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1 Convolutional Coding in Hybrid Type-II ARQ Schemes on Wireless Channels Sorour Falahati, Tony Ottosson, Arne Svensson and Lin Zihuai Chalmers Univ. of Technology, Dept. of Signals and Systems, Communication Systems Group, SE Goteborg, Sweden. fsorour.falahati, tony.ottosson, arne.svensson, Abstract In our previous work, we have proposed a hybrid type- II Automatic Repeat request (ARQ) scheme (Scheme 5) based on Rate Compatible Punctured Convolutional (RCPC) codes which performs signicantly better than all the other ARQ schemes in all the channel conditions. In this work, we study the performance of Scheme 5 based on High Rate Optimized RCPC (HRO-RCPC) codes combined with simple repetition codes in more detail. The performance is investigated for dierent packet sizes and constraint lengths of the convolutional encoder in various fading environments. We have also studied the performance of the HRO- RCPC codes combined with optimized Rate Compatible Repetition Convolutional (RCRC) codes as another alternative for channel coding in hybrid type-ii ARQ schemes. The corresponding simulation results show that these codes with high parent code rate perform almost as good as the HRO-RCPC codes with simple repetition codes at lower parent code rate. Introduction In data communications, ARQ schemes can be employed in order to provide almost error free reception. By applying channel coding in ARQ schemes, data bits can be protected against channel impairment. However, on time varying channel, it is reasonable to adapt the redundancy bits to the channel variations. Therefore, transmission of unnecessary redundancy bits can be avoided. This scheme is called hybrid type-ii ARQ where RCPC codes are well-known candidates for the channel coding [{8]. We have proposed in [8], a hybrid type-ii ARQ scheme named Scheme 5. It was shown that Scheme 5 to be superior to all the other considered schemes in [8], including simple ARQ scheme, by providing a signicantly better improvement in the performance in all the channel conditions. In this work we examine the performance of Scheme 5 based on the HRO-RCPC codes combined with simple repetition codes for dierent constraint lengths and information packet sizes. The system performance is measured as the throughput of the system dened as the inverse of the average number of transmitted symbols per error free detected data bit [, 2]. We have also studied the performance of the HRO-RCPC codes combined with optimized RCRC codes as another alternative for channel coding in hybrid type-ii ARQ schemes. The following two Sections 2 and 3, contain brief explanations of the system model and the dierent ARQ schemes. Section 4 and 5 describe the hybrid type-ii ARQ schemes based on HRO-RCPC codes in combination with simple repetition codes and optimized RCRC codes, respectively. The numerical results of Sections 4 and 5 are presented in Section 6. Finally some conclusions are drawn in Section 7. 2 System Model The hybrid ARQ schemes that are presented in this paper are based on the RCC codes [6]. The candidate code word for transmission is antipodal modulated where the energy per coded bit is denoted by E c. Then channel symbols are interleaved and packed into blocks of xed length, which are referred to as channel blocks, and then transmitted over a Rayleigh fading channel. The channel block length is denoted by L c. The statistical characteristics of the channel are assumed to be independent for each channel block transmission. The fading process is generated by the Jakes model [9]. The received symbols are deinterleaved and fed into the maximal ratio combining receiver [] which is assumed to have perfect channel state information (CSI). Soft decoding is then performed by a Viterbi decoder. More details can be found in [8]. 3 Description of Hybrid Type- II ARQ Schemes We assume a selective repeat (SR) ARQ protocol with innite buers in the transmitter and the receiver.

2 The feed-back channel is assumed to be error free. In all the schemes the erroneously received channel blocks are not discarded but combined in an optimum way with the newly received block(s) for a given packet. The considered schemes are briey described in the following. The relation between the data packets and the channel blocks are shown in Fig.. Scheme is a simple ARQ scheme in the sense that no channel coding for error correction is applied. Each information block is concatenated with n p parity bits for error detection to form the code word C with length L which is transmitted over the channel. The received packet is combined with the previously erroneously received packets, if they exist. In case of error detection a retransmission is requested until an error free reception is achieved. Schemes 2 to 5 are type-ii hybrid ARQ schemes. In all the hybrid schemes C with length L is the input to the convolutional encoder. It contains the information bits, n p parity bits for error detection and a zero tail corresponding to the memory of the convolutional encoder. By puncturing or repeating the encoded bits in a proper manner [5, 6] rate compatible code words at higher or lower code rates can be obtained, respectively. As it is shown in Fig., each hybrid scheme uses a set of rate compatible codes with corresponding code rates R k where k. The code rates are given in decreasing order such that R k > R k+. C k contains the incremental redundancy bits in the code word at the rate R k which are not included in the code words of the higher rate codes and sometimes is referred as incremental code word. The interleaving is done over the incremental code word. Schemes 2 and 3 apply codes of rates 2=3 and =3. Scheme 4 uses codes of rates, =2 and =3. The code rates, 2=3, =2, 2=5 and =3 are used in Scheme 5 where the step between two consecutive code rates is smaller compared to the other schemes. The encoder input C is twice as large in Schemes 3 and 5 compared to Schemes 2 and 4. In Scheme 2 each incremental code word has a length of :5L c which is transmitted in one channel block. In Scheme 3 each incremental code word after interleaving is however divided into three channel blocks of length L c. All the incremental code words in Scheme 4 and Scheme 5 have a length of L c except for the rst code word C in Scheme 5 which has a length of 2L c that is divided into two channel blocks after interleaving. In all the hybrid schemes, the transmission starts with the code of the highest rate. If the code word is received in error, a retransmission is requested and the transmitter sends the incremental code word of the next lower rate. This code word is combined at the receiver with previously erroneously received incremental code words. Then it is decoded and checked for existence of a detectable error. This procedure repeats until error free recep- Figure : The diagrams of the data blocks and channel blocks in the ARQ schemes. tion is achieved. If the received code word of rate =3 still fails in correcting all the detectable errors, the code of lower rates are used. The details about the code words at lower rates at this stage are more claried in Sections 4 and 5. 4 HRO-RCPC Codes in Hybrid Type-II ARQ Schemes As a result of extensive computer search, good convolutional codes for parent rates =2; =3; =4 and constraint length 3? 5 are obtained and presented in []. These codes are based on the Optimum Distance Spectrum (ODS) criterion where the codes have the maximum free distance and provide a low information error weight on each error path. A convolutional code of rate =n and constraint length K can be punctured periodically with period p to provide codes of higher rates. These codes will be rate compatible if the symbols of the lower rate codes are used in the higher rate codes as well. By searching for an optimum puncturing pattern which satises both the ODS and rate compatibility criteria, optimum RCPC codes with higher rates are found. This search can be done in two directions, i.e. from low rate to high rate and vice versa which are referred as Low Rate Optimized RCPC (LRO-RCPC) and High Rate Optimized RCPC (HRO-RCPC) codes, respectively. The HRO-RCPC codes of parent code rate =3 and puncturing period equal to 2 is given in Table 2

3 where K denotes the constraint length of the encoder, (g ; g ; g 2 ) represents the generator polynomials in octal form and P Rk is the puncturing matrix for the corresponding code rate R k. The P Rk elements are zeros or ones corresponding to deleting or keeping the corresponding bits. We would like to mention that since the RCPC codes with rate are catastrophic, the ODS criterion can not be used for optimizing P. Due to the rate compatibility criterion, there are two choices for P. Therefore we have used both of them in our simulations for hybrid ARQ schemes and selected the one which provided higher throughput in the most cases. The results are given in Section 6. If the received code word of parent rate =3 is still not able to correct all the detectable errors, the already transmitted incremental code words are repeated in the same order until no error is detected. Obviously at this stage the received code words are simple repetition code words. Therefore in this method both the punctured codes and simple repetition codes are employed. 5 Optimum RCRC Codes in Hybrid Type-II ARQ Schemes As it is said in Section 4, HRO-RCPC codes combined with simple repetition codes are good candidates for channel coding in hybrid type-ii ARQ schemes. Another approach is applying optimized RCRC codes instead of simple repetition codes at rates below the parent code rate. Since the RCRC codes are rate compatible with the code of parent rate, they are rate compatible with the corresponding RCPC codes of higher rates. Thus the two families of RCPC codes and RCRC codes form a family of RCC codes. The repetition matrix denoted by Q, is similar to the puncturing matrix P. The elements of Q are either one or greater than one, indicating the number of duplications of the corresponding symbol [5]. Table : HRO-RCPC codes of parent code rate =3 K (g ; g ; g 2 ) P 2=3 P =2 P 2=5 3 (5; 7; 7) 4 (3; 5; 7) 5 (25; 33; 37) 6 (47; 53; 75) 7 (33; 65; 7) 8 (225; 33; 367) 9 (575; 623; 727) Optimization of the RCRC codes are based on the ODS and rate compatibility criteria. The optimum RCRC codes are obtained from a computer search for dierent parent code rates [5,2]. We would like to mention that in the search method for RCRC codes proposed by Kallel in [5], the maximum dierence among the elements in repetition matrix is zero or one. However this search limitation is removed in [2]. In Table 2 the search results for the puncturing/repetition patterns for the RCC codes of parent code rate =2 and puncturing or repetition period 2 are given. 6 Numerical Results The constraints on simulations are as follows. We assume that an equal number of parity bits for error detection for a given size of the data blocks should be used for all the channel conditions in order not to have any undetected errors. Therefore the shortest parity bits which fulll this requirement are selected. BPSK modulation and block interleaving over the incremental code words are applied. The Rayleigh fading process with unit average power for the fading envelope is generated by using the Jakes model [9]. Furthermore the maximum ratio combining receiver with perfect CSI and soft decisions uses the Viterbi algorithm decoder at the parent code rate. Each simulation was continued until data blocks were received correctly. Some simulations results of the Sections 4 and 5 are given in the Subsections 6. and 6.2, respectively. 6. Numerical Results based on HRO- RCPC Codes In our previous study [8], hybrid type-ii ARQ schemes based on LRO-RCPC codes of the parent code rate =3 are investigated. The results show that Scheme 5 provides the highest throughput in all the cases. Here, we study the performance of Scheme 5 based on HRO-RCPC codes of the same parent code rate. We mainly focus on the eects of the data packet size and the constraint length of the encoder on the performance. The code table for HRO-RCPC codes is given in Table. Also we choose n p = 8 for L = f8; 6g, n p = 2 for L = f24; 32g and n p = 6 for L = f4; 48; 56; 64g. Table 2: The RCC codes of parent code rate =2, K = 7, (33; 7) P 2=3 P =2 Q 2=5 Q =3 Q 2=7 Q =

4 Normalized throughput K = 3 K = 4 K = 5 K = 6 K = 7 K = 8 K = 9 Freq. of correct reception per information packet L = 8 L = 6 L = 24 L = 32 L = 4 L = 48 L = 56 L = The length of the convolutional encoder input L Figure 2: Simulated normalized throughput for Scheme 5 with normalized Doppler frequency.. Solid and dash-dotted lines corresponds to 5 db and 5 db SNR respectively Number of transmitted code words Figure 3: Simulated histograms of the number of transmitted code words for Scheme 5 with constraint length 7, 5 db SNR and normalized Doppler frequency.. The simulations have been done for slow and fast fading environments with the normalized Doppler frequencies : and : respectively. Some of the results are presented in Figs. 2 and 3. The comparisons of throughput performance of the proposed Scheme 5 for dierent packet sizes and constraint lengths for slow fading environment is given in Fig. 2. The solid lines show that the dierence in the throughput due to the dierent constraint lengths at fairly high SNR, e.g. 5 db, is small. We can see that by increasing the packet length, the tail overhead becomes less eective on the throughput than the strength of the code given by the constraint length. Also the dashed-dotted lines in Fig. 2 show that at lower SNR, where more of the received symbols are in error, codes with larger constraint length perform better than the codes with smaller constraint length, even for large packets. However the dierence becomes smaller for short packets due to the advantage of having short packets in fading environments. Furthermore, the codes with constraint length 6 or 7 perform equally well as the codes with the constraint length 9. Hence the tail overhead can be reduced without loosing throughput. The histograms of the number of transmitted code words at 5 db SNR and slow fading are given in Fig. 3 for K = 7. It can be seen that most of the transmissions at rate are not successful except for short packets. The largest probability of error free receptions happens at rate 2=3 and the probability increases with increasing packet length. Most of the packets are received correctly at rate =3. These and similar simulation results are given in more detail in [3]. 6.2 Numerical Results based on Optimum RCRC Codes Instead of using the RCPC codes combined with the simple repetition codes, the RCPC codes combined with the optimized RCRC codes can be used. Furthermore, the analytical and numerical results given in [8, 3] and Section 6. show that most of the data packets are received correctly at the parent rate =3 and retransmission at this rate is seldom requested. Therefore we decided to apply the HRO-RCPC codes combined with the optimized RCRC codes at parent code rate =2, and compare them with the HRO-RCPC codes combined with the simple repetition codes at the parent code rate =3, in our hybrid schemes. The constraint length of both codes is 7. The code table for the RCC codes at parent rates =3 and =2 are given in Tables and 2, respectively. The simulated throughput for L = 28, n p = 2 and normalized Doppler frequency : are given in Fig. 4 where the throughput of the hybrid type-ii schemes based on the RCRC codes (solid lines) and RCPC codes (dash-dotted lines) are compared. We can see that the dierence in throughput is small but with some advantage to the RCPC/simple repetition codes for fast fading environments. Similar results are found for slow fading channels [3]. 7 Conclusion This work is a continuation of the previous studies for hybrid type-ii ARQ schemes on Rayleigh fading channels [8] where a hybrid scheme named Scheme 5, 4

5 Normalized throughput Scheme 2 Scheme 3 Scheme 4 Scheme Average E c /N o (db) Figure 4: Simulated normalized throughput for 28 bits information blocks, 2 CRC bits and normalized Doppler frequency.. Solid and dash-dotted lines correspond to RCC codes at parent rate /2 and /3 respectively. was proposed. It was shown that Scheme 5 performs very well compared to the other hybrid schemes and provides the highest throughput in all the examined situations. In this work the performance of Scheme 5 based on HRO-RCPC codes in combination with simple repetition codes for dierent constraint lengths and packet sizes are investigated. The simulation results show that the HRO-RCPC codes with intermediate constraint lengths perform equally well as the ones with longer constraint lengths. Furthermore, we show that in most cases, short packets provide higher throughput in a fading environments. Our results also show that by applying HRO- RCPC codes in hybrid type-ii ARQ schemes for long packets, good performance in fading environments can be still obtained. We have also studied the performance of the HRO- RCPC codes combined with optimized RCRC codes as another alternative for channel coding in hybrid type-ii ARQ schemes. The simulation results show that these codes with high parent code rate almost perform equally well as the HRO-RCPC codes with simple repetition codes at lower parent code rate. We would like to mention that tailbiting decoding with a Circular Viterbi Algorithm (CVA) [4] is also examined in this study. It is shown that the improvement in the performance due to tailbiting is noticeable only for short packets, otherwise the suboptimality of CVA decoding compared to the General Viterbi Algorithm (GVA) decoding degrades the performance considerably. The interested reader can nd the corresponding results in [3]. References [] S. Lin and D. J. Costello Jr., Error Control Coding: Fundamental and Applications, Prentice-Hall, Englewood Clis NJ, 983. [2] S. Wicker, Error control systems for digital communication and storage, Prentice-Hall, Englewood Clis, NJ, 995. [3] Y. M. Wang and S. Lin, \A Modied Selective Repeat Type-II Hybrid ARQ System and its Performance Analysis," IEEE Transactions on Communications, vol. 3, pp. 593{68, May 983. [4] S. Kallel, \Analysis of a type II hybrid ARQ scheme with code combining," IEEE Transactions on Communications, vol. 38, no. 8, pp. 33{37, Aug. 99. [5] S. Kallel and D. Haccoun, \Generalized type II hybrid ARQ scheme using punctured convolutional coding," IEEE Transactions on Communications, vol. 38, no., pp. 938{946, Nov. 99. [6] J. Hagenauer, \Rate-compatible punctured convolutional codes (RCPC codes) and their applications," IEEE Transactions on Communications, vol. 36, no. 4, pp. 389{4, Apr [7] H. Lou and A. S. Cheung, \Performance of Punctured Channel Codes with ARQ for Multimedia Transmission in Rayleigh Fading Channels," Proc. IEEE Vehicular Technology Conference, vol. 46, pp. 282{286, May 996. [8] S. Falahati and A. Svensson, \Hybrid type II ARQ schemes for Rayleigh fading channels," in Proc. International Conference on Telecommunications, Porto Carras, Greece, June 998, vol., pp. 39{44. [9] W. C. Jakes, Microwave mobile communications, John Wiley and Sons, Inc., New York, 974. [] J. G. Proakis, Digital Communications, McGraw- Hill, New York, 3rd edition, 995. [] P. Frenger, P. Orten, T. Ottosson, and A. Svensson, \Multi-rate convolutional codes," Tech. Rep. 2, Communication Systems Group, Department of Signals and Systems, Chalmers University of Technology, Sweden, Apr [2] Lin Zihuai, \Rate Compatible Convolutional (RCC) Codes and Their Applications to hybrid ARQ/FEC transmission," M.S. thesis, School Electrical and Computer Engineering, Chalmers University of Technology, 998. [3] S. Falahati, T. Ottosson, A. Svensson, and L. Zihuai, \Hybrid type-ii ARQ Schemes based on Convolutional Codes in Wireless Channels," in Proc. FRAMES Workshop, Delft, the Netherlands, Jan. 999, pp. 225{233. [4] Richard V. Cox and C. E. Sundberg, \An ecient adaptive circular viterbi algorithm for decoding generalized tailbiting convolutional codes," IEEE Transactions on Vehicular Technology, vol. 43, no., pp. 57{68, Feb