On Turbo-Schedules for LDPC Decoding

Transcription

1 On Turbo-Schedules for LDPC Decoding Alexandre de Baynast, Predrag Radosavljevic, Victor Stolpman, Joseph R. Cavallaro, Ashutosh Sabharwal Department of Electrical and Computer Engineering Rice University, MS Main Street, Houston, TX Nokia Incorporation 6000 Connection Drive, Irving, TX Corresponding author: Alexandre de Baynast Department of Electrical and Computer Engineering Rice University, MS Main Street, Houston, TX Telephone: Fax: Abstract The convergence rate of LDPC decoding is comparatively slower than turbo code decoding: 25 LDPC iterations versus 8-10 iterations for turbo codes. Recently, Mansour proposed a turbo-schedule to improve the convergence rate of LDPC decoders. In this letter, we first extend the turbo-scheduling principle to the check messages. Second, we show analytically that the convergence rate of both turbo-schedules is about twice as fast as the standard message passing algorithm for most LDPC codes. EDICS Category: CL1.3.1 This work was supported in part by Nokia Corporation and by NSF under grants ANI , EIA

2 On Turbo-Schedules for LDPC Decoding 1 Alexandre de Baynast, Predrag Radosavljevic, Victor Stolpman, Joseph R. Cavallaro, Ashutosh Sabharwal Abstract The convergence rate of LDPC decoding is comparatively slower than turbo code decoding: 25 LDPC iterations versus 8-10 iterations for turbo codes. Recently, Mansour proposed a turbo-schedule to improve the convergence rate of LDPC decoders. In this letter, we first extend the turbo-scheduling principle to the check messages. Second, we show analytically that the convergence rate of both turbo-schedules is about twice as fast as the standard message passing algorithm for most LDPC codes. I. INTRODUCTION The low-density parity-check (LDPC) codes are commonly decoded by using the iterative standard messagepassing (SMP [1]) algorithm. About 25 iterations are necessary to match the performance of a turbo-decoder which requires only 8 10 iterations. Therefore, increasing the convergence speed of the LDPC decoder is essential. A turbo-schedule for the message passing algorithm was introduced in [2]. Simulations show that the turbo-schedule significantly improves the convergence rate of the LDPC decoders. In this letter, we first propose to extend the turbo-scheduling principle to the check messages. Second, in Section III, we determine upper bounds on the convergence rate for the turbo-schedules for any Gallager code. We show that the convergence rate of the turbo-schedules is approximatively twice as fast as SMP thereby making LDPC decoding almost as fast as turbo code decoding.

3 2 II. THREE DIFFERENT MESSAGE PASSING SCHEDULES The LDPC codes can be represented by a bipartite graph that consists of two types of nodes: the bit nodes and the check nodes [1]. Each coded bit is represented by a bit node, whereas each parity check equation represents a check node. The Belief Propagation (BP) algorithm is used to decode the LDPC codes iteratively [1]. Bit nodes and check nodes exchange messages according to a pre-determined schedule. In this section, we successively describe three different schedules. A. Standard Message Passing (SMP) Schedule Throughout the paper, we assume an AWGN transmission channel. The channel log-likelihood ratio LLR j related to the bit node j, j = 1,...,N is equal to 2r j /σ 2 where r j is the j-th noisy received sample, j = 1,...,N and σ 2 is the noise variance of the channel. We define the Psi-function as: Ψ(x) = log[tanh( x /2)] ([1]). sgn(x) is the sign-function: 1 if x < 0, 1 otherwise. N(m) and M(j) denote the subset of the indices corresponding to the location of the 1s in the m-th row and the j-th column respectively, of the parity-check matrix (PCM) of the LDPC code. The SMP schedule is a two-phase iterative schedule (phases P 1,P 2 ). After initializing the bit-node messages L(q mj ), j,m, j = 1,...,N, m M(j) to LLR j, the decoding algorithm is [2]: P1: Do for every row m = 1,...,M (a): Update check-equation messages: R m = (b): Update check-node messages j N(m): sgn(l(q mj )) Ψ j N(m) j N(m) Ψ(L(q mj )) (1) R mj = sgn(r m ) sgn(l(q mj )) Ψ [Ψ(R m ) Ψ(L(q mj ))] (2) P2: Do for every column j = 1,...,N (a): Update a posteriori messages: L(q j ) = R mj + LLR j (3) m M(j)

4 3 (b): Update bit-node messages m M(j): L(q mj ) = L(q j ) R mj (4) Tentative decoding: Stop if all parity-check equations are verified or if a maximum number of iterations is reached. Otherwise, go to step P1 (next decoding iteration). We can summarize this message-passing schedule in the following compact form where (1), (2), (3) and (4) correspond to Equations 1-4 in the decoding algorithm described above. { m : (1) { j : (2)}} { j : (3) { m : (4) }} SMP schedule B. Row Message Passing (RMP) Schedule In [2], an improved message passing schedule is presented. We refer it as the Row Message Passing (RMP) schedule since the turbo-effect is applied on the rows of the PCM. For a given row m, m = 1,...,M, simultaneous updating of all messages R mj, L(q j ) and L(q mj ), j = 1,...,N(m) is performed (one sub-iteration) and the processing is repeated for all rows (one super-iteration): m : { j : (4)} (1) { j : (2)} { j : (3)} RMP schedule C. Column Message Passing (CMP) Schedule As for RMP, the turbo message passing effect can be also achieved by updating messages from column to column. A similar approach has been independently proposed in [3]. For all check nodes that participate in the current column j, simultaneous updating of all corresponding messages is performed:

5 4 j : { m : (2)} (3) { m : (4)} { m : (1)} CMP schedule Since the messages R mj and L(q mj ), j, j = 1,...,N(m), are updated before R m for this schedule, Equation 1 can be simplified as: R m = sgn(r mj ) sgn(l(q mj )) Ψ [Ψ(R mj ) + Ψ(L(q mj ))]. III. CONVERGENCE ANALYSIS Density Evolution Analysis with Gaussian Approximation (DEA-GA) has been proposed in [1]. It predicts the average behavior of BP through the iterations. DEA-GA is readily applicable for RMP and CMP schedules since the equations involved during the decoding remain the same. A. Upper bound on the convergence rate for RMP and CMP schedules In this section, we derive an upper bound on the convergence rate based on an approximation of DEA-GA. We introduce the following metric: Definition 1: where P (i) e α (i) RMP/CMP = arg{ min P (αi) e,smp P e,rmp/cmp} (i) (5) α s.t. P (i) (αi) e,rmp/cmp P e,smp represents the targeted probability of bit error at the end of the i-th iteration. The coefficient α represents the convergence accelerating factor (CAF) of the turbo-schedule, i.e. how many times it is required to increase the number of iterations with the basic SMP schedule in order to reach the same probability of error at the end of the iteration i with turbo-schedule (RMP or CMP). Theorem 1: We assume a regular Gallager code of infinite length (no cycles) with w c 1s per column and w r 1s per row. The CAF for the RMP schedule is upper bounded as: α (i) RMP log { [( 2+w c ) 0 0] [ A iwc ( mv0 + (A I wc ) 1 b ) (A I wc ) 1 b ] + 1 } log (m u0 w r 1) ilog (w c 1) (6)

6 5 TABLE I CONVERGENCE ACCELERATING FACTOR (CAF) DETERMINED THROUGH DENSITY EVOLUTION ANALYSIS FOR SEVERAL REGULAR GALLAGER CODES FOR A BIT ERROR RATE OF IN REGULAR FONT, α FROM DEA; IN BRACKETS, UPPER BOUNDS FROM EQUATIONS 6 AND 7; IN ITALIC, α FROM SIMULATIONS. IN SIMULATIONS, ALL CODEWORD LENGTHS ARE EQUAL TO 10K. iter. Sched. (3, 6) (3, 9) (4, 6) (4, 8) 2 15 RMP 2.00 (2.00) (2.00) (2.00) (2.00) 2.00 CMP 2.00 (2.00) (2.00) (2.00) (2.00) 2.00 RMP 1.93 (2.07) (2.07) (2.13) (2.13) 2.00 CMP 1.93 (2.07) (2.07) (2.13) (2.13) 2.00 where m u0 = 2/σ 2 with σ 2 noise variance; the matrix A and the vectors m v0 and b are defined in appendix Section A. I wc is the w c w c identity matrix. The CAF for the CMP schedule is upper bounded as: α (i) CMP log { [( 2+w c ) 0 0] [ C i(wr 1) ( m v 0 + (C I wr ) 1 d ) (C I wr ) 1 d ] + 1 } log (m u0 w r 1) ilog (w c 1) where the matrix C and the vectors m v 0 and d are defined in appendix Section B. The proofs of Equations 6 and (7) 7 are given in the appendix. Remark 1: Since the matrix C is always positive definite, RMP asymptotically converges faster than CMP. A 0 wc (w r w c 1) 0 (wr w c 1) (w r 1) B. Results of DEA-GA for RMP and CMP schedules The results obtained through DEA for a Bit Error Rate of 10 7 are plotted in Table I. In the simulations, the codeword length is equal to 10K for all codes and the construction of PCMs is random. CAFs calculated from Equations 6 and 7 match with the simulations for all considered codes. Performance of RMP and CMP are similar, and both outperform SMP.

7 6 IV. CONCLUSION In this paper, we first extended the turbo-schedule principle [2] for the check messages. By introducing a new metric, the convergence accelerating factor, we analyzed the convergence rate of the three schedules of Belief Propagation algorithm for LDPC decoding. RMP and CMP schedules outperform SMP by about a factor of two for all considered codes. Simulations and density evolution analysis validated our approach. APPENDIX PROOF OF TH. 1 A. Proof of the inequality in Equation 6 We adopt the same notation as in [1]. Since the probability of bit error P e at the output of the decoder is a monotonically decreasing function with respect to the average value of the check-node messages m (i) u at any iteration, the CAF can be expressed as: α (i) IMP = arg min α s.t. m (i) u,imp m(αi) u,smp m (i) u,imp m (αi) u,smp. According to [1], page 662, m (i) u,smp is solved by the following recursive sequence m (i) u,smp = ξ(m u0 +m (i 1) u,smp,w r 1) where m u0 = 2/σ 2, m (0) u,smp = 0 and ξ(x,a) = φ 1 {1 [1 φ(x)] a }. The φ-function is defined in [1], page 660. By taking the expectation in Equations 2 and 4, m u,rmp, (i,l) l = 1,...,w c can be expressed as: v,rmp = m (i,l 1) v,rmp + m (i,l 1) u,rmp m (i 1,l) u,rmp u,rmp = ξ( v,rmp,w r 1) Then, we lower bound m (i) u,smp and we upper bound m (i) u,rmp and therefore calculate the corresponding CAF which clearly upperbounds α (i) RMP. We loosely lower and upper bound the function ξ(x,a) as (first order of the Taylor series): max(0,x a 2) ξ(x,a) x, x > 0. For SMP, we have: m (i) u,smp m u0 w r 1 + (w c 1)m (i 1) u,smp with m (0) u,smp = 0, i.e.: m (i) u,smp ( ) wc m u0 w r 1 w c 1 (w c 1) i m u 0 w r 1. w c 1 For RMP, we have: v,rmp 2m (i,l 1) v,rmp m (i 1,l) v,rmp + w r + 1 u,rmp v,rmp

8 7 A solution to this system for any tuple (w c,w r ) is: (m (0,0) v,rmp = m u0 ) [ u,rmp [1 0 0] A (i 1)wc+l ( m v0 + (A I wc ) 1 b ) ] (A I wc ) 1 b. The w c w c matrix A is defined as (wc 2) 1 I wc 1 0 (wc 1) 1 ; m v0 and b are w c 1 vectors with all zeros components save the first one which is equal to m u0 and w r + 1, respectively. Therefore, an upper bound for CAF is the solution of: m (i,wc) u,rmp = m (αi) u,smp. B. Proof of the inequality in Equation 7 For the CMP schedule, we follow the same procedure that we used for the RMP schedule, i.e. we find an upper bound for the average value of the check-node messages m (i) u,cmp and then an upper bound for the CAF. By using { the fact that φ 1 1 } [ N n=1 [1 φ(x n)] φ {1 1 1 φ( 1 )]} N N 1 x n, m u,cmp, (i,l) l = 1,...,w r is upper bounded by the following recursive sequence: u,cmp w r 1 m=1 m(i,l m) v,cmp v,cmp = m u0 + (w c 1) u,cmp A solution to this system for any tuple (w c,w r ) is: (m (0,l) v,cmp = m u0, l = 0,...,w r ) [ u,cmp [1 0 0] C (i 1)wr+l ( m v 0 + (C I wr ) 1 d ) ] (C I wr ) 1 d where the w r w r matrix C is defined as w c 1 w r 1 w c 1 w r 1 I wr 1 w c 1 w r 1 0 (wr 1) 1. All components of the w r 1 vector m v 0 are set to m u0 ; d is a w r 1 all-zero vector except for its first component which is equal to m u0. Therefore, an upper bound for CAF is the solution of: m (i,0) u,cmp = m (αi) u,smp.

9 8 REFERENCES [1] S. Chung, T. Richardson, and R. Urbanke, Analysis of sum-product decoding of low-density parity-check codes using a Gaussian approximation, IEEE Transactions on Information Theory, vol. 47, no. 2, pp , [2] M. Mansour and N. Shanbhag, High-throughput LDPC decoders, IEEE Transactions on Very Large Scale Integration (VLSI) Systems, vol. 11, no. 11, pp , [3] Juntan Zhang and M. Fossorier, Shuffled iterative decoding, IEEE Transactions on Communications, vol. 53, no. 2, pp , 2005.