Future Network and MobileSummit 2010 Conference Proceedings Paul Cunningham and Miriam Cunningham (Eds) IIMC International Information Management Corporation, 2010 ISBN: 978-1-905824-16-8
Link Adaptation with Retransmissions for Non-Binary LDPC Codes Stephan Pfletschinger, Monica Navarro Centre Tecnlol`ogic de Telecomunicacions de Catalunya (CTTC) Av. Carl Friedrich Gauss 7, 08860 Castelldefels, Spain Tel: +34 93 6452927, Email:
[email protected] Abstract: We develop a link adaptation scheme based on non-binary LDPC codes for two scenarios: a frequency-selective quasi-static channel and a time-selective flat fading channel with very limited channel state information at the transmitter. For both cases, we present a unified characterization of the coding performance including puncturing and repetition coding which is based on the accumulated mutual information per code symbol. In the following, we apply this description for the adaptation of modulation and code rate. For frequency-selective channels, the modulation is adapted per subchannel while the same code is applied to all subchannels of a link. For fading channels with limited channel knowledge at the receiver, basically the same characterization of the code performance is applied, but in this case the knowledge of the accumulated mutual information is obtained at the receiver and fed back to the transmitter in order to adapt the size of the retransmission units. This approach leads to an adaptive retransmission scheme which maintains the throughput to a great extent although there is no reliable channel state information available. Both schemes can be combined in a straightforward manner and may be embedded into a multi-user multi-carrier system. Keywords: Adaptive coding and modulation (ACM), non-binary LDPC codes, hybrid automatic repeat request (HARQ)
1.
Introduction
In modern wireless systems, link adaptation to the highly variable channel conditions is an essential ingredient of any transmission scheme. The main difficulties for link adaptation stem from the frequency and time selectivity of the channel, in addition to the dynamic allocation of transmission units by an adaptive, channel-aware scheduler. In this paper, we present a link adaptation scheme which includes frequency-selective adaptive modulation and adaptive retransmissions in a hybrid automatic repeat request (HARQ) protocol. The transmission scheme is based on non-binary LDPC coding, whose noteworthy performance has been shown recently [1, 2]. In the following, we show how adaptive coding and modulation (ACM) can be efficiently employed over frequency-selective channels and how it can be combined with adaptive HARQ based on incremental redundancy. The presented algorithm for frequency-selective adaptation is based on the performance description of the channel coding scheme by the mutual information (MI), thus exploiting the benefits of the nonbinary channel code to full extent. On the other hand, if due to the lack of accurate channel state information at the transmitter (CSIT), exact adaptation becomes difficult, the importance of a retransmission scheme with HARQ increases. In this context, we present a simple method to adapt modulation and code rate both at the initial as well as at all retransmissions which achieves a surprisingly high throughput at very moderate delay. c The authors Copyright
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Figure 1: Block diagram of the downlink of an adaptive OFDMA system
2.
System Model and Characterization of the Channel Code
2.1 System model We consider the modulation, coding and retransmission scheme of one link, which may be part of a multi-carrier multi-user system as depicted in Fig. 1. Although ACM and HARQ only consider the link of a single user, we must take into account some of the key features of the multi-user system, such as a given packet length and the variability of the transmission unit, such that the link adaptation scheme can be seamlessly embedded into it. For adaptation, we employ modulations with R1 ∈ {1, 2, 4, 6} bits per symbol, namely BPSK, QPSK, 16-QAM and 64-QAM. As channel codes, we use the non-binary LDPC codes defined in the DAVINCI project (www.ict-davinci-codes.eu), which have shown superior performance compared to the codes employed in current IMT-Advanced candidate systems [1, 2]. These codes are defined over GF (64), hence one code symbol corresponds to 6 bits and application to 64-QAM is particularly simple. In the following, we employ a mother code with rate Rm = 21 and K = 240 message symbols (i.e. 1440 bits), which is punctured in steps of 12 GF(64) symbols. 2.2 Description of coding performance For the design of a link adaptation scheme which includes channel coding, a sufficiently accurate description of the error rate as a function of the channel characteristics is required. While for AWGN or flat fading channels, the relationship between word error rate (WER) and channel SNR can be obtained by simulations, for a frequencyadaptive system as depicted in Fig. 1 this is not possible since different modulations might be applied to the same codeword. In this context, an important observation has been discovered recently by several authors [3, 4, 5]: the word or bit error rate of a coded modulation scheme depends nearly exclusively on the mutual information per code symbol. In other words, it is (approximately) sufficient to know the MI of the equivalent channel from the encoder output to the decoder input in order to estimate the error rate. This observation also holds for the non-binary LDPC codes, which we apply in the following and is illustrated in Fig. 2 (a): the WER depends mainly on the code rate, while it is nearly independent of the modulation. The curves in Fig. 2 (a) have been obtained by simulations over the AWGN channel, but with a Rayleigh fading channel, we obtained very similar results. c The authors Copyright
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Figure 2: (a) Word error rate as a function of the mutual information per code symbol for different code rates and modulations. (b) Required accumulated mutual information as a function of the word length While this has been observed first for binary LDPC and for turbo codes, we can apply this idea to non-binary codes as well. Instead of using the MI per coded bit, we apply the MI per coded symbol, which has the additional benefit that it is simpler to compute and is slightly higher than the corresponding bit-wise information. For the AWGN channel, the MI per code symbol is given by the symmetric capacity with 2m -QAM (see e.g. [6, 7]) " # X |w + x − z|2 − |w|2 1 X E ld exp − (1) Cm (γ) = m − m 2 x∈X w N0 z∈X where w ∼ CN (0, N0 ) is the complex-valued, circularly symmetric Gaussian noise, X denotes the QAM constellation and γ = ES /N0 is the SNR. Since a code symbol in GF (64) corresponds to 6 bits, the MI per code symbol is given by Ic (γ) =
6 · Cm (γ) m
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For an adaptive scheme with a pre-defined WER constraint, we easily obtain a relation between the MI per code symbol and the required code rate. In the following, we apply the values obtained for QPSK since for this modulation the highest MI is required to achieve a certain rate. This has been done for a target WER of 0.01 and 20 codes rates R2 = 40−p , with p = 0, 1, . . . , 19. For each code rate, the associated word length is given by N = K/R2 = 480 − 12 · p and the accumulated mutual information (ACMI) by Iw = N · Ic (The same parameter has been used in the context of HARQ by Cheng [8]). We hence obtain a direct relation between the word length and the required ACMI to achieve the given target error rate. This is illustrated in Fig. 2 (b) and is the basis for the adaptation algorithms for frequency-selective link adaptation with and without retransmissions.
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3.
Frequency-Selective Adaptation
The characterization of the coding performance by the MI per code symbol facilitates the design of a frequency-selective ACM scheme, also known as bit-loading algorithm. With constant power allocation, this leads to even simpler schemes than the classical bit-loading algorithms for uncoded QAM and has been exploited by Stiglmayr and independently by Li and Ryan [4, 5]. In this section, we present a variation of Stiglmayr’s algorithm which is especially suited for combination with IR-HARQ, while maintaining the low computational complexity. In a first step, the modulation is adapted per subchannel such that the minimum MI per code symbol exceeds the required value for the mother code and QPSK modulation, here Ic,m = 3.53, with corresponds to the SNR γm = 1.33 dB. This ensures that the overall code rate does not fall below the mother code rate. Now, for each subchannel the MI per code symbol Ic is given and in a second step, the ACMI and the word length are incremented per subchannel: Iw := Iw + Ic(i) · N (i) N := N + N (i)
(3)
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where Ic and N (i) denote the MI and the number of code symbols of subchannel i, respectively. This process results in an adaptation trajectory, which is depicted in Fig. 2 (b) and continues until the required ACMI fw (N ) is reached. The condition (i) Ic > Ic,m , enforced by the adaptive modulation step, guarantees that this process terminates before the word length reaches 2K, the length of the mother code. In Fig. 3, the achieved rate and obtained word error probability of this algorithm and that of Stiglmayr’s MI-ACM algorithm [4] are shown for a frequency-selective Rayleigh fading channel. Throughput and error rates are comparable, with a slight advantage for the ACMI algorithm for high SNR which is due to a saturation effect of the highest allowed code rate in the MI-ACM algorithm. The main advantage of the ACMI vs. the MI-ACM algorithm is that it allows to start transmission of a partial codeword and its seamless combination with HARQ.
4.
Link Adaptation with Adaptive Retransmissions
4.1 Unified characterization of puncturing and repetition coding The idea of increasing the word length until the necessary ACMI is reached can be applied in a very similar way to ACM with retransmissions, where also the length of the retransmission unit (RTU) can be adapted. In the following, we employ HARQ with cyclic incremental redundancy, which is superior to simple packet repetition (Chase combining), provided a set of rate-compatible codes is available. In this scheme, the modulation and code rate for the initial transmission are adapted by rate-compatible puncturing in the same way as described above. If decoding fails after the first transmission, in the second transmission the transmitter sends additional parity symbols and, depending on the initial code rate and the RTU size, repeats code symbols from the first transmission. This is called cyclic incremental redundancy since, after running out of parity symbols, the transmitter starts repeating systematic symbols. The implementation of this scheme is straightforward: denote by cm = (c0 , c1 , . . . , c2K−1 ) the
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Figure 3: (a) Achieved rate with ACMI and MI-ACM algorithms. (b) Word error rates for both algorithms. mother code word. Then, the transmitted code symbols are given by ct = cπ(0) , cπ(1) , . . . , cπ(N (ν) −1) with π(i) = mod (i0 + i, 2K) ,
for i = 0, 1, . . . , N (ν) − 1
Here, i0 denotes the index to the next code symbol to be transmitted, which is i0 = 0 for the initial transmission, and N (ν) is the number of transmitted code symbols at the ν-th transmission. In order to apply the same procedure as before, we need to extend the description of the coding performance to cyclic repetitions. We can find a relation between the code rate R2 and the MI Ic for repetitions of entire codewords by noting that in these cases the received SNR sums up. If for the mother code, an SNR of γm is required, for an ν-fold repetition, the necessary SNR is γ = ν1 γm . The √ symmetric capacity for an AWGN channel with QPSK modulation is given by 2J(2 2γ), with the J-function defined in [9]. The MI per GF (64) code symbol for ν repetitions is then r p γm Ic = 6J 2 = 6J 8γm R2 (4) ν 1 = 2ν . where we used the word length N = ν · 2K, which gives the code rate R2 = K N This holds strictly only for integer values of ν, but by allowing for real values with ν ≥ 1, we obtain an interpolation for all code rates R2 ∈ (0, 0.5]. This interpolation for repetition coding is depicted in Fig. 4 and allows to extend the adaptation trajectory of Fig. 2 (b) to any value N ≥ K.
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Figure 4: Unified description of performance of punctured and repeated codes. (a) Code rate as a function of the mutual information per code symbol. (b) Accumulated mutual information as a function of the word length. 4.2 Adaptation of the RTU size HARQ is essential in situations when CSIT is not reliable, which is the typical case in broadband wireless systems. Even if the channel could be estimated perfectly well at the receiver, there is an intrinsic feedback delay which in conjunction with dynamic scheduling makes it impossible to dispose of accurate CSI at the transmitter. In the absence of exact CSIT, the only possibility to maintain a comparable throughput to perfect channel knowledge is to introduce a retransmission protocol which combines the information and energy of all retransmission. Therefore, we do not consider type I (H)ARQ which discards information of unsuccessful transmissions. The link adaptation scheme with adaptive RTU is based on the following premises • the transmitter never obtains accurate CSI • the receiver, in turn, can estimate the channel accurately and obtain accurate CSI of all past transmissions The adaptation trajectory hence is built successively: while the transmitter at most has a rough estimate of the MI per code symbol Iˆc , the receiver obtains the true value Ic and updates the values for Iw and N according to (3). In case of decoding error, the value of Iw is fed back to the transmitter, which again uses his inaccurate estimate Iˆc for the next retransmission. Hence, the decoding trajectory, which is known at the transmitter side, is built by exact sections except the last one. 4.3 Simulation results for two adaptive schemes In this section, we consider the rather extreme case that the channel is i.i.d. block Rayleigh fading and the only CSI at the transmitter is the average SNR. In other words, the channel coefficient h is assumed constant during each slot and independent for the c The authors Copyright
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Figure 5: Throughput and delay for HARQ with adaptive RTU size, in comparison to ACM with perfect CSIT without retransmissions. next slot, i.e. h is i.i.d. complex normal distributed. Consequently, the instantaneous SNR γ is exponentially distributed with pdf p(γ) = γ1¯ exp (−γ/¯ γ ) for γ ≥ 0. 1. With these assumptions, the transmitter estimates the MI per code symbol based γ ) and uses this value for the adaptation on the average SNR γ¯ , i.e. Iˆc = m6 Cm (¯ of the codeword length, in the first as well as in all retransmissions. If decoding is not successful, retransmissions are triggered until the codeword is decoded successfully, i.e. there is no residual error. The obtained throughput (in correct bits per channel use) and delay (given as the average number of retransmissions) is depicted in Fig. 5. 2. Another, simpler, strategy is to fix the modulation to the natural constellation size of the code symbol: since the employed channel code is defined over GF (64), we use a 64-QAM and adapt only the code rate. This strategy naturally cannot compete with the fully adaptive version, but for medium to high SNR it achieves a throughput which is close to that of the first scheme while the average delay is even lower for average SNRs higher than 5 dB. The comparison to the case of perfect CSIT without retransmissions shows that HARQ allows to nearly maintain the same throughput. This is a very positive observation since without retransmissions the degradation of a system due to the lack of CSIT is significant. It is also worth noting that this is achieved with a very low number of retransmissions.
5.
Conclusions
In this paper, we used the observation that the error rate of non-binary LDPC codes may be characterized by the mutual information per code symbol in order to devise adaptive schemes for frequency-selective and fading channels. Both schemes can be c The authors Copyright
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combined straightforwardly and embedded into a highly adaptive multi-user system. The presented algorithm for frequency-selective adaptation has the advantage that it allows to transmit an encoded packet in various slots, irrespective of the size of the transmission unit and permits a seamless combination with HARQ based on incremental redundancy. Simulation results showed that with adaptive retransmissions, the lack of CSI at the transmitter causes a very moderate throughput reduction.
Acknowledgment This work is supported by INFSCO-ICT-216203 DAVINCI (Design And Versatile Implementation of Non-binary wireless Communications based on Innovative LDPC Codes), funded by the European Commission under the Seventh Framework Programme (FP7). It has also been partially funded by the Spanish Government under grant TEC200806327-C03-03/TEC.
References [1] S. Pfletschinger, A. Mourad, E. L´opez, D. Declercq, and G. Bacci, “Performance evaluation of non-binary LDPC codes,” in ICT-MobileSummit, (Santander, Spain), June 2009. [2] I. Gutierrez, G. Bacci, J. Bas, A. Bourdoux, H. Gierszal, A. Mourad, and S. Pfletschinger, “DAVINCI non-binary LDPC codes: Performance and complexity assessment,” in Future Network & Mobile Summit, (Florence, Italy), June 2010. [3] S. ten Brink, “Iterative decoding trajectories of parallel concatenated codes,” in 3rd IEEE/ITG Conference on Source and Channel Coding, (Munich, Germany), Jan. 2000. [4] S. Stiglmayr, M. Bossert, and E. Costa, “Adaptive coding and modulation in OFDM systems using BICM and rate-compatible punctured codes,” in European Wireless, (Paris, France), April 2007. [5] Y. Li and W. E. Ryan, “Mutual-information-based adaptive bit-loading algorithms for LDPC-coded OFDM,” IEEE Trans. Wireless Commun., vol. 6, no. 5, pp. 1670– 1680, 2007. [6] G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Trans. Inform. Theory, vol. 28, pp. 55–67, Jan. 1982. [7] U. Wachsmann, R. F. H. Fischer, and J. B. Huber, “Multilevel codes: Theoretical concepts and practical design rules,” IEEE Trans. Inform. Theory, vol. 45, pp. 1361– 1391, July 1999. [8] J.-F. T. Cheng, “Coding performance of hybrid ARQ schemes,” IEEE Trans. Commun., vol. 54, pp. 1017–1029, June 2006. [9] S. ten Brink, Design of Concatenated Coding Schemes based on Iterative Decoding Convergence. PhD thesis, Institut f¨ ur Nachrichten¨ ubertragung, University of Stuttgart, April 2001. Aachen: Shaker Verlag.
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