Anbuselvi et al., International Journal of Advanced Engineering Technology E-ISSN

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1 Research Paper ANALYSIS OF A REDUED OMPLEXITY FFT-SPA BASED NON BINARY LDP DEODER WITH DIFFERENT ODE ONSTRUTIONS Anbuselvi M, Saravanan P and Arulmozhi M Address for orrespondence, SSN ollege of Engineering Rajalakshmi ollege of Engineering, Tamilnadu, India. ABSTRAT Low density parity check (LDP) codes are the linear block codes with excellent error correcting performance, near to Shannon s limit. Non-binary LDP codes are more suitable for applications of short and moderate block length than binary LDP codes. The criticality in achieving a better decoding capacity with reduced hardware complexity has been addressed in several research ventures of LDP decoder design. The computation complexity of the decoder increases linearly with the order of Galois field, in par with the compromise on the decoding performance. In this paper, a variable threshold based technique and three different matrix construction methods are proposed, attaining a better tradeoff between the decoding performance and computation complexity. The implications of these proposed methods on IEEE 8.n based application, are analysed and validated. In this work, the proposed modifications on the construction of the codes and handling are inculcated in Non-binary Quasi-cyclic Irregular LDP codes on the base of Fast Fourier Transform (FFT) Sum Product (SP) decoding algorithm. With the inclusion of the two variations, the computation strength of the decoder reduces by 55% on average and the decoding performance improves by db. KEYWORDS Non-binary LDP codes, FFT-SPA, Matrix construction, Message, Decoder design. INTRODUTION Gallager introduced a class of linear block codes namely Low density parity check (LDP) codes []. LDP codes defined by a sparse parity check matrix having lesser number of s are called as binary LDP. ompared to other codes, LDP codes have better decoding performance for larger block lengths []. This makes the LDP codes, applicable in different communication standards namely DVB-S and IEEE 8.(WLAN). However these kind of binary LPD codes are not found to be suitable for other applications such as space communication, involving short or moderate block length. To meet out such kind of applications, Non binary LDP codes are constructed over the higher order finite field []. The decoding performance of the Nonbinary LDP codes increases with the increase in the field order [4]. Accordingly, the computation and hardware complexity increases with the Non-binary LDP codes. The different complexity techniques are reviewed in the literature, focusing on the decoding algorithms. Non-binary LDP codes based Min-sum algorithm [5] and Min-max algorithm [6] were proposed with reduced computations. A significant approach on the complexity is the method, addressed in the literature [7]. These works also focus on optimization of FFT-SPA based Non-binary LDP decoder, with reduced computation complexity and nominal decoding performance. The Another research direction in the design of Nonbinary LDP decoder is code construction. The key element behind the performance of this code is the parity check matrix (PM). The formation of the PM decides the decoding ability and computation strength of the code. onventional LDP codes work on the random PM representing the framework for generating possible set of codewords. Sparsity, the number of non-zero elements in the PM, effects on the computation strength of the decoder. Different methods of code construction are addressed in the literature [8-] satisfying the properties of PM, row-column constraint [] and girth [] of good codes. These methods of framing the PM are quite complex and results on the decoding performance. The authors contribute in framing a variable threshold based method, which effectively reduces the memory requirement and thereby the hardware complexity of the decoder architecture. However this method affects the decoding ability of the codes, increasing the Bit Error Rate (BER). This negative effect has been compensated with the various code construction methodologies. This paper also presents a varied structure of the three different PM namely, Lower Diagonal PM (LDM), Doubly Diagonal PM (DDM) and Hierarchically Diagonal PM (HDM). The effectiveness of these two methods has been evaluated in terms of the decoding capability and hardware complexity of the code. NON-BINARY LDP DEODING ALGORITHM Belief propagation algorithm The traditional LDP decoder works on the belief propagation algorithm, transacting the probability values between the two computation nodes namely check node and variable node. n Q m Qm ( a), Rm ( a) R m R m ~ H ^ Fig.. Flow diagram of the decoding algorithm The check node is the horizontal processing step, verifying the parity check constraints of the codeword. The variable node is the vertical processing step, representing the codeword. Sumproduct algorithm and variations of sum-product are formulated for binary LDP codes. The computation strength of these algorithms lies on the product term. In check node processing step, probabilities of a symbol are multiplied resembling the convolution operation. Fast Fourier Transform can be incorporated in the convolution to reduce the computation complexity. For hardware realization,

2 Hadamard matrix based computation can be made in the check node processing step []. A flow diagram of the FFT-SP algorithm is presented in Fig., in which each step of the algorithm is depicted with the corresponding input and output variables. The variable node is updated with the prior channel information during the first iteration. From the next iteration, the computations for the variable node update are carried out with the inputs from the check node update unit. The validation of the codeword is performed in the decision block and the iteration count is verified. If the codeword is not valid, the number of iterations is increased for the next cycle of the decoding procedure. If the maximum iteration is attained and the codeword is not validated, then decoding failure is declared, otherwise the cycle repeats. Message Reduction Technique The template is used to format your paper and style the text. All m In passing algorithm, the s are transferred from check node (NU) to variable node (VNU) and vice versa. ompared with variable node, check node has larger number of computations. An approach for complexity is the s between the two nodes which claim the level of computations with respect to the particular decoding algorithm. In the hardware perspective, more the number of s, larger memory requirement for intermediate storage and complex set of interconnections thereby increase in routing complexity. By reducing the configuration set, the s are reduced and hence the hardware complexity is reduced. The various methodologies for the of the configurations with respect to the decoding algorithm are proposed [4]. onfigurations define the computation strength of the decoding algorithm. onfiguration set is the possible set of input symbol satisfying the parity constraints with respect to the PM. The number of configurations depends on the order of Galois field (GF) and the code length. The computation of the algorithm increases with the increase in the configuration set. At the other perception, configuration can be viewed as the of the channel probability values, considered for the consequent computations. This method is also called as the technique. A variable threshold method for identifying the (reduced set of channel probabilities) step for subsequent processing is proposed in this work. The block diagram (Fig.) shows the procedural way of made in this work. This process involves sorting, and formation of the matrix structure for the computation at the check node unit. Initially, the channel probabilities or the set corresponding to the each symbol of the codeword is sorted in an ascending order. Then the set of s having the lesser probabilities are formed as a reduced set having the mean value as the threshold. This step is called as. Next the truncated sets are mapped to the PM matrix with the corresponding non-zero values appropriately after permutation. The significant variation in this method is that the threshold is fixed based on the channel probabilities. Hence this approach of is dynamic and configurable. The in the s/channel probabilities reflects on the number of non-zero elements on the matrix formation at the variable node processing and check node processing. Fig.. Flow diagram of proposed method Let, represent aprior channel information for the n n th symbol. For a given GF (q), aprior information having columns and q rows is defined as, l ( q ) ( q) ( q) l The Q-matrix is framed as follows. For each non-zero element in H-matrix, place the corresponding column vector from. The elements that are placed should be permuted according to the symbol q. The partition of the Q matrix represents the likelihoods that the nth received symbol belonging to. The received probabilities of s in are arranged in descending order. First half of the values from the list are segregated and the thresholds are defined for framing the M matrix that contains the reduced number of elements from. The framing of reduced configuration set, ψ is given by the equation. The range of thresholds T, is fixed to frame the M matrix. The formation of the M matrix is as follows. M { T Matrix of r n M (q) (q) (q) (q) n n Non-binary LDP code with the code rate of ½, and (4,648) over GF (4) is designed. The LDP code with these specifications has been followed in IEEE 8.n standard [5]. With the proposed method of the configurations set, the FFT-SPA based non-binary LDP decoder is analyzed. The dependent parameter which decides on the of configuration set is the threshold and its range. The threshold varies with respect to the channel probabilities, which in return depends on the SNR values. The decoding performance of the Non-binary LDP decoder based on FFT-SPA, with and without method is shown in Fig.. It is inferred that, with the proposed configuration method, the decoding performance is marginally degraded. The computation blocks

3 involved in FFT-SPA algorithm based non-binary LDP decoder are the multiplier and adder blocks. Table, the formulation of the computations using FFT-SPA based on the row weight, column weight and the field order is presented Fig.. omparison of decoding performance of FFT-SPA based Non-binary LDP decoder TABLE I. Formulation of omputations Formulati omputations on unit Multiplications additions Without qnw NU qmw c(w c- r(w r-) ) VNU (q-)qmw r (q-)w cn With q NU rmw r(w r-) q rnw c(w c- ) VNU (q r-)q rmw r (q r-)w cn Where (n, m) represents the matrix size and (w r, w c ) corresponds to the row and column weights respectively. q is the order of Galois field and Ψ corresponds to the number of configurations defined. Table presents the computations without and with for the matrix size of x 4. TABLE II. OMPUTATIONS ANALYSIS FOR X 4 MATRIX Formulation Without With omputations unit Multiplications additions NU VNU 88 7 NU 96 VNU 8 8 Non-binary LDP decoder using FFT-SPA is designed for the specification of IEEE 8.n standard, with the proposed method of (also called as configuration ). It is clear that, with the proposed method of, the number of multiplications gets reduced by 6% on average. The number of additions is also reduced by % on average. This analysis proves that the Non-binary LDP decoder designed with the proposed method reduces the overall computation by 46 % on average. The marginal performance degradation of the decoder with this can be compensated by various code construction methods. However, the proposed has been proved to be computationally efficient for the design of Nonbinary LDP decoder. The author also contributes in proposing three different code construction methodologies, which improves the decoding capability, as elaborated in the next section. ODE ONSTRUTION The computation complexity of the decoder is based on the construction of the PM and thereby has the significant variation in the decoding performance. The improved construction techniques enhance the hardware architecture in terms of the reduced memory requirement and faster computations. Here in this paper, the other contribution is in terms of the different construction methods of PM, analyzing effects in the decoding performance and evaluating its computational strength, collating the method. Lower Diagonal Parity heck Matrix Sparsity is one of the significant metric for the analysis of the PM. The percentage of non-zero entries in the PM is called as sparsity. LDP codes are profound with the sparse PM. The increased sparsity of the matrix reflects in the number of row processing unit and column processing unit. These units are also called check node unit (NU) and variable node unit (VNU) respectively. Earlier methods of code construction have been worked on the random PM. Later with the systematic codes, the structure of the PM is defined with set formation. The trivial pattern is the Quasi-cyclic structure, where only the diagonal elements are non-zero. With the advantages of the systematic codes, here three structures of PM with defined formation are presented. The randomly generated standard PM is modified to obtain LDM [6]. The second half of the random matrix is an array masked [7] with the identity masked. Similarly with the first half of the PM, the lower part is array masked of the identity matrix. The matrix below shows an example of the LDM for 4, called as a base matrix. For the LDP decoder with the block length of 648, array dispersion technique [7] has been applied with the masking matrix of size 7 7. H LDM = Sparsity is the measure of the density of the parity check matrix. As the sparsity of the matrix has the direct influence on the different issues like memory requirement and computation complexity, the construction of the PM accounts for the increased sparsity. The above matrix reflects the increased sparsity, thereby supports in reducing the computation complexity of the decoder. Using array dispersion method, the matrix of has been framed with the sparsity of 97 non-zero elements. In other way, the matrix density is.46. Girth is the shortest cycle of the processing nodes in the tanner graph representation of the PM. Higher the girth value better the convergence of the decoding procedure. The proposed LDM based construction method has the girth value of more than 4, providing the earlier decoding convergence. Doubly Diagonal Parity heck Matrix In the second kind of code construction, the array masking with the doubly diagonal identity matrix has been applied to the first half of the random matrix. The second part has been retained as the identity matrix as a quasi-cyclic structure [6]. The different matrix parameters have been analysed for this Doubly Diagonal Parity heck Matrix (DDM) also.

4 Sparsity of 96 and density of.9 has been attained with this matrixc structure. Girth of this matrix is also found to be more than 4. Thus, the codes constructed using the above matrix structure validates for the category of good codes. The framework of the matrix of the above kind is as follows: H DDM = Hierarchically Diagonal Parity heck Matrix A Hierarchically Diagonal Matrix (HDM) is constructed by dividing random matrix into two halves and the second half is an identity matrix. The first half is again divided into four submatrices of equal size. Now the diagonals of the submatrices are masked with zero. The pattern has been constructed for matrix size 4 [8]. Now circular permutation is applied to the pattern. Each element of 4 matrix (masked with zero) is replaced by quasi cyclic sparse circulant matrix of size 7 7. A circulant matrix is a square matrix which has its rows cyclically shifted. The consecutive rows of the matrix are built by cyclically shifting the previous row by one place to the right. Thereby the matrix is enlarged to size H HDM=. The properties of this HDM structure follow closer to the DDM structure. Sparsity of the matrix is 99 and density of.94. This matrix structure also agrees with the higher girth. These matrix structures satisfy the basic row-column constraints of the PM. The validation of these structures on the Non-binary LDP decoder can be done by evaluating its coding performance. The next section presents the performance analysis of the decoder with these three different matrix structures equations consecutively. PERFORMANE AND OMPUTATION ANALYSIS Figure 4 Decoding performances without The method has been applied for LDM, DDM and HDM and their performance and computation complexity are analyzed. Figure 4 shows the decoding performance of the Non binary LDP decoder based on FFT-SPA without configuration for LDM, DDM and HDM. It is inferred that comparing LDM and DDM, there is relatively an improvement in the decoding performance. omparing these two matrices with HDM, it is evident that there is a greater development in the decoding performance (greatly reduced bit error rate). It has been validated that compared with random PM structures, the proposed three matrices improves the decoding capability. The effectiveness of these matrices has also been analyzed in terms of the computation complexity. The major computation blocks are the NU and VNU, framed with the set of adders and multipliers. Table shows the evaluation of the computation strength of the different matrix structures; LDM, DDM and HDM based decoder without the effect of method. Table omputation analysis of LDM, DDM, HDM structures without Processing unit omputations NU Multiplications additions VNU Multiplications additions The studies on the different matrixes conclude that, with the increased computations from LDM, DDM to HDM has the impact on the improved decoding performance. Hence the tradeoff between the decoding performance and computation complexity holds for these matrix structures also. Understanding the development of the decoder with the different matrix constructions, the analysis is extended by imposing the technique on these matrixes based decoder. Figure 5 Decoding performances with Figure 5 shows the decoding performance of the Non binary LDP decoder based on FFT-SPA with for LDM, DDM and HDM. The decoding performance of these different matrixes based codes maintains the same level of improvement, as in the previous case. However analyzing the computation part of the different matrixes with, projects different inferences. Table 4 shows the computation strength of the different processing units with the inclusion of LDM, DDM and HDM based matrix structures. Table 4 omputation analysis of LDM, DDM, HDM structures with Processing unit omputations NU Multiplications additions VNU Multiplications

5 Processing unit omputations additions Among the three, LDM has lesser number of computations. Based on the NU computations, HDM based codes have more computation strength and on the other side, with respect to VNU computations, DDM based codes have larger computations. It is inferred that approximately 68% of multiplications and 4% of additions can be reduced with the inclusion of method with different matrix structures. With technique, memory requirement decreases and speed of computation increases in hardware perspective ONLUSION Non-binary LDP decoder using FFT-SP algorithm is designed for IEEE 8.n. The performance of the decoder is marginally degraded, but with the proposed method, the computations are greatly reduced, making it suitable for efficient hardware implementation. It is evaluated that the proposed method reduces the computations by 46%. The three different PM structures are combined with the method and the decoding performance of the Non-binary LDP decoder are evaluated. For the improved decoding, with a compromise on the computation strength HDM structures are found to be appropriate. For the reduced computations, LDM based decoder are appreciable with nominal decoding degradation REFERENES [] Gallager R, Low-Density Parity-heck odes, ambridge, MA: MIT Press, 96 [] MacKay D.J. and Neal R, Near Shannon-limit performance of low-density parity-check codes, Electronic letters, Vol., No.6, pp: , 997 [] Davey M.. and Mackay D.J, Low density parity check codes GF (q), IEEE ommunication Letters, Vol., No.6, pp:65-67, 998. [4] El Hassani S., et al, A omparison Study of Binary and Non-Binary LDP odes Decoding, Proc in IEEE international conference, software, telecommunications and computer networks (softom), pp: 55 59, [5] David Declereq, Marc Fossorier, Extended Min sum algorithm for decoding LDP codes over GF(q), Information Theory, ISIT,pp: , 5. [6] Savin V, Min Max decoding for non binary LDP codes, in Proc.IEEE International Symposium on Information Theory, Toronto, ON, anada, pp: , 8 [7] Xinmiao Zhang and Fang ai, Reduced-complexity Extended Min-sum check node processing for Non-binary LDP decoding, 5 rd IEEE International Midwest Symposium on ircuits and Systems (MWSAS), pp: 77-74,. [8] Lingqi Zeng, Lan Lan, Ying Yu Tai, Bo Zhou, Shu Lin and Abdel-Ghaffar, onstruction Of Nonbinary yclic, Quasi-yclic And Regular LDP odes: A Finite Geometry Approach, IEEE Transactions on ommunications, Vol. 56, No., 8 [9] Lingqi Zeng, Lan Lan, Ying Yu Tai, Shumei Song, Shu Lin and Abdel-Ghaffar, onstruction Of Nonbinary Quasi-yclic LDP odes: A Finite Field Approach, IEEE Transactions on ommunications, Vol. 56, No.4, pp: 84-9, 8. [] Shumei Song, Bo Zhou, A Unified Approach To The onstruction Of Binary And Non-Binary Quasi-yclic LDP odes Based On Finite Fields, IEEE Transactions on ommunications, Vol.57, No., 9 [] K. Lally and P. Fitzpatrick, Algebraic structure of quasicyclic codes, Discrete Applied Mathematics, Vol., No. -, pp. -75,. [] ] Jose M.F. Moura, Jin Lu, Haotian Zhang, Structured Low-Density Parity-heck codes, IEEE Signal Processing Magazine, pp. 4-55, January 4 [] Rolando et al., 9, Non-binary Error ontrol oding for Wireless ommunication and Data Storage, John Wiley, ISBN [4] Xiao Ma et al., Low complexity X-EMS algorithms for Non-binary LDP codes, IEEE Transactions on ommunications, Vol.6, No., pp: 9-,. [5] IEEE 8. Wireless LANs WWiSE Proposal (4): High Throughput Extension to the 8. Standards, IEEE. [6] Aruna S and Anbuselvi M, FFT-SPA based non-binary LDP decoder for 8.n standard, International onference on ommunications and Signal Processing (ISP), pp: ,. [7] Bo Zhou, Li Zhang, Jingyu Kang, Qin Huang, Shu Lin, and Khaled Abdel-Ghaffar, Array Dispersions of and onstructions of Quasi-yclic LDP odes Over Non-Binary Fields, IEEE International Symposium on Information Theory, pp. 58-6, 8. [8] Arulmozhi M and Anbuselvi M, Improvements on construction of Quasi cyclic Irregular Non Binary LDP odes, International conference on Information, ommunications and Embedded Systems, 4.