2998 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 7, NOVEMBER 1998 Error Probability for Block Codes Over Channels with Block Interference Marco Chiani, Member, IEEE Abstract—A methodology is presented to evaluate analytically the error probability for block codes over block interference channels. The proposed analysis is based on the knowledge of the moments of the bit-error probability over the interference, thus allowing, for instance, fast performance evaluation of block- coded Slow Frequency Hopping (SFH) systems with antenna diversity over fading channels. As an example of application, Slow Frequency Hopping Multiple Access (SFHMA) systems with nonideal interleaving are analyzed in the presence of fading, cochannel interference, and additive Gaussian noise. Index Terms—Block codes, block interference channels, fading channels, frequency hopping, interleaving. I. INTRODUCTION I N many real communication systems the time-variant im- pairments affecting the transmission oblige us to consider channel models with memory. An important class, and one that can be mathematically dealt with, is that of channels with block interference, which was introduced in [1]. These channels are characterized by the fact that the noise severity remains constant in blocks of consecutive transmitted symbols but is independent from block to block. Some examples of communication systems that match this definition are Slow Frequency Hopping systems (in this case we will also speak of “block fading channel”), optical communication systems using direct detection and pulse-position modulation, concatenated coding systems with a memoryless inner channel, and an inner block code. For example, in time-division multiple- access (TDMA) transmission over fading channels, in order to let the forward error correction work efficiently the code symbols should be suitably time-interleaved [2]; however, due to delay and/or memory constraints, often the interleaver produces groups of time-adjacent symbols, that, in relation with the “fastness” of the fading process (a measure of which is the coherence time of the fading), could give rise to a block fading model. As an example, in the GSM and DCS1800 mobile radio systems groups of 456 bits derived from a mixture of encoded and unprotected speech bits are interleaved over eight time slots, so that there are 57 bits per time slot. Assuming a constant channel within each time slot and perfect (time- slot) interleaving and/or frequency hopping, the situation can Manuscript received January 31, 1997; revised February 23, 1998. This work was performed under contract with MURST and CNR (Rome, Italy). The material in this paper was presented in part at the Communication Theory Mini Conferemce-GLOBECOM’96, London, U.K., November 1996. The author is with DEIS, University of Bologna, 40136 Bologna, Italy (e- mail: mchiani@deis.unibo.it). Publisher Item Identifier S 0018-9448(98)06896-5. Fig. 1. Block coded slow frequency hopping. be described by a block fading channel, with 57 bits per independent block. For many other systems this channel model can also be used with a certain degree of approximation. In [1] this class of channels have been thoroughly analyzed in terms of capacity and cutoff rate. In this paper the problem of fast analytical evaluation of the performance of block-coded transmission with hard-decision decoding operating in channels with block interference is in- vestigated. The methodology presented is very simple, starting from the moments of the bit error probability with respect to interference. As an example, the performance of block-coded SFH systems over Rayleigh fading channels is investigated, thus extending previous work on the same subject [3]–[6]. The paper is organized as follows. In Section II, we present the general methodology to derive the error probability for block codes over block interference channels. In Section III the methodology is applied to evaluate the performance of SFH systems over Nakagami, Ricean, and Rayleigh channels, with and without antenna diversity. Another application ex- ample is provided in Section IV, where the Slow Frequency Hopping Multiple Access technique is investigated in terms of codeword error probability. Finally, some numerical results are presented in Section V. II. ERROR PROBABILITY EVALUATION The analysis applies to Forward Error Correcting (FEC) block codes such as Bose–Chauduri–Hocquenghem (BCH) and Reed–Solomon (RS) codes [7] over block interference channels. In the following will indicate that a code- word is composed of symbols, of which are information symbols. The code rate is, therefore, . The decoder 0018–9448/98$10.00  1998 IEEE