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Signal Processing for Data Storage Using Data Dependent Noise to Aid Error Correction Manchester Conference Centre 14 th July 2005 Dr. Mohammed Zaki Ahmed email : mahmed@plymouth.ac.uk url : http://www.plymouth.ac.uk/staff/mahmed Centre for

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Signal Processing for Data Storage

Using Data Dependent Noise to Aid Error Correction

Manchester Conference Centre 14th July 2005

  • Dr. Mohammed Zaki Ahmed

email : mahmed@plymouth.ac.uk url : http://www.plymouth.ac.uk/staff/mahmed

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Centre for Research in Information Storage Technology

July 14, 2005

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Signal Processing for Data Storage

Acknowledgements

  • Prof Barry Middleton (Manchester)
  • Dr Achim Fahrner (Ulm, Germany)
  • Prof Hisashi Osawa (Ehime, Japan)

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Centre for Research in Information Storage Technology

July 14, 2005

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Signal Processing for Data Storage

Presentation Overview

  • Introduction and Motivation
  • System Overview, Sources of Noise and Typical Detection/Decoding
  • Modifications to the Viterbi and MAP Decoders : Asymmetric Decoders

(AD)

  • Results and Future Work
  • Conclusion

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Centre for Research in Information Storage Technology

July 14, 2005

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Signal Processing for Data Storage

1 Introduction and Motivation

  • The classical Viterbi Algorithm[1] and Maximum A Posteriori (MAP)[2]

detectors assumes that all symbols are equally likely and noise is data independent.

  • Information storage channels are perturbed by data independent noise

(Additive White Gaussian Noise, AWGN) and many data dependent sources of noise (signal jitter, media saturation, soft underlayer spike noise, etc) Extending and simplifying the proof of the classical Viterbi and MAP decoders to include channel data dependence was investigated. Results are presented in this talk.

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Centre for Research in Information Storage Technology

July 14, 2005

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Signal Processing for Data Storage

2 System Overview and Sources of Noise

For simulation purpose, a perpendicular magnetic recording channel is assumed.

  • The sources of noise include AWGN and Sampling Jitter, with

Sampling Jitter seen as an additional (noisy) signal.

  • The channel is equalised using Partial Response (PR) polynomials,

and uses a Log-MAP decoder to decode the PR sequence.

  • User data are recovered after the application of Error Correction

Codes, in this particular case Low Density Parity Check (LDPC) codes are used[3].

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Centre for Research in Information Storage Technology

July 14, 2005

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Signal Processing for Data Storage

3 Block Diagram - Perpendicular Recording

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Centre for Research in Information Storage Technology

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Signal Processing for Data Storage

Jitter Noise

  • 1
  • 0.5

0.5 1

  • 3
  • 2
  • 1

1 2 3 Normalised Readback Voltage Sample Period Ideal Transition Half Sample Jitter

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Centre for Research in Information Storage Technology

July 14, 2005

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Signal Processing for Data Storage

  • For the perpendicular channel model, we assume a hyperbolic tangent

readback signal from an isolated transition u(t), given by u(t) = tanh

  • ln(3) t

D50

  • where D50 is the normalised user density[4].
  • The error correction code (ECC) is a (4096,3072) LDPC code.
  • The received signal r(t) can be described by the following equation

r(t) = h(t) ∗ (s(t) + j(t) + n(t)) , where h(t) is the impulse response of the PR equaliser, s(t) is the channel readback waveform (with ISI), j(t) is the transition jitter noise and n(t) is AWGN, or the electronics noise and ∗ denotes convolution.

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Signal Processing for Data Storage

  • Noise prediction is used within the decoder, resulting in the decoded

signal being described as r(t) = h(t) ∗ s(t) + j(t) + n(t) . Without noise prediction, the data dependence of the noise is lost.

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Signal Processing for Data Storage

4 Modifications to Trellis Decoders - Asymmetric Decoders (AD)

Typical expression in the Viterbi and Log-MAP decoders include an add–compare–select branch. For Viterbi decoders expressions like min{Sa + (ri − sj)2, Sb + (ri − sk)2}

  • ccur in abundance, where Sa and Sb are state metrics, ri is the received

sample and sj and sk are ideal possibilities for ri. This can be written using 1 multiplication, 2 additions and 1 compare as min{Sa, Sb + ri · k1 + k2} where k1 = 2(si − sj) and k2 = (s2

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July 14, 2005

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Signal Processing for Data Storage The modification required to this computation is min{Sa + (ri − sj)2, Sb + [(ri − sk)2 · α + β]} where α and β can be precomputed depending on the statistics of the data dependent noise, and are constant as long as the statistics of the data dependent noise are constant. This can also be written as min{Sa, Sb + ri · (ri · k1 + k2) + k3} where k1 = α − 1, k2 = 2(si − α · sj) and k3 = α · s2

j − s2 i + β

This increases the computation by 1 additional multiplication and 1 additional addition for every metric computation compared to the classical Viterbi.

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Signal Processing for Data Storage Similar computations can also be done for the Log-MAP decoder. α depends on the statistics of the noise and β depends on the noise statistics and the probability of the ideal symbols si, sj. A crucial aspect of the decoder is the dependence of α and β on the decoder output, so additional storage of all possibilities are required before any decision is made.

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Signal Processing for Data Storage

5 Results and Future Work

  • The results compare fixed electronics SNR, and varying the maximum

transition jitter within a sector of 4096 bits are shown for GPR[0.74,0.83,0.33,0.08,0.01] at different recording densities.

  • Results without ECC are for SNR of 16dB and with ECC are at 12dB.
  • Results are shown for increasing recording frequency.

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Signal Processing for Data Storage

Results Before Error Correction - Low D50 = 1.0

1.0e-05 1.0e-04 1.0e-03 1.0e-02 1.0e-01 1.0e+00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Bit Error Rate Percentage Transitions in Sector BER with Fixed Electronics Noise AD tmax/T = 0.0 AD tmax/T = 0.5 BCJR tmax/T = 0.0 BCJR tmax/T = 0.5

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Signal Processing for Data Storage

Results Before Error Correction - High D50 = 1.4

1.0e-05 1.0e-04 1.0e-03 1.0e-02 1.0e-01 1.0e+00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Bit Error Rate Percentage Transitions in Sector BER with Fixed Electronics Noise AD tmax/T = 0.0 AD tmax/T = 0.5 BCJR tmax/T = 0.0 BCJR tmax/T = 0.5

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Signal Processing for Data Storage

6 Results After Error Correction - D50 = 1.0

1.0e-05 1.0e-04 1.0e-03 1.0e-02 1.0e-01 1.0e+00 0.0 0.1 0.2 0.3 0.4 0.5 Bit Error Rate Percentage Jitter tmax/T BER with Fixed Electronics Noise AD BCJR

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Signal Processing for Data Storage

  • Results show promise especially in the presence of error correction.
  • The effects of Inter–Symbol Interference in spreading the data

dependance of sampling jitter at higher recording densities is very interesting and solvable, but computationally prohibitive at present.

  • There appears to be pattern dependant errors at high frequency,

possibly linked to ISI.

  • The effect of new algorithms that produce near ML decoding using

iterative algorithms[5] is also being investigated.

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Signal Processing for Data Storage

References

[1] G.D. Forney, Jr. The viterbi algorithm. Proceedings of IEEE, 61(3):268–278, March 1973. [2] L.R. Bahl, J. Cocke, F . Jelenik, and J. Raviv. Optimal decoding of linear codes for minimising sybmol error rate. IEEE Transactions in Information Theory, 20:284–287, March 1974. [3] Mohammed Zaki Ahmed, Achim Fahrner, and Purav Shah. Asymmetric map decoding for perpendicular magnetic recording with data dependent noise. In International Symposium on Physics of Magnetic Materials, Singapore, September 2005. [4] Y. Okamoto, H Osawa, H Saito, H Muraoka, and Y Nakamura. A study

  • n 3/4 MTR coded PRML systems in perpendicular recording using

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Signal Processing for Data Storage double layer medium. Technical Report of IEICE, paper number MR2000–8:1–6, July 2000. [5] E. Papagiannis, C. Tjhai, M. Ahmed, M. Ambroze, and M. Tomlinson. Improved iterative decoding for perpendicular magnetic recording. In International Symposium on Communications Theory and Applications, Ambleside, UK, July 2005.

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Signal Processing for Data Storage

Please email me for copies of the presentation at mahmed@plymouth.ac.uk

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