802                                                                              IEEE TRANSACTIONS ON COMMUNICATIONS, VOL....
804                                                                                  IEEE TRANSACTIONS ON COMMUNICATIONS, ...
806                                                                             IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. ...
808                                                                              IEEE TRANSACTIONS ON COMMUNICATIONS, VOL....
Sandberg v deetzen_trcomm_2010
Sandberg v deetzen_trcomm_2010
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  1. 1. 802 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 3, MARCH 2010 Design of Bandwidth-Efficient Unequal Error Protection LDPC Codes Sara Sandberg and Neele von Deetzen Abstract—This paper presents a strategy for the design of to rejection of the packet or even a crash of the source decoder,bandwidth-efficient LDPC codes with unequal error protection. while errors in the payload are often tolerable.Bandwidth efficiency is obtained by appropriately designingthe codes for higher order constellations, assuming an AWGN In order to allow for bandwidth-efficient transmission, itchannel. The irregularities of the LDPC code are designed, is desirable to use higher order modulation techniques, suchusing the Gaussian approximation of the density evolution, to as -QAM, -PSK, 2, or more advanced constel-enhance the unequal error protection property of the code as lations. Modulation with higher order constellations (HOCs)well as account for the different bit error probabilities given may imply that different bits in the symbol have differentby the higher order constellation. The proposed code designalgorithm is flexible in terms of the number and proportions of error probabilities, i.e., the modulation already provides someprotection classes. It also allows arbitrary modulation schemes. UEP. Coded modulation is a well-known strategy to opti-Our method combines the design of unequal error protection mize the coding scheme given the modulation to improveLDPC codes for the binary input AWGN channel with the code the performance of transmission systems in terms of overalldesign for higher order constellations by dividing the variable bit-error rate (BER), [1]–[4]. In multilevel coding [5], thenode degree distribution into sub-degree distributions for eachprotection class and each level of protection from the modulation. modulation alphabet is successively partitioned into smallerThe results show that appropriate code design for higher order subsets, where each partitioning level is assigned a label.constellations reduces the overall bit-error rate significantly. These labels are protected by separate channel codes withFurthermore, the unequal error protection capability of the code certain protection capabilities. In such a strategy, the codesis increased, especially for high SNR. have to be designed carefully, depending on the modulation Index Terms—LDPC, unequal error protection, higher order scheme and its partitioning or labeling strategy. However,constellations, bandwidth efficiency. applying a shorter code on each level may have drawbacks (e.g. higher BER) compared to designing one long code whose I. I NTRODUCTION output bits are appropriately assigned to the levels. Therefore,M ULTIMEDIA applications require large amounts of we employ the latter, taking advantage of the longer code. data to be transmitted through networks and wireless This paper focuses on low-density parity-check (LDPC)transmission systems with reasonable delay and error perfor- codes, originally presented by Gallager in [6]. They exhibitmance. Therefore, resources like power, time, or bandwidth a performance very close to the capacity for the binary-have to be used economically. Multimedia data usually have input additive white Gaussian noise (BI-AWGN) channel, [7].heterogeneous sensitivity against transmission errors. They The close-to-optimal performance of LDPC codes for the BI-often contain a header for data management in higher layers, AWGN channel suggests the use of LDPC codes also for othersome essential payload as well as additional payload used channels. The aim of this paper is to design LDPC codes forfor enhanced quality. Hence, the transmission system should transmission using HOCs in applications where the source bitsprovide unequal error protection (UEP) in order to account have different sensitivities to errors and UEP is desired. Asfor the different properties. Suitable UEP may increase the far as we know, such a code design has not been proposedperceived performance for applications where different bits before. However, a code design using turbo codes for codedhave different sensitivity to errors, such as transmission of modulation with UEP has recently been suggested, [8].progressively encoded images or a frame with header infor- LDPC codes are block codes with a sparse parity-checkmation. For packet-based transmissions with no possibility of matrix of dimension ( − ) × , where = / denotesretransmission, an error in the header is critical and may lead the code rate and and are the lengths of the information Paper approved by A. K. Khandani, the Editor for Coding and Information word and the codeword. The codes can be represented by aTheory of the IEEE Communications Society. Manuscript received January bipartite graph, called a Tanner graph [9], which facilitates a25, 2008; revised September 17, 2008. decoding algorithm known as the message-passing algorithm S. Sandberg is with the Department of Computer Science and Elec-trical Engineering, Luleå University of Technology, Sweden (e-mail: [10]. The graph consists of two types of nodes, variable nodessara.sandberg@ltu.se). and check nodes, which correspond to the bits of the codeword N. von Deetzen was with the School of Engineering and Science, Jacobs and to the parity-check constraints, respectively. A variableUniversity Bremen, Germany. She is now with Silver Atena Electronic Sys-tems Engineering GmbH, Germany (e-mail: n.vondeetzen@silver-atena.de). node is connected to a check node if the bit is included This work is part of the FP6 / IST project M-Pipe and is co-funded by in the parity-check constraint. For regular LDPC codes, allthe European Commission. Furthermore, it has been funded by the DFG variable nodes have the same degree and all check nodes(Deutsche Forschungsgemeinschaft). The material in this paper was presentedin part at the IEEE ICC’07 conference. have another common degree. However, irregular LDPC codes Digital Object Identifier 10.1109/TCOMM.2010.03.0800352 are known to approach capacity more closely than regular 0090-6778/10$25.00 ⃝ 2010 IEEE c Authorized licensed use limited to: Werner Henkel. Downloaded on July 02,2010 at 18:43:04 UTC from IEEE Xplore. Restrictions apply.
  2. 2. SANDBERG and VON DEETZEN: DESIGN OF BANDWIDTH-EFFICIENT UNEQUAL ERROR PROTECTION LDPC CODES 803LDPC codes. The irregular variable node and check node gree distributions into sub-distributions. Such sub-distributionsdegree distributions may be defined by the polynomials [7] ∑ ∑ have also been employed for systems without UEP capability() = =2 −1 and () = =2 −1 to account for HOCs [14], [15], [24]. In [14], the differentrespectively, where is the maximum variable node amount of protection for each modulation level is taken intodegree and is the maximum check node degree of the account in the initialization of the density evolution algorithmcode. The coefficients of the degree distributions describe the that is employed to optimize the sub-degree distributions ofproportion of edges connected to nodes with a certain degree. the code. Both [15] and [24] designed LDPC codes for a setIn order to optimize the degree distribution of an irregular of parallel subchannels.LDPC code, the decoding behavior has to be investigated. In this paper, we propose a UEP-LDPC code design forUsing a message-passing algorithm, the messages along the HOCs that is based on optimizing the variable node degreeedges of the graph are updated iteratively. The messages distribution (). Apart from designing the code to accountat the input of a variable node and a check node at each for the UEP provided by the modulation itself and therebyiteration represent mutual information and can be computed reducing the overall BER, the aim is to provide a flexibleby means of density evolution using a Gaussian approximation design method that can use the UEP from the modulation to[11]. Thereby, the decoding behavior of a code ensemble create a code with other UEP properties which are usuallywith a specific degree distribution may be predicted. Density specified by the source coding unit. We design UEP-LDPCevolution is an asymptotic tool, but is commonly used to codes using sub-degree distributions both for the protectiondesign codes of finite length. classes and for the classes of bits with different protection LDPC codes have been designed for larger constellation resulting from the modulation. This allows for a very flexiblesizes in previous works. In [12], separate codes were designed code design where any conventional modulation scheme, likefor each level in a multilevel coding scheme. On the other -QAM or -PSK as well as more complex schemes likehand, bit-interleaved coded modulation (BICM), [13], [14] hierarchical constellations [25], may be used. The separationemploys only one code for all levels of the modulation. A of the variable node degree distribution into sub-degree dis-design method for discrete multitone modulation (DMT) is tributions significantly increases the number of design param-proposed in [15]. This method may also be directly applied eters. Therefore it is important to note that the code designto HOCs and is very similar to the design approach suggested is solved by iterative linear programming (LP), which enablesin [14]. In [14] and [15], the codes are designed to have an efficient optimization of the sub-degree distributions. Ourlocal properties that match the HOCs and bit positions of code design is based on the design method for UEP-LDPCthe modulation scheme are assigned to the codeword bits. codes suggested in [22] that employs iterative LP.Reliability mappings of bits from the modulation to codeword The paper is organized as follows. Section II presents thebits have also been suggested, [16], [17]. overall system and modulator models. Section III contains the In many applications, the desired UEP properties are low main part of this paper which introduces the code optimizationBER within one or several classes of bits, while the per- of UEP capable codes used with HOCs and provides anformance of the remaining classes should be comparable to algorithm for the code design. In Section IV, some simulationnon-UEP codes. In the following we address such codes as results for 8-PSK and 64-QAM are presented and comparedcodes with good UEP capability. UEP is commonly provided to BPSK-optimized codes.by multilevel coding or adapted code rates, for example bypuncturing. These methods have in common that different II. S YSTEM M ODELcodes are used for each level of protection. However, formost applications the more important bits are fewer than the A. Model Descriptionless important bits. This implies that even if the code rate Each codeword bit is assigned to a protection class andused for the more important bits is low, this codeword will a modulation class. We apply only one UEP-LDPC code,usually be short. To avoid the use of short codewords (or providing protection classes at its output (see Fig. 1). Thea very long delay), LDPC codes that provide UEP within independent and identically distributed (i.i.d.) information bitsone codeword may be designed instead. Such codes can for are divided into −1 protection classes 1 . . . −1 . Theexample be constructed by an algebraic method based on least protected class contains the parity bits. Each bit isthe Plotkin construction, [18]. However, since it is widely protected by the UEP-LDPC code according to the protectionobserved that the connection degree of the variable nodes class it belongs to, where 1 has the best protection. Theaffects the bit-error rate for a limited number of decoder sizes of the protection classes (they do not have to be of equaliterations, it is more typical to design the variable and/or check size) as well as the allocation of information bits to protectionnode degree distribution of the code in an irregular way using classes are usually defined by the source coding unit.density evolution, [19]–[23]. In this case the codeword bits are The bits of the protection classes are remultiplexed and as-divided into several protection classes with different protection signed to certain bit positions of the modulator, correspondingdepending on the connection degrees of their bits. In [21], to modulation classes 1 , . . . , . A bit that belongs tothe check node degree distribution is adapted, keeping the modulation class will be mapped to the bit position invariable node degree distribution fixed, whereas the authors of the symbol that corresponds to that modulation class, i.e., it[22] optimize the irregular variable node degree distribution will be naturally protected according to modulation class .while keeping the check node degree distribution fixed. The Bits belonging to one modulation class are affected by thebasic idea in [20]–[23] is to achieve UEP by dividing the de- same (channel) bit error probability. The bit assignment will Authorized licensed use limited to: Werner Henkel. Downloaded on July 02,2010 at 18:43:04 UTC from IEEE Xplore. Restrictions apply.
  3. 3. 804 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 3, MARCH 2010 C1 011 0 1 M1 1 0 1 0 UEP− 010 0 1 1 0 001 u i ∈ Ck 1 0 REMUX s 1 0 1 0 k = 1, . . . , Nc − 1 LDPC CNc −1 MNs C Nc 11 00 11 00 0 000 1 11 00 1 0 1 0 110Fig. 1. Schematic description of the proposed scheme. The source bits areencoded by a UEP-LDPC code and the coded bits are assigned to modulation 1 0 1 0 1 0classes before modulation. 1 0 1 0 100 111 11 00 1 0 1 0 11 00 1 0 1 0 11 00 1 0 d3 d2 d1 101be described in Section III-A. In the following, we assume anAWGN channel with noise variance 2 . Fig. 2. 8-PSK modulation with Gray labeling.B. Modulation assume their behavior being equal to BPSK signaling. From Let us assume a modulation scheme with = 2 the known bit-error probabilities of the binary componentsymbols, labeled by binary vectors d = ( , . . . 2 , 1 ). In channels, it is easy to determine their equivalent noiseorder to design codes for HOCs, we investigate the error 2 variances which are required for density evolution laterprobabilities of the individual bit positions in the symbol. De- on. We define the noise vector 2 = [1 . . . ] to be a 2 2pending on the signal constellation and the labeling strategy, vector that contains the equivalent noise variances for eachthe error probabilities of the individual bit positions may be separate bit-error rate, i.e., for each modulation class, ordereddetermined. with the lowest variance first. We assume that there are One may use traditional constellations like 8-PSK or 64- distinct equivalent noise variances, where ≤ .QAM as well as non-uniform constellations, so-called hier-archical modulation [26] or multiresolution modulation [25]. Example 2.1 (8-PSK): 8-PSK modulation with Gray label-When UEP is desired, non-uniform constellations have the ing is shown in Fig. 2. Considering only neighboring signaladvantage that they can provide greater differentiation in error points, the approximate symbol-error rate expression for thisprobabilities of the bit positions in the symbol than traditional modulation is given as [30],constellations. Furthermore, the design methods in the above (√ )references allow for a controllable amount of UEP through ,8− = 2 6 /0 sin , (1) 8variations of the distances between the signal points. where (⋅) is the Gaussian probability Q function. The Generally, one can approximate the symbol-error probabil- ˜ average bit-error rate is ≈ / log2 ( ).ity as well as the bit-error probabilities ,1 . . . , The expressions for the bit-error probabilities of the indi-of a constellation using the union bound. For Gray labeling, vidual bits in the symbol arethe average bit-error probability is often assumed to be equal (√ ˜for all bit positions, i.e., , ≈ log 1 ⋅ . However, it is ) 2 () ,1 ≈ 6 /0 sin , (2)important to be aware of the different bit-error probabilities 8 1 (√ )when designing codes for HOCs. In this work we only ,2 = ,3 ≈ 6 /0 sin . (3)consider Gray labeling, since the bit-errors have low statistical 2 8dependencies and can thus be assumed to be independent Note that the above expressions are obtained by applying[27]. This is very important for the message-passing decoder. approximations. The approximations are assumed to beHowever, for labelings other than Gray, the differences in BER appropriate for our purposes, but can be replaced by moremay be more significant. If such labelings are of interest, we exact formulas. □suggest the use of the improved decoding algorithm proposedin [28] that combines decoding and demodulation and takes The equivalent noise variances of each modulation classinto account the statistical dependencies among bit errors may be determined from the different bit-error probabilitiesoriginating in the same symbol. by means of an equivalent BPSK channel, In [5], it was stated that a symmetric channel with 1 2an input alphabet of size 2 can be represented by = ( )2 . (4) −1 ( , )equivalent binary-input channels. Depending on the labeling,the equivalent channels may not be symmetric. Since the Usually, LDPC codes are designed for one noise variancesymmetry of the channel is an important property for the and it is assumed that all bits are transmitted over thedensity evolution of LDPC code ensembles, we use a method corresponding channel. In this paper we design codes using thecalled i.i.d. channel adapters [29]. In order to force symmetric equivalent noise variances from (4). To compare the standardcomponent channels, i.i.d. binary vectors t = [1 . . . ] are code design with our new design in a fair way, we calculate an 2added modulo-2 to the codeword. At the receiver, the equivalent average BPSK noise variance, denoted by .log-likelihood ratio is multiplied by −1 if = 1. Since This is the noise variance of a BPSK channel that would give athe equivalent component codes are now symmetric, one can bit-error probability equal to the average bit-error probability Authorized licensed use limited to: Werner Henkel. Downloaded on July 02,2010 at 18:43:04 UTC from IEEE Xplore. Restrictions apply.
  4. 4. SANDBERG and VON DEETZEN: DESIGN OF BANDWIDTH-EFFICIENT UNEQUAL ERROR PROTECTION LDPC CODES 805 d1 III. UEP-LDPC C ODES FOR H IGHER O RDER ˜ 2 Pb ∼ σBP SK C ONSTELLATIONS d2 A. Optimization of the Degree Distribution for HOCs d3 The mutual information messages from a check node to a a) Standard code design variable node ( ) and from a variable node to a check node ( ) at iteration , computed by means of density evolution 2 using the Gaussian approximation [11], are given by Pb,d1 ∼ σ1 d1 ∑ (−1) = 1 − (( − 1) −1 (1 − (−1) )) , (5) 2 Pb,d2 ∼ σ2 =2 d2 ∑ ∑ ∑ 2 2 Pb,d3 ∼ σ3 () = , ( 2 + ( − 1) −1 (−1) ( )) , (6) d3 =1 =1 =2 b) Design for HOCs with (⋅) computing the mutual information = () by () = 1 − {log2 (1 + − )} (7)Fig. 3. Channel assumptions for a) standard code design b) design for HOCs. ∫ 1 − (−)2 − 4 = 1− √ log2 (1 + ) ⋅ 4 ℝof the HOC. Thus, we compare our new design for HOCs for a consistent Gaussian random variable ∼ (, 2) 2with a standard code design with and use both with with mean and variance 2. In the special case with onlyHOCs. Fig. 3 shows a schematic explanation of the channel one protection class and one modulation class (correspondingassumptions made by the receiver. to standard LDPC code design), i.e., = = 1, the update rule for the messages from variable nodes to check nodes is In the following, we claim that approximating the HOC given bychannel by equivalent BPSK channels together with i.i.d.channel adapters meets our requirements. ∑ 2 () = ( + ( − 1) −1 ((−1) )) . (8) =2 2 The update rule for the messages from check nodes to variableC. Notations nodes is not affected by the division of the variable node degree distribution into sub-distributions since the check node We consider a UEP-LDPC code with protection degree distribution is constant for all protection classes andclasses. The proportions of each information class, = modulation classes. Equations (5) and (6) can be combined to[1 , . . . , −1 ], are given by the normalized lengths of yield the mutual information evolution of the LDPC codeeach class corresponding to the information bits. equals the () = (, , 2 , (−1) ) . (9) number of bits belonging to protection class divided by the () (−1) (−1)total number of information bits . The proportion distribution If for any , then and describeof the bits in the codeword belonging to the protection classes a code ensemble for which density evolution converges foris thus given by p = [1 , . . . , −1 , (1 − )]. the noise variance vector 2 . Since the variable node degreeis the number of different bit-error rates for the bits in a distributions give proportions of edges connected to variablesymbol. We will describe the bits with a distinct bit-error rate nodes of certain degrees, the constraintas belonging to one modulation class , = 1, . . . , . ∑ ∑ ∑ = [1 , . . . , ] defines the proportion of bits in the code- , = 1 (10) word that belongs to each modulation class. =1 =1 =2 The vector contains the overall variable node de- must be fulfilled.gree distribution, both for different protection classes When optimizing a degree distribution, there are differentand different modulation classes. Let , be the pro- strategies. One could for example maximize the code rateportion of edges connected to variable nodes of de- given a certain threshold, where the threshold is defined as thegree that belong to modulation class and pro- lowest /0 for which density evolution converges. In thistection class . Define = [ ,2 , . . . , , ] paper, we choose to minimize the threshold given a fixed code [ ] 1 1 rate. In order to do so, we rerun a linear programming routineand = 1 , . . . , 1 , . . . , , . . . , , while increasing the SNR, until the SNR is high enough sowhere (⋅) denotes the transpose. is a column vector that it is possible to find a degree distribution for which densityof length − 1 and is a column vector of length evolution converges.( − 1) . The vector = [2 , . . . , ] de- The aim of the code design is to reduce the BERs of thescribes the check node degree distribution. We define 1 to protection classes by taking the different error probabilitiesbe an all-ones vector of appropriate length. of the modulation levels into account. The design algorithm Authorized licensed use limited to: Werner Henkel. Downloaded on July 02,2010 at 18:43:04 UTC from IEEE Xplore. Restrictions apply.
  5. 5. 806 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 3, MARCH 2010should also give the possibility to trade overall BER for UEP variable node degree as in [22].capability. The natural way of assigning bits from modulation Repeated optimization with the target function (11) resultsclasses to protection classes to achieve UEP, is to use the in a degree distribution with UEP capability. However, thisbest protected bits from the modulation, that is, modulation target function does not account for the fact that variableclass 1 , for protection class 1 and continue like that until nodes connected to the first modulation class have lowerall bits have been assigned to a protection class. However, error probability after demodulation than those from worsethis assignment is not guaranteed to give a degree distribution modulation classes. Therefore, we introduce a scaling factorwith the lowest possible threshold. As discussed later, there for each modulation class which decreases with increasingis always a trade-off between a low threshold and good modulation class index, i.e., 1 2 . . . 0.UEP capability. By distinguishing not only between degree distributions of different protection classes but also of different ∑ ∑ max , (12)modulation classes, linear programming may be used to assign =1 =2bits from the modulation classes to the protection classes. It is well-known that a higher connectivity of a variable The choice of this scaling factor affects how valuable a betternode leads to better protection. Thus, the optimization target modulation class is compared to a modulation class withis to find a variable node degree distribution for the whole code higher equivalent noise variance. We choose = − + 1.that maximizes the average variable node degree of the class This choice of has the effect that the linear programmingbeing optimized. UEP capability may be obtained by running algorithm, if possible while fulfilling all constraints, will usethe optimization algorithm sequentially, one protection class at modulation classes with low noise variance for the protectiona time, for an /0 slightly higher than the threshold. When class being optimized.the proportion of edges in the Tanner graph that are associated Besides the optimization constraints given in (9) and (10),with a specific class is high, many messages will be passed there are some more constraints which have to be fulfilledand the local convergence of the graph is fast. However, in the by the optimized . The variable node degree distribution islimit when the number of decoder iterations tends to infinity, connected to the check node degree distribution and the codeall messages have transited the whole graph and there should rate by ∑be no UEP capability left, [22]. This implies that the UEP =2 / = 1 − ∑ . (13)capability theoretically should be decreasing with increasing / =2number of iterations. However, our results show that for areasonable number of iterations the UEP capability is not Furthermore, the proportion vectors and impose theaffected much by the number of decoder iterations. It has following two constraints on , where and denotebeen shown that the choice of code construction algorithm the total number of variable nodes belonging to protectionis critical for remaining UEP capability after a reasonable class and to modulation class , respectively,number of iterations, [31]. Figure 8 in Section IV shows the = ⋅ ⋅ = 1, . . . , − 1 , and (14)BER as a function of the number of decoder iterations for the = ⋅ = 1, . . . , . (15)code design and code construction algorithm suggested in thispaper. Moreover, and are connected to by The target function for protection class can be formu- ⎛ ⎞lated as ∑ ∑ , ∑ ∑ ∑ = ⋅ ⋅ (1 − )/ ⎝ ⎠ (16) max , . (11) =1 =2 =2 =1 =2 andThis target function only concerns the bits of the current ⎞ ⎛protection class. The optimization constraints, such as the con- ∑ ∑ , ∑ = ⋅ ⋅ (1 − )/ ⎝ ⎠ , (17)vergence constraint (9) or the proportion distribution constraint =1 =2 =2(10), will involve the whole set of code bits. In every step, thedegree distributions of lower protection class bits are fixed and respectively.only degree distributions of higher protection classes may be When designing good LDPC code ensembles, the stabilitychanged, [22]. The target function will ensure a maximized condition which ensures convergence of the density evolutionaverage variable node degree, since a high value of the above for mutual information close to one should be fulfilled [7].sum implies that a large proportion of the edges is connected The stability condition gives an upper bound on the numberto variable nodes belonging to class . In [22], the authors of degree-2 variable nodes. For a BPSK scheme, where allshow that it is not only the average connection degree to bits are affected by the same noise variance 2 , we have [7]be maximized but also the minimum variable node degree. ∫ 1 1Therefore, the second optimization target of our code design − = 0 ()− 2 = − 22 (18) ′ (0)′ (1) ℝis the maximization of the minimum variable node degreeof each protection class. Therefore, the optimization is first with 0 () being the message density corresponding toperformed for a high minimum variable node degree. If no the received values and ′ () and ′ () being the deriva-degree distribution for which density evolution converges can tives of the degree polynomials. It is straightforward to ∑ ∑ be found, the optimization is repeated for a lower minimum see that, for our scheme, ′ (0) = =1 =1 ,2 and Authorized licensed use limited to: Werner Henkel. Downloaded on July 02,2010 at 18:43:04 UTC from IEEE Xplore. Restrictions apply.
  6. 6. SANDBERG and VON DEETZEN: DESIGN OF BANDWIDTH-EFFICIENT UNEQUAL ERROR PROTECTION LDPC CODES 807 ∑ ()′ (1) = =2 ⋅ ( − 1). In our case, the bits are af- 1) Initialization = fected by channel noise with different equivalent noise vari- 2) While optimization failure 2ances (see (4)) and thus different densities 0, (). The a) Optimizedensity of the whole set of bits is equivalent to the (weighted) ∑ ∑ ∑sum of the individual densities, i.e., ⋅0, (). This gives max , (20) the stability condition =1 =2 ∫ ∑ ∑ under the constraints [1 ] − [6 ]. − 12 − = ⋅ 0, ()− 2 = ⋅ 2 . (19) [1 ] Rate constraint, see (13) ℝ =1 =1 ∑ ∑ ∑ , 1 ∑ All the above given constraints are rewritten to contain only = (21), , , , 2 , and , i.e., without using and . This makes =1 =1 =2 1 − =2 the algorithm independent of the code length. [2 ] Proportion distribution constraintsB. Optimization Algorithm i) See (10) The optimization algorithm proposed here is a modification ∑∑ of the hierarchical optimization algorithm presented in [22] 1 = 1 (22)for HOCs. The optimization is performed at /0 = + , =1 =1which will be the threshold of the optimized code, where is ii) ∀ ∈ {1, . . . , − 1}, see (14) and (16)the lowest possible threshold in dB for the given and ,and is an offset from the lowest threshold that provides more ∑ ∑ , ∑ flexibility in the choice of . An irregular LDPC code always = (23) =1 =2 1 − =2 provides some inherent UEP capability. This is achieved bymapping higher degree variable nodes to the more protected iii) ∀ ∈ {1, . . . , − 1}, see (15) and (17)classes. For = 0 this is the only UEP capability available.On the other hand, by introducing 0, we allow for more ∑ , ∑ 1 ∑ = (24)freedom in the code design. The LP algorithm will assign 1 − =2 =1 =2higher degrees or larger fractions of edges to the high degreesof the first protection class and accept worse properties for [3 ] Convergence constraint, see (9)the less important classes. Hence, 0 leads to increased (, , 2 , ) (25)UEP capability compared to the inherent UEP capability ofthe ’minimum threshold code’. [4 ] Stability condition, see (18) and (19) The algorithm can be divided into an inner and an outer ⎡ ⎤−1 ∑∑ ∑ 2 ∑loop. For a given /0 , the outer loop runs over the ,2 ⎣ −1/2 ⋅ ( − 1)⎦protection classes, starting with the first. At this point, the =1 =1 =1 =2degree distribution of the code is designed while optimizing (26)the corresponding protection class. As mentioned before, there [5 ] Minimum variable node degree constraintare two target functions to be maximized, i.e., the average ∀ () , ∀ : , = 0 (27) connection degree and the minimum connection degree ofthe class’ variable nodes. Since we are using LP with only a [6 ] Previous optimization constraintssingle-objective function, we choose it to be the maximization of the average connection degree. The maximization of the ∀ ′ , ∀ : ′ is fixed (28)minimum variable node degree is performed by the inner loop () () b) If failure, = − 1which runs over different values for the minimum degree,starting from some maximum value and successively reducing End (While)it during the procedure. Once a valid solution is found forthe current protection class, the next protection class is C. Code Constructionoptimized. At this point, the classes 1 , . . . , −1 have When the optimal degree distribution of the variable nodes already been optimized and 1 , ..., −1 , ∀, are fixed. is found, a parity-check matrix is constructed by a modifica-The optimization algorithm can be stated as follows. tion of the approximate cycle extrinsic (ACE) message degree I) Fix /0 = + and calculate 2 . algorithm [32]. The ACE algorithm constructs a parity-check II) Find by performing the inner optimization loop for matrix following a given variable node degree distribution each protection class. while selectively avoiding small cycle clusters that are isolated For the optimization of class , a linear-programming from the rest of the graph. For irregular LDPC codes, the ACEroutine is executed. It requires the definition of the check node algorithm has good performance in the error-floor region.degree distribution , /0 = + , the maximum variable The original ACE algorithm [32] only ensures a certainnode degree , the code rate , and the proportion variable node degree distribution. However, the design algo-vectors and . rithm optimizes the variable node degree distribution given a certain check node degree distribution and a parity-check Authorized licensed use limited to: Werner Henkel. Downloaded on July 02,2010 at 18:43:04 UTC from IEEE Xplore. Restrictions apply.
  7. 7. 808 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 3, MARCH 2010 TABLE I TABLE II D EGREE DISTRIBUTIONS OF THE UEP SCHEME . D EGREE DISTRIBUTIONS OF THE HOC-UEP SCHEME . 1 2 3 1 2 3 = 0 dB 7 = 0.0799 3 = 0.1790 2 = 0.2103 = 0 dB 1 9 = 0.1703 3 = 0.1673 2 = 0.1240 8 = 0.0948 6 = 0.0737 3 = 0.0181 10 = 0.0555 30 = 0.3029 7 = 0.0414 30 = 0.1811 = 0.1 dB 11 = 0.1783 3 = 0.2041 2 = 0.1841 2 30 = 0.0854 4 = 0.0225 2 = 0.0878 12 = 0.1184 4 = 0.0393 3 = 0.0575 5 = 0.0738 3 = 0.0022 30 = 0.2183 7 = 0.0117 4 = 0.0183 = 0.1 dB 1 12 = 0.3290 3 = 0.1070 2 = 0.1585 30 = 0.1782 2 3 = 0.0396 2 = 0.0547 4 = 0.1026 3 = 0.0137matrix with degrees given by and is desired. The modified 5 = 0.0165ACE construction of the parity-check matrix used here also = 0.2 dB 1 15 = 0.4840 3 = 0.0848 2 = 0.1733ensures that the check node degree distribution equals . 30 = 0.0327 2 3 = 0.0782 2 = 0.0414Whereas the ones in a column are located at random positions 4 = 0.0940 3 = 0.0115in the original ACE algorithm, we allow only those positions = 0.3 dB 1 16 = 0.4993 3 = 0.0268 2 = 0.2163where a one does not violate the defined check node degree 2 16 = 0.0345 3 = 0.1849 2 = 0.0092 4 = 0.0291distribution. Generally, the recently proposed ACE constrainedprogressive edge growth (PEG) code construction algorithm TABLE III HOC DESIGN WITHOUT UEP SIMILAR TO [14].[33] has been shown to perform well compared to the ACEalgorithm, especially in the error-floor region. Despite this Information bits Parity bits = 0 dB 1 3 = 0.1474 4 = 0.0008 2 = 0.1237good performance, it is unsuitable for UEP applications, 5 = 0.0021 6 = 0.0008 3 = 0.0154since it entirely looses its UEP capability after a few decoder 7 = 0.0023 8 = 0.0005iterations. 9 = 0.2204 10 = 0.0040 11 = 0.0008 12 = 0.0004 13 = 0.0002 14 = 0.0002 15−25 = 0.0013 26 = 0.0002 IV. S IMULATION R ESULTS 27 = 0.0003 28 = 0.0004 29 = 0.0011 30 = 0.1751 In this section, simulation results for examples with 8- 2 3 = 0.0019 4 = 0.0509 2 = 0.0881PSK and 64-QAM are presented. We denote our scheme by 5 = 0.0581 6 = 0.0049 3 = 0.0005 7 = 0.0008 8 = 0.0057HOC-UEP, which is a UEP-LDPC code optimized for the 9 = 0.0017 10 = 0.0006 2different from the modulation. We compare the HOC-UEP 11 = 0.0003 12 = 0.0002scheme to two other schemes. The first one is a UEP-LDPC 13 = 0.0001 14 = 0.0001 15−25 = 0.0008 26 = 0.0001code optimized for BPSK [22] which is denoted by “UEP”, 27 = 0.0002 28 = 0.0003designs the code for a BPSK channel with the comparable 29 = 0.0005 30 = 0.0869 2 (see Section II-B) and assigns the best bits from themodulation to the first protection class and vice versa. Thesecond comparison is a design without UEP similar to the Finite-length codeword simulations with = 4096 and 50design proposed in [14]. The variable node degree distributions decoding iterations are performed using the equivalent BPSKare optimized for = 1/2, = 3, = [0.3, 0.7], channels. A soft demapper provides the message passing = 30, and () = 0.007497 +0.991018 +0.001509, decoder with the channel log-likelihood ratios (LLRs) whichwhich is found by numerical optimization in [7] to be a good 2 are computed using the appropriate noise variances of thecheck node degree distribution for = 30. modulation classes. Fig. 4 shows the overall BER after 50 decoder iterationsA. Results for 8-PSK averaged over all protection classes for both = 0 dB and For Gray-labeled 8-PSK, the noise vector 2 is calculated = 0.1 dB. It is seen that the overall BERs of the HOC-UEPaccording to (4), with = 2 and = [2/3, 1/3]. Table I codes are lower than for the UEP codes. For an overall BERshows the degree distributions of the UEP scheme. We choose of 10−3 , there is a gain of around 0.8 dB by the HOC-UEP 2 = 0.1 dB to allow for some increased UEP capability. scheme. This is due to the consideration of the different 2The resulting degree distributions are given for each in contrast to only assuming the average . The overallprotection class . For comparison, the degree distribution BER for the design similar to the approach in [14] (HOCfor the minimum threshold, that is = 0 dB, is also shown. without UEP) is shown for comparison, even though thisThis degree distribution corresponds to the code design with scheme has no UEP capability. It is expected that the overallinherent UEP only. The degree distributions of the HOC-UEP BERs for the codes with 0 dB are higher than for thescheme, , are given in Table II. corresponding codes with = 0 dB, because the thresholds of For comparison, we also optimize a code for HOCs but the codes are increased in order to allow an increased averagewithout UEP capability, which is similar to the approach variable node degree of the most protected classes. However,in [14]. Note that we separate the degree distributions for at high /0 the HOC-UEP scheme shows a slightly lowerinformation bits and parity bits, which is not done in [14]. overall BER for the case with = 0.1 dB compared to = 0The degree distributions are shown in Table III. dB. Reasons for this are discussed later in this section. Authorized licensed use limited to: Werner Henkel. Downloaded on July 02,2010 at 18:43:04 UTC from IEEE Xplore. Restrictions apply.
  8. 8. SANDBERG and VON DEETZEN: DESIGN OF BANDWIDTH-EFFICIENT UNEQUAL ERROR PROTECTION LDPC CODES 809 −1 −1 10 10 HOC−UEP C ε=0 dB 1 HOC−UEP C ε=0 dB −2 −2 2 10 10 HOC−UEP C ε=0.1 dB 1 HOC−UEP C ε=0.1 dB −3 −3 2 10 10Overall BER BER −4 −4 10 10 −5 −5 10 UEP ε=0 dB 10 UEP ε=0.1 dB −6 HOC−UEP ε=0 dB −6 10 10 HOC−UEP ε=0.1 dB Approach similar to [14] −7 −7 10 10 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 Eb/N0 (dB) E /N (dB) b 0 Fig. 4. Overall bit-error rate performance of the UEP scheme and the HOC- Fig. 6. Bit-error rate performance of protection class 1 and 2 for the UEP scheme for = 0 dB and = 0.1 dB. HOC-UEP scheme and 8-PSK. −1 10 at high /0 . Because of the large difference in BER of the −2 protection classes and also the knowledge of the proportion of 10 bits in the classes, we know that it is mainly 2 that governs the performance in terms of the overall BER. By comparing 10 −3 also with Fig. 4, it is seen that the overall BER of the approach similar to [14] without UEP is almost equal to the overall BER of the HOC-UEP scheme with = 0 dB and worse than theBER −4 10 HOC-UEP scheme with = 0.1 dB. When comparing these schemes, remember also that the HOC-UEP scheme provides 10 −5 UEP C1 ε=0 dB a BER significantly lower than the overall BER for 1 . UEP C2 ε=0 dB Fig. 7 shows the effect of different threshold offsets in the −6 UEP C ε=0.1 dB HOC-UEP scheme, with BER as a function of for /0 = 10 1 UEP C ε=0.1 dB 2.4 dB. It shows that = 0.1 dB is a good choice for our 2 example, since the BERs for both classes are lower than for −7 10 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 code design without increased UEP capability ( = 0 dB). Eb/N0 (dB) For higher values of , the first class is assigned even more edges and the second and third classes have concentrated low Fig. 5. Bit-error rate performance of protection class 1 and 2 for the degrees. With increasing , which implies increasing the code UEP scheme and 8-PSK. threshold, the global convergence of the code becomes worse and this affects the performance of all classes. Please note that The BER performance of the individual protection classes for short-length codes, the degree distribution with the lowest 1 and 2 for the UEP scheme are shown in Fig. 5. Note that threshold may not yield the best performance [7]. we do not show protection class 3 because it contains parity Fig. 8 shows the BER of the HOC-UEP scheme as a bits only and is of little interest for possible applications. As function of the number of decoder iterations for /0 = 2.4 expected, the UEP capability is increased with increasing . dB. The UEP capability is not affected much with increasing Fig. 6 shows the BER performance of protection classes number of decoder iterations, at least up to a maximum of 1 and 2 for the HOC-UEP scheme. The results show that 200 iterations. the HOC-UEP = 0.1 dB code has lower BER for both information classes than the HOC-UEP = 0 dB code. Note B. Results for 64-QAM that the differences in BER are partly balanced by protection In this section, we give an example of a UEP-LDPC code class 3 which is not shown here. By comparing Fig. 5 and designed for and used with 64-QAM. We design a code with Fig. 6, we see that the BERs of the HOC-UEP scheme are the parameters , , , , , , and as in the previous much lower than those of the UEP scheme at high /0 . section. 64-QAM yields three modulation classes of equal size, The gain of the HOC-UEP scheme compared to the UEP and thus, = 3 and = [1/3, 1/3, 1/3]. Fig. 9 shows scheme is 0.8 dB for protection classes 1 and 2 , at a BER of the BER performance of protection classes 1 and 2 of the 10−6 and 10−4 , respectively. We also see that the difference 64-QAM HOC-UEP code compared to a UEP code designed in performance between the protection classes is higher for for BPSK, both with = 0.1 dB and transmitted over the the HOC-UEP scheme than for the UEP scheme, especially 64-QAM modulated channel. The plot shows that for very Authorized licensed use limited to: Werner Henkel. Downloaded on July 02,2010 at 18:43:04 UTC from IEEE Xplore. Restrictions apply.