Application of High-Rate Tail-biting Codes to Generalized ... - CiteSeerX

Application of High-Rate Tail-biting Codes to Generalized Partial Response Channels M. T¨uchlery , C. Weißy , E. Eleftheriouz , A. Dholakiaz, and J. Hagenauery

y Institute for Communications Engineering, Munich University of Technology, Arcisstr. 21, D-80290 M¨unchen, Germany z IBM Research, Zurich Research Lab., S¨aumerstr. 4, CH 8803, R¨uschlikon, Switzerland Abstract—The performance of high-rate tail-biting convolutional codes serially concatenated with generalized partial response channels is studied. The effect of precoders on the overall performance is investigated. Extrinsic information transfer charts are used to guide the selection of appropriate tail-biting codes and precoders. Simulation results for a magnetic recording system modeled as a serial concatenation of tail-biting codes with a generalized partial response channel are presented. In particular, rate-8/9 and -16/17 short and long block length tail-biting codes are studied. In the former case, hard-decision decoded interleaved Reed-Solomon (RS) codes are used as the outer-most code, whereas in the latter case the sector-size tail-biting codes replace the RS codes traditionally used in storage systems. The results indicate that high-rate tail-biting codes deliver significant performance gains when used in conjunction with a rate-1 precoder and iterative detection/decoding. Furthermore, the results also show that long tail-biting codes can outperform hard-decision decoding of RS codes by approximately 2.25 dB at a sector error rate of 10 4 .

I. I NTRODUCTION Turbo codes [1] and low-density parity-check (LDPC) codes [2, 3] are recent breakthroughs in coding theory that have been shown to approach the capacity of the additive white Gaussian noise (AWGN) channel with moderate decoding complexity. The original turbo codes used two or more parallel concatenated convolutional codes with an interleaver and were decoded via an iterative procedure. Very soon it became clear that similar performance could also be achieved with serial concatenation of the convolutional code components [4]. The remarkable performance achieved by concatenation of simple component codes and decoders should be attributed to the “turbo” principle, namely, iterative a-posteriori probability (APP) estimation of the symbols with successively refined a-priori distributions [5]. This principle can be used in many communications applications including magnetic recording. Recent work in investigating iterative decoding schemes for the magnetic recording channel based on serial as well as parallel concatenated turbo codes and LDPC codes has been reported in [6, 7] and references therein. The high-rate convolutional codes used in the turbo code schemes are exclusively punctured from low-rate optimal codes. We use a class of rate- (n 1)=n convolutional codes with a decoding performance significantly better than that of punctured codes with a comparable decoding complexity. However, due to their encoding principle the overhead using trellis termination is large. This overhead can be avoided using tail-biting encoding [14]. A soft-input/softoutput (SISO) decoder for the considered class of codes based on a modified BCJR algorithm [8] generates the likelihood information needed between the component decoders of the serially concatenated system. It should be emphasized that to apply iterative decoding schemes to magnetic recording systems, high code rates must be considered. It was shown in [11] that for overall rates of less than 0.8, there are diminishing returns in performance owing to the more severe code-rate penalty on the magnetic recording channel than on the AWGN channel. Furthermore, with data

rates on high-end hard-disk drives exceeding 1 Gbit/s, high-rate codes are again favorable because the internal clock rate scales inversely with the code rate. Finally, low-rate codes require that the data be recorded at very high normalized linear densities, which in practice would render the recording process unreliable because of the thermal instabilities in the media. Clearly, the introduced codes offer an alternative way to obtain powerful high-rate, short- and long-block-length codes, and to our knowledge have not yet been considered for magnetic recording applications. In this paper, we investigate serial concatenation of inner generalized partial response (PR) channels and outer high-rate tail-biting convolutional codes of various block sizes. In the literature, such a configuration is referred to as turbo equalization [6, 9–11]. In particular, we concentrate on rate-8/9 and -16/17 tail-biting codes having block lengths of approx. 500 and 4000 bits. In the former case, hard-decision decoded interleaved Reed-Solomon (RS) codes are employed as the outermost code, whereas in the latter the long tail-biting code replaces the traditionally used RS codes. In both cases, the effects of various rate-1 precoders on the overall performance and potential error floors are investigated. Extrinsic information transfer charts are used to guide the selection of appropriate tail-biting codes and precoders. Finally, simulation results for a magnetic recording system modeled as a serial concatenation of tail-biting codes with a generalized PR channel are presented. II. S YSTEM C ONFIGURATION A block diagram of the recording channel, modeled as a generalized PR intersymbol interference (ISI) channel, together with a serially concatenated detector/decoder configuration is shown in Fig. 1. User data, typically organized into sectors of 512 bytes, are fed to a RS encoder that operates on 8-bit symbols. Each sector is encoded into nRS RS codewords that are then byte-interleaved to a depth of nRS bytes. The interleaved stream bn 2 f0; 1g is fed to the tail-biting encoder, which generates length N blocks , [ 0 1 ::: N 1 ℄ of code bits n 2 f0; 1g using tail-biting codes of rate-(n 1)=n. The encoder output is interleaved using the permutation ( ) to 0 . The deinterleaver  1 (
) reverses (
). The block 0 is passed through a rate-1 precoder represented by the polynomial p(D) = 1+ p1 D + ::: + pP DP , pi 2 f0; 1g, yielding the bits 00n = 0n  p1 00n 1  :::  pP 00n P , where  denotes addition modulo 2. The precoded stream is mapped into bipolar symbols +1 and 1, which are then written onto the disk by a write-head. Adopting a linear model for the write/read process on a magnetic disk and assuming the presence of thermal noise only, the magnetic recording channel is represented by a Lorentzian channel model characterized by a single parameter, namely, the normalized linear density pw50 =T . The output of the recording







2

User Data

User Data

RS Encoder

RS Decoder

Interleaver

De-Interleaver

^bn

bn

Encoder

Q

(dec)

n

Q

Noise Coloring Filter

ln

ln

Q

1

l0

n

0n

Precoder

Fig. 2. Terminated and tail-biting trellis of a convolutional code.

MAP Decoder

wn

0(dec)

ln

00n

Generalized

zn

PR Channel

MAP Detector (Inverse precoder)

Fig. 1. The magnetic recording system with iterative detection/decoding. 2 . The data channel is corrupted by AWGN wn with variance w signal is then read back via a low-pass filter (LPF) and shaped such that the overall discrete-time transfer function, including the head/disk-medium characteristics, the analog LPF, and the sampler, closely matches a generalized PR polynomial f (D). In particular, polynomials of the form f (D) = (1 D2 )s(D), where s(D)=1+s1 D+ ::: +sS DS , si 2 R, is a noise whitening filter render the total noise at the input of the detector approximately white, provided the equalizer and the predictor filter are sufficiently long [12, 13]. In practice the memory of the generalized PR polynomial is typically four and the noise is slightly colored. The channel output samples zn are then fed into a 2S +2 state maximum a-posteriori probability (MAP) symbol detector, where S +2= deg(f (D)). Because we use turbo equalization, the detector has available a-priori information L( 0n ) about the code bits 0n , where L( ) , ln(P ( = 0)=P ( = 1)) is the log-likelihood ratio (LLR) of . Initially (first detection), we have L( 0n )=0, 8n. The detector outputs the a-posteriori LLR minus the a-priori LLR [25]

Ldet ( 0n ) , L( 0n jz) L( 0n ) , ln

z

P ( 0n =0 j z) L( 0n ): P ( 0n =1 j z)

where , [z0 z1 :::zN 1 ℄. Inverse precoding is included in the detection procedure, too (only the trellis branch labeling has to be adjusted). For that reason, we will use precoder polynomials p(D) of degree P  (L +2). After deinterleaving, the LLRs Ldet ( n ) are considered a-priori information L( n ) for the decoder, which computes ^bn of the transmitted data bits and

Ldec ( n ) , L( n jy) L( n )=ln

y

P ( n =0 j y) L( n ); P ( n =1 j y)

where , [L( 0 ) L( 1 ):::L( N 1 )℄. After interleaving, the LLRs Ldec ( n ) are considered a-priori information L( 0n ) for the detection task in the subsequent iteration. After meeting a suitable stopping criterion, the tail-biting decoder produces hard decisions ^bn , which are deinterleaved and passed to the RS decoder. Finally, the RS decoder corrects a prespecified number of byte errors in each estimate of an RS codeword, and the output is delivered to the user. In our investigations, we have explored the possibility of replacing the RS code by just a long tail-biting code implying a larger interleaver which has shown to be beneficial for all systems relying on the turbo principle. 1 In this case, the user sector 1 Clearly, since tail-biting codes do not suffer from a rate loss irrespective of

data is directly fed to the tail-biting encoder. In the case of short tail-biting codes, the RS code and byte-interleaving are necessary to meet the stringent performance requirements as well as to eliminate the potential error floors seen when using only the short tail-biting codes. We call the latter option that involves the use of RS codes “mode 1” and the former “mode 2”. In Fig. 1, the blocks drawn with dashed lines distinguish between the two modes of operation and also show the optional use of a precoder. III. C HANNEL C ODING A. Tail-Biting Codes Using tail-biting instead of zero termination avoids the rate loss due to termination without suffering from a degraded error performance [14]. Now, the state of the encoder at the beginning of the encoding process is not necessarily the all-zero state but can also be any of the other states. The idea behind tailbiting is to control the encoder such that it starts and ends the encoding process in the same state (tail-biting boundary condition). In the trellis representation of a tail-biting code only those paths that start and end in the same state are valid code words. This suggests to wrap around the trellis and join the starting and ending states–like a snake biting its own tail–which is why this encoding method is called tail-biting. Fig. 2 shows a terminated and a tail-biting trellis of a convolutional code consisting of 5 sections. The tail-biting codes considered in this paper are represented by an (n 1)n generator matrix (D)=[ (D)℄, where is the (n 1)  (n 1) identity matrix and

G

Ir

I

r(D) = 1=q(D)
[g (D) g (D) ::: g (D)℄ 1

2

N

1

T

1)  1 vector containing n 1 numerator polynomials gi (D) and a common feedback (recursive) polynomial q (D). This class of codes gives rise to rate-(n 1)=n tail-biting codes. is a (n

For feedforward encoders, satisfying the tail-biting boundary condition is trivial, since the last information bits determine the state of the encoder at the end of the encoding process. For example, a memory m code of the considered class of rate(n 1)=n codes with q(D) = 1 can be encoded by initializing the encoder with the last (n 1)
m information bits. When the encoder has feedback, e.g. for non-trivial q (D), the correct initial state has to be computed employing the state space representation of the encoder [15]. Using a terminated trellis, the overhead due to termination would be m
n, which is rather large given the rates and memories of the codes considered in this paper. their block length, we could achieve the same effect by concatenating several short tail-biting codes to fill a long interleaver opening up, e.g., the possibility of parallel processing.

3

B. SISO Decoding of High Rate Tail-Biting Codes Since the invention of “turbo” codes [1], SISO decoders have turned out to be one of the key ingredients for the excellent performance of iterative decoding schemes. The optimum SISO decoder is based on the symbol-by-symbol MAP decoding rule, which is to use the sign of L( n j ) to estimate the transmitted

n . We point out that the SISO decoder actually communicates L( n j ) itself (soft information) rather than the hard-decided code bits. Since in a serially concatenated scheme the observation already consists of a-priori LLRs on the code bits n , we can write

y

y y

P = ln P 2C P Q 2C = L( ) + ln P 2C Q

P ( n =0 j y) L( n jy) = ln P ( n =1 j y) n

2C : n =1

1

e N 1 e j =0;j 6=n

N

: n =0

p( ; y) : n =1 p( ; y)

: n =0

L( j ) j

6

j =0;j =n

L( j ) j

(1)

;

where C denotes the set of codewords. In practical systems, it is advantageous to work in the logarithmic domain as opposed to the probability domain because of the inherent normalization of LLRs. As the codewords of a tail-biting code that start in a particular trellis state show the same structure as that of a terminated convolutional code (see Fig. 2), efficient trellis-based MAP decoding based on the BCJR algorithm [16] can also be applied to tail-biting code. In principle, the BCJR algorithm has to be performed for every initial state [8]. The complexity of the BCJR algorithm depends heavily on the number of branches leaving each state of the trellis. For example, from each state of a trellis for one of the rate- (n 1)=n codes considered here originate 2n 1 branches. Therefore, a direct application of the BCJR algorithm to high-rate convolutional codes, i.e., codes with rate larger than 1=2, is often too complex. One alternative are suboptimal MAP decoding algorithms that approximate (1), e.g., the suboptimal tail-biting MAP decoder proposed in [17]. Exact decoding with low complexity is possible based on the Poisson summation formula, which enables us to equivalently compute L( n j ) based on the codewords ? of the dual code C ? of C [18]:

y

L( n jy) = L( n )+

P

IV. EXIT C HART A NALYSIS

Q

N

1

6

n

j

j

j =0;j =n N j

;j

n

j

(2)

j

This formula results in huge savings, since the dual codes of high-rate codes contain significantly fewer codewords than the original code2 , e.g., the dual codes of the considered rate-(n 1)=n codes have rate-1=n and, hence, only 2 branches leave each state of the corresponding trellis. Eq. (2) can be computed efficiently by performing forward and backward recursions on the trellis of the dual code as proposed in [19]. Note that the complexity savings using the dual trellis usually cannot be generalized to suboptimum MAP decoding algorithms for tail-biting codes. However, in [8] it has been shown that any suboptimal MAP decoding algorithm that performs the same operations as the BCJR algorithm on a section basis can 2 The code dual to a tail-biting code is again a tail-biting code.

OF

S YSTEM PARAMETERS

We use extrinsic information transfer (EXIT) charts [20, 21] to select appropriate p(D) and (D) for a magnetic recording system utilizing turbo equalization. The EXIT charts are a useful tool to evaluate the performance of an iteratively decoded system assuming infinite block length and perfect interleaving. Applying the EXIT analysis, the detector is modeled as a device mapping a sequence of input LLRs L( n ) interleaved to L( 0n ) to a new sequence of output LLRs Ldet ( n ) (after deinterleaving). The decoder maps L( n ) to Ldec ( n ). Given the code bit

n 2 f0; 1g, both the LLRs Ldet ( n ) and Ldec ( n ) are assumed to be mutually independent and identically distributed (i.i.d.) random variables governed by the single parameter probability density function

G

fx

(l)= p 1 2 exp( 2

l x
 2 =2 22 );

x=

(

+1; 1;

n =0;

n =1:

As both Ldet ( n ) and Ldec ( n ) are input to decoder and detector, respectively, it is clear that the input and the output LLRs of detector and decoder are distributed according to fx (l). Therefore, both detector and decoder can be regarded as devices that map an input  2 2 [0; 1) to an output  0 2 2 [0; 1). This idea is also applied in [22] and [24], where the conditional mean or the error rate, respectively, of the output LLRs is observed. For the EXIT analysis, the mutual information between an LLR and a code bit is taken as the parameter that specifies the behavior of the iterative algorithm. To obtain the chart, uniformly distributed bits n and corresponding a-priori LLRs L( n ) distributed with fx (l) for a given  2 are fed separately into the detector and the decoder. For the detector, the n are transmitted over the channel and the received zn are fed into the detector, too. Instead of  2 , the mutual information

Iin ,

X Z1

2f+1

x

tanh(L( )=2) ? ln P ? Q 1 tanh(L( )=2) ? :

? 2C ? ( 1) =0 6=

? 2C ?

be transferred to the dual domain. Clearly, combining suboptimum decoding strategies with the computation in the dual domain results in huge complexity savings for high-rate codes.

;

fx (l) log2

g 1

1

fx (l) f+1 (l)+ f

1

(l) dl;

Iin 2 [0; 1℄;

which is uniquely determined from  2 , is taken as the input parameter. At the output, the mutual information

X Z 1^ f^ (l) f (l) log Iout , dl; f^ (l)+ f^ (l) 2f g 1 x

x

x

+1; 1

2

+1

1

Iout 2 [0; 1℄;

is computed directly from an approximation f^x (l) of the output distribution using a histogram of the output LLRs Ldet ( n ) or Ldec ( n ), respectively. This approach is more accurate than 2 measuring  0 at the output and mapping it to Iout as it is done (det) (det) at the input. The EXIT chart is the mapping from Iin to Iout (dec) (dec) (detector) or Iin to Iout (decoder) for several parameters  2 of the input LLR distribution. Fig. 3 depicts detector EXIT charts for precoders ( p(D) = 1 equals no precoding) at an SNR of 11:6 dB, where SNR is equal to w 2 . The selected normalized linear density corresponds to

4

0.95

0.95 (Det)=I (Dec)

1

0.9

0.9

In

IOut(Det)=IIn(Dec)

1

= Decoding

0.8

= Detection 0.75

0.85

Out

0.85

0.8

0.75

0

0.2

0.4 0.6 IIn(Det)=IOut(Dec)

0.8

1

pw50 =T =3:2. In addition, Fig. 3 shows a decoder EXIT chart (reflected on the x = y axis) using the rate-8/9, memory 4 code g1 g2 g2 g3 g3 g4 g4 ℄ ; q =1+ D + D + D + D4 ; g1 =1+ D2 + D3 + D4 ; g2 =1+ D + D2 + D4 ; g3 =1+ D2 + D4 ; 3 4 g4 =1+ D + D + D : 8=9

2

1

T

3

All charts were obtained using 107 input LLRs. Performing (det) detection (vertical arrows yielding Iout ) and decoding (hori(dec) zontal arrows yielding Iout ) iteratively at this SNR, we have a steady significant gain in mutual information between the circulating LLRs and the encoded bits. The ultimate goal is to (dec) reach Iout = 1, which is the case if the signs of the output LLRs Ldec ( n ) always yield the correct code bit and the decoder yields a zero bit error rate P (bn 6= ^bn ), too. We observe (det) that without precoder the detector cannot achieve Iout = 1 (det) when Iin = 1 [25], which causes an error floor, because (dec) Iout = 1 cannot be reached even after an infinite number of iterations. In contrast, using a recursive precoder, the detec(det) (det) (dec) tor achieves Iout = 1 for Iin = 1 and thus Iout = 1 can be achieved. However, Fig. 3 shows that with precoding the detector looses mutual information especially during the initial phase of the iterative process. Finally, we have found that the precoder p(D)=1+D has by far the best properties in terms of convergence rate of the iterative process as well as error floors and that all other precoder polynomials of degree less than 5 are significantly worse. For all considered detectors, lowering the SNR results nearly in a parallel downward shift of the transfer curve. The EXIT charts are also suitable for selecting appropriate codes. Fig. 4 shows detector charts for p(D)=1, p(D)=1+ D and decoder chart for three different rate-8/9 codes. The system parameters are chosen as before. The chart of Code 1 belongs to the code given above. Code 2 is the memory 2 code:

r = [g

0

0.2

0.4 I (Det)=I In

Fig. 3. Detector and decoder EXIT charts.

r = 1=q
[g

Detector: p(D)=1 Detector: p(D)=1+D Decoder: Code 1 Decoder: Code 2 Decoder: Code 3

I

p(D)=1 p(D)=1+D p(D)=1+D+D2 p(D)=1+D+D4 rate 8/9 code

g1 g1 g1 g2 g2 g2 g2 ℄T =q; q =1+ D + D2; g1 =1+ D2 ; g2 =1+ D 1

and Code 3 is punctured from the rate-1/2 optimum free dis-

0.6 (Dec)

0.8

1

Out

Fig. 4. Decoder EXIT charts for several rate- 8=9 codes.

tance memory 5 code:

G(D) = [1+ D + D + D + D + D 1+ D + D ℄: 2

3

4

5

2

5

Obviously, extensive puncturing of “good” mother codes yields “poor” codes, e.g., Code 2 with only 4 states in the trellis performs nearly as well as the punctured Code 3 with 32 states in terms of the error floor governed by the intersection of the detector and the decoder transfer curve. An increasing trellis complexity lowers the error floor using a system without pre(dec) coding. With precoding, however, Iout = 1 is possible for all codes. The performance differs only in the number of iterations required to achieve this optimum. Assuming an infinite block length, the optimal system thus uses precoding and a code, where the decoder transfer curve is as parallel as possible to the detector curve, since in this case the SNR can be lowered as much as possible while maintaining a bottleneck between the two transfer curves. Fig. 5 depicts the trajectory of an iteratively decoded precoderless system given the Code 1, pw50 =T = 3:2, 11:6 dB SNR, and various block lengths N . Here, Iin and Iout are computed using histograms of the output LLRs obtained using the transmitted data for each iteration. It can be seen that the EXIT charts are most accurate as the block length N approaches infinity. For finite N , the transfer functions are accurate for some initial iterations and provide an upper performance bound with increasing number of iterations. This shows that the design rules obtained using the EXIT charts alone must be applied carefully. For example, the EXIT chart predicts that Code 1 and 2 perform equally well in a precoded system after an sufficient amount of iterations, whereas a real system might not be able to provide the required independence assumptions for such a number of iterations and thus a system using Code 1 performs better than a system using Code 2. V. S IMULATION R ESULTS The error rate performance of a magnetic recording system employing tail-biting codes serially concatenated with a generalized PR channel has been studied via computer simulations. The target polynomial is (1 D2 )(1+s1 D+s2 D2 ) which gives rise to a 16-state trellis. The read-back signal is first filtered

5

1

In

0.9

I

Out

50000 552

0.85

MAP detector, no precode MAP decoder: rate 8/9 (code 1) 0.8

0

0.2

0.4 I (Det)=I In

0.6 (Dec)

0.8

1

Out

Fig. 5. Trajectory of a serially concatenated system using varying block sizes.

by a 5th order low-pass Butterworth filter and is then fed to a minimum-mean-squared error (MMSE) PR4 equalizer with ten taps. The coefficients s1 and s2 are obtained by applying MMSE prediction to the noise component of the signal at the output of the PR4 equalizer. The SNR is defined as the ratio of 2 the energy of the encoded data symbol and the variance w of the AWGN at the input of the low-pass filter. In the performance evaluation, we have considered the two basic modes of operation described in Section II. In mode 1, a sector of 4096 data bits or 512 data bytes, together with 4 extra bytes used for CRC, is split into 3 words of 172 bytes each, which are encoded using a (184; 172)256 RS code. The encoded bytes are matrix-interleaved to 8 blocks of 552 bits. Each 552 bit block is encoded using a tail-biting code, which is either the rate-8=9 Code 1 given by 8=9 from Section IV or the rate-16=17 code

r

r

= 1 [g1 g1 g2 g2 g2 g3 g3 g3 g4 g4 g4 g4 g5 g5 g5 g5 ℄ ; q =1+ D2 + D3 + D4 + D5 ; g1 =1+ D2 + D4 + D5 ; g2 =1+ D + D3 + D5 ; g3 =1+ D3 + D4 + D5 ; 2 3 5 g4 =1+ D + D + D + D ; g5 =1+ D + D3 + D4 + D5 : 16=17

T

VI. C ONCLUSIONS

q

In mode 2, we directly encode the entire sector of 4096 data bits using the rate-8=9 or -16=17 tail-biting code. The permutation (
) is given by an S-random construction with optimal S [23]. The reference system is an uncoded system operating at the normalized linear density pw50 =Tuser =3:0 and using a 16-state detector. Table I shows the normalized channel linear densities taking into account the aforementioned modes of operation and code rates. In our simulations, the detector and decoder employ the optimum MAP algorithm, and the iterative process is terminated after a maximum of eight iterations. We have observed that four to five iterations were sufficient to obtain most of the performance gain. Performance results are shown for a precoderless system and a system that uses the precoder p(D)=1+D which, based on the EXIT chart analysis, provides the best convergence. Figs. 6 and 7 compare the bit error rates of the rate-8/9 and -16/17 tail-biting codes, respectively, on a magnetic recording TABLE I N ORMALIZED CHANNEL LINEAR DENSITIES.

mode code rate 8=9 code rate 16=17

1 3.6387 3.4365

2 3.375 3.1875

We have investigated the performance of magnetic recording systems modeled as serial concatenation of generalized PR channels with high-rate tail-biting codes. Various performance measures such as the bit error rate at the output of the tail-biting decoder as well as the block error rate or sector error rate have been used to evaluate the system performance under various modes of operation. Our results indicate that the use of a precoder and an inverse precoder combined with the MAP detector is essential in an iterative detection/decoding scheme. By employing the EXIT chart analysis, we have found that the p(D) = 1+ D precoder provides the best convergence and the highest performance gain −1

10

uncoded: p(D)=1 uncoded: p(D)=1+D Mode 1: rate 8/9, p(D)=1 Mode 1: rate 8/9, p(D)=1+D Mode 2: rate 8/9, p(D)=1 Mode 2: rate 8/9, p(D)=1+D

−2

10

−3

10

Bit Error Rate

(Det)=I (Dec)

4608 0.95

system using the previously described modes of operation. In both cases, the precoded mode 1 system gains very little compared to a precoderless mode 1 system. For the long tail-biting codes, i.e., mode 2, the use of precoder has a significant impact on performance. As can be seen in Figs. 6 and 7, a 1.5 dB gain is achieved at a bit error rate of 10 7 over a precoderless system. Furthermore, we observe that the bit error rate curves f or both rate-8/9 and -16/17 long tail-biting codes with precoding change slope at an SNR of around 12.5 dB. Figs. 8 and 9 compare the sector error rates of the rate-8/9 and -16/17 tail-biting codes, respectively. In addition, Fig. 9 shows the performance of a 3-way interleaved magnetic recording system using the (184; 172)256 RS code with hard-decision decoding and without any inner code. As can be seen, the precoded mode 1 system achieves a small gain of 0.25 to 0.5 dB in sector error rate compared to a precoderless mode 1 system. Furthermore, a precoded mode 1 system is no better than a recording system that uses RS coding and no inner code, demonstrating that the use of short inner tail-biting codes in conjunction with outer-most RS codes does not offer any performance benefit. On the other hand, for long tail-biting codes, the use of a precoder has again a significant impact on the sector error rate performance. The results show that the long tailbiting code outperforms hard-decision decoding of RS codes by approx. 2.25 dB at a sector error rate of 10 4 . Finally, it is conjectured that the slope at high SNR indicates the dominant influence of the minimum distance of the overall structure consisting of the concatenation of the magnetic recording channel, the interleaver and the channel code. This is a topic of further study.

−4

10

−5

10

−6

10

−7

10

−8

10

11

12

13

14

15

16

17

18

19

20

SNR

Fig. 6. Bit error rate performance of magnetic recording systems.

6

−1

0

10

10

uncoded: p(D)=1 uncoded: p(D)=1+D Mode 1: rate 16/17, p(D)=1 Mode 1: rate 16/17, p(D)=1+D Mode 2: rate 16/17, p(D)=1 Mode 2: rate 16/17, p(D)=1+D

−2

10

uncoded: p(D)=1 uncoded: p(D)=1+D only RS code Mode 1: rate 16/17, p(D)=1 Mode 1: rate 16/17, p(D)=1+D Mode 2: rate 16/17, p(D)=1 Mode 2: rate 16/17, p(D)=1+D

−1

10

−3

Sector Error Rate

Bit Error Rate

10

−4

10

−5

10

−2

10

−6

10

−3

10 −7

10

−8

10

11

−4

12

13

14

15

16

17

18

19

10

20

SNR

10

uncoded: p(D)=1 uncoded: p(D)=1+D Mode 1: rate 8/9, p(D)=1 Mode 1: rate 8/9, p(D)=1+D Mode 2: rate 8/9, p(D)=1 Mode 2: rate 8/9, p(D)=1+D

−1

Sector Error Rate

10

[7] [8] [9]

−2

10

[10] −3

10

[11] [12]

−4

12

13

14

15

16

17

18

19

20

SNR

Fig. 8. Sector error rate performance of magnetic recording systems.

in conjunction with tail-biting codes. Furthermore, simulation results have shown that the use of short inner tail-biting codes in conjunction with outer RS codes do not offer any performance benefits. On the other hand, iterative detection/decoding of long tail-biting codes that use the p(D) = 1+D precoder can outperform hard-decision decoding of RS codes by approx. 2.25 dB at a sector error rate of 10 4 . Although the performance results of a long tail-biting code serially concatenated with a generalized PR channel are impressive, there remain open issues for future research. The change of slope in the error rate curves observed at high SNRs needs further investigation with a focus on code, interleaver, and precoder design as well as the influence of the code’s free distance on the overall system performance. R EFERENCES [1] [2] [3] [4] [5] [6]

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Fig. 9. Sector error rate performance of magnetic recording systems.

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Fig. 7. Bit error rate performance of magnetic recording systems.

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