Advanced Soft-Reliability Information-Based Post-Viterbi Processor Jun Lee and Kees A. Schouhamer Immink, Fellow, IEEE

Abstract — This paper proposes a new soft-reliability complexity trade-off offered is very attractive and affordable. information-based post-Viterbi processor with advanced The above approach has been widely studied for magnetic noise-robustness for reducing probability of miss-correction recording channels, and for systems [1-10]. and no correction of a conventional soft-reliability-based In conventional soft-reliability information-based post- post-Viterbi processor. Among all likely error starting Viterbi processor in conjunction with an error detection code positions for prescribed error events, the two schemes are [2], when the syndrome is non-zero, the error detection code equal to attempt to correct error-type corresponding to a generates a set of all likely error starting positions for position with minimum one only if there exist positions where prescribed error events. Then, the scheme computes the soft- a soft-reliability estimate is negative. The main difference reliability values over the set of likely error starting positions, between the two schemes is how they acquire the soft- and it outputs an error starting position and its error- reliability estimate. The soft-reliability estimate of the new type associated with minimum one only if there exist likely scheme is obtained through the elimination of the noise- error starting positions that generate the negative soft- sensitive component from the log-likelihood ratio of the reliability estimate. Finally, based on the position and its posteriori probabilities, which is the soft-reliability estimate error-type, the scheme performs the error correction. Note that of conventional scheme. As a result, the new scheme is based on more reliable soft-reliability information so reducing the the scheme attempts to perform error correction only if probability of miss-correction and no correction. positions with negative soft-reliability estimate exist. A soft-reliability estimate for the conventional scheme is Index Terms — Error detection code, dominant error event, given as the log-likelihood ratio of the posteriori probabilities soft-reliability information-based post-Viterbi processor, cyclic [2], which includes a noise-sensitive component. The noise- redundancy check code. sensitive component is the main source of detrimental factors such as miss-correction and no correction. For the actual error  I. INTRODUCTION starting position, we found that the scheme leads to no The demand for high-density digital data storage systems correction or miss-correction because the soft-reliability has been growing steadily. Although technological estimate of the position fails to give negative value due to a innovations in the design of recording media and heads are specific noise pattern. Note that the soft-reliability for the key to achieving high density recording systems, the role of actual error starting position usually gives a negative value, sophisticated coding and signal processing techniques for data but sometimes a positive value is yielded only if the noise recovery is increasingly becoming crucial in supporting and characteristic roughly matches to error signal samples of augmenting these advancements. actual error event. There has been a growing interest in error detection codes This paper introduces a new soft-reliability information- with error correction properties [1]-[10]. Unlike conventional based post-Viterbi processor with improved noise-robustness. read channels, where the error correction code (ECC) is In new and conventional schemes, the procedures for finding expected to correct all the errors at the output of the a set of likely error starting positions and performing soft- constrained decoder, dominant error events are corrected by reliability information-based error-correction over the set are applying a low redundancy error detection code. This error the same. The main difference between the two schemes is detection code is an inner ECC that can correct dominant error how they obtain the soft-reliability estimate. The new scheme events at the output of the channel detector by using only a obtains more reliable soft-reliability information by removing few parity bits. In this way, the correction capacity loss of the the noise-sensitive component from the log-likelihood ratio of outer ECC is significantly reduced and the error propagation a posteriori probabilities. The new scheme can give minimum of the modulation decoder is also minimized. The approach negative soft-reliability value at actual error starting position using error detection codes, referred to as post-Viterbi even when the channel noise is severe and consequently, the processor (in other words, maximum likelihood (ML) post- probability of miss-correction and no correction is processor), has found wide acceptance since the performance- significantly reduced. The performance has been evaluated for the magnetic recording channel. With only a few alterations, 1 Jun Lee is with Data and Storage R &D Laboratory, LG Electronics in the technique can be applied to optical storage systems. Korea (e-mail: [email protected]). Kees A. Schouhamer Immink is with the Institute for Experimental The paper is organized as follows. Section II introduces the Mathematics, Ellernstrasse 29-31, Essen, (e-mail: immink@turing- new technique. In Section III, simulation results are given, machines.com). and finally, conclusions are given in Section IV.

II. A NEW POST-VITERBI PROCESSOR lh 1 rhbnsnkikikkk   , (1) This section firstly overviews conventional soft-reliability i0 information-based post-Viterbi processor. Secondly, a new soft-reliability information-based post-Viterbi processor will where bk is a k-th bipolar coded bit of b , hk is a k-th element be described. lh 1 of a channel target response h0  [hh01 ,...,l  ] of length lh , A. Conventional soft–reliability information -based Post- h n is an AWGN sample, and s is a k-th signal sample Viterbi Processor k k generated by the convolution of the bipolar coded sequence b The performance of a partial response maximum likelihood lh 1 (PRML) system can be improved by employing a soft- and the channel target response h0 . By definition, the ML reliability information based post-Viterbi processor based on ˆ ˆˆ detector selects a bipolar coded sequence b [bb01 ,...,N  ] that an error detection code that can correct a dominant error event minimizes the Euclidean metric at the output of the channel detector. In the soft-reliability information-based post-Viterbi processor based on error N 1 2 detection code [2], an error detection decoder computes a  rskk ˆ , (2) syndrome to check for the presence of errors in the estimated k0 codeword, which is found at the output of the channel detector. When the syndrome is non-zero, the scheme is activated. where sˆk is a k-th signal sample yielded the convolution of the Based on the syndrome, the scheme generates a set of all output sequence of the ML detector bˆ and the channel target likely error starting positions for prescribed error events and response hlh 1 , i.e., then it attempts to find a position and its error-type with 0 minimum one only if there exist positions that yield negative l 1 soft-reliability estimate over the set. Note that the number of h ˆ sˆkiki hb . (3) position with negative soft-reliability value is mostly one, i0 but rarely it is more than one because of severe channel impairments. If positions that produce negative soft-reliability We try to design a decoding scheme that corrects one of estimate do not exist, the scheme does not make any prescribed error events occurred at the output of the ML correction. However, it is observed that the scheme makes no detector that dominates the other error events. Let us assume correction or miss-correction with high probability even when that one of the prescribed error events error occurs in the estimated codeword. As a result, the e()i ,{1,,}iE   occurs in a codeword, where E is the number of bit errors increases which can be a significant detrimental factor to the detection performance. We have number of prescribed error events. Then, Li likely error observed that the problem happens when noise characteristic ()i starting positions  pij ,{1,,}and1,, E j Li  for approximately matches to error signal samples of actual error event. For solving the problem, we propose a new soft- e()i ,{1,,}iE   from a syndrome computed by error reliability information with better noise-robustness. The soft- detection code are given, where L is the number of likely reliability information of the conventional scheme is given as i error starting positions for a prescribed i-th error event. If the the log-likelihood ratio of the posteriori probabilities, but that ()i ()i of the new scheme is obtained by adding a correction length of the error event e is l , and its error starting ()i ()i component that reduces the sensitivity of channel noise to log- position is pm among  pjj ,1,,and1 Lii mL , likelihood ratio of the posteriori probabilities. Then, the new ()i information becomes more robust to noise compared to that of then the error event e is given by conventional scheme and consequently, the new scheme has a e()i  ee()ii,,, () e () i smaller bit error rate performance than the conventional one. 01 l()i 1  ˆˆ ˆ  (4) B. A New Soft-reliability Information-based Post-Viterbi bb()ii,,, () b () ii () bb () ii ,,, () b () ii ()  pp11 pl pp 11 pl  Processor  mm m mm m  ()ii () () ii () plmm11ˆ pl  bb()ii () We describe a sub-optimum post-Viterbi error correction ppmm scheme based on the new soft-reliability information. We assume that a bipolar codeword b [bb01 ,...,N  ] of length N, and the error signal vector, which is the convolution of the generated by an error detection encoder, is transmitted over a ()i lh 1 error event e and the channel target response h0 , is partial response channel and corrupted by additive white expressed as Gaussian noise (AWGN).

The k-th input signal of the ML detector, rk is expressed as ()ii () ()i pllmh2 ()ii () () ii () e plmmh 111ˆ pl l sbbh()ii () 0 ()i ppmm pm 

()ii () () ii () pll()i ()i 2 22 pllmh22 pll mh  jh ssˆ  ()i rsˆˆ rs ()ii () . (5) kp kk kk ppmm j  . (9) ()i ()i ()ii2 () plljh2 ee ()i   ssn20 e kp ()i  kkk   Here, each error signal sample sk is j 

l 1 e()i h ˆ Actually, a conventional post-Viterbi scheme based on error sbbhk  kikii i0 correlation filter [1] is easily derived from the soft information , (6) ()i l 1 e h ()i ()iii () () estimate in (7). The error signal ek between the equalizer  ehpkpll()i imfor m h 2 kpm i i0 output rk and the signal sample sˆk is

()i ()i where e j  0 if j  0 or jl1. Let sˆk be a k-th signal ()ii () lh 1 esnee  e()i  hn. (10) kkkkp()i i ik sample produced by the convolution of the flipped ML i0 m ()i ˆˆˆˆ pm 1 detector output sequence bb[bb01 ,...,N  ] [0 , ()ii () Then, the error signal is convolved with a bank of error ˆˆplm 1 N 1 ()i -b()i ,] b ()ii () , according to an error event e of length pm plm  correlation filters to find the most probable error starting ()i ()i positions and their error types of each prescribed error event. l and corresponding likely error starting positions pm for Let us assume that an error event e()i in estimated codeword lh 1 1 mLi , and the channel target response h0 , i.e., ()i occurs at pm , which is one among lh 1 ˆˆ . ()i skkii bh pj,1,,and1 L mL. Then, for the error i0  j ii In [2], we derived a criterion of conventional soft-reliability correlation filter corresponding to e()i with length l()i , the information-based post-Viterbi processor that finds the most e()i ()i probable error starting position among all likely error starting filter output f at pm becomes in vector notation ()i positions pij ,1,...,and1,, E j Li for the prescribed   T pll()ii ()22 pll () ii  ()  ()i ()ii ()mh () i mh error events e ,1,,iE   is fsee   s e ()ii  () ppmm  T , (11)   pll()ii () 2 ()i mh pll()ii () 2 ()i ()i e mh  plljh2 22 sn ˆ  p()i Pargmin ()i rsˆˆ rs 0. (7) ()i m  kp kk kk  pm all piE{1,..., } j   jL{1,...,i }     Soft-reliability information  (Log-likelihood ratio of posteriori probabilities) ()ii () pllmh 2 where n ()i is the AWGN vector. Basically, equation pm Equation (7) means that the error correction is performed (11) is supposed to give the maximum output compared to based on an error starting position and its error-type e()j associated with minimum one only when it exist likely error other error correlation filter outputs f corresponding to the starting positions with negative soft-reliability value. For the most likely error starting positions of each prescribed error ˆ ()i ()i most likely error starting position P=pm among event e()j , where j  1,...,Eji and  . But, f e often fails to ()i pi()i ,1,...,and1,, E j Lfor e ,1,,iE  , soft- ()j  j i    give the maximum value, so that f e for j 1,...,Eji and reliability estimate in (7) is gives the maximum and the miss-selection occurs. Moreover, ()j e()i e ()ii () 22 although f yields the maximum compared to f for pllmh2 rsˆˆ rs  ()i kk kk ()i kp m  j  1,...,Eji and  , it has been often observed that f e can 2 . (8) ()ii () ()ii ()  pllmh2 ssnee 20   produce the incorrect error position (miss-positioning, kp ()i  kkk   m  pj()i ,1,,and L jm). The main reason of the miss-  ji For the other likely error starting positions ()ii () T ()i pllmh2 ()ii () e pllmh 2 ()i correction is because of s  n p ,1,...,and1,,()iEjLjm , soft-reliability value p()i  ji p()i m m in (7) is in (11). We can roughly express the severe AWGN sequence,

when the error event e()i occurs, as

()ii () ()i 2 ()ii ()()i pllmh2 () ii () pll()i 2  ()ii ()  pllmh22e pll mh  jhee ns()ii  ε () , (12)  ()i ssnkkk2 pp()i kp    mm    j pm   . (15) ()i ()i ()ii2 () plljh2  ee ()ii ()  ss2  pllmh2 kp ()i  kkk   where ε ()i is a vector that consists of any real j   pm e()i numbers. When we plug (12) into (11), then f becomes For non-error starting positions, the soft-reliability information (15) can produce positive value because of T ()ii () () ii () 2 ()ii ()pllmh22 () i pll mh  ()ii () ()i ee  e ee e fs  s sskkk2  even if 20skk . However, for ()ii  ()       ppmm T the actual error starting position, the soft-reliability pll()ii ()22 pll () ii  () ()iimh () mh information (14) may fail to give negative value ssee  ()ii () 2 pp  ()ii () mm because ee . Note that the soft- . (13) sskkk  20   ()ii () T  ()i pllmh2 (ii)() e pm llh 2  s ε ()i reliability information for the actual error starting position ()i pm pm usually gives negative value. Sometimes a positive value is T pll()ii () 2 yielded if the AWGN characteristic roughly equals ()i mh pll()ii () 2 e mh ()ii () s ε ()i ()i pllmh 2 ()i pm e pm  s . Clearly, compared to the other likely error p()i m starting positions, the soft-reliability information for the actual e()i So, it is highly probable that the filter output f cannot yield error starting position is a smaller positive value. Then, the the maximum output, so that it is prone to miss-correction. If threshold for correction of an actual error event may be not this is the case, then we have to check soft-reliability estimates exactly zero, but slightly larger. Since determination of (8) and (9). Assuming severe AWGN sequence (12), we can whether the estimated error starting position is correct rewrite the soft-reliability estimates (8) and (9). For the most depends upon the threshold zero, we need to have more ˆ ()i probable error starting position P=pm among reliable soft-reliability estimate. ()i ()i For a more reliable soft-reliability estimate, we try to pij ,1,...,and1,, E j Li for e ,1,,iE  , the     e()i eliminate 2 skk n (noise-sensitive component) in (14) AWGN can be sometimes assumed as (12). Accordingly, k   equation (8) is ()ii () eeˆ by adding 22()skknsrs k  kk (correction kk   ()ii () 22 pllmh2   ()i rskkˆˆ rs kk component) to the conventional soft-reliability estimate, kp m  which is the log-likelihood ratio of posteriori probabilities. As 2 ()ii () ()ii ()  pllmh2 ssnee 2  a result, a new criterion for finding the most probable error kp ()i  kkk   m  ()i . (14) starting position among pij ,1,...,and1,, E j Li  for 22 pll()ii () 2 eee()iii () ()  mhsss 22   ()i kp ()i kkkk   e ,1,,iE  is m      2 ()ii () ()ii ()  pllmh2 ssee2  kp ()i  kkk   m   22 rskkˆˆ rs kk pll()i ()i 2  ˆ jh ()i For the other likely error starting positions Pargmin ()i e 0. (16) iE{1,..., } kp j 2(rssˆ ) ()i all p kkk   p ,1,...,and1,,()iEjLjm , the AWGN can be jL{1,...,i }   ji Correction term  ()i ()iii () () A new soft-reliability information plljh22 pll mh   assumed as n ()i  ε ()i because there are p j pm ˆ ()i no errors in the likely error positions. Accordingly, equation For the most likely error starting position P=pm among (9) is pi()i ,1,...,and1,, E j Lfor e()i ,1,,iE  , soft-  j i    ()i ()i plljh2 22 reliability estimate in (16) is  ()i rskkˆˆ rs kk kp j 

()ii () 22 ()i pllmh2 e  ()i rskkˆˆ rs kk2( rss kkk ˆ ) kp m  2 ()ii () ()ii ()  pllmh2 ns2 ee n2 ns kp ()i kk k kk m  . (17) 2 ()ii () ()ii () () i  pllmh2 nn2222 nsee s ns e kp ()i kk kk k kk m  2 ()ii ()  ()i   pllmh2 se  0 kp ()i  k   m  

For the other likely error starting positions ()i  p ji,1,...,and1,,()iEjLjm  , soft-reliability value in (16) is

()i ()i ()i plljh22 2 e  ()i rskkˆˆ rs kk2( rss kkk ˆ ) kp j 

()i 2 pll()i 2 ()iii () ()  jhsneee n2 2 sns kp ()i  kk k kkk j Fig. 1. Comparison of BERs of post-Viterbi processors under N50.

2 ee()ii () 22 ssnnnkkkkk2    . (18) pl()i  ()i l 2     j h  kp ()i 2 j ()ii () 22ssnee  kkk

()i 2 pll()i 2 ()ii ()  jh34ssnee 0 kp ()i  kkk j 

With the modified soft information, the soft-reliability information for an actual error starting position becomes equal to the square sum of the corresponding error signal. So, there is no ambiguity to decide the actual error starting position. In other words, the soft-reliability information (17), for the actual error starting position, becomes negative value irrespective of noise characteristic and the soft-reliability estimates (18), for the other non-error starting positions, are always positive

()ii2 () value because of ee. Therefore, the new 34skkk  sn  soft-reliability information becomes more reliable and Fig. 2. Comparison of BERs of post-Viterbi processors under N100. consequently, the probability of miss-correction and no correction of the new scheme is significantly reduced dominant error events (E=6) at user density 1.4 are listed in compared to that of conventional scheme. [1][9][10]. As a reference, the BER of a PRML (1, 6, 7, 2) system is also shown at the same user density 1.4. For the user III. SIMULATION RESULTS density 1.4, the corresponding channel densities for the PRML (1, 6, 7, 2) system and post-Viterbi processors are 1.4 (=D A (203, 200) cyclic redundancy check (CRC) code u generated by a generator polynomial g(x) = 1+x2+x3 is used as /R=1.4/1) and 1.42 (Du/R =1.4/(200/203)), respectively. The an error detection code [11] and the code can detect the signal-to-noise ratio (SNR) has been defined in [10] as the dominant error events for perpendicular recording [9][10]. ratio of the energy of the first derivative of the transition response E and the noise spectral density. In our simulations, The bit error rates (BERs) of post-Viterbi processors are dt simulated and compared at user density Du=1.4 with the the noise parameter N50 or N100 in the SNR definition signifies 3 channel target response of h0  [1,6,7,2](h(D) = 1 + 6D + a mixture of 50% AWGN and 50% jitter noise or 100% 7D2 + 2D3 , where D is one-symbol delay) over perpendicular AWGN, respectively. Fig. 1 shows the BERs of conventional, advanced and new post-Viterbi processors under N . In Fig. recording. The user density Du is defined by Du = R  Dc, 50 where R is the code rate and Dc is the channel density. The 1, the legend “conventional scheme” [1] corresponds to an error correlation filter-based post-Viterbi processor that [2] J. Lee and K.A.S. Immink, “A New Post-Viterbi Processor Based on determines the most probable error starting position and its Soft-Reliability Information, ” IEEE Trans. on Consumer Electronics, vol. 57, no. 2, pp. 842-847, May. 2011. error-type based on a likelihood value, “advanced scheme” [2] [3] T. Conway, “A new target response with parity coding for high density means conventional soft-reliability information-based post- magnetic recording channels,” IEEE Trans. on Magn., vol. 34, no. 4, pp. Viterbi processor that chooses the most likely error starting 2382–2386, July 1998. [4] K. Cai and K.A.S. Immink, “A General construction of constrained position based on soft-reliability estimate, and “new scheme”, parity-check codes for optical recording,” IEEE Trans. on Commun., which is proposed in the paper, is a new soft-reliability vol. 55, no. 7, pp. 1070-1079, July 2008. information-based post-Viterbi processor that supplies more [5] R. D. Cideciyan, J. D. Coker, E. Eleftheriou, and R. L. Galbraith, “Noise predictive maximum likelihood detection combined with parity-based reliable soft-reliability estimate than “advanced scheme. All post-processing,” IEEE Trans. on Magn., vol. 37, no. 2, pp. 714–720, post-Viterbi processors produce considerable performance Mar. 2001. gains compared to conventional PRML (1, 6, 7, 2) system. [6] W. Feng, A. Vityaev, G. Burd, and N. Nazari, “On the performance of parity codes in magnetic recording systems,” in Proc. IEEE Among the post-Viterbi processors, the “conventional GLOBECOM 2000, pp. 1877–1881. scheme” results in worst performance because the scheme [7] K. Saeki and Z. Keirn, “Optimal combination of detection and error does not have any criterion for judging whether an estimated correction coding for magnetic recording,” IEEE Trans. on Magn., vol. error starting position is correct. Thus, the scheme 37, no. 2, pp. 708–713, Mar. 2001. [8] Z. A. Keirn, V. Y. Krachkovsky, E. F. Haratsch, and H. Burger, “Use of accomplishes error correction based on an error starting redundant bits for magnetic recording: Single-parity codes and Reed position and its error-type corresponding to the maximum Solomon error-correcting code,” IEEE Trans. on Magn., vol. 40, no. 1, likelihood value whenever the syndrome of a codeword is pp. 225–230, Jan. 2004. [9] J. Moon, J. Park and J. Lee, “CRC-based high-rate error detection code non-zero. In the “advanced scheme”, the miss-correction of for perpendicular recording,” IEEE Trans. on Magn., vol. 42, no. 5, pp the dominant error events is reduced compared to 1626-1628, May 2006. “conventional scheme” because the scheme attempts to [10] J. Moon and J. Park, “Detection of prescribed error events: Application to perpendicular recording,” in Proc. IEEE ICC 2005, vol. 3, pp. 2057– perform error correction only if there exist error starting 2062, May 2005. positions whose the soft-reliability estimates are negative. The [11] S. Lin and D. J. Costello, Error control coding : Fundamentals and “new scheme” yields the best performance among the schemes application (second edition), Prentice Hall, 2004. considered here. As shown in Fig. 1, while “advanced scheme” suffers from no correction and miss-correction due to Jun Lee received his B.S. and M.S. degree from channel noise, the newly proposed soft-reliability information- Dongguk University, Seoul, Korea in 1998 and based scheme overcomes the problem by using a more reliable 2000, respectively. Since March 2000, he has been a Ph.D. student in Dept. of Electronic Engineering at soft-reliability estimate. Fig. 2 shows a comparison of the Dongguk University. In 2003, In 2003, he BERs of conventional, advanced, and new post-Viterbi received Ph. D. degree and he joined the faculty of Samsung Advanced Institute of Technology processors under N100 . The result shows the same (SAIT), Suwon, Korea, and he is currently working performance trend as shown in Fig. 1, irrespective of the noise with LG Electronics. His research interests are distribution. signal processing and coding for storage systems and communication theory. He received, with Kees Schouhamer Immink, the Chester Sall Award for the 1st place best paper in the IEEE Transactions on IV. CONCLUSION Consumer Electronics 2009 We have investigated a new soft-reliability information- based post-Viterbi processor that reduces the probability of Kees Schouhamer Immink received his PhD miss-correction and no correction of conventional scheme. degree from the University of We have found that miss-correction and no correction of the Technology. He was with Research Labs in Eindhoven from 1968 till 1998. He founded and conventional scheme result from the soft-reliability became president of Turing Machines Inc. in 1998. information with noise-sensitive component, which is given as He is, since 1994, an adjunct professor at the the log-likelihood ratio of posteriori probabilities. We have Institute for Experimental Mathematics, Essen proposed more reliable soft-reliability information with better University, Germany. Immink designed coding techniques of virtually all consumer-type digital noise-robustness obtained by adding correction component to audio and video recording products, such as the log-likelihood ratio of posteriori probabilities. By , CD-ROM, CD-Video, Digital Audio Tape recorder, Digital simulation, we have found that probability of miss-correction Compact Cassette system, DCC, Digital Versatile Disc, DVD, Video Disc Recorder, and Blu-ray Disc. He received widespread recognition for his many and no correction of the new scheme is considerably reduced contributions to the technologies of video, audio, and data recording. He compared to that of conventional scheme. As a result, we received a Knighthood in 2000, a personal ‘Emmy’ award in 2004, the 1996 conclude that the new technique is a good candidate for high- IEEE Masaru Ibuka Consumer Electronics Award, the 1998 IEEE Edison Medal, 1999 AES Gold Medal, the 2004 SMPTE Progress Medal, and with capacity storage systems. Jun Lee, the Chester Sall Award for the 1st place best paper in the IEEE Transactions on Consumer Electronics 2009. He was named a fellow of the REFERENCES IEEE, AES, and SMPTE, and was inducted into the Consumer Electronics Hall of Fame, and elected into the Royal Academy of Sciences [1] J. Lee and K.A.S. Immink, “Advanced Signal Processing Technique for and the US National Academy of Engineering. He served the profession as Storage Systems, ” IEEE Trans. on Consumer Electronics, vol. 56, no. 4, President of the Audio Engineering Society inc., New York, in 2003. pp. 2373-2379, Nov. 2010.