International Zurich Seminar on Communications (IZS), March 2 4, 2016

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1 Resource-Aware Incremental Redundancy in Feedbac and Broadcast Richard D. Wesel, Kasra Vailinia, Sudarsan V. S. Ranganathan, Tong Mu University of California, Los Angeles, CA {wesel,vailinia, sudarsanvsr Dariush Divsalar Jet Propulsion Laboratory Cal. Inst. of Tech., Pasadena, CA Abstract This paper reviews recent results from the UCLA Communication Systems Laboratory on the use of incremental redundancy. For channels with ACK/ACK feedbac, this paper reviews how the transmission lengths used for communicating incremental redundancy should be optimized under the constraint of a limited number of incremental redundancy transmissions. For broadcast channels, this paper reviews optimization of the trade-off between pacet-level erasure coding and physical-layer channel coding in the context of bloc fading with diversity that grows with boclength. I. ITRODUCTIO This invited tal reviews two results [, [2 optimizing the use of incremental redundancy. In systems with feedbac, incremental redundancy adapts the coding rate to the the accumulated information density the "instantaneous capacity" of the channel allowing the Shannon limit to be approached at much shorter average bloclengths than those required for the accumulated information density to concentrate around the Shannon capacity [3, [4, [5, [6, [7, [8. In systems without feedbac, incremental redundancy can provide a "fountain" of information from which a receiver need only "drin" what is needed to reliably identify the desired message [9, [0, [. For each of these two scenarios, this paper reviews optimization techniques that improve performance. II. TRASMISSIO LEGTHS FOR ACK FEEDBACK ACK/ACK feedbac is non-active in the sense that the feedbac does not change what is transmitted but rather only indicates whether additional transmissions are needed. For channels with ACK/ACK feedbac, the sequential differential optimization SDO approach of [ optimizes the transmission lengths used to communicate incremental redundancy. This optimization maximizes throughput under the constraint of a limited number of incremental redundancy transmissions. A. The ormal Approximation SDO utilizes the power of the normal approximation introduced in [3 that characterizes the behavior of the rate that a This material is based upon wor supported by the ational Science Foundation under Grant umbers 6250 and Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the ational Science Foundation. This research was carried out in part at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with ASA JPL Tas Plan Cumulative Distribution Function BLDPC Simulation Gaussian Approximation Rate R S Fig.. Empirical complementary cumulative distribution function c.d.f. and the Gaussian approximation Q-function corresponding to the rate R S at which the B-LDPC code of [ was able to decode successfully. channel can support at finite bloclength. Following [3, define information density ix, Y as f Y X y x ix, Y = log 2. f Y y The expected value of ix, Y is the capacity of the channel. For the example of a BI-AWG channel with noise z, ix, Y = log 2 + e 2z+/2 = iz. The accumulated information density I n at the receiver after n symbols is n I n = iz. 2 = As pointed out by [3, 2 is a sum of independent random variables that will converge quicly to a normal distribution according to the central limit theorem, leading to the normal approximation of [3. A ey result of [ is that a normal approximation also accurately describes the rate at which actual variable-length codes with incremental redundancy will successfully decode. Fig. shows that for the B-LDPC code used in [ the empirical complementary cumulative distribution function on the rate at which decoding is successful is very closely approximated by a normal distribution for this example of the BI-AWG channel with SR of 2 db. We have similarly confirmed the accuracy of the normal approximation to predict the rate at which decoding is successful for B-LDPC codes in higher-sr AWG channels that require larger constellations and in fading channels with channel state information nown at the receiver. 63

2 Fig. shows that R S is well-approximated by a Gaussian with mean µ S = ER S and variance 2 S = VarR S: f RS r = e r S 2 2 S π 2 S The c.d.f. of the bloclength S at which decoding is successful is F S n = P S n = F RS /n. Taing the derivative of F S using the Gaussian approximation of F RS produces the following reciprocal-gaussian" approximation for p.d.f. of S : f S n = n 2 e n S πS 2 S. 4 B. Sequential Differential Optimization SDO uses the tight Gaussian approximation discussed above to optimize the sequence of bloclengths {, 2,..., m } to maximize the throughput. Suppose that the number of incremental transmissions is limited to m. An accumulation cycle AC is a set of m or fewer transmissions and decoding attempts ending when decoding is successful or when the m th decoding attempt fails. If decoding is not successful after the m th decoding attempt, the accumulated transmissions are forgotten and the process starts over with a new transmission of the first bloc of symbols. From a strict optimality perspective, neglecting the symbols from the previous failed AC is sub-optimal. However, the probability of an AC failure is sufficiently small that the performance degradation is negligible. eglecting these symbols greatly simplifies analysis. The cumulative bloclength j at the jth stage is simply the sum of the first j increment lengths. Using the p.d.f. of S from 4 we can compute the probability that the decoder will need a particular incremental transmission. For j < j+, the probability of a successful decoding attempt at bloclength j+ but not at j is j+ j+ f S ndn = j j n 2 2πS 2 rj+ = Q Q e n S 2 2 S 2 dn 5 rj, 6 where r j = / j. Define the throughput as R T = E[K E[, where E[ represents the expected number of channel uses and E[K is the effective number of information bits transferred correctly over the channel. The expression for E[ is E[ = Q 7 + m j [Q j Q j=2 + m [ Q m j 8. 9 The first term 7 shows the contribution to the expected bloclength from successful decoding on the first attempt. Q S is the probability of decoding successfully with the initial bloc of. Similarly, the terms in the summation of 8 are the contributions to the expected bloclength from decoding that is first successful at total bloclength j for j 2 at the j th decoding attempt. Finally, the contribution to expected bloclength from not being able to decode even at m is 9. Even when the decoding has not been successful at m, the channel has been used for m channel symbols. The expected number of successfully transferred information bits E[K is E[K = Q m, 0 m where Q S is the probability of successful decoding. ote that E[K depends only on and m. In fact, E[K and is not sensitive to the specific choice of m for reasonably large values of m. The initial bloclength is and we see the optimal bloclengths {, 2,..., m } to maximize the throughput. Over a range of possible values, the SDO technique introduced in [ selects { 2,..., m } to minimize E[ for each fixed value of by setting derivatives to zero as follows: E[ = 0, j =,..., m. j For each j {2,..., m}, the optimal value of j is found by setting E[ j = 0, yielding a sequence of relatively simple computations. In other words, we select the j that maes our previous choice of j optimal in retrospect. E[ For j > 2, j = 0 depends only on { j2, j, j } as follows: E[ j =Q + j jq j j2 Q. j Thus we can solve for j as Q j + jq j = Q j j Q j2. 2 For each possible value of, SDO can be used to produce an infinite sequence of j values that solve for any choice of m. The sequence does not depend on m, only. Each such sequence is an optimal sequence of increment lengths for a given density of decoding attempts on the time axis. As increases, the density of decoding attempts decreases, lowering system complexity. Using SDO to compute the optimal m decoding points is equivalent to selecting the most dense SDOoptimal sequence that when truncated to m points still meets the frame-error-rate target. 64

3 R T Ideal non-active feedbac m= Ideal non-active feedbac m=2,3,..., m Fig. 2. Throughput as a function of the number m of incremental transmissions permitted. C. Approaching Capacity at Short Bloclengths with Feedbac Fig. 2 shows the resulting throughputs obtained by using SDO to find the optimal increment lengths for values of m in the range of 2 < m < 20 for the target FER of 0 3 for the B-LDPC code of [ for = 96 message bits. Fig. 2 illustrates that with m = 0 decoding points, a system can closely approach the performance of a system that has m =, which is the limiting case where decoding is attempted and feedbac of an ACK/ACK is required after every received symbol. Using SDO, variable-length codes with average bloclengths of around 500 symbols can closely approach capacity in theory and in practice as demonstrated in [. Fig. 3 illustrates the example of a binary-input BI additive white Gaussian noise AWG channel with frame error rate FER required to be less than 0 3. For a system transmitting symbols at an average bloclength of λ, the throughput R t is defined by R t = /λ. For reference, Fig. 3 shows the curves of possible throughput R t as a function of λ for some values of. The performance characterization for fixed-bloclength codes is from [3 and is based on the normal approximation, which is shown in [3 to be accurate for bloclengths as small as 00 symbols. The computation of the random coding lower bound on the performance of variable-length codes with feedbac is based on the analysis in [4. Fig. 3 shows curves from [, [5, [6 that show simulation results that approach or exceed the performance promised by [4 in the range of average bloclengths below 500 bits. For values of = 6, = 32, = 64, and = 89 these throughput results exceed Polyansiy s random coding lower bound. As the average bloclength becomes larger, the random coding lower bound is more predictive. ote that variable-length codes with feedbac approach capacity at very short bloclengths. In Fig. 3, the randomcoding lower bound for a system with feedbac is 0.27 db from the Shannon limit for = 280 with a bloclength of less than 500 bits. Looing at implemented codes for = 280 in Fig. 3, the m = non-binary LDPC B-LDPC code is 0.53 db from Shannon limit. Using SDO, the B-LDPC non-active feedbac system in Fig. 3 that uses ten rounds of single-bit feedbac to operate within 5 db of the Shannon limit with an average bloclength of less than 500 bits. Expected Throughput R t BIAWG Channel, SR = 2dB, Capacity=422 =6 =32 =64 =89 =96 =84 =280 BIAWG capacity Randomcoding lower bound [2 Fixed bloclength, no feedbac [ 024 CC, active feedbac, m=5 [5 64 TBCC, nonactive, m= [3 B LDPC, active, m= [4 B LDPC, nonactive, m=0 R t =/λ =6,32,64,89,84, Average Bloclength λ Fig. 3. Approaching capacity at short bloclengths using feedbac. III. PACKET-LEVEL VS. PHYSICAL LAYER REDUDACY Consider a broadcast setting that uses a hybrid of pacetlevel erasure coding and physical layer coding to provide a stream of information with the goal of each receiver decoding the desired message at the earliest opportunity. There is a trade-off between using available redundancy for additional pacets in a pacet-level erasure code or simply for additional physical-layer code symbols. As the amount of available redundancy grows, the wor of [2 shows that in several different bloc fading scenarios the physical layer coding rate decreases ultimately to zero while the pacet-level erasure coding rate does not. This indicates that at some point incremental redundancy should be directed to the physical layer rather than additional pacet-level erasure coding. This paper reviews the recent wor [2 that studies the this hybrid coding approach using a proportional diversity bloc fading model in which diversity increases linearly with bloclength. A. The Channel Model and Optimization Problem Consider a transmitter and a receiver communicating over a fading channel [3. The one-dimensional channel is modeled as Y = HX + Z where X is the transmitted symbol, Y is the received symbol, H is the fading coefficient, and Z is i.i.d. additive white Gaussian noise AWG with variance 2 and mean 0. We assume the channel is Rayleigh with E [ H 2 =, Z has unit variance, i.e. 2 =. Let the average transmit power be E [ X 2 = P. Then, the instantaneous signal-tonoise ratio SR when H = h is h 2 P. For this Rayleigh fading channel, SR denoted γ is exponentially distributed 65

4 with parameter P that depends only on the average transmit power. ote that, γ has a mean of P. A message consisting of m pacets with nats of information per pacet is to be transmitted with a low probability of message error q; this is the probability that the receiver fails to recover all the m pacets. The transmitter uses the channel for T units of time for an overall code rate of m T. It performs erasure coding across the m pacets at a rate R E and codes each resultant pacet at a channel-coding rate such that m T = R E. 3 That is, the m pacets are first coded using an erasure code at rate R E to yield m R E pacets. ote that, for erasure coding, R E has to satisfy R E. To transmit each pacet, the transmitter uses a channel code at rate [nats/channel-use so that the resultant codeword bloc-length of each pacet is. For a fixed average transmit power, our objective is to pic the value of and thus R E that optimizes an objective function. The unit of channel-coding rate is nats/channel-use" for convenience. The receiver is assumed to now the fading coefficient H while the transmitter does not. The proportional diversity PD model introduce a parameter l f, which describes the length of a fade. With the bloc-length being, the number of bloc fades F P in a transmitted codeword of a system with PD bloc fading of fade lengths l f is F P = l f. 4 With PD bloc fading, long codewords benefit from an inherent increase in diversity. For this wor, we assume that each bloc-fading event is independent, i.e. H assumes i.i.d. values across different bloc fades. The receiver sees m R E = T pacets from the channel. The number of pacets that the decoder of the erasure code requires to recover the message, denoted ˆm m, depends upon the erasure code. For Reed-Solomon erasure codes, ˆm = m; for fountain codes such as a Raptor code, ˆm > m typically. Thus, the probability of message error q can be written using the binomial distribution as q = ˆm i=0 RC T i p e i p RCT i e. 5 In the above expression, p e denotes the probability that a pacet is not decoded successfully and declared an erasure upon reception from the channel; this is called the probability of pacet erasure. Owing to our assumption that the channel codes in the system operate close to capacity with zero blocerror probability when the Shannon capacity exceeds the attempted rate, p e constitutes only one event: fading outage [4. The binomial sum in 5 can be computed numerically only for small values of T. Hence, we approximate the random variable that denotes the number of pacets successfully decoded by the channel decoder using the Central Limit Theorem CLT, and obtain the Gaussian approximation for q [2 as [ ˆm T p e q Φ, 6 RC T p e p e where Φx is the value of the c.d.f. of the standard normal random variable at x R. To summarize, the objective is to minimize the messageerror probability q in 5 via 6, where p e is also a function of. Writing the minimization problem in terms of, R E can be obtained as R E = m T. Hence, the optimization problem is as follows: min Φ [ s.t. p e = P ˆm T p e RC T p e p e l f l f i= l f l f l f, C γ i + C γ last < + ɛ, ˆm T l f, T. 7 ote that, minimizing Φ is equivalent to minimizing its argument, and the value of q need not be explicitly computed. We have specified the dependence of p e on here for clarity. As noted in [2, [5, and many previous wors, the evaluation of p e for the bloc-rayleigh fading channel or for its PD version is not a straightforward tas. One can use [5 or similar wors for the bloc-rayleigh fading channel to compute the outage probability p e with a minuscule error. But, our fading model complicates it further as we have a sum of two random variables that are not identically distributed in the expression for p e in 7. We first expand and rearrange the terms in p e for our one-dimensional PD bloc-rayleigh fading channel with capacity-achieving codes to obtain p e = P l f i= W i + l f l f W last < c l f, 8 where c = 2 + ɛ, W i = log + γ i, W last = log + γ last. B. Gaussian Approximations of the Optimization Problem Based on Gaussian approximations of p e in 8, as inspired by [2, [2 presents four approximations to the optimization problem 7. For numerical-search based results, [2 uses a very low value of the margin, say ɛ = 0.05, to obtain c. Gaussian Approximation Approx. : Ignoring the contribution of W last in 8, we get p e = P l f i= W i < c l f. 9 66

5 The above can be approximated using the Gaussian CDF as c l p e = Φ f l f µp. 20 l f VarP The values of µp and VarP, which denote the mean and variance of log + γ with γ Exponential P, can be computed as stated in [2. By ignoring the flooring function, we get Gaussian approximation Approx., which is an adaptation of 9 in [2 to PD bloc-rayleigh fading: [ c µp p e = Φ. 2 l f VarP 2 Gaussian Approximation 2 Approx. 2: For Approx. 2, we evaluate 20 directly. The approximation to p e that is being made here is imprecise in the sense that, 20 evaluates to the same value for a range of values; the reason being the presence of the flooring function. 3 Gaussian Approximation 3 Approx. 3: This approximation is the evaluation of 8 with a constrained search space that limits such that both m R E and l f are positive integers. 4 Gaussian Approximation 4 Approx. 4: The Gaussian approximation that we mae here considers both the terms in 8, maing it the most appropriate. Once we find out l f µp and VarP, we assume that i= W i is Gaussian and also that l f l f W last is Gaussian. Thus, their linear sum is another Gaussian random variable denoted W G, which stands for Gaussian approximation of weighted average mutual information, with meanw G = µp, l [ f 2 VarW G = VarP +. l f l f l f 22 Thus, p e for this approximation Approx. 4 is [ c l p e = Φ f meanw G. 23 VarWG C. Results and conclusions Fig. 4 shows an example of the optimal values of and R E obtained from Approximations and 4 as the overall code rate m T goes to 0. As observed by Courtade and Wesel [2 for the fixed diversity bloc-fading channel, for the PD blocfading model that the optimal channel-coding rate goes to 0. However, where the optimal value of R E approached a nonzero constant less than for fixed diversity in [2, under PD bloc-fading it is approaching. With sifficient overall redundancy, pacet-level erasure coding is unnecessary in a bloc fading channel with proportional diversity. ote that rate-compatibility in this scenario is challenging because the rate R E increases for a sufficiently low overall rate. Optimal values R C, R E 0.8 m = 50, = 20, l f = 0, P = 5dB 0.4 RC Approx. RE 0.2 Approx. RC Approx. 4 RE Approx Overall code rate m/t Fig. 4. Result of optimization problem 7 as a function of overall code rate for an example system with m = 0, = 20, l f = 0 and P = 5 db. REFERECES [ K. Vailinia, A. R. Williamson, S. V. S. Raganathan, D. Divsalar, and R. D. Wesel. Feedbac systems using non-binary LDPC codes with a limited number of transmissions. In IEEE Information Theory Worshop, pages 67 7, Hobart, Tasmania, Australia, ovember 204. [2 S. V. S. Ranganathan, T. Mu, and R. D. Wesel. Optimality and ratecompatibility for erasure-coded pacet transmissions when fading channel diversity increases with pacet length. In arxiv.org/abs/ , 206. [3 Y. Polyansiy, H. V. Poor, and S. Verdú. Channel coding rate in the finite bloclength regime. IEEE Trans. Inf. Theory, 565: , May 200. [4 Y. Polyansiy, H. V. Poor, and S. Verdú. Feedbac in the non-asymptotic regime. IEEE Trans. Inf. Theory, 578: , Aug. 20. [5 A. R. Williamson, T.-Y. Chen, and R. D. Wesel. Variable-length convolutional coding for short bloclengths with decision feedbac. IEEE Trans. Commun., 637: , July 205. [6 A. R. Williamson, T.-Y. Chen, and R. D. Wesel. Firing the genie: Two-phase short-bloclength convolutional coding with feedbac. In IEEE Inf. Theory and Applicat. Worshop, pages 6, San Diego, CA, February 203. [7 K. Vailinia, T.-Y. Chen, S. V. S. Raganathan, A. R. Williamson, D. Divsalar, and R. D. Wesel. Short-bloclength non-binary LDPC codes with feedbac-dependent incremental transmissions. In IEEE Int. Symp. Inf. Theory, pages , Honolulu, Hawaii, June 204. [8 A. R. Williamson, T.-Y. Chen, and R. D. Wesel. Reliability-based error detection for feedbac communication with low latency. In IEEE Int. Symp. Inf. Theory, pages , Istanbul, Turey, July 203. [9 M. G. Luby, M. Mitzenmacher, M. A. Shorallahi, and D. A. Spielman. Efficient erasure correcting codes. IEEE Trans. on Info. Th., 472: , Feb [0 M. Luby. LT codes. In The 43rd Annual IEEE Symposium on the Foundations of Computer Science, pages , Atlanta, Georgia, ov [ A. Shorollahi. Raptor codes. IEEE Trans. on Info. Th., 526: , June [2 T.A. Courtade and R.D. Wesel. Optimal allocation of redundancy between pacet-level erasure coding and physical-layer channel coding in fading channels. IEEE Trans. on Comm., 598:20 209, Aug. 20. [3 A. Goldsmith. Wireless Communications. Cambridge University Press, ew Yor, [4 E. Biglieri. Coding for Wireless Channels. Springer, [5 A. Yilmaz. Calculating outage probability of bloc fading channels based on moment generating functions. IEEE Trans. Commun., 59: , ovember