A Constituent Codes Oriented Code Construction Scheme for Polar Code-Aim to Reduce the Decoding Latency

Transcription

1 A Constituent Codes Oriented Code Construction Scheme or Polar Code-Aim to Reduce the Decodin Latency arxiv: v3 [cs.it] 20 Sep 2017 Abstract This paper proposes a polar code construction scheme that reduces constituent-code related decodin latency. Constituent codes are the sub-codewords with speciic patterns. They are used to accelerate the successive cancellation decodin process o polar code with neliible perormance deradation. We modiy the traditional construction approach to yield increased number o desirable constituent codes that speeds the decodin process. For (n, k) polar code, instead o directly settin thek best and(n k) worst bits to the inormation bits and rozen bits, respectively, we swap the locations o some inormation and rozen bits careully accordin to the qualities o their equivalent channels. We conducted the simulation o 1024 and 2048 bits lenth polar codes with multiple rates and analyzed the decodin latency or various lenth codes. The numerical results show that the proposed construction scheme enerally is able to achieve at least around 20% latency deduction with an neliible loss in ain with careully selected optimization threshold. I. INTRODUCTION Recently, polar code [1] attract more and more research interests due to that it is the irst code which provably achieves the channel capacity and its low codin complexity. Such property makes it very promisin or real scenario such as wireless communication and storae. Successive cancellation (SC) [1], list successive cancellation (LSC) [2] and belie propaation (BP) [3] are the three most widely known decodin alorithms. Amon those, SC and LSC receive more attention due to their simpler hardware complexity compared with that o BP. Due to its serial property, SC decoder suers rom the hih decodin latency. LSC considered as an extension o SC, similarly, has the same problem. The latency reduction o SC decoder is able to beneit the LSC decoder as well. Thus, a lot o eorts have been done on the SC decoder to reduce the latency rom both hardware and alorithm aspects. C. Leroux [4] proposed both tree and line architecture o SC decoder. For lenth n polar code, it takes (2n 2) clock cycles to decode. Later on, he [5] proposed a semiparallel architecture or both tree and line SC decoder, which makes a trade-o between hardware complexity and latency. C. Zhan [6] proposed a low latency SC decoder with precomputin and overlapped architecture. Pre-computin technoloy reduces the latency to (n 1) clock cycles, and the overlap scheme siniicantly the throuhput with multiple rame situation. B. Yuan [7] proposed an architecture with applyin the 2-bit decodin at the last stae and some ate level optimizations, this urther reduce the latency to(3/4n 1) clock cycles. Alamdar-Yazdi [8] proposed the simpliied SC Tiben Che and Gwan Choi Department o Electrical and Computer Enineerin Texas A&M University, Collee Station, Texas {ctb47321, wanchoi}@tamu.edu (SSC) which can siniicantly reduce the latency via some certain pattern sub-codewords. These kinds o sub-codewords are also called constituent codes. This is the irst time the concept o constituent codes has been mentioned. Inspired by this, Sarkis [9] proposed the ast-ssc which can urther reduce the latency by explorin more kinds o constituent codes. T. Che [10] proposed the hardware architecture o constituent codes based polar codes decoder. It allows the decodin processes are compatible with both conventional and constituent codes based polar codes. P. Giard [11] proposed an unrolled architecture with ast-ssc is able to achieve 237 Gbps throuhput. P. Giard [12] also proposed a low-complexity decoder or low rate polar code. In that work, he urther utilized the potential o constituent codes by chanin the rozen and inormation sets. Additionally, the idea o o constituent codes also beneits other decodin alorithm. J. Xu [13] applied the constituent concept to the BP decodin which siniicantly reduces the computin complexity. G. Sarkis [14] proposed an constituent codes based LSC decodin. T. Che [15] also pointed that the constituent codes can beneit the overlapped LSC architecture in term o hardware eiciency. Introducin constituent code is an approach to reduce decodin latency. Most o the aorementioned works stress on the decodin sides. In this paper, we explore the potential o constituent codes rom an opposite anle. We stress on the construction scheme to make the codeword more constituent-coderiendly. By adjustin the traditional construction approach, more expected types o constituent codes are manually produced or decodin. For (n, k) polar code, instead o directly settin the k best and (n k) worst bits to the inormation bits and rozen bits, respectively, we thouhtully swap the locations o some inormation and rozen bits accordin to the qualities o their equivalent channels. The constituent codes oriented polar code construction alorithm is described. We conducted the simulation o 1024 and 2048 lenth polar codes with multiple rates and analyzed the decodin latency or various lenth codes. The numerical result shows that the proposed construction scheme typically achieves 4-20% latency reduction with neliible loss in decodin perormance with careully selected optimization threshold. Some relevant discussions are also presented. This paper is oranized as ollows. The relative backround is reviewed in section II. Then, the proposed construction scheme are described in section III. Ater that, the numerical results and relevant discussions are presented in section IV. Finally, this paper is concluded in section V.

2 u1 u2 u3 u4 u5 u6 u7 u8 x1 x2 x3 x4 x5 x6 x7 x8 Stae 0 Stae 1 Stae 2 Stae αl β 3 l βr α αr β Fi. 1. Encoder o (8,4) polar code, Tree presentation o (8,4) SC decoder A. Polar code II. BACKGROUND As described by E. Arikan [1], a polar code is constructed by successively perormin channel polarization. Polar codes are linear block codes o lenth n = 2 m. The coded codeword x (x 1,x 2,,x n ) is computed by x = ug where G = [ F m ], and F m is the m-th Kronecker power o 1 0 F =. Each row o G corresponds to an equivalent 1 1 polarizin channel. For an (n, k) polar code, k bits that carry source inormation are called inormation bits. They are transmitted via the k best channels. While the rest n k bits, called rozen bits, are set to zeros and are placed at the n k worst channels. Fi. 1a shows an example o the construction o 8-bit polar code, where inormation bits and rozen bits are denoted by black nodes and white nodes on the most let side, respectively. Polar codes can be decoded by recursively applyin successive cancellation to estimate û i usin the channel output y0 n 1 and the previously estimated bits û i 1 0. This method is reerred as successive cancellation (SC) decodin. Actually, SC decodin can be rearded as a binary tree traversal as described in Fi. 1b. The number o bits o one node in stae m(m = 0,1,2...) is equal to 2 m. α stands or the sot reliability value, typically is lo-likelihood ratio (LLR). Each let and riht child nodes can calculate the LLR or current node via and unctions, respectively [5]. However, in order to compute unction, a eedback β l rom let child o the same parent node is needed. This kind o eedback is called partial sum. Actually, this serial property o eedback operation limits the throuhput o SC decodin. B. Constituent codes based SC decodin The recursive processin o ettin the partial sum rom each node siniicantly constrains the decodin speed. Thus, in order to obtain partial sum directly without perormin tree traversal, constituent codes based SC decodin has been proposed [8], [9]. Certain patterns in the codewords allows us to decode the sub-codewords and et their correspondin partial sums immediately, which siniicantly reduces the partial-sumconstrained latency. N 0, N 1, N SPC and N REP are the our most common constituent codes. N 0 and N 1 only contain either rozen bits Fi. 2. Ɲ 0 Ɲ 1 Ɲ REP SC decodin tree simpliied by constituent codes or inormation bits, respectively. N SPC and N REP contain both rozen bits and inormation bits. In the N SPC codes, only the irst bit is rozen. It makes the lenth n constituent codes as a rate (n 1)/n sinle parity check (SPC) code. In the N REP codes, only the last bit is inormation bit. In this case, all the correspondin partial sums should be the same since they all are the relections o the last inormation bit. All the above our constituent codes can be decoded very quickly. Accordin to T. Che s implementation [10], the latency o lenth n constituent code can be reduced rom 2n 2 to 1, 1, lo 2 n+1 and lo 2 n or N 0, N 1, N SPC and N REP codes, respectively. Fi. 2 shows an example o how constituent code can simpliy the SC decodin tree. III. Ɲ SPC PROPOSED CONSTRUCTION SCHEME As described beore, constituent codes are decoded aster than conventional polar codes. Thus, in order to reduce the decodin latency, we expect more constituents codes, especially constituent codes with lare lenth. Accordin to the deinition o constituent codes, the initial distribution o constituent codes is determined by the location o inormation and rozen bits. I we can chane the locations, we are able to manually produce expected constituent codes. However, the location o rozen and inormation bits are very sensitive, and random chanes may cause neative inluence on the codin perormance. Thus, a thouhtul construction scheme to produce more expected constituent code is on demand. The division o inormation and rozen bits is decided by the qualities o the equivalent channels which are correspondin to each bit. Based on the channel model, the equivalent channel qualities can be calculated accordinly [1] [16] [17] [18]. This ives us a hint that i we swap some inormation and rozen bits those with similar channel qualities, this miht only incur a very sliht perormance loss. The numerical simulation results in the ollowin section prove this idea. In this work, binary erasure channel (BEC) is used as our channel model, and thus Bhattacharyya parameter is used as the metric or equivalent channel quality. This method can be extended to any other kind o channel model. Now, we have the idea about how to chane the division o constituent codes. Next, we need to consider what kind o chanes are desired. For any lenth n polar code, it can be rearded as a combination o the ollowin our types o subcodewords. Type-I: All the bits are rozen bits. This is also N 0 constituent code. Type-II:All the bits are inormation bits. This is also N 1 constituent code.

3 Fi. 3. Type-III REP Ɲ Ɲ 0 REP Ɲ Type-III Type-IV Examples o constituent codes division optimization Type-III: Only one bit is inormation bit, the rest are rozen bits. This can be rearded as the combination o one N REP and multiple N 0 constituent codes. Type-IV: Only one bit is rozen bit, the rest are inormation bits. This can be rearded as the combination o one N SPC and multiple N 1 constituent codes. Accordin to T. Che s results [10], there is only one clock cycle needed or decodin N 0 and N 1 node. Thus, it is unnecessary to optimize type-i and type-ii codes since they are already ully optimized. Our taret should be ocused on the type-iii and type-iv. These two types are similar, they all only have one node with dierent type with others. There are two situations we need to deal with. The irst situation is the optimization or sinle type-iii or type-iv sub-tree. We can swap the dierent node with the irst or last one to make it a N SPC or N REP nodes or type-iv or type-iii sub-tree, respectively. The second situation is that the optimization or a combination o type-iii and type-iv sub-trees. For this case, we can swap the dierent node between the two types to make them became one type-i and one type-ii sub-trees. For the irst situation, suppose we have one Type-III node at stae m+1. It consist o one N REP and one N 0 node at stae m. This is shown in Fi 3a. Totally, it needs 1+lo 2 2 m = m+1 to inish decodin. I we move the place o the inormation bit to make it a N REP constituent code, the new latency or decodin should be lo 2 2 m+1 = m+1. There is no chane i we do this modiication. This is also similar to type-iv situation. For the second situation, suppose we have one type-iii and one type- IV nodes at stae m as shown in Fi. 3b, the totally latency or decodin should be 2m + 1. I we swap the inormation bit in type-iii and rozen bit in type-iv, we et one N 0 and one N 1 nodes. The total should be reduced to 2. This makes a hue dierence. Thus, our taret should be the swap operation between type-iii and type-iv nodes. Based on above discussion, the constituent code oriented polar code construction alorithm is proposed in alorithm (1). Ater we et the inormation and rozen bits positions usin the traditional way, we apply this alorithm to adjust the location o some inormation and rozen bits to make more desired constituent code. This is an enhancement to the traditional construction method. IV. SIMULATION RESULTS AND DISCUSSIONS Fi. 4 shows the simulation results o proposed construction scheme. They are the vs Eb/N0 perormance or 1024 and 2048 lenth at dierent rate with multiple optimization thresholds. The polar codes is constructed based on the Ɲ 0 Ɲ 1 Alorithm 1 constituent code oriented polar code construction Input: The set o Bhattacharyya parameter,ǫ n ; the initial set o bit property (rozen or inormation), L; the threshold or bit swappin,t h. Initialize: index = 1. 1: et the sub-codeword-type look up table T rom L; this table ives the sub-codeword type inormation and its index 2: i T[index] is type-iii sub-codeword then 3: et the index i o the inormation bit in T[index], search all the next type-iv sub-codewords and ind the one whose rozen bit s Bhattacharyya parameter has the minimum dierence with ǫ n [i]. Recode the index o that. 4: i ǫ n [i] ǫ n [] < T h then 5: swap the property o L[i] and L[], update T. 6: end i 7: end i 8: i T[index] is type-iv sub-codeword then 9: et the index o the rozen bit in T[index], search all the next type-iii sub-codewords and ind the one whose inormation bit s Bhattacharyya parameter is has the minimum dierence with ǫ n []. Recode the index i o that. 10: i ǫ n [i] ǫ n [] < T h then 11: swap the property o L[i] and L[], update T. 12: end i 13: end i 14: increase index by 1, repeat to 2, until to the end o T BEC channel and with taret erasure rate. Table I shows the decodin latency or each threshold and the comparison with constituent codes decodin without any optimization and other state-o-the-art decoders. We even calculated the latencies or the codes with very lon lenth such as Accordin to Fi. 4 and Table I, we note that the proposed construction scheme enerally is able to achieve at least around 20% latency reduction with an neliible ain loss. For a certain code lenth, we can ind that the acceptable threshold is increasin alon with the code rate. This is due to the nature o channel polarization. There are more rozen and inormation bits mixed in the ront and middle part o codeword or lower rate codes, which ives more lexibility durin the optimization. However, this does not indicate that this codin scheme is not workin well on hih rate. The interestin part is that the perormance on hih rate is as ood as the low rate accordin to Fi. 4. It s possibly attributable to the ollowin two reasons. First is that the codin perormance o hih rate itsel is much worse than that o low rate. This causes the dierence ater optimization is not so obvious. The second reason is that the simulated code lenth is still not lon enouh due to the limitation o simulation environment. This is per codin theory that suests the loner polar code will enerally yield hiher polarization. For a certain code rate, we can ind the acceptable threshold is decreasin alon with the code lenth. Since the loner codes are more polarized, the dierence o each equivalent channel is becomin smaller and smaller as the lenth is increasin. This works ine with low and medium lenth but not so obvious with hih lenth. This also can be explained by the two reason presented beore.

4 Fi. 4. lenht=1024,rate= ber th=1e 13 ber th=5e 12 ber th=1e 11 er th=1e 13 er th=5e 12 er th=1e 11 lenht=1024,rate=0.5 ber th=1e 4 ber th=5e 4 ber th=1e 3 er th=1e 4 er th=5e 4 er th=1e 3 lenht=1024,rate= ber th=0.1 ber th=0.2 ber th=0.4 er th=0.1 er th=0.2 er th= lenht=2048,rate= ber th=1e 18 ber th=1e 17 ber th=1e 16 er th=1e 18 er th=5e 17 er th=5e 17 lenht=2048,rate=0.5 ber th=1e 6 ber th=5e 5 ber th=1e 4 er th=1e 6 er th=5e 5 er th=1e 4 lenht=2048,rate= ber th=0.1 ber th=0.2 ber th= er th=0.1 er th=0.2 er th= The ber vs Eb/N0 perormance or proposed construction scheme We compared the latency o proposed desin with the decoder in [7]; this is the astest non-constituent-codes-based polar code decoder to the best o our knowlede. We can see even constituent codes decoder without any optimization is much aster than that. Our proposed construction scheme is capable o achievin 20% or more latency reduction. This is very siniicant especially or very lon code lenths. Compared with [12] in which also chanes the rozen and inormation sets to beneit the decodin, our work has two main dierences. The irst one is that we taret a dierent decoder architecture, which results in dierent demandin o TABLE I. LATENCY REDUCTION decoder lenth rate threshold latency reduction(%) no optimization 303 1e e e no optimization e e e no optimization no optimization 576-1e e e no optimization proposed e e e no optimization no optimization e e e no optimization e e e no optimization [7] desirable constituent codes combination. The second dierence is that the construction alorithm is dierent. V. CONCLUSION This paper presented a novel polar code construction scheme which reduces the decodin latency. The proposed constituent codes oriented polar code construction alorithm can automatically produce more types o constituent codes which are desirable. The simulation results show that the proposed construction scheme enerally is able to achieve at least around 20% latency deduction with neliible decodin perormance loss. Besides, compared with non-constituentcodes-based decoder, the constituent codes based decoder has a measurable advantae in term o latency. Our construction scheme is able to urther enhance the timin perormance. REFERENCES [1] E. Arikan, Channel polarization: A method or constructin capacityachievin codes or symmetric binary-input memoryless channels, Inormation Theory, IEEE Transactions on, vol. 55, no. 7, pp , [2] I. Tal and A. Vardy, List decodin o polar codes, in Inormation Theory Proceedins (ISIT), 2011 IEEE International Symposium on. IEEE, 2011, pp. 1 5.

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