Adaptive Incremental Redundancy for HARQ Transmission with Outdated CSI

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1 Adaptive Incremental Redundancy for HARQ Transmission with Outdated CSI Leszek Szczecinski, Pierre Duhamel, and Moshiur Rahman INRS-EMT, Montreal, Canada CNRS, Laboratory of Signals and Systems, Gif-sur-Yvette, France University of Trento, Italy Abstract We analyze the throughout achievable in transmission over block-fading channels when the instantaneous channel state information (CSI) is not available at the transmitter. Assuming operation with incremental redundancy hybrid ARQ (HARQ), we propose to adapt the transmissions rates using the outdated CSI, i.e., the one experienced by the receiver in the past transmissions that resulted in a packet decoding failure. We show that, even if the CSI is fully outdated, i.e., in independently block-fading channel, the adaptation provides notable gains over a non-adaptive HARQ and for high SNR, a few transmissions are necessary to approach closely the ergodic capacity. I. INTRODUCTION In this paper we analyze the hybrid automatic repeat request (HARQ) protocol for communication over block-fading channel. We assume that the channel state information (CSI) that the transmitter may obtain from the receiver through the feedback channel, is fully outdated at the moment the subsequent transmissions are carried out. Since this means that the actual CSI is not known at the transmitter, error-free transmission cannot be guaranteed. In such a situation, the achievable transmission rate should be analyzed in the average sense and the relevant performance criterion is the throughput defined as the average number of correctly received bits over the average number of channel uses (or symbols employed to convey information) [] [2] [3]. Our analysis is based on quite general assumptions to simplify and/or to make the analysis tractable Random channel codes are used and the resulting codewords can be punctured into sub-codewords of arbitrarily chosen length. The receiver applies a maximum likelihood decoding so the information-theoretic analysis we use is valid in the asymptotic sense, that is, when the transmission length goes to infinity. An error-free return channel exists that allows the receiver to inform the transmitter about the decoding failure/success (the so-called ACK/NACK messages of HARQ process). The work was supported by the 7th framework program of European Community FP7/27-23 under the grant #23668 and by the government of Quebec, under grant #PSR-SIIRI-435. Part of this work was done when L. Szczecinski was on sabbatical leave with CNRS, Laboratory of Signals and Systems, Gif-sur-Yvette, France. Channel is constant during the transmission of a subcodeword but varies independently between transmission attempts and we know its the probabilistic law. The channel state is known (estimated) at the receiver but the instantaneous CSI is unknown at the transmitter. HARQ is truncated which means that only a limited number K of transmission attempts is allowed. All transmissions attempts of the same data block convey independently generated sub-codewords, i.e., HARQ with incremental redundancy (IR) is considered. Finally, we assume that the decoding failure notification (NACK) sent over the return channel may be accompanied by the channel state information (CSI) experienced at the receiver during the transmission attempts that failed. That is, the feedback channel must carry more than one bit (required for ACK/NACK messages) but the overhead due this additional signalling is considered negligible comparing to the data load. This work is concerned with evaluation of the HARQ transmission strategies that use the received, yet outdated CSI, to adapt the number of channel uses (or, equivalently the rate) in the subsequent transmission attempts. This makes our work different from the analysis of HARQ shown, e.g., in [4] [5] or [6] which was not adaptive, i.e, the transmission parameters (rates) were set independently of the CSI returned by the receiver. The idea of adapting the transmission rate using partially outdated CSI studied in [7] [8] [9] considered predefined modulation/coding schemes. In our work we consider random coding and ML decoding establishing the limit for any practical communication scheme. We also lift the assumption of correlated block-fading using in [7] [8] and consider independently fading channels which is the most challenging scenario. More importantly, however, we propose rate-adaptation policies that result in compact and tractable formulas for the throughput, which can be then easily maximized. This is important, as [7] [8] used the adaptation strategies that may be considered local, i.e., optimized on a per-transmission basis, and [9] considered two transmission to avoid the complex optimization. The use of outdated CSI, called a multi-level feedback, was also analyzed in [] [] where, the power (and not the transmission rates, as in our case) was adapted. We emphasize, that the transmitter remains unaware of the instantaneous CSI (at the time the transmission is carried out) because it is independent of the CSI sent by the receiver. In

2 2 such a case, the conventional adaptive modulation and coding [3] cannot be implemented thus, the outdated CSI is useless if retransmissions are not possible. On the other hand, as we will show, a fully outdated CSI can be used to improve considerably the throughput if HARQ is implemented. II. SYSTEM MODEL In the transmission system under study, information bits are separated into packets each containing N b bits which are encoded into codeword of N s complex symbols x, x 2,..., x Ns. The symbols are drawn randomly from the zero-mean complex Gaussian distribution with unitary variance. We assume that N s can be arbitrarily large, that is, we are able to construct the code with arbitrarily low rate. Any subset of the symbols x j is called a sub-codeword. We assume that the transmission with incremental redundancy is implemented, thus these subsets are disjoint and the sub-codeword are composed of different symbols x j. The ARQ process for each packet starts sending the subcodeword x composed of N s, < N s symbols. Using the feedback channel (which is assumed error-free), the receiver sends back (to the transmitter) a one-bit message required by the ARQ process. If the packet is not decoded correctly, the NACK message is sent. Then, the transmitter knowing that the first sub-codeword was not decoded correctly, sends a subcodeword x 2 composed of N s,2 symbols. After unsuccessful decoding, another NACK message is generated to which the transmitter responds sending the codeword x 3. This continues till the maximum allowed number of transmission attempts K is reached (truncated HARQ) or until an ACK message, denoting a successful decoding, is received. For convenience, we normalize the values of N s,k via ρ k = N s,k /N b that is interpreted as a redundancy (measured by the number of channel uses per information bit). The rate of the k-th transmission attempt is given by R k = /ρ k and since they, in general, may change with k, we talk about variablerate transmission. If R k R, k we obtain the fixed-rate transmission considered before in [2] or [2]. The channel remains constant during transmission of the kth sub-codeword k =,..., K and the received signal is given by y k = γ k x k + η k () where η k is the vector of zero-mean complex, unitary-variance uncorrelated Gaussian variables (modelling noise) with unitary variance. The signal-to-noise ratio (SNR) γ k defines the channel state information (CSI) which is perfectly known/estimated at the receiver, but unknown to the transmitter. Although CSI does not change during the transmission of one sub-codeword, it varies independently from one sub-codeword to another. This corresponds to a practical scenario where subsequent subcodewords are sent in non-adjacent time instants, which are sufficiently separated to make the SNR independent for all practical purpose. The receiver can determine if the decoding error occurs using an outer error-detection code which introduces some overhead that we neglect for simplicity of the analysis. The channel gains γ are Rayleigh distributed, so the SNR is characterized the probability density function (pdf) p γ (γ) = exp( γ/γ) (2) γ where γ is the average SNR; the cumulative density function is given by F γ (γ) = exp( γ/γ). The coding scheme is revealed to the receiver, which, after the kth transmission, implements a maximum likelihood decoding using the observations ỹ k = [y,...,y k ]. The decoding is successful after k transmission attempts if the average accumulated mutual information at the receiver is larger than the overall transmission rate. For the incremental redundancy, this condition can be expressed as k N b k N C l N s,l > k s,l N s,l I k = k C l ρ l, (3) where C l = C(γ l ) is the mutual information per symbol that the receiver acquires after transmission with SNR γ l. Since the symbols x j have Gaussian distribution, C(γ) = log 2 ( + γ) and it is possible to find a code such that, when (3) is satisfied, the probability of decoding error goes to zero when N b [3] [4]. Obviously, for a fixed-rate HARQ (i.e., ρ l ρ = /R), condition (3) translates into k C(γ l) R [2]. The system-level implementation of the variable-rate HARQ described above deserves some comments. It may be assumed that each transmission contains only one sub-codeword. In such a case, the duration of transmission attempts must vary, which might be a valid approach for a single-user communication when the transmitter and the receiver can negotiate the transmission time for each sub-codeword. On the other hand, it may be a questionable strategy in multiuser communications, where sharing the requirement for a variable-length transmission with all the users is not practical. Even if it might be possible to assign the resources (time) independently of the varying transmission length, it would lead to the bandwidth loss (sub-codewords shorter than the assigned transmission time slot) or to collisions (sub-codewords longer than the assigned time slot). To avoid such a conceptual difficulty, we assume that the sub-codewords corresponding to different packets are gathered in frames that have a constant duration of N F symbols. Such an assumption, also used in [5], [6] allows us to deal with variable-length codewords to fill up the frame and corresponds to TDMA-type communication, where users are provided with a fixed transmission time (frame). The example is shown in Fig.. Before analyzing the proposed HARQ scheme we recall the lower and upper bounds on the achievable throughput. Namely, if CSI is perfectly known, it is possible to transmit at the rate R(γ) = C(γ) without any packet loss. This provides the maximum throughput achievable when transmitting with constant power, i.e., the ergodic capacity [3], [7] C = C(γ)p(γ)dγ. (4)

3 3 P 6 P 8 P P 5 P 4 N F P 3 P 2 P P 7 P 6 P 5 P 4 P 3 P 2 P γ 2 P 9 P 8 P 7 P 6 P 5 P 4 P 3 P 2 P γ 3 Figure. Example of the structure of three frames sent over channels with corresponding SNRs γ, γ 2, and γ 3 when delivering data packets denoted by P l, l =,...,. The sub-codewords corresponding the first, second, and third transmission attempts are shown, respectively, in blue, yellow, and green. When transmitting this frame with SNR γ the data packets P P 6 are not correctly decoded so a NACK is sent to the transmitter. The next frame carries thus six sub-codewords of length N s,2 and since N s, > N s,2, the empty space is filled with two sub-codewords of the length N s, corresponding of the packets P 7 and P 8. After the transmission of the second frame the receiver, again, is not able to decode the packets so six sub-codewords of length N s,3 ( packets P P 6 ) are sent together with two of length N s,2 (packets P 7 and P 8 ). Since the length of the sub-codeword in the second transmission attempt depends on the SNR in the first transmission, the length N s,2 for the packets -6 is not the same as for the packets 7-8. We note also that the residual time is filled with the sub-codeword corresponding to the packet P 9, P and the relative loss due to unshaded/unfilled space can be made arbitrarily small loading the frame with many sub-codewords. We also know from [2], that the throughput of a well designed HARQ, approaches C when we let the maximum allowed number of transmission K. If, on the other hand, we do not know the channel state and only one transmission is allowed (K = ), the maximum achievable throughput is calculated as [3], [8] { } ν(ρ) η = max (5) ρ ρ where ν(ρ) = Pr { C(γ) ρ < } = F γ ( 2 /ρ ) is the outage probability when transmitting the codewords with rate ρ. III. HARQ WITH ADAPTIVE INCREMENTAL REDUNDANCY (HARQ-AIR) On top of the conventional signalling between the transmitter and the receiver (ACK/NACK messages), we also habilitate the receiver, to send back to the transmitter, after each unsuccessful transmission attempt, the CSI which is entirely defined through the SNR γ k (for the kth transmission attempt). We want to exploit the knowledge of γ,...,γ k to adjust the number of channel uses in the kth transmission attempt of the same packet. In general, due to unavoidable processing and communication delays, the CSI at the moment of transmission is outdated, i.e., not the same as the CSI obtained from the receiver. If both CSIs were correlated, the transmitter might use the outdated one to predict the actual CSI. In our case, though, the CSIs are assumed independent which arguably represents the worst case from the point of view of usefulness of the CSI. In particular, such a fully outdated CSI cannot be used for the conventional adaptive modulation and coding, cf., [3] [9]. Since, at first, it may thus appear surprising that the outdated CSI can be useful for HARQ, consider a simple example to get an intuition why the knowledge of outdated CSI can γ be beneficial. Assume that K = 2 and then, after the first unsuccessful transmission attempt, the NACK sent by receiver is also accompanied by γ or C = C(γ ) so the transmitter also knows C ρ = ǫ < where ǫ. If ǫ is very small, according to (3), only a small additional value C 2 ρ 2 > ǫ is necessary to ensure a successful decoding. Although C 2 is random, we can easily see that knowing that ǫ decreases, ρ 2 may be also decreased (i.e. adapted using CSI C ) without changing the probability of losing the packet after the second (and the last) transmission. Since this also translates into a shorter transmission time N s,2, some time is left free within the frame for transmission of sub-codewords of other packets (that we assume continuously available). In general, the redundancy ρ k in the kth transmission attempt may depend on all the previous CSI γ, γ 2,..., γ k (or equivalently C,..., C k ) which are known at the transmitter thanks to the feedback we habilitated. But from (3), we know that the decoding in the k-th transmission will be successful if I k = I k +ρ k C k >. The decoding success/failure depends thus on I k that we know, and on C k, which is random and cannot be predicted from previous CSI C,..., C k (due to assumed independence of SNR in the block-fading channel model). Thus, we do not need to use C,..., C k as I k is the only parameter to be considered when adapting the redundancy ρ k [] via a scalar function ρ k = ρ k ( Ik ), k = 2,...,K (6) whose argument satisfies I k [, ) because the k-th transmission is necessary only if the k -th was unsuccessful. Our objective is to evaluate the gains of such an adaptive incremental redundancy (AIR), i.e, when adjusting the redundancy ρ k via (6), and compare it with the non-adaptive HARQ (where the redundancy does not depend on the previous CSI). The relevant performance criterion is the throughput [2], which, according to the reward-renewal theorem [], is the ratio between the expected number of correctly received bits N b and the expected number of channel uses N s required by the HARQ protocol to deliver the packet in up to K transmission attempts where η AIR = N b N s. (7) N b = N b ( f K ) (8) and f K is the probability of decoding failure after K transmissions (maximum number of attempts allowed). Let the space of the events c,..., c k that result in k consecutive NACK messages be defined as D NACK,k = {c,..., c k : I <,...,I k < } { = c,..., c k : c <, c 2 < I ρ ρ 2 (I ), (9)..., c k < I } k ρ k (I k )

4 4 where I k I k (c,...,c k ) = k c lρ l. Then f K = Pr { } C,...,C K D NACK,K = p C (c )... p C (c K )dc... dc K = c,...,c K D NACK,K ρ p C (c )dc I ρ 2 (I ) p C (c 2 )dc 2... I K ρ K (I K ) p C (c K )dc K () where p C (c) is the pdf of C(γ). I Note that the integration limit k ρ k (I k ) for the kth inner integral depends on c,..., c k via, thus, in general, a multidimensional integration is required to calculate (). The expected number of channel uses required in (7) is given by K N s = N s,k, () k= where N s,k the expected number of channel uses in the kth transmission attempt, is obtained averaging the number of channel uses in the kth transmission, ρ k (I k )N b, over all the events that provoked the kth transmission, i.e., averaging is done over the space D NACK,k. This leads to N s,k = N b ρ k (I k ) p C (c )... = ρ c,...,c k D NACK,k p C (c )dc I ρ 2 (I ) I k 2 ρ k (I k 2 ) p C (c k )dc... dc k (2) p C (c 2 )dc 2... ρ k (I k )p C (c k )dc k (3) To evaluate the throughput (7), we need to calculate (3) and (). While this can be done relatively simply via multidimensional integrals if ρ k (I k ) are given, the problem of optimizing the functions ρ k (I k ) seems to be much more difficult. Therefore, to proceed, we opt for simplifications and will optimize a particular class of these functions. A sound a priori requirement on ρ k (I k ) is that it decreases monotonically with I k. The rational for this is that for a larger mutual information I k at the receiver, shorter sub-codewords (less redundancy) are needed to decode the packet. In the limit, when I k approaches unity, the redundancy ρ k (I k ) should approach zero. Similar considerations were also outlined in [7] and agree with the following heuristic choice of ρ k (I k ) ρ k (I k ) = ( I k ) ρ k, k = 2,...,K. (4) Now, one parameter ρ k defines the whole function ρ k (I k ) whose form, as we will see, matches well the problem we solve and simplifies the expression for the throughput. Even with this simple, heuristic choice of ρ k (I k ) significant gains over non-adaptive HARQ can be yield, provided we select appropriately the parameters ρ k. Of course, an optimal function ρ k (I k ), i.e., not restricted to the linear form (4), can only improve upon the results we present. The immediate consequence of using (4) is that the upper limits in () become independent of the integration variables, which yields f K = K ρ l p C (c l )dc l = K ν( ρ l ) (5) where ν(ρ), defined after (5), has an analytical form. To calculate (3), we obtain from (3) a recursive relationship I k = I k c k ρ k ( I k ) = ( I k ) ( c k ρ k ), (6) letting us factorize the function (4) as k ρ k (I k ) = ρ k ( c l ρ l ). (7) which used further in (3) yields N s,k N b = ρ k k ρ l ( c l ρ l )p C (c l )dc l k = ρ k ξ( ρ l ) (8) where ξ(ρ) = ν(ρ) ρ h(ρ) and h(ρ) = /ρ c p C (c)dc can be obtained via numerical integration. The throughput can be now expressed as K k= η AIR = ν( ρ k) K k= ρ k k ξ( ρ l), (9) where, for notational convenience, we defined ρ ρ ; note that, unlike ρ 2,..., ρ K which characterize the redundancy adaptation functions defined in (4), ρ corresponds the rate of the first transmission. We denote the maximum achievable throughput as ˆη AIR = max ρ,..., ρ K {η AIR }. It is quite interesting to note that replacing ξ( ρ k ) with ν( ρ k ) in the denominator of (9), we obtain the expression for the throughput of HARQ-I [6] (i.e., when the repetition coding is used instead of incremental redundancy) and whose maximization yields nothing but η, cf. (5). IV. OPTIMIZATION AND NUMERICAL RESULTS The throughput (9) is a non-linear and possibly a nonconvex function of the optimization variables ρ k, k =,...,K. The gradient-based optimization methods are unreliable in such a case and we opt for an approximation which turns out to be very simple and robust. First, we start reformulating the optimization problem into its equivalent form ˆη AIR = max X max ρ,..., ρ K QK ξ( ρ l)=x K k= ν( ρ k) K k= ρ k k ξ( ρ l). (2)

5 5 η K = 2, AIR K = 3, AIR K = 4, AIR K = 2, VR K = 4, VR K = 8, VR 2 η, K = C, K γ[db] 25 3 Figure 2. Throughput: HARQ-AIR when K = 2, 3,4; the throughput of non-adaptive, variable-rate HARQ is also shown for comparison. ρk C 2.5 K = 2 K = 3 K = γ Figure 3. The values of ρ k, k =,..., K for K = 2, 3,4. For a given K, ρ k < ρ k+, e.g, ρ < ρ 2 < ρ 3 < ρ 4. where the outer, scalar maximization provides the global optimum using an inner, constrained optimization. Next, we take advantage of the fact that, in the vicinity of the optimum, the outage probability f K = K k= ν( ρ k) is very small (in practice, it will be at the order of.), so the most significant improvement of the throughput is yield decreasing the value of the denominator (2). Thus, instead of maximizing jointly the fraction in (2) we will minimize its denominator and use the results in the numerator ˆη AIR max X K k= ν( ρ k (X) ) V K (X) (2) where ρ k (X), k =,...,K are the solutions of the constrained optimization { K } k V K (X) = ρ k ξ( ρ l ). (22) min ρ,..., ρ K QK ξ( ρ l)=x k= The main reason for using the approximation (2) is that (22) can be reformulated as { K k V K (X) = min ρ k ξ( ρ l ) ρ K { = min ρ K ρ K min ρ,..., ρ K Q K k= ξ( ρ k)= X ξ( ρ K ) X ξ( ρ K ) + V K k= ( X ξ( ρ K ) K + ρ K )} ξ( ρ l ) } (23) which lets us to solve it recursively, which is characteristic of the dynamic programming [2]. To implement (23), we discretize X and for each of the discrete values, a -D search is carried out. Starting with V (X) = ξ (X) = ρ, we recursively repeat the optimization (K times) to obtain V 2 (X),...,V K (X). This is, of course, much simpler that K-dimensional optimization required in (2). We show, in Fig. 2, the throughput of HARQ-AIR for K = 2, 3, 4 with the lower bound η given by (5) (i.e., when K = ) and the upper bound C by (4). Observe that for high SNR the throughput of HARQ-AIR gets very close to the ergodic capacity when K = 4 (this is notable as we recall that the transmitter has no knowledge of the instantaneous CSI). The gains are less pronounced for lower SNR where we expect the power adaptation suggested in [] to improve significantly the throughput. We also show the throughput of the variable-rate nonadaptive HARQ scheme (HARQ-VR) shown previously in [6], which is based on incremental redundancy transmission but where the rates are optimized for the given average SNR and do not depend on the feedback during the transmission attempts. Note that values of K are not the same in both HARQ schemes and are chosen to cover well the throughput gap between η and C so comparison can be done: for example, for high average SNR, HARQ-AIR with K = 3 performs equally well as HARQ-VR with K = 8. The optimal values ρ k were always monotonically increasing with k (i.e., ρ k ρ k ) and we show them in Fig. 3 after convenient scaling by C. They allow one to reproduce the results from Fig. 2 and provide an interpretation of HARQ- AIR behaviour. From Fig. 3 we note that while ρ K C >, we also have ρ k C < for k =,...,K. This means that during the transmission attempts k =,...,K the > ρ > C are larger than the ergodic capacity. The rate in the last transmission, R K = ρ K(I K ) depends on the mutual information accumulated at the receiver, i.e., on I K. In particular, when I K is close to rates R k = ρ k (I k ) zero, R K ρ K < C. So, the HARQ process starts with high transmission rates (which translate into shorter transmission) and in the case the SNRs γ,..., γ K are small and the decoding fails, the rate of the final transmission K is set much lower to increase the probability of successful decoding. This transmission strategy of HARQ-AIR is clearly different from what was observed in HARQ-VR [6], where the first

6 6 Kavg K = 2, VR K = 4, VR K = 8, VR K = 2, AIR K = 3, AIR K = 4, AIR γ Figure 4. Average number of tranmission attempts K avg for HARQ-AIR and HARQ-VR. transmission s rate was always close to C, and the rates R k, k = 2,...,K higher than C. The effect of using high rates in the transmissions k =,...,K must also affect the average number of transmissions k K K avg = + k= K f k = + k= ν( ρ k ) (24) which we show in Fig. 4 for HARQ-AIR comparing it to the average transmission number required by HARQ-VR. Indeed, for the same K, the average number of transmission attempts of HARQ-AIR is greater than when using HARQ-VR. We can see that in the range of SNR we studied, K avg > K, thus there is a strong bias towards the decoding in the last transmission attempt. Despite this particular behaviour, for high average SNR, HARQ-AIR outperforms HARQ-VR in terms of the throughput and the average delay, e.g., comparing HARQ-AIR with K = 4 and HARQ-VR with K = 8 we gain for all γ > 5[dB]. V. CONCLUSIONS In this work we analyzed the HARQ transmission with incremental redundancy where transmission attempts are carried over independently varying channels. We propose to enhance HARQ signalling allowing the receiver to inform the transmitter about the experienced CSI. We demonstrate that, although the CSI is outdated (and cannot be used for the conventional adaptive modulation and coding), it can be exploited by the HARQ protocol to adapt the amount of redundancy used in the subsequent transmission attempts. We formulate the general expression for the throughput when the redundancy is adapted to the outdated CSI. Further, we propose a simple adaptation policy leading to tractable expression for the throughput, and we use dynamic programming approach to optimize its parameters. The proposed HARQ with adaptive incremental redundancy (HARQ-AIR) yields notable improvement of the throughput when compared to non-adaptive HARQ. REFERENCES [] M. Zorzi and R. 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