Integrated Code Design for a Joint Source and Channel LDPC Coding Scheme

Transcription

1 Integrated Code Design for a Joint Source and Channel LDPC Coding Scheme Hsien-Ping Lin Shu Lin and Khaled Abdel-Ghaffar Department of Electrical and Computer Engineering University of California Davis CA USA {hsplin shulin ghaffar}@ucdavis.edu Abstract This paper presents a design of LDPC codes for a joint source-channel (JSC) coding system. In the construction of such an LDPC code the source compression matrix and the channel code parity-check matrix are designed jointly. The integrated matrix is used for both JSC encoding and decoding. Experimental results show that the codes constructed not only perform well in the waterfall region but also achieve low errorfloors. I. INTRODUCTION Application of LDPC (low-density parity-check) codes 1 for JSC coding was first proposed in 2 3 in which two JSC-LDPC coding schemes were proposed. The first scheme consists of two LDPC codes in serial concatenation the outer code is for source compression and the inner code is for channel coding. The Tanner graphs 4 of the two component codes are combined into a single global graph for joint decoding of the untransmitted source symbols and the received channel code symbols. In decoding the source variable nodes (VNs) and channel VNs exchange information iteratively through the source check nodes (CNs). This scheme is referred to as the JSC-LDPC concatenated coding scheme (JSC-LDPC- CC) scheme. The second scheme uses the Burrows-Wheeler transform (BWT) 3 and a single LDPC code for both JSC encoding and decoding which is referred to as the JSC-BWT- LDPC coding scheme. In spite of the introduction of the two JSC-LDPC coding schemes the authors in 2 3 did not give any code design. Design of LDPC codes for the JSC-LDPC-CC scheme was first reported in 5. However the JSC-LDPC-CC system using the LDPC codes designed in 5 showed very poor (or high) error-floor which is undesirable in practical applications for systems where relatively low-error rates are required. To improve the error-floor performance of the JSC-LDPC-CC scheme protograph-based LDPC codes were proposed for the JSC-LDPC-CC scheme in 6. Using two existing protographbased LDPC codes the authors of 6 demonstrated that the error-floor performance of the JSC-LDPC-CC scheme was indeed improved but still relatively high. The weakness of the JSC-LDPC-CC scheme is that the information exchange between the untransmitted source symbols and the received channel code symbols in the JSC-decoding is very limited. In each decoding iteration the information exchange between the Tanner graph of the source code and the Tanner graph of the channel code is only through the edges connecting the CNs of the source code Tanner graph and the VNs of the channel code Tanner graph that correspond to the compressed information symbols. Each source CN is connected only to one channel VN that corresponds to a compressed information symbol. This information exchange limitation is a major cause of high error-floor and slow decoding convergence. To overcome the weakness of the JSC-LDPC-CC scheme an enhanced JSC-LDPC-CC scheme was proposed and analyzed in 7 and 8 in which the source symbols are doubly encoded. This double encoding of source symbols increases the linkage between the untransmitted source symbols and channel output code symbols. Consequently extra edges are added between source VNs in the source Tanner graph and the CNs in the channel Tanner graph for the JSC-decoding. These added extra edges enhance the information exchange between untransmitted source symbols and channel output code symbols. In 7 and 8 the authors demonstrated that the enhancement of information exchange significantly improves the error-floor p erformance o f t he J SC-LDPC-CC scheme. Since the source symbols are doubly encoded we refer to this enhanced JSC-LDPC-CC scheme as a doubly encoded (DE) JSC-LDPC-CC (DE-JSC-LDPC-CC) scheme. In the proposed DE-JSC-LDPC-CC scheme multi-edge type LDPC codes 9 were used. A JSC code optimization was proposed in 7. The optimization is under the restriction such that in the joint Tanner graph for decoding each VN in the source code graph is connected to only one CN in the channel code graph. The optimization is applied to the source code and the channel code separately. No explicit description of code construction was given. To remove cycles of length 4 in the joint Tanner graph for decoding an information shortening strategy was introduced in 8. The information shortening may result in a reduction of information exchange between the channel VNs and the source VNs and hence may slow down the decoding convergence and cause a higher error floor. This paper presents an integrated design of LDPC codes for the DE-JSC-LDPC-CC scheme. In the design the source compression and the channel coding matrices are designed jointly to produce an integrated LDPC parity-check matrix denoted by H dejsc whose associated Tanner graph has girth at least 6. The integrated LDPC parity-check matrix H dejsc is used for both JSC encoding and decoding. JSC-decoding based on the integrated parity-check matrix H dejsc not only provides a good waterfall error performance for the DE-JSC-LDPC-CC

2 coded system but also results in a low error-floor. The codes designed for the DE-JSC-LDPC-CC system are constructed based on finite fields and they are quasi-cyclic (QC). The rest of this chapter is organized as follows. Section II gives a brief description of the DE-JSC-LDPC-CC scheme proposed in 7. Sections III presents a specific algebraic construction of quasi-cyclic (QC) LDPC codes whose Tanner graphs have girth at least 6. In Section IV we present an integrated design of QC-LDPC codes for the DE-JSC-LDPC- CC system based on the results presented in Section III. In Section V two examples are given to show that the designed JSC-LDPC codes for the DE-JSC-LDPC-CC system perform very well compared with comparable codes designed in 7 8 especially in error-floor performance. Section VI concludes the paper with some remarks. Through this paper the source is assumed to be binary and memoryless with symbol probabilities Pr(s = 0) and Pr(s = 1) respectively. II. DE-JSC-LDPC CODING SYSTEM In the DE-JSC-LDPC-CC system proposed in 7 the encoding consists of two steps. First a sequence s of m source symbols called a message is systematically encoded into codeword (b s) of length m + l in an outer code C outer given by the null space of an l (m + l) sparse parity-check matrix H outer = I H s where H s is the source compression matrix. The l symbols in b are regarded as the compressed information symbols. Then the codeword (b s) is encoded into codeword u in a channel code C c the inner code of the system given by the null space of an (n m l) n LDPC paritycheck matrix H c. The codeword u has length n and consists of three parts the source sequence s the compressed information vector b and the parity-check part p i.e. u = (p b s). The parity-check part p consists of n m l parity-check symbols. In transmission the source output sequence s is punctuated from u and the punctured vector v = (p b) of length n m consisting of the compressed information and the channel coding parity-check symbols is transmitted. Hence for the DE-JSC-LDPC-CC system the compression rate R comp is l/m the channel code rate R c is (m + l)/n transmission rate R t is l/(n m) and the total code rate of the system R jsc is m/(n m). At the receiver end of the DE-JSC-LDPC-CC system the untransmitted source symbols and the received code symbols are decoded jointly based on the following (n m) n paritycheck matrix: H H dejsc = c (1) 0 I H s where 0 is an l (n m l) zero matrix and I is an l l identity matrix. The matrix H dejsc is composed of the source compression matrix H s and channel code parity-check matrix H c as submatrices. Since both H s and H c are sparse matrices the combined matrix H dejsc given by (1) is also a sparse matrix whose null space gives an LDPC code C dejsc called a JSC-LDPC code. Let G dejsc be the Tanner graph associated with H dejsc (or the Tanner graph of C dejsc ). It follows from the structure of H dejsc that G dejsc is composed of the Tanner graph of the source code C s and the Tanner graph of the channel code C c as subgraphs as shown in Figure 1. C 1c channel Tanner graph C 2c l compressed information symbols n m transmitted symbols C n m lc V 1 V n l m V n l m+1 V n m source Tanner graph V n m+1 Fig. 1. The Tanner graph of G dejsc C 1s C 2s Cls m source symbols In G dejsc each source VN is not only connected to compressed source symbol VNs through the source CNs of the source Tanner graph G s but also connected to channel paritycheck symbol VNs through the CNs of the channel code Tanner graph G c. As a result in each JSC-decoding iteration a source VN-s receives extrinsic information from more VNs in the channel code Tanner graph (i.e. more information from the channel) than the JSC-LDPC-CC system. Hence the information exchange between the untransmitted source symbols and the channel output code symbols is enhanced. This enhancement of information exchange between the source Tanner graph and the channel code Tanner graph improves the performance of the JSC-LDPC-CC system. The modified sum-product algorithm (SPA) proposed in 5 can be applied to decode the untransmitted source symbols and the received channel code symbols jointly based on the joint Tanner graph G dejsc or the joint parity-check matrix H dejsc given by (1). In the modified SPA the initial LLR used for each untransmitted source symbol s is given by log(pr(s = 0)/Pr(s = 1)). In 7 and 8 the authors demonstrated that the DE-JSC- LDPC coding system outperforms the JSC-LDPC-CC system significantly as shown in Figure 3 of 7. In 7 multi-edge type LDPC codes 9 were designed for the DE-JSC-LDPC-CC system. A JSC code optimization was presented. The optimization is under the restriction such that in the joint Tanner graph for decoding a source VN in the source Tanner graph G s is only connected to one CN in the channel code graph G c. The restriction puts a limit on the message exchange between the two graphs even though not much as the JSC-LDPC-CC scheme. Furthermore the optimization is applied to the source code and the channel code separately not jointly. Note that even if the Tanner graphs G s and G c associated with the source and channel parity-check matrices H s and H c are cycle-4 free the Tanner graph G dejsc associated with the overall parity-check matrix H dejsc given V n

3 by (1) may not be cycle-4 free. To remove cycles of length 4 in the joint Tanner graph G dejsc for decoding an information shortening strategy was introduced in 8. The information shortening may result in a reduction of information exchange between the channel CNs and the source VNs. The restriction on linking the source VNs in the source Tanner graph G s and the CNs in the channel code Tanner graph G c design of the source and channel codes separately and the information shortening to remove cycles of length 4 limit the performance improvement of the DE-JSC-LDPC-CC scheme over the JSC-LDPC-CC scheme proposed in 5. The performance of the DE-JSC-LDPC-CC scheme can be improved if: 1) The source and channel codes are designed jointly; 2) The Tanner graph G dejsc associated with the joint parity-check matrix H dejsc is free of cycles of length 4 without information shortening; and 3) There is no restriction on linking the source VNs to channel CNs. III. AN ALGEBRAIC CONSTRUCTION OF QUASI-CYCLIC LDPC CODES In this section we will present an algebraic method for constructing binary low-density parity-check (LDPC) matrices whose associated Tanner graphs have girth at least 6. The construction of such an LDPC matrix is based on a specific expansion of a base matrix over a nonbinary field that satisfies a certain constraint. The base matrices constructed in this section will be used in the next section to construct paritycheck matrices for JSC encoding and decoding in the DE- JSC-LDPC system. Tanner graphs associated with these paritycheck matrices have girth at least 6. Furthermore the codes given by the null space of these matrices are QC-LDPC codes. Let α be a primitive element of GF(q). Then the powers of α α 0 = 1 α α 2... α q 2 give all the nonzero elements of GF(q). For 0 j q 2 we represent the element α j by a binary circulant permutation matrix (CPM) denoted by A(α j ) with columns and rows labeled from 0 to q 1 of size (q 1) (q 1) whose generator (top row) has the unit-element 1 of GF(q) as its single nonzero component at position j. It is clear that the representation of the element α j by the binary CPM A(α j ) of size (q 1) (q 1) is unique and the mapping between α j and A(α j ) is one-to-one. This matrix representation of α j is referred to as the binary CPM-dispersion of α j. The 0 element of GF(q) is represented by a (q 1) (q 1) zero-matrix (ZM). Let B(k t) = b ij 0 i < k 0 j < t be a k t matrix over GF(q). Disperse each nonzero entry of B(k t) into a binary CPM of size (q 1) (q 1) and a zero entry into a ZM of size (q 1) (q 1). This results in a k t array H(k t) of binary CPMs and/or ZMs of size (q 1) (q 1). The array H(k t) is a k(q 1) t(q 1) matrix over GF(2) and is called the binary CPM-dispersion of the matrix B(k t) denoted by CPM(B(k t)) i.e. H(k t) = CPM(B(k t)). The matrix B(k t) is called the base matrix for dispersion. If q is large enough H(k t) will be a sparse matrix whose null space gives a QC-LDPC code. A matrix in which no two rows (or two columns) have more than one location where they both have nonzero entries is said to satisfy the row-column (RC) constraint. Such a matrix is called an RC-constrained matrix. If the parity-check matrix H of an LDPC code C satisfies the RC-constraint then the Tanner graph G of C has girth at least The following theorem gives the necessary and sufficient condition on a base matrix B(k t) whose CPM-dispersion H(k t) as a k(q 1) t(q 1) matrix satisfies the RC-constraint 11. Theorem 1. Let B(k t) be a k t matrix over GF(q) and H(k t) be the binary CPM-dispersion of B(k t). A necessary and sufficient condition for the array H(k t) to satisfy the RCconstraint is that every 2 2 submatrix in B(k t) contains at least one zero entry or is non-singular. For simplicity we refer to the necessary and sufficient condition given in Theorem 1 as the 2 2 SM-constraint. A base matrix B(k t) that satisfies the 2 2 constraint is called a 2 2 constrained matrix. It follows from Theorem 1 that the CPM-dispersion H(k t) of a 2 2 constrained base matrix B(k t) satisfies the RC-constraint. Consequently the Tanner graph associated with H(k t) has girth at least 6. There are many algebraic methods for constructing base matrices that satisfy the 2 2 SM-constraint (also see their references). Among these methods the most flexible one is the method proposed in 14. The construction is based on two arbitrary sets of a chosen field. In the following we describe this method for later use in integrated design of JSC-LDPC codes. Let α be a primitive element of GF(q). For 1 k t q let S 0 = {α i0 α i1... α i k 1 } and S 1 = {α j0 α j1... α jt 1 } be two arbitrary non-empty subsets of elements in GF(q) of size k and t respectively with i e and j l in L = { q 2} i 0 < i 1 <... < i k 1 and j 0 < j 1 <... < j t 1. Let η be any nonzero element in GF(q). We form the following k t matrix B(k t) over GF(q): B(k t) = ηα i0 + α j0 ηα i0 + α j1... ηα i0 + α jt 1 ηα i1 + α j0 ηα i1 + α j1... ηα i1 + α jt (2) ηα i k 1 + α j0 ηα i k 1 + α j1... ηα i k 1 + α jt 1 Matrix B(k t) given by (2) has the following structural properties: 1) all the entries in a row (or a column) are distinct elements of GF(q); 2) each row (or each column) contains at most one zero entry; 3) no two rows (or two columns) have identical entries in any position. It was proved in 14 that B(k t) satisfies the 2 2 SM-constraint. Hence it can be used for constructing an RC-constrained matrix by CPMdispersion. The CPM-dispersion of B(k t) gives a k t array H(k t) of binary CPMs and/or ZMs of size (q 1) (q 1) a k(q 1) t(q 1) matrix over GF(2) which satisfies the RCconstraint. The null space of H(k t) gives a binary QC-LDPC code C whose Tanner graph has girth at least 6. In the next section the 2 2 constrained base matrix of the

4 form given by (2) will be used for the integrated design and construction of JSC-LDPC codes for the DE-JSC-CC system. In the integrated design of a JSC-LDPC code besides the construction of a 2 2 constrained base matrix and its CPM-dispersion another ingredient needed is masking. Given a base matrix B(k t) that satisfies the 2 2 SM-constraint a shaping operation can be performed on B(k t) to obtain a new base matrix. The matrix shaping operation is known as masking which was first introduced in 16 and has been used to construct LDPC codes with large girth and small numbers of short cycles in their Tanner graphs Suppose 10 we replace a nonzero entry in B(k t) by the zeroelement of GF(q). In the CPM-dispersion H(k t) of B(k t) this replacement results in replacing a (q 1) (q 1) CPM in H(k t) by a (q 1) (q 1) zero matrix which is referred to as masking. Let µ be a nonnegative integer less than the number of nonzero entries in B(k t). The replacement of µ nonzero entries in B(k t) by µ zeros amounts to replacing µ CPMs by µ ZMs at the corresponding locations in H(k t). Masking µ CPMs in H(k t) amounts to removing µ(q 1) edges from the Tanner graph G associated with the array H(k t). Removing these edges from G may break many short cycles. As a result the resultant Tanner graph denoted by G mask may have a smaller number of short cycles or a larger girth (larger than 6) or both. The subscript mask" of G mask stands for masking". Masking the base matrix B(k t) = b ij 0 i < k 0 j < t can be modeled mathematically as a Hadamard matrix product 17. Let Z(k t) = z ij 0 i < k 0 j < t be a k t matrix with the zero and unit elements of GF(q) as entries. Define the following product of Z(k t) and B(k t): B mask = Z(k t) B(k t) = z ij b ij 0 i < k 0 j < t where z ij b ij = b ij if z ij = 1 and z ij b ij = 0 if z ij = 0. In this matrix product the nonzero entries in B(k t) at the locations corresponding to zero entries in Z(k t) are replaced by zeros. The binary CPM-dispersion of B mask (k t) gives a k t masked array denoted by H mask (k t) of CPMs and ZMs of size (q 1) (q 1). We call Z(k t) and B mask (k t) the masking matrix and the masked base matrix respectively. Since B mask (k t) is a submatrix of the 2 2 SM-constrained matrix B(k t) B mask (k t) must also satisfy the 2 2 SM constraint. Hence the null space of H mask (k t) also gives a binary QC-LDPC code denoted by C mask whose Tanner graph G mask has a smaller number of short cycles than the unmasked Tanner graph G. Binary CPM-dispersions of properly chosen base matrices in conjunction with masking can result in QC-LDPC codes with good error probability performance as well as low error-floor. Masking is a key ingredient in our integrated design of JSC- LDPC codes for the DE-JSC-LDPC-CC system which will be presented in the next section. IV. AN INTEGRATED CODE DESIGN FOR THE DE-JSC-LDPC-CC SYSTEM In this section we present an integrated design of JSC- LDPC codes for the DE-JSC-LDPC-CC system presented in Section II. Our approach is first to design an RC-constrained JSC parity-check matrix H dejsc in the form given by (1). Then the upper submatrix H c of H dejsc serves as the paritycheck matrix of the channel LDPC code and the submatrix H s at the lower right corner of H dejsc serves as the source compression matrix. The design of H dejsc is carried out in an integrated manner. A. General Structure of a DE-JSC-LDPC Matrix Consider the (n m) n JSC parity-check matrix H dejsc for the DE-JSC-LDPC-CC system given by (1). We partition this matrix into three submatrices as follows: with H 1 = Hc1 0 H dejsc = H 1 H 2 H 3 (3) Hc2 H 2 = I Hc3 H 3 = (4) H s where H c1 H c2 and H c3 form a partition of the parity-check matrix H c of the channel code C c i.e. H c = H c1 H c2 H c3. The sizes of H c1 H c2 and H c3 are (n m l) (n m l) (n m l) l and (n m l) m respectively. The columns of the first submatrix H 1 of H dejsc correspond to the channel code parity-check symbols the columns of the second submatrix H 2 of H dejsc correspond to the compressed information symbols and the columns of the third submatrix H 3 of H dejsc correspond to the source symbols. In the design of H dejsc the submatrices H c1 H c2 and H c3 of the channel code parity-check matrix H c have different column weight distributions. The submatrix H c2 has the largest column weights designed for protecting the transmitted compressed information symbols (output of source encoder). The submatrix H c1 has the next largest column weights designed for protection of the transmitted parity-check symbols. Since the source symbols are not transmitted they don t need protection against the channel noise. However in JSCdecoding their corresponding VNs in the joint Tanner graph G dejsc need to be connected to the VNs which correspond to both the compressed information and the channel parity-check symbols to enhance the information exchange between them and the transmitted code symbols. Hence the submatrix H c3 has the least column weights. However they must be large enough to provide sufficient information exchange between the untransmitted source symbols and the transmitted code symbols in the JSC-decoding based on H dejsc (or G dejsc ). Each column of H c3 must have weight at least 1. The source compression submatrix H s of H dejsc has low constant column weights. B. An Integrated Design of the JSC Parity-Check Matrix: Design Steps In the following we propose an integrated design of a JSC parity-check matrix H dejsc for both JSC encoding and decoding in the DE-JSC-LDPC-CC system. The design and construction of such a JSC-LDPC matrix take the following five steps: 1) Design a 2 2 SM-constrained base matrix B over a chosen field GF(q); 2) Construct a masking matrix Z

5 of the same size as that of the base matrix B which has the same form as the JSC parity-check matrix H dejsc given in (1) (3) and (4) with three submatrices Z 1 Z 2 Z 3 ; 3) Design column weight distributions for the submatrices of Z; 4) Mask the base matrix B with Z to produce a masked matrix B mask which has the same column weight distribution as that of the masking matrix Z; and 5) Take the CPM-dispersion of the masked base matrix B mask to produce the JSC parity-check matrix H dejsc which has the same column weight distribution as that of the masking matrix Z. The design of column weight distributions for the three submatrices Z 1 Z 2 Z 3 of the masking matrix Z is carried out such that the three submatrices H 1 H 2 H 3 of the JSC paritycheck matrix H dejsc have the column weight distributions as described in Section IV-A. The null space over GF(2) of H dejsc gives a JSC-LDPC code C dejsc for the DE-JSC- LDPC-CC system. C. Design and Construction of a Section-Wise Regular JSC Parity-Check Matrix Suppose our objective is to design a DE-JSC-LDPC-CC system that achieves a compression rate R comp = l/m a channel code rate R c = (m + l)/n a transmission rate R t = l/(n m) and a total system rate R jsc = m/(n m). With our proposed construction of the JSC parity-check matrix H dejsc based on the CPM-dispersion of a 2 2 SM-constrained and masked base matrix as described above the parameters m l and n must be multiples of q 1 (the dispersion factor). Let m = m 0 (q 1) l = l 0 (q 1) and n = n 0 (q 1) respectively. In the construction of the JSC-LDPC matrix H dejsc of a DE-JSC-LDPC code C dejsc the base matrix is an (n 0 m 0 ) n 0 matrix B(n 0 m 0 n 0 ) in the form given by (2) which is constructed based on two subsets S 0 and S 1 of a chosen field GF(q) with sizes n 0 m 0 and n 0 respectively. Assume that all entries in B(n 0 m 0 n 0 ) are nonzero (just for simplicity but not necessary). Then B(n 0 m 0 n 0 ) has constant column weight n 0 m 0 and constant row weight n 0 respectively. The masking matrix Z(n 0 m 0 n 0 ) is an (n 0 m 0 ) n 0 matrix over GF(2) of the following form: Zc (n Z(n 0 m 0 n 0 ) = 0 m 0 l 0 n 0 ) (5) 0 I Z s (l 0 m 0 ) where 0 is an l 0 (n 0 m 0 l 0 ) zero matrix I is an l 0 l 0 identity matrix Z c (n 0 m 0 l 0 n 0 ) is an (n 0 m 0 l 0 ) n 0 matrix and Z s (l 0 m 0 ) is an l 0 m 0 matrix. Partition the upper submatrix Z c (n 0 m 0 l 0 n 0 ) of Z(n 0 m 0 n 0 ) into three submatrices Z c1 (n 0 m 0 l 0 n 0 m 0 l 0 ) Z c2 (n 0 m 0 l 0 l 0 ) and Z c3 (n 0 m 0 l 0 m 0 ). Then Z(n 0 m 0 n 0 ) can be expressed in the following form: where Z(n 0 m 0 n 0 ) = Z 1 Z 2 Z 3 (6) Zc1 (n Z 1 = 0 m 0 l 0 n 0 m 0 l 0 ) 0 Zc2 (n Z 2 = 0 m 0 l 0 l 0 ) I Zc3 (n Z 3 = 0 m 0 l 0 m 0 ) Z s (l 0 m 0 ). (7) From (5) (6) and (7) we see that the masking matrix Z(n 0 m 0 n 0 ) has exactly the same form as the parity-check matrix H dejsc of the DE-JSC-LDPC code C dejsc given in (1) (3) (4). In constructing Z(n 0 m 0 n 0 ) the column weight assignments for the three submatrices of Z(n 0 m 0 n 0 ) can be carried out either in the form of section-wise regular or based on an optimal VN-degree distribution designed using the density evolution 18 (or an EXIT-chart 19). For a section-wise regular column weight assignment the column weight of each submatrix of the upper submatrix Z c (n 0 m 0 l 0 n 0 ) of the masking matrix Z(n 0 m 0 n 0 ) is constant. Let w 1 w 2 and w 3 be the column weights assigned to submatrices Z c1 (n 0 m 0 l 0 n 0 m 0 l 0 ) Z c2 (n 0 m 0 l 0 l 0 ) and Z c3 (n 0 m 0 l 0 m 0 ) of Z(n 0 m 0 n 0 ) respectively. These column weights are chosen such that w 2 > w 1 > w 3. The 3-tuple (w 1 w 2 w 3 ) is referred to as the section-wise column weight distribution of Z c (n 0 m 0 l 0 n 0 ). Let w s be the column weight of the submatrix Z s of Z(n 0 m 0 n 0 ). Then the section-wise column weight distribution of the masking matrix Z(n 0 m 0 n 0 ) is (w 1 w w 3 + w s ) where w 1 w and w 3 + w s are the column weights of the submatrices Z 1 Z 2 Z 3 of Z(n 0 m 0 n 0 ) respectively. The masked base matrix B mask (n 0 m 0 n 0 ) obtained by masking B(n 0 m 0 n 0 ) with Z(n 0 m 0 n 0 ) has the same form as the masking matrix Z(n 0 m 0 n 0 ) as shown in (5) (6) and (7) i.e. B mask (n 0 m 0 n 0 ) = Bc (n 0 m 0 l 0 n 0 ) 0 I non B s (l 0 m 0 ) = B 1 B 2 B 3 (8) where B c (n 0 m 0 l 0 n 0 ) has the form of B c1(n 0 m 0 l 0 n 0 m 0 l 0 ) B c2 (n 0 m 0 l 0 l 0 ) B c3 (n 0 m 0 l 0 m 0 ) and Bc1 (n B 1 = 0 m 0 l 0 n 0 m 0 l 0 ) 0 Bc2 (n B 2 = 0 m 0 l 0 l 0 ) I non B 3 = Bc3 (n 0 m 0 l 0 m 0 ) B s (l 0 m 0 ) (9). (10) The masked base matrix B mask (n 0 m 0 n 0 ) has the same column weight distribution as that of the masking matrix Z(n 0 m 0 n 0 ). Note that I non is an l 0 l 0 diagonal matrix with nonzero entries on its main diagonal and zero entries elsewhere. These nonzero entries are elements in GF(q). Since B(n 0 m 0 n 0 ) satisfies the 2 2 SM-constraint the masked

6 matrix B mask (n 0 m 0 n 0 ) also satisfies the 2 2 SMconstraint. The CPM-dispersion of B mask (n 0 m 0 n 0 ) denoted by CPM(B mask (n 0 m 0 n 0 )) gives the following RCconstrained matrix over GF(2): CPM(B mask (n 0 m 0 n 0 )) CPM(B = c (n 0 m 0 l 0 n 0 )) (11) CPM(0) CPM(I non ) CPM(B s (l 0 m 0 )) = CPM(B 1 ) CPM(B 2 ) CPM(B 3 ) (12) where CPM(B c (n 0 m 0 l 0 n 0 )) = CPM(B c1(n 0 m 0 l 0 n 0 m 0 l 0 )) CPM(B c2 (n 0 m 0 l 0 l 0 )) CPM(B c3 (n 0 m 0 l 0 m 0 )) (13) and CPM(Bc1 (n CPM(B 1 ) = 0 m 0 l 0 n 0 m 0 l 0 )) CPM(0) CPM(Bc2 (n CPM(B 2 ) = 0 m 0 l 0 l 0 )) CPM(I non ) CPM(Bc3 (n CPM(B 3 ) = 0 m 0 l 0 m 0 )) CPM(B s (l 0 m 0 )). (14) Note that CPM(0) is an l 0 (q 1) (n 0 m 0 l 0 )(q 1) zero matrix denoted by 0 q 1. CPM(I non ) is an l 0 l 0 diagonal array with binary CPMs of size (q 1) (q 1) lying on its main diagonal and ZMs elsewhere. The CPMarray CPM(I non ) can be put into an l 0 (q 1) l 0 (q 1) identity matrix denoted by I q 1 by a simple row permutation denoted by π. Applying the row permutation π to the rows of the lower submatrix CPM(0) CPM(I non ) CPM(B s (l 0 m 0 )) of CPM(B mask (n 0 m 0 n 0 )) we obtain a row permuted matrix 0 q 1 I q 1 π(cpm(b s (l 0 m 0 ))). Replacing the lower submatrix CPM(0) CPM(I non ) CPM(B s (l 0 m 0 )) of CPM(B mask (n 0 m 0 n 0 )) given in (11) by the row permuted matrix 0 q 1 I q 1 π(cpm(b s (l 0 m 0 ))) we obtain the following RC-constrained LDPC parity-check matrix: CPM(B H dejsc (n 0 m 0 n 0 )) = c (n 0 m 0 l 0 n 0 )) 0 q 1 I q 1 π(cpm(b s (l 0 m 0 ))) (15) Let H c1 = CPM(B c1 (n 0 m 0 l 0 n 0 m 0 l 0 )) H c2 = CPM(B c2 (n 0 m 0 l 0 l 0 )) H c3 = CPM( B c3 (n 0 m 0 l 0 m 0 ) H c = CPM(B c (n 0 m 0 l 0 n 0 )) = H c1 H c2 H c3 H s = π(cpm(b s (l 0 m 0 ))) Hc1 H 1 = 0 q 1 H 3 = H 2 = Hc2 I q 1 H c3 π(cpm(b s (l 0 m 0 ))) Then the matrix H dejsc (n 0 m 0 n 0 ) given by (15) has. exactly the form given by (1) (3) and (4) and is an (n 0 m 0 ) n 0 array of CPMs and ZMs of size (q 1) (q 1) an (n 0 m 0 )(q 1) n 0 (q 1) matrix. The submatrix H c = CPM(B c (n 0 m 0 l 0 n 0 )) is the parity-check matrix of the channel code and H s = π(cpm(b s (l 0 m 0 ))) is the source compression matrix. It follows from column weight assignments of the upper submatrix Z c (n 0 m 0 l 0 n 0 ) of the masking matrix Z(n 0 m 0 n 0 ) that H c1 H c2 and H c3 have column weights w 1 w 2 w 3 respectively with w 2 > w 1 > w 3 and the column weight of source compression matrix H s = π(cpm(b s (l 0 m 0 ))) is w s. The null space over GF(2) of H dejsc (n 0 m 0 n 0 ) gives a QC-JSC-LDPC code C dejsc of length n 0 (q 1) for the DE-JSC-LDPC coding system with a total system code rate R jsc = m 0 /(n 0 m 0 ). The QC structure of C dejsc simplifies the implementation of the JSC-decoder. The Tanner graph of C dejsc has girth at least 6. The matrix H dejsc (n 0 m 0 n 0 ) is used for both JSC encoding and decoding. The Tanner graph G dejsc associated with H dejsc (n 0 m 0 n 0 ) contains both the source Tanner graph G s and the channel Tanner graph G c as subgraphs which are connected by edges. There are two types of connections between the source VNs and the channel VNs. In the type-1 connection each source VN-s in G s is connected to a group of channel parity-check and compressed information VNs in G c through w 3 channel code CNs in G c by paths of length 2. In the type-2 connection each source VN-s in G s is connected to a group of compressed information VNs in G c through w s CNs in G s by paths of length 2. In both types of connections the source VNs in G s are connected to the compressed information VNs in G c. The total number of paths of length 2 that connect the source Tanner graph G s and channel code Tanner graph G c is a measure of connectivity of the two subgraphs which is a measure of the amount of information exchange between the transmitted code symbols and the untransmitted source symbols in each decoding iteration of an iterative decoding algorithm. A large connectivity between G s and G c not only makes the JSC-decoding based on H dejsc (n 0 m 0 n 0 ) converging fast but also gives a low error-floor performance of the DE-JSC-LDPC-CC system which will be shown in two. examples given in the next section. V. EXAMPLES In the following we give two examples to illustrate the construction of QC-JSC-LDPC codes for the DE-JSC-LDPC- CC system. The parity-check matrices of these codes have section-wise constant column weights. In the examples we demonstrate the designed JSC-LDPC codes outperform the multi-edge type JSC-LDPC codes constructed in 7 for the DE-JSC-LDPC-CC system with comparable message lengths code lengths compression rates transmission rates and total system rates. Example 1. In this example we design a QC-JSC-LDDC code for the DE-JSC-LDPC-CC system to achieve a total system rate R jsc = 2 for transmitting each message of m = 3302

7 source symbols. One such design is to set m 0 = 26 l 0 = 6 and n 0 = 39. Let GF(2 7 ) be the field for code construction. In this case the CPM-dispersion factor is 127 and hence m = 127 m 0 = 3302 l = 127 l 0 = 762 and n = 127 n 0 = With these chosen parameters the designed DE-JSC-LDPC-CC system achieves an information compression rate R comp = l 0 /m 0 = a transmission rate R t = l 0 /(n 0 m 0 ) = a channel code rate R c = (m 0 + l 0 )/n 0 = and a total system code rate R jsc = m 0 /(n 0 m 0 ) = 2. Let α be a primitive element of GF(2 7 ). We construct a base matrix B(13 39) over GF(2 7 ) in the form given by (2). The construction is carried out by setting η = 1 and using the following subsets of GF(2 7 ): S 0 = {α 0 α 1... α 12 } and S 1 = {α 1 α 2... α 39 }. The base matrix B(13 39) satisfies the 2 2 SM-constraint and all its entries are nonzero elements in GF(2 7 ). Next we construct a masking matrix in the form given by (5) Z(13 39) = Z c (7 39) 0 I Z s (6 26) (16) The upper submatrix Z c (7 39) = Z c1 (7 7) Z c2 (7 6) Z c3 (7 26) of Z(13 39) consists of three submatrices Z c1 (7 7) Z c2 (7 6) and Z c3 (7 26) of sizes and 7 26 respectively. The column weights w 1 w 2 w 3 designed for these three submatrices are w 1 = 3 w 2 = 7 and w 3 = 1. The weight w s assigned to the submatrix Z s (6 26) is 2. When we construct each submatrix of Z(13 39) we keep weights of all the rows equal or close to each other. With the above column weight assignments the column weight distribution of Z(13 39) is (3 8 3). Masking the base matrix B(13 39) with the designed masking given by (16) we obtain a masked matrix B mask (13 39) which has the same form and the same column weight distribution as the masking matrix Z(13 39). The CPM-dispersion of B mask (13 39) gives a JSC parity-check matrix H dejsc (13 39) which is a array of CPMs and ZMs of size a matrix with section-wise column weight distribution (3 8 3). The array H dejsc (13 39) consists of two subarrays the upper and the lower ones. The upper subarray H c (7 39) of H dejsc (13 39) serves as the parity-check matrix of the channel code which is a matrix with section-wise column weight distribution (3 7 1). The lower subarray of H dejsc (13 39) is 0 I H s (6 26) a 6 39 array of CPMs and ZMs of size The subarray H s (6 26) is a matrix with constant column weight 2 and two row weights 8 and 9 which serves as the source compression matrix. The null space of H dejsc (13 39) gives a ( ) QC- JSC-LDPC code C dejsc for the DE-JSC-LDPC-CC system. The parity-check matrix H dejsc (13 39) is used for both JSC encoding and decoding of both the untransmitted source and received code symbols in the DE-JSC-LDPC-CC system. With the above designed DE-JSC-LDPC-CC system 3302 source symbols are compressed into 762 information symbols based on the source compression matrix H s for transmission. Hence the compression rate is The channel code C c given by the null space of H c (7 39) is a ( ) QC-LDPC code with rate The number of transmitted code symbols for each message (3302 source symbols) is 1651 which consists of 762 compressed information symbols and 889 parity-check symbols. So the transmission rate R t and the total system rate R jsc are 762/1651 = and 3302/1651 = 2 respectively. The block error rate (BLER) performance (a block of 3302 source symbols) of the designed DE-JSC-LDPC-CC system over the binary AWGN channel using BPSK signaling decoded based on the JSC parity-check matrix H dejsc (13 39) of the QC-JSC-LDPC code C dejsc is shown in Figure 2. The decoding is carried out with 50 iterations of modified SPA given in 5. The source symbol probabilities are set Pr(s = 0) = 0.03 and Pr(s = 1) = 0.97 (the same probabilities used in 7). In decoding the initial LLR used for each untransmitted source symbol s is given by log(pr(s = 0)/Pr(s = 1))= Also included in Figure 2 is the BLER performance of a multiedge type JSC-LDPC code designed in 7 for the DE-JSC- LDPC-CC system with message length 3200 and total code rate We see that the JSC-LDPC code designed above outperforms the JSC-LDPC code given in 7 significantly in the DE-SJC-LDPC-CC system. The JSC-LDPC code given in 7 has a very high error-floor however the JSC-LDPC code constructed above has no visible error-floor down to the BLER of The bit error rate (BER) performance (per source symbol) of the designed DE-JSC-LDPC-CC system is also shown in Figure 2. BLER/BER 10 0 Simulation result BLER Beltrão Neto-Henkel code BLER section-wise (3 8 3)-regular code BER section-wise (3 8 3)-regular code E b /N 0 (db) Fig. 2. The BLER and the BER performances of the DE-JSC-LDPC systems designed in Example 1 and 7. Example 2. Suppose we replace the code construction field GF(2 7 ) used in Example 1 by the field GF(2 8 ) without changing the parameters i.e. m 0 = 26 l 0 = 6 and n 0 = 39. Using GF(2 8 ) as the code construction field the CPM-dispersion factor is 255. In this case m = 6630 l = 1530 and n = The base matrix B(13 39) is constructed based

8 on GF(2 8 ) in the form given by (2) by setting η = 1 and using the two subsets of GF(2 8 ) S 0 = {α 0 α 1... α 12 } and S 1 = {α 1 α 2... α 39 } where α is a primitive element of GF(2 8 ). All the entries in B(13 39) are nonzero. Construct a masking matrix Z(13 39) which has the same structure as the masking matrix constructed in Example 1 but the section-wise column weight distribution of its upper submatrix Z c (7 39) = Z c1 (7 7) Z c2 (7 6) Z c3 (7 26) is set to (2 7 1). The column and row weight distributions of the submatrix Z s (6 26) in the lower submatrix 0 I Z s (6 26) of Z(13 39) are the same as that of the submatrix Z s (6 26) constructed in Example 1 i.e. constant column weight 2 and two different row weights 8 and 9 respectively. Using the new base and masking matrices B(13 39) and Z(13 39) and following the same construction steps as described in Example 1 we construct the parity-check matrix H dejsc (13 39) of a JSC-LDPC code C dejsc. The parity-check matrix H dejsc (13 39) is a matrix a array of CPMs and ZMs of size The upper submatrix H c (7 39) of H dejsc (13 39) has section-wise column weight distribution of (2 7 1). Hence the section-wise column weight distribution of H dejsc (13 39) is (2 8 3). The JSC-LDPC code C dejsc is a ( ) QC-LDPC code whose Tanner graph has girth at least 6. The DE-JSC-LDPC-CC system using the above JSC-LDPC parity-check matrix H dejsc for JSC encoding and decoding achieves a compression rate R comp = transmission rate R t = and a total system code rate R jsc = 2 same as the system designed in Example 1. A message of 6630 source symbols is compressed into a sequence of 1530 information symbols for transmission. Figure 3 shows the BER and BLER performances of the DE-JSC-LDPC-CC system using the ( ) JSC-LDPC code over the binary AWGN channel decoded based on the parity-check matrix H dejsc designed above. The decoding is carried out with 50 iterations of the modified SPA given in 5. Again the source symbol probabilities are set Pr(s = 0)= 0.03 and Pr(s = 1)= In decoding the initial LLR used for each untransmitted source symbol s is given by log Pr(s = 0)/Pr(s = 1) = Also included in Figure 3 is the BLER performance of a comparable DE-JSC-LDPC- CC system using a multi-edge type JSC-LDPC code designed in 7 (with message length 6400 compression rate transmission rate total system code rate 2.03) decoded with 50 iterations of the modified SPA given in 5. We see that the DE-JSC-LDPC-CC system using the JSC-LDPC code C dejsc designed above outperforms the DE-JSC-LDPC-CC system using the multi-edge type JSC-LDPC code designed in 7. The DE-JSC-LDPC-CC system designed in 7 starts to have an error floor below the BLER of but the DE-JSC-LDPC-CC system designed in this example has no visible error-floor down to the BLER of VI. CONCLUSION AND REMARKS In this paper we presented an integrated design of JSC- LDPC codes for the DE-JSC-LDPC coding system proposed BLER/BER 10 0 Simulation result BLER Beltrão Neto-Henkel code BLER section-wise (2 8 3)-regular code BER section-wise (2 8 3)-regular code E b /N 0 (db) Fig. 3. The BLER and the BER performances of the DE-JSC-LDPC systems designed in Example 2 and 7 of length 6k. by Beltrão Neto and Henkel 7. The proposed integrated design provides a systematic method for constructing JSC- LDPC codes with quasi-cyclic structure whose Tanner graphs have girth at least 6. The construction of these codes is based on finite fields. In two examples given in Section V we showed that the JSC-LDPC codes designed with the method given in Section IV outperform the multi-edge type JSC-LDPC codes of comparable message and code lengths compression rate transmission rate and total system rate designed in 7. Code construction based on this design is quite flexible. With the same choice of parameters m 0 l 0 and n 0 the same masking matrix Z(n 0 m 0 n 0 ) and the same construction of the base matrix B(n 0 m 0 n 0 ) we can construct JSC-LDPC codes of various lengths while maintain the compression rate the transmission rate and the total system code rate the same by simply using different code construction fields. In the construction of a JSC-LDPC code presented in Section IV the masking matrix is designed with a section-wise constant column weight distribution. However an irregular masking matrix can be designed and used for constructing an irregular JSC-LDPC code as well. In this case each of the three submatrices of the masking matrix has multiple column weights. The overall column weight distribution of the masking matrix can be designed based on an optimal VN degree distribution γ opt (X) of a Tanner graph for a given rate using the density evolution 18 (or EXIT-chart 19). Since the optimal VN degree distribution γ opt (X) is designed for a code of infinite length of a given rate when it is applied to a code of finite length (short) of the same rate it is not optimal anymore. In this case the VN degree distribution γ opt (X) must be adjusted. The adjusted VN degree distribution γ(x) is then applied to the construction of a masking matrix that satisfies the column weight requirements for its three submatrices. To achieve this we first divide γ(x) into three partial VN degree distributions denoted by γ 1 (X) γ 2 (X) and γ 3 (X) respectively with γ(x) = γ 1 (X) + γ 2 (X) + γ 3 (X). The

9 partial distribution γ 2 (X) consists of a set of highest degrees in γ(x) the partial distribution γ 1 (X) consists of a set of next highest degrees in γ(x) and the partial distribution γ 3 (X) consists of a set of the lowest degrees in γ(x). Then we construct the submatrices Z 1 Z 2 Z 3 of Z(n 0 m 0 n 0 ) with column weight distributions specified by γ 1 (X) γ 2 (X) and γ 3 (X) respectively. In the construction of Z 3 all the column weights of its lower submatrix Z s (l 0 m 0 ) must be nonzero and small preferably of constant weight 2. Once the masking matrix Z(n 0 m 0 n 0 ) has been constructed we mask the base matrix B(n 0 m 0 n 0 ) to obtain an irregular masked base matrix B mask (n 0 m 0 n 0 ). The CPM-dispersion of B mask (n 0 m 0 n 0 ) gives an (n 0 m 0 )(q 1) n 0 (q 1) irregular JSC parity-check matrix H dejsc (n 0 m 0 n 0 ) with column weight distribution specified by the VN degree distribution γ(x) (or close to it). Then the null space of H dejsc (n 0 m 0 n 0 ) gives an irregular QC-JSC-LDPC code C dejsc. As pointed out in Section III there are other algebraic methods for constructing 2 2 SM-constrained matrices over finite fields Matrices constructed by these methods can also be used as base matrices for constructing QC-JSC- LDPC codes for the DE-JSC-LDPC-CC coding system. REFERENCES 1 R. G. Gallager Low density parity-check codes IRE Trans. Inf. Theory vol. IT-8 no. 1 pp Jan G. Caire S. Shamai and S. Verdu Almost-noiseless joint source channel coding-decoding of sources with memory in 5th International ITG Conference on Source and Channel Coding pp G. Caire S. Shamai and S. Verdu A new data compression algorithm for sources with memory based on error correcting codes in Proc. IEEE Information Theory Workshop pp R. M. Tanner A recursive approach to low complexity codes IEEE Trans. Inf. Theory vol. IT-27 no. 5 pp Sep M. Fresia F. Perez-Cruz and H.V. Poor Optimized concatenated LDPC codes for joint source-channel coding in Proc. IEEE Int. Symp. Inf. Theory (ISIT) pp J. He L. Wang and P. Chen A joint source and channel coding scheme based on simple protograph structured codes in Proc. Communications and Information Technologies (ISCIT) pp H. V. Beltrão Neto and W. Henkel Multi-edge optimization of lowdensity parity-check codes for joint source-channel coding in Proc. 9th International ITG conference on Systems Communications and Coding pp H.V. Beltrão Neto and W. Henkel Information shortening for joint source-channel coding schemes based on low-density parity-check codes in Proc. 8th International Symposium on Turbo Codes and Iterative Information Processing (ISTC) pp T. Richardson Multi-edge type LDPC-codes presented at the workshop honoring Professor R. McEliece on his 60th birthday California Institute of Technology CA. May W. E. Ryan and S. Lin Channel Codes: Classical and Modern. New York: Cambridge Univ. Press Q. Diao Q. Huang S. Lin and K. Abdel-Ghaffar A matrix theoretic approach for analyzing quasi-cyclic low-density parity-check codes IEEE Trans. Inf. Theory vol. 58 no. 6 pp Jun Y. Y. Tai L. Lan L Zeng S. Lin and K. A. S. Abdel-Ghaffar Algebraic construction of quasi-cyclic LDPC codes for the AWGN and erasure channels IEEE Trans. Commun. vol. 54 no. 10 pp Oct L. Lan L. Zeng Y. Y. Tai L. Chen S. Lin and K. Abdel-Ghaffar Construction of Quasi-Cyclic LDPC Codes for AWGN and Binary Erasure Channels: A Finite Field Approach IEEE Trans. Inf. Theory pp J. Li K. Liu S. Lin and K. Abdel-Ghaffar Algebraic quasi-cyclic LDPC codes: Construction low error-floor large girth and a reducedcomplexity decoding scheme IEEE Trans. Commun. vol. 62 no. 8 pp J. Li S. Lin K. Abdel-Ghaffar W. E. Ryan and D. J. Costello Jr. LDPC Code Designs Constructions and Unification. UK: Cambridge Univ. Press J. Xu L. Chen I. Djurdjevic S. Lin and K. Abdel-Ghaffar Construction of Regular and Irregular LDPC Codes: Geometry Decomposition and Masking" IEEE Trans. Inf. Theory vol. 53 no. 1 pp Jan R. A. Horn and C. R. Johnson Matrix Analysis. Cambridge U.K.: Cambridge Univ. Press T. J. Richardson and R. L. Urbanke The capacity of low-density paritycheck codes under message-passing decoding IEEE Trans. Inf. Theory vol. 47 no. 2 pp Feb S. ten Brink Designing iterative decoding schemes with the extrinsic information transfer chart AEU Int. J. Electron. Commun. vol. 54 pp Dec