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New Min-sum LDPC Decoding Algorithm Using SNR-Considered Adaptive Scaling Factors
New Min-sum LDPC Decoding Algorithm Using SNR-Considered Adaptive Scaling Factors
ETRI Journal. 2014. Jun, 36(4): 591-598
Copyright © 2014, Electronics and Telecommunications Research Institute(ETRI)
  • Received : July 23, 2013
  • Accepted : November 15, 2013
  • Published : June 01, 2014
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About the Authors
Yongmin Jung
Yunho Jung
Seongjoo Lee
Jaeseok Kim

Abstract
This paper proposes a new min-sum algorithm for low-density parity-check decoding. In this paper, we first define the negative and positive effects of the received signal-to-noise ratio (SNR) in the min-sum decoding algorithm. To improve the performance of error correction by considering the negative and positive effects of the received SNR, the proposed algorithm applies adaptive scaling factors not only to extrinsic information but also to a received log-likelihood ratio. We also propose a combined variable and check node architecture to realize the proposed algorithm with low complexity. The simulation results show that the proposed algorithm achieves up to 0.4 dB coding gain with low complexity compared to existing min-sum-based algorithms.
Keywords
I. Introduction
Low-density parity-check (LDPC) code, which is defined by a sparse parity-check matrix, has received much attention in wireless communication systems because of its excellent performance in error correction. Originally, a codeword was decoded by using a standard iterative decoding algorithm — namely, a sum–product (SP) algorithm [1] . However, the high computational complexity of its check node operation results in high hardware overhead in the design of an LDPC decoder. To replace the SP algorithm, min-sum (MS)-based algorithms, such as an original MS and a normalized MS (NMS), were proposed [2] [3] . Unlike the SP algorithm, the minimum function of the check node of the MS algorithm [2] generates an approximated message. Thus, degradation in the performance of error correction is observed. To compensate for this performance degradation, the existing NMS algorithm normalizes the approximated message by using a fixed scaling factor (SF) [3] . In many literatures, MS-based decoding algorithms were modified to improve error correction performance [4] [11] . In previous works, the SF was found for the given LDPC code by using density evolution [3] [4] , [11] . The found SF can achieve reasonable error correction performance at a very low signal-to-noise ratio (SNR) near to the theoretical SNR threshold. This SF has a fixed value, and the effect of the received SNR is not considered in the decision of this SF.
In this paper, we propose an SNR-considered adaptive NMS (SANMS) algorithm considering the effect of the received SNR, and we present the adaptive SFs for the proposed algorithm. We also propose a combined variable and check node architecture to realize the proposed algorithm with low complexity. The simulation results show that the proposed algorithm achieves improved performance with low hardware complexity compared to the existing MS-based algorithms.
The remainder of this paper is organized as follows. In section II, the iterative decoding algorithms, such as the SP-based and MS-based, and the modified MS-based algorithms are introduced. In section III, the SANMS algorithm is proposed to improve the error correction performance by applying various SFs depending on the received SNR. Section IV proposes the combined variable and check node architecture to perform the proposed SANMS algorithm with low complexity. Section V shows the error correction performance of the proposed SANMS algorithm and demonstrates that the proposed algorithm can improve the error correction performance compared to the existing MS-based algorithms. Concluding remarks are given in section VI.
II. Related Works
The check node and variable node operations of the SP algorithm are expressed as follows. The check node generates a check to the variable (C2V) message. The C2V message rij , which is generated at the i th check node and delivered to the j th variable node, is expressed as [1]
r ij =  j`N( i )\j sign( q j`i )×Φ( j`N( i )\j Φ(| q j`i |)),
where N ( i ) denotes the set of variable nodes connected to the i th check node, and qij denotes the variable to check (V2C) the message going from the j th variable node to the i th check node. The function Φ( x ) is ln(( ex + 1)/( ex − 1)). Since Φ( x ) is a non-linear function, the computational complexity of Φ( x ) is very high. The V2C message qij , which is generated at the j th variable node and delivered to the i th check node, is expressed as [1]
q ji = 2 y j / σ 2 + i`M( j )\i r i`j ,
where yj is the j th message of the transmitted codeword, 2 yj / σ 2 is the received log-likelihood ratio (LLR) of the j th bit, σ 2 is the noise variance computed from the received SNR, and
∑ i`∈M(j)\i r i`j
is the sum of the incoming C2V message. The set of check nodes connected to the jth variable node is denoted by M ( j ). In the rest of this paper, rij and
∑ i`∈M(j)\i r i`j
represent the extrinsic information of C2V and V2C messages, respectively.
The MS algorithm significantly reduces the computational complexity of the check node operation by replacing the non-linear operation with a simple linear operation, that is, the minimum function [2] . The C2V message of the MS algorithm is expressed as [2]
r ij =  j`N( i )\j sign( q j`i )  ×   min j`N(i)\j | q j`i |,
where
min j`∈N(i)\j | q j`i |
represents to find the minimum value from among the incoming V2C messages. The variable node operation of the MS algorithm is the same as that of the SP algorithm.
Since the minimum function of the MS algorithm generates an approximated message compared to the SP algorithm, performance degradation in error correction is observed. The NMS algorithm normalizes the C2V message by using an SF to compensate the performance degradation [3] . The C2V message of the NMS algorithm is expressed as
r ij = α×  j`N( i )\j sign( q j`i )  ×   min j`N(i)\j | q j`i |,
where α denotes the SF of the check node operation. The value of α is less than 1. Other NMS algorithms [4] [5] normalize the V2C message instead of the C2V message as
q ji =2 y i / σ 2 +β ×  i`M( j )\i r i`j ,
where β is the SF of the variable node operation. The value of β is less than 1. Due to the low complexity and the improved performance, the NMS algorithm [3] is usually employed in the implementation of LDPC decoders [6] [8] .
To improve the error correction performance, modified NMS algorithms were proposed [8] [10] . A modified NMS algorithm in [8] applied two different SFs to the first minimum value and the second minimum value. An adaptive NMS algorithm in [9] adjusted the SF α depending on the distributions of the received LLR value. Another adaptive NMS algorithm in [10] applied different SFs depending on the state of the check node. These modified NMS algorithms [8] [10] achieved a performance improvement in error correction just by using the adaptive SF α in the existing NMS algorithm [3] . A two-dimensional NMS (TDNMS) algorithm [11] significantly improved the error correction performance by normalizing both the C2V and V2C messages with SFs α (in (4)) and β (in (5)), respectively. These SFs were obtained by using the density evolution [3] [4] , [11] , or experimental results [8] [10] ; the SFs of all the aforementioned MS-based algorithms are less than one.
III. Proposed SNR-Considered Adaptive Normalized MS Algorithm
Since the received SNR affects the decoding performance, the SNR effect should be considered in the decision of SF values of the MS-based algorithms. However, in existing MS-based algorithms, this has not yet been the case. In this section, we propose a new MS-based algorithm that considers the SNR effect. The proposed algorithm uses the adaptive SFs. In decision of these SFs, the SNR effect is mainly considered.
In the MS-based decoding algorithm, the check node selects the lowest among the incoming V2C messages. At the variable node, the corrupted received bit is corrected by summing the received LLR and the incoming C2V messages. In other words, the variable node corrects the corrupted bit by summing its received LLR and the outgoing V2C messages of other variable nodes that share a common check node. Thus, V2C messages are very important information in MS-based decoding. To improve the performance of error correction, the value of a V2C message is adjusted by using the SF in (5).
In adjusting the value of a V2C message at the variable node, we define two types of effects from the point of view of the received SNR. This is because the received SNR is more important than the theoretical SNR threshold in real implementations of an LDPC decoder. First, when the received SNR is very low, many received LLRs are corrupted. At the first iteration, these LLRs are delivered to the neighboring check nodes as V2C messages. Corrupted V2C messages cause incorrect C2V messages, as shown in Fig. 1 . These incorrect C2V messages hinder error correction at the variable node. At every iteration, these corrupted LLRs and incorrect incoming C2V messages cause the wrong bit decision and the wrong outgoing V2C messages at the variable node. In this paper, this effect is called the negative effect of wrong or low-reliability messages.
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Negative effect of low-reliability messages.
Second, when the received SNR is very high, the number of correct received LLRs is much greater than the number of corrupted received LLRs. Thus, many check nodes satisfy the parity-check equation and deliver many correct C2V messages to the variable nodes. At the variable nodes, the correct incoming C2V messages can then correct the corrupted received LLRs, even if there are only a few wrong incoming C2V messages, as shown in Fig. 2 . In this paper, this effect is called the positive effect of the correct or high-reliability messages.
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Positive effect of high-reliability messages.
Based on these two types of effects, we propose an SANMS algorithm for LDPC decoding that gives high priority to the effect of the received SNR. Since not only is the received LLR included in the V2C message at every iteration, but also its reliability is directly affected by the received SNR; thus, the proposed algorithm applies two different SFs; one to the received LLR and the other to the extrinsic information in the V2C message.
Therefore, the proposed SANMS algorithm is expressed as
q ji =  β LLR  ×  2 y j σ 2 +  β ext  ×  i`M( j )\i r i`j ,
where βLLR and βext are the SFs applied to the received LLR and the extrinsic information, respectively. Unlike (5), an additional SF, βLLR , is applied to the received LLR in (6).
The above two types of effects are considered in the decision of the SF values. When the received SNR is low, SFs less than one are mainly considered to reduce the negative effect. On the other hand, SFs greater than one are mainly considered to emphasize the positive effect when the received SNR is high. To verify that the SF considering the received SNR improves error correction performance, an experimental simulation was carried out. In this simulation, (972, 1,944) irregular LDPC code was used, which is defined in the IEEE 802.11ac standards. The maximum number of iterations was set to 10 because only a small number of iterations is considered in the real implementation of an LDPC decoder [6] [8] . Figure 3 shows that the number of wrong V2C messages is affected by the different SF values for the given received SNR at every iteration. The SFs greater than one reduce the number of wrong V2C messages occurring at a high SNR, and the SFs less than one reduce the number of wrong V2C messages occurring at a low SNR. Figure 3 shows that the adaptive SFs, which reduce the negative effect or emphasize the positive effect depending on the received SNR, are able to improve the performance of error correction in the proposed algorithm.
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Experimental analysis of SF effects at low and high SNRs.
IV. Proposed Low Complexity Variable and Check Node Architecture for SANMS Algorithm
Compared to the TDNMS or NMS algorithms, the proposed algorithm requires one or two more multiplications. To solve this problem of complexity, we also propose a low complexity hardware architecture (shown in Fig. 4 ). The proposed combined variable and check node architecture is divided into two processes. The initial process, which multiplies the received LLR by the SF βLLR , is performed once before the iterative decoding because the received LLR does not change in the decoding process. The iterative process, which multiplies the extrinsic information by the SFs α and βext , is performed at every iteration. The iterative process combines the variable and check node operations. Since the variable and check node operations are carried out at different clock cycles, the multiplication of the extrinsic information by SFs α and βext can share one multiplier. The multiplier mul 1, which is used in the initial process, can be reused in the iterative process, as shown in Fig. 4 . Therefore, the proposed algorithm can be performed with just one multiplier by using the proposed combined variable and check node.
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Low complexity combined variable and check node architecture.
The SFs α , βLLR , and βext are stored in the memory. The adaptive SFs βLLR and βext are selected depending on the received SNR. Compared to the overall hardware area of the LDPC decoder, the amount of this memory can be ignored.
V. Simulation Results
The adaptive SF values applied to the proposed SANMS algorithm for the given received SNR were obtained by the experimental simulations, which achieved improved performance compared to the existing MS-based algorithms for the given LDPC codes. Since the goal of the proposed SANMS algorithm is to improve the error correction performance by using the adaptive SFs depending on the received SNR in the realized LDPC decoder, the theoretical analysis of the derivation of the SFs is not covered in this paper. Like the NMS algorithm, the SF α of the check node operation was set to 0.8 to compensate for the performance degradation caused by the minimum function. Tables 1 , 2 , and 3 show the adaptive SF values with regard to the received SNR. The SNR-considered SFs listed in Tables 1 , 2 , and 3 were obtained by the experimental simulations for (324, 648), (648, 1,296), and (972, 1,944) irregular LDPC codes, which are defined in the IEEE 802.11ac standards. The maximum iteration number was set to 10 by considering that the maximum iteration number is usually set somewhere between 10 and 15 in the implementation of an LDPC decoder [6] [8] . A codeword is transmitted over an additive white Gaussian noise channel. From Tables 1 , 2 , and 3 , we can observe that the effective SF values are greater than one for high SNRs, and the effective SF values are less than one for low SNRs. We can also observe that emphasizing the positive effect of the received LLR, by applying the βLLR (that is, βLLR > 1), results in the improved performance for high SNRs, and reducing the negative effect of the extrinsic information, by applying the βext (that is, βext < 1), results in the improved performance for low SNRs. To compare the error correction performance of the proposed algorithm with the existing decoding algorithms, not only the proposed SANMS algorithm but also the SP algorithm [1] and the MS-based algorithms (that is, MS and NMS algorithms [2] [3] ), were simulated. The TDNMS algorithm (that is, the modified MS-based algorithm achieving the significant improvement) was also simulated. Figures 5 , 6 , and 7 show the comparison results of the bit error rate (BER) performances for the SP, MS, NMS, TDNMS, and the proposed SANMS algorithms. In the proposed algorithm, the SFs of Tables 1 , 2 , and 3 were used. The maximum iteration numbers were set to 10 and 30.
Adaptive SFs with respect to SNR for (324, 648) LDPC code.
SNR (dB) LLR, βext) SNR (dB) LLR, βext)
0.8 (0.90, 0.80) 2.0 (1.20, 0.95)
1.0 (0.95, 0.75) 2.2 (1.30, 0.95)
1.2 (1.00, 0.85) 2.4 (1.35, 0.95)
1.4 (1.05, 0.85) 2.6 (1.35, 1.00)
1.6 (1.10, 0.90) 2.8 (1.40, 1.00)
1.8 (1.10, 0.90) 3.0 (1.40, 1.00)
Adaptive SFs with respect to SNR for (648, 1,296) LDPC code.
SNR (dB) LLR, βext) SNR (dB) LLR, βext)
0.8 (0.80, 0.80) 2.0 (1.30, 1.00)
1.0 (1.05, 0.80) 2.2 (1.30, 1.00)
1.2 (1.10, 0.90) 2.4 (1.30, 1.05)
1.4 (1.10, 0.90) 2.6 (1.30, 1.10)
1.6 (1.10, 0.95) 2.8 (1.40, 1.10)
1.8 (1.25, 0.95) 3.0 (1.40, 1.10)
Adaptive SFs with respect to SNR for (972, 1,944) LDPC code.
SNR (dB) LLR, βext) SNR (dB) LLR, βext)
0.8 (0.80, 0.75) 2.0 (1.30, 1.05)
1.0 (1.05, 0.85) 2.2 (1.30, 1.05)
1.2 (1.10, 0.90) 2.4 (1.30, 1.10)
1.4 (1.20, 0.90) 2.6 (1.30, 1.10)
1.6 (1.25, 1.00) 2.8 (1.30, 1.15)
1.8 (130, 1.00) 3.0 (1.30, 1.15)
In Figs. 5 , 6 , and 7 , the black curves and the red curves are the BER performances when the maximum iteration numbers are set to 10 and 30, respectively. When the maximum iteration number is 10, the proposed SANMS algorithm outperforms the existing MS-based algorithms and obtains a coding gain of up to 0.4 dB. Moreover, the proposed SANMS algorithm achieves a BER performance close to the SP algorithm with low complexity. From these simulation results, we can demonstrate that the proposed algorithm is effective in both short and long LDPC codes and can provide the best error correction performance with low complexity at the realized LDPC decoder.
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BER performance comparison for (324, 648) LDPC code.
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BER performance comparison for (648, 1,296) LDPC code.
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BER performance comparison for (972, 1,944) LDPC code.
To verify that the proposed algorithm maintains the coding gain when the maximum iteration number is large, the experimental simulations with the large iteration number were also performed. The maximum iteration number was set to 30. As shown in Fig. 7 , the proposed SANMS algorithm maintains a coding gain for the long LDPC code. Even though all the decoding algorithms except for the MS algorithm show similar performances for the short LDPC codes (as shown in Figs. 5 and 6 ), the proposed SANMS algorithm achieves the better performance compared to the existing MS-based algorithms.
In LDPC decoding, the iterative decoding is performed on each codeword block. If the decoded codeword block does not satisfy the parity-check equation, this codeword block has to be retransmitted. Thus, a block error rate (BLER) performance is an important indicator for evaluating the performance of the LDPC decoding algorithm. Figures 8 , 9 , and 10 show the comparison results of the BLER performances for the SP, MS, NMS, TDNMS, and the proposed SANMS algorithms. In Figs. 8 , 9 , and 10 , the black curves and the red curves are the BLER performances when the maximum iteration numbers are set to 10 and 30, respectively. From these simulation results, we can demonstrate that the proposed SANMS algorithm can provide the best BLER performance compared with the existing MS-based algorithms in the waterfall region regardless of the maximum iteration numbers and the codeword lengths. The proposed algorithm obtains a coding gain of up to 0.4 dB. Unlike the BER performance comparison, the proposed algorithm maintains the coding gains for large maximum iteration numbers.
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BLER performance comparison for (324, 648) LDPC code.
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BLER performance comparison for (648, 1,296) LDPC code.
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BLER performance comparison for (972, 1,944) LDPC code.
Recently, LDPC codes have been employed as error correction codes in wireless communication systems such as IEEE 802.11ac and IEEE 802.16e. These wireless communication systems already have a prefix to estimate the received SNR and the channel responses. The estimated SNR is used to generate soft-information in the wireless communication systems because the general LDPC decoder requires the soft-information in decoding. Thus, the proposed algorithm does not require additional costs compared to the existing LDPC decoding algorithms. Since the proposed SANMS algorithm adjusts the SFs βLLR and βext depending on the received SNR, an accurate SNR estimator is required. The accuracy of the received SNR can be guaranteed by using the SNR estimator proposed in [12] [13] . The SNR estimation algorithms presented in [12] [13] achieved very low mean square error in the SNR estimation. Since the accurate SNR estimation algorithm is not a primary concern of this paper, the details of the SNR estimation technologies are omitted.
Even though the received SNR is imperfectly estimated, only little BER and BLER performance degradation is observed at the waterfall region because the adaptive SFs for the over- or underestimated received SNR are almost the same as the adaptive SFs for the actually received SNR. For example, assume that the actually received SNR is 2.8 dB and the imperfectly estimated received SNR is either 3.0 dB or 2.6 dB. When the received SNR is overestimated, the adaptive SFs for 3.0 dB received SNR are used in decoding. For (972, 1,944) LDPC code, there is no performance degradation because the adaptive SFs for 3.0 dB received SNR are equal to the adaptive SFs for 2.8 dB received SNR, as shown in Table 3 . When the received SNR is underestimated, the adaptive SFs for 2.6 dB received SNR are used in decoding. For (972, 1,944) LDPC code, the adaptive SFs for 2.6 dB received SNR are almost the same as the adaptive SFs for 2.8 dB received SNR, as shown in Table 3 . In the experimental simulations, the proposed SANMS decoding with the adaptive SFs for 2.6 dB received SNR and with the adaptive SFs for 2.8 dB received SNR obtained 9.26 × 10 −7 BER and 8.57 × 10 −7 BER, respectively. From the experimental simulation results, we can demonstrate that only little performance degradation is observed at the waterfall region even if the received SNR is imperfectly estimated.
VI. Conclusion
We proposed the SANMS algorithm for LDPC decoding, which adjusts the received LLR value as well as the extrinsic information value by applying the adaptive SFs. We first defined the negative and positive effect of the received SNR. Based on these SNR effects, the SNR-considered adaptive SFs were obtained. The proposed algorithm with the adaptive SFs achieved the performance improvement compared to the existing MS-based algorithms. We also proposed the combined variable and check node architecture to realize the proposed algorithm with low complexity. In conclusion, the proposed decoding algorithm and architecture can be well applied to the real implementation of LDPC decoders.
This work was supported by the IT R&D pr ogram of MOTIE/KEIT (10035389, Research on high speed and low power wireless communication SoC for high resolution video information mining).
BIO
Yongmin Jung received his BS (summa cum laude), MS, and PhD degrees in electrical and electronic engineering from Yonsei University, Seoul, Rep. of Korea, in 2007, 2009, and 2014, respectively. Since 2014, he has been a senior engineer in the Mobile Communication Division, Samsung Electronics Co., Ltd., Suwon, Rep. of Korea. He received the best paper award in the 2012 International SoC Design Conference. His research interests include error correction encoding/decoding algorithms and SoC/VLSI implementation; wireless communication system algorithms and SoC/VLSI implementation; mobile and video communication algorithms and SoC/VLSI implementation; and image signal processing.
Yunho Jung received his BS, MS, and PhD degrees in electrical and electronic engineering from Yonsei University, Seoul, Rep. of Korea, in 1998, 2000, and 2005, respectively. From 2005 to 2007, he was a senior engineer in the Wireless Device Solution Team, Communication Research Center, Telecommunication Network Division, Samsung Electronics Co. Ltd., Suwon, Rep. of Korea. From 2007 to 2008, he was a research professor at the Institute of TMS Information Technology, Yonsei University, Seoul, Rep. of Korea. He is currently an associative professor at the School of Electronics, Telecommunication, and Computer Engineering, Korea Aerospace University, Goyang, Rep. of Korea. His research interests include signal processing algorithms and SoC/VLSI implementation for wireless communication systems; and image processing systems.
Seongjoo Lee received his BS, MS, and PhD degrees in electrical and electronic engineering from Yonsei University, Seoul, Rep. of Korea, in 1993, 1998, and 2002, respectively. From 2002 to 2003, he was a senior research engineer at the IT SoC Research Center and the ASIC Research Center, Yonsei University, Seoul, Rep. of Korea. From 2003 to 2005, he was a senior engineer in the Core Tech Sector, Visual-Display Division, Samsung Electronics Co., Ltd., Suwon, Rep. of Korea. He was a research professor at the IT Center and the IT SoC Research Center, Yonsei University, Seoul, Rep. of Korea from 2005 to 2006. He is currently an associate professor in the Department of Information and Communication Engineering at Sejong University, Seoul, Rep. of Korea. His current research interests include SoC design for high-speed wireless communication systems; and image processing systems.
Jaeseok Kim received his BS degree in electronic engineering from Yonsei University, Seoul, Rep. of Korea, in 1977; his MS degree in electrical and electronic engineering from KAIST, Daejeon, Rep. of Korea, in 1979; and his PhD degree in electronic engineering from RPI, NY, USA, in 1988. From 1988 to 1993, he was a member of the technical staff at AT&T Bell Labs, USA. He was a director of the VLSI Architecture Design Lab of ETRI from 1993 to 1996. He was a director of the IT SoC research center from 2001 to 2009. He is currently a professor at the School of Electrical and Electronic Engineering at Yonsei University, Seoul, Rep. of Korea and a director of the System IC 2015, National Project, Rep. of Korea. His current research interests include communication VLSI design, high-performance digital signal processor VLSI design, multimedia VLSI design, and CAD S/W.
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