576 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 49, NO. 2, MARCH 2000 Throughput Evaluation for ARQ Protocols in Finite-Interleaved Slow-Frequency Hopping Mobile Radio Systems Marco Chiani, Member, IEEE Abstract—The errors distribution for packet transmission over mobile radio systems employing slow-frequency hopping (SFH) with nonideal interleaving is analytically studied. A new analytical methodology is then presented to evaluate automatic repeat request (ARQ) techniques, taking into account forward error correction (FEC) and nonideally interleaved SFH over Rayleigh fading channels. The selective-repeat (SR) ARQ and the type I hybrid ARQ/FEC techniques are compared when varying the number of bits per hop, thus gradually passing from the noninterleaved non-SFH case to the perfect interleaved case. It is found that the use of SFH and/or interleaving deteriorates the efficiency of the transmission: in other words, when ARQ is applied to slow fading channels the best performance is obtained without SFH and interleaving. The ARQ/FEC technique analyzed here gives marginal improvements with respect to pure SR only when perfect interleaving and SFH are applied. Index Terms—Automatic repeat request, data communications, error correction codes, fading channels. I. INTRODUCTION I N MOBILE radio systems, it is known that speech trans- mission can be improved by using slow-frequency hopping suitably joined with forward error correction (FEC) codes in order to exploit the inherent diversity of multipath fading chan- nels [1]. For this reason, the SFH technique has been proposed in many cellular systems such as GSM, IS-54, and DCS-1800, where some advantages in terms of better system quality are ex- pected from interference and frequency diversity [1], [2]. However, the services that can be offered by mobile radio systems are not only concerned with voice, but also with data transmission. Data transmission has inherently different requirements with respect to voice communication. In fact, data transmission is usually of a nonreal-time nature and hence can tolerate delays in transmission. Moreover, it should be highly reliable, imposing a low bit error rate (in the order of 10 or less). To achieve these qualities in a wireless channel environment, automatic repeat request (ARQ) strategies could be applied [3]–[7]. For this reason, it is of some interest to establish if the error control and transmission techniques adopted for voice are also suitable for data transmission. In particular, here we analyze the Manuscript received November 24, 1997; revised January 19, 1999. This work was supported by MURST and CNR (Rome). This paper was presented in part at Globecom’97, Phoenix, AZ, November 1997. The author is with DEIS, University of Bologna, 40136 Bologna, Italy (e-mail: mchiani@deis.unibo.it). Publisher Item Identifier S 0018-9545(00)02569-X. effect of SFH on ARQ by using a realistic channel model, taking into account thermal noise, fading, nonideal interleaving, and forward error correction. In the literature, the joint use of ARQ and SFH with finite interleaving over fading channels such as Rayleigh or Rice has not been analyzed, due to the difficulties in finding the perfor- mance of error correcting codes over SFH and fading. In fact, the only contribution in this direction is in [8] where, however, a Gilbert–Elliott channel model is assumed. In this paper, we will refer the analysis to those wireless radio systems that are best described by the Rayleigh or Rice fading channel model. We will start by studying the errors distribution for packet transmission over SFH fading channels, following the analytical approach presented in [9]. Then, two important ARQ techniques, known as selective-repeat (SR) and hybrid type I ARQ/FEC (ARQ/FEC-I) will be compared in terms of throughput. The paper is organized as follows. In Section II, the errors distribution for SFH is analytically derived. In Section III, the performance of ARQ over the mobile radio channel is investi- gated. In Section IV, some numerical results are presented, fol- lowed by conclusions in Section V. II. ERRORS DISTRIBUTION FOR PACKET TRANSMISSION OVER SFH FADING CHANNELS We consider a typical mobile radio scenario, where time-divi- sion multiple access (TDMA) is combined with slow-frequency hopping to efficiently share the radio resource available. Note that this scenario is well suited to describe some third-genera- tion cellular TDMA-based systems. In order to study the performance of block codes joint with SFH, we make the hypothesis that the separation between the FH carriers is sufficiently high with respect to the coherence bandwidth of the radio channel, so that different carriers give rise to independent frequency-nonselective fading channels. The analysis applies to FEC block codes such as Bose–Chauduri–Hocquenghem (BCH) and Reed–Solomon (RS) codes [3]. In the following, will indicate that a codeword is composed of symbols, of which are infor- mation symbols. The code rate is therefore . The decoder is supposed to work with hard decisions, correcting up to symbol errors. The SFH structure is described in Fig. 1; each packet, com- posed by a codeword, is split in several subblocks, each inserted in a TDMA slot that is transmitted in one hop. 0018–9545/00$10.00 © 2000 IEEE