Constrained Codes for PRML

1. Constrained Codes for PRML Panu Chaichanavong December 14, 2000 • Partial Response Channel • Maximum Likelihood Detection • Constraints for PRML • Examples • Conclusion

2. 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]

3. Partial Response (PR) Interleavedprecoding and where

4. Partial Response (PR) Ideal PR4 transition response

5. Maximum Likelihood (ML) We can simplify y(t) to be Therefore the sequence y after the A/D converter is

6. 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

7. Maximum Likelihood (ML) Trellis diagram of the even interleave To reduce the memory of the detector, we don’t want a long run of 0’s

8. Constraints for PRML No more than consecutive 0’s No more than consecutive 0’s in each subsequences This is denoted by constraint

9. Lattice of States Let g be the number of 0’s since the last 1 in the global string b be the number of 0’s in the substring containing the last bit a be the number of 0’s in the other substring We have the following relation:

10. 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

11. Examples

12. (0,3/3) Constraint By using this rule, state1 is less than state2 if state2 is below and to the left of state1

13. (0,3/3) Constraint

14. (0,4/4) Constraint

15. (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

16. 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