Article

# Statistical properties of selected recording codes

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## Abstract

Most recording systems encode their data using binary run-length-limited (RLL) codes. Statistics such as the density of 1s, the probabilities of specific code strings or run lengths, and the power spectrum are useful in analyzing the performance of RLL codes in these applications. These statistics are easy to compute for ideal run-length-limited codes, those whose only constraints are the run-length limits, but ideal RLL codes are not usable in practice because their code rates are irrational. Implemented RLL codes achieve rational rates by not using all code sequences which satisfy the run-length constraints, and their statistics are different from those of the ideal RLL codes. Little attention has been paid to the computation of statistics for these practical codes. In this paper a method is presented for computing statistics of implemented codes. The key step is to develop an exact description of the code sequences which are used. A consequence of the code having rational rate is that all the code-string and run-length probabilities are rational. The method is illustrated by applying it to three codes of practical importance: MFM, (2, 7), and (1, 7).

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... Since the capacity of (d, k)-constrained sequences is irrational (see Ashley and Siegel [2]), it is clear that code implementations which, by necessity, operate at rates of the form m / n, where mand nare integers, can never attain 100%' of the capacity. It was noted by Howell [3] that implemented codes differ from maxentropic, "ideal, " sequences bY,the addition of certain constraints, which he called incidental contraints. He found that certain bit patterns, which could readily be used as a basis for constructing sync patterns, are absent in sequences generated by the popular (l, 7) and (2, 7) codes. ...
... It is, therefore, clear that code implementations can never attain 100% of the capacity . It was noted by Howell [3] that implemented codes differ from maxentropic, "ideal, " sequences by the addition of a few constraints, which he called incidental constraints, Howell [3] described in detail the incidental constraints ofthree popular (d, k) codes, namely the rate 2/3, (1, 7) code [15], the rate 1/2, (2, 7) code [16], and the rate 1/2, (I, 3) code. He found that eertain bit pattems do not occur in sequenees generated by these codes. ...
... It is, therefore, clear that code implementations can never attain 100% of the capacity . It was noted by Howell [3] that implemented codes differ from maxentropic, "ideal, " sequences by the addition of a few constraints, which he called incidental constraints, Howell [3] described in detail the incidental constraints ofthree popular (d, k) codes, namely the rate 2/3, (1, 7) code [15], the rate 1/2, (2, 7) code [16], and the rate 1/2, (I, 3) code. He found that eertain bit pattems do not occur in sequenees generated by these codes. ...
... The characteristic equation pertaining to the recursion (17) equals (18) The difference between the largest root of (18) and (10) in the working range which means that the loss in capacity resulting from the truncation of the weights can be simply approximated by (19) A numerical comparison of the above result with the outcome of the method of computing the cycle time revealed that (19) can serve as a reliable rule of thumb. ...
... The characteristic equation pertaining to the recursion (17) equals (18) The difference between the largest root of (18) and (10) in the working range which means that the loss in capacity resulting from the truncation of the weights can be simply approximated by (19) A numerical comparison of the above result with the outcome of the method of computing the cycle time revealed that (19) can serve as a reliable rule of thumb. ...
... In this subsection, we will compare codes contructed by the new coding technique with traditional sliding-block codes: the rate , code and the rate , code [19], which have been used in magnetic disk drives. The maximum runlength constraint is imposed to restrict the maximum time between two consecutive transitions in the recorded signal. ...
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... That means the process of imposing the incidental constraints will involve discarding states rather than discarding branches between states. There are other ways to arrive at the desired results, but this method was preferred because it facilitated the use of the computational tools that were available [2], [4]. ...
... A one followed by a run of 6 zeros must start with the one in position a or c. There are other choices of incidental constraints that would result in the same restrictions, but with different labels attached to the individual bits of the output symbols [2], [5], [61. For these other choices, the individual bits of the output symbols would be labeled (bca) and (cab). ...
... The statistical properties of a code are embodied in the runlength constraint graph. Because this code is based on the same incidental constraints as the standard IBM (1,7) code, the set of sequences which can be generated is the same [2]. Although the mapping of user bits to channel bits is different, the statistical properties are exactly the same. ...
... spectrum resulting from the graph is shown in Fig. 11 along with the power spectrum for the maxentropic case. It turns out that I. INTRODUCTION the constraint graph in Fig. 5 corresponds exactly to one of the unused components of the cubed (1,7) constraint graph in [2]. ...
... Quite recently [2], a new algorithm for generating zero-disparity codewords was presented by Knuth. The method is based on a simple correspondence between the set of all m-bit source words and the set of all (m + p)-bit balanced codewords. ...
... Most schemes for generating dc-balanced sequences use lookup tables, and are therefore restricted to codewords of medium size. An alternative and easily implementable encoding technique for zero-disparity codewords that is capable of handling (very) large blocks was described by Knuth [2]. The method is based on the idea that there is a simple correspondence between the set of all m-bit binary source words and the set of all (m + p&bit codewords. ...
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... (Note that the SNR = 22 db corresponds to the "peak-shift" probability 2Q(A /2a) = 3.2 x lo-".) Popular rate 2/3 (1,7) codes are the Jacoby code [141 and the AHM (IBM) code [15], [16]. For high signal-to-noise ratios, their rate is much lower than our lower bound on C,(2,8, a21 (Fig. 5). ...
... For high signal-to-noise ratios, their rate is much lower than our lower bound on C,(2,8, a21 (Fig. 5). Similar comparisons can be made for rate l/2 IBM [II, [21, [16] and Zerox [13] (2,7) codes (Fig. 6). ...
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... As is well known for the case of q = 2 (see, e.g., [14], [36]), its characteristic polynomial is ...
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A computer model of a peak detecting magnetic recording channel has been implemented and used for channel design and performance evaluation. The model predicts raw error rate, ontrack and off-track, as a function of linear density, run-length-limited (RLL) modulation code, write precompensation rules, and tapped-delay-line (TDL) equalizer. It assumes noise additivity and validity of linear superposition. and it bases calculations on a measured disk/electronics noise spectrum and digitized isolated transition readback signals from the data track and adjacent tracks. Details of the model are described, and illustrative applications to RLL (d,k) code selection and pulse slimming equalizer design for a specific channel are discussed.
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The power spectral density (PSD) is the average power per unit frequency of encoded random data transmitted over a perfect channel. The one-sided PSDs of a number of channel codes of recent interest in digital magnetic recording are calculated from codeword dictionaries and state diagrams. Given here are:
Article
A few new modulation codes(or run-length-limited codes) in digital magnetic recording is presented. Linear density limits of the new codes and the existing codes are evaluated for a typical recording channel. One of the new codes shows the highest linear density limit among the existing codes and is improved over, for example, MFM and 4/5-rate-NRZI codes by about 20% to 30% in the density limit for the channel. A method for constructing the new codes is presented. Copyright © 1976 by The Institute of Electrical and Electronics Engineers, Inc.
Article
It is proven that 100-percent efficient fixed-rate codes for run-length-limited (RLL) (d,k) and RLL charge-constrained (d, k; c) channels are possible in only two eases, namely (d,k; c)=(0,1;1) and (1,3;3) . Specifically, the binary Shannon capacity of RLL (d, k) constrained systems is shown to be irrational for all values of (d, k),0 leq d < k . For RLL charge-constrained systems with parameters (d, k;c) , the binary capacity is irrational for all values of (d, k; c),0 leq d < k,2c geq k + 1 , except (0,1; 1) and (1,3;3) , which both have binary capacity 1/2 .
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Method and Apparatus for Generating a Noiseless Sliding Block Code for a (1,7) Channel with Rate 2/3
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Algorithms for Sliding Block Codes Theory &lt;b&gt;IT-29&lt