CASE 0:16-cv-02891-WMW-SER Document 40-3 Filed 11/11/16 Page 1 of 3
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`3992
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`IEEE TRANSACTIONS ON MAGNETICS, VOL. 32, NO. 5 , SEPTEMBER 1996
`
`te
`
`Jaekyun Moon and Barrett Brickner
`Center for Micromagnetics and Information Technologies
`Department of Electrical Engineering, University of Minnesota
`Minneapolis, Minnesota 55455
`A b s t r a c t - A new code is presented which
`im-
`proves
`the minimum distance
`properties of se-
`quence detectors
`operating at high linear densities.
`This code, which
`is called the maximum transition
`run code, eliminates data patterns producing three
`or more consecutive
`transitions while
`imposing
`the usual k-constraint necessary
`for timing recov-
`ery. The code possesses
`the simikr distance-gain-
`ing property of the (1,k) code, but can be imple-
`mented with considerably higher rates. Bit error
`rate simulations on fixed delay tree search with de-
`cision
`feedback and high order partial response
`maximum likelihood detectors confirm
`large coding
`(0,k) code.
`gains over the conventional
`
`E2PR4ML. Note that a traditional (1 ,k) runlength limited (RLL)
`code eliminates all eight transition patterns shown in Fig. 1
`[4][5], but the rate penalty is typically too large to see any
`coding gain unless the linear density is very high. The idea of
`MTR coding is to eliminate three or more consecutive
`transitions, but allow
`the dibit pattern in
`the written
`magnetization waveform. Thus, with MTR coding, the error
`events of the form f ( 2 -2 2) will still be prevented as with (1,k)
`coding, but the rate penalty is significantly smaller than that of
`the typical (1,k) RLL code. Notice that with the MTR constraint,
`the write precompensation efforts can be directed mainly on dibit
`transitions, unlike in conventional (0,k) coded systems. An
`independent study also suggests that removing long runs of
`consecutive transitions improves the offtrack performance in
`some PRML systems [6]. There exist other types of code
`constraints that can offer similar distance-enhancing properties
`for high order PRML systems [7].
`
`I. INTRODUCTION
`N this paper, we present a new code designed to improve the
`distance properties of sequence detectors operating at relatively
`high linear densities. The basic idea is to eliminate certain input
`bit patterns that would cause most errors in sequence detectors.
`More specifically, the code eliminates input patterns that contain
`three or more consecutive transitions in the corresponding
`current waveform, and, as a result, the performance of any near-
`optimal sequence detector improves substantially at high linear
`densities [ 1][2]. This code constraint, designated the maximum
`transition-run (MTR) constraint, can be realized with simple
`fixed-length block codes with rates only slightly lower than the
`conventional (0,k) code. Bit error rate (BER) simulation results
`with fixed delay tree search with decision feedback (FDTS/DF)
`detection and high order partial response maximum likelihood
`(PRML) detection confirm a large coding gain of the MTR codes
`over the conventional (0,k) code.
`
`11. CODING METHODS
`Investigation of high density error patterns in FDTS/DF
`detection reveals that errors arise mostly due to the detector's
`inability to distinguish the minimum distance transition
`patterns, four pairs of which are shown in Fig. 1. These pairs of
`magnetization waveforms give rise to an NRZ input error pattern
`of e,=+(2 -2 21, assuming input data take on +l's and -1's. The
`proposed approach is to remove data patterns allowing this type
`of error pattern through coding. The potential improvement in
`the FDTS detection performance using this approach can be
`estimated by computing the increase in the minimum distance
`between two diverging lookahead tree paths after removing the
`paths that allow the +(2 -2 2) error events [3]. A simple
`minimum distance analysis for PRML systems reveals that this is
`also a critical error pattern in high order PRML systems such as
`
`Manuscript received March 4, 1996. This work was supported in part
`by Seagate Technology and the National Storage Industry Consortium
`(NSIC).
`
`Fig. 1: Pairs of write patterns causing most errors in sequence
`detection at high linear densities.
`
`Fig. 2 shows the state diagram of the MTR code based on the
`NRZI convention, where 1 and 0 represent the presence and
`absence, respectively, of a magnetic transition. Also included is
`the usual k-constraint for timing recovery. The capacity of the
`code can be obtained by finding the largest eigenvalue of the
`adjacency matrix for the given state diagram [8]. The capacities
`for different k values are given in Table 1.
`
`h
`0
`Fig. 2: State transition diagram for the MTR code with k=6
`
`Table 1: Capacities for MTR codes.
`
`While state-dependent encoders and sliding-block decoders can
`be designed for the MTR constraint (which can be easily
`generalized to limit any runs of consecutive transitions), we
`observe that simple fixed-length block codes can be realized with
`
`001 8-9464/96$05.00 0 1996 IEEE
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`CASE 0:16-cv-02891-WMW-SER Document 40-3 Filed 11/11/16 Page 2 of 3
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`good rates and reasonable k values. A computer search is utilized
`to first find all n-bit codewords that are free of an NRZI 11 1 string
`or k+l consecutive NRZI 0's. Then, in order to meet the MTR
`constraint at the codeword boundaries, words that start or end
`with an NRZI 11 string are removed. Also, the k constraint is
`satisfied at the boundary by removing the words with k, + I
`leading 0's or k, + I trailing O's, where k, +k, = k . Finally, if the
`number of the remaining codewords is greater than or equal to 2m,
`then those codewords can be used to implement a rate m/n block
`code. Table 2 shows important code parameters for representative
`block codes obtained through computer search. The efficiency
`was found by dividing the code rate m/n by the capacity computed
`for the given value of k and the MTR constraint. As an example
`of an MTR block code, 16 codewords required to implement the
`rate 4/5 code with k=8 are given in Table 3.
`
`n
`
`m
`
`k eff. No. avail. No. needed
`codewords
`codewords
`8
`5
`16
`16
`.91
`4
`282
`256
`.92
`10 6
`8
`514
`512
`.94
`1 1 6
`9
`1 2 8
`1,066
`1,024
`.95
`10
`.95
`1 7 6
`18,996
`16,384
`1 4
`65,536
`69,534
`.96
`1 9 7
`16
`2 8 8
`2 4
`.98 17,650,478 16,777,216
`Table 2: Parameters for MTR block codes.
`
`00010
`00100
`
`01000
`01001
`
`01101
`10000
`
`10100
`10101
`
`Table 3: A rate 4/5 MTR block code with k=8.
`
`111. MODIFIED DETECTION AND DISTANCE INCREASE
`To realize the coding gain at the detector output, the detector
`has to be modified. In the case of PRML systems, this amounts to
`removing those states and state transitions that correspond to the
`illegal data patterns from the trellis diagram. For the FDTSIDF
`detector, the code-violating lookahead paths must be prevented
`from being chosen as the most-likely path, a technique similar to
`the one used in the (1,7) coded FDTS/DF channel [9]. To illustrate
`the idea, consider Fig. 3 that shows a 2=2 lookahead tree utilized
`in FDTS/DF detection. By utilizing the past decision, an illegal
`path, which contains three consecutive transitions, can be
`identified as indicated by either the solid (when the past decision
`is -1) path or the shaded (when the past decision is 1) path. The
`complexity of the FDTS/DF detector can also be reduced
`considerably with the MTR code, as elaborated in a companion
`paper [IO].
`
`Fig. 3: Modified FDTS detection with MTR coding
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`3993
`
`With this modification in FDTS/DF detection, the squared
`minimum Euclidean distance between any two diverging paths,
`denoted by p:,,, is given by 4.(1+fL2 + fZ2 + ... +f,') for7
`greater than or equal to 2, where f k represents the equalized dibit
`response (at the output of thle forward equalizer). For example, the
`effective SNR gain of the 7=2 FDTS/DF over the decision
`feedback equalization (DFH) channel, assuming the same MTR
`code, is given by 1O.log,,~(l+ fi' + f 2 ' ) dB.
`The distance gain with MTR coding is also significant for high
`order PRML systems such as E2PR4. When the critical NRZ error
`pattern is +(2 -2 2), the minimum distance for the E2PR4
`response { 1 2 0 -2 -1) is 6&. With MTR coding, the worst case
`error pattern becomes a single bit error pattern of +{2}, and the
`corresponding channel output distance is simply the square root
`of the energy in the equalized dibit response, or lo&. This
`increase in the minimum distance is equivalent to an SNR gain of
`2.218 dB. When the code rate penalty is small, the overall coding
`gain is significant.
`IV. BER SIMULATION RESULTS
`To verify the coding gain, FDTS/DF detection was simulated
`with the rate 4/5 and rate 16/19 MTR codes as well as with a rate
`8/9 (0,k) code. The BERs were first obtained as a function of
`readback SNR for different tree depths. The BER of the PR4ML
`detector was also simulated for comparison. The Lorentzian
`transition response was assumed, and the user density, defined as
`PW50 over the user bit interval, is fixed at 2.5 for all codes. The
`SNR value required to achieve an error rate of
`was then
`recorded for each depthkode: combination.
`The results are summarized in Fig. 4, where the effective SNR
`improvement of each system over PR4ML is shown. The
`performance advantage of MTR codes is clear. With the rate
`16/19 MTR code, for example, the depth7 FDTS/DF performs as
`well as the depth 5 FDTS/I)F used with the conventional (0,k)
`code, yielding a 2.5 dB gain over the PR4ML. When the 4/5 MTR
`code is used, FDTS/DF with a tree depth of 2 outperforms the
`depth 5 FDTS/DF with the 8/9 (0,k) code; For a given tree depth,
`the rate 16/19 MTR code yields a 1.5 - 2 dB coding gain over the
`conventional 8/9 (0,k) code.
`Also shown are the SNR performances of PRML systems with
`and without MTR coding. The coding gain is obvious with
`E2PRML and E3PRML, in which the minimum distance is
`improved with the MTR code. However, with EPR4ML the
`performance advantage of the MTR code is small since the MTR
`code does not improve the minimum distance in the EPR4
`system. This is because the rninimum distance error pattern in an
`EPR4 system is of the form +{2}, which is not affected by the
`MTR constraint. The MTR 'code does, however, eliminate non-
`minimum distance error patterns of the form rt(...2 -2 2...},
`resulting in a small performance improvement over the (0,k)
`coded EPR4 system when the code rate is sufficiently high as with
`the 16/19 code.
`Comparisons also can be made between the PRML systems and
`FDTS/DF systems. For example, the depth 2 FDTSDF with the
`rate 4/5 MTR code improves more than 1 dB over EPR4ML with
`the rate 8/9 (0,k) code. At this density and with a Lorentzian
`transition response, EPR4ML has a 1.5 dB advantage over
`PR4ML. Of the PR targets, the EPR4 appears to provide a best fit
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`CASE 0:16-cv-02891-WMW-SER Document 40-3 Filed 11/11/16 Page 3 of 3
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`3994
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`to the natural channel as indicated by the superior performance of
`EPR4ML over even higher order PRML systems. Large enough
`FIR filters are used for equalization for both PRML and FDTS/DF
`systems so that the performances are not degraded by imperfect
`equalization.
`In Fig. 5, similar plots are presented for a modeled MR head
`response. The trends are similar to the Lorentzian case, except
`that within the PRML family the performance improves as the
`order of the PR polynomial increases. Also, the MTR coding gain
`is larger than in the case of the Lorentzian response for all
`detectors. The depth 2 FDTS/DF channel with the rate 4/5 MTR
`code provides a 2.5 dB SNR gain over the EPR4ML channel with
`the rate 8/9 (0,k) code. With the particular MR head response used
`here, EPR4ML already has a 4 dB advantage over PR4ML at this
`linear density.
`Since the MTR code eliminates data patterns with crowded
`transitions, the overall transition noise, as measured per unit
`length of track, is expected to be reduced. Fig. 6 shows the
`simulation results similar to those presented in Fig. 5 , except
`random transition position jitter and transition width variations
`are included in the read waveform construction process [ 111. The
`rms values of both transition noise parameters are set at 4.4 % of
`the user bit interval. The SNR reflects only the additive noise
`component. As is evident from the figure, the coding gain of the
`MTR code over the (0,k) code is much larger in the presence of
`transition noise. For example, with 7=2 FDTS/DF detection, the
`SNR difference is 6 dB between the rate 4/5 MTR code and the rate
`8/9 (0,k) code which allows long runs of consecutive transitions.
`Although the results are not shown here, we have also observed
`that the MTR code tends to reduce the relative frequencies of long
`error events in DFE and FDTS/DF systems.
`
`E
`" 0
`
`RLL(0A). rate 8/Y
`
`MM,km8, rate 415
`
`M R M ,
`
` rate 16/19
`
`DFE
`
`&u=l
`tau=2
`FDTWDF Tree Depth
`
`tau=?
`
`Fig. 6: Summary of FDTS/DF performances with and without
`MTR codes (MR head response and mixed noise).
`
`V. CONCLUSION
`A simple coding scheme is presented which improves the
`performance of FDTS/DF and high order PRML systems operating
`at relatively high linear densities. The code eliminates three or
`more consecutive transitions while allowing the k-constraint for
`timing purposes. The code can be implemented as simple block
`codes with reasonable rates such as 4/5, 8/10 and 16/19. BER
`simulations on FDTSlDF and PRML systems confirm large
`coding gains over the conventional (0,k) code.
`
`REFERENCES
`[l] B. Brickner and J. Moon, "Coding for increased distance with
`a d=O FDTS/DF detector," Seagate Internal Report, May 1995;
`Also see, J. Moon and B. Brickner, "Coded FDTS/DF,"
`presented at the Annual Meeting of the National Storage
`Industry Consortium, Monterey, CA, June 1995.
`[2] J. Moon and B. Brickner, "MTR codes for Data Storage
`Systems," Invention Disclosure No. 96025, University of
`Minnesota, September 1995.
`[3] B. Brickner and J. Moon, "A signal space representation of
`FDTS for use with a d=O code," Globecom'95, Singapore,
`November 1995.
`[4] K. A. S. Immink, "Coding techniques for the noisy magnetic
`recording channel," IEEE Trans. Commun., vol. 37, no. 6,
`May 1989.
`[SI J. Moon and J.-G. Zhu, "Nonlinearities in thin-film media and
`their impact on data recovery," IEEE Transactions on
`Magnetics, vol. 29, No. 1, Jan. 1993.
`[6] E. Soljanin, "On-track and off-track distance properties of
`class4 partial response channels," SPIE Conference,
`Philadelohia. PA. Oct. 1995.
`[7] R. Kara6ed and P H Siegel, "Coding for high order partial re-
`sponse channels," SPIE Conference, Philadelphia, PA, Oct
`1995
`[SI P H Siegel, "Recording codes for digital magnetic storage,"
`IEEE Transactions on Magnetics, vol MAG-21, no 5, pp
`1344 - 1349, Sept. 1985.
`[9] J Moon and L. R Carley, "Perfosmance Comparison of
`in Magnetic Recording," I E E E
`Detection Methods
`Transactions on Magnetics, vol. 26, no. 6, Nov. 1990
`[IO] B Brickner and J. Moon, "A high dimensional signal space
`implementation of FDTS/DF," presented at Intermag '96,
`Seattle, Washington, April 1996
`[ 111 J Moon, "Discrete-time modeling of transition-noise-
`dominant channels and study of detection performance," IEEE
`Transactions on Magnetics, vol 27, no. 6, Nov. 1991
`
`EPRML
`
`E3PRML
`
`DFE
`
`mu=2
`
`~du=Lz
`
`9 RLL(O.4) rate 819 + MTRk=E rate415 e MTR:k=7 rate 16/19
`
`Fig, 4: Summary of PRML and FDTSmF performances with and
`without MTR codes (Lorentzian response and additive noise).
`
`EPRML ElPRML
`4 RLL(0 4) rrfe 819
`.iF MTR k 8 raB 415
`
`DFE
`
`mu=2
`
`l a u d
`
`.6- MTR k=7 ratc 16/19
`
`Fig. 5: Summary of PRML and FDTS/DF performances with and
`without MTR codes (MR head response and additive noise).
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`LSI Corp. Exhibit 1012
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