420 IEEE COMMUNICATIONS LETTERS, VOL. 5, NO. 10, OCTOBER 2001 Analysis and Design of Interleavers for CDMA Systems A. Tarable, G. Montorsi, Member, IEEE, and S. Benedetto, Fellow, IEEE Abstract—In the last years, the Turbo-principle has been ap- plied to multiuser receivers in CDMA systems. However, the role of interleavers in these Turbo-receivers has not been studied yet. In this letter, we give a notion of optimality and of mutual optimality and a way to test it. Moreover, limiting to congruential interleavers and convolutional codes, we give sufficient conditions to construct a set of mutually optimal interleavers. Also, we provide some sim- ulations to support our results. Index Terms—Code division multiaccess, interleaver design, it- erative multiuser receivers, permutations, turbo codes. I. INTRODUCTION I T IS widely known that the multiaccess channel can be viewed basically as an encoder. From this considera- tion, many authors, induced by the powerful properties of Turbo-codes, thought of a Turbo-like iterative receiver for coded CDMA systems [1]–[3]. In this type of receiver, a user-separating soft-input soft-output block (called hereafter US-SISO) first attenuates the multiaccess interference (MAI) of the received signal and passes its output to a bank of single-user SISO channel decoders. These decoders, in their turn, provide some additional feedback information to the US-SISO, used, in the successive iteration, to improve the user separation, and so on. Simulations clearly show that the users should interleave their coded bit streams, before transmitting, in order to obtain good performance. However, the role of interleavers for these receivers, and so their design, is quite different from that of Turbo codes. In this letter, we propose a study on how these interleavers should be designed. In particular, we give a definition of optimality of an interleaver, and an easy way to test the distance from optimality. In order to give a practical rule of design, we consider the class of congruential interleavers, which are easy to study. For this subset, we identify certain sufficient conditions to obtain optimal interleavers and find the maximum number of users supported by such a system. Finally, some numerical simulations give a direct confirmation of the design approach. II. SYSTEM DESCRIPTION Let us consider users transmitting on the same AWGN channel. The th user, , feeds the encoder with the information word and obtains the codeword Manuscript received April 20, 2001. The associate editor coordinating the review of this letter and approving it for publication was Prof. A. Haimovich. This work was supported by Qualcomm Inc., San Diego, CA, USA. The authors are with Department of DELEN, Politecnico di Torino, I 10129 Turin, Italy (e-mail: tarable@polito.it). Publisher Item Identifier S 1089-7798(01)09858-1. , belonging to the code , the same for all users, which can be a block or a terminated convolutional code. Each codeword is then interleaved according to the permutation , the sequence of codewords is BPSK modulated, spread and sent to the channel. We suppose all users are received word-syn- chronous. At the receiver, after a bank of matched filters, the US-SISO, which treats the bits as uncoded, performs some fil- tering and outputs streams, each entering a single-user SISO channel decoder. See [5] for more details. The observables in the th stream have the following expression: (1) where is the th user’s th modulated coded bit; is the corresponding amplitude after the filtering performed by the US-SISO; and is a Gaussian random variable with zero mean and a given variance. The decoders perform a BCJR algorithm, which is nonoptimal in this case, since it assumes that the inputs are affected by uncorrelated noise, while the s, being part of a codeword, have correlation. The purpose of the interleavers is to destroy this correlation, and to make the in- terferers’ coded bit streams “as memoriless as possible.” In the following sections, we will clarify this concept. III. -OPTIMAL INTERLEAVERS Let be a binary linear block code with rate and denote with and its generator and parity-check matrix, respectively. Consider interleavers acting on codewords, con- stituted by permutations of the integers . Let be, with a slight abuse of notation, the equivalent code obtained by permuting with all the codewords in and define . It is easy to prove that is a subcode of . The following definition can now be stated. Definition 1: The interleaver is called -optimal if and only if (2) The rationale of this definition lies in the operation performed by the decoders. In fact, they accept the input as if coming from a single-user memoryless channel, while the other users’ coded signals give rise to an interference with memory. Suppose . If is -optimal, the fraction of code- words in that are also in is , by Definition 1. But this is also the probability of belonging to , for an -dimen- sional word output by a binary memoriless source. If , then contains only the all-zero codeword, always present in a linear code and obviously resistant to all permutations. 1089–7798/01$10.00 © 2001 IEEE