Precoded PRML, serial concatenation, and iterative (turbo) - CiteSeerX

Precoded PRML, serial concatenation, and iterative (turbo) decoding for digital magnetic recording Laura L. McPheters and Steven W. McLaughlin Georgia Institute of Technology, Atlanta, Georgia 30332 USA Krishna R. Narayanan Texas A&M University, College Station, Texas, 77843 USA

Abstract|We show that a high rate serially concatenated code used in conjunction with a precoded partial response equalized magnetic recording channel and iterative decoding can provide similar performance to a turbo code on the same channel. The precoded partial response maximum-likelihood (PRML) read channel is incorporated as the inner code of the concatenated coding scheme and the outer code is a high rate convolutional code. Gains of 4.8 dB above conventional PRML at a bit error rate of 10?5 for a rate 13/14 code can be achieved. Index Terms|Iterative decoding, precoded PRML, serial concatenated codes, turbo codes. I. Introduction

Recently [1], [2], [3], have shown that high rate turbo codes [4] can provide 4-6 dB of coding gain when applied to PR4 and EPR4 equalized magnetic recording. In this paper we show that this outstanding performance is achievable without the use of turbo codes per se. That is, we show that a high rate serially concatenated code used in conjunction with a conventional NRZI precoded channel and iterative decoding can provide eectively identical performance. This alternative and simpler con guration shows that existing PRML systems already have the potential for getting some of the turbo code gains by virtue of the precoder. Others have also independently addressed this topic [5], [6]. II. Concatenated Codes and Iterative Decoding

A. Channel Model Fig. 1 shows a conventional digital magnetic recording channel that uses a partial response maximum-likelihood (PRML) read channel. {c k}

Precoder

Channel (1-D)(1+D)

Equalizer

n

Viterbi Detector

{c^ k}

PR Channel

Fig. 1. A conventional partial response system.

The channel is equalized to some target intersymbol interference pattern, which for magnetic recording is usually

of the form (1 ? )(1+ )n ,  1. The precoder used for the PR4-equalized channel is 1 (1  2 ). This precoded PR channel is what will be used as the inner code of the concatenated coding system. The Viterbi detector is replaced by an a posteriori probability (APP) soft-input, soft-output module so that it may share information with the outer decoder and AWGN is assumed for simplicity. D

D

n

=

D

B. General Structure for Concatenated Code A concatenated coding system for a partial response magnetic recoding channel is shown in Fig. 2. This is similar to known concatenated coding, except for a (pseudorandom) interleaver separating the outer and inner codes and a feedback portion used for the iterative decoding which are characteristic of serial concatenation [7]. {u k} Outer High {c k} {c~ } Inter- k Rate Code leaver

"Inner Code"

Precoder

extrinsic information

{rk}

^~ {c } {c^k} Outer Code APP Channel k De-interDecoder Detector leaver

{u^ k}

Fig. 2. Concatenated code and iterative decoding for PR system.

The precoded PR4 channel lends itself naturally as a rate 1 inner code since it has properties similar to recursive convolutional codes as will be described later. The outer code in this setting need not be a turbo code, rather any good, high rate block or convolutional code will suf ce. This con guration has been studied outside of the context of magnetic recording by [8], [9], [10]. The decoder for the serial concatenated system consists of two constituent a posteriori probabilities APP (soft-in, soft-out) decoders, one for the precoded PR channel and one for the outer code much like that in [1]. The trellis of the precoded PR channel estimates the APPs and passes \extrinsic" information to the trellis (APP) decoder for the outer code. This iterative decoding is performed until a suitable termination criterion is met. III. Serial Concatenation in PR Setting

De ne an input error sequence f k g, at the input to the inner code as the dierence sequence between the transmitted codeword, f 0k g, of the outer code and another e

Manuscript received March 5, 1999. This work was supported in part by NSF Grant NCR-9702024.

PR + Channel {y k} {rk} {n k} Interleaver

c

codeword, say f 1k g such that 1k = 0k + k . f k g has Hamming weight ( ) and produces an error event, f k g, with Euclidean distance from a reference path de ned by ( ). Benedetto et al. [7] indicate the need for the inner code of a serial concatenated system to be a recursive convolutional (RC) code due to its minimum input error min ) = 2 . It turns out that sequence min k , of weight, ( RC codes also have an additional necessary property of error events accumulating weight through the use of an interleaver. Consider having just a precoder of the form 1 (1  ) as the inner code for a serial concatenation system in comparison to having a precoded 1 ? channel as the inner code. The precoder by itself is a simple recursive convolutional code and will have the necessary properties to make a good inner code. The precoded 1 ? channel on the other hand is nonlinear and with the precoding in eect canceling the channel (mod 2), it does not t into the standard RC code form. However, it too has the necessary properties to achieve interleaving gains as will be shown. c

c

c

e

e

w e

E

RC codes generally meet both design rules, however it is not so obvious that the nonlinear precoded PR channel does.

d E

e

=

w e

C. Role of Precoded PR Channel The trellis for a precoded PR channel with aggregate (1?D ) is shown in Fig. 4. (1D ) State 1

D

0/0 1/2

0/0 1/2

0/0 1/2

D

D

A. Simple Recursive Convolutional Code The trellis for a 1=(1  D) rate 1 RC code is shown in Fig. 3 where branches are labeled as c~k =yk where c~k is the input to the inner code and yk is the output (see Fig. 2). State 1

0/1

0/1

0/1

1/0

1/0

1/0

1/1 State 0

1/1

0/0

1/1

0/0

0/0

Fig. 3. Trellis for simple RC code.

The minimum input error sequence, f min k g = \...0 0 0 1 1 0 0 0...," of this code has has ( min ) = 2. Since the code is linear, a reference sequence f 0k g can be assumed to be the all zeros sequence. If the output of the outer code is the weight two sequence f 1k g = \...0 0 0 1 1 0 0 0...," on average the interleaver will throw the ones far apart such that f ~1k g for example would be \...1 0 0 0 ... 0 0 0 1 ..." Observe on the trellis that the further apart the ones are, the greater the distance ( ) relative to the all zeros codeword that is accumulated in the error event. The resulting error codeword would be further in distance from the transmitted codeword. This in turn can reduce the multiplicity of the low weight error events which leads to interleaver gain as we now describe. e

w e

c

c

c

d E

B. Design Rules to Achieve Interleaver Gain Interleaver gain for turbo codes and serial concatenated codes is de ned as the gain associated with an increase in interleaver size providing a reduction in bit error rate. In order to achieve interleaver gain two criteria need to be met:  The minimum weight of an input error sequence must be two.  The distance, d(E ) of an input error event sequence needs to increase as the separation between the ones increases.

2

2

1/+ State 0

1/+

0/0

2

1/+

0/0

0/0

Fig. 4. Trellis for precoded PR channel.

Observe that the minimum input error sequence has weight two. Let the transmitted codeword be f 0k g = \...0 0 0 0 0 0..." If the output of the outer code is f 1k g = \..0 0 0 1 1 0 0 0...," the interleaver will on average throw the two ones far apart. However no matter how far apart the ones are, ( ) will always be eight relative to the all zeros codeword and therefore no weight accumulation is possible. Since the inner code in this case is nonlinear, the all zeros reference cannot be assumed. Instead, assume that the reference path has at least some ones in it which is a reasonable assumption if the input sequence to the outer linear code is i.i.d. In this case, if the two ones of the input error sequence are spread apart by the interleaver, weight accumulation on the error event is achievable. Therefore, assuming an i.i.d. sequence, both design criteria are met with the precoded PR channel as the inner code and interleaver gain can be achieved. c

c

d E

D. Error Enumerating Function View A transfer function can be derived for a linear code which uses an all zeros reference to enumerate the incorrect output sequence paths. To enumerate the incorrect paths for a nonlinear system, all possible reference sequence need to be used. We will apply conventional error state diagram and transfer function methods that are used for linear systems to determine an error enumerating function that takes into consideration all possible references, but will modify the weights along the edges as will be described. E0

4 D L

E1

4 D L

E2

16 1/2(1 + D )L

Fig. 5. Error state diagram.

In Fig. 5, which shows the error state diagram for a precoded 1 ? channel, the state 0 denotes the time instant at which the erroneous path diverges from the reference sequence and 1 denotes all time instants when the paths remain diverged. The state 2 denotes the time instant when the paths re-merge. On each edge are labels that indicate the accumulated distance and length of the error event assuming the inputs are i.i.d. The exponent of is the squared Euclidean distance and the exponent D

E

E

E

D

of is the length and weight of the error event. The edge labels take into account the i.i.d. assumption. For example, there are two possible cases for the paths remaining diverged which is represented by staying in 1 , as seen in Fig. 4. An input of 0 results in a distance of 0 and an input of 1 results in a distance of 16. The two inputs are then each equally weighted by 1 2 to obtain the label 21 (1 + 16 ) . The edge from 0 to 1 has the label 1 ( 4 + 4 ) = 4 corresponding to diverging from ei2 ther state 0 or state 1 in Fig. 4. 1 to 2 is the same but for the error event re-merging. The average error enumerating function, where the average is taken over all possible input sequences, is then given by L

E

=

D

D

L

E

E

E

8

T

=

8

D L

2

10

E

2

avg (D; L) = 1 ? 1 (1 + D16 )L 2 D L

[1 + 12 (1 +

D

D

L

D

−2

10

(1) 16

) + L

1 16 2 2 22 (1 + ) + + 21k (1 + 16 )k k + ] The minimum input error sequence has weight two as expected and error events of length = 2 have distance eight. As the length of the error event increases, the coecients of the outer terms of the binomial expansion of 21k (1+ 16 )k decrease rapidly. We can therefore see that the exponent of 16 approximately increases linearly with , suggesting that as the error event length increases, the distance also increases. The transfer function for the all zeros reference can be found by replacing the weight of the edge from 1 to 1 with instead of 21 (1 + 16 ) , resulting in

Uncoded PR4

−3

10

:::

L

::: :

r=16/17 bound −4

10

r=13/14 bound

k

k

−5

10

D

D

k

E

L

D

8

E

L

2

(2)

avg (D; L) = 1 ? L D L

T

= 8 2 [1 + + 2 + + k + ] This shows that error events of length have Euclidean distance or output weight of only eight for all . Weight accumulation is possible in the rst case (1) whereas no weight accumulation is possible in the all zeros example (2) which is consistent with the previous description. D L

L

L

:::

L

[1] [2] [3]

::: :

k

[4]

k

IV. Simulation and Results

The system simulated uses a rate 13/14 outer convolutional code punctured from a 1/2 rate code with generator (561 753)8 and an interleaver size of N=10010. The inner code is an NRZI 1 (1  2 ) precoded PR4 channel. The performance after 5 iterations is shown in Fig. 6 in comparison to the rate 16/17 turbo code performance in [1]. A gain of 4.8 dB at a bit error rate (BER) of 10?5 over a conventional PRML system is achieved and it is about 1.5 dB from capacity. This new con guration in Fig. 2 has comparable performance to the turbo code on the same channel. An outer code with a shorter constraint length was also used, speci cally a 16 state code with generator (31 33)8. This code performs about 0.8dB worse than the 256 state code and exhibits an error oor at a BER

[5] [6] [7]

;

=

;

D

r=13/14 conv code, G(561,753) r=13/14 conv code, G(31,33) r=16/17 turbo code

−1

D L

BER

D

L

above 10?5 as can be seen in Fig. 6, while the 256 state code does not. In a non-concatenated setting, a shorter constraint length code can perform as well or better than a longer constraint length code at very low SNR. This is also the case when used in concatenated code structure with iterative decoding, however a lower error oor may result as indicated.

[8] [9] [10]

3

4

5

6

7 Eb/No (dB)

8

9

10

Fig. 6. Code performance on a precoded PR4 channel. References L. McPheters, S. McLaughlin, and E. Hirsch, \Turbo Codes for PR4 and EPR4 Magnetic Recording," Proc. of the 1998 Asilomar Conf. on Computers and Commun., Paci c Grove, CA, pp. 1778-1782, Nov. 1998. W. Ryan, L. McPheters and S. McLaughlin, \Combined Turbo Coding and Turbo Equalization for PR4-Equalized Lorentzian Channels," Proc. of Conf. Inform. Sci. and Sys., Princeton, NJ, pp. 489-493, Mar. 1998. W. Ryan, \Performance of High Rate Turbo Codes on a PR4Equalized Magnetic Recording Channel," Proc. 1998 Inter. Conf. Commun., pp. 947-951, June 1998. C. Berrou, A. Glavieux, and P. Thitimajshima, \Near Shannon limit error- correcting coding and decoding: Turbo codes," Proc. 1993 Inter. Conf. Commun., pp. 1064-1070, May 1993. M. O berg and Paul H. Siegel, \Performance Analysis of Turbo-Equalized Dicode Partial-Response Channel," Proc. 36th Allerton Conf. on Commun., Control and Computing, Monticello, IL. Sept. 1998. T. Souvignier, A. Friedmann, M. O berg, P. Siegel, R. E. Swanson, and J. K. Wolf, \Turbo Codes for PR4: Parallel Versus Serial Concatenation," accepted to Proc. 1999 Inter. Conf. Commun., June 1999. S. Benedetto, G. Montorsi, D. Divsalar, and F. Pollara, \Serial Concatenation of Interleaved Codes: Performance Analysis, Design, and Iterative Decoding," TDA Progress Report 42126, JPL, Pasadena, California, Aug. 15, 1996. D. Divsalar and F. Pollara, \Serial and Hybrid Concatenated Codes with Applications," Proc. Inter. Symp. on Turbo Codes and Related Topics, pp. 80-87, Sep. 1997. K. Narayanan and G. Stuber, \A Serial Concatenation Approach to Iterative Demodulation and Decoding," Proc. Commun. Th. Mini Conf., GLOBECOM 1998, pp. 155-160, Nov. 1998. I. Sason and S. Shamai, \Improved Upper Bounds on the Performance of Parallel and Serial Concatenated Turbo Codes via their Ensemble Distance Spectrum," Proc. 1998 Inter. Symp. on Inform. Th., Oct. 1998.