Iterative Encoding with Gauss-Seidel Method for Spatially-Coupled Low-Density Lattice Codes

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1 202 IEEE International Symposium on Information Theory Proceeings Iterative Encoing with Gauss-Seiel Metho for Spatially-Couple Low-Density Lattice Coes Hironori Uchikawa, Brian M. Kurkoski, Kenta Kasai an Kohichi Sakaniwa Dept. of Communications an Integrate Systems, Tokyo Institute of Technology, Tokyo, Japan. {uchikawa, kenta, School of Information Science, Japan Avance Institute of Science an Technology JAIST, Ishikawa, Japan. Abstract While it is known that spatially-couple low-ensity lattice coes SC-LDLC have better ecoing performance than conventional non-couple LDLC lattices, in this paper it is shown that their encoing complexity is also lower. Since nonzero elements are mainly in lower triangular entries of the sparse inverse generator matrix of SC-LDLC, iterative encoing with the Gauss-Seiel metho performs well. The convergence spee of iterative encoing is evaluate by both the mean square error MSE an the symbol error rate between a given integer vector b an the inversely generate integer vector from the coewor of b. Numerical experiments show that the convergence of encoing for SC-LDLC is 3 times faster than that of the conventional LDLC, at an MSE of 0 0 for imension n = I. INTRODUCTION The aitive white Gaussian noise AWGN channel is an important channel both from theoretical an practical points of view. In 959, Shannon foun the capacity of the AWGN channel using ranom coing []. Due to the lack of structure, ranom coes are not practical as channel coes. Almost 40 years later, it was prove that lattice coes achieve the channel capacity [2], [3]. Although lattice coes have elegant structure, they are not efficiently ecoable coes, i.e. their ecoing complexity is not linear in the imension. In 2008, Sommer et al. propose efficiently ecoable lattice coes calle low-ensity lattice coes LDLC, by efining a sparse inverse generator matrix [4]. Since Sommer et al. i not use shaping methos, the term LDLC lattices is use in this paper. Although LDLC lattices can be ecoe efficiently using belief-propagation BP algorithm, capacity-achieving LDLC lattices have not so far been constructe. The noise threshol of LDLC lattices in [4] appeare within 0.5 B of the capacity of the unconstraine-power AWGN channel. In [5], we evelope a new LDLC lattice construction, calle spatially-couple SC LDLC lattices, base upon spatial coupling principles [6]. Evaluation was performe using Monte Carlo ensity evolution using a single-gaussian approximate BP ecoer [7]. While the conventional lattice construction leaves a gap of 0.5 B to capacity, the SC-LDLC lattice construction reuces this gap to 0.33 B from capacity, Previously we reporte 0.22 B gap from capacity [5], but the correct value is 0.33 B. of the unconstraine power channel. Although the ecoer s computational complexity is only On, generating lattice points requires computational complexity of On 2 in general. In [4], Sommer et al. suggeste that linear computational complexity encoing can be accomplishe using the Jacobi metho, which is an iterative algorithm for etermining the solution of a system of linear equations. Sommer et al. escribe that the convergence of the metho was guarantee by the nature of the sparse inverse generator matrix of LDLC lattices. However, they i not investigate convergence spee of such iterative encoing methos. Since processing over many iterations is time- an power-consuming, not only convergence conitions but also convergence spee is important for practical applications. In this paper, we evaluate the convergence spee of encoing using the Gauss-Seiel metho, another iterative algorithm [8]. Numerical experiments show that the SC-LDLC encoer with the Gauss-Seiel metho has significantly faster convergence spee than the LDLC encoer with the Gauss-Seiel metho because of the spatially-couple structure. II. LDLC AND SC-LDLC LATTICES A. Lattices An n-imensional lattice Λ is efine by an n-by-n generator matrix G. The lattice consists of the iscrete set of points x =x,x 2,...,x n T for which x = Gb, where b =b,...,b n T is from the set of all possible integer vectors, b i Z. The transpose of a vector x is enote x T. Lattices are linear, in the sense that x + x 2 Λ if x an x 2 are lattice points. It is assume that G is n-by-n an full rank note SC-LDLC lattices allow G to have aitional rows which are linearly epenent. The ecoing performance of lattices are evaluate over the unconstraine-power AWGN channel [4], [5]. Let the coewor x be an arbitrary point of the lattice Λ. This coewor is transmitte over an AWGN channel, where noise z i with noise variance σ 2 is ae to each coewor symbol. Then the receive sequence y = y,y 2,...,y n T is /2/$ IEEE 747

2 y i = x i + z i, for i =, 2,...,n. The maximum-likelihoo ecoer selects ˆx Λ as the estimate coewor, an a ecoing error is eclare if ˆx x. The capacity of this channel is the maximum noise power at which a maximumlikelihoo ecoer can recover the transmitte lattice point with error probability as low as esire for sufficiently large block length. In the limit that n becomes asymptotically large, there exist lattices which satisfy this conition if an only if [9]: σ 2 etg 2/n. 2 2πe In the above etg is the volume of the Voronoi region, which is inversely proportional to lattice ensity. B. LDLC lattices An LDLC lattice is a lattice which has a sparse inverse generator matrix H. Since the inverse generator matrix H = G of LDLC lattices is sparse, LDLC lattices can be ecoe using BP [4]. An n, α, LDLC lattice is efine by the n- by-n matrix H which has row an column weight, where each row an column has one entry of weight ± an entries with weight which epens upon α. More precisely, the matrix H is efine as: H = P + w P i, 3 where i=2 P i = S i P i. 4 S i enotes a ranom sign change matrix, P i enotes a ranom permutation matrix, an α w =. We choose 0 α<, so that BP ecoing of LDLC lattices will converge exponentially fast [4]. The permutation matrices are not chosen in a totally ranom manner but so as to generate H having exactly one ± an exactly ±w s in each column an row. The ranom sign change matrix S i is a square, iagonal matrix, where the iagonal entries are + or with probability /2. C. Spatially-Couple LDLC lattices We efine an N,α,,L SC-LDLC lattice as a imension NL + lattice with an NL NL inverse generator matrix H [L] as escribe by Eq. 7. The structure of H [L] is similar to the parity check matrix of tail-biting convolutional coes. In Eq. 7, H l = P l +w i=2 P l i is an inverse generator matrix of an N,α, LDLC lattice for l {,...,L}, an each P l i represents a istinct matrix of the form of Eq. 4, for istinct l an i. In this construction, N integers are set to 0. The integer vector of form: [ ] b b =, 0 N is use, so that if the N,α,,L SC-LDLC lattice has a lattice point x = x,...,x l,...,x L T an x l = x l,...,xl N, then H [L]x = b, where b = b,...,b l,...,b L + T an b l =b l,...,bl N,is an information integer vector, an, 0 N is the all zero column vector of length N. The inverse matrix of H [L] is efine as G [L]. Since we set 0 N to the last sections of b, The sub-matrix G [L] which is from column to NL + of G [L] can be use for generating lattice points. Therefore a lattice point in the imension NL + SC-LDLC lattice is generate as x = G [L] b = G [L] b. 5 The imension of the SC-LDLC lattice is, therefore, less than the number of elements in x, which is n = NL. Dimension ratio is efine as R L = NL + NL = L. 6 The ratio R L converges to with increasing L, withagap of O/L. Therefore, this imension loss is negligible for sufficiently large L. We observe that the noise threshol of the N, 0.8, 7, L SC-LDLC lattices, with sufficiently large L, is very close to the theoretical limit [5]. III. ENCODING WITH THE GAUSS-SEIDEL METHOD Generally, a lattice encoer generates a lattice point x using Eq.. For LDLC lattices, G is not sparse in general, an so the encoer requires computational complexity an storage of On 2. However, b = Hx is a system of linear equations which can be solve using iterative methos. One such metho, the Jacobi metho, can be escribe as a parallel approach, fining a caniate solution xt on iteration t,using the previous solution xt. This approach was suggeste by Sommer et al., observing that encoing complexity is On, since H is sparse [4]. The Gauss-Seiel metho is another iterative metho. The Gauss-Seiel metho is a serial approach; in each iteration, each element of the caniate solution is compute in turn. To perform computations serially, the Gauss-Seiel metho ecomposes the matrix into upper- an lower-triangular parts. In this section, LDLC an SC-LDLC encoers with the Gauss- Seiel metho are escribe. The Gauss-Seiel metho has faster convergence spee than the Jacobi metho. Moreover the SC-LDLC encoer with the Gauss-Seiel metho performs well because of the spatially-couple structure. A. LDLC Encoer with the Gauss-Seiel Metho Before explanation of the Gauss-Seiel Metho, we efine the matrix H as a row-permute version of the sparse inverse generator matrix H, such that H has ± in the iagonal entries non-zero iagonal elements are require by the Gauss-Seiel metho. For example, a pictorial view of a matrix H for the 00, 0.8, 7 LDLC lattices is shown in Fig.. Also the vector b is a permute version of the integer vector b, such 748

3 H [L] = P wp L +2 wp 2 P 2.. wp wp 2 wp L L wp P L wp L + wp L +2 P L P L wp L + 7 The element-wise equation of Eq. 8 is as follows x i t += b h i h i,j x j t + i,i j J i {j j<i} j J i {j j >i} h i,j x j t, Fig.. Pictorial view of a row-permute matrix H of the 00, 0.8, 7 LDLC lattices. Re resp. blue ots represent positive resp. negative coefficients. Dark resp. light ots represent ± resp. ±w coefficients. that the ith element of b equals the element of b for which the corresponing row of H has ± at the ith location. Using the Gauss-Seiel metho [8], an LDLC encoer calculates several iterations of the form: xt +=L + D b Uxt, 8 where D is the iagonal matrix with the iagonal elements of H, an L resp. U has the lower resp. upper triangular elements without the iagonal elements of H. Therefore, H is ecompose as: H = L + D + U. Denote t as the inex of iteration an x0 is initialize with 0 n. The convergence properties are well-stuie, an is relate to the spectral raius of the matrix L+D U [8]. However, convergence may still occur even if properties are not satisfie, an arguments concerning convergence are beyon the scope of this paper. where x i t + is the ith entry of xt +, h i,j is the i, j entry of H, an J i is the column inex set of the non-zero elements in the row i of H. The computation of x i t+ uses the elements of xt + that have alreay been compute, an the elements of xt that have not yet been compute at iteration t +. Consier a sparse inverse generator matrix H [L] of the SC-LDLC lattices. Non-zero elements are mainly in lower triangular entries of H [L] without the first N rows, see Eq. 7. Since the Gauss-Seiel metho uses alreay-compute elements of xt + at iteration t + for x i t + in the case that the non-zero elements are in J i {j j <i}, it is expecte that the convergence spee of the encoing of the SC-LDLC lattices with the Gauss-Seiel metho is faster than that of the LDLC lattices. B. SC-LDLC Encoer with the Gauss-Seiel Metho An SC-LDLC encoer with the Gauss-Seiel metho generates a coewor vector x l t + at section l an iteration t + by Eq. 9. The matrix P l is row-permute version of the P l, such that H [L] has ± in the iagonal entries. For example, a pictorial view of a matrix H [L] for a 5, 0.8, 7, 20 SC-LDLC lattice is shown in Fig. 2. The vector b l is a permute version of the integer vector b l, such that the ith element of b l equals the element of b l for which the corresponing row of P l has ± at the ith location. All elements of x0 are initialize with 0. From Eq. 9, it is observe that the SC-LDLC encoer uses the elements of xt l compute at the previous iteration for the first sections. For remaining sections, the SC-LDLC encoer uses only the elements of xt + l compute at the current iteration. Intuitively it is expecte that the convergence spee of the encoing of SC-LDLC lattices with the Gauss-Seiel metho is faster than that of LDLC lattices. 749

4 x l t + T = P l P l T b l T w T b l T w l m= m= P l m m+ x l m t + T P l m m+ x l m t + T w m=l T P L m+l m+ x L m+l t for l<, for l L 9 We generate 000 sparse inverse generator matrices for each lattice ensemble an evaluate 00 ranomly generate integer vectors 2 for each matrix to compute the average. Mean square error MSE at iteration t is calculate by Fig. 2. Pictorial view of a row-permute matrix H [L] of the 5, 0.8, 7, 20 SC-LDLC lattices. Re resp. blue ots represent positive resp. negative coefficients. Dark resp. light ots represent ± resp. ±w coefficients. Due to the spatially-couple structure of H [L], the SC- LDLC encoer can be implemente using a transversal filter architecture. The SC-LDLC encoer architecture is shown in Fig. 3. The encoer consists of transversal filter with ring FIFOs. This encoer is simple because the encoer oes not nee to manage routing connections with all the elements of x for computations ifferently from the LDLC encoer. Fig. 3. Architecture of the iterative encoer of SC-LDLC lattices. Note that section inex l points at l + L if the l is negative. IV. SIMULATION RESULTS Convergence spee of the encoing with the Gauss-Seiel metho is evaluate using Monte-Carlo simulations. MSEt = b HxtT b Hxt, n where n is the length of b. Fig.4showsMSEt for 000, 0.8, 7 an 0000, 0.8, 7 LDLC lattices, an 50, 0.8, 7, 20 an 500, 0.8, 7, 20 SC-LDLC lattices. 000, 0.8, 7 LDLC an 50, 0.8, 7, 20 SC-LDLC resp. 0000, 0.8, 7 LDLC an 500, 0.8, 7, 20 SC-LDLC lattices are the same imension n = 000 resp. n = It is observe that convergence spee becomes faster with increasing imension n. Convergence of the 500, 0.8, 7, 20 SC-LDLC lattices is almost 3 times faster than that of the 0000, 0.8, 7 LDLC lattices at an MSE of 0 0. It is also observe that the slopes of the MSEt curves for the imension n = 000 both LDLC an SC-LDLC lattices change aroun an MSE of 0 5. We conjecture that this is cause by small cycles in the H, similar to error floors of low-ensity parity-check coes. Section size N oes not seem to affect the convergence of SC-LDLC lattices because there are small ifferences in the MSEt between the 50, 0.8, 7, 20 an 500, 0.8, 7, 20 SC- LDLC lattices. Fig. 5 shows MSEt for the 50, 0.8, 7, L SC-LDLC lattices for L =20, 00, 200 an 500. We observe that convergence spee accelerates with increasing coupling factor L. From the results in Fig. 5, we can obtain sufficiently accurate coewors for transmission with few iterations if L is sufficiently large. We also show symbol error rate SER performance in Fig. 6. The SER at iteration t is calculate by n i= SERt = I[b i Hxt i ], n where I[ ] is an inicator function that returns if an argument is true, otherwise returns 0, an i enotes ith element of an argument vector. The imension of lattices in Fig. 6 is 000, therefore 8 iterations is sufficient for SC-LDLC lattices with the Gauss-Seiel metho to vanish encoing symbol error from expectations. On the other han, 2 iterations is necessary for LDLC lattices. In aition, the Gauss-Seiel metho converges slightly faster than Jacobi metho for conventional LDLC lattices, an significantly faster for SC-LDLC lattices. 2 In the simulations, b i is uniformly istribute over { 0,, 0} for i {,,n}. 750

5 Mean square error ,0.8,7 LDLC 0000,0.8,7 LDLC 50,0.8,7,20 SC-LDLC 500,0.8,7,20 SC-LDLC Symbol error rate ,0.8,7 LDLC with GS 000,0.8,7 LDLC with Jacobi 50,0.8,7,20 SC-LDLC with GS 50,0.8,7,20 SC-LDLC with Jacobi Fig. 4. versus mean square error of b. 000, 0.8, 7 LDLC an 50, 0.8, 7, 20 SC-LDLC resp. 0000, 0.8, 7 LDLC an 500, 0.8, 7, 20 SC-LDLC lattices are the same imension n = 000 resp. n = The convergence of SC-LDLC lattices is much faster than that of LDLC lattices at each imension Fig. 6. The number of iterations versus symbol error rate of b. The Gauss- Seiel metho GS an the Jacobi metho Jacobi are evaluate for both 000, 0.8, 7 LDLC an 50, 0.8, 7, 20 SC-LDLC lattices. The Gauss-Seiel metho converges slightly faster than Jacobi metho for conventional LDLC lattices, an significantly faster for SC-LDLC lattices. Mean square error ,0.8,7,20 SC-LDLC 50,0.8,7,00 SC-LDLC 50,0.8,7,200 SC-LDLC 50,0.8,7,500 SC-LDLC Fig. 5. The number of iterations versus mean square error of b. We observe that the convergence spee of SC-LDLC lattices accelerates with increasing coupling factor L. V. DISCUSSION We evaluate the convergence performance of iterative encoing with the Gauss-Seiel metho for LDLC an SC- LDLC lattices. Numerical experiments showe that the convergence of SC-LDLC lattices is significantly faster than that of LDLC lattices because of the spatially couple structure. The Gauss-Seiel metho separates the matrix into lower an upper triangular parts. The sparse inverse generator matrix of SC-LDLC lattices is ominate by the lower-triangular components, an since the upper-triangular part is smaller, this may help explain the higher convergence rates for the SC-LDLC lattices. Note that if SC-LDLC lattices existe where H is purely lower-triangular, then encoing woul be particularly straightforwar an coul be implemente by a simple forwar- substitution. As in many previous papers, shaping was not consiere here because of the unconstraine power scenario i.e. Poltyrev setting. It shoul be further note that if a purely lower-triangular matrix is available, then a power constraint can be easily introuce by using cubic shaping, see [0]. REFERENCES [] C. E. Shannon, Probability of error for optimal coes in a Gaussian channel, Bell System Technical Journal, vol. 38, pp , May 959. [2] R. Urbanke an B. Rimoli, Lattice coes can achieve capacity on the AWGN channel, IEEE Transactions on Information Theory, vol. 44, pp , Jan [3] U. Erez an R. Zamir, Achieving /2 log +SNR on the AWGN channel with lattice encoing an ecoing, IEEE Transactions on Information Theory, vol. 50, pp , Oct [4] N. Sommer, M. Feer, an O. Shalvi, Low-ensity lattice coes, IEEE Transactions on Information Theory, vol. 54, pp , Apr [5] H. Uchikawa, B. M. Kurkoski, K. Kasai, an K. Sakaniwa, Threshol improvement of low-ensity lattice coes via spatial coupling, in Proc. 202 Int. Conf. on Computing, Networking an Communications ICNC, pp , Jan Hawaii, USA. [6] S. Kuekar, T. Richarson, an R. Urbanke, Threshol saturation via spatial coupling: Why convolutional LDPC ensembles perform so well over the BEC, IEEE Transactions on Information Theory, vol. 57, pp , Feb. 20. [7] B. M. Kurkoski, K. Yamaguchi, an K. Kobayashi, Single-Gaussian messages an noise threshols for ecoing low-ensity lattice coes, in Proc IEEE Int. Symp. Inf. Theory ISIT, pp , July [8] Y. Saa, Iterative Methos for Sparse Linear Systems. New York: Society for Inustrial an Applie Mathematic SIAM, 2n e., [9] G. Poltyrev, On coing without restrictions for the AWGN channel, IEEE Transactions on Information Theory, vol. 40, pp , Mar [0] N. Sommer, M. Feer, an O. Shalvi, Shaping methos for low-enisty lattice coes, in Proc IEEE Information Theory Workshop ITW, pp , Oct