sec. II presents related results available in the literature. In Sec. III we derive the approximate capacity results for the WFFD-NCSI channel while, in Sec. IV, we derive the approximate capacity results for the WFFD-RCSI channel. Finally, Sec. V concludes the paper. I. CHANNEL MODEL In the GP channel [] the output of a point-to-point channel is obtained as a random function of the channel input and a state sequence. The encoder has non-causal knowledge of the state sequence and can thus design the channel input as to minimize the adverse effects of the state on the channel output. The WDP channel [] is the GP channel in which the output is obtained as a linear combination of the channel input, the state sequence and a Gaussian noise sequence. We consider a variation of the WDP channel in which the state sequence is multiplied by a iid fading sequence. The state is assumed to be Gaussian distributed, while the fading can have any distribution, discrete or continuous, P A, with Var[A i ] =. The channel output is therefore obtained as: Y N = X N +ca N S N +Z N, () for E [ X N ] NP and Z i,s i N(0,), i.i.d. and A P A i.i.d. and i [...N], c R + and Var[A i ] =. In the following we consider two distinct channel models: the channel in which the fading sequence is unknown at either the transmitter or the receiver and the channel in which it is available only at the receiver: the Writing onto Fast Fading Dirt with No Channel Side Information (WFFD-NCSI) channel for which Y NCSIN = Y N, () for Y N in (). the Writing onto Fast Fading Dirt with Receiver Channel Side Information (WFFD-RCSI) channel for which Y RCSIN = [Y N A N ], (3) for Y N and A N in (). A graphical representation of the WFFD-NCSI channel and the WFFD-RCSI channel is depicted in Fig. : the switch in the figure indicates whether the fading sequence is known at the receiver (WFFD-RCSI) or not (WFFD-NCSI). In the study of the WFFD channel, standard definitions of code, achievable rate and capacity are employed. Definition. Code. A ( NR ;N) code for the WFFD channel in Fig. consists of the message set W = [... NR ], an

5 5 Z N X N Y N W Enc. + + Dec. Ŵ S N A N RCSI vs NCSI Fig.. The Writing onto Fast Fading Dirt (WFFD) channel with and without channel side information (RCSI/NCSI respectively). encoding and a decoding function respectively defined as: X N : W S N X N, X N = X N (W,S N ), Ŵ : Y N W, Ŵ = Ŵ(Y N ), where K N is the support of the Random Variable (RV) K N. The rate R is achievable if there exists a sequence of codes of rate R such that the probability of decoding error goes to zero as the block-length N goes to infinity. Definition. Achievable rate. Let the probability of decoding error for a ( NR ;N) code be defined as ] P e ( NR ;N) = Pr[Ŵ(Y N ) W, (4) then, a rate R R + is said to be achievable if there exists a sequence of codes such that Note that the error probability in (5) is averaged over all state sequences and messages. lim P e( NR ;N) = 0. (5) N Definition 3. Capacity and approximate capacity. The capacity C is the supremum of all the achievable rates. The rate R IN is said to be an inner bound to the capacity if R IN C. Similarly, the rate R OUT is an outer bound to capacity if R OUT C. Inner and outer bounds for which R OUT R IN, (6) for some constant R + are said to characterize the capacity to within an additive gap of bits per channel use (bpcu). At first sight, the model in () seems to be quite restrictive but actually encompasses a large class of channel models. To observe this, consider a more general formulation than that of () where Y N = h X N +c A N S N +Z N, (7) for some Z N,S N iid Gaussian distributed while E[ X N ] NP. The noise variance and input gain in (7) can be reduced to one by scaling the channel output and the channel input and accordingly adjusting the input power constraint. The fading-

6 6 times-state term c A N S N can be rewritten (dropping the superscript N for convenience) as: c A S = (8) c ((S µ S )(A µ A )+S µ A +A µ S µ A µ S ). Note now that µ A µ S in (8) is constant term that can be subtracted from the channel output. The term µ S A N can be subtracted from the output in the WFFD-RCSI channel while it is equivalent to additive noise in the WFFD-NCSI channel. For these reasons it is possible to assume µ S = 0 with only a small loss of generality. Note also that c A S = c σ A σ S A σ A S σ S, (9) which shows that the coefficient c in () can be used to normalize the variance of both fading and state to one. Finally, we can assume c > 0 since the state distribution is symmetric. Since we are interested in determining the capacity to within a small gap, we shall actually assume c >, since capacity for c < can be attained to within one bpcu by treating the fading-times-state term as noise. II. RELATED RESULTS In this section, we briefly introduce some classic results available in the literature which are relevant in the study of the WFFD-NCSI channel and the WFFD-RCSI channel. The Gel fand-pinsker (GP) channel. The capacity for the GP channel is a classic information theoretic result. Theorem II.. Capacity of the GP channel [, Th. ]. The capacity of the GP channel is obtained as C = max P U,X S I(Y;U) I(U;S), (0) This expression in (0) is convex in P X S,U for a fixed P U S which implies that X can be chosen to be a deterministic function of U and S. On the other hand, (0) is neither convex nor concave in P U S for a fixed P X S,U. Given the fact that (0) contains an auxiliary RV U and its convexity properties, determining a closed-form solution for the maximization in (0) is extremely challenging. Additionally, the lack of tight bounds on the cardinality of U makes in very hard to obtain numerical approximations of the optimal value. The Writing on Dirty Paper (WDP) channel. One of the few channel models for which the maximization in (0) is known in closed-form is the WDP channel for which the assignment X N(0,P), X S in (0) yields C = /log(+p), regardless of the distribution of S N. U = X + P S, () P +

7 7 The Carbon Copying onto Dirty Paper (CCDP) channel. In the CCDP channel [3], the channel outputs in a M-user compound channel are obtained as a linear combination of the input, Gaussian noise and one of M possible state sequences. The transmitter has knowledge of all of the M state sequence and must guarantee correct decoding at all the receivers. More specifically, the channel output at the m th receiver is obtained as: Ym N = XN +csm N +ZN m, m [...M], () where { Sm, N m [...M] } is a jointly Gaussian iid sequence with any covariance matrix. In [3] the authors derive inner and outer bound for this model, although no characterization of capacity is shown. The following theorem summarizes the relevant inner and outer bounds for the case of M = and independent channel states with equal, unitary variance. Theorem II.. Outer bound for the two-user CCDP with independent channel states[3, Th. 3]. The capacity C of the -user CCDP channel with independent channel states with equal, unitary variance is upper bounded as ( log +P+c + ) P +c / c ( C R OUT = log +P+c +c ) P c/ [ ( + 4 log c P+)] c > (3) and lower bounded as C R IN = ( ) log + P c /+ ( log P+c /+ ) (c + 4 log c ) c / c / < P + 4 log(p +) c / P + (4) The outer bound in (3) is derived through a clever bounding of the entropy terms obtained from Fano s inequality which we later improved in [9]. The inner bound in (4) is obtained by sending the superposition of two codewords: the base codeword treats the states as additional noise while the top codeword is pre-coded against a combination of the state sequences. In [0] we show that the inner bound in (4) is to within a small gap from capacity. Writing onto Fast Fading Dirt with Receiver Channel Side Information (WFFD-RCSI) channel. In [], the RHS (0) is optimized for the WFFD-RCSI channel in which S,U,A and X are restricted to be jointly Gaussian RVs. Theorem II.3. Achievability with jointly Gaussian signaling [, Sec. IV] [7, Th. ]. Consider the WFFD-RCSI channel for S,A N(0,), A S, let ρ = (ρ XS,ρ US,ρ UX ) and let K denote the region ρ t < t {XS,US,UX} K = +ρ XS ρ US ρ XS ρ US ρ UX = 0, (5) for K [,] 3 then, the following rate is an inner bound to capacity of this model: C R IN = max ρ K E θ[r Γ (ρ,θ) a = θ], (6)

8 8 for R Γ (ρ,θ) = ((P log +c +θρ XS c ) P +)( ρ US) log( P( ρ UX )+c ( ρ US ) +θc(ρ XS (ρ UX ρ US )) ) P +. (7) The result in Th. II.3 attempts to generalize the result of [] to the fast fading case although, in all likelihood, one needs to consider a wider class of joint distributions P UX S than the jointly Gaussian distribution, even when all the variables in (0) are Gaussian distributed. III. THE WRITING ONTO FAST FADING DIRT WITH NO CHANNEL SIDE INFORMATION (WFFD-NCSI) CHANNEL. The WFFD-NCSI channel is the WDP channel in which the state sequence S N is multiplied by a fast fading realization A N, unknown at both the transmitter and at the receiver. In this section we show that, when A has large entropy, capacity can be approached by ignoring the randomness in the fading and instead pre-coding the channel input against the sequence cµ A S N. Intuitively, when the entropy of the fading process is sufficiently high, then the value of the fading-times-state is too unpredictable and the best coding option is to pre-code against the average fading value times the state sequence. Although conceptually simple, this result makes it possible to obtain a tight characterization of capacity for a number of canonical fading distributions, such as Gaussian, log-normal and Rayleigh fading. Note that the capacity of the WFFD-NCSI channel is implied by Th. II., but no closed-form expression of the maximization in (0) is known. Theorem III.. Outer bound and approximate capacity for the WFFD-NCSI with a continuous fading distribution. Consider the WFFD-NCSI channel and let A have a continuous distribution with entropy h(a) = log(πeα) for some α (0,], then capacity C is upper bounded as C R OUT = max P X S I(Y;X S) log ( P + c α + α and the exact capacity C is to within a gap of max{g NCSI,} bpcu from R OUT in (8b) for (8a) ) +, (8b) G NCSI = log(α). (9) Proof: The proof can be found in App. A and only an outline of the proof is provided here. The outer bound in (8) is obtained by providing the state to the receiver as a genie-aided side information. Since neither the transmitter or the receiver have knowledge of the fading value, this genie-aided model is equivalent to a fading point-to-point channel in which the fading-times-state term acts additive noise with random but known variance. In the inner bound, the transmitter pre-codes against the sequence cµ A S N while treating the term c(a N µ A )S N as additional noise. Note that, since in Costa pre-coding the input is independent from the state, this additive noise is also independent from the input. The gap from capacity in Th. III. can be readily evaluated for some standard fading distributions.

9 9 Lemma III.. Gap from for some fading distributions. Uniform distribution. When A is uniformly distributed over the set [µ A 3,µ A + 3], the capacity of the WFFD-NCSI channel is to within a gap G NCSI U bpcu, from the outer bound in (8b) for α = 0.7. Gaussian distribution. When A is Gaussian distributed with mean µ A and unitary variance, the capacity of the WFFD-NCSI channel is to within a gap G NCSI N = bpcu, from the outer bound in (8b) for α =. Rayleigh distribution. When A is Rayleigh distributed, i.e. A = U +V for U,V N(0,/(4 π)) and independent, the capacity of the WFFD-NCSI channel is to within a gap G NCSI R.35 bpcu, from the outer bound in (8b) where γ is the Euler-Mascheroni constant and, thus, α = Log-normal distribution. When A is log-normal distributed, i.e. A = e Z for Z (µ Z,σZ ) and for µ Z = σ Z ln ( e σ Z ), (0) to obtain a unitary variance,then the capacity of the WFFD-NCSI channel is to within a gap G NCSI log = log(e)µ Z log(σ Z ) bpcu, () for µ Z in (0), from the outer bound in (8b). Note that G NCSI log for this distribution is so that α log 0 as σ Z. is not finite for all values of σz and is actually increasing with σ Z. In particular, we have that α in (8b) α NCSI log = σ Z e σ Z e σ Z, () The result in Th. III. is substantially a negative result since it shows that, for a number of typical fading distributions, the best strategy is to Costa pre-code against the mean value of the fading-times-state and treat the remaining randomness as additional noise. Note, additionally, that when the fading has zero mean, state knowledge at the transmitter is useless. The gap between inner and outer bounds in Lem. IV.5 is evaluated using the outer bound in (8b): a tighter characterization of capacity can be obtained by numerically evaluating the outer bound in (8a) for the different distributions in Lem. IV.5. Similarly, it is possible to obtain larger inner bound by explicitly evaluating the attainable region for each different output

10 OUT PP R 3.5 OUT GA R Rate.5 R IN N R IN R R INU RIN log IN GNCSI R N c Fig.. Inner and outer bounds to the capacity of the WFFD-NCSI channel with Gaussian state and various fading distributions for P = 00 and c [0,0]. distributions. This tighter characterization of capacity is presented in Fig. for P = 00, c [0,0] and where: R OUT PP is the point-to-point capacity /log(+p) which is a trivial outer bound to the capacity of any WFFD-NCSI channel, R OUT GA is the genie-aided outer bound in (8a), R IN N is the achievable rate in Th. III. explicitly evaluated for the case of i Gaussian distributed fading with µ A = as in Lem. IV.5, R IN R, R INU and R IN log are the achievable rates for the Rayleigh, uniform and log-normal distributions (the last two with µ A = ) respectively. IN GNCSI R N is the loose inner bound used to obtain the gap in (9). IV. THE WRITING ONTO FAST FADING DIRT WITH RECEIVER CHANNEL SIDE INFORMATION (WFFD-RCSI) CHANNEL The WFFD-RCSI channel is obtained from the WFFD-NCSI channel by additionally providing the receiver with the fast fading sequence A N. As for the WFFD-NCSI channel, the capacity of the WFFD-RCSI channel is a special case of the GP capacity in Th. II. but obtaining a closed-form expression is very challenging. In actuality, the study of the WFFD-RCSI channel is more arduous than that of the WFFD-NCSI channel because of the distributed way in which transmitter and receiver can cooperate in managing the fading-times-state term. As an illustrative example, consider the WFFD-RCSI channel with no additive noise and in which the state and the input are restricted to take values in {,+}, while A has any distribution in R\{0}, i.e. Y N = X N +A N S N, X N,S N {,} N. (3) Given the cardinality of the input, the capacity of this channel is at most bpcu and can be attained by setting Pr[X = ] = Pr[X = +] = /, X S and U = XS in (0) which also implies Pr[X = ] = Pr[X = +] = /, U S, regardless

11 of the distribution of S. With this assignment, U can be recovered from the squared channel output, in fact: Y A=a = X +a S +axs = +a +au, so that the receiver can reconstruct U as U = Y a, (4) a regardless of the distribution of A. This simple example shows that the maximization in (0) for the WFFD-RCSI channel might yields some unexpected results and that jointly Gaussian assignments are, in general, far from optimal. Another interesting observation is with regard to the effectiveness of DPC in the WFFD-RCSI channel: since the receiver has knowledge of the fading, the transmitter could choose a fading value k A and pre-code the transmitted codeword against the fading-times-state realization cks N. The receiver would then attempt to decode the transmitted codeword from those channel outputs in which the fading realization is sufficiently close to the value k. This transmission strategy attains [ ( (+P)(+c R IN = E A log a )] +P) Pc (a k) +P +c a (5a) + = log(+p) [ ( Pc E (a k) )] A log +c a +P +. (5b) From the expression in (5) we observe that fading values such that a [k /c,k+/c] can be decoded without incurring in much rate loss from the point-to-point capacity. It then follows that the value k should be chosen so as to maximized Pr[A [k /c,k+/c]] and thus the best choice of k might not corresponds to the mean or the mode of the distribution P A. Let then denote I as the interval of length proportional to /c which contains most of the probability in P A : the expression in (5) can then further be bounded as R IN P A(I) P A(I) log(+p) κ [ ( P E c (a k) ) A log P ++c a +P + log(+p) κ [ ( ) (a k) E A log a + A / I ] A / I (6a) ], (6b) where κ is a constant proportional to the length of I. Note that the negative logarithmic term in the RHS of (6) can be considered small for all practical purposes (as the decoder could disregard the channel outputs corresponding to problematic fading realizations) so the overall attainable rate is P A (I) times the point-to-point capacity. This implies that, from a high level perspective, DPC as in the WDP channel only attains a fraction of the point-to-point capacity corresponding to the probability that the fading value falls in chosen interval of size proportional to /c. When the transmit power grows to infinity, then, a positive transmit rate can be attained for all fading distributions, although this strategy might be far from optimal. In the next subsection we investigate the case of discrete fading distributions and, in the following subsection, extend these results to the case of continuous fading distributions. The approximate capacity results for discrete fading distributions are For simplicity, we assume that P A in well-behaved distribution that takes values on a dense support if discrete and is smooth if continuous.

12 obtained by adapting and improving on the outer bounding techniques for the CCDP in [3]. The outer bound in [3] holds for a compound channel and is adapted to the WFFD-RCSI channel by letting each fading sequence in the typical set of P A be a compound user. Since the distribution of the typical sequences is roughly uniform over the typical set, it is possible to adapt the compound channel approach to the fast fading scenario, although one needs to carefully account for the fact that the number of virtual compound users grows with the block-length N. Another novel ingredient which allows for a crucial tightening of the outer bounds in [3] is obtained from the following observation. Lemma IV.. The capacity of the WFFD-RCSI channel is decreasing in c. The capacity of the WFFD-RCSI channel is decreasing in the parameter c, the gain of the fading-times-state. Proof: The proof is presented in App. B. Although seemingly straightforward, Lem. IV. is key in tightening the outer bounds in presented in the remainder of the section. A. WFFD-RCSI channel with a discrete fading distribution In this subsection we first derive the approximate capacity for the case in which A is uniformly distributed in {,+} to introduce the main inner and outer bounding techniques. This result is then extended to the case in which A has a mode larger than a half and for the case in which A is uniformly distributed over a set of increasingly spaced points. ) WFFD-RCSI channel with antipodal, uniform fading: we start by showing that, when A is a uniform antipodal sequence, one can adapt the result in (3) for the -CCDP channel to obtain a an outer bound for the WFFD-RCSI channel. This is despite of the fundamental difference between the two models, being the CCDP channel a slow fading model while the WFFD-RCSI channel is a fast fading model: in the former model a fading value is randomly chosen and kept fixed throughout the transmission block-length while, in the latter model, the fading value is randomly drawn at each channel transmission. Theorem IV.. Outer bound for the WFFD-RCSI channel with antipodal uniform fading. Consider the WFFD-RCSI channel in which A is uniformly distributed over the set A {,+}, then the capacity C is upper bounded as C R OUT = log(p +c +) 4 log( c ). (7) Proof: The proof is provided in App. C. The result in Th. IV. is obtained by adapting the derivation in [3, Th. 5] by letting every fading sequence A N = a N correspond to a compound user in the CCDP channel. In doing so, the number of compound user becomes a function of N: despite of this the outer bound derivation can be adequately adapted and, remarkably, one obtains substantially the same outer bound for these two models. The outer bound in Th. IV.4 can be tightened using the observation in Lem. IV., since the expression in (7) is not monotonically decreasing in c for all P as show in Fig. 3. For this reason it is possible to improve the outer bound in Th. IV.

13 3.55 R [bpcu].5.45 Outer bound in Th. IV..4 optimized outer bound P + c Fig. 3. The outer bound in (7) for P = 0 and c [0,0]: although capacity is decreasing in c, the outer bound expression has a minimum in P +. by considering a channel with a parameter c = min{ P +,c} c. Such a channel has a larger capacity than the original channel but provides a tighter outer bound to the capacity of the original channel. Theorem IV.3. Outer bound and approximate capacity for the WFFD-RCSI channel with antipodal uniform fading. Consider the WFFD-RCSI channel where A is uniformly distributed over the set {, +}, then the capacity C is upper bounded as C R OUT = log(p +)+ c 4 log(p +) c P + log(p +c +) 4 log(c ) < c < P + (8) and the exact capacity C is to within a gap of G RCSI A = / bpcu from R OUT in (8). Proof: The proof is a continuation of the proof of Th. IV. in App. C and can be found in App. D. Capacity in Th. IV. can be approached by the inner bound represented in Fig. 4 using the notation in [], in which the channel input is comprised of the superposition of two codewords: let α [0,] and α = α then The bottom codeword X N PAS, at power αp, conveys the message W PAS at rate R PAS by treating the fading-times-state sequence ca N S N as additional noise. The top codewordx N SAN, at power αp, conveys the message W SAN at rate R SAN and is pre-coded against the state sequence cs N as in the WDP channel. Since the codeword XPAS N treats the fading-times-state as noise, it attains the rate R SAN (α) log (+ ) αp +αp +c. (9)

14 4 X N PAS (W PAS) cs N X N SAN (W SAN) Fig. 4. The inner bound approaching capacity in Th. IV., represented using the notation in []: nodes represent codewords, solid arrows superposition coding and dotter arrows binning. On the other hand, since A N is equal to one for half of the time on average, the codeword XPAS N is correctly pre-coded against ca N S N for half of the time as well. Accordingly we have R PAS (α) 4 log(αp +), (30) as argued in (6) and the total transmission rate is R IN (α) = R SAN (α)+r PAS (α). (3) The expression in (3) can be optimized over the power allocation ratio α: this optimization yield an expression identical to that of (8) minus / bpcu. As for the result in Th. III., the gap in Th. IV.3 can be tightened by numerically evaluating inner and outer bounds for the given fading distribution, as shown in Fig. 5. In Fig. 5 we also include the performance of different Gaussian coding strategies inspired by the result in [] which illustrate the decay in performance of jointly Gaussian strategies in the presence of fading. As in Th. II.3, one can consider the class of strategies in which P UX S in (0) is restricted to jointly Gaussian distributions. This class of strategies attains the rate R IN LA RCSI = log( ρ US) (H(U X +Z)+H(U X +cs +Z)), (3) for H(U X +Z) = ( log ρ UX P ) P + H(U X +cs +Z) = ( log (ρ ) UX P +cρus ). P +c ++cρ XS P and any ρ US,ρ UX and ρ XS in (5). More precisely, Fig. 5 depicts the point-to-point capacity R OUT PP, the outer bound R OUT CC (CC is in Carbon Copy in reference to [3]) in (8), the inner bound R IN SAN (SAN as in State As Noise ) in which the fading-times-state is treated as noise, the inner bound R IN GA (LA as in Gaussian Assignment ) for jointly Gaussian [X U S] in (3) and finally the inner bound R IN Costa is the optimal assignment in (3) when ρ XS = 0, that is, when the input is independent from the

15 5 3.5 OUT PP R OUT CC R R [bits] IN SAN R R IN Costa R IN GA R IN SA P Fig. 5. Inner and outer bounds to the capacity of the WFFD-RCSI channel with Gaussian state and antipodal uniform fading for c = 0 and P [0...50]. state as in []. The result in Th. IV.3 can be extended beyond the case of binary support and beyond the case of uniform distribution: both these extensions are considered in the remainder of this subsection. In must be noted that the outer bounding technique utilized in Th. IV.3 relies on the fact that P A is a discrete distribution for which one can define typical sets using strong typicality. As we shall see, extensions to the case of continuous fading distributions are generally harder to tackle. ) WFFD-RCSI channel with a discrete fading distribution with mode larger than half: Although valid only for one discrete fading distribution, the result in Th. IV.3 is the first characterization of capacity for the WFFD-RCSI channel. By building upon this result, in this section we derive the approximate capacity for the WFFD-RCSI channel in which P A is discrete distribution with mode larger than one half. From a high-level perspective, for this class of fading distributions the transmission strategy in Fig. 4 is still as effective as in Th. IV.3 since full state cancellation can be attained for more than half of the time by having the transmitter pre-code against the mode of A times the state, Theorem IV.4. Outer bound for the WFFD-RCSI channel with a discrete fading distribution with mode larger than half. Consider a WFFD-RCSI channel in which A has a discrete distribution with mode larger than a half, that is m A, s.t. P A (m), (33) define moreover P m = P A (m), P m = P A (m) and G RCSI m G RCSI m then the capacity C is upper bounded as = P m = P m E ( E[log((a m) ) a m] ) +3 (34a) [ ( ) ] (a m) log a + a m +, (34b) C R OUT = (35)

16 6 log(+p)+ P m P m c (+µ A ) P m log(+p) + Pm log( Pc (+µ A )) P m < P m c (+µ A ) +G RCSI m P m c (+µ A ) P m(p +) P m log(+p) +G RCSI m P m c (+µ A ) > P m(p +) and the exact capacity C lies to within a gap of min{g RCSI m Proof: The proof can be found in App. E. +G RCSI m,} bpcu from R OUT in (35). Note that the gap between inner and outer bound in Th. IV.4 depends only on the fading distribution and is otherwise independent on the other channel parameters, P and c. The inner bound in Th. IV.4 is a variation of the attainable scheme in Fig. 4 in which the top codeword is pre-coded against the mean fading cms N. It is interesting to notice that the result in Th. IV.4 depends on the mean of the fadingµ A : one might expect a better attainable strategy could be attained by pre-coding the base codeword against the codeword cµ A S N so as to remove the dependency of (35) from µ A. Unfortunately this is, in general, not the case as argued at the beginning of the section, when discussing the performance of DPC in the WFFD-RCSI channel. The result of Th. IV.4 can be readily evaluated for some simple discrete distributions. Lemma IV.5. Gap from capacity for some discrete fading distributions. Geometric distribution. Consider the case in which A is distributed according to a geometric distribution, i.e. P A (k a +j ) = p j p, j N, (36) for some p [0,] with p = p, > 0, p = p (to obtain a unitary variance) and k a = A ( p)/p (to obtain zero mean). For this fading distribution, Th. IV.4 can be applied when p A /: the gap from capacity depends on the values G RCSI G and G RCSI G for which obtained as G RCSI G = G RCSI G = log(n +)p( p) n n= p log log n= 4ka ( p), ( n ) (k a + n) + p n A p G RCSI G G RCSI G p A (k a +log ). (37)

17 7 4.5 p = / p = 3/4 p = 7/8 OUT CC R 4 R [bits] R IN + R IN c Fig. 6. Inner and outer bounds to the capacity of the WFFD-RCSI channel with Gaussian state and geometric fading for P = 00 and c [...0]. 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, (38) for n [ N...+N], Np( p) = (to obtain unitary variance) and k a = N p (to obtain zero mean). Note that the case N = and p = / recovers the result in Th. IV.3. Using a simple approximation of the central binomial coefficient we obtain that the theorem is applicable (sufficient condition) when 4 N 4N (pp) N. For this distribution we can approximate G RCSI B can be easily evaluated from the fact that the distribution s mode is in zero: using the Gaussian approximation of the binomial distribution while G RCSI B G RCSI B G RCSI B = P m. = log(πe)+o ( N ) Fig. 6 and Fig. 7 present the results in Lem. IV.5 for = and different values of p in (36) and (38) respectively. More specifically: the outer bound R OUT CC in the outer bound in (35), the inner bound R IN is the inner bound in Th. IV.4 and, finally the inner bound R IN + is the inner bound obtained by a more precise evaluation of the achievable rate which depends on the chosen fading distribution.

18 8 4 p = /4 p = /5 p = / OUT CC R R [bits] R IN + R IN c Fig. 7. Inner and outer bounds to the capacity of the WFFD-RCSI channel with Gaussian state and binomial fading for P = 00 and c [...0]. A simplification and tightening of the result in Th. IV.4 is possible when the transmit power P is larger than the maximum value of c a : Lemma IV.6. Outer bound and approximate capacity for the WFFD-RCSI channel with a discrete fading distribution with mode larger than half. Consider a WFFD-RCSI channel in which A has a discrete distribution with mode larger than a half as in Th. IV.4 and assume moreover than then the capacity C is upper bounded as P + c max{a}, (39) a A C R OUT = E A [ log( P +c a + )] P m logc +G RCSI m, (40) and the exact capacity is to within a gap of bpcu from the outer bound in (40). Proof: The proof is provided in App. F. 3) WFFD-RCSI channel in the strong fading regime: Another extension of the result in Th. IV.3 is the case in which the fading takes one of M possible values with equal probability. For this model, we establish the approximate capacity for the regime in which the points in the support are increasingly spaced apart, that is a i+ ca i (4) We refer to the condition in (4) as strong fading condition. For this regime, we show that the encoder cannot simultaneously pre-code the input against multiple fading realization and thus it can effectively cancel the channel state only by pre-coding

19 9 Fig. 8. The linear deterministic approximation of the WFFD channel in (4). the input against the state sequence cks N for some k A. This strategy, as shown in (6), can only attain a fraction /M of the point to point capacity. The condition in (4) does not seem to offer itself to any intuitive or operational interpretation. Indeed, the deterministic binary linear approximation of Gaussian networks in [3] can be used to glean some important intuitions on the nature of this result. The deterministic approximation technique of [3] focuses on the interaction between different the components in the channel outputs through a powerful visualization: we briefly introduce the model here, solely for the illustrative purposes: more details can be found in [3] and it the related literature. Consider the channel Y N m = Sk npn X N k +Sk namn S N k, (4) where S is a binary matrix with S ij = δ i,j for (i,j) [...m] and X N k and S N k are the first k bits of the binary expansion of X N and S N respectively. Also let n p = log(p) and n am = log(ca m ) and k = max{n p,n am } so that the multiplication by S k np erases all but the n p most significant bits of the binary vector X k, k [...N]. Similarly, the multiplication S k nam S k erases all but the n am most significant bits of S k. The channel in (4) is also represented Fig. 8: from a high-level perspective, it approximates the behaviour of a Gaussian channel with a binary, linear and deterministic channel by removing the effect of the additive noise and approximating the sum over R with the XORing of binary vectors. Through the approximation in Fig. 8 we provide an intuitive explanation of the strong fading condition in (4): consider Fig. 9 which represents, in vertical sections, the output for each possible fading realization in the linear deterministic approximation.each possible channel output corresponds to a different set of high-level bits in the vector S k : for larger values of a k, more bits appear above the noise floor. Accordingly, the way in which the channel input sums to the fading-times-stat term is different in each channel output. When two coefficients a k and a j are close, the transmitter can hope to code for two fading realization simultaneously and thus boost the probability of correct decoding. Unfortunately though, if the two fading realizations are such that the channel input collides with two independent portions of the binary state vector, then, no coding

20 0 Fig. 9. The linear deterministic approximation of the 4-WFFD channel in the strong fading regime. across fading realization is possible. This is the intuitive interpretation of the condition in (4) as illustrated in Fig. 9: (4) implies that the shift of the binary state vector from two successive fading values be larger than c, and thus the, for each fading realization, the input collides with an independent set of bits of the state vector. When this occurs, then, the channel substantially reduces to a WDP channel in which the state is randomly chosen among a set of M independent sequences and the capacity of this model is /M the capacity of the point to point channel. This result is similar in spirit to our result in [0] for the WFFD channel with slow fading that is, for the channel in which a fading coefficient is constant through for the transmission block-length. Theorem IV.7. Outer bound and approximate capacity for the WFFD-RCSI channel in the strong fading regime. Consider a WFFD-RCSI channel in which A is uniformly distributed over a discrete set A for some /(c ) < a < a <... < a M, while a i+ a i > /c then, if a i+ ca i, the capacity C is upper bounded as C R OUT = log(+p)+grcsi s M log(+p) + M M log( (+µ A )c) +G RCSI s M log(+p)+grcsi s M M (+µ A )c M M M < (+µ A )c M (+µ A )c M > M M (P +) M M (P +) (43) and the exact capacity lies to within a gap of max{g RCSI s +G RCSI s,} bpcu from R OUT where G RCSI s = log(+µ A)+ 3 [ = min E k A log G RCSI s ( (a k) a + (44a) ) ] a k, (44b) Proof: The proof is provided in App. G. The achievability proof in Th. IV.7 again relies on the superposition of two codeword: one treating the fading-times-state

21 as noise and one pre-codes against one realization of the fading times the state sequence, as for the proof of Th. IV.3 and Th. IV.4. The pre-coding done by the top codeword allows for perfect state pre-cancellation only for a portion /M of the time, but no better pre-coding is possible under the conditions in (4). Remark IV.8. The result in Th. IV.7 can be generalized to the case in which a i+ κca i for some κ R +. In this case the gap between inner and outer bound becomes a function of κ and such gap increases as κ decreases. We choose to focus on the case κ = for the sake of clarity and ease of exposition. Lemma IV.9. Gap from capacity for a fading distribution. As an example of the result in Th. IV.7, consider the case in which A is uniformly distributed over the set A(M) = {,c,c,...,c M } (45) where = V (46a) V = M c M ( c M ( c M ) ). (46b) ( c) Note that (46b) is chosen as to guarantee a unitary variance. The result in Th. IV.7 can be applied when > /(c ): note that in (46b) is an increasing function in M for every fixed c, so that the condition > /(c ) is verified for all M when it is verified for M = 3, which yields the condition c >.887. B. WFFD-RCSI channel with a continuous fading distribution The results derived so far are limited to the case of discrete fading distributions: although relevant from a theoretic standpoint, this case is not particularly meaningful in practical applications. In this subsection we investigate the how the bounding techniques developed so far can be applied to the case of the continuous fading distributions. The results obtained so far all rely on a rather intuitive attainable strategy in which part of the transmitter power is used to send a codeword that treats the fading-times-state term as additional noise while the remaining power is use to pre-code against one realization of the fading times the state. We conjecture that this strategy also approaches capacity in the case of a continuous distribution and we thus focus on the derivation of outer bounds to match this inner bound expression. We first derive an outer bound along the same lines as the outer bound in Th. IV.3 and argue that this outer bound is tight when the loss from binning is relatively small. A more complex set of outer bounds is then derived which relies on upper bounding the capacity of a WFFD-RCSI channel with continuous fading with the capacity of a WFFD-RCSI channel with discrete fading. The gap between the capacity of the two models can be made small by letting the discrete distribution be an appropriately quantized version of the continuous distribution. More specifically, the capacity of the original channel is enhanced by providing a genie-aided side information at

22 the receiver of the form E N = c(a N A N )S N, (47) where A N is a quantized version of the fading sequence A N so that E N has the form of a quantization error. Theorem IV.0. Outer Bound and approximate capacity for the WFFD-RCSI channel with a continuous fading distribution. Consider the case in which A has a continuous distribution, then the capacity C is upper bounded as log(+p)+ (+µ A )c R OUT 4 = log(+p)+ < (+µ A )c P log( P(+µ A )c) +G RCSI c 4 log(+p)+grcsi c (+µ A )c > P + (48) and the exact capacity lies to within a gap of max{g RCSI c G RCSI c G RCSI c +G RCSI c,} bpcu from R OUT where [ = log(+µ A )E A A log(a a ) ], = max k A log(+µ A ) log ( Pc (a k) P +c a + + (49a) ) + (49b) Proof: The proof is provided in App. H. Note that the expression in (49a) is equal to infinity for all discrete distributions and might, similarly, be infinity or not defined for continuous distributions. Additionally, the term in (49b) is small only when most of the probability is concentrated in a small interval of size /c. For this reason we wish to develop and alternative approaches to the derivation of the approximate capacity for the case of continuous fading distributions. Using the genie-aided bounding technique in (47) we can upper bound the capacity of the WFFD-RCSI channel for some canonical continuous fading distributions. In particularly, we consider different quantization strategies to relate the results in the previous subsections to some relevant fading distributions. More specifically, we use the result in Th. IV.3 to study the continuous uniform fading case, the result in Th. IV.4 to study the Gaussian distributed fading case and, finally the result in Th. IV.7 for Rayleigh fading case. Although not specifically developed for this purpose, the theorems above naturally lend themselves to the analysis of these continuous distributions. The following lemma establishes the gap between the capacity of the original channel and the channel with a discrete fading distribution. Lemma IV.. Gap between continuous and quantized fading. Consider the WFFD-RCSI channel with a continuous fading distribution with capacity C c and let A i be have any discrete distribution P A A for which, using Th. IV.3, Th. IV.4 or Th.