Constrained Codes for PRML

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Constrained Codes for PRML
Panu Chaichanavong
December 14, 2000

• Partial Response Channel
• Maximum Likelihood Detection
• Constraints for PRML
• Examples
• Conclusion

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Sources

• Fisher et al, PRML detection boosts hard-disk
  drive capacity, IEEE Spectrum November 1996
• Wang and Taratorin, Magnetic Information
  Storage Technology, Academic Press (1999)
• Chapter 1 of the text
• Discussion with Brian yesterday
• Marcus et al, Finite-State Modulation Codes
  for Data Storage, IEEE J. Sel. Areas Comm.,
  Vol.10, no.1, January 1992 MSW92

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Partial Response (PR)
Interleavedprecoding
and
where
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Partial Response (PR)
Ideal PR4 transition response
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Maximum Likelihood (ML)
We can simplify y(t) to be
Therefore the sequence y after the A/D converter
is
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Maximum Likelihood (ML)
It turns out that an odd sample depends only on
odd data bits, and vice versa Furthermore, If
is 0 then is also 0 If is 1 then
is 2 if the last nonzero sample in its
subsequence is 2 and vice versa This means that
we can treat odd and even subsequences separately
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Maximum Likelihood (ML)
Trellis diagram of the even interleave
To reduce the memory of the detector, we dont
want a long run of 0s
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Constraints for PRML
No more than consecutive 0s No more than
consecutive 0s in each subsequences
This is denoted by constraint
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Lattice of States
Let g be the number of 0s since the last 1 in
the global string
b be the number of 0s in the substring
containing the last bit a be the
number of 0s in the other substring
We have the following relation
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Lattice of States
Denote each state by given that a and
b are valid
i.e.
and
Then the representation is given by
If is valid
Form the lattice of states by
If
Place state at the coordinate
If
Place state at the coordinate
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Examples
(0,G/I) Capacity Rate Efficiency () Encoder States Decoder Look-ahead (bits)
(0,4/4) (0,4/3) (0,3/6) (0,3/5) (0,3/4) (0,3/3) 0.961366 0.939505 0.944539 0.941533 0.934253 0.915723 8/9 8/9 8/9 8/9 8/9 8/9 92.4 94.6 94.1 94.4 95.1 97.0 1 3 1 2 3 4 0 0 0 0 8 7
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(0,3/3) Constraint
By using this rule, state1 is less than state2 if
state2 is below and to the left of state1
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(0,3/3) Constraint
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(0,4/4) Constraint
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(0,4/4) Constraint
Adjacency matrix is
(0,2) (2,1) (0,2) 27
298 (2,1) 28 269
Number of codewords of length 9 generated from
each state
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Conclusion

• PRML performs better than peak detection
  because it chooses the most probable sequence
  rather than a single sample values
• constraint is required for timing control
• constraint reduces decoding delay and thus
  decoder memory
• A state can be denoted by a pair of number
  and can be placed in the lattice to show the
  partial ordering
• Number of states of the encoder can be easily
  predicted from the lattice of states