Classifications

• • H—ELECTRICITY
  • H03—ELECTRONIC CIRCUITRY
  • H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
  • H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
  • H03M13/03—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words
  • H03M13/05—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words using block codes, i.e. a predetermined number of check bits joined to a predetermined number of information bits
  • H03M13/11—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words using block codes, i.e. a predetermined number of check bits joined to a predetermined number of information bits using multiple parity bits
  • H03M13/1102—Codes on graphs and decoding on graphs, e.g. low-density parity check [LDPC] codes
  • H03M13/1148—Structural properties of the code parity-check or generator matrix
• • H—ELECTRICITY
  • H03—ELECTRONIC CIRCUITRY
  • H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
  • H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
  • H03M13/35—Unequal or adaptive error protection, e.g. by providing a different level of protection according to significance of source information or by adapting the coding according to the change of transmission channel characteristics
  • H03M13/356—Unequal error protection [UEP]
• • H—ELECTRICITY
  • H03—ELECTRONIC CIRCUITRY
  • H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
  • H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
  • H03M13/61—Aspects and characteristics of methods and arrangements for error correction or error detection, not provided for otherwise
  • H03M13/618—Shortening and extension of codes

Definitions

• the present invention relates to a method and a corresponding apparatus for unequal error protection for the transmission of data, in particular source-encoded audio and/or video data, allowing to adapt the error correction and coding gains to requirements and sensitivity of the data, using irregular Low-Density Parity-Check (LDPC) codes.
• LDPC Low-Density Parity-Check
• the present invention relates further to an encoding method and apparatus for encoding an input signal and to a corresponding decoding method and apparatus for decoding a received signal which has been encoded by such an encoding method.
• the present invention relates further to a signal which has been encoded by an encoding method according to the invention, to a record carrier carrying such a signal and to a computer program for implementing the methods of the present invention on a computer.
• UEP codes are a suitable tool to protect data according to quality requirements or importance levels.
• Low-Density Parity-Check codes are generally known to be (almost) capacity-achieving. The common understanding was that they can be constructed to offer UEP properties by using different connection degrees at the bit-node side of the describing bipartite Tanner graph. Involving a bit node into more checks, i.e., connecting it to more check nodes would improve the protection. However, the reliabilities of all bits grow with the number of iterations and the UEP properties finally disappear with the number of iterations.
• the object of the invention is to make available another option to establish UEP properties in an LDPC construction by choosing the check profile, i.e., the connection degree on the check-node side of the Tanner graph to be chosen according to the protection requirements of the connected bit nodes.
• the check profile i.e., the connection degree on the check-node side of the Tanner graph to be chosen according to the protection requirements of the connected bit nodes.
• One possible practical realization to obtain the uneven degree distribution is outlined which uses a precoding leading to a sub-code of a chosen mother LDPC code, which may not have UEP properties, yet.
• the pruning can be simplified by just setting information to fixed known values.
• the present invention relates to a method for unequal error protection for the transmission of data of different sensitivities in accordance with claim 1 . It realizes unequal error protection by an unequal distribution of connections (edges) to check nodes, representing the parity-check equations in the so-called Tanner graph.
• Fig. 1 shows the Tanner graph of a regular LDPC code with a constant number of connections
• Fig. 2 shows the pruning procedure to obtain a check-irregular code from a regular one
• Fig. 3 shows a block diagram of an apparatus for pruning represented by precoding
• Fig. 4 shows a flow-chart of an embodiment of the pruning method according to the present invention
• Fig. 5 shows a block diagram of the encoding and decoding scheme according to the present invention
• Fig. 6 shows EXIT curves for different error-protection classes of almost concentrated and non-concentrated check-irregular codes
• Fig. 7 shows bit-error rates of different protection classes of almost concentrated and non-concentrated check-irregular codes
• Figs. A.1 to A.19 show further figures for illustrating the present invention.
• Figure 1 shows the Tanner graph of a regular LDPC code with a constant number of connections to variable and check nodes.
• 10 represent the variable (bit) nodes of the codeword.
• the parity checks defined by the rows of the parity-check matrix 40 are represented by the check nodes 20 together with the incoming edges 30.
• Figure 2 sketches an example of the procedure (pruning) according to claim 6 leading to the desired unequal distribution of edges in the graph at the check-node side.
• variable nodes can have a regular or irregular distribution.
• data will not be of the same importance or sensitivity.
• the present invention assumes iterative decoding of the LDPC code.
• the coded bits are arranged into sensitivity classes C k - One of them will typically be reserved for the parity symbols, since there is no special protection requirement.
• ⁇ (k) the relative portion of bits devoted to class C ⁇ as ⁇ (k), i.e.,
• V " rxik) — 1 Nc denotes the number of classes including the parity check bits class.
• the desired non-concentrated check-degree distribution realizing the unequal protection of the different classes according to claim 3 can be realized by pruning a mother LDPC code.
• the parameters of the mother code be (/Vo, Ko), the length and the number of information symbols (bits), respectively.
• the pruning algorithm is performed in an iterative manner with respect to an optimum order in which the bits are pruned.
• the optimization of the pruning algorithm takes into account two key parameters of the Mh class, and mn
• connection degree d that belong to class C k -
• the upper limit t is the maximum rmax possible check degree.
• the protection in class C k can be improved by minimizing the average check connection degree d (Ck > , which requires to minimize mn as well.
• Any pruned bit must not be linked with a check node of degree already identical to the lower limit of a priori chosen degree distributions.
• V V :; p c? - ⁇ > where p y denotes the proportion of edges of the graph connected to check nodes of degree /.
• mn may be further reduced after ensuring that these listed constraints are fulfilled (if the lower limit of allowed degrees is not yet reached).
• a further pruning process is used to reduce d (Ck > .
• Minimizing the average check connection degree d (Ct) can be shown to increase the difference between the average mutual information of messages from check nodes of one class C k , Xcv C ⁇ k ⁇ and the average mutual information of messages from check nodes of the whole graph, x cv , which is given as a possible measure for the quality of one class relative to the average in claim 4.
• pruning (as illustrated in Figure 3) can be represented by a precoder P 70 with K 1 inputs and ⁇ T 0 outputs, delivering the input to the original mother encoder G
• a permutation matrix Eli can be used to formulate H mpmn as
• H 1 must be of full rank to ensure that P is of full rank. It can be further rephrased
• H 1 is of full rank then A(N 0 - ⁇ T 0 + 1 : N O ,:) is invertible.
• a tool is available to determine P , and the additional requirement in the pruning procedure according to claims 5 and 6 to ensure that H 1 is full rank.
• a still flexible low-complexity realization of pruning is obtained by just omitting information bits from the input to the encoder of the LDPC mother code and setting it to a fixed value (e.g. to zero) also known at the receiver.
• This is performed by making use of so-called systematic LDPC codes, that is LDPC codes for which the parity check matrix has an upper (or lower) triangular structure.
• the pruning is then performed by simply omitting an information bit of the mother code, or equivalents by removing the corresponding column in the information part of the parity check matrix (the part which is not upper triangular). By doing so, the dimensions of H 5 and G s will be
• N 0 - (K 0 - K 1 ) N 0 - (K 0 - K 1 ) mother code need to be known at the transmitter and receiver in order to be able to encode and decode the pruned code.
• the method of claim 8 hence describes a pruning without a preprocessing matrix, just realized by omitting input bits to the encoder.
• Figure 4 illustrates a flow-chart of the pruning method according to the present invention.
• Figure 5 shows the principal block structure of a transmission of source (e.g. video-, audio-) encoded data with different sensitivity to errors and its corresponding unequal protection.
• source e.g. video-, audio-
• Figure 5 shows the principal block structure of a transmission of source (e.g. video-, audio-) encoded data with different sensitivity to errors and its corresponding unequal protection.
• Well-known coding methods adapted to heterogeneous sensitivity of data often focus on average performance over the whole codeword.
• UEP could be achieved by puncturing or pruning convolutional codes to adapt the code rate without changing the decoder.
• UEP properties can also be obtained within the same codeword and thus, the shown different coding gains can appear in the same codeword.
• UEP unequal error protection
• LDPC Codes achieved by irregularity on the check node profile, the bit node profile is set to be regular.
• a UEP coding scheme could be useful in the transmission of multi-media content (voice, fixed image, or video) whose characteristics have heterogeneous sensibility to errors.
• the code stream of source-encoded blocks is hierarchically structured and contains typically:
• compressed data delivered from the source codei e.g. speech encoder coefficients, image texture, or motion vectors.
• the parameters of the protection should be adaptable dynamically with minimum changes in the encoder and/or the decoder.
• variable and check blocks with respect to 7r, which are the usual polynomials for conventional representation:
• xi v (d) and x[ c (b) be the mutual information between the input of the channel and the messages from check nodes of degree d to any bit node at the Uh iteration, and from bit nodes of degree b to any check node, respectively.
• Equation (1) we observe that the smaller d is, the greater is the mutual information of messages coming out of check nodes of degree d, i.e. the faster is the local convergence.
• Equation (2) we see on Equation (2) that the mutual information of messages coming out of bit nodes of degree 6 is larger when 6 is larger. This is what we are going to exploit to optimize the local convergence speeds of UEP LDPC codes.
• a sensitivity class by a set of information bits in the codeword that will have the same protection, i.e., approximately the same error probability at a given number of iterations.
• the sensitivity classes are defined by the source encoder.
• B and D can either be the sets of the degrees over the whole graph, and then Equation (1) describes the usual Gaussian approximation of density evolution, or the sets of the degrees inside the kth sensitivity class called C k .
• a check node will belong to a class C k if it is linked to at least one bit node of this class. Consequently, a check node can belong to several sensitivity classes.
• the average mutual information of messages coming out of the check nodes of class C k . to the bit nodes of this class can be expressed as
• the first solution to limit the degradation of the overall convergence threshold is to limit the range of the check irregularity around jf c ⁇ in the optimization. Or we could check after the optimization process wether the non-concentrated code has a threshold that is not too far from the optimum (concentrated code) threshold. We have also verified by simulations that for short block lengths, the UEP designed codes have similar global performance as the concentrated code.
• Pruning is a well-known method foi convolutional codes [I], but not so much for LDPC codes, for which it has been applied to reduce the influence of stopping sets [I]. Pruning away some bits of the codeword means to consider them deterministic, i.e. fixing the pruned bits, e.g., to zero. Consequently, we do not transmit these bits that disappear from the graph of the code since their messages are equal to infinity. Besides, since the edges connected to the pruned bits disappear, the girth (minimum cycle length) of the subcode can only be increased. Thus, the columns of the parity matrix that correspond to these bits are removed.
• H m and G m denote the parity- check and generator matrices, respectively, of the mother code of dimension KQ and length NQ.
• H m and G m denote the parity- check and generator matrices, respectively, of the mother code of dimension KQ and length NQ.
• TO construct a subcode oF dimension K ⁇ we prune away KQ — Ki columns of H m , and obtain the parity-check matrix H s of the subcode.
• the next section deals with our sequential prun ing procedure. 4.2 The Sequential Pruning Procedure
• the N c sensitivity classes to be optimized are defined by the proportions a(k) for k ⁇ N c — 1.
• the optimization focuses on the two important quantities in the bound (6) : p (Cfc ' and d ⁇ , and is composed of two main stages. For a given class G k -'
• H mprun will denote H m whose pruned columns are replaced by zero columns. Then we have
• H s and G s are obtained by removing columns in H m , and the corresponding ones in G m which are columns of the identity part of G 1n . The corresponding rows of G 1n are also removed. Then H s and G s are of size M 0 X NQ - (Ko - Ki) and K ⁇ x N 0 - (K 0 - Ki), respectively. They are both of full rank and the code rate of the subcode is the target rate:
• Table 2 Comparison of degree distributions for the different classes of the imconcentrated code.
• Figure (6) shows the EXIT curves defined in equation (5) for each class of almost concentrated and non-concentrated check irregularity codes.
• the intermediate classes are quite equivalent whereas the last class of the non-concentrated code has a slower convergence than the corresponding one in the concentrated one.
• Figuie(7) repiesents bit error rates of the UEP almost concentrated and non-concentrated codes after 30 decoding iterations.
• the check irregularity is a mean to achieve LJEP at low number of iterations (accelerating the convergence), but also at a high number of iterations since the differences between classes are still visible after 30 decoding iterations when Looking at the bit error rates.
• stretching the check degrees allows a stronger difference in the error protection , without degradation of total average bit error probability at this code length, compared with the concentrated code.
• Source coding in this block, source data are compressed and reshaped. Some parts of the source si ⁇ gon* al are more vulnerable than others.
• the code rate R is den ned as the ratio between tran smitted information bits and the number of tran smitted bits.
• the source can be en coded in an uniform way or in a heterogeneous one in order to take into account the properties of the source signal (unequal error protection (UEP) techniques).
• UEP unequal error protection
• Modulation the order and the type of the modulation are man aged here (PSK, ASK, QAM, multicarrier modulation), the power of transmitted symbols, or the spreading when we have to deal with a multiple access system by code spreading.
• the physical chan nel disturbances are introduced.
• the chan nel leads to inter symbol interferences (ISI caused by fading chan nel) due to multi-paths, multiuser inteferences (MUI), and adds (e.g. white gaussian or impulse) n oise.
• ISI inter symbol interferences
• MUI multiuser inteferences
• adds e.g. white gaussian or impulse
• the receiver is composed of correspondin g blocks, that can be described as follows:
• Demodulation/Despreadin g used to find bits from received symbols and to separate users in a multiple access system.
• Chan nel decodin g corrects remainin g errors in the previous obtained binary sequence.
• Source decodin g recon struction of the emitted data by decompression of the sequence goin g out from chan nel decoder.
• LDPC Low-Density Parity-Check
• LDPC codes have iterative decodin g, that allows to reach bit error probabilities of 10 ⁇ 5 — 10 ⁇ 6 , for a wide range of signal to n oise ratios. These are the required orders for sen sible application s such as fixed picture or video tran smission s. A delay caused by the interleaver must be tolerated. Therefore LDPC codes can be an alternative to Turbo Codes for UEP target multimedia tran smissions.
• LDPC codes are low den sity linear block codes, introduced by Gallager [8] in 1963.
• parity matrix can be regular or not.
• a code is regular if the number of non zero elements in every rows (respectively column s) is constant. Irregular if these numbers are n ot con stant.
• a regular LDPC code with its three parameters (N, t c , t r ) is defined by a matrix with exactly t c and t r ones per column and row, respectively.
• Those three parameters define a family of regular codes, and one code among this family is given by a particular realization of the parity-check matrix.
• an LDPC code can be represented by a bipartite graph, called factor graph [12], or Tan ner graph, made of two kinds of nodes: variable n odes representing bits of codeword, and check n odes associated to parity-check functions.
• Those two kinds of vertices are linked with each other by edges indicatin g to which parity equation variable nodes, i.e. the associated bits, take part in .
• the degree of con nection of a bit n ode (the same for a check n ode) is the number of edges linked to this n ode.
• a node is said i con nected or of degree i if it is con nected to i edges.
• One code corresopnds to one particular realization of the interleaver. 2.1.1.2 Irregular LDPC Codes
• a code is irregular if it is not regular.
• the usual parameterization of irregular LDPC codes is done by means of polynomials:
• ⁇ is the proportion of edges of the graph connected to bit nodes of degree i
• t cmax is the maximum number of edges linked to a bit node.
• p ⁇ is the proportion of edges of the graph connected to check nodes of degree j
• t rmax is the maximum number of edges linked to a check node
• codes should be systematic: information bits are directly copied into the codeword.
• the generator matrix G from H is n ot too easy. Nevertheless it is possible to encode usin g the parity matrix.
• a sub-optimum decodin g algorithm kn own as Sum- Product algorithm or Belief Propagation (BP) algorithm is used in stead. It spreads along edges messages forwardin g probabilities or logarithmic likelihood ratios (LLR). To each branch two messages are associated, one for each direction .
• the principle of BP is Bayes rule applied locally (on every bit of the codeword) and iteratively to estimate a posteriori probabilities (APP) of every bit.
• LLR logarithm likelihood ratio
• c is the bit value of the node and y denotes all the information available to the node up to the present iteration obtained from edges other than the one carrying v.
• c is the bit value of the variable node that gets the message from the check node up to the present iteration obtained from edges other than the one carrying u.
• ⁇ m is the message (LLR) over the mth edge coming out of a bit node.
• the messages u k are the LLR coming out of a check node and u 0 is the LLR of the channel observation.
• every messages U k are equal to zero.
• u k is the message (LLR) over the fcth edge coming out of a check node.
• the messages v m are the LLR coming out of a bit node.
• P ⁇ (z) is the average probability of the codes of the family, such that sub-jacent graph be a tree.
• the zero codeword is transmitted (since for a chan nel and the BP decoder fulfilling symmetry condition , error probabilities at the Zth iteration do n ot depend on the transmitted codeword).
• Therfore u 0 is Gaussian N(2/ ⁇ 2 , A/ ⁇ 2 ) which is con sistent according to Def. (2.4).
• Theorem 3 of [19] (p.628) asserts that con sistency is kept alon g iterations for a given binary-input memoryless output- symmetric chan nel.
• v be a message such that v ⁇ N( ⁇ m, 2m) that is the output of a binary-input Gaussian channel.
• the mutual information between v and input c of the virtual channel is given by:
• Equation (2.10) is rewritten as:
• J is continuous and a strictly mon oton ous function , so J "1 exists and permits to compute the mean of messages from the mutual information .
• x is the random variable describin g the codeword bit value associated to the variable node a
• y is the random variable describin g all the information incorporated into this message.
• the stability condition is very important because it controls the mutual information behavior at very low error probabilities (or equivalently, when the mutual information is near to 1).
• ⁇ * ⁇ ⁇ g mm(-l— ⁇ F( ⁇ , x, ⁇ 2 ) > Z 5 VzG [O 5 I]) (3.1)
• the code stream of source-encoded blocks is hierarchically structured and contains:
• Compressed data delivered from the source coder e.g. speech encoder coefficients, image texture, or movement vectors.
• Irregular punctured/pruned systems Puncturin g con sists of n ot emitting some bits of the codeword, thereby decreasin g the initial code rate R.
• the receiver kn ows the puncturin g pattern , and con siders n ot tran smitted bits as erasures. This technique worsen s the performance of the code allowin g to obtain a wide ran ge of rates.
• Unequal error protection can then be achieved by applyin g different code rates to each part of the source data, according to the required robustness.
• An other way of addin g irregularity is usin g a pre-processin g block before the code, in order to prune it. Puncturin g and prunin g will be the chosen method to realize UEP in our work, and has been further studied in [24] for Turbo Codes.
• the local minimum distance associated to each bit of the codeword determines the maximum number of errors in the whole codeword, still allowin g this bit to be corrected.
• the local minimum distance can be greater than the global one, which mean s that the rth bit can be corrected even if the whole codeword is n ot recovered by MLD. That explain s the interest of such codes under MLD, when considerin g in the previously mentioned JPG tran smission , for example. Con struction methods of such codes have been presented, but a big problem is the poor control that we can have over the proportion s of the classes, which can be very disturbing for the latest application .
• An other family of such irregular coding systems is multi-level coded modulation .
• Each bit of a symbol is associated to a given code, which differs from others by its code rate.
• the protection level of bits depends on the code, and on the position in con stellation labellin g, which mean s that two kinds of irregularities can be exploited
• LDPC codes can be punctured [9] in order to create average irregularity. Puncturing in fluences the code rate: average performances differ between two codewords encoded with different puncturing patterns. Nevertheless it is more suited to make use of an irregularity that leads to unequal error protection of bits in side a codeword: most con nected bits will have lower error probability. This has been highlighted in [6], and applied for optimization for several tran smit chan nels. The optimization for AWGN done by Poulliat in [17] will be presented in the next chapter.
• the first con siders LDPC code as a linear block code and optimizes the code according to the local minimum distances [4, 16].
• the second approach is an asymptotic optimization for BP decodin g and is based on pruning and puncturin g of a mother code.
• Equation (4.2) is obtained by adding the mutual information comin g into each class of bitn odes since there is no overlap between the classes. We then can derive convergence and stability conditions from the fact tha 4.1.2 Cost Function for such UEP
• X 11 be the random variable whose distribution is N(O, 1).
• X is Gaussian consistent for any iteration according to the symmetry of the chan nel and the conservation of the con sistence alon g the iteration s:
• X den ote improperly X ⁇ bit 0, but this will make the expression s clearer.
• J "1 (xh-l ) is an increasin g function of I. Since Q is a decreasing function , 4.3 shows that at a given number of iteration s, the more a bit is con nected, the more it is protected, con sidering the associated error probability (the convergence of this node is faster).
• the derived linear programming algorithm is meant to achieve a joint optimization of ⁇ * ) and P 0 J 11n , under the con straints of proportion , code rate, convergence, stability, and hierarchical constraints (since the optimization is sequential, the irregularity profile of already optimized classes must n ot be modified by the current optimization ).
• the degree of independence of the ith column of the parity-check matrix of the code is the minimum number of columns that are included in a linear combination that equals zero, with a coefficient one at the ith column.
• Lemma 4.1 The local minimum distance d ⁇ of the ith bit of the codeword is the the degree of independence of the ith column of the parity -check matrix of the code.
• the protection level f ⁇ of the ith bit of a codeword is the maximum number of errors in the codeword that still allows the correction of this bit.
• the local minimum distance associated to each bit of the codeword determines the maximum number of errors in the whole codeword that still allows the correction of this bit.
• the local minimum distance can be greater than the global one, which mean s that the «th bit can be corrected even if the whole codeword can n ot be restored by MLD.
• Those algebraic properties can be linked to Majo ⁇ ty Logic Decoding presented in [3] which works on a poorer difmition of local minimal distance to simplify the decoding.
• Classes are n ot defined by their proportion s at the begin ning, which is another drawback of the linear codin g approach.
• n ot intend to construct an arbitrary linear block code, but a subcode of a mother code from which we choose the right column s to be removed in the parity-check matrix in order the resultin g parity-check matrix be the matrix definin g a code with the required properties.
• parameters of the optimization are the parameters of the optimization :
• U is the vector where indexes of columns of H to be pruned away are stored (length K 0 - K 1 ).
• w lmtt is the initial w ⁇ vector, ordered in decreasin g order, before optimization of a selected column .
• a cycle of length 2d is a set of d variable nodes and d constraint nodes connected by edges such that a path exists that travels through every node in the set and connects each node to itself without traversing an edge twice.
• Definition 4.5 (C d Cycle set)
• a set of variable nodes in a bipartite graph is a G d set if (l) it has d elements, and (2) one or more cycles are formed between this set and its neighboring constraint set.
• a set of d variable nodes does not form a C d set only if no cycles exist between these variables and their constraint neighbors.
• variable node set is called an S d set if it has d elements and all its neighbors are connected to it at least twice.
• Ld Linearly dependent set A variable node set is called an Ld set if it is comprised of exactly d elements whose columns are linearly dependent but any subset of these columns is linearly independent.
• the code is decoded in an optimal way, in the sense of the minimum distance.
• the code can have UEP properties due to its local minimum distances, associated to some cycles in the graph, that can be different from each other.
• the UEP properties are then dependent on the realization of the H matrix.
• the local properties of the code are taken into account by the MLD.
• the Belief Propagation is sub-optimum decodin g, and quite "global" in the sen se that it does n ot take into account local properties randomly created with the H matrix. Local differences will be created by the local sub-optimalities of BP decoding at finite code len gth, and some of these sub-optimalities are associated to small local minimum distances.
• Belief Propagation decodin g is the Maximum Likelihood Decodin g.
• the minimum distance tends to in finity and the length of the smallest cycle, called the girth, too. Therefore, all local minimum distances tend to in finity too and UEP properties den ned by the two mean s of decodin g tend to be the same.
• UEP properties depend on the code and also on the way that it is decoded: the optimization must be done as a function of the chosen decodin g method. This is, of course, practically determined by the code len gth since at low N (N ⁇ 500), MLD will be used, otherwise BP.
• Theorem 3 in [11] states that under local tree assumption of depth 2T and some other constraints, for any 1 ⁇ I ⁇ T, the distribution functions Q ⁇ (d) of messages originating from check nodes of degree d and Pi (b) of messages originating from bit nodes of degree b are equal to
• ⁇ ® ⁇ d i-j[(d- I)J- 1 ( i - ⁇ ⁇ (b, d) x «l(b) (4.13) b&B
• LLR LLR ⁇ M., and 0 ⁇ ta.nh(LLR) ⁇ 1. At a high number of iteration s, many LLRs are high.
• n ode At a bit n ode, the important LLRs are the highest because they are summed up. At a given high number I of iterations, we decide that a message comin g out of a bit n ode is of bad quality if the correspondin g LLR is below a fixed threshold that does n ot depend on the con sidered bit node or on the number of iterations. At a high en ough number I of iteration s, a bit n ode produces bad message (i.e. a low LLR) if the number of incomin g high LLRs is below a fixed number that we choose in terms of the number of iterations, i.e.
• This ratio is a con stant. It does n ot depend on q ⁇ , i.e., on the number of iteration s, for high en ough number of iteration s. The behaviors of different check n odes remain s different even at high number of iterations, i.e., at a low bit-error rate. This explains that the UEP created by irregularities over check n odes remain s at a high number of iterations which we exploit in this work.
• Den sity problem according to Gallager's result, den sest codes have the lowest gap to capacity. At given code rate, there is one optimum average con nectivity of check nodes ⁇ p that minimizes the gap to the capacity Fig. (A.7) (for in finite code length and infinite number of iteration s). This key parameter of the code, linked with t cm ⁇ x , determines the den sity of the code. The denser is the graph, the higher have to be the con nectivity ratio. If the value of ⁇ p is moved from the optimum, the value of t cm ⁇ x must be chan ged too.
• the required p can be achieved wether with a concentrated degree distribution at check n ode side, or with an unconcentrated one.
• Our optimization by removing bit n odes, decreases ⁇ p while t cmax is kept. The UEP less den se code must have higher threshold.
• tran smission has to be achieved, even with poor quality, we allow big amplitude on degrees of check nodes. For example if one wants to tran smit a JPG picture even with bad quality, puttin g headers and very low frequency DCT coefficients in most protected classes en sures the transmission , even if the resultin g picture is quite fuzzy.
• n ot be a problem if the maximum degree of con nection of bit n odes is adapted, i.e. in a joint optimization . It could raise a problem if the check optimization is proceeded after the bit n odes optimization , as a second stage. If it is done before, a con straint on t cmax should be added in the optimization of bit node profile if one wants to keep the best convergence threshold.
• Figure (A.17) shows the coding scheme that we use as a startin g point.
• H and G be the parity-check (size M 0 x N 0 ) and generator (size K 0 x N 0 ) matrices of the mother code and assume that they are in a systematic form (i.e. full rank).
• R 0 be the code rate of the mother code.
• the subcode has a given number of info bits : K 1 .
• K 1 the code rate of the mother code.
• P a preprocessin g generator matrix
• This preprocessin g matrix is n ot needed if we prune away only columns of information of the H matrix, and choose the K 1 best protected columns amon g the information column s of the H matrix, which reduces a lot the possible UEP configuration s.
• H 3 and G 3 are the parity-check and generator matrices of the subcode, are obtained by removin g column s to prune away in H, and the correspondin g ones, which are column s of the identity, in G where we remove also corresponding rows (i.e. the row where there was the one). Since the best protected columns are chosen as bein g information column s, they are already made of the identity. Then H s and G s are of size M 0 x N 0 — (K 0 — Ki) and K 1 x N 0 — (K 0 — K 1 ), respectively.
• the obtained rate is the desired one.
• Definition 4.11 A matrix is in a reduced row echelon form if it is made of a triangular upper part of size the rank of the matrix, after linear combinations of its rows, and then permutation of the columns.
• Definition 4.12 A matrix will be said in a reduced row form if the previous manipulations on its rows have been made, but without permutting its columns at the end.
• Theorem 4.1 A necessary and sufficient condition on G that allows to compute P that fulfills Eq. (4.21) is: rank(G') > K 1 (4.22)
• Equation (4.23) can be represented by
• G' fulfills Condition (4.22), a solution for P 2 exists, and if rank(G') ⁇ K 0 , then we have degrees of freedom for P 2 , and then also for P.
• a constraint calledCode rate constraint in the optimization algorithm, en sures that the parity-check matrix of the subcode, i.e. the matrix of the mother code without the pruned column s, will have a code rate of N , ⁇ 1 1 _ ⁇ y or that equivalently ran k(H m otherpruned)-
• ⁇ pruned is not anymore the parity matrix of the subcode since an other parity equation s are added.
• the subcode is den ned by:
• G s P G -. K 1 X N 0 H 5 : [N 0 - K 1 ) x N 0
• H 5 is made of H mot/>ercorfe an d the H p parity matrix of the generator preprocessing matrix P.
• H p is of size (Ko — K 1 ) x K 0 :
• I K0 - K1 is the identity associated to redundancy column s of the precode P, and HL (K0 - K1) XK1 are associated to information bits of the subcode.
• H of the mother code is the identity associated to redundancy column s of the precode P, and HL (K0 - K1) XK1 are associated to information bits of the subcode.
• the K 0 bits of the codewor of the precode P are directly copied into the K 0 information bits of the mother code.
• H s of the subcode is n ot in a systematic form in Eq. (4.31), and then we can n ot distinguish column s of redundancy and column s of information of the subcode in this form.
• H Ssys H Ssys , where the last K 1 column s are associated to the K 1 information bits of the subcode, and the K 0 — K 1 pruned column s are taken among the N 0 — K 1 columns, which are the column s of a squarred upper trian gular matrix.
• condition (4.22) be fulfilled to be able to compute the P matrix and have a code rate of the subcode equal to the one desired, even if we choose column s to be pruned away and best protected column s amon g redundancy of the mother code.
• Computation of the preprocessing matrix After having verified that we can choose the K 0 — K 1 bits to be pruned away and the K 1 best protected amon g the TV 0 bits of the mother code, we are goin g to explain how the P matrix is computed.
• H 5 Another possibility is to consider the P matrix as some addition al parity-check equation s, as showed in expression of H 5 .
• H p an arbitrary H p , for example such that it improves the UEP properties of interesting bits by choosing its irregularity accordingly, or as a part of H mother to decrease the required memory.
• the user will have to choose the con straints on the optimization and so the stren gth of UEP, according to his available memory and processing power.
• H p (K 0 — Ki) x K 0 whose elements are h(i,j) and P r : K 0 X Ki whose elements are d(i,j)
• N 0 (Ut) denotes the set of check nodes linked to variable node bit.
• N ⁇ (bit) is the set of bit nodes linked to each check node belonging to No(bit).
• bit pruned arg max ⁇ (di(6i£)) under:
• This condition is automatically fulfilled in the case of a regular mother code.
• the mother code has parameters (2000,3,6).
• the decoding is done bu using only the pruned parity-check matrix of the mother code.
• Fig. (A.18) shows EXIT curves defined in Eq. (4.16) for each class of almost concentrated and unconcentrated check irregularity codes.
• the intermediate classes are quite equivalent whereas the last class of the unconcentrated code has a slower convergence than the corresponding one in the concentrated one.
• Figure (A.19) shows the behavior at low bit-error rates, which cannot be seen from an EXIT curve. This would be near the (1,1) point in the EXIT chart, i.e. at a high number of iterations. Here for 30 iterations.
• UEP properties remain also at a high number of iterations, which constitutes a huge difference from UEP properties generated by irregularities over bit nodes, which induces convergence speed differences.
• the check optimization would be a means to achieve UEP at low number of iterations (accelerating the convergence), and at a high number. This behavior can be explained by Fig. (A.14) and the comments following it in the first section.
• the puncturin g could be a method to realize UEP by increasin g the code rate and worsening certain bits, but without the possibility to improve some others.
• the quality of the messages coming to interestin g checks i.e. belonging to one class
• the quality of the messages coming to interestin g checks would be more important than the degrees of these checks.
• erasure messages i.e. with LLR, den ned in Def . (2.2), that equals zero
• prunin g i.e. LLR equals infinity that makes the bitn ode and the linked edges disappear from the graph).
• G ⁇ is the set of bit nodes of degree i linked to check nodes of degree j.
• Tr 4 is the proportion of puntured symbols in G hJ before decoding.
• ⁇ (i, j) and p(i, j) are the proportion of bit nodes of degree i among bit nodes linked to check nodes of degree j, and the proportion of check nodes of degree j among check nodes linked to bit nodes of degree i, respectively.
• bitnode is of degree i and linked to check of degree j )
• the design goal optimal puncturin g defined in [9] is to maximize the puncturin g fraction p ⁇ for a given E b /N 0 , such that Eq. (4.40) is fulfilled.
• a computer program may be stored/distributed on a suitable medium, such as an optical storage medium or a solid-state medium supplied together with or as part of other hardware, but may also be distributed in other forms, such as via the Internet or other wired or wireless telecommunication systems.
• a suitable medium such as an optical storage medium or a solid-state medium supplied together with or as part of other hardware, but may also be distributed in other forms, such as via the Internet or other wired or wireless telecommunication systems.

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