Abstract:
A decoding method in a concatenated low-density generator matrix (LDGM) code-based transmission system for detecting a signal using a parity check matrix including a systematic bit part mapped to systematic bits and a parity check part mapped to parity bits. The decoding method includes generating an outer code parity check matrix with a predetermined size using a pseudorandom algorithm; generating an inner code parity check matrix using the outer code parity check matrix; and decoding a received signal using the inner code parity check matrix.

Description:
PRIORITY  
       [0001]     This application claims the benefit under 35 U.S.C. §119(a) of an application entitled “Method for Encoding/Decoding Concatenated LDGM Code” filed in the Korean Intellectual Property Office on Apr. 6, 2005 and assigned Serial No. 2005-28573, the entire contents of which are incorporated herein by reference.  
       BACKGROUND OF THE INVENTION  
       [0002]     1. Field of the Invention  
         [0003]     The present invention relates generally to channel encoding in a communication system, and in particular, to a low-density generator matrix (LDGM) code encoding/decoding method for minimizing encoding/decoding complexity to improve its performance.  
         [0004]     2. Description of the Related Art  
         [0005]     In general, a turbo code has a very low encoding complexity but has a high decoding complexity, whereas a low-density parity check (LDPC) code has a high encoding complexity but has a very low decoding complexity, compared with the turbo code.  
         [0006]     The LDPC code, which is a linear code having a parity check matrix including a small number of ‘1’s, uses a probabilistic iterative decoding algorithm and exhibits performance approaching the Shannon&#39;s channel capacity limit. An advantage of the LDPC code over the turbo code consists in the parallel decoder structure, capable of high-speed decoding. However, the LDPC code is much higher than the turbo code in encoding complexity because its encoding process includes complex matrix multiplication.  
         [0007]     A low-density generator matrix (LDGM) code is superior to the standard LDPC code and turbo code in terms of complexity. In particular, the LDGM code, because of the sparse structure of its generator matrix, is linear with respect to block size like the turbo code, for the throughput required in the encoding process.  
         [0008]     In addition, the LDGM code, which actually is a subset of the LDPC code, can perform decoding using the same method and with the same complexity as those of the standard LDPC code, because the parity check matrix of a structured LDGM code is also sparse.  
         [0009]     Equation (1) and Equation (2) below show a parity check matrix and a generator matrix of a standard LDGM code, respectively.  
             H   =     [         1       0       1       1       0       1       1       0       0           1       1       0       1       1       0       0       1       0           0       1       1       0       1       1       0       0       1         ]             (   1   )               G   =     [         1       0       0       0       0       0       1       1       0           0       1       0       0       0       0       0       1       1           0       0       1       0       0       0       1       0       1           0       0       0       1       0       0       1       1       0           0       0       0       0       1       0       0       1       1           0       0       0       0       0       1       1       0       1         ]             (   2   )             
 
         [0010]     As shown in Equation (1) and Equation (2), the parity check matrix and the generator matrix of the LDGM code are sparse in that most of their elements are ‘0’. Therefore, a decoder for the LDGM code can be constructed in the method used for designing a decoder for the LDPC code. An encoder for the LDGM code also has a very low complexity because the generator matrix has a small number of ‘1’s.  
         [0011]      FIG. 1  is a bipartite graph expressing the parity check matrix of Equation (1). In  FIG. 1 , a first check node  111  is connected to first, third, fourth and sixth bit nodes  121 ,  123 ,  124  and  126 , and a first coded bit node  131  constituting a parity part of the parity check matrix. A second check node  112  is connected to first, second, fourth and fifth bit nodes  121 ,  122 ,  124  and  125 , and a second coded bit node  132 . A third check node  113  is connected to second, third, fifth and sixth bit nodes  122 ,  123 ,  125  and  126 , and a third coded bit node  133 .  
         [0012]     However, because there are bit nodes connected only to one check node as shown in the bipartite graph of the parity check matrix, an error floor occurs causing a dramatic reduction in bit error rate (BER) performance. In order to address the disadvantage, a concatenated LDGM code has been proposed which uses two different LDGM codes as an inner code and an outer code, thereby showing performance approaching the Shannon&#39;s channel capacity limit.  
         [0013]     Further, the concatenated LDGM code increases encoding and decoding complexity because it uses two encoders and two decoders.  
       SUMMARY OF THE INVENTION  
       [0014]     It is, therefore, an object of the present invention to provide a concatenated LDGM code encoding/decoding method for reducing complexity of a decoder.  
         [0015]     According to one aspect of the present invention, there is provided a decoding method in a concatenated low-density generator matrix (LDGM) code-based transmission system for detecting a signal using a parity check matrix including a systematic bit part mapped to systematic bits and a parity check part mapped to parity bits. The decoding method includes generating an outer code parity check matrix with a predetermined size using a pseudorandom algorithm; generating an inner code parity check matrix using the outer code parity check matrix; and decoding a received signal using the inner code parity check matrix.  
         [0016]     Preferably, the outer code parity check matrix generating step includes generating a first systematic bit part with a predetermined size using the pseudorandom algorithm; and adding a first parity check part in the form of an identity matrix with the same row size, to the first systematic bit part.  
         [0017]     Preferably, the inner code parity check matrix generating step includes extending each row of the first systematic bit part of the outer code parity check matrix to a partial matrix with a predetermined row size; extending the first parity check part using the pseudorandom algorithm; generating a second systematic bit part of the inner code parity check matrix by arranging the extended partial matrixes and the extended first parity check parts; and adding a second parity check part in the form of an identity matrix with the same row size, to the second systematic bit part.  
         [0018]     According to another aspect of the present invention, there is provided an encoding method in a concatenated low-density generator matrix (LDGM) code-based transmission system for detecting a signal using a parity check matrix including a systematic bit part mapped to systematic bits and a parity check part mapped to parity bits. The encoding method includes generating an outer code parity check matrix with a predetermined size using a pseudorandom algorithm; generating an inner code parity check matrix using the outer code parity check matrix; generating a generator matrix using the inner code parity check matrix; and concatenatively-encoding a transmission signal using the generator matrix.  
         [0019]     According to further another aspect of the present invention, there is provided a low-density generator matrix (LDGM) decoder in a concatenated LDGM code-based transmission system using an inner LDGM code and an outer LDGM code, wherein the LDGM decoder groups a plurality of parity check nodes constituting a parity check matrix of the inner LDGM code and uses the parity check node groups as check nodes of the parity check matrix.  
         [0020]     Preferably, the parity check matrix of the outer LDGM code includes a first systematic bit part with a predetermined size, generated using a pseudorandom algorithm; and a first parity check part having the same row size as that of the first systematic bit part.  
         [0021]     Preferably, the parity check matrix of the inner LDGM code includes a second systematic bit part including first partial matrixes generated by extending each row of the first systematic bit part in a predetermined size and second partial matrixes generated by extending the first parity check matrix using a pseudorandom algorithm; and a second parity check part in the form of an identity matrix having the same row size as that of the second systematic bit part. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0022]     The above and other objects, features and advantages of the present invention will become more apparent from the following detailed description when taken in conjunction with the accompanying drawings in which:  
         [0023]      FIG. 1  is a bipartite graph expressing a parity check matrix of a general LDGM code;  
         [0024]      FIG. 2  is a conceptual diagram of a process of generating a parity check matrix in an LDGM code encoding/decoding method according to the present invention;  
         [0025]      FIG. 3A  is a bipartite graph for a first row of the systematic bit part of the outer parity check matrix shown in  FIG. 2 ;  
         [0026]      FIG. 3B  is a bipartite graph for the matrix generated by extending the first row of the systematic bit part of the outer parity check matrix shown in  FIG. 2 ;  
         [0027]      FIG. 3C  is a bipartite graph for a description of a process of integrating the check codes extended in the bipartite graph of  FIG. 3B ;  
         [0028]      FIG. 3D  is a bipartite graph given after integration of the check nodes extended in the bipartite graph of  FIG. 3B ;  
         [0029]      FIG. 4  is a conceptual diagram illustrating an improved belief-propagation algorithm applied to an LDGM code encoding/decoding method according to the present invention;  
         [0030]      FIG. 5  is a block diagram schematically illustrating a structure of a concatenated LDGM decoder implemented by applying an LDGM code encoding/decoding method according to the present invention;  
         [0031]      FIG. 6  is a graph comparing performance in a simulation between the proposed concatenated LDGM code and the conventional concatenated LDGM code; and  
         [0032]      FIG. 7  is a graph comparing performance in a simulation between the proposed concatenated LDGM code and the conventional concatenated LDGM code. 
     
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS  
       [0033]     A method for encoding/decoding a concatenated LDGM code according to the present invention will now be described with reference to the accompanying drawings.  
         [0034]     The concatenated LDGM code encoding/decoding method according to the present invention generates an inner LDGM code by extending an outer LDGM code and uses an inner LDGM decoder in decoding the outer LDGM code. The inner LDGM decoder can be modified when necessary.  
         [0035]     The present invention defines an outer LDGM code as an (n 1 , n 1 -k, p) regular LDGM code, and defines an inner LDGM code as an (n 2 , n 2 -k, rp) regular code. Herein, n 1  and n 2  denote lengths of the outer LDGM code and the inner LDGM code, respectively; n 1 -k and n 2 -k denote the number of parity check equations for the outer LDGM code and the inner LDGM code, respectively; and p and rp denote the number of edges of each bit node in a systematic part of a parity check matrix for the outer LDGM code and the inner LDGM code, respectively. Herein, k denotes the number of systematic bits, and r and s=n 1 /(n 1 -k) are natural numbers. The concatenated LDGM code encoding/decoding method according to the present invention first generates a parity check matrix of an outer LDGM code with a pseudorandom algorithm, and then generates a parity check matrix of an inner LDGM code by extending each row of the parity check matrix of the outer LDGM code to s rows. Herein, the number of edges of each bit node increases r-fold. For convenience of description, the parity check matrix of the outer LDGM code will be referred to as an “outer parity check matrix” and the parity check matrix of the inner LDGM code will be referred to as an “inner parity check matrix.” 
         [0036]      FIG. 2  is a conceptual diagram for a description of an LDGM code encoding/decoding method according to the present invention. An embodiment of the present invention will be described with reference to a (9, 3, 2) outer LDGM code and an (18, 9, 4) inner LDGM code.  
         [0037]     In  FIG. 2 , an outer parity check matrix is divided into a systematic bit part corresponding to systematic bits in the left-hand side of a dotted line and a parity check part corresponding to parity bits in the right-hand side of the dotted line. Each row of the systematic bit part of the outer parity check matrix is extended to 3 rows. In this case, the number of edges of each bit node increases twofold. In this extension process, a progressive edge growth (PEG) algorithm can be used for performance maximization. The other region of the systematic bit part, constituting a parity check matrix of an inner LDGM code, can be generated with the pseudorandom algorithm.  
         [0038]      FIG. 3A  is a bipartite graph for a first row of the systematic bit part of the outer parity check matrix shown in  FIG. 2 , and  FIG. 3B  is a bipartite graph for the matrix generated by extending the first row of the systematic bit part of the outer parity check matrix shown in  FIG. 2 .  
         [0039]     In  FIG. 3A , a check node  312  is connected to 1 st , 2 nd , 4 th  and 5 th  bit nodes  321 ,  322 ,  324  and  325 , and one coded bit node  332 . According to the present invention, if a first row of the systematic bit part is extended to a 3-row matrix component and the matrix component generated through extension is expressed with a bipartite graph, three check nodes  341 ,  342  and  343  are generated and one of coded bit nodes  351 ,  352  and  353  is connected to each of the check nodes as shown in  FIG. 3B . The first check node  341  is connected to the 1 st , 2 nd  and 5 th  bit node  321 ,  322  and  325 ; the second check node  342  is connected to the 2 nd  and 4 th  bit nodes  322  and  324 ; and the third check node  343  is connected to the 1 st , 4 th  and 5 th  bit nodes  321 ,  324  and  325 .  
         [0040]     In the decoding process, if the extended check nodes constituting the parity check matrix of the inner LDGM code are integrated into a new check node, the result becomes equal to a check node of a parity check matrix of the outer LDGM code.  
         [0041]      FIGS. 3C and 3D  are conceptual diagrams of a process of integrating the check codes extended in the bipartite graph of  FIG. 3B . If one dummy check node  361  is added and coded nodes are combined with one coded bit node  371  as shown in  FIG. 3C  and then the extended check nodes are integrated into the dummy node  361  as shown in  FIG. 3D , the result becomes equal to the bipartite graph of  FIG. 3A .  
         [0042]     If a belief-propagation algorithm is modified by designing an inner code from a given code as described above, an inner decoder can be used during decoding of an outer code.  FIG. 4  is a conceptual diagram illustrating an improved belief-propagation algorithm applied to an LDGM code encoding/decoding method according to the present invention.  
         [0043]     Generally, a check node message of the belief-propagation algorithm is updated in accordance with Equation (3) through Equation (5).  
               T   m     =       ∏       n   ′     ∈     N   ⁡     (   m   )           ⁢       1   -     exp   ⁡     (     z     mn   ′       )           1   +     exp   ⁡     (     z     mn   ′       )                     (   3   )                 T   mn     =       T   m     ×         1   -     exp   ⁡     (     z   m     )           1   +     exp   ⁡     (     z   m     )           /       1   -     exp   ⁡     (     z   mn     )           1   +     exp   ⁡     (     z   mn     )                       (   4   )                 L   mn     =     ln   ⁢       1   -     T   mn         1   +     T   mn                   (   5   )             
 
         [0044]     In Equation (3) through Equation (5), N(m) denotes a set of bit nodes connected to a check node m except for a bit node with a column weight of 1, z mn  denotes a priori probability of a bit node n associated with the check node m, expressed in log-likelihood ratio, and z m  denotes a priori probability of the bit node with a column weight of 1 at the check node m, expressed in log-likelihood ratio. A bit node message update rule can be expressed as Equation 6:  
               z   mn     =       F   n     +       ∑       m   ′     ∈       M   ⁡     (   n   )       ⁢   \   ⁢           ⁢   m         ⁢     L       m   ′     ⁢   n                   (   6   )             
 
 where F n  denotes a priori probability received at a receiver of a bit node n, expressed in log-likelihood ratio (LLR), and M(n)\m denotes a set of check nodes connected to a bit node n, except for a check node m. 
 
         [0045]     To decode a concatenated LDGM code, the present invention first decodes an inner code using an inner decoder and then decodes an outer code after applying a slight modification to the same decoder.  
         [0046]     If a set of inner check nodes extended from a check node j of an outer code is denoted by S(j) and a new check node (dummy check node) included in S(j) is denoted by C j , then Equation (7) can be derived from Equation (3) and Equation (4) at C j .  
               T   j   ′     =       (       ∏     m   ∈     S   ⁡     (   j   )           ⁢     T   m       )       1   /   r               (   7   )             
 
         [0047]     In Equation (7), because the number of edges between inner bit nodes is r times larger than the number of edges between outer bit codes, the same message propagated from the bit nodes to the dummy check node C j  is multiplied by a square of r. Therefore, an r th  route given in Equation (5) must be taken at the dummy check node C j . Thereafter, Equation (4) and Equation (5) are equal to each other except that T m  is replaced with T j ′. For the bit node message update rule, because the same messages from the dummy check node C j  are added at every bit node r times, Equation (6) is modified as Equation (8).  
               z   mn     =       F   n     +       1   r     ×       ∑       m   ′     ∈       M   ⁡     (   n   )       ⁢   \   ⁢           ⁢   m         ⁢     L       m   ′     ⁢   n                     (   8   )             
 
         [0048]     As described above, application of the LDGM code encoding/decoding method according to an embodiment of the present invention can reduce the decoder complexity. It is preferable to install a bit interleaver between an inner encoder and an outer encoder to improve performance of the LDGM code.  
         [0049]      FIG. 5  is a block diagram schematically illustrating a structure of a concatenated LDGM decoder implemented by applying an LDGM code encoding/decoding method according to the present invention.  
         [0050]     Unlike the conventional concatenated LDGM decoder which includes an inner decoder and an outer decoder, a new LDGM decoder according to the present invention includes an inner decoder and an interleaver. As illustrated in  FIG. 5 , an output bit stream of an inner decoder  510  is input back to the inner decoder  510  after being interleaved by an interleaver  520 . The interleaved output is equal to an input to the conventional outer decoder. As a result, the present invention decodes both outer and inner codes with one decoder. By applying the structure of the decoder to the encoder in the same way, it is possible to implement an encoder with low complexity.  
         [0051]      FIGS. 6 and 7  are graphs comparing performance in a simulation between the proposed concatenated LDGM code and the conventional concatenated LDGM code.  
         [0052]     For the simulation, a (20000, 10000, 6) inner code with a coding rate of 0.5 and a (10000, 500, 3) outer code with a coding rate of 0.95 were used in an additive White Gaussian noise (AWGN) channel environment, and the same outer code was used for both the conventional concatenated LDGM code and the proposed concatenated LDGM code. The inner code for the conventional concatenated LDGM code was created using the PEG algorithm, and the proposed inner code was created by extending the outer coder according to the present invention.  
         [0053]     It can be noted from  FIG. 6  that when interleaving is applied at a Bit Error Rate (BER) of 10 −5 , the proposed concatenated LDGM code and the conventional concatenated LDGM code show almost similar performance.  
         [0054]      FIG. 7 , which is a performance curve for a concatenated LDGM code with a codeword length of 2000, shows a similar result to that of  FIG. 6 . As a result, application of the proposed concatenated LDGM code can obtain the almost similar performance to that of the conventional concatenated LDGM code using only one decoder.  
         [0055]     As can be understood from the foregoing description, the concatenated LDGM code encoding/decoding method according to the present invention can decode an outer code with one inner decoder during implementation of a decoder because it generates an inner LDGM code from an outer LDGM code by extending each row of a parity check matrix of the outer LDGM code, thereby contributing to a reduction in decoding complexity.  
         [0056]     While the invention has been shown and described with reference to a certain embodiment thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims.