IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-29, NO. 4, JULY 1983 521 Hybrid ARQ Error Control Using Sequential Decoding ALEXANDER DRUKAREV, MEMBER, IEEE, AND DANIEL J. COSTELLO, JR., SENIOR MEMBER, IEEE Abstract-An important feature of ARQ sequential decoding is that a very low undetected error probability can be achieved without increasing significantly the complexity of decoding. Several ARQ sequential decoding algorithms based on the stack algorithm are considered. Analysis is done for a memoryless channel with noiseless feedback, and the emphasis is on evaluating the undetected error probability and the maximum throughput attainable with each algorithm. A time-out algorithm is analyzed and the parameters optimizing the performance of this algorithm are found. A new algorithm called the slope control algorithm, capable of achieving a better throughput than the time-out algorithm, is proposed. The algorithm is analyzed using random coding arguments, and the parameters maximizing the throughput for various conditions are found. All theoretical results are verified by computer simulation for a binary symmetric channel. I. INTRODUCTION E RROR control strategies used in data transmission systems can be classified as forward error control (FEC), pure automatic repeat request (ARQ), and hybrid ARQ, the latter one being a combination of FEC and ARQ strategies [l]. Since the error-detecting capabilities of a code exceedits error-correcting capabilities, ARQ systems have a lower error probability than FEC systems of ap- proximately the same level of complexity. This is achieved at the expense of lower throughput for ARQ systems.’ Most of the work in the area of ARQ error control has been done using block codes [2]-[6]. However, convolu- tional codes have a number of features that make them attractive for use in systemswith repeat request: 1) 2) Convolutional codes have good error-correcting and error-detecting capabilities. Decoding of a convolutional code is a sequential process, and hence a retransmission can be requested before reaching the end af a codeword (message),2 which is not the case when a block code is used. Manuscript received June 4, 1981; revised March 25, 1982. This work was supported by the National Science Foundation under Grants ENG78-05665 and ECS 8103746. This paper was presented in part at the IEEE International Symposium on Information Theory, Los Angeles, CA, February I98 I. This paper formed part of a dissertation submitted by A. Drukarev to the Department of Electrical Engineering, Illinois Institute of Technology, Chicago, IL, in partial fulfillment of the requirements for the Ph.D. degree. A. Drukarev is with the Bell Telephone Laboratories, Naperville-Wheaton Road, Naperville, IL 60566. D. J. Costello, Jr., is with the Department of Electrical Engineering, Illinois Institute of Technology, Chicago, IL 60616. ‘Whereas this observation is true for block codes and the Viterbi decoding of convolutional codes, it is interesting to note that when sequent&l decoding is used, ARQ methods can achieve both lower error probability and higher throughput (see Sections II and III). *In the context of this paper, a codeword is equivalent to a message. 3) 4) A have Itoh If path memory truncation is used, then only a por- tion of a message will have to be retransmitted in responseto a retransmission request, and hence the system throughput will increase. Sequential decoding of convolutional codes appears to be an especially promising method of ARQ error control. number of ARQ schemeswith convolutional codes been proposed. Fang [7] and recently Yamamoto and [8] suggested ARQ schemes based on the Viterbi ^^ _ algorithm. Both schemes suffer the same drawback as the original Viterbi algorithm; namely, the complexity of de- coding grows exponentially with the code constraint length K. Because of that, the schemes cannot achievea very low undetected error probability. In earlier works with sequen- tial decoding [9], [lo], [l l] attempts were made to use repeat requests to improve the performance of a system. Insufficient knowledge of the behavior of sequential decod- ing algorithms at the time when the research was con- ducted did not allow the authors to perform a complete analysis of the suggestedschemes,or to determine their optimal parameters. More recently, Kahn et al. [12] de- scribed an ARQ technique that uses sequential decoding with a time-out condition. A detailed analysis of this method will be presentedin Section III. In all the schemes consideredabove the original message and its retransmissions are processed independently. Ancheta [ 131 and Metzner [ 141 suggested interesting schemes in which the retransmitted messages consist of parity check bits. As a result, the original message together with the retransmissions form a variable rate convolutional code. A disadvantageof both schemes is their complexity. Besides, it is not quite clear what improvement (if any) these schemescan offer compared to conventional ARQ schemes. Analysis of some hybrid-ARQ sequential decoding pro- ceduresfor memorylesschannelswith noiseless feedback is the subject of this paper. In light of the potential speed advantage of the stack algorithm and the relatively low (and continually decreasing) cost of computer memory, the stack algorithm is of primary interest here. Moreover, as we will see, the memory requirements for the stack algo- rithm when ARQ error control is used are less stringent than those for FEC error control. After presenting some background concepts and defini- tions in Section II, we will analyze a time-out algorithm in 001%9448/83/0700-0521$01.00 01983 IEEE