K. Kusume, G. Bauch, "Simple Construction of Multiple Interleavers: Cyclically Shifting a Single Interleaver," in IEEE Transactions on Communications, vol. 56, no. 9, pp. 1394-1397, September 2008.

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 9, SEPTEMBER 2008

Simple Construction of Multiple Interleavers: Cyclically Shifting a Single Interleaver Katsutoshi Kusume, Member, IEEE, and Gerhard Bauch, Senior Member, IEEE

(k)

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Paper approved by A. Anastasopoulos, the Editor for Iterative Detection, Estimation and Coding of the IEEE Communications Society. Manuscript received November 7, 2006; revised May 17, 2007. This paper was presented in part at the International Symposium on Turbo Codes & Related Topics, April 2006, and at the IEEE Global Telecommunications Conference on Communications, November 2006. The authors are with DOCOMO Euro-Labs, Landsbergerstr. 312, 80687 Munich, Germany (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2008.060420.

Πk

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INGLE interleaver construction rules for turbo codes have been relatively well-studied in the last years and many proposals can be found in literature. Recently, multiple interleavers attracted more and more attention in many research areas such as code division multiple access (CDMA), multi-dimensional concatenated codes, and interleave division multiple access (IDMA), and so on. In most cases, multiple interleavers are generated in a completely random manner and only a few papers on the generation of multiple interleavers can be found in literature, e.g., [1]–[3]. A typical way to get K random interleavers of size N is independently repeating a random interleaver generation procedure K times. And the generation of individual random interleaver is drawing N pseudo-random numbers from a uniform distribution, sorting them, and the resulting sorting pattern is used as an interleaver. Simulation results in literature are often the performance averaged over a large number of interleaver realizations. This is a convenient way to evaluate systems when the interleaver design itself is not the main focus. In practical systems, however, the above procedure is too complex and storing multiple interleavers may require a prohibitively high amount of memory, particularly if variable block lengths have to be supported. Hence, we wish to have a simple construction rule for interleavers which minimizes the required memory. In this article we focus on IDMA, e.g., [4], which has a close relation to CDMA, but users are separated only by user-distinct interleavers. Hence, multiple interleavers are essential system components for IDMA. We will show that the conventional interleavers are insufficient for the user

Convolutional encoder

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Index Terms—Interleaver design, multiple interleavers, permutations, interleave division multiple access (IDMA), iterative multiuser detection.

I. I NTRODUCTION

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Abstract—Utilizing multiple interleavers recently attracted increasing attention in many research areas. Interleaver generation should be simple in order to avoid huge memory requirements for storing interleaving patterns. We propose to derive multiple interleavers by cyclically shifting and self-interleaving a common mother interleaver in a few steps. Our focus is on the good user separation in interleave division multiple access systems. The proposed method may also find other application areas such as multi-dimensional concatenated codes.

System model of IDMA.

separation in IDMA. A new practical method to generate multiple interleavers is necessary. II. S YSTEM M ODEL We consider the system model of IDMA in Fig. 1. In the following, we briefly summarize the system model and the iterative processing at the receiver. A detailed description in a more general setting can be found, e.g., in [4]. At the (k) transmitter, Nb information bits b of user k, k = 1, . . . , K, are encoded by a rate Rc convolutional code followed by a rate Rr repetition code. The resulting N code bits are (k) interleaved by the user-distinct interleaver Πk to get cj,i . (k) (k) The pairs of bits cj,1 and cj,2 are mapped onto the complex (k) symbols xj , which are elements of a QPSK constellation (k) j { √12 + √2 , √12 − √j2 , − √12 + √j2 , − √12 − √j2 }. The symbols xj are then transmitted over an AWGNchannel. The received (k) K + ηj where signal can be expressed as yj = k=1 xj the noise ηj is zero-mean complex Gaussian distributed with variance N0 /2 per real dimension. The receiver applies iterative detection and decoding. The multiuser detector (MUD) computes a posteriori log(k) likelihood ratio (LLR) about the code bits L(m) (cj,i ) = (k)

log

P (cj,i =+1 | yj ) (k)

P (cj,i =−1 | yj )

based on the received values yj and on (m)

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the a priori LLRs La (cj,i ) which are sent from the K a posteriori probability (APP) decoders and are initialized (m) (k) (k) to zeros. The extrinsic LLRs Le (cj,i ) = L(m) (cj,i ) − (m) (k) La (cj,i ) are deinterleaved and sent to the decoder as the (d) (k) a priori LLRs La (cj,i ). For the QPSK modulation, the (m) (k) (m) (k) as Le (cj,1 ) = computation of Le (cj,i ) can be simplified √ √ (k) (m) (k) (k) 2 2 Re{¯ yj } and Le (cj,2 ) = 2(k)22 Im{¯ yj } where (k) 2 σj σj  (k) 2 (k ) (k) (k) (k) = + N0 , ρj = E[|xj − x ˜j |2 ] = σj k =k ρj

c 2008 IEEE 0090-6778/08$25.00 

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KUSUME and BAUCH: SIMPLE CONSTRUCTION OF MULTIPLE INTERLEAVERS: CYCLICALLY SHIFTING A SINGLE INTERLEAVER

 (k) (k) (k ) (k) (k) 1 − |˜ xj |2 , y¯j = yj − k =k x˜j , x˜j = E[xj ] = √ (m) (k) (m) (k) (tanh(La (cj,1 )/2) + j tanh(La (cj,2 )/2))/ 2, and E[·] denotes expectation. The APP decoder computes improved (k) (k) a posteriori LLRs L(d) (cj,i ) about code bits and L(d) (b ) = (k)

P (b

=+1)

 log about information bits taking into account the (k) P (b =−1) code constraints. The APP decoding is a standard function (d) (k) and will not be described. The extrinsic LLRs Le (cj,i ) = (k) (d) (k) L(d) (cj,i ) − La (cj,i ) are sent to the MUD as the a priori (m) (k) LLRs La (cj,i ) after interleaving. After some iterations, (k) taking the sign of L(d) (b ) gives the estimation of the information bits.

III. P REVIOUS W ORKS In [1] 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 permutation rule of a congruential interleaver of size N reads as: πk (n) = sk + npk mod N

for n = 0, . . . , N − 1,

(1)

where sk is a starting integer index often set to 0, and pk is an integer increment which 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 pk (and sk if non-zero). The idea in [1] is to find user-specific congruential interleavers (i.e., to choose pk ’s) for a given convolutional code (a code is common for all users) such that the resulting interleavers yield good asymptotic distances between the effective code words after interleaving. Unfortunately, the method has a strict limitation on block size and cannot always find proper interleavers for a given block length N . This is particularly a problem for low rate codes since the large free distance causes the algorithm to demand a large minimum required block size (cf. Theorem 2 in [1]). In [2] 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, only symbol interleavers are considered, 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” [2]. Similar to [1], there is a strict limitation on the symbol interleaver’s block length N  = N/n0 , which must be a prime number. “Power interleavers” proposed in [3] are generated from a common interleaver and are particularly relevant to this work. Given a “master interleaver” Π (permutation matrix), K power interleavers are generated as Πk = Πk . A major drawback is 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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(b) common interleaver Π and cyclic shift τk(c)

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Πk (c) repeat (b) with multiple cyclic shifts τk,1 ,..., τk,D (c)

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Fig. 2. (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.

large delays of interleaver generation for large K. Unequal delays for different k may be unfavorable in a certain system as well. IV. C YCLICALLY S HIFTED M ULTIPLE I NTERLEAVERS Our design is inspired by some observations regarding impacts of multipath channels and user asynchronism in IDMA [5]. It turned out that user-asynchronism, i.e., different user transmission delays, has a positive impact on the user separation. Even when all users share a single common interleaver in IDMA systems (in fact, it is not “interleave division” any more), user separation is still possible to some extent if user-distinct transmission delays are present. That is due to the fact that the interleaver cycle in a received sequence is shifted according to a user delay so that the deinterleaving operation of certain user does not recover the original sequence orders of other users, thus other users’ signals are effectively decorrelated. The observation leads us to a very simple strategy for generating multiple interleavers as summarized in Fig. 2. The aforementioned scenario is depicted in Fig. 2 (a) where τk denotes a user delay. Since user separation is still possible when user delays are present, a common interleaver Π together with a user delay τk can be virtually seen as a simple form of a user-distinct interleaver. However, intentional user transmission timing control (to introduce user-asynchronism) is not preferred from a system design point of view and it is even hard to realize in some scenarios such as decentralized systems. Therefore, we propose to exploit a cyclic shift τk(c) (the superscript ‘(c)’ indicates cyclic) as illustrated in Fig. 2 (b) because of its analogy to the user delay. Then, each pair of the common interleaver, which we call mother interleaver, and the cyclic shift τk(c) can be regarded as a newly generated interleaver. The choice of cyclic shifts is completely independent of the presence of user delays, then it is not required to properly control user delays aiming at good user separation. The idea can be easily generalized by using D (c) (c) , . . . , τk,D , i.e., by cascading D pairs of the cyclic shifts, τk,1 cyclic shift and the common mother interleaver as illustrated in Fig. 2 (c). In other words, each interleaver is derived by cyclically shifting and self-interleaving the common mother interleaver in D steps.

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 9, SEPTEMBER 2008 0

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Fig. 3. BER performance of IDMA on an AWGN channel after 16 iterations for K = 2 users.

Fig. 4. BER performance of IDMA on an AWGN channel after 14 iterations for K = 6 users.

A set of cyclic shifts for K interleavers may be optimized for a given scenario depending on a channel, user delays, an FEC code, a block length, the number of users, and so forth. However, we propose to choose them randomly. No optimization effort is necessary if the performance is sufficiently good when using randomly chosen cyclic shifts. As we will see later, only a few randomly chosen cyclic shifts are sufficient for good user separation. This is a desirable feature since each interleaver is determined by only a few cyclic shifts and there is no need to store a large sets of interleavers where each of them may be optimized for a particular scenario. Furthermore, each interleaver can be even characterized by only a single parameter of a random seed, from which a few cyclic shifts are easily generated.

rather poor (labeled as “Tarable”). The performance using the interleavers from [1] (labeled as “Br¨uck”) is also poor. The computer search according to the algorithm in [1] found p1 = 1 and p2 = 13 (s1 = s2 = 0). Since both of these two conventional interleavers in [1], [2] are congruential interleavers, randomly chosen congruential interleavers are also reasonable to be considered and are evaluated. All the possible values for pk can be found that are relatively prime to the block size N . Each user randomly picks one value out of all the possible values of pk which is updated for every transmission frame. Moreover, the starting index sk is randomly chosen and updated in order to introduce further randomness. No effort is made to ensure that pk and sk are distinct for the two users. The high error floor can be observed in Fig. 3. Now, we evaluate the performance of interleavers based on our proposal. The UMTS turbo interleaver [6] is used as the mother interleaver, just as an example2. D = 2 cyclic shifts are randomly chosen and updated for every transmission. There is no guarantee that the cyclic shifts are distinct for users. We see that the performance is as good as using completely randomly generated interleavers. As the second scenario, we compare the performance of K = 6 users where Nb = 128 information bits are encoded by the rate Rc = 1/2 memory 4 standard convolutional code ([31, 27]8, the trellis is terminated) followed by the rate Rr = 1/4 repetition code. The resulting interleaver size is N = 1056. The BER performance after 14 iterations is averaged over all users and is plotted in Fig. 4. The block size N  = 139 of symbol interleaver is found by the computer search according to the algorithm in [2] (N = 1112 close to our target N = 1056). Any integer 1 ≤ pk < 139 can be chosen for congruential interleavers in (1), and here pk = k

V. S IMULATION R ESULTS Simulation results in two different scenarios are presented to compare the performance of IDMA systems using interleavers generated from different schemes. In the first scenario, we consider K = 2 users where information bits are encoded by a rate Rc = 1/2 memory 2 standard convolutional code with the generator polynomial [7, 5]8 . The trellis of the encoder is terminated. No repetition code is applied. The BER performance averaged over two users for the block size N = 8000 after 16 iterations is illustrated in Fig. 3. For comparison, we also plot the single user bound as well as the performance using completely randomly generated interleavers (a usual case in literature). As explained in Section III, symbol level congruential interleavers with certain constraint on the block size are considered in [2]. The block size N  = 4003 (N = 8006) of symbol interleavers is found by computer search. This value is chosen to be close to the current target N = 8000. Then, any integer 1 ≤ pk < N  can be chosen for congruential interleavers in (1). Here, p1 = 1 and p2 = 113 (the first user has no interleaving as it is the case in [2]) and sk = 0 for both users. The performance using these interleavers is

2 UMTS turbo interleaver is defined for 40 ≤ N ≤ 5114. The definition is extended such that the inter-row permutation pattern in [6] is chosen as T (j) = P at1 for N > 5114, just to obtain a deterministic interleaver for convenience.

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KUSUME and BAUCH: SIMPLE CONSTRUCTION OF MULTIPLE INTERLEAVERS: CYCLICALLY SHIFTING A SINGLE INTERLEAVER

is chosen and sk = 0 for all users. It can be observed that the performance using these interleavers, labeled “Tarable”, shows the high error floor. We note that the algorithm does not take advantage of the repetition code since no optimization is provided in [2] for the case of a convolutional code followed by a repetition code. For the given coding parameters, the algorithm in [1] identifies the required minimum block size that is 20800 in this example. In order to find interleavers for 6 users, the block size must be even larger, and with N = 21104 we found p1 = 1, p2 = 105, p3 = 525, p4 = 2625, p5 = 8125, and p6 = 12979. The starting index is set sk = 0 for all users. In spite of the required long block size (roughly 20 times larger than the current target block size), users cannot be separated (see the curve labeled “Br¨uck”). Using random congruential interleavers also results in the high error floor. The performance of power interleavers Πk = Πk with the UMTS turbo interleaver Π is almost as good as that of random interleavers. With a random choice of the exponent 1 ≤ mk ≤ 100 for Πk = Πmk , however, we observe the high error floor (see the curve labeled as “random power interleavers”). Note that no effort is made to avoid interleaver collisions. Such random power interleavers may be desired in a decentralized network or in a cellular network where there is no cooperation among base stations regarding the interleaver assignment. Increasing the maximum value of mk can naturally reduce the collision probability, but larger delays for generating interleavers result. Finally, we evaluate the performance of interleavers based on our proposal. In particular, we are interested in the impact of the number of cyclic shifts on the performance. The BER performance is plotted for 1, 2 and 3 randomly chosen cyclic shifts being used together with the common UMTS turbo interleaver. Relatively high error floors can be observed for one and two random cyclic shifts. With three random cyclic shifts, no error floor can be observed in error rates of practical interest.

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VI. C ONCLUSIONS We proposed a simple method for generating multiple interleavers which are derived from a common mother interleaver with only a few randomly chosen cyclic shifts and selfinterleaving. Any interleaver may be used as the common mother interleaver. It turned out that only a few randomly generated cyclic shifts are sufficient for good user separation in IDMA systems and no further optimization effort is necessary. Although our focus was on the user separation in IDMA systems, interleavers generated by the proposed scheme may be applied in many other application areas such as multidimensional concatenated codes. ACKNOWLEDGMENT The authors thank anonymous reviewers for their constructive critics and suggestions, and also for pointing out the reference [3] which has particular relevance to this work. R EFERENCES [1] S. Br¨uck, U. Sorger, S. Gligorevic, and N. Stolte, “Interleaving for outer convolutional codes in DS-CDMA systems,” IEEE Trans. Commun., vol. 48, no. 7, pp. 1100–1107, July 2000. [2] A. Tarable, G. Montorsi, and S. Benedetto, “Analysis and design of interleavers for iterative multiuser receivers in coded CDMA systems,” IEEE Trans. Inform. Theory, vol. 51, no. 5, pp. 1650–1666, May 2005. [3] H. Wu, L. Ping, and A. Perotti, “User-specific chip-level interleaver design for IDMA systems,” IEE Electron. Lett., vol. 42, no. 4, pp. 233– 234, Feb. 2006. [4] L. Ping, L. Liu, K. Y. Wu, and W. K. Leung, “Interleave-division multipleaccess,” IEEE Trans. Wireless Commun., vol. 5, no. 4, pp. 938–947, Apr. 2006. [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), Sept. 2005. [6] 3G TS 25.212, “3rd generation partnership project; technical specification group radio access network; multiplexing and channel coding (FDD) (release 1999),” Mar. 2000.

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