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

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]

Partial Response (PR) Interleaved precoding and where

Partial Response (PR) Ideal PR4 transition response

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

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

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

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

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:

Lattice of States Denote each state by given that a and b are valid i.e.and Form the lattice of states by: IfPlace state at the coordinate IfPlace state at the coordinate Then the representation is given by If is valid

Examples (0,G/I)CapacityRateEfficiency (%) Encoder States Decoder Look- ahead (bits) (0,4/4) (0,4/3) (0,3/6) (0,3/5) (0,3/4) (0,3/3) /

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

(0,3/3) Constraint

(0,4/4) Constraint

Adjacency matrix is (0,2) (2,1) (0,2) (2,1) Number of codewords of length 9 generated from each state

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