Cap(S) = logλ( D(G)),
COMBINED CONSTRAINTS FOR PERPENDICULAR RECORDING CHANNELS Ismail Demirkan 1, University of California, San Diego, 9500 Gilman Dr., La Jolla, CA, 92093-0401 Yuan Xing Lee, Hitachi Global Storage Technologies, 5600 Cottle Road, CUY/028, San Jose, CA, 95193 The maximum transition run (MTR) and running digital sum (RDS) constraints have wide variety of applications in magnetic recording systems. MTR constraints eliminate consecutive runs of transitions larger than j so that certain common error events can be avoided. RDS constraints exhibit null at DC frequency and suppress low frequency contents of the input signal in NRZ domain. If the RDS of an NRZ sequence is lower and upper bounded by N1 and N 2 , respectively, the digital-sum-variation (DSV) of the sequence is defined as N = N1 + N 2 1. The RDS (N ) constraint contains all bipolar sequences having DSV N . The maximum-likelihood sequence detectors are typically employed in contemporary magnetic recording systems. The error rate performance of the system is determined by the error events with minimum and near minimum squared-Euclidean distances when the signal-to-noise (SNR) ratio is high. For perpendicular recording channels, some typical error events with small distances are (+) , (+) , (+ +) , (+ +) , (+0+) , (+00+) , (+000+) , and their inverse correspondences. The error rate performance of perpendicular recording channels can be improved by combining the properties of MTR constraints and parity checking [1]. For example, the MTR (j=3) constraint with 2-bit parity checking eliminates (+ +) and corrects (+) , (+) , (+ +) , and (+00+) error events. The timevarying MTR (j=3) constraint can be utilized to improve code efficiency, since the capacity of the TMTR (j=3) constraint, .9613, is larger than that of the MTR (j=3) constraint, .9468. The TMTR (j=3) constraint can also eliminate the error event (+ +) . Contrary to longitudinal recording channels, perpendicular recording channels exhibit low frequency contents. The targets of channel detector may be chosen as DC-free to balance signal energy and noise power properly at low frequencies. However, DC-free targets sharply suppress low frequency noise and signal energy, resulting in losses in SNR. Since RDS constrained data sequences are DC-free and have attenuated low-frequency contents, they may be used in conjunction with DC-free targets to improve the spectral SNR at low frequencies. In designing practical codes for the magnetic recording channels, it is common to implement several constraints into the same code. Even though the individual properties of the constraints are satisfied in a code, it is usually not known how good codes achieve. The capacity of a constraint S represented by a graph G is given by Cap(S) = log (D(G)) , where D(G) is the connection matrix of G and (D(G)) is the largest eigenvalue of D(G) . The graph of a combined constraint can be obtained by using the fiber product of graphs of its constituent constraints [2]. We propose a new class of codes, which combine both constraints of MTR (or TMTR) and RDS, named RDS-MTR (or RDS-TMTR) in short. The capacity values of some RDS-TMTR constraints are shown in Table I. For large values of N and j the capacity of the RDS-TMTR constraints approaches to that of its constituent RDS and TMTR constraints. The same holds for RDS-MTR constraints. The characteristics of the RDS-MTR (or RDS-TMTR) constraints are better understood in the frequency domain. Similar to the RDS constraint, RDS-TMTR constraints have a null at DC frequency. A method for measuring suppression near DC frequency is the cut-off frequency, w 0 , which is defined as H (w 0 ) = 1 / 2 , where H is the power density spectrum. When N increases, the cutoff frequencies of the RDS-TMTR constraints decrease so that the constraint become less effective to remove the frequency content near DC frequency. Since the capacity of the TMTR (j=3) constraint is about 0.9613, a rate 20 / 21 0.9524 block code can be designed based on this constraint. The TMTR (j=3) has to be satisfied not only inside of the codewords but also at the codeword boundaries. We modify the TMTR (j=3) constraint such that the pattern of codeword positions in which four consecutive 1's can start has period of 21. The following pattern is one of the best patterns resulting in larger number of codewords: [3,4,3,4,3,4,3,4,3,4,3,3,4,3,4,3,4,3,3,4,3]. The codewords that do not satisfy this pattern are removed from the codebook. In order to satisfy the constraint over the codeword boundaries, the codewords having certain prefixes and suffixes are excluded from the codebook. The number of codewords starting and ending with at most two consecutive 1's is more than the required number of codewords: 1, 072, 089 > 2 20 , giving 23,513 excessive codewords to implement the RDS constraint.
1 Ismail Demirkan, Center for Magnetic Recording Research, University of California, San Diego, 9500 Gilman Dr, La Jolla, CA, 92093-0401 Phone: 858-534-6542, [email protected]
The TMTR constraints are localized in the sense that only certain neighbor bits are related to each other. However, RDS constraints are not localized so that the encoder has to keep the RDS information from the beginning to end of whole sector. Except for some low rate codes, such as zero disparity code [3], it is not possible design a high rate RDS block code that guarantees DC-free sequences. Let z denote the global RDS value for whole sector. The codewords in NRZI domain are converted to NRZ domain by assuming the previous NRZ level being -1. For example, the codeword c=[1,1,0,0,0] is transformed to d=-1[+1, -1, -1, -1, -1], which changes the global RDS value by z = 3 . In the case of previous NRZ level being +1, every digit in d is flipped to result in z = 3 . Therefore every codeword with non-zero z can change global RDS value in both ways according to previous NRZ level that cannot be controlled. The codewords with | z |> 11 are discarded to obtain better spectral performance, dropping the number of excessive codewords to 12149. Five blocks of the rate 20/21 block code are used in parallel to obtain the rate 100/105 block code as shown in Fig 1. In order to prevent short error events, two bit parity bits are embedded into the code in such a way that the both RDS and TMTR constraints are still valid after inclusion of parity check bits. A method for eliminating short error events using parity check codes has been proposed in [4]. The range of the global RDS value is roughly [-55, +55] for each codeword. However, the global RDS value has to be limited to a certain range along the entire sequence so as to obtain DC-free property. This can be done by means of inserting a redundant polarity bit before each 107-bit codeword. The polarity bit is not decoded and it only affects the polarity of write current waveform, not the information bits in NRZI domain.
N\j 4 8 12 16 20 24 32 40 Cap(MTR)
TABLE I Capacity Values for RDS-TMTR Constraints 1 2 3 4 5 6 0.3471 0.5234 0.6060 0.6420 0.6657 0.6763 0.6702 0.8051 0.8611 0.8850 0.8978 0.9036 0.7352 0.8638 0.9142 0.9362 0.9471 0.9521 0.7593 0.8857 0.9339 0.9553 0.9655 0.9703 0.7708 0.8963 0.9434 0.9645 0.9744 0.9791 0.7772 0.9022 0.9487 0.9696 0.9794 0.9840 0.7837 0.9082 0.9541 0.9748 0.9844 0.9890 0.7868 0.9110 0.9567 0.9773 0.9868 0.9913 0.7925 0.9163 0.9613 0.9818 0.9912 0.9957
Cap(RDS) 0.6942 0.9103 0.9575 0.9752 0.9838 0.9886 0.9935 0.9958 1.000
Fig 1. Block diagram of the rate 100/108 block code References: [1] R. D. Cideciyan and E. Eleftheriou, “Codes satisfying maximum transition run and parity-check constraints,” in IEEE International Conference on Communications, pp. 635–639, Jun. 2004. [2] B. Marcus, R. M. Roth, and P. H. Siegel, “Constrained systems and coding for recording channels,” in Handbook of Coding Theory. V.S. Pless and W.C. Huffman, Eds. Amsterdam, The Netherlands: Elsevier, 1998, pp. 1635-1764. [3] K. A. S. Immink, Codes for Mass Data Storage Systems. The Netherlands: Shannon Foundation Publishers, 1999. [4] R. D. Cideciyan and E. Eleftheriou, “Combined modulation/parity codes for magnetic recording,” in Proc. IEEE International Symposium on Information Theory, p. 48, 2003.
1 Ismail Demirkan, Center for Magnetic Recording Research, University of California, San Diego, 9500 Gilman Dr, La Jolla, CA, 92093-0401 Phone: 858-534-6542, [email protected]
The TMTR constraints are localized in the sense that only certain neighbor bits are related to each other. However, RDS constraints are not localized so that the encoder has to keep the RDS information from the beginning to end of whole sector. Except for some low rate codes, such as zero disparity code [3], it is not possible design a high rate RDS block code that guarantees DC-free sequences. Let z denote the global RDS value for whole sector. The codewords in NRZI domain are converted to NRZ domain by assuming the previous NRZ level being -1. For example, the codeword c=[1,1,0,0,0] is transformed to d=-1[+1, -1, -1, -1, -1], which changes the global RDS value by z = 3 . In the case of previous NRZ level being +1, every digit in d is flipped to result in z = 3 . Therefore every codeword with non-zero z can change global RDS value in both ways according to previous NRZ level that cannot be controlled. The codewords with | z |> 11 are discarded to obtain better spectral performance, dropping the number of excessive codewords to 12149. Five blocks of the rate 20/21 block code are used in parallel to obtain the rate 100/105 block code as shown in Fig 1. In order to prevent short error events, two bit parity bits are embedded into the code in such a way that the both RDS and TMTR constraints are still valid after inclusion of parity check bits. A method for eliminating short error events using parity check codes has been proposed in [4]. The range of the global RDS value is roughly [-55, +55] for each codeword. However, the global RDS value has to be limited to a certain range along the entire sequence so as to obtain DC-free property. This can be done by means of inserting a redundant polarity bit before each 107-bit codeword. The polarity bit is not decoded and it only affects the polarity of write current waveform, not the information bits in NRZI domain.
N\j 4 8 12 16 20 24 32 40 Cap(MTR)
TABLE I Capacity Values for RDS-TMTR Constraints 1 2 3 4 5 6 0.3471 0.5234 0.6060 0.6420 0.6657 0.6763 0.6702 0.8051 0.8611 0.8850 0.8978 0.9036 0.7352 0.8638 0.9142 0.9362 0.9471 0.9521 0.7593 0.8857 0.9339 0.9553 0.9655 0.9703 0.7708 0.8963 0.9434 0.9645 0.9744 0.9791 0.7772 0.9022 0.9487 0.9696 0.9794 0.9840 0.7837 0.9082 0.9541 0.9748 0.9844 0.9890 0.7868 0.9110 0.9567 0.9773 0.9868 0.9913 0.7925 0.9163 0.9613 0.9818 0.9912 0.9957
Cap(RDS) 0.6942 0.9103 0.9575 0.9752 0.9838 0.9886 0.9935 0.9958 1.000
Fig 1. Block diagram of the rate 100/108 block code References: [1] R. D. Cideciyan and E. Eleftheriou, “Codes satisfying maximum transition run and parity-check constraints,” in IEEE International Conference on Communications, pp. 635–639, Jun. 2004. [2] B. Marcus, R. M. Roth, and P. H. Siegel, “Constrained systems and coding for recording channels,” in Handbook of Coding Theory. V.S. Pless and W.C. Huffman, Eds. Amsterdam, The Netherlands: Elsevier, 1998, pp. 1635-1764. [3] K. A. S. Immink, Codes for Mass Data Storage Systems. The Netherlands: Shannon Foundation Publishers, 1999. [4] R. D. Cideciyan and E. Eleftheriou, “Combined modulation/parity codes for magnetic recording,” in Proc. IEEE International Symposium on Information Theory, p. 48, 2003.
