International Journal of Future Computer and Communication, Vol. 2, No. 4, August 2013 Quadratic Permutation Polynomial Interleavers for LTE Turbo Coding Ching-Lung Chi Abstract—Interleaver is an important component of turbo min δ, (2) , code and has a strong impact on its error correcting performance. Quadratic permutation polynomial (QPP) δ, is the Lee metric between points interleaver is a contention-free interleaver which is suitable for , π and p j,πj parallel turbo decoder implementation. To improve the performance of LTE QPP interleaver, largest-spread and , || | | (3) maximum-spread QPP interleavers are analyzed for Long Term Evolution (LTE) turbo codes in this paper. Compared where with LTE QPP interleaver, those algorithms have low computing complexity and better performances from || mod , mod (4) simulation results. The quadratic polynomials which lead to the largest spreading factor for some interleaver lengths are given in Index Terms—Quadratic permutation polynomial (QPP) interleaver, Long Term Evolution (LTE), largest-spread QPP [2]. An algorithm for faster computation of is also interleaver, maximum-spread QPP interleaver. presented. It is based on the representatives of orbits in the representation of interleavercode. Section II gives a brief review of QPP interleaver. Largest-Spread and Maximum-Spread QPP interleavers are I. INTRODUCTION the topics of Section III. Section IV presents the bit error The selection of turbo coding was considered during the rates (BER) resulted from simulations for LTE standard, LS study phase of Long Term Evolution (LTE) standard to and MS QPP inter leavers length 512. Section V concludes meet the stringent requirements. Following deliberations the paper. within the working group, the quadratic permutation polynomials (QPP) were selected as interleavers for turbo codes, emerging as the most promising solutions to the LTE II. QPP INTERLEAVERS requirements. The QPP interleavers for the LTE standard [1] involve 188 different lengths. In this paper we take two A. Permutation Polynomial(PP) over Integer kinds of methods to improve the LTE QPP interleaver. Given an integer2, a polynomial The polynomial interleavers offer the following benefits , where , , , , and are [2]: special performance, complete algebraic structure, and nonnegative integers, is said to be a permutation polynomial efficient implementation (high speed and low memory over when permutes 0, 1, 2, , 1. requirements). A QPP interleaver of length L is defined In this paper, all the summations and multiplications are as: modulo unless explicitly stated. We further define the π x mod ,01 (1) formal derivative of the polynomial to be a where and are chosen so that the quadratic polynomial such that 23 polynomialin (1) is a permutation polynomial (i.e. the . For any integer , whether polynomial set π0,π1,,π1 is a permutation of the is a permutation polynomial can be determined by set 0,1, , 1 and determines a shift of the the following theorem [3]. permutation elements. Theorem 1: Let 23 In the following we only consider quadratic polynomials be a polynomial with integer coefficients. is a permutation polynomial over the integer ring 2 if with free term 0 , as for the QPP interleavers in the and only if 1) is odd, 2) is even, and LTEstandard. If 0,1, , 1 , then the 3) is even. permutation function is: . The spread factor D is defined as Theorem 2: is a permutation polynomial over the integer ring2 if and only ifis a permutation polynomial over Z and 0modulo for all integersx. Manuscript received October 25, 2012; revised November 25, 2012. Theorem 3: For any ∏ , where are Ching-Lung Chi is with the Dept. of Computer and Communication SHU-TE University Kaohsiung City 824, Taiwan (e-mail: distinctprime numbers, is a permutation polynomial [email protected]) modulo if and only if is also a permutation DOI: 10.7763/IJFCC.2013.V2.186 364 International Journal of Future Computer and Communication, Vol. 2, No. 4, August 2013 polynomial modulo , . TABLE I: EXAMPLES OF MAXIMUM-SPREAD QPP INTERLEAVER ζ ζ 4 128 15 32 17 32 16 2 2 64 5 512 31 64 33 64 35 4 3 128 6 2048 63 128 65 128 64 8 4 256 7 8192 127 256 129 256 128 12 7 512 8 32768 255 512 257 512 256 32 12 1024 9 131072 511 1024 513 1024 512 64 23 2048 Because the function of interleaver is to permute search metric on choosing better coefficients, before this we 0,1, , 1 , we can apply the construction of present some definitions firstly: permutation polynomial over integer to Turbo 1) Distance interleavers by properly choosing and polynomial coefficients . The first-degree polynomial , | | | | (5) (linear interleaver) is the simplest one among all where|| minmod , mod . permutation polynomials. However, the linear interleaver has very bad input weight 4 error event characteristics, 2) Spread Factor causing a higher error floor for medium or long frame size min , , (6) cases. And quadratic permutation polynomial can overcome where, and . these defects, this leaves it the next in the chain. When 2, becomes . Notice 3) Nonlinearity Metric of QPP Interleavers that the coefficient just corresponds to a cyclic rotation ⁄ gcd2, (7) in the interleaved sequence. It does not appear in the conditions for to be a permutation polynomial and it Obviously, QPP interleaver can be viewed as a linear does not affect the performance. We ignore for interleaver that is “disturbed” by quadratic simplification, so becomes quadratic polynomial interleaver , the periodicity of the disturbance is at most.This disturbance can overcome a higher error floor of .. linear interleaver at a certain extent. B. QPP Interleavers Since the minimum distance of a turbo code is now QPP Interleaver completes interleaving by making QP known to grow at most logarithmically, therefore, the spread (Quadratic Polynomial) satisfy some conditions. In order factor that controls the effective free distance should be to makebe a QPP, coefficients and have to meet “rewarded” at most logarithmically. The nonlinearity is some special conditions. Then the following 2 corollaries expected to have a proportional reduction in the deduced from theorem 1 and 2 are given. multiplicities of low-weight code words so it is reasonable Corollary 1 [3]: A quadratic polynomial of the to leave it. So we can get a simple metric of searching form is a permutation polynomial coefficients of QPP interleaver [4]. over if and only if 0and 0 mod. , Corollary 2 [4]: Let∏ , denote divides maxΩ ln (8) by | and denote can not divides by . The Further choosing coefficients and by necessary and sufficient condition for a quadratic maximizationΩ, we can get some coefficients with good polynomial to be a permutation performances. Sincewill grow fast as grows, it is not polynomial can be divided into two cases. good for improving decoding performance, so it must be 1) 2|and4 (i.e., , 1) modified. We define a new modified nonlinearity metric , is odd, gcd,⁄ 2 1, and ∏ , which can avoid above defect. So we have a corresponding refined metric , 1, psuch that2and, 1. maxΩ ln (9) 2) Either 2 or 4| (i.e., , 1 ) gcd , 1and ∏ , , 1, psuch , where mod |0,1,,1 that, 1. Through this refined metric, we search coefficients more Apply above corollaries, we can find some available than one couple, since the up-bound of interleavers spread coefficients and in QPP interleavers by computer factor is √2 , in order to avoid not searching. However, the interleavers with different becoming so small when frame sizeis small that we limit coefficients have different performances, so we have to the search process under the condition of √2 , choose better based on these coefficients. Then we discuss 365 International Journal of Future Computer and Communication, Vol. 2, No. 4, August 2013 where 0.45. While 2000, must decrease to 0.30 Relationships (19) and (20) could also be obtained as properly. follows. From (12) we have ⁄ 2 (21) III. LARGEST-SPREAD AND MAXIMUM-SPREAD QPP ⁄ (22) INTERLEAVERS 2 A. Largest-Spread QPP Interleaver Then, from (14) we get In this section we present a theorem which states · (23) sufficient conditions to be satisfied by the coefficients of Or, taking into account (21), (22) and (13), two quadratic polynomials [4], so that the resulting ⁄ interleavers are identical. For the LTE standard, the · 2· (24) interleaver’s length is always an even number. Or, from (15), Theorem 4: Consider two QPP interleavers described by ·2⁄· · (25) the following polynomials (the free term is considered zero): From here, (19) and (20) result immediately. mod , x 0,1, , L 1 (10) Because 0,1, , 1, 1 and 1are mod , x0,1,,L1 (11) divisible by 2, i.e. then, there are, which verify relation (13). Ifis even, then , and the following relation is fulfilled: The theorem shows that two QPPs generate identical permutation functions, if the coefficients are at the same ⁄ 2 (12) distance 2⁄. Therefore, we can only consider The two quadratic polynomials lead to identical coefficients 0,1,,2⁄1in polynomial searching, permutations. because the interleaver mod is the Proof: same as theinterleaver ⁄ 2 For the two QPP to lead to identical permutations, it is 2⁄ mod , if ⁄ 2, or as the interleaver described required that by the permutation ⁄ 2 , 0,1,,1 (13) 2⁄ mod , if ⁄ 2. As the number of searched We denote QPPs is halved, so is the search time, therefore speeding up the search process. · (14) B. Maximum-Spread QPP Interleaver · (15) Theorem 5: The following is an infinite sequence of QPPs where , .Under the conditions above we have to that generate maximum-spread interleavers: Show that there are , , 0,1, , 1 , which verify relationship (13). Subtracting (14) from (15) 2 1x2mod2 k 1,2,3, and considering (13), we have: (26) Strictly, we have QPP interleavers only when 3.