A New Finite Word-length Optimization Method Design for LDPC Decoder

Transcription

1 A New Finite Wod-length Optimization Method Design fo LDPC Decode Jinlei Chen, Yan Zhang and Xu Wang Key Laboatoy of Netwok Oiented Intelligent Computation Shenzhen Gaduate School, Habin Institute of Technology HIT Campus of Shenzhen Univesity Town, Shenzhen, China Abstact: -A new wod-length optimization method based on Monte Calo simulation is poposed. The wodlength of the check node extinsic message is also futhe optimized in this pape. In the poposed optimization method, and in the pocess of optimizing the wod-length of the channel data, the statistical distibution esults of vaiable node s posteio pobability data and check node s extinsic message ae also obtained. The optimized wod-length of vaiable node s posteio pobability data and check node s extinsic message is concluded by the statistical distibution esult and the BER (Bit Eo Rate) cuves. Compaed to the pue Monte Calo simulation, the poposed method could educe the amount of simulation wok by moe than 50%, and have the same wod-length optimization esults. Key-Wods: - LDPC decode, wod-length optimization, Monte Calo simulation, min-sum. 1 Intoduction Low density paity check (LDPC) code was fist poposed by Gallage in 1960 [1], and it was ediscoveed by MacKay and Neal in 1996 [2]. Due to the excellent decoding pefomance, LDPC code has been widely used in many communication systems, such as wieless local aea netwok (802.11n) [3], digital video boadcasting second geneation (DVB-S2) [4] and wold inteopeability fo micowave access (802.16e) [5-6]. Punctued LDPC codes used in coheent optical OFDM systems is also studied in [7]. Belief-popagation (BP), o called sum-poduct [2], is one of the best LDPC decoding algoithms, but it is not suitable fo hadwae implementation because of the exponent computation. Min-sum algoithm [8], which decodes LDPC only by compaisons and additions, simplifies the decoding pocess geatly with acceptable decoding pefomance loss. Modified min-sum algoithms wee poposed in [9-11], nomalized facto and offset facto ae used to get a bette pefomance in decoding. Layeed decoding method was poposed in [12]. Check node and vaiable node ae both updated simultaneously in this method to educe the decoding latency half without pefomance loss. In [13], a fast-convegence algoithm using layeed decoding was poposed and about 1/6 iteation numbes deceased fo LDPC codes used in DVB- S2. The finite wod-length (o the quantization scheme) of the data deeply affects the decoding pefomance and the total aea of the LDPC decode in hadwae implementation. So the finite wodlength optimization method should be able to balance between the decoding pefomance and the complexity of the LDPC decode. Monte Calo simulation method is a widely used method in finite wod-length optimization [14]. In LDPC decoding simulation, andom numbes ae used as the message bits of the code wod in LDPC codes simulation. These andom numbes, geneated by a pseudo andom numbe algoithm, ae independent and equally likely to be 0 o 1. BER (Bit Eo Rate) is used to measue the LDPC decoding pefomance ove vaious finite wodlengths. The expected BER is always below 10-6 in LDPC decoding, so it costs a long time to obtain the finite wod-length of all the tems by using Monte Calo simulation method. In this pape, we poposed a new finite wod-length optimization method. In this method, the finite wod-length of channel message is obtained by Monte Calo simulation, and the finite wod-length of othe messages ae obtained by the statistical distibution esults in Monte Calo simulation. The poposed method saves the E-ISSN: Issue 4, Volume 12, Apil 2013

2 simulation wok moe than 50% to obtain the same esults compaed to the oiginal Monte Calo simulation method. The est of this pape is oganized as follows. Section 2 descibes the LDPC decoding algoithm, nomalized min-sum algoithm and off-set min-sum algoithm. Layeed decoding scheme, wod-length of fix-point data and Monte Calo Simulation method ae also discussed in this section. Section 3 poposes a new wod-length optimization method. Section 4 poposes the method to futhe optimize the wod-length of the check node extinsic message. Simulations esults and discussions ae given in section 5. Section 6 concludes the pape. 2 Backgound 2.1 Min-Sum Algoithm Min-sum algoithm is one of the most popula appoaches of BP [9], and it educes the hadwae complexity geatly. Both nomalized min-sum and offset min-sum algoithm ae based on the min-sum algoithm. Min-sum algoithm is expessed as follows: At the beginning of decoding, all vaiable node messages q ae installed by (4). In each iteation, i (1), (2) and (3) ae pocessed seially, and a guess of the codewod is obtained by the sign of (0 fo 0, 1 fo 0) in (3), if the codewod fits all the paity check o the iteation exceeds the pedefined maximum iteation time, the decoding stops. 2.2 Modified min-sum Decoding Algoithm Although it is easy to implement the min-sum algoithm, it esults in degadation in decoding pefomance. Both nomalized min-sum and offset min-sum ae modified vesions of the min-sum algoithm, the fist one with a nomalized facto and the second with an additive coection facto, and these algoithms achieve almost the same pefomance as that of the BP algoithm. In the nomalized min-sum algoithm, check node updating opeation uses nomalization constant smalle than 1, and (1) is changed to: i a min q (5) V \ i V \ i i ' i a min q (1) V \ i V \ i i ' In offset min-sum algoithm, the check node updating opeation is given as follows: In (1), q i LLR LLR ' C i \ ' C i ' i (2) ' i (3) P( x 0 y) LLR log (4) P( x 1 y) i is the check-to-vaiable message passed fom check node i to vaiable node, q is the vaiable-to-check message passed fom vaiable node to check node i ', a is the sign of q, V \ i is the set of the vaiable nodes which connect to check node without node i, in (2), C \ is the set of the check nodes which connect to i vaiable node i without check node, in (3), is the log likelihood atio (LLR) fo vaiable node, C i is the set of all the check nodes which connect to vaiable node i, in (4), x is the tansmitted bit and y is the message eceived fom channel. i V \ i a max (6) min q, 0 V \ i In (5) and (6), both coection factos ae used to decease the magnitude of i, and thei pefomance is analyzed in [12]. Compaed to offset min-sum, nomalized min-sum algoithm has bette decoding pefomance but the multiplication inceases the implementation complexity. In this pape, nomalized min-sum algoithm is used in the poposed LDPC decode. 2.3 Layeed Decoding Scheme In layeed decoding scheme, the paity check matix can be viewed as hoizontal layes, and each laye can epesent a component code [13]. The code is composed of all layes and thei intesections. In C-LDPC code, the ows and columns ae natually blocked by the pemutation matix, so each block ow can be indicated as a laye. As each laye stats decoding, the inputs of vaiable node contain channel inputs and the extinsic messages of the check nodes fom the pevious layes. Iteations within a laye ae called sub-iteations and the oveall pocess is labelled as supe-iteations. E-ISSN: Issue 4, Volume 12, Apil 2013

3 In each sub-iteation, the vaiable-to-check message q is fistly computed by: ' i q i ' ' ' (7) Whee i ' is the check-to-vaiable message in the pevious supe-iteation of this laye, and ' is the LLR esult of vaiable node fom the pevious sub iteation. The i ' is computed by (1) o (5) o (6), and is computed by: i i q ' (8) When supe-iteation is finished, The codewod is obtained by the sign of. In geneal, the decoding convegence speed of the layeed decoding scheme is two times faste than that of the two-phase scheme, and in layeed decoding, only and i ae stoed in memoies because the vaiable-to-check message computed by (7). 2.4 Wod-length of Fix-point Data i q i ae The diffeence of fix-point data and floating-point data is that fo fix-point data, the position of the decimal point is fixed. The wod-length of intege pat and the faction pat ae constant. Fo fix-point data, we have: WL IWL FWL WL is the wod-length of the fix-point data, IWL is the wod-length of the intege pat, FWL is the wod-length of the faction pat. A signed numbe is always expessed by two s complement fom, in this case, IWL contains a sign bit: 0 fo positive numbe, 1 fo negative numbe, and in this case: WL IWL FWL 1 So the ange of the signed fix-point data is IWL IWL FWL [ 2, 2 2 ], the pecision of the data is 2 FWL. In LDPC decode hadwae implementation, we use the notation ( WL : FWL ) to epesent a quantization scheme. As intoduced peviously, WL bits ae used fo total bit size and FWL bits ae used fo factional values. 2.5 Monte Calo Simulation Method The quantization scheme of LLR significantly affects the decoding pefomance and the total decode complexity, and the wod-length of othe tems in the decoding algoithm ae also depended on it, so the quantization scheme of LLR should be detemined fistly. Though lage wod-length has good decoding pefomance, it causes hadwae ovehead fo the buffes and a lage numbe of hadwae fo the iteative decoding computation. A small wod length may esult in vey poo pefomance. Hence, the quantization scheme should balance the decoding pefomance and the hadwae complexity. Thee ae two steps in Monte Calo simulation method. Fist, the quantization scheme of channel message is obtained. Second, in two-phase decoding scheme, the finite wod-length of extinsic message of check node and vaiable node ae obtained. In layeed decoding scheme, the finite wod-lengths of vaiable node message and the extinsic message of check node ae obtained sequentially. In [15-17], the wod-lengths of vaiable node message and the extinsic message of check node ae equal. 3 An Impoved Wod-length Optimization Method In LDPC decoding, the expected BER is always below 10-6, and that means 10 8 message bits should be tansmitted in Monte Calo simulation. It will cost seveal days to obtain the quantization scheme of all the tems using Monte Calo method. In the poposed method, days of time will be saved because of moe than half Monte Calo simulations ae omitted. The poposed finite wod-length optimization method is expessed as follows: Step 1, the quantization scheme of channel message is obtained using Monte Calo simulation. Meanwhile, the statistical distibution esult of vaiable node s posteio pobability message is also obtained. Step 2, by analyzing the statistical distibution esult, the quantization scheme of is achieved. At last, the wod-length of i equals to the wodlength of. This method could be used in any min-sum based decoding algoithm. E-ISSN: Issue 4, Volume 12, Apil 2013

4 4 Futhe Wod-length Optimization of the Check Node Extinsic Message In layeed decoding scheme, only i is stoed in the extinsic memoy. So the wod-length of i deeply affects the aea of the decode. In [15-17], the wodlengths of i and ae equal. But in min-sum based LDPC decoding algoithm, as function (1) shows, the check node extinsic message i is the minimum value of the input messages q. So thee i ' is a chance that the wod-length of i could be futhe optimized. In this section, we popose methods to futhe optimize the wod-length of both in Monte Calo method and the poposed i method. 4.1 In Monte Calo Method Let LLR and use the quantization scheme obtained in Monte Calo simulation method as intoduced in section 2.5. The appopiate wodlength of i is obtained by analyzing BER cuves of i with vaious wod-length. Moe Monte Calo simulations ae needed in the wod-length optimization pocess of i, and that means moe time is needed in Monte Calo simulation method to obtain the quantization scheme of all the tems. 4.2 In the Impoved Optimization Method In the impoved method, the pocess of optimizing the wod-length of i is simila as the pocess of. The statistical distibution esult of i is obtained in step 1, and the esult is used to choose the appopiate wod-length of i. In the poposed method, the pocess of optimizing the wod-length of i doesn t bing in any exta Monte Calo simulation, so the time of optimizing the wod-length of all the tems is nealy the same as in section 3. 5 Simulations and Discussions 5.1 Monte Calo Method In this section, we use layeed offset min-sum algoithm as the decoding algoithm, all the BER cuves ae obtained by tansmitted 100,000 code wods, the max iteation numbe is 50, the modulation method is BPSK and the channel module is AWGN. BER esults ove Monte Calo Simulation ae used to compae the diffeent pefomance of vaious quantization scheme of LLR, and i. The pefomances of the (1944, 972) LDPC code in IEEE n with floating point, (7:4), (6:3) and (5:2) quantization schemes of LLR ae shown in Fig. 1. It shows that (7:4) quantization scheme has the best pefomance of the thee fix-point quantization schemes, and the diffeence of decoding pefomance between (7:4) and (6:3) quantization scheme is less than 0.05dB. The (5:2) quantization scheme has the wost decoding pefomance. Thus it tuns out that using the (6:3) scheme of LLR seems to be the optimal tadeoff between hadwae complexity and decoding pefomance. This quantization scheme has a 3 pecision of with a maximum value of and a minimum value of 2 4. Let LLR use the (6:3) quantization scheme, the decoding pefomances of the (1944, 972) LDPC code in IEEE n with vaious wod-length of (which is denote by W ) ae shown in Fig. 2. As shown in Fig. 2, W 9 has nealy the same pefomance as W, and when W 8, the BER cuve has an eo floo at So at last, we choose W 9. The pecision of is the same as that of LLR. Let LLR use (6:3) quantization scheme, uses (9:3) quantization scheme, the decoding pefomances of the (1944, 972) LDPC code in IEEE n with vaious wod-length of i ae shown in Fig. 3. When W 7, the BER cuves is coincide with the cuve of W, and thee is only little diffeence between W 6 and W 7. Fo example, when SNR=1.9dB, BER of W 7 is , and BER of W 6is But the diffeence between W 5 and W 6 is quite significant. So at last, we choose W 6. And the pecision of i is also the same as that of LLR. E-ISSN: Issue 4, Volume 12, Apil 2013

5 Figue 1. BER Cuves of LLR unde Layeed Offset Min-Sum Algoithm with Diffeent uantization scheme. Figue 2. BER Cuves of unde Layeed Offset Min-Sum Algoithm with Diffeent uantization E-ISSN: Issue 4, Volume 12, Apil 2013

6 Figue 3. BER Cuves of i unde Layeed Offset Min-Sum Algoithm with Diffeent uantization Table 1: uantization Scheme of Offset Min-Sum Decoding Algoithm Message uantization scheme Range Pecision LLR (6:3) (-4, 3.875) (9:3) (-32, ) (6:3) (-4, 3.875) i The quantization scheme of all the tems in offset min-sum algoithm is summaized in table 1. In [15-17], the wod-length of i would be the same as, so in the poposed method, the wod-length is futhe optimized by 1/ The Impoved Optimization Method We also use the example in 5.1. The statistical distibution esult of and i in the simulation of Fig. 1 is shown in Fig. 4 and Fig. 5. The assumed SNR of AWGN channel is 1.9dB, and in this case, the aveage iteation is In Fig. 4 and Fig. 5, the x-axis data is quantized. Fo example, the numbe 5 in x-axis equals to As shown in Fig. 4, the distibution ange of inceases as the iteations incease, and most values of is in the ange of (-256, 256), so the quantization scheme of is (9:3). Fom Fig. 4, we can see that the aea of the cuves decease as the iteations incease. It means the amount of deceases with the iteations, and that because in each iteation, many code wods ae decoded, and the numbe of un-decoded codes deceases with the inceasing of iteations. Fom Fig. 4, we can also see that the statistical distibution cuve of the channel input data LLR is the supeposition of two Gauss cuves, and the symmety axis of the two cuves ae x 8 and x 8. That because in the simulation, the modulation system is BPSK, and in BPSK, bit 0 is changed to -1, and bit 1 is unchanged, and the channel module in the simulation is AWGN, the data eceived at each time is equal to the sent data plus Gaussian noise, so the statistical distibution cuve of LLR is the supeposition of two Gauss E-ISSN: Issue 4, Volume 12, Apil 2013

7 Figue 4 Statistical Distibution of Figue 5 Statistical Distibution of i Figue 6. BER Cuves of unde Layeed Offset Min-Sum Algoithm with Diffeent uantization E-ISSN: Issue 4, Volume 12, Apil 2013

8 Figue 6. BER Cuves of unde Layeed Offset Min-Sum Algoithm with Diffeent uantization cuves. The quantization scheme of LLR is (6:3), so the symmety axis of the two cuves ae x 8 and x 8. As shown in Fig. 5, the distibution ange of i inceases with the iteations, and most values of i is in the ange of (-32, 32), so the quantization scheme of i is (6:3). In Fig. 5, thee is a wave cest aound 0 in the statistical distibution cuve of the fist iteation, and the cest disappeas as the iteations incease. That because in the fist few iteations, the signs of many ae uncetain o the eliabilities of ae small, and fom function (1), we can conclude that in the fist few iteations, most values of i ae aound 0. When the numbe of iteations inceases, moe codewod is decoded, and fo the un-decoded codewod, the eliability of and the ange of i would inceases. So the wave cest aound 0 disappeas with iteations. Fig. 6 shows the decoding pefomance of the (1944, 972) LDPC code in IEEE n with floating-point and the final quantization scheme. It is shown that the decoding pefomance loss using the poposed quantization scheme compaed with floating point is less than 0.1dB. In this example, by using Monte Calo method, 12 BER cuves ae needed to obtain the final quantization scheme. In the impoved finite wodlength optimization method, only 3 BER cuves and 2 statistical distibution cuves ae needed. So in this example, the impoved finite wod-length optimization method educes the simulation wok by 75%. Geneally speaking, in Monte Calo simulation method, fo layeed decoding scheme, at least 9 BER cuves ae needed to obtain the final quantization scheme (3 cuves fo LLR, 3 cuves fo and 3 cuves fo i ), fo two-phase decoding scheme, at least 6 BER cuves ae needed to obtain the final quantization scheme(3 cuves fo LLR and 3 cuves fo the extinsic messages of vaiable nodes). In the poposed finite wod-length optimization method, only 3 BER cuves ae needed to obtain the final quantization scheme fo both layeed decoding scheme and two-phase decoding scheme. So the poposed method can educe the simulation wok by moe than 50% 6 Conclusion In this pape, we poposed a new wod-length optimization method and futhe optimized the wod-length of the check node extinsic message. In E-ISSN: Issue 4, Volume 12, Apil 2013

9 the poposed method, the wod-length of vaiable node s posteio pobability data and check node s extinsic message is concluded by the statistical distibution esult and the BER (Bit Eo Rate) cuves. Compaed to the pue Monte Calo simulation, the poposed method could educe the amount of simulation wok by at least 50%, and has the same esults. Refeences: [1] R.G.Gallage, Low-density paity-check codes, IEEE Tansactions on Infomation Theoy, Jan.1962, pp [2] D. J. C. MacKay and R. M. Neal, Nea Shannon limit pefomance of low-density paity-check codes, Electonic Lette, vol. 32, Aug 1996, pp [3] IEEE Std n-2009, Wieless LAN Medium Access Contol (MAC) and Physical Laye (PHY) Specifications: Enhancements fo Highe Thoughput, IEEE P802.11n, Sep [4] Digital video boadcasting (DVB); Second geneation faming stuctue, channel coding and modulation systems fo boadcasting, inteactive sevices, news gatheing and othe boad-band satellite applications, ETSI, EN V1. 1.1, [5] IEEE Std e, Ai Inteface fo Fixed and Mobile Boadband Wieless Access Systems, IEEE P802.16e, Feb [6] M. Khan, R. Caasco and I. Wassell, application of high ate LDPC codes fo IEEE d o WiMAX (fixed boadband) systems, WSEAS Tansactions on Communications, vol. 6, iss. 1, Jan. 2007, pp [7] J. Chang, K. Takeshi, F. Hisato and H. Kazuhisa, Influence of lase linewidth and modulation level fo coheent optical OFDM with punctued LDPC codes, WSEAS Tansactions on Communications, vol. 10, iss. 6, pp [8] J. Chen and M. P. C. Fossoie, Nea optimum univesal belief popagation based decoding of low-density paity check codes, IEEE Tansactions on Communications, vol. 50, Ma. 2002, pp [9] J. Chen and M. P. C. Fossoie, Decoding lowdensity paity-check codes with nomalized APP-based algoithm, Poceedings of IEEE Globecom, San Antonio, TX, Nov. 2001, pp [10] E. Eleftheiou, T. Mittelholze, A. Dholakia, Reduced-complexity decoding algoithm fo low-density paity-check codes, IEEE Electonic Lettes, vol. 37, Jan. 2001, pp [11] M.P.C. Fossoie, M. Mihalevic, H. Imai, Reduced complexity iteative decoding of low density paity check codes based on belief popagation, IEEE Tansactions on Communications, vol. 47, no. 5, May 1999, pp [12] D.E. Hoceva, A Reduced Complexity Decode Achitectue via Layeed Decoding of LDPC Codes, IEEE wokshop on Signal Pocessing Systems, Oct. 2004, pp: [13] J. Xie, L. Yin, N. Ge and J. Lu, Fast Convegence Algoithm fo Decoding of Low Density Paity Check Codes, WSEAS Tansactions on Communications, volume.8, issue7, July 2009, pp [14] N. Metopolis, S. Ulam, The Monte Calo Method, Jounal of the Ameican Statistical Association, vol. 44, no. 247, pp [15] V. K. K. Sinivasan, C. K. Singh, P. T. Balsaa, A Geneic Scalable Achitectue fo Min- Sum/Offset-Min-Sum Unit fo Iegula/Regula LDPC Decode, IEEE Tansactions on VLSI, vol. 18, no. 9, Aug., 2010, pp [16] T. C. Kuo, A. N. Willson, A flexible Decode IC fo WiMAX C-LDPC Codes, IEEE Intenational Confeence on Custom Integated Cicuits, Sept. 2008, pp [17] C. H. Liu, S. W. Yen, C. L. Chen and H. C. Chang, An LDPC Decode Chip Based on Self- Routing Netwok fo IEEE e Applications, IEEE Jounal of Solid-State Cicuits, vol. 43, n. 3, Ma. 2008, pp E-ISSN: Issue 4, Volume 12, Apil 2013

10 Biogaphies Jinlei Chen bon in He eceived the B.S. degee fom Jilin Univesity in micoelectonics in 2005 and M.S. degee in micoelectonics fom the Shenzhen Gaduate School, Habin Institute of Technology, Shenzhen, China, in Since 2008, he has been a PhD candidate in micoelectonics. His main eseach inteests include LDPC codes and VLSI hadwae design. Yan Zhang bon in He has been pofesso of the Shenzhen Gaduate School, Habin Institute of Technology since His main eseach inteests ae application specific instuction set pocesso design, including medical image pocessing chips and wieless communication baseband chip. Wang Xu bon in Received the M.A s. degees in micoelectonics fom the Shenzhen Gaduate School, Habin Institute of Technology, Shenzhen, China, in Since 2008, he has been a PhD candidate in micoelectonics. His main eseach inteests include image pocessing and embedded DSP pocesso design. E-ISSN: Issue 4, Volume 12, Apil 2013