K. Kusume and G. Bauch, "Cyclically Shifted Multiple Interleavers," in Proc. IEEE Global Telecommunications Conference (GLOBECOM 2006), (San Francisco, California, USA), November/December 2006.

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Katsutoshi Kusume http://kusume.googlepages.com/

Cyclically Shifted Multiple Interleavers Katsutoshi Kusume and Gerhard Bauch DoCoMo Euro-labs, Landsbergerstr. 312, 80687 Munich, Germany Email: {kusume,bauch}@docomolab-euro.com

I. I NTRODUCTION In the last years iterative methods based on the so-called turbo principle have been studied in many research areas such as detection/decoding, equalization problems, and many others. Our focus in this paper is iterative multiuser detection which has been intensively studied, especially in the area of code division multiple access (CDMA), e.g. [1], [2]. Iterative multiuser detection techniques comprise a multiuser detector (MUD) and a bank of independent a posteriori probability (APP) decoders for the forward error correction (FEC) codes of multiple users. Multiple access interference (MAI) as well as inter symbol interference (ISI) is mitigated in an iterative manner. Both MUD and MAP decoder are soft-in soft-out (SISO) blocks which exchange soft information where interleaving and deinterleaving operations are performed between those SISO blocks. In such an iterative receiver, an interleaver plays an important role. It has been noted in literature that the performance of CDMA systems generally improves when each user interleaves a bit-stream after channel encoding using a user-specific interleaver. In this paper we particularly focus on interleave division multiple access (IDMA), e.g. [3]–[6], which has a close relation to CDMA, but users are separated only by userdistinct interleavers. No user-specific spreading code is applied in IDMA. Hence, multiple interleavers are key system components for IDMA. Despite its importance, usually interleavers for such systems are randomly chosen and the performance is

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Abstract— We propose a simple strategy to generate multiple interleavers. As suggested in literature, using user-distinct interleavers is an effective means to separate multiple users, if applied in addition to user-specific spreading codes in code division multiple access (CDMA). In this paper we particularly focus on interleave division multiple access (IDMA), which has a close relation to CDMA, but users are separated only by userdistinct interleavers. Hence, multiple interleavers are essential system components for IDMA. Despite its importance, usually interleavers for such systems are randomly chosen and there are only few papers on the generation of multiple interleavers. We show that the conventional multiple interleavers proposed for CDMA are not sufficient for the user separation in IDMA. Moreover, in order to minimize memory requirements and signaling overheads to store and exchange interleavers, a simple interleaver construction rule is desirable in practical systems. Therefore, we propose to derive multiple interleavers from a single interleaver, common for all users, with only a few userdistinct cyclic shifts. Although this simple design is empirical and no optimality is claimed, simulation results show sufficiently good performance. We also proposed a simple procedure to exchange the information of interleavers.

L( b (k ))

Receiver for K users Fig. 1.

System Model of IDMA.

averaged over a large number of interleaver realizations. To the authors’ knowledge, there are very few papers on the design of multiple interleavers, e.g. [7], [8] for CDMA, although there have been a relatively large number of research results on the single interleaver design for turbo codes, e.g. [9], [10]. It is important to note that the conventional interleavers in [7], [8] developed for CDMA do not perform sufficiently good for the user separation in IDMA, as we will show by means of computer simulations. A new method to generate multiple interleavers is necessary. Moreover, it should be also noted that a simple interleaver construction rule is desirable in practical systems so that multiple interleavers can be generated on the fly. Then, we can avoid to store interleaver indices that demands a large amount of memory for a number of users and also for different block sizes. At the same time the number of parameters that determine each interleaver should be as small as possible to minimize signaling overheads. We propose to derive multiple interleavers from a single interleaver which is common for all users. Motivated by some observations, we propose to use a few user-specific randomly chosen cyclic shifts to generate different interleavers from a single common interleaver. Any well-studied single interleaver may be chosen as a single common interleaver such as the standardized UMTS turbo internal interleaver [9]. Although our simple design is empirical and no optimality is claimed, simulation results show the good performance; at least as good as the case of using completely randomly chosen interleavers. II. S YSTEM M ODEL We consider the system model of IDMA in Fig. 1. At the transmitter, information bits b(k) of user k, k = 1, . . . , K, are encoded by a rate Rc convolutional code followed by a (k) rate Rr repetition code. The coded bits cn
are interleaved (k) by the user-distinct interleaver Πk to get cn which are then (k) mapped on modulation alphabets. The signals xj from K

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users are transmitted over a channel. The received signal yj
K
νk (k) (k) can be expressed as yj = k=1 =0 g xj−τk − +ηj where (k) τk is a user delay, g and νk are, respectively, complex valued channel taps and a memory of a multipath channel of user k. A complex valued noise ηj is zero-mean Gaussian distributed and has a variance of N0 /2 per real dimension. The receiver applies iterative multiuser detection and decoding. The MUD computes a posteriori log-likelihood ratio (LLR) about the coded bits based on the received values yj and on the a priori LLRs which are fed back from the decoders. After subtracting the a priori LLR from the a posteriori LLR, the extrinsic information about the coded bits is deinterleaved and sent to the decoder. The algorithm of the MUD for IDMA is rather simple and omitted here, for it is not our main focus of this paper. A detailed description of the MUD for IDMA can be found, e.g. in [3], [5], [6]. Since information bits are encoded by a convolutional code followed by a repetition code, APP decoding comprises two tasks: the decoding of the repetition code, which is simply summing up every 1/Rr LLRs [11], and the decoding of the convolutional code by a standard decoder described, e.g. in [12]. The soft output from the APP decoder of convolutional code is repeated 1/Rr times due to the repetition code and sent to the MUD after subtracting the a priori LLR and interleaving. In the following, our focus is on the generation of user-distinct multiple interleavers Πk . III. P REVIOUS W ORKS The single interleaver construction for turbo codes has been relatively well-studied in the last years and many proposals can be found in literature, e.g. [9], [10]. On the other hand, despite the increasing attention of multiple interleavers, only very few papers about multiple interleavers can be found in literature, e.g. [7], [8]. Some research areas, where multiple interleavers play an important role, include CDMA [13], [14], a coded spatial multiplexing [15], and IDMA, e.g. [3]–[5], and so on. In those systems multiple interleavers are usually generated in a completely random manner. A typical way to get random interleavers is to first generate N pseudo-random numbers, sort them, and the resulting indices are used as permutation indices. This procedure is independently repeated K times to get a set of interleavers. Simulation results in literature are often the performance averaged over a large number of interleaver realizations. It is a convenient way to evaluate different systems when the interleaver design itself is not the main focus, however, it is too complex in practical systems. In [7] the authors proposed a criterion to find multiple interleavers for convolutionally coded CDMA systems where the interleavers are limited to congruential interleavers due to its mathematical tractability. The indices of a congruential interleaver of length N are calculated as: πk (n) = sk + nck mod N (k)

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must be relatively prime to N to ensure that each element is read out once and only once. Therefore, each interleaver is determined by the single parameter ck (and sk if non-zero). The idea in [7] is to find user-specific congruential interleavers (i.e. to choose ck ’s) for a given convolutional code (a code is common for all users) such that the resulting interleavers lead to good asymptotic distances between the effective codes after interleaving. Unfortunately, the method has a strict limitation on block lengths and cannot always find proper interleavers for a given block length N . A large block length is often required. Such unfortunate situations typically occur for low rate codes with a large free distance since the search algorithm is tightly related to a free distance of codes (cf. Theorem 2 in [7]). Furthermore, as we will see later, even when interleavers are found for certain block length, the performance is very poor; users in IDMA cannot be separated sufficiently well. In [8] some heuristic rules were proposed for the generation of a set of interleavers. Those heuristics are aiming at either: (1) improving the maximum-likelihood (ML) bound or (2) minimizing the “inter-iteration gain reduction” (IIGR). However, as the authors stated, no recipe has been provided for a deterministic construction of interleavers in general; neither for improving the ML bound nor for minimizing the IIGR. A deterministic method was provided only in the particular subset of congruential interleavers defined in (1) for an (n0 , k0 , m) terminated convolutional code1 . More specifically, the authors consider only symbol interleavers, i.e. “interleavers that take n0 output bits in a single trellis step as a symbol drawn from GF(2n0 ) and do permutation at the symbol level” [8]. Similar to [7], there is a strict limitation on the symbol interleaver’s block length
N
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−1 N that i=0 Z i = 1+Z 1+Z must be an irreducible polynomial over GF(2) [8]. The resulting N
is always a prime number that might be an unacceptable restriction in practice. Moreover, we will show that users cannot be separated well in IDMA with interleavers generated from [8]. IV. C YCLICALLY S HIFTED M ULTIPLE I NTERLEAVERS Our approach is an empirical design which has been derived from several observations. We start our investigation with an analysis of the impact of using a single interleaver common for all users. Simulation parameters are described as follows. Information bits of frame length 128 are encoded by the rate Rc = 1/2 memory 4 standard convolutional code with the generator polynomial [31, 27]8 where the octal notation has the least significant bit on the left. The trellis of the convolutional encoder is terminated with 4 additional termination bits, resulting in 264 coded bits. The coded bits are further encoded by the rate Rr = 1/4 repetition code that gives 1056 coded bits. Thus, the resulting interleaver size is N = 1056. Note that, in order to have a balanced number of zeros and ones, the repetition code yields alternating bits, namely 0 → 0101 and 1 → 1010 which are common for all users. 1 k and n are the number of input bits and of output bits, respectively, 0 0 at each trellis step. m is the constraint length.

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10 4 6 8 10 0 2 4 6 8 10 Eb/N0 in dB Eb/N0 in dB Fig. 2. BER performance of IDMA on an AWGN channel after 6 iterations using user-distinct multiple random interleavers or a single random interleaver common for all users. K = 4 users are either synchronous (left figure) or asynchronous with the user delays τk = k − 1 (right figure). 0

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Fig. 2 shows the BER performance on an AWGN channel after 6 iterations. The coded bits are BPSK modulated. K = 4 users are either synchronous (left figure) or asynchronous with user delays of τk = k − 1 (right figure). In each scenario, the performance is plotted for two cases: either using (1) user-distinct multiple random interleavers or (2) a single random interleaver common for all users. Multiple random interleavers for K users are generated independently of each other. Random interleavers are newly generated for every transmission frame. The performance is averaged over a large number of interleaver realizations and also over all users. It can be observed that the performance does not improve by iterations using a single common interleaver in the usersynchronous case (left figure) while the performance is as good as user-distinct multiple interleavers in the user-asynchronous case with the user delays of τk = k − 1 (right figure). Plotted in Fig. 3 is the BER performance on a multipath channel after 4 iterations. The channel delay is set as νk = 7 for all K = 4 users. The channel taps are generated from zero mean Gaussian distribution with uniform power delay profile, (k) i.e. E[|g |2 ] = 1/(νk + 1) = 1/8 and are constant over each transmission frame. In contrast to the previous example on an AWGN channel (left figure in Fig. 2), users are still separable to some extent, even using a single interleaver common for all users in the user synchronous case (left in Fig. 3). That is due to the user-independent multipath channels. However, an error floor can be observed. The performance improves with the user-distinct delays of τk = k − 1 (right figure in Fig. 3). We conclude from these observations that certain userdistinct delays can help the user separation. That is due to the fact that the interleaver cycle is shifted so that the deinterleaving operation of a user does not recover the original sequence orders of other users, thus other users’ signals are effectively decorrelated. That leads us to a very simple strategy to generate multiple interleavers which is summarized in Fig. 4. The system with a common interleaver Π and a user

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10 4 6 8 10 0 2 4 6 8 10 Eb/N0 in dB Eb/N0 in dB Fig. 3. BER performance of IDMA on a multipath channel after 4 iterations using user-distinct multiple random interleavers or a single random interleaver common for all users. K = 4 users transmit either synchronously (left figure) or asynchronously with the user delays of τk = k − 1 (right figure). 0

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(c) repeat (b) with multiple cyclic shifts τk,1 ,..., τk,D Fig. 4. (a) A common interleaver Π and a user delay τk , (b) a user cyclic shift τk(c) , analogous to τk in (a), and (c) generation of interleaver Πk by (c) exploiting multiple cyclic shifts τk,d and a single common interleaver Π.

delay τk is illustrated in Fig. 4 (a). This corresponds to a user asynchronous scenario with a single interleaver common for all users, for example the performance curves with circle markers in the right figures of Figs. 2 and 3. Although certain user delay can help the user separation, it is generally hard to control users’ transmission delays in many scenarios such as the uplink of cellular systems and decentralized systems like ad hoc networks. Therefore, a cyclic shift τk(c) (the superscript ‘(c)’ indicates cyclic) may be introduced as an alternative to the user delay as depicted in Fig. 4 (b). The idea can be easily (c) (c) , . . . , τk,D , generalized by using multiple D cyclic shifts, τk,1 i.e. by cascading D pairs of a cyclic shift and a common interleaver as illustrated in Fig. 4 (c). As we will see later, only 2 or 3 cyclic shifts are sufficient for the good user separation. Only a single cyclic shift may not be sufficient as we have observed error floors on a multipath channel in the user asynchronous scenario (i.e. with only “a single user delay” τk = k − 1 analogous to a cyclic shift) using a single common interleaver in Fig. 3. Cyclic shifts might be optimized for a given scenario depending on a channel, a coding rate, a block length, the number of users, and so on. However, we propose to choose them randomly. No optimization effort is

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4 6 8 10 0 2 4 6 8 10 Eb/N0 in dB Eb/N0 in dB Fig. 5. BER performance of IDMA on an AWGN (left figure) and on a multipath channel (right figure) for K = 4 synchronous users.

4 6 8 10 0 2 4 6 8 10 Eb/N0 in dB Eb/N0 in dB Fig. 6. BER performance of IDMA on an AWGN channel after 14 iterations for K = 6 synchronous users.

necessary if the performance is good enough using randomly chosen cyclic shifts. As discussed later in Section V, the random choice of cyclic shifts can be easily integrated with an information exchange of interleavers. To evaluate the proposed strategy, computer simulations have been performed with the same parameter set as in Figs. 2 and 3. In principle, any good interleaver can be used for the single common interleaver. Here, just as examples, we use the UMTS turbo internal interleaver [9] and also the dithered golden interleaver described in Appendix I. Fig. 5 shows the BER performance on an AWGN (left figure) and on a multipath channel (right figure) for K = 4 synchronous users. Two user-specific cyclic shifts are used to generate multiple interleavers where the cyclic shifts are independently and randomly generated from a uniform distribution between 0 and N − 1. These cyclic shifts are updated for each transmission frame. As we can observe from Fig. 5, the performance using interleavers based on our proposed method is as good as using completely randomly generated multiple interleavers. The role of interleavers becomes more important for highly user-loaded scenarios since more iterations are necessary to mitigate the severe MAI. Fig. 6 shows the BER performance after 14 iterations on an AWGN channel where K = 6 users are synchronous. The coding parameters are the same as Figs. 2 and 3. Coded bits are QPSK modulated. In the left figure of Fig. 6 we compare the performance using interleavers generated by conventional techniques. As comparisons we also plot the single user bound as well as the performance using completely randomly generated interleavers. The performance curve with square markers, indicated as “Tarable” in Fig. 6, is obtained using interleavers according to [8]. As explained in Section III, symbol level congruential interleavers are considered with certain restriction on the block length. The block length
N
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N −1 i is chosen such that i=0 Z is an irreducible polynomial over GF(2), and then due to the 4 termination bits we have Nb = 135 (N = 1112) which is close to our current parameter

Nb = 128 (N = 1056). Then, any integer 1 < ck < 139 can be chosen for congruential interleavers in (1) where ck = k is chosen here (the first user has no interleaving as it is the case in [8]) and sk = 0 for all users. As can be observed the performance is very poor. The performance curve with triangle markers, labeled as “Brueck” in Fig. 6, is for the case of using congruential interleavers from [7]. For given coding parameters the algorithm first identifies a required minimum block length, which is 20800 in this example. In order to find interleavers for 6 users, the block length must be longer, and with N = 21104 we found c1 = 1, c2 = 105, c3 = 525, c4 = 2625, c5 = 8125, and c6 = 12979. The starting index is set sk = 0 for all users. In spite of the required long block length (roughly 20 times longer than the current target block length), users cannot be separated and there is a high error floor. The above conventional interleavers are both congruential interleavers. We also tested randomly chosen congruential interleavers. For the block length of N = 1056, 319 ck ’s can be found as [5, 7, 13, . . . , 1051, 1055] which are relatively prime to the block length. Each user randomly picks one value out of the 319 possible ck ’s which are updated for every transmission frame. At the same time the starting index sk is also randomly chosen and updated for every frame in order to introduce further randomness. A high error floor (the performance curve with circle markers) can be observed and the performance is far from what is achievable by completely randomly chosen interleavers. Next, we evaluate the performance of interleavers based on our proposal. In the right figure of Fig. 6 the BER performance after 14 iterations is plotted for three cases, where 1, 2 and 3 randomly chosen cyclic shifts are used to generate multiple interleavers together with the common UMTS turbo interleaver. As can be observed, there are relatively high error floors using one and two random cyclic shifts. With three random cyclic shifts, the error floor cannot be observed in the error rate of practical interests.

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Fig. 7. BER performance of IDMA on an AWGN channel for K = 110 synchronous users. Nb = 256 information bits are coded by only repetition code of rate Rr = 1/64 and then BPSK modulated. The performance is plotted after 5, 15, 20, and 30 iterations.

We are also interested in a system with a large number of users. Fig. 7 shows the BER performance on an AWGN channel for K = 110 synchronous users. Nb = 256 information bits are coded only by the rate Rr = 1/64 repetition code and therefore, the interleaver size is N = 16384. The coded bits are BPSK modulated. Interleavers are generated from 3 randomly chosen cyclic shifts and the common UMTS turbo interleaver. The BER performance is plotted after 5, 15, 20, and 30 iterations and is compared with the case of using completely randomly generated multiple interleavers. We can observe that the performance of the interleavers generated by our simple strategy is as good as that of the completely randomly generated multiple interleavers. V. I NFORMATION E XCHANGE OF I NTERLEAVERS In decentralized networks it is hard to achieve the perfect coordination of interleaver usages among network nodes. A careful management of interleavers would result in a large signaling overhead. As explained in the previous section, with our proposed method each interleaver is identified with only a few randomly chosen cyclic shifts. This method can be nicely integrated to the simple information exchange procedure described as follows. A transmitter generates a random seed s and exchanges with a receiver a three tuple (s, f, n) which determines a rule to generate interleavers. The same n cyclic shifts are generated from the common random seed s. f and n denote, respectively, a frequency to update the interleaver and the number of cyclic shifts. Updating interleaver would further decrease the probability of interleaver collision. It is suggested that the receiver decides whether the transmission request should be accepted based on certain criteria. The criteria are not limited to the collision of the random seeds. For example, the interference level at the receiver or the receiver’s capability to handle interferences can be taken into account. If the transmission request is rejected, the transmitter may retry the procedure with a new random seed.

The construction of the dithered golden interleaver [10] is briefly summarized here followed by the concrete parameter description used in this paper. The interleaver has a regular structure with some additional randomness. The indices of interleaver size N are calculated by sorting the following values: p(n) = s + nc + d(n) mod N for n = 0, . . . , N − 1 where s is any starting index, d(n) is the n-th dither component which is randomly generated from a uniform distribution between 0 and N W , where W is a normalized width of the dither distribution. m The real √ valued c is computed as c = N (g + q)/r where g = ( 5 − 1)/2 ≈ 0.618 is the golden section value, m is any positive integer greater than zero, r is the index spacing between nearby elements to be maximally spread, and q is any integer modulo r [10]. The following parameters are used in our simulations: s = 0, m = 1, q = 0, r = 1, W = 0.01 as suggested in [10] for a typical implementation. R EFERENCES [1] X. Wang and H. V. Poor, “Iterative (Turbo) soft interference cancellation and decoding for coded CDMA,” IEEE Transactions on Communications, vol. 47, no. 7, pp. 1046–1061, July 1999. [2] P. D. Alexander, A. J. Grant, and M. C. Reed, “Iterative detection in code-division multiple-access with error control coding,” European Trans. on Telecommunications, vol. 9, pp. 419–425, Sep./Oct. 1998. [3] L. Ping, L. Liu, K. Y. Wu, and W. K. Leung, “Interleave-division multiple-access,” IEEE Trans. on Wireless Commun., vol. 5, no. 4, pp. 938–947, April 2006. [4] H. Schoeneich and P. A. Hoeher, “Adaptive interleave-division multiple access - a potential air interface for 4G bearer services and wireless LANs,” in Proc. 1st IEEE and IFIP Int. Conf. on Wireless and Optical Communications and Networks (WOCN ’2004), June 2004, pp. 179–182. [5] K. Kusume and G. Bauch, “CDMA and IDMA: Iterative multiuser detections for near-far asynchronous communications,” in Proc. IEEE Int. Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC 2005), September 2005. [6] ——, “A simple complexity reduction strategy for interleave division multiple access,” to appear in IEEE VTC 2006 fall. [7] S. Br¨uck, U. Sorger, S. Gligorevic, and N. Stolte, “Interleaving for outer convolutional codes in DS-CDMA systems,” IEEE Transactions on Communications, vol. 48, no. 7, pp. 1100–1107, July 2000. [8] A. Tarable, G. Montorsi, and S. Benedetto, “Analysis and design of interleavers for iterative multiuser receivers in coded CDMA systems,” IEEE Trans. on Info. Theory, vol. 51, no. 5, pp. 1650–1666, May 2005. [9] 3G TS 25.212, “3rd generation partnership project; technical specification group radio access network; multiplexing and channel coding (FDD) (release 1999),” March 2000. [10] S. Crozier, J. Lodge, P. Guinand, and A. Hunt, “Performance of turbocodes with relative prime and golden interleaving strategies,” in Proc. of the International Mobile Satellite Conference, 1999, pp. 268–275. [11] D. Divsalar and F. Pollara, “Hybrid concatenated codes and iterative decoding,” California Institute of Technology, Pasadena, California, Tech. Rep. TDA Progress Report 42-130, 1997. [12] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Transactions on Information Theory, vol. 20, no. 2, pp. 284–287, March 1974. [13] M. Moher, “An iterative multiuser decoder for near-capacity communications,” IEEE Transactions on Communications, vol. 46, no. 7, pp. 870–880, July 1998. [14] R. H. Mahadevappa and J. G. Proakis, “Mitigating multiple access interference and intersymbol interference in uncoded CDMA systems with chip-level interleaving,” IEEE Transactions on Wireless Communications, vol. 1, no. 4, pp. 781–792, October 2002. [15] M. Sellathurai and S. Haykin, “Turbo-BLAST for wireless communications: Theory and experiments,” IEEE Transactions on Signal Processing, vol. 50, no. 10, pp. 2538–2546, October 2002.

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