Multi-edge Optimization of Low-Density Parity-Check Codes for Joint Source-Channel Coding

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1 Multi-ege Optimization of Low-Density Parity-Check Coes for Joint Source-Channel Coing H. V. Beltrão Neto an W. Henkel Jacobs University Bremen Campus Ring 1 D Bremen, Germany {h.beltrao, w.henkel}@jacobs-university.e Abstract We present a novel joint source-channel coing system base on low-ensity parity-check coes where the amount of information about the source bits available at the ecoer is increase by improving the connection profile between the factor graphs that compoun the joint system. Furthermore, we propose an optimization strategy for the component coes base on a multi-ege-type joint optimization. Simulation results show a significant improvement in the performance compare to existent joint systems base on low-ensity parity-check coes. I. INTRODUCTION The separation principle between source an channel coing states that there is no loss in asymptotic performance when source an channel coing are performe separately. It is though wiely observe that for communication systems transmitting in the non-asymptotic regime with limite elay constraints, the separation principle may not be applicable an gains in complexity an fielity may be obtaine by a joint esign strategy [1]. In this paper, we investigate a joint system which performs linear encoing of sources by means of error-correcting coes. The strategy of such schemes is to treat the source output u as an error pattern an perform compression calculating the synrome generate by u, i.e., the source encoer calculates s = uh T, where H is the parity-check matrix of the linear error-correcting coe being consiere as source encoer, an the synrome s represents the compresse sequence. Herein we consier only binary memoryless sources, since it is wiely known that linear source coes achieve the entropy rate for this kin of sources. Nevertheless, the optimality of linear source compression can be extene to very general sources with memory an nonstationarity [2]. Compression schemes base on synrome encoing for binary memoryless sources were evelope in the context of variable-to-fixe length algorithms in [3] an [4]. Afterwars, Ancheta [5] evelope a fixe-to-fixe linear source coe base on synrome formation. Due to the limitations of the practical error-correcting coes known at that time, this line of research was left asie by the avent of Lempel-Ziv coing. Nevertheless, ue to a lack of resilience of state-of-the-art ata compressors to transmission errors an to the fact that such This work was fune by the DFG project number HE-3654/16-1. compression algorithms just have an efficient performance with block sizes much longer than the ones typically specifie in some moern wireless stanars, there are state-of-the-art applications that o not apply ata compression. In orer to cope with such limitations of some moern ata compression algorithms, the authors in [2] propose the use of synrome-source compression by means of low-ensity parity-check (LDPC) coes together with belief propagation ecoing, which was further extene in [6] to cope with a noisy channel. In contrast to general linear coes, an LDPC coe has a sparse parity-check matrix an can thus be use as a linear compressor with linear complexity in the block length. In aition, synrome source-coing schemes can be naturally extene to joint source-channel (JSC) encoing an ecoing configurations. One of the schemes propose in [6] for JSC consists of a serial concatenation of two LDPC coes, where the outer coe works as a synrome-source compressor an the inner coe as the channel coe. The coewor resulting from such a concatenation is then jointly ecoe using the source statistics an channel information by means of the belief propagation algorithm applie to the joint source-channel factor graph. In spite of its introuction in [6], it was in [7] that this scheme was first stuie for a JSC application. However, the bit error-rate curves presente in [7] showe consierably high error floors for source output sequences with moerate block length. The propose solutions to cope with such high error floors were either to reuce the source compression rate or to increase the coewor size, but such solutions have the following rawbacks. First of all, increasing the size of the coewor woul unermine one of the avantages of the JSC scheme, namely the possibility of a better performance in a non-asymptotic scenario. Secon, reucing the compression rate is also not esirable, since it pushes the system performance away from its asymptotically achievable capacity. In this paper, we propose the construction of an LDPC-base joint source-channel coing scheme which significantly lowers the error floor resulting from the compression of source sequences that correspon to uncorrectable error patterns of the LDPC coes use as synrome-source encoer.

2 II. LDPC-BASED JOINT SOURCE-CHANNEL SYSTEM In [6], the authors propose a configuration for a joint source-channel encoing system using LDPC coes for both source compression an channel coing. This structure is base on a serial concatenation of two LDPC coes where the outer an the inner coes perform synrome-source compression an channel coing, respectively. In this concatenate approach, a coewor c is efine by c = s G cc = u H T sc G cc, where G cc is the l m LDPC generator matrix of the channel coe, H sc is the l n parity-check matrix of the LDPC coe applie for source coing, s is the 1 l source compresse sequence, an u is the 1 n source output. Consiering a binary memoryless source an performing stanar belief propagation ecoing, the simulation results in [7] showe the presence of error floors in the error-rate curves, which are a consequence of the fact that some output sequences emitte by the source form error patterns that cannot be correcte by the LDPC coe use as source compressor. Our iea to cope with this problem is to improve the amount of information about the source bits available at ecoing. We o it by inserting new eges connecting the check noes of the channel coe to the variable noes of the source coe in the factor graph that represents the serial concatenate system introuce in [6]. The reasoning of this strategy is that such an ege insertion will provie an extra amount of extrinsic information to the variable noes of the source LDPC which will significantly lower the error floor ue to uncorrectable source output patterns. We epict this iea in Fig. 1, where the new inserte eges in the concatenate JSC system of [6] are represente by the ashe lines. Fig. 1. Joint source-channel factor graph with inserte eges. The variable an the check noes of the source LDPC (left) represent the source output an the compresse source sequence, respectively. Since we will consier only binary sources, the variable noes represent binary symbols. In this system, each check noe of the source LDPC is connecte to a single variable noe of the channel coe (right) forming the systematic part of the channel coewor. We consier that only m variable noes are transmitte (the n source output symbols are puncture prior to transmission). Thus, the overall rate is n/m. Furthermore, L sc v an L cc v enote the log-likelihoo ratios representing the intrinsic information receive by the source (v = 1,...,n) an channel (v = n + 1,...,n + m) variable noes, respectively. In this work, we will limit our investigation to memoryless binary sources. A. Encoer To unerstan our propose serial encoing strategy, consier the representation of the factor graph epicte in Fig. 1 by a m (n+m) matrix H. This matrix can be written as where H sc is the l n source encoer parity-check matrix, H cc is the (m l) m parity-check matrix of the channel coe, I is an l l ientity matrix, an L is an (m l) n matrix, to which we will refer as linking matrix. The linking matrix L represents the connections among the check noes of the channel coe to the variable noes of the source coe. The encoing of our propose system iverts slightly from the serial approach of [7]. The ifference lies in the fact that the message to be encoe before the transmission is forme by the concatenation of the source output u an its synrome s compute by the source coe, i.e., a coewor c is efine by c = [u,s] G L = [u,u H T sc] G L, (1) where G L is an (n + l) (n + m) matrix constructe such that the row space of G L is the null space of [L,H cc ], i.e., G L is the generator matrix of a linear systematic coe whose parity-check matrix is given by the horizontal concatenation of the matrices L an H cc. In the following, we show that every coewor of the coe spanne by G L is a coewor of the coe spanne by the null space of H. Proposition 1: Let H = [ [H sc,i,0] T,[L,H cc ] T ] T enote the matrix whose factor graph representation correspons to the joint system epicte in Fig. 1, H L = [L,H cc ], an [u,s] be the concatenation of the source output u an its synrome-compresse sequence s. A coewor c forme by the encoing of the vector [u,s] by the linear coe spanne by the null space of the matrix H L is also a coewor of the linear coe spanne by the null space of H. Proof : Let G L enote the systematic generator matrix of the null space of the matrix H L. Since the coe spanne by the rows of G L is systematic, its coewors can be written as c = [,p], where is the systematic part of the coewor. Let = [u,s], then we can write c = [u,s,p], where u = [u 0,...,u n 1 ] represents the source output, s = [s 0,...,s l 1 ] enotes the synrome compresse sequence, an p = [p 0,...,p m l 1 ] is a vector whose elements are the parity bits generate by the inner prouct between [u,s] an G L. For every coewor c, we have c H T L = c [L,H cc ] T = 0. (2) Recall now that accoring to our compression rule, an since our operations are efine over GF(2), we can write [u 0,...,u n 1 ] H T sc = [s 0,...,s l 1 ] [u 0,...,u n 1 ] H T sc +[s 0,...,s l 1 ] I = 0, (3) where I is an l l ientity matrix, an 0 is a vector whose elements are all equal to zero. Note that Eq. (3) can be written as [u 0,...,u n 1,s 0,...,s l 1 ] [H sc,i] T = 0. (4)

3 Consier now the l (n+m) matrix [H sc,i,0]. Accoring to Eq. (4), for every vector p = [p 0,...,p m l 1 ], we can write [u 0,...,u n 1,s 0,...,s l 1,p 0,...,p m l 1 ] [H sc,i,0] T = 0, i.e., c [H sc,i,0] T = 0. (5) Finally, consier the inner prouct c H T = c [ [H sc,i,0] T,[L,H cc ] T ] = [ c [H sc,i,0] T,c [L,H cc ] T ]. (6) Substituting eqs. (2) an (5) into Eq. (6), we have c H T = 0, i.e., a coewor c of the coe spanne by the null space of H L is also a coewor of the coe spanne by the null space of H. The encoing algorithm of our propose joint sourcechannel system can be summarize as follows: 1) Given a source output vector u, compute s = u H T sc. 2) Compute v = [u, s], i.e., the horizontal concatenation of vectors u an s. 3) Generate the coewor c = v G L. 4) Transmit c after puncturing its first n bits. Steps 1 an 3 are the source an channel encoing steps, respectively. Since H sc is sparse, the source encoing has a complexity that is linear with respect to the block length. Furthermore, applying the technique presente in [8] for encoing LDPC coes by means of their parity-check matrix, the complexity of the channel encoing can be mae approximately linear. B. Decoer The ecoing of the LDPC-base joint source-channel system is one by means of the belief propagation algorithm applie to the factor graph of Fig. 1, whose structure is known to both the encoer an the ecoer. We assume that the ecoer knows the statistics of the source. Herein, we assume that the source is a memoryless Bernoulli source with success probability p v, an that the transmission takes place through a binary input AWGN ( channel. Within this framework, we can write L sc v = log 1 pv ) p v an L cc v = 2yv σ where y n 2 v is the receive BPSK moulate coewor transmitte through an AWGN with noise variance σn 2 (consequently L cc v has variance σch 2 = 4/σ2 n). III. MULTI-EDGE NOTATION A. Multi-ege-type LDPC coes Multi-ege-type LDPC coes [9] are a generalization of irregular an regular LDPC coes. Diverting from stanar LDPC ensembles where the graph connectivity is constraine only by the noe egrees, in the multi-ege setting, several ege classes can be efine, an every noe is characterize by the number of connections to eges of each class. Within this framework, the coe ensemble can be specifie through two noe-perspective multinomials associate to variable an check noes, which are efine respectively by [9] ν(r, x) = ν b, r b x an µ(x) = µ x, (7) where b,, r, an x are vectors which are explaine as follows. First, let m e enote the number of ege types use to represent the graph ensemble an m r the number of ifferent receive istributions. The number m r represents the fact that the ifferent bits can go through ifferent channels an thus, have ifferent receive istributions. Each noe in the ensemble graph has associate to it a vector x = (x 1,...,x me ) that inicates the ifferent types of eges connecte to it an a vector = ( 1,..., me ) referre to as ege egree vector which enotes the number of connections of a noe to eges of type i, where i (1,...,m e ). For the variable noes, there is aitionally the vector r = (r 0,...,r mr ), which represents the ifferent receive istributions an the vector b = (b 0,...,b mr ), which inicates the number of connections to the ifferent receive istributions (b 0 is use to inicate a variable noe with no available intrinsic information at the ecoer). We use x to enote me i=1 xi i an r b to enote m r i=0 rbi i. Finally, the coefficients ν b, an µ are non-negative reals such that, if n is the total number of variable noes, ν b, n an µ n represent the number of variable noes of type (b,) an check noes of type 1, respectively. B. Multi-ege notation for joint source-channel factor graphs In orer to being able to quantify the amount of information exchange by the iniviual factor graphs representing the channel an source coes uring ecoing, we efine herein a multi-ege framework for the JSC system. Within this framework, we efine four ege types within the corresponing graph, i.e.,m e = 4. Aitionally, now we also have two ifferent receive istributions corresponing to the source statistics an channel information, respectively. Figure 2 epicts the four ege types an two receive istributions. The soli an ashe lines epict type-1 an type-2 eges, respectively. The type-3 an type-4 eges are epicte by the ash-otte an otte lines, respectively. Moreover, the receive istributions of the source an channel variable noes are epicte by soli an ashe arrows, respectively. Note that the source an channel coe factor graphs exchange information solely through type-3 an type-4 eges. Since the variable noes have access to two ifferent observations, the vector r = (r 0,r 1,r 2 ) has three components. The first component (r 0 ) represents a bit with no available intrinsic information, secon component (r 1 ) correspons to the observation accessible to the n source LDPC variable noes, an the thir component (r 2 ) enotes the channel observations, which are available only to the m channel LDPC variable noes. Furthermore, since each variable noe has access to either the source statistics or the channel observation, we can write b = (0,1,0) for the source an b = (0,0,1) for the channel variable noes, respectively. 1 We will frequently refer to noes with ege egree vector as type noes.

4 {1,4}, we can write I (j) = λ (j) J BSC(( j 1)[J 1 (I (j) 1 )] + s j with the function J BSC efine as [7] s [J 1 (I (s) 1 )],p v), (10) Fig. 2. Multi-ege joint source-channel factor graph. IV. ASYMPTOTIC ANALYSIS In this section, we erive the multi-ege-type mutual information evolution equations for LDPC-base joint sourcechannel coing systems. We will use the ege-perspective egree istributions λ (j) (r,x) an ρ (j) (x) to escribe the evolution of the mutual information between the messages sent through type-j eges an the associate variable noe values. The ege-perpective multi-ege egree istributions can be written as λ (j) (r, x) = ν x j (r, x) ν xj (1, 1), ρ(j) (x) = µ x j (x) µ xj (1), (8) where ν xj (r, x) an µ xj (x) are the erivatives of ν(r, x) an µ(x) with respect to x j, respectively. Note that, since we are ealing with synrome-source encoing (a framework where the source output is analogous to an error pattern) of memoryless Bernoulli sources with a probability of emitting a one equal to p v, we can moel the receive istributions of the source coe variable noes as the istribution of the output of a BSC with crossover probability p v [5]. Let I (j) (I (j) ) enote the mutual information (MI) between the messages sent through type-j eges at the output of variable (check) noes at iteration l an the associate variable noe value. Assuming Gaussian approximation [10] of the messages exchange through the joint factor-graph uring BP ecoing, we can express the mutual information equation for the channel coe variable noes, i.e., for j {2,3} as I (j) = λ (j) J(σ2 ch +( j 1)[J 1 (I (j) 1 )] + s j s [J 1 (I (s) 1 )]), (9) where σch 2 is the variance of the receive channel message, λ (j) is the probability of a type-j ege being connecte to a variable noe with ege egree vector, an the function J( ) relates all the MI quantities to the variance of LLR messages an is efine as [11] J(σ 2 ) = 1 e (ξ σ 2 /2) 2 2σ 2 2πσ 2 log 2 [1+e ξ ]ξ. In aition, for the source coe variable noes, i.e., for j J BSC (σ 2,p v ) = (1 p v )I(x v ;L (1 pv) )+p v I(x v ;L (pv) ), where x v enotes the corresponing bitnoe variable, L (1 pv) N( σ2 2 +Lsc v,σ 2 ), an L (pv) N( σ2 2 Lsc v,σ 2 ). Finally, the mutual information between the messages sent by a check noe through a type-j ege an its associate variable value for both source an channel LDPC coes (i.e., for all j) can be written as I (j) =1 (j) cmax i=1 : j=i ρ (j) J(( j 1)[J 1 (1 I (j) )]+ s [J 1 (1 I (s) )]), s j (11) where ρ (j) is the probability of a type-j ege being connecte to a check noe with ege egree vector, an (j) c max is the maximum number of type-j eges connecte to a check noe. In orer to limit the search space of the optimization algorithm, we consier only check-regular source an channel LDPC coes. Furthermore, the check noes of source an channel LDPC coes are consiere to have ege egree vectors = ( c1,0,1,0) an = (0, c2,0,1), respectively. As a consequence, the multi-ege check noe egree istributions of the source an channel LDPC coes are given by ρ (1) (x) = x c an ρ (2) (x) = x c 2 1 2, respectively. A. Source coe mutual information evolution For the source coe factor graph, the variable noes only have connections to type-1 an type-4 eges, i.e., all source coe variable noes have an ege egree vector = ( 1,0,0, 4 ) where 1 {2,..., (1) }, an 4 {0,1}. We can summarize the set of mutual information evolution equations as follows: variable noes messages upate: I (1) = λ (1) J BSC(( 1 1)[J 1 (I (1) 1 ())] check noes messages upate: I (1) () = [J 1 (I (4) 1 ())],p v) (12) J(( c1 1)[J 1 (1 I (1) )]+[J 1 (1 I (3) ())]) (13) source to channel ecoer messages upate: I (4) () = 4 J BSC ( 1 [J 1 (I (1) 1 ())],p v) (14) I (3) = 1 J( c1 [J 1 (1 I (1) )]) (15)

5 channel ecoer messages upate: I (3) () = T v(i (3) 1,I(4) 1 ()) (16) I (4) () = T c(i (3),I(4) ()) (17) where T v ( ) an T c ( ) are the transfer functions of the channel ecoer, which is consiere to be fixe. Given the channel coe egree istribution λ (2) (r, x) an ρ (2) (x), those functions can be explicitly compute by means of eqs. (9) an (11) for every ege egree vector 2. In the computation of T v ( ) an T c ( ), the rightmost sum in Eq. (11) will be zero if I (4) () = 0, since the corresponing check noe is not receiving any information through type-4 eges in this case. Combining eqs. (12) - (17) we can summarize the mutual information evolution for the source coe as a function of the mutual information in the previous iteration, the source statistics, the channel conition, an the egree istributions: I (1) = F 1 (λ, c,i (1) 1,p v,σ ch ), (18) where c = [ c1, c2 ], an λ = [λ (1),λ (2) ] with λ (j) enoting the sequence of coefficients λ (j) for all an j {1,2}. The initial conitions are I (4) v,0 () = I(4) c,0 () = I(1) c,0 () = 0, an I (3) c,0 = 0. By means of Eq. (18), given a channel LDPC coe, we can preict the convergence behavior of the iterative ecoing for the source coe an then optimize the multi-ege egeperspective variable noe egree istributions λ (1) (r, x) uner the constraint that the mutual information must be increasing as the number of iterations grows. V. OPTIMIZATION In the propose algorithm herein, we first compute the rate optimal channel LDPC coe assuming a transmission over an AWGN channel with noise variance σn. 2 This is a stanar irregular LDPC optimization [12] an since we are not consiering any connection to the source coe in this first step, it can be one by means of eqs. (9) an (11) with = (0, 2,0,0) an 2 {2,..., (2) }, where (j) enotes the maximum number of type-j eges connecte to a variable noe. The optimize egree istribution obtaine at this step will be enote as λ (2) 0 (r, x). After having optimize the channel coe variable noes egree istribution, we assign the variable noes of higher egree to the message bits. This is one in orer to better protect the compresse message transmitte through the channel, since the more connecte a variable noe, the better its error error rate performance [13]. This can be one as follows, 1) Given λ (2) 0 (r, x), compute the noe-perspective multiege egree istribution ν 0 (r, x) = (2) λ 0 (r,x)x λ(2) 0 (r,x)x2 2) Assign a fraction R cc of noes (the ones with higher egree) to the systematic part of the coewor, where 2 For the computation [ of T v( ), note that by means of Eq. (8) we can write ] λ (3) (2) (r, x) = λ (r,x) 10, where f λ (2) (r,x) x enotes the partial erivative of f x 3 with respect to x. R cc is the rate of the channel coe. This is one by turning a variable noe with ege egree vector = (0, 2,0,0) into a variable noe with ege egree vector = (0, 2,1,0). This gives rise to a moifie noeperspective egree istribution ν(r, x), where a fraction of R cc noes have one connection to type-3 eges. 3) Given ν(r, x), compute the new ege-perspective multiege variable noe egree istribution λ (2) (r, x) = ν x2 (r,x) ν x2 (1,1). Once we have optimize the channel coe, we optimize (maximizing its rate) the source LDPC coe consiering its connections to the channel LDPC coe graph. Let vmax = [ (1), (2), (3), (4) ] be a vector whose components (j) represent the maximum number of connections of a single variable noe to type-j eges. Also, recall that the components of the vector c = [ c1, c2 ] efine the number of connections of the source coe check noes to type-1 eges ( c1 ) an the number of connections of the channel coe check noes to type-2 eges ( c2 ). Aitionally, λ (j) enote the sequence of the coefficients ofλ (j) (r, x). Given vmax, c, p v, an σn 2 = σch 2 /4, the optimization problem can be written as shown in Algorithm 1. Algorithm 1 Joint source-channel coe optimization 1) Optimize the rate of the channel LDPC coe without consiering the connections to the factor graph of the source LDPC coe. Save the obtaine the egree istribution λ (2) 0 (r, x). 2) Compute λ (2) (r, x) by assigning as systematic bits a fraction of the variable noes with higher egrees of the optimize channel LDPC coe. 3) Consiering λ = [λ (1),λ (2) ], maximize (1) vmax s=2 : 1 =s λ(1) /s uner the following constraints, C 1 : λ(1) = 1, C 2 : F 1(λ, c,i,p v,σ ch ) > I, I [0,1), C 3 : : 1 =2 λ(1) < C 4 : : 4 >0 1 2 p v(1 p v) 1 ( c1 1), λ (1) λ 1 = 1/( c1 c2 : (2) 3 =1 2 ). where C 1 an C 2 are the proportion an convergence constraints, respectively. Since we are consiering the convergence only through eges of type-1, the stability conition C 3 remains the same as for stanar LDPC coes ensembles with coewors transmitte over a BSC with transition probability p v [9]. Furthermore, the rate constraint C 4 must be consiere ue to the fact that the number of type-4 eges connecte to the source coe variable noes must be equal to the number of channel coe check noes (since we assume that every channel coe check noes is connecte to only one type-4 ege). For given λ (2), c, p v, an σ ch, the constraints C 1, C 2, C 3, an C 4 are linear in the parameter λ (1). This means that the optimization of both source an channel coes can be solve by linear programming. For a given channel conition, every ifferent set of vectors vmax, c will give rise to systems with a ifferent overall rate. In practice, we fix the vector vmax an vary c1 an c2 to obtain the joint system with maximum overall rate for a binary symmetric source with

6 transition probability p v an an AWGN with noise variance σ 2 n. VI. SIMULATION RESULTS Herein, we present simulation results obtaine with an LDPC-base JSC coing system constructe accoring to the egree istributions optimize by the algorithm previously propose. We optimize a system with the following parameters: p v = 0.03, σ 2 n = 0.95, vmax = [30,30,1,1], an c = [22,6]. For such source an channel conitions, the asymptotically optimal Shannon limit is C/H(S) 2.58 source symbols per channel use, where C is the channel capacity an H(S) is the source entropy. The compression rate obtaine for the source coe was R sc = , an the transmission rate obtaine for the channel LDPC coe was R cc = giving an overall coing rate of R over = R cc /R sc In orer to show the merits of the propose optimization, we compare the performance of our propose system with the LDPC-base JSC systems with the same overall rate R over = 2.03 introuce by Caire et al. in [6] to which we will refer as System I. This system has the eges between the check noes of the source LDPC coe an the systematic variable noes of the channel LDPC coe as the only connections between the factor graphs of the source an channel coes. Furthermore, it consists of a source coe jointly optimize with a fixe channel coe previously optimize for the AWGN channel. Every simulation point presente was obtaine consiering BPSK moulate signal transmitte over an AWGN channel an a total of 50 ecoing iterations. Since we are intereste in an almost-noiseless compression scenario, we chose the wor-error rate (WER) as figure of merit. The simulation results for our optimize system (referre to as JSC opt) an System I with source message of lengths n = 3200 an n = 6400 are epicte in Fig. 3. As mentione previously, the results for System I show an very high error floor for high SNR s which are a consequence of the compression of source coewors that form error patterns not correctable by the source LDPC coe. Figure 3 shows that our propose system manage to significantly lower this error floor while keeping the overall rate constant. VII. CONCLUDING REMARKS We propose an LDPC-base joint source-channel coing scheme an, by means of a multi-ege analysis, propose an optimization algorithm for such systems. Base on a synrome source-encoing, we presente a novel configuration where the amount of information about the source bits available at the ecoer is increase by improving the connection profile between the factor graphs of the source an channel coes that form the joint system. The presente simulation results show a significant reuction of the error floor cause by the encoing of messages that correspon to uncorrectable error patterns of the LDPC coe use as source encoer in comparison to existent LDPC-base joint source-channel coing systems. The next step will be to improve our esign by placing infinite reliability on some source variable noes. This was WER WER System I n=3200 WER System I n=6400 WER JSC opt n=3200 WER JSC opt n= E /N (B) b 0 Fig. 3. Performance of joint source-channel coe systems with R over = 2.03 for n = 3200 an n = one in the context of pure source compression in [2] an will be the subject of our future investigations in orer to lower even more the error floor presente in our simulations. REFERENCES [1] J. L. Massey, Joint source an channel coing, Communications Systems an Ranom Process Theory, vol. 11, pp , Sijthoff an Norhoff, [2] G. Caire, S. Shamai, an S. Verú, A new ata compression algorithm for sources with memory base on error correcting coes, in IEEE Workshop on Information Theory, Paris, France, Mar Apr , pp [3] P. E. Allar an A. W. Brigewater, A source encoing technique using algebraic coes, in Proc. Canaian Computer Conference, 1972, pp [4] K. C. Fung, S. Tavares, an J. M. Stein, A comparison of ata compression schemes using block coes, in Proc. IEEE Int. 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