Irregular Designs for Two-State Systematic with Serial Concatenated Parity Codes

rregular Designs for Two-State Systematic with Serial Concatenated Parity Codes Jordan Melzer and Keith M. Chugg Electrical Engineering Department, University of Southern California Los Angeles, California 90089-2565 Email: {jmelzer, chugg}@usc.edu Abstract The rregular Repeat Accumulate (RA) codes introduced by Jin et al. [1] are structured Low Density Parity Check (LDPC) codes that are known to perform as well as the best unstructured LDPCs. RAs reach rates that are quite close to capacity on the additive white Gaussian noise (AWGN) channel, can be both encoded and decoded in linear time, are perceived to be low complexity codes, and have been adopted into recent standards. Systematic with Serially Concatenated Parity (S-SCP) codes were recently introduced in [2], [3] as a class of turbo like codes with members that exhibit good performance over a wide range of block sizes, code rates, and target error probabilities. n this paper we introduce irregular designs for Systematic with Serially Concatenated Parity (r-s-scp) codes and show that both for near-capacity and more practical lower-complexity designs, irregular two-state S-SCP codes can be designed to achieve the performance of RA codes at lower complexity.. NTRODUCTON Repeat Accumulate (RA) codes [4] and rregular Repeat Accumulate (RA) codes [1] are a code family that can be viewed either as structured Low Density Parity Check (LDPC) codes [5] or as concatenated codes. RA codes are known to perform as well as the best unstructured LDPC codes, reaching rates approaching capacity on the Additive White Gaussian Noise (AWGN) channel. They are perceived to be low-complexity, offer linear-time encoding, and have been adopted into recent standards, most notably the DVB- S2 digital television standard [6]. Systematic with Serially Concatenated Parity (S-SCP) codes were introduced in [2], [3] as a class of codes with universally good performance. For these codes, a single decoder can provide performance close to theoretical limits over a wide range of block sizes, code rates, and desired frame error rates. rregular design [7] is known to lower the E b /N 0 threshold the lowest E b /N 0 that will give perfect error correction in the limit of infinite block size and an infinite number of iterations and has been proposed as a way to increase the coding gain of a range of turbo-like codes. The improvement in performance that irregular design allows comes at the cost of extra computation. For a fixed decoding complexity, the code with the lowest threshold usually does not give the best performance. n employing irregular design, it is more useful to find a good trade-off between complexity and performance than to create a code with the best asymptotic performance. Despite its importance, this trade-off is not frequently explored in the literature. n this paper we continue the design of S-SCP codes begun in [3]. No codes were presented in [3] with r < 1 2 and many codes with superior r = 1/2 performance, though higher complexity, have been published. n this paper we extend the design of S-SCP codes to low rates and demonstrate designs that offer an attractive mix of low-complexity and high-performance. rregular design was suggested in [3] as a method to increase the coding gain of S-SCP codes at the expense of complexity. Here we present design techniques for irregular S-SCP (r-s- SCP) codes aimed at producing codes with good performance vs. complexity trade-offs and demonstrate them in designing r-s-scp codes with structures similar to those of [3] and RA codes. We show that, both for large and moderate block-size codes, r-s-scp codes achieve lower block error rates than RA codes with similar complexities. A second way to achieve larger coding gains, one that also usually increases the per-bit decoding complexity, is to reduce the rate of a code. We illustrate low-rate designs for r-s- SCP codes as well as an incremental redundancy scheme that exhibits performance vs. complexity tradeoffs very close to those of r-s-scp point designs. n section we describe irregular S-SCP codes, in section we discuss optimization techniques for r-s-scps based on EXT (EXtrinsic nformation Transfer) charts, and in section V we compare the complexity vs. performance tradeoffs of RA and r-s-scp codes. n section V we introduce low-rate r-s-scp codes and an incremental redundancy scheme for which the complexity and performance of the code are, at each rate, similar to those of an optimized r-s-scp point design.. RREGULAR S-SCPS An irregular Systematic with Serial Concatenated Parity (r-s-scp) code is an S-SCP code with varying inner or outer codes. For many modern codes, irregularity has been successfully used to reduce E b /N 0 requirements to quite near the constrained capacity [8] [1] [9]. Heuristic optimization techniques based on density evolution [8] or EXT charts [10] are used as an aid to irregular design. These methods simulate the performance of the code in the limit of infinite block size and, usually, infinite iterations. n this paper we limit our attention to r-s-scp codes with an inner code comprising a punctured accumulator (G(D) = 1 1+D ) and an irregular outer code comprising a duo-binary

(a) General S-SCP structure b d Q Q/J P = kq/j j pm i outer nner Parity parity bits code Generator (PG) kq bits k systematic bits b i (b) rregular Repeat Accumulate (c) Example 2-state r-s-scp b i 1+D = Q(i) P/S d j PG = d Accumulator b P/S d j P = kq/j pm i repetition parity bits code 1/(1+D) J:1 (=) Q(i) k systematic bits b i PG = d Accumulator 1/(1+D) J:1 P = kq/j parity bits p m k systematic bits Fig. 1. (a) General r-s-scp code (b) RA code (c) Example 2-state r-s-scp code encoder (G(D) = 1 + D) with outputs that are repeated a variable number of times. We use this example because it is simple and similar in structure to the Flexible LDPC S- SCP code presented in [3]: a two-state regular S-SCP code with the inner and outer code polynomials of our example code and outer code outputs repeated twice. Similar two-state S-SCP codes are the Accumulate Repeat Accumulate codes of [11]. The design methods presented here are general, and other designs may prove to have superior properties. Figure 1 illustrates the general r-s-scp code structure, the structure of the two-state r-s-scp codes considered in this paper, and the structure of the systematic RA codes to which our r-s-scp codes are compared. Note that, though RA codes may be seen as a sub-class of S-SCP codes with G(D) = 1 outer codes, in this paper we refer to example codes with outer G(D) = 1+D by S-SCP and outer G(D) = 1 by RA. We call the number of times that a bit i from the outer code is repeated Q(i), and the average repetition factor for a code Q. Much of our design work will be aimed at producing distributions on Q with desirable properties. We call the rate of the puncturer after the inner code J, and calculate it as the ratio between the number of output bits from the outer code and the number of parity bits. The rate of such an r-s-scp is r = J J+Q.. EXT CHART BASED OPTMZATON OF R-S-SCP CODES Because of the similarity between S-SCP and seriallyconcatenated codes, the optimization heuristics of [9] and [12] for irregular serially-concatenated codes may be used, with slight modification, to design irregular S-SCP codes. Both methods rely on EXT (EXtrinsic nformation Transfer) curves [10] characterizations of component codes based on the single variable of mutual information between the extrinsic likelihoods and their true values for the bits passing b i through the interleaver of a concatenated code. When applied to irregular designs, these methods exploit the property that the EXT curve of a mixture of codes is (excluding interaction through trellis state at points where the code used is varied) a linear mixture of the EXT curves of the individual codes. n both methods a fixed inner code is assumed and optimization is performed by using an EXT chart of the EXT curve of the inner code and the EXT curve of mixtures of outer codes to predict performance. n particular, for a code to be decodable the EXT curve of the inner code must lie above the EXT curve of the outer code. For a serial code, only the inner code is directly in contact with the channel, so the EXT charts of the outer code are not a function of channel SNR. With S-SCPs, both inner and outer codes are directly in contact with the channel, requiring that EXT curves for both codes be created at each signal-to-noise ratio (SNR) of interest. n [9], the optimization proceeds as follows. An initial mixture of codes expressed as a vector of positive fractions of coded bits from each code is repeatedly projected onto constraints on the rate of the outer code and the existence of a decoding tunnel between the EXT curves of the two codes. f a code mixture is found that satisfies those constraints, the technique uses a gradient search to find the curve with lowest mean-square distance between the EXT curves which satisfies the constraints. f no code mixture is found that satisfies the constraints, the code is determined to be below threshold. n [12], an optimization heuristic for fixed-iteration performance is described. Here the constraint requiring the existence of a decoding tunnel is relaxed and a gradient search is performed for the code with the largest mutual information after a fixed number of iterations on the EXT chart. We opted to perform the optimization for finite iterations using the Nelder-Mead Simplex non-gradient method [13]. n our formulation of this problem as a standard simplex optimization, we chose the dimensions of the simplex as the smallest number needed to meet the rate constraint: the number of individual or even mixtures of two outer codes (expressed as pairs of positive weights that sum to one) that have the desired rate at the interleaver. The goal of the optimization is, then, to select a unit-sum, positive vector in this space with the largest mutual-information. To meet the constraint that the selected fractions sum to one, we exclude one dimension from the optimization. Using an objective function that returns less than zero when the inner and outer curves meet and the mutual information on the log-likelihood ratios at the end of the iteration otherwise forms the problem as a standard Nelder Mead Simplex maximization. This method can be easily extended to designs with irregular inner codes. V. COMPLEXTY VS. PERFORMANCE TRADE-OFF OF RA AND R-S-SCP CODES terative decoders for turbo-like codes are sub-optimal and exhibit a trade-off between computational complexity and error-rate. n this section we compare the complexity vs. performance trade-offs of several RA and r-s-scp code designs. We express this trade-off in terms of block error rate

(BLER) and 2-state trellis sections, and give operation counts for each design that justify the use of trellis sections as a measure of complexity. Following [3, section V], the computation done in the decoder for RA and two-state S-SCP codes on each iteration is expressed as g( ) and addition operations, where 1 g(x, y) = min (x, y) min (0, x + y) (1) and min is a log-sum: min (x, y) = log[exp(x) + exp(y)] = min(x, y) ln[1 + exp( x y )] n table V, we list this per-decoded-bit, per-iteration complexity as {C g, C + } for the codes considered in this section. For these codes, each two-state trellis-section requires three g( ) operations and 2 2.5 additions. Since the complexity of the g( ) operation, though architecture dependent, is several times that of an addition, the average complexity per trellis section of each of the codes is almost identical, justifying the use of trellis sections as a clear, single-dimensional measure of complexity. The complexity in two-state trellis sections per decoded bit is listed as C T S in table V. For RA codes, C T S = Q while for our example S-SCP codes C T S = Q + 1. For both codes, smaller Q gives lower per-iteration complexity. As demonstrated for our example codes in table V, the typical Q of a good r-s-scp code is smaller than that of a good RA code. For our example S-SCP codes, the outer and inner code are inverse operations: G(D) for the inner code is 1/G(D) for the outer code. Since the soft-in, soft-out (SSO) decoder performs both probabilistic encoding and decoding, the SSOs of a function and its inverse are identical, just with the sense of which variables are coded and which are uncoded switched. Thus, for this example code, the SSO(s) used for decoding the inner code during one half of an iteration may be used to decode the outer code in the other half of an iteration. For some hardware architectures, the area of decoders for this code will scale directly with Q. To demonstrate the low-threshold performance as well as the complexity advantage offered by r-s-scp codes, we first consider a pair of rate-1/2 codes designed to give low thresholds for large block sizes. We compare the RA with average repetition Q = 8 and a minimum degree of three found in [1] with a Q = 4 irregular S-SCP designed for low threshold using the EXT chart method of section and the regular two-state S-SCP code of [3]. The two irregular codes were simulated using repetition profiles of period 1024. n table V, the Q column lists repetition factors while the repetitions column for each code indicates the number of inputs, out of each 1024, that are repeated by each repetition factor. Table V also lists the complexity per input bit for each code in terms of g( ) and + operations as {C g, C + } and in terms of trellissections as C T S. The last row in table V, labelled ( E b N 0 ) 1 For min-sum decoding, changing min*( ) in (1) to min( ) yields the appropriate decoder. (2) Regular Low Low Low Low S-SCP Threshold Complexity Threshold Complexity r-s-scp r-s-scp Jin RA RA Q 2 4 3 8 5 Q repetitions repetitions 2 1024 704 444 0 0 3 0 0 462 685 574 4 0 80 30 0 190 5 0 0 26 0 0 6 0 0 1 0 40 7 0 0 25 0 24 8 0 184 16 0 25 9 0 0 7 0 38 10 0 0 5 0 32 11 0 0 1 57 18 12 0 0 2 221 5 13 0 0 0 0 28 14 0 0 1 0 18 15 0 0 0 0 18 16 0 56 1 0 14 17 0 0 1 0 0 21 0 0 1 0 0 23 0 0 1 0 0 46 0 0 0 35 0 48 0 0 0 26 0 C T S 3 5 4 8 5 C G, C + 9, 7 15, 11.3 12, 9.6 24, 17 15, 11 ( E b N 0 ) db 0.88 0.35 0.69 0.35 0.6 TABLE CODE PARAMETERS FOR FVE RA OR 2-STATE S-SCP CODES db, lists the approximate E b /N 0 threshold of each code, as calculated on an EXT chart. The EXT threshold calculated for both irregular codes is 0.35 db and agrees closely with the threshold obtained via density evolution for the RA code in [1]. This threshold is similar to those for unstructured Low Density Parity Check codes [14]. n figure 2, block error rate (BLER) and bit error rate (BER) vs. E b /N 0 curves are given for the low threshold r-s-scp and RA and the regular S-SCP codes at a complexity of 400 trellis sections per input bit, a large complexity figure equivalent to, respectively, 80, 50, and 133 iterations at a block size k = 16384 input bits. Note that this block size is one of the larger ones used in current communication systems. At this fixed complexity, the r-s-scp shows a 0.05 db gain over the RA and a 0.2 db gain over the regular code. n figure 3, BLER and BER vs complexity curves are given for the same codes and block size at E b /N 0 = 0.9 db. Here the low-threshold r-s-scp code reaches the performance of the low-threshold RA code with approximately 60% of the computational complexity. This is also, roughly, the difference in per-iteration complexity: 5/8. r-s-scp codes also offer complexity advantages for the practical case of moderate complexity and block size. Here design heuristics based on an infinite number of iterations may no longer be appropriate. Using the finite-iteration method of [12] and the technique outlined in section, we designed low-complexity RA and r-s-scp codes targetting rate- 1 2 and k = 1024. The repeat profiles of these codes are listed in

Fig. 2. Large block size BER and BLER vs E b /N 0 for low threshold RA, r-s-scp, and regular S-SCP codes with, respectively, 50, 80, and 133 iterations (400 2 state trellis sections), k = 16384, r = 1 2. Fig. 4. Moderate block size BER and BLER vs E b /N 0 for regular S- SCP, low-complexity S-SCP and RA, and low threshold r-s-scp and RA codes with 33, 25, 20, 20, and 13 iterations (100 2 state trellis sections), k = 1024, r = 1 2. Fig. 3. Large block size BER and BLER vs. computational complexity in 2 state trellis sections for low threshold RA, r-s-scp, and regular S-SCP codes at E b /N 0 = 0.9 db, k = 16384, r = 1 2. Fig. 5. Moderate block size BER and BLER vs complexity for regular S- SCP, low-complexity R-S-SCP and RA and low threshold r-s-scp and RA codes at E b /N 0 = 1.5 db, k = 1024, r = 1 2 table V. BLER and BER vs E b /N 0 for these low-complexity designs as well as for the low threshold designs and the regular code are given at a fixed complexity of 100 trellis sections in figure 4. Here the low-complexity r-s-scp code shows a BLER advantage of approximately 0.05 db over both the low-complexity irregular repeat accumulate code and the lowthreshold irregular S-SCP code and more than 0.3 db over the low-threshold RA code of [1]. Figure 5 gives BLER and BER vs. complexity curves for these codes at E b /N 0 = 1.5 db. Here the low-complexity r-s-scp code shows a computational complexity advantage in BLER of approximately 75% similar to the per-iteration complexity advantage of 4/5 over both the low complexity RA and low-threshold r-s-scp codes and much larger complexity gains over the low threshold RA code. The regular S-SCP gives the lowest BLER until a complexity of 60 trellissections per bit but shows little iteration gain beyond that. V. R-S-SCPS FOR NCREMENTAL REDUNDANCY WTH NCREMENTAL COMPLEXTY ncremental redundancy is a hybrid-arq technique being deployed in current cellular systems. With this method a highrate codeword is first sent. f a receiver fails to decode this codeword, each retransmission is composed of additional bits that, together with all previously transmitted bits, form a codeword of a lower rate code. Here the codewords for higher rates can be seen as punctured versions of codewords from lower rates. An incremental redundancy scheme based on uniform puncturing of parity bits from a systematic mother code has been adopted into CDMA2000 1XEV-DO [15]. With uniform

Q min (i) High Rate 1+D = 1/(1+D) Low Rate Q (i) P/S J:1 P=kQ/J parity bits k systematic bits Fig. 6. Progressive transmission encoder with a two-component interleaver for a G o(d) = 1 + D, G i (D) = 1 r-s-scp family 1+D Bit ndex Pattern a) b) c) xxxo... xoxo... 2048 high rate 4096 low rate xoxooo......xxxo 2048 4096 xxxx......xoxo xxxx......xxxx...xxxx...xoxooo 6144 Fig. 7. Example puncturer for a progressive transmission scheme. x s represent punctured bits, o s transmitted bits, and underlined o s new bits to transmit: (a) r = 2/3, Q = 2, J = 4, 2048 trellis sections (b) r = 1/3, Q = 4, J = 2, 4096 trellis sections (c) r = 1/5, Q = 6, J = 6, 6144 trellis sections puncturing of parity bits from a turbo-like code, regardless of the rate of the code the decoder must process each trellis section of the mother code. Each punctured parity bit allows the decoder to perform two fewer additions per iteration, but this saving is small and may not translate into an area saving or speed increase in a hardware implementation. For S-SCP codes, using uniform puncturing of parity bits to provide incremental redundancy would create a family of codes for which the Q distribution is fixed by the Q distribution of the lowest rate code and the J parameter increases with the rate. Fixing the Q distribution both fixes the per-iteration complexity of the code regardless of the code rate and allows for an optimal design only at one rate. n contrast, we present a technique whereby at any rate an optimal Q and J and a near optimal Q distribution may be selected, allowing for a better performance vs. complexity trade-off over a wide range of rates. As the rate increases, the optimal Q reduces, reducing the size of the interleaver and the number of trellis sections of the inner code. To reduce Q through puncturing requires using puncturing to remove sections from the end of the inner trellis and, with them, the corresponding bits from the end of the codeword. Without any additional structure to the code, reducing the size of the interleaver will quickly tend to produce Q s of 1 and 0, leading to a loss in interleaver gain and poor BER and BLER performance [3, section V]. An interleaver structured to produce a desirable Q distribution for the highest rate codes will produce interleaver gain for all lower rates. Such a structured interleaver for an incremental redundancy code with incremental complexity may be constructed of two or more parallel interleavers. The encoder for such a progressive transmission S-SCP code with two interleaver components is shown in figure 6. Sample puncture patterns for this scheme are shown in figure 7 for three rates 1/5, 1/3, and 2/3 from an example code with k = 1024, component interleavers of 2048 and 4096 bits, and a Q ranging from 6 to 2. The Q distributions of the interleaver and its components are listed in table. The code parameters for each rate are listed in table. Note that in these tables indicates the size of the interleaver which is also the complexity in trellis sections of the inner code. Performance curves for this code at six rates are shown in figure 8. These curves are within 0.2 db of E b /N 0 of those for good r-s-scp point designs. One component of this interleaver is designed for the highest code rate to be considered, and at this highest rate the puncturer selects only bits output from this component. n figure 6, this interleaver component is labelled high rate and the repetition profile on the inputs to it is labelled Q min (i). The puncture pattern used for this high-rate code is listed as (a) in figure 7, and in this example Q min (i) = 2, r = 2/3, and the code has 2048 inner trellis sections. f the r-s- SCP code has been designed to satisfy the design rules in [3, section V], this high-rate interleaver component ensures interleaver gain at all rates. At the lowest rate, the puncturer selects bits from all sections of the composite interleaver. Any interleaver components beyond the highest rate interleaver may be designed such that the Q distribution of the composite interleaver is optimal. For the two-component interleaver in figure 6, the difference distribution Q (i) which describes the inputs to the second interleaver component marked low rate is designed so that Q min (i) + Q (i) is drawn from the optimal Q distribution for the lower rate. pattern (c) in figure 7 illustrates the puncture pattern for the example lowest-rate, a r = 1/5, Q = 6 code which has 6144 inner trellis sections. At intermediate rates, the puncturer operates at the J of the optimal Q distribution for that rate. There should be, at most, a small mismatch between the actual Q distribution and this desired Q distribution. The puncture pattern used for the example intermediate rate r = 1/3 code is listed as (b) in 7. This rate has Q = 4 and 4096 inner trellis sections. This design achieves rate compatible puncturing, so is suitable for an incremental redundancy scheme. At each rate, there is flexibility in the per-iteration complexity that allows a near-optimal trade-off between complexity and performance. n contrast to the uniform puncturing scheme used in the CDMA2000 1xEV-DO code and other turbo-codes, this incremental redundancy scheme allows for substantial reductions in complexity at higher rates. Since only a small fraction of the blocks transmitted under an incremental redundancy scheme will be decoded at their lower-rates while all have decoding attempts made at higher rates, the improvement that r-s-scps offer at higher rates is especially attractive for incremental redundancy schemes. The EXT chart design method described in section assumes that there is little interaction between the trellissections corresponding to bits that are repeated a different number of times. For smaller block sizes, grouping high

E Fig. 8. Plot of BER and BLER vs. b for k = 1024, r = N 0 1/5, 1/3, 1/2, 2/3, 4/5, 8/9, Q = 6, 4, 3, 2, 2, 2 progressive transmission with progressive complexity r-s-scp code. The 6144 bit interleaver is derived from a 2048 bit interleaver for rates 2/3 and a 4096 bit interleaver. The repetition profile after the outer code is scrambled. min full 2048 4096 6144 Q 2 4 6 Q Rep., Period=128 0 38 2 128 53 38 4 53 6 18 8 18 14 17 16 17 30 2 32 2 TABLE REPETTON PROFLES FOR NCREMENTAL REDUNDANCY R-S-SCP Rate Q J Pattern 8/9 2048 2 16 xxxxxxxxxxxxxxxo 4/5 2048 2 8 xxxxxxxoxxxxxxxo 2/3 2048 2 4 xxxoxxxoxxxoxxxo 1/2 3072 3 3 xxxoxxxoxoxoxxxo 1/3 4096 4 2 xoxoxoxoxoxoxoxo 1/5 6144 6 1.5 xoooxoxoooxoxooo One period of each puncture pattern is bolded. ncremental bits for each rate are underlined. TABLE CODE PARAMETERS FOR NCREMENTAL REDUNDANCY R-S-SCP repetition degree trellis sections on the outer code into small interleaver components may increase the difficulty of avoiding short cycles. For these block sizes, a designer may wish to use few interleaver components and avoid grouping high repetition degree outer trellis sections together. This mixing of different repetition degree trellis sections will further reduce the accuracy of the already approximate EXT chart design method. A more accurate EXT chart design method would model the repetition code separately from the outer code, describing the soft-information passing from the outer code to the repetition code using a single variable or one variable per interleaver component. V. CONCLUSON rregular S-SCP designs bring additional good properties to the Systematic with Serial Concatenated Parity code family. r- S-SCP codes can be designed to achieve computational complexity advantages over well-designed RA codes in reaching moderate to low thresholds. These designs allow for low-rate codes with low thresholds. For codes with smaller block sizes, the finite-iteration design method of [12] is a practical method. Although not discussed in detail in this paper, S-SCP designs also exhibit similar reduction in memory requirements relative to RA codes because of their smaller interleavers. For the example two-state designs, both inner and outer codes may be decoded with the same two-state SSOs, allowing for single SSO decoder units to be used on both halves of an iteration with little additional logic [3]. n these examples, a single decoder implementation may support rates from at least 1/5 to 19/20 with fine granularity over a wide range of block sizes and even support progressive transmission with performance complexity-tradeoffs that are competitive with point designs of other popular codes. ACKNOWLEDGMENT This work was supported in part by NSF grant CCF- 0428940 and TrellisWare Technologies nc. REFERENCES [1] H. Jin, A. Khandekar, and R. McEliece, rregular repeat-accumulate codes, in Proc. 2nd nternational Conference on Turbo Codes, Brest, France, 2000. [2] K. M. 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