p J Data bits P1 P2 P3 P4 P5 P6 Parity bits C2 Fig. 3. p p p p p p C9 p p p P7 P8 P9 Code structure of RC-LDPC codes. the truncated parity blocks, hig

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1 A Study on Hybrid-ARQ System with Blind Estimation of RC-LDPC Codes Mami Tsuji and Tetsuo Tsujioka Graduate School of Engineering, Osaka City University , Sugimoto, Sumiyoshi-ku, Osaka, Japan tsuji@comm.info.eng.osaka-cu.ac.jp, tsujioka@info.eng.osaka-cu.ac.jp Abstract -Compatible LDPC (RC-LDPC) codes have much attention because of their flexibility and simplicity to provide various performances by changing the number of parity blocks. The authors have considered several blind rate estimation techniques for RC-LDPC codes based on likelihood information of the sum-product decoders and have evaluated their performance. However, in practical systems, reliable transmission must be required. Then, we have to consider how to apply the blind estimation techniques to re-transmission systems. In this paper, the authors focus on a Type-2 Hybrid-ARQ system and propose a novel reliable re-transmission system with blind rate estimation. Some results of numerical analysis are reported. By computer simulations, we evaluated throughput when RC-LDPC codes with information bits of 1536 and 3072 are employed. It can be obtained that the proposed re-transmission system has good performance compared to conventional Hybrid-ARQ schemes. Freq-3 Freq-1 TX Freq-2 RX Even when a transmitter TX sends information to a receiver RX, the TX can estimate the channel status like SNR by detecting adjacent frequency simultaneously. According to estimated SNR, the TX can select error correcting code rate to satisfy required BER performance. Thus, an adaptive code selection system. If some errors occur, our efficient Hybrid ARQ approach recovers the errors. Fig. 1. Concept of adaptive rate selection system. I. INTRODUCTION -compatible LDPC (RC-LDPC) codes are suitable for providing various code rates to be flexible to error tolerance by changing the number of parity blocks [1], [2]. RC-LDPC codes have been investigated by many researchers [3], [4]. The authors have considered several blind rate estimation techniques for RC-LDPC codes based on likelihood information calculated by values of bit nodes of the sum-product decoders and have reported its superiority [5], [6]. The blind rate estimation is used in a adaptive rate selection system. Figure 1 shows the concept of adaptive rate selection system. To optimize performance in fast time-vary channels, modulation/error correcting codes should be selected at a transmitter. Even when the transmitter is sending information to a receiver, the receiver can estimate channel status like SNR by monitoring adjacent frequencies simultaneously. According to estimated SNR, the transmitter can select code rate to satisfy the required BER performance as shown in Fig. 2. However, high reliable transmission is required in the practical communication systems. So, an efficient re-transmission control scheme combined with the blind rate estimation technique is expected to be developed. In this paper, the authors focus on a Type-2 Hybrid-ARQ (HARQ) system as re-transmission control having superior throughput characteristics, and propose an HARQ system with blind estimation of RC-LDPC codes. Bit Error Low Code (Good BER Performance) Required BER Fig. 2. High Code (Poor BER Performance) Adaptive Selection SNR Adaptive rate selection. II. RC-LDPC CODES AND BLIND ESTIMATION TECHNIQUES A. Codeword of RC-LDPC Codes We show the code structure of RC-LDPC codes in Fig. 3. The longest code is C 9 in this example. The lengths of data and parity part are given by Jp and 9p, both of them are multiple value of p, where J and p are code parameters. We can divide parity part of C 9 into 9 blocks of p bits and can truncate some parity blocks to construct shorter codewords. Increasing 46

2 p J Data bits P1 P2 P3 P4 P5 P6 Parity bits C2 Fig. 3. p p p p p p C9 p p p P7 P8 P9 Code structure of RC-LDPC codes. the truncated parity blocks, high code rate is obtained instead of performance degradation. After that, we can get shorter version of codes, C 1 C 8 from C 9. From the different point of view, we can say that a longer codeword of C 2 is constructed by adding a parity block P2 to a shorter codeword of C 1, and so on. In other words, if transmission of codeword of C 1 failed, there is high possibility of decoding as C 2 having high error tolerance at a receiver by sending just only a parity block P2. In this paper, the proposed HARQ utilizes this characteristic of the RC-LDPC codes. B. Blind Estimation Technique In a sum-product decoder for 2-ary LDPC codes, each value of bit nodes converges to +1 or 1 by increasing of the number of message passing times, and the decoding process goes to be stable. If received signal has uncorrectable errors due to low SNR, by arranging the codeword structure in reverse order depicted in Figs. 5 and 6, each value of the bit nodes does not converge and it goes to around 0. If we pay our attention to this property and use likelihood information value of γ = all bit nodes f( z i ), comparing likelihood information values when a receiver tries to decode received signal as C 1 to C 9 as possible sent codes, we can perform a blind rate estimation. Here, z i is a value of bit node i, f( ) is a monotonous function for evaluation. The blind rate estimation based on likelihood information of decoders is shown in Fig. 4. At a transmitter, monitoring the channel status, an encoder is selected according to it and an encoded message is transmitted to a receiver. Even if the receiver does not know which encoder is selected, it can estimate the used encoder or code rate by comparing likelihood information values of decoders. III. PROPOSED HYBRID-ARQ SCHEME In this paper, we propose HARQ system combined with blind rate estimation. At the re-transmission phase, only parity blocks for being longer codes are sent as additional information. It provides good blind rate estimation even if the transmitter changes code rate at random. We show an example of block diagram of transmitter and receiver for the proposed HARQ system in Fig. 5. The transmission sequence is shown in Fig. 6. An ARQ frame at the 1st-phase is encoded using flexibly selected code from C 1 to C 6 according to channel status. In this example, at the 1st transmission, Block-1 Block-F in the ARQ frame are encoded to C 1, C 3, C 6,, C 1, respectively. Here, F is the ARQ frame size in blocks (codewords). The HARQ is finished when all of the rate estimation of blocks in the ARQ frame are successful and a frame ACK is replied via a reverse channel. If at least one rate estimation fails, a frame NAK is returned and it goes to the next re-transmission phase. We suppose that no error occurs in the reverse channel. At the 2nd-phase, the transmitter sends a frame consisting of the F parity blocks for being longer codewords, where each additional parity is concatenated with the previously received block and the receiver construct one rank longer codeword sequence to be estimatedand decoded. Similarly, the transmitter continues to repeat the re-transmission of the parity whether the transmission of a message succeeds or till the number of transmission times reaches a limit value of k max. If it reaches the longest (lowest rate) code for encoding, zero gap is re-transmitted as a parity block in this paper s evaluation. We show the example of block diagram of transmitter and receiver for the proposed HARQ in Fig. 5. TX Buffer-1 is a buffer to retransmit from 1st-phase when transmission number of times reaches the limit value of k max. Unlike Fig. 4, the transmitter encodes it by the longest codeword and it stores the parity in TX Buffer-2 that we did not transmit in 1st-phase and reuses it. At the receiver, received signal is stored into different RX Buffers every transmission phase and the signals in RX Buffers are used for forward rate estimation processing while rebuilding the received signal sequence by deinterleaving by deinterleaver. The data after decoding are stored away once by RX Decoded Buffer, and it is output as the received data after all blocking of the frame succeeds. If there is an error at 1 block in a frame, we destroy the signal in RX Decoded Buffer and restart re-transmission phase from the first parity block if necessary. We showed Fig. 7 about processing of deinterleaver. In the 1st-phase, we advance decoding and a rate estimation processing with the same way in Reference[5], [6]. After the 2nd-phase, we push forward a rate estimation by 1 block consecutively by inserting the parity information that we received into the head of the estimation buffer. It is important that the parity is located the head of codeword. The short codeword is a subset of the long codeword. Supposing the parity was located at the end of codeword. The performance of rate estimation would be degraded due to narrow margin of likelihood information of decoders for possible sent codes. Therefore we find a large difference of the likelihood value at the each decoder by arranging it in inverse order like a figure. IV. NUMERICAL ANALYSIS The transmitter can make a code selection according to the channel status like SNR. Suppose we use h codes of from C 1 to C h for encoding at the first transmission. The ARQ frame includes F blocks (codewords). In the frame, there are l 1 blocks of C 1, l 2 blocks of C 2,, l h blocks of C h, h respectively. Here, l i = F is satisfied. The average probability of distribution of l 1, l 2,,l h in the frame is given by multinomial distribution P usage (l 1, l 2,, l h ) = F! h l i! h q li i, (1) where q i is the average probability of code usage of C i. 47

3 Transmitter Receiver input ENC-1 ENC-2 AWGN channel DEC-1 DEC-2 output ENC-6 DEC-6 selector Likelihood estimator Fig. 4. Blind estimation based on likelihood information of decoders. input Transmitter SEL selector ENC-6 ARQ controller 1st-trans SEP SEL Unsent Re-trans parity parity Receiver Frame ACK/NAK ARQ controller AWGN channel SW control DE INT DEC-1 DEC-2 DEC-6 output Clear if failed Likelihood estimator TX Buffer-1 (for data) TX Buffer-2 (for parity) RX Buffer-1 RX Buffer-2 (for 1st trans) (for parity) Estimation Error RX Decoded Buffer Block Code# 1 2 C3 3 C6 4 C5 F Parity P2 P3 P4 P5 P6 P4 P5 P P P2 P3 P4 P5 P6 Fig. 5. Example of block diagram of transmitter and receiver for the proposed HARQ. # $! " " $ Fig. 6. Transmission sequence of the proposed HARQ. 48

4 We define the probability of rate estimation error when C i is transmitted as P EE (C i ). The probability that the rates of all blocks in the frame are estimated perfectly after k transmissions (it requires k transmissions until success) is given by P success (k) = h {1 P EE (C k+i 1 )} li. (2) When we assume code length of C i, L codeword (C i ), the number of the bits transmitted by the first transmission of a frame message is given as follows. L phase (1) = L codeword (C i )l i. (3) Assuming parity length of C i, L parity (C i ), the number of the bits transmitted from the second to the kth transmission is given as follows. L phase (k) = {L parity (C i+k 1 ) L parity (C i+k 2 )}l i. (4) The probability to succeed at the kth transmission is given as follows k 1 P phase (k) = P success (k) {1 P success (i)}. (5) The probability that the tramsmitter continues failing in until the kth transmission is given by. P fail (k) = k {1 P success (i)}. (6) Therefore, the average number of transmission bits, D, for all patterns when the transmitter send a message in the k max times (maximum transmission times) is given as follows. D = P usage (l 1,, l h ) l 1+ +l h =F [ { k max P fail (k max )L phase (k max )+ P phase (k) k=1 }] k L phase (l). k In addition, a relation of l=1 L phase(l) = h L codeword(c i+k 1 )l i is concluded. When we define data length of C i is represented as L data (C i ), the average number of the information data bits, K, is given by K = F l=1 (7) q i L data (C i ). (8) Therefore, the throughput η is found by η = K D [1 P fail(k max )]. (9) Estimation Process for 1st-Phase Starting position for estimation Received sequence at phase-1 P1 Data-1 Estimation Process after 2nd-Phase Received sequence at phase-2 P2 P4 P7 P2 P1 Data-1 C2 estimated successfully P4 Fig. 7. C4 estimated successfully Movement of deinterleaver. TABLE I CODE PARAMETERS. P7 P6 P5 P4 C7 estimated successfully. Zero gap is retransmitted, if it reaches lowest rate. L data = 1536 L data = 3072 Code L codeword L parity L codeword L parity C C C C C C C C C V. PERFORMANCE EVALUATION By computer simulations, we evaluate the performance of the adaptive rate selection of RC-LDPC codes for HARQ system. We construct RC-LDPC codes based on the (J, L) QC codes[6] with the parameters of (p, F ) = (192, 5) or (384, 10), J = 8 and L = 9. The code parameters are shown in Table I. Set the number of selectable codes h = 6 and the maximum iteration of Sum-Product decoding of I max = 2. BPSK modulation and the AWGN channel are assumed in this evaluation. As a conventional HARQ, typical Type-2 HARQ is evaluated, in which the first transmission is the same as the proposed HARQ and the re-transmissions are made by sending the parity alternating with the information data. Figures 8 and 9 show the results of throughput of rate estimation. Here, the throughput of rate estimation means the probability that rate estimation of all blocks in each ARQ frame are successful after re-transmission in the HARQ sequence. For k max = 1, the performance is the same because there is no re-transmission. The proposed HARQ performs well for k max = 2, 3, 4. The larger k max we set, the better performance the proposed HARQ provide. Also, degradation of throughput is not so large even if k max = 4 is given. Comparing Figs 8 49

5 Taiwan-Japan Conventional kmax=1 (No ARQ) Fig. 8. Comparison of throughput versus SNR for RC-LDPC code (L data = 1536, h = 6, F = 5, k max = 1, 2, 3, 4). Fig. 10. Comparison of throughput versus SNR for RC-LDPC code (L data = 1536, h = 6, F = 10, C min = 1, 2, 3, 4, 5). kmax=1 (No ARQ) Conventional Fig. 9. Comparison of throughput versus SNR for RC-LDPC code (L data = 3072, h = 6, F = 5, k max = 1, 2, 3, 4). Fig. 11. Comparison of throughput versus SNR for RC-LDPC code (L data = 3072, h = 6, F = 10, C min = 1, 2, 3, 4, 5). and 9, it performs well when L data is large in spite of the same code rates. Finally, we consider the case that codes C Cmin to C Cmin+h 1 are employed for adaptive rate selection instead of C 1 to C h, where C min is the minimum code number used due to bad channel status. Set h = 6 again. The simulation results of L data = 1536 and L data = 3072 are shown in Figs. 10 and 11, respectively. Obviously, the proposed HARQ performs better than the conventional HARQ. Especially, in the case of small L data, improvement of throughput is large. It seems that the proposed HARQ works effectively in various conditions of L data = 1536, 3072 and C min = 1, 2,, 5. So, we can say that L data and C min can be set flexibly according to estimated SNR of channel. VI. CONCLUSION In this paper, the blind rate estimation of RC-LDPC codes for HARQ is discussed. The proposed HARQ outperforms the conventional Type-2 HARQ in various L data and C min. ACKNOWLEDGMENT This research was partly supported by the ICOM Electronics Communication Engineering Promotion Foundation. REFERENCES [1] R. G. Gallager, Low-density parity-check codes, Research Monograph Series, Cambridge MIT Press, [2] D. J. C. MacKay, Good error-correcting codes based on very sparse matrices, IEEE Trans. Inform. Theory, vol. 45, pp , March [3] M. Fossorier, Qusai-cyclic low density parity check codes, Proc. ISIT2003, pp. 150, Yokohama, Japan, June-July [4] W. Matsumoto, H. Imai, A study on rate compatible LDPC codes, Proc. ISITA2004, Parma, Italy, Oct [5] T. Yoshikawa, T. Tsujioka, H. Sugiyama, M. Murata, Comparison of Estimation Techniques for -Compatible LDPC codes, Proc. HISC2005, pp , Hawaii, USA, May [6] T. Yoshikawa, T. Tsujioka, H. Sugiyama, S. Hara, M. Murata, Estimation Techniques for -Compatible LDPC Codes, IEICE Trans. on Fundamentals, vol. J89-A, no. 12, pp , Dec