on the application of low complexity iterative decoding ...

ON THE APPLICATION OF LOW COMPLEXITY ITERATIVE DECODING SCHEMES TO FADING CHANNELS Wei Zhang, Xiaowei Jin, Hongzhi Chen, Adrish Banerjee, Daniel J. Costello Jr., and Thomas E. Fuja Department of Electrical Engineering University of Notre Dame Notre Dame, IN 46556, U.S.A. email:{wzhang2, xjin, hchen1, banerjee.5, costello.2, tfuja}@nd.edu ABSTRACT Bit-interleaved coded modulation (BICM) combines coding and modulation in a way that is well-suited to the requirements of fading channels. If turbo codes or low density parity check (LDPC) codes are employed in a BICM scheme, it is possible to achieve very good performance at signal-to-noise ratios near capacity. In this paper, we investigate the application of BICM to fading channels using low complexity (2- and 4-state) multiple turbo codes and simple, algebraically constructed LDPC codes. Specifically, we consider modulation schemes operating at 1 bps/Hz and 2 bps/Hz with rate-1/2 coded QPSK and 16-QAM modulation, respectively. I. Introduction Error control coding is an indispensable tool for achieving efficient and reliable communication over a noisy channel. The last decade has seen the development of powerful techniques, such as turbo codes and low density parity check (LDPC) codes, that can achieve near-Shannon-limit performance when used in conjunction with iterative decoding. However, these codes as originally formulated focused on low rate applications with binary modulation; as a result, to achieve good performance they require bandwidth expansion, which limits their use to applications that do not require high spectral efficiency. On bandwidth-constrained channels, the combination of error control coding and multilevel modulation known as coded modulation can provide both good performance and high spectral efficiency. For transmission in multipath environments such as those encountered in mobile communications, the channel can be modeled as being impaired by multiplicative fading with coefficients described by a Rayleigh distribution. The primary design criterion of a coded modulation scheme for the Rayleigh fading channel is its “diversity order” - i.e., the minimum symbol-wise Hamming distance between any two valid modulated sequences. Bit interleaved coded modulation (BICM), as first proposed by Zehavi [1], uses a convolutional code followed by a bit interleaver and a mapper to increase the diversity order of the modulated signal; the bit-interleaver spreads out the bit-wise Hamming distance provided by the convolutional code over multiple modulated This work was supported in part by U.S. Army Grant DAAD16-02-C-0057, NSF Grants CCR00-75514 and CCR02-05310, and NASA Grant NAG5-10503.

symbols, thereby increasing the diversity order and so providing better coding gains on fading channels than those realized by conventional trellis coded modulation (TCM). BICM using turbo codes was first proposed by Le Goff, Glavieux, and Berrou [2]. They used a binary turbo encoder whose output bits are interleaved, grouped into mtuples, and then mapped to an M = 2m -ary signal set. The turbo encoder in [2] is constructed by concatenating in parallel two identical (symmetric) 16-state recursive systematic convolutional constituent encoders. Both coherent additive white Gaussian noise (AWGN) channels and channels with independent Rayleigh fading were considered. Subsequently, several other bandwidth efficient schemes using turbo codes have been developed. The recent paper by Costello et al. [3] discusses many of these approaches and shows that BICM using turbo codes provides a practical and flexible solution well suited to most applications requiring a combination of high bandwidth efficiency and near-capacity achieving performance. In [4], Banerjee et al. presented some BICM designs based on asymmetric turbo codes. (Asymmetric turbo codes use non-identical constituent encoders [5].) Also, in [6], Banerjee et al. introduced bandwidth efficient code designs based on multiple turbo codes. (Multiple turbo codes use three or more parallel concatenated constituent encoders connected by different interleavers [7]). Both symmetric and asymmetric configurations were considered. On AWGN channels, the multiple turbo code designs outperformed the original design from [2] not only in the waterfall region but also in the error floor. Moreover, compared to the original design, the constituent encoders were simpler (2- or 4- state), thus resulting in less decoding complexity. LDPC codes are linear codes described by a sparse parity check matrix. They have a graph-theoretic interpretation – i.e., the parity check matrix can be represented by a bi-partite graph consisting of “variable nodes” and “constraint nodes”, with an edge connecting a variable node to a constraint node if and only if that parity constraint includes that variable. Decoding is carried out by iteratively passing “messages” (log-likelihood ratios) along the edges of the graph. LDPC codes were originally constructed using pseudo-random techniques. Recently, however, some algebraic construction schemes have surfaced [8], [9], [10]. The algebraic structure makes these codes better suited for analysis than randomly constructed codes, thus guaranteeing a

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certain level of performance. Also, the expense of storing a large parity check matrix needed in a random construction is reduced. Finally, the algebraic structure lends itself to a straightforward encoding procedure. All these features make algebraically constructed LDPC codes potentially attractive alternatives to turbo codes in BICM applications. A particular algebraic LDPC construction first proposed by Tanner [8] (and generalized by Sridhara et al. [11]) uses a parity check matrix composed of cyclically-shifted identity matrices. The resulting code is quasi-cyclic and has a sparse graph structure well suited to message passing decoding. It was shown in [11] that, for short-to-moderate blocklengths, these codes perform comparably to randomly constructed LDPC codes on the BPSK-modulated AWGN channel. Furthermore, the quasi-cyclic nature of these codes permits a convolutional representation, thus allowing for the possibility of continuous encoding and decoding. The use of these codes in a BICM context for spectral efficiency on AWGN channels was investigated in [12]. In this paper we extend the use of the BICM schemes proposed for AWGN channels in [6] and [12] to the case of independent and correlated Rayleigh fading channels. Section II describes the proposed systems and the code designs. In Section III, simulation results are presented. Finally, a short summary and discussion in Section IV concludes the paper. II. System Design The BICM system first proposed in [1] is shown in Figure 1. The information sequence u is encoded into a code sequence c. Any kind of channel encoder, such as convolutional, turbo, or LDPC, can be selected. The coded bits c from the output of the encoder are interleaved at the bit level into v = π(c), where π represents the interleaving pattern. The output stream of the interleaver is broken into m-bit words, which are then mapped to points in the signal constellation X ⊆ C, where |X | = 2m and C is the set of complex (2-dimensional) signals. Then the signal sequence s is sent over a non-frequency selective Rayleigh fading channel. The received sequence is r = a · s + n, where n is additive white Gaussian noise and a is the fading parameter. We assume that the channel state information a is fully known at the receiver. The demapper calculates the channel log-likelihood ratios LLR(v|r) before they are deinterleaved into LLR(c|r). The decoder then performs iterative decoding based on the LLR(c|r)’s and produces ˆ for the input information bits. hard-decision estimates u If the code rate is R, then the spectral efficiency is Rm bps/Hz. We consider Gray mapping (see Figure 2) only, since it yields the best performance for BICM [13] and gives the best initial extrinsic estimates for iterative decoding. Also, since no additional gain is achieved with joint iterative demapping and decoding using Gray mapping [14], we separate demapping and decoding in this paper. A coherent receiver with perfect channel state information is considered. Let rt denote the received signal at time t, and let its in-phase and quadrature components, rI,t and

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rQ,t respectively, be given by rI,t = at · sI,t + nI,t rQ,t = at · sQ,t + nQ,t . (1) Here, sI,t and sQ,t are the in-phase and quadrature components of the transmitted symbol at time t. The terms nI,t and nQ,t are two independent Gaussian noise samples with 2 . If we assume an ideal symbol zero mean and variance σN interleaver is used prior to transmission, then {at } consists of i. i. d. Rayleigh distributed random variables. In a more realistic model, the at ’s are temporally correlated within the channel coherence time Td . (See Section III.) Assume that each codeword consists of n bits, and denote the channel LLR of bit vtm+j as LLR(vtm+j ). Here, t indexes which 2m -ary symbol conveys the bit (0 ≤ t ≤ (n/m) − 1) and j indexes the bit within the symbol (0 ≤ j ≤ m − 1). Then   P (vtm+j = 0 | rt , at ) LLR(vtm+j ) = log P (vtm+j = 1 | rt , at ) "P # xt ∈X0j p(rt | xt , at ) = log P xt ∈X1j p(rt | xt , at ) (equally likely P (xt ) is assumed) "P # 1 2 2 krt − at · xt k } xt ∈X0j exp{− 2σN = log P , 1 2 xt ∈X1j exp{− 2σ2 krt − at · xt k } N

(2) b ∈ {0, 1}, consists of those signal points in X where with a binary label (under the mapping) with a value of b in position j . In this paper, Gray mapping is assumed.

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Xbj ,

A. Low Complexity Turbo Codes It has recently been shown that the parallel concatenation of a bank of simple encoders separated by several interleavers, i.e., multiple turbo codes (see Figure 3), can perform as well as a more complex conventional turbo code (with one interleaver) at the same rate [15], [16]. The constituent codes (CC’s) in Figure 3 can be the same, resulting in a symmetric configuration, or they can be different, resulting in an asymmetric configuration. To achieve variable spectral efficiencies using a specific encoder and signal constellation, the outputs of the turbo encoder can be punctured with different puncturing patterns before bitinterleaving and mapping. u CC1 Interleaver1

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As an example, consider a simple 4-state [1 1+D+D 1+D ] Big Numerator Accumulator (BNA) constituent encoder in a symmetric parallel concatenation configuration with two random interleavers [15]. We refer to this as the 3BNA turbo code. The rate of the unpunctured code is 1/4, so to bring the overall rate to 1/2 we use the puncturing pattern shown in Figure 4(a) for both 16-QAM and QPSK1 . The rate 1/2 coded bits are mapped to a 16-QAM signal set with 4 bits per symbol and to a QPSK signal set with 2 bit per symbol, so the spectral efficiencies are 2 bps/Hz with 16-QAM and 1 bps/Hz with QPSK. We use the Gray mapping rules shown in Figures 2(a) and 2(b). It should be noted that for the puncturing pattern in Figure 4(a), some of the information bits are deleted as well as some of the parity check bits; this is referred to as a partially systematic code.

The multiple turbo code design can include more than three constituent encoders (and, as a result, more than two interleavers). However, to achieve an overall rate of 1/2 in this case, more bits must be punctured, possibly including all of the information bits. Nonsystematic multiple turbo codes result if no information bits are transmitted[17]. (See the puncturing pattern in Figure 4(b).) As an example, we use an asymmetric multiple turbo code with four encoders and three interleavers [16]. It includes a parallel concate1 ] encoders and nation of three accumulator (ACC) [1 1+D one feedforward (FF) [1 1 + D] encoder. So the constituent encoders all have two states. The rate of the unpunctured code is 1/5, and we refer to this as the 3ACC-1FF turbo code. To make the overall rate 1/2, we use the puncturing patterns shown in Figure 4(b) for 16-QAM and QPSK. The mappings are the same Gray mappings used for the 3BNA scheme above. In the decoding of multiple turbo codes, we use the maximum a posteriori probability (MAP) decoding algorithm in each constituent decoder. In one decoding iteration, the computational complexity is proportional to the total number of encoder states. Since each encoder has few states (2 or 4), decoding complexity is effectively reduced. There are several choices for the decoding structure in terms of how the constituent decoders exchange information [18]. This paper uses the extended serial structure (see Figure 5), since it yields the best performance for a given number of iterations [18]. In the extended serial structure, the most recent extrinsic LLR’s from the other decoders are added to form the a priori LLR for the current decoder.

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Fig. 5. Extended serial decoding structure.

B. Algebraically Constructed LDPC Codes SYS: X

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Fig. 4. Puncturing patterns for multiple turbo codes 1 Rate-1/2 QPSK is not normally considered “bandwidth efficient,” but we include it here to indicate the wide range of spectral efficiencies that can be achieved using BICM.

The approach to algebraic LDPC code design that is considered in this paper was introduced in [8] and subsequently generalized in [11]. The construction is based on permutation matrices. The parity check matrix H consists of a j × k array of p×p permutation matrices, where p is a prime. Each of the sub-matrices is an identity matrix with rows cyclically shifted by a specific amount that is determined by the structure of a multiplicative group. The resulting H matrix is a jp × kp sparse matrix with column weight j , row weight k, and code rate R ≥ 1 − (j/k). Compared with randomly constructed LDPC codes, the LDPC code thus constructed has a more structured parity check matrix, and the resulting code is quasi-cyclic. (A quasi-cyclic code has the property that any codeword cyclically shifted by s places results in another codeword. A cyclic code is a quasi-cyclic code for which s = 1.) The algebraic structure of these codes facilitates encoder and decoder implementation.

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The construction is now presented in more detail. For a given prime p, the column weight j and row weight k are selected from the prime factors of p − 1. Furthermore, two nonzero elements α and β are chosen from GF(p) such that their multiplicative orders are k and j , respectively. Then the matrix P is given as   1  β     2 k−1 β2   P =   .  1 α α ··· α  ..  β j−1   1 α α2 ··· αk−1  β αβ α2 β ··· αk−1 β    =  . (3) .. . . . . .  .  ··· . . . j−1 j−1 2 j−1 k−1 j−1 β αβ α β ··· α β where all multiplications are carried out modulo p. The parity check matrix H is obtained from P by replacing the (s, t)th element of P, denoted Ps,t , 1 ≤ s ≤ j, 1 ≤ t ≤ k, with a p × p identity matrix with rows cyclically shifted by an amount Ps,t . Figure 6 illustrates the cyclic shifting of the rows of an identity matrix. The circulant nature of the shifted identity matrices results in the quasi-cyclic structure of the H matrix. One can observe the quasi-cyclic nature of these codes by reordering the columns of H. Ps,t 1

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Fig. 6. Cyclic shifting of the rows of an identity matrix.

As an example, for p = 31 we choose the row weight k = 5 and column weight j = 3 with generators α = 2 and β = 5. Using these values, the matrix P can be calculated as " # 1 2 4 8 16 5 10 20 9 18 . P = (4) 25 19 7 14 28 Then, by replacing each element of P with a 31×31 shifted identity matrix, a 93 × 155 parity check matrix H with rank 91 is obtained. In this way, a (155, 64) LDPC code with a code rate of R ≈ 0.4129 is constructed. This code is quasi-cyclic (when the codewords are appropriately reordered) since it is closed under cyclic shifts of length 31. To facilitate mapping coded bits to an M = 2m -ary signal constellation, the code can be shortened to a multiple of m and each codeword mapped to n/m signal points, where n ≤ 155 is the (shortened) blocklength. For the (155, 64) LDPC code, a shortened (152, 61) code is obtained for 16QAM modulation, and a (154, 63) code can be used for QPSK modulation.

III. Simulation Results The BICM systems employing low-complexity turbo codes and algebraically constructed LDPC codes were simulated over channels impaired by Rayleigh fading and additive white Gaussian noise. Both i.i.d. Rayleigh fading and correlated Rayleigh fading were investigated; in each case the mean-square value of the fading coefficients was normalized to unity (i.e., E[a2t ] = 1 for all t). To represent the correlated fading process, an improved Jakes’ model [19] was used. For these simulations, the carrier frequency is 900 MHz and the vehicle speed is 60 mph, meaning the maximum Doppler frequency shift fd is approximately 80 Hz. The symbol period Ts is chosen so that Ts ∗ fd = 0.01. This implies that fading coefficients separated in time by no more than (approximately) 25 symbol periods are correlated. A. Low Complexity Turbo Codes Figures 7 and Figure 8 display the simulation results for 2 bps/Hz 16-QAM and 1 bps/Hz QPSK bit-interleaved coded modulation, respectively, when the channel code is a turbo code. Specifically, three turbo codes are considered: • the 3GPP standard turbo code using two identical 1+D+D 3 [1 1+D 2 +D 3 ] convolutional codes, punctured to rate 1/2 using the 3GPP puncturing pattern; • the 3BNA turbo code punctured to rate-1/2 as described in Section 2.1; • the 3ACC-1FF turbo code, also punctured to rate-1/2 as described in Section 2.1. For each turbo code a block of 2048 information bits were encoded in each frame, and 30 decoder iterations were executed. From the simulation results, we see that the 3BNA turbo code achieves improved performance compared to the more conventional 3GPP in i.i.d. fading at both 1 bps/Hz and 2 bps/Hz. There is a similar improvement in correlated fading at 2 bps/Hz; for correlated fading at 1 bps/Hz the 3GPP has a slight performance edge. Significantly, the multiple turbo code performance is achieved with less decoding complexity. (The complexity savings of the multiple turbo code design are analyzed in [20].) B. Algebraically Constructed LDPC codes To achieve a spectral efficiency of 2 bps/Hz, a BICM scheme with a rate 1/2 LDPC code and 16-QAM modulation is used. The rate 1/2 LDPC code is obtained by shortening a (5299, 3030) algebraically constructed LDPC code with column weight 3, row weight 7, and 757 × 757 permutation matrices; the shortened code has a blocklength of 4536. This same code was also used to achieve a spectral efficiency of 1 bps/Hz with QPSK. A maximum of 100 iterations was used in the message passing decoding algorithm. (Note: At a BER of 10−5 , the average number of iterations required for convergence was ten.) The results are plotted in Figures 9 and 10. Also plotted are comparable results for randomly-designed LDPC codes. The algebraically designed codes enjoy a slight coding gain

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Turbo Codes with 16−QAM on Fading Channels

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Fig. 9. Performance of BICM using LDPC codes and 16-QAM at a spectral efficiency of 2 bps/Hz.

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Fig. 8. Performance of BICM using turbo codes and QPSK at a spectral efficiency of 1 bps/Hz.

over the random codes – and, once again, the algebraically designed codes have clear implementation advantages relative to the randomly designed codes. IV. Conclusions This paper presented new designs for bandwidth efficient (1 and 2 bps/Hz) bit interleaved coded modulation in fading environments. The new schemes exhibit performance comparable or superior to that of conventional BICM turbo/LDPC codes. Moreover, the new codes’ low complexity offer implementation advantages. These results suggest that BICM using low complexity turbo codes and algebraically constructed LDPC codes represents a promising approach to achieving good performance and high spectral efficiency on wireless channels.

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Fig. 10. Performance of BICM using LDPC codes and QPSK at a spectral efficiency of 1 bps/Hz.

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“Iterative decoding of non-systematic turbo codes,” in Proceedings of the IEEE International Symposium on Information Theory, Sorrento, Italy, June 2000, pp. 172. [18] J. Han and O. Y. Takeshita, “On the decoding structure of multiple turbo codes,” in Proceedings of the IEEE International Symposium on Information Theory, Washington, D.C., June 2001, p. 98. [19] M.F. Pop and N.C. Beaulieu, “Limitations of sum-ofsinusoids fading channel simulators,” IEEE Transactions on Communications, vol. 49, no. 4, pp. 699–708, Apr. 2001. [20] D. J. Costello, Jr., A. Banerjee, C. He, and P. C. Massey, “A Comparison of Low Complexity TurboLike Code Designs,” in Proceedings of the 36th Annual Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, Nov. 2002.

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