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We report on block-coding techniques for partial-response channels
with transfer function (1∓D<sup>m</sup>), m=1, 2, ... . We
consider various constructions of block codes with prescribed minimum
Euclidean distance. Upper and lower bounds to the size of a code with
minimum squared Euclidean distance greater than unity are furnished. A
table is presented of cardinalities of codes of small length with
prescribed minimum squared Euclidean distance

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... Wolf and Ungerboeck 2] considered trellis coding techniques to improve performance, Karabed and Siegel 3] used matched spectral-null codes to reduce the complexity from the ML solution without any loss in performance. Recently Tolhuizen et al. 4] described block codes for PRS channels as another method. Finally Isaka and Imai 5] and Nasari-Kenari et al. 6] described the concatenation of a convolutional encoder with the PRS channel. ...

This paper describes the operation of an iterative receiver for use on partial response signalling (PRS) channels. The data is convolutionally encoded and interleaved before being transmitted across the channel. An iterative receiver based on the turbo decoding principle is used to recover the data. Analysis and simulations confirm the exceptional performance of the receiver, with complexity independent of the memory of the channel. I. Introduction PRS channels have been found to be very useful for magnetic storage channels. The receiver size, however, grows exponential with the memory of the channel, adding complexity constraints. The concatenation of channel coding with the PRS channel has been previously described e.g. [1] as a method of improving performance due to coding gain. In this work we include channel coding not only to achieve coding gain improvements but to assist with equalisation of the PRS channel. Motivated by a recent iterative multi-user DS-CDMA receiver design [2...

We show that block coded sequences are cycloergodic, and based on this property, we introduce a new nonprobabilistic formula to calculate the average power spectral density of these sequences. We present a new sufficient condition to construct codes with an arbitrarily high-order spectral-null at zero frequency. Given this condition, we outline two new coding schemes and use them to generate new classes of efficient high-order spectral-null sequences

We present a new construction of block codes for the (1-D)-PR channel with a minimum squared Euclidean distance of six. Except for small code lengths, the new codes have larger rates than other known codes

When a block modulation code is concatenated with an error-correction code (ECC) in the standard way, the use of a modulation code with long block lengths results in error propagation. This article analyzes the performance of modified concatenation, which involves reversing the order of modulation and the ECC. This modified scheme reduces the error propagation, provides greater flexibility in the choice of parameters, and facilitates soft-decision decoding, with little or no loss in transmission rate. In particular, examples are presented which show how this technique can allow fewer interleaves per sector in hard disk drives, and permit the use of more sophisticated block modulation codes which are better suited to the channel

We present a new construction of block codes for the (1-D)-PR
(partial response) channel. The codewords in the code correspond to
constant-sum subsets of a difference set. It is shown that at the output
of a noiseless (1-D)-PR channel; the minimum squared Euclidean distance
of such a code is at least six, compared to two for the uncoded system.
This construction yields larger code rates than previously known codes
with the same minimum distance for large code lengths. The construction
technique also imposes upper bounds on the decoding complexity of the
codes

This memory can be due to time dispersion, as in the ISI channel or a convolutional code. In DS/CDMA with non-orthogonal users the memory in the channel is due to users themselves. This is described in Chapter 2

Since the early 1980s we have witnessed the digital audio and video revolution: the Compact Disc (CD) has become a commodity audio system. CD-ROM and DVD-ROM have become the de facto standard for the storage of large computer programs and files. Growing fast in popularity are the digital audio and video recording systems called DVD and BluRay Disc. The above mass storage products, which form the backbone of modern electronic entertainment industry, would have been impossible without the usage of advanced coding systems.
Pulse Code Modulation (PCM) is a process in which an analogue, audio or video, signal is encoded into a digital bit stream. The analogue signal is sampled, quantized and finally encoded into a bit stream. The origins of digital audio can be traced as far back as 1937, when Alec H. Reeves, a British scientist, invented pulse code modulation \cite{Ree}. The advantages of digital audio and video recording have been known and appreciated for a long time. The principal advantage that digital implementation confers over analog systems is that in a well-engineered digital recording system the sole significant degradation takes place at the initial digitization, and the quality lasts until the point of ultimate failure. In an analog system, quality is diminished at each stage of signal processing and the number of recording generations is limited. The quality of analog recordings, like the proverbial 'old soldier', just fades away. The advent of ever-cheaper and faster digital circuitry has made feasible the creation of high-end digital video and audio recorders, an impracticable possibility using previous generations of conventional analog hardware.
The general subject of coding for digital recorders is very broad, with its roots deep set in history. In digital recording (and transmission) systems, channel encoding is employed to improve the efficiency and reliability of the channel. Channel coding is commonly accomplished in two successive steps: (a) error-correction code followed by (b) recording (or modulation) code. Error-correction control is realized by adding extra symbols to the conveyed message. These extra symbols make it possible for the receiver to correct errors that may occur in the received message.
In the second coding step, the input data are translated into a sequence with special properties that comply with the given "physical nature" of the recorder. Of course, it is very difficult to define precisely the area of recording codes and it is even more difficult to be in any sense comprehensive. The special attributes that the recorded sequences should have to render it compatible with the physical characteristics of the available transmission channel are called channel constraints. For instance, in optical recording a '1' is recorded as pit and a '0' is recorded as land. For physical reasons, the pits or lands should neither be too long or too short. Thus, one records only those messages that satisfy a run-length-limited constraint. This requires the construction of a code which translates arbitrary source data into sequences that obey the given constraints. Many commercial recorder products, such as Compact Disc and DVD, use an RLL code.
The main part of this book is concerned with the theoretical and practical aspects of coding techniques intended to improve the reliability and efficiency of mass recording systems as a whole. The successful operation of any recording code is crucially dependent upon specific properties of the various subsystems of the recorder. There are no techniques, other than experimental ones, available to assess the suitability of a specific coding technique. It is therefore not possible to provide a cookbook approach for the selection of the 'best' recording code.
In this book, theory has been blended with practice to show how theoretical principles are applied to design encoders and decoders. The practitioner's view will predominate: we shall not be content with proving that a particular code exists and ignore the practical detail that the decoder complexity is only a billion times more complex than the largest existing computer. The ultimate goal of all work, application, is never once lost from sight. Much effort has been gone into the presentation of advanced topics such as in-depth treatments of code design techniques, hardware consequences, and applications. The list of references (including many US Patents) has been made as complete as possible and suggestions for 'further reading' have been included for those who wish to pursue specific topics in more detail.
The decision to update Coding Techniques for Digital Recorders, published by Prentice-Hall (UK) in 1991, was made in Singapore during my stay in the winter of 1998. The principal reason for this decision was that during the last ten years or so, we have witnessed a success story of coding for constrained channels. The topic of this book, once the province of industrial research, has become an active research field in academia as well. During the IEEE International Symposia on Information Theory (ISIT and the IEEE International Conference on Communications (ICC), for example, there are now usually three sessions entirely devoted to aspects of constrained coding. As a result, very exciting new material, in the form of (conference) articles and theses, has become available, and an update became a necessity.
The author is indebted to the Institute for Experimental Mathematics, University of Duisburg-Essen, Germany, the Data Storage Institute (DSI) and National University of Singapore (NUS), both in Singapore, and Princeton University, US, for the opportunity offered to write this book. Among the many people who helped me with this project, I like to thank Dr. Ludo Tolhuizen, Philips Research Eindhoven, for reading and providing useful comments and additions to the manuscript.
Preface to the Second Edition
About five years after the publication of the first edition, it was felt that an update of this text would be inescapable as so many relevant publications, including patents and survey papers, have been published. The author's principal aim in writing the second edition is to add the newly published coding methods, and discuss them in the context of the prior art. As a result about 150 new references, including many patents and patent applications, most of them younger than five years old, have been added to the former list of references. Fortunately, the US Patent Office now follows the European Patent Office in publishing a patent application after eighteen months of its first application, and this policy clearly adds to the rapid access to this important part of the technical literature.
I am grateful to many readers who have helped me to correct (clerical) errors in the first edition and also to those who brought new and exciting material to my attention. I have tried to correct every error that I found or was brought to my attention by attentive readers, and seriously tried to avoid introducing new errors in the Second
Edition.
China is becoming a major player in the art of constructing, designing, and basic research of electronic storage systems. A Chinese translation of the first edition has been published early 2004. The author is indebted to prof. Xu, Tsinghua University, Beijing, for taking the initiative for this Chinese version, and also to Mr. Zhijun Lei, Tsinghua University, for undertaking the arduous task of translating this book from English to Chinese. Clearly, this translation makes it possible that a billion more people will now have access to it.
Kees A. Schouhamer Immink, Rotterdam, November 2004

We consider trellis-coding techniques for improving the reliability of digital transmission over noisy partial-response channels. Such channels are commonly encountered in digital communication systems, and also play a role in devices for data recording. Concentrating on the channels with characteristics (1 mp D) , we study methods to obtain codes which increase free Euclidean distance between permitted sequences of channel outputs and avoid the occurrence of unlimited runs of identical outputs at the expense of some loss in data rate. One technique employs the concept of set partitioning. The other is based on using convolutional codes designed for maximum free Hamming distance in conjunction with a precoder. Both methods lead to essentially equivalent codes.

A new family of codes is described that improve the reliability of digital communication over noisy, partial-response channels. The codes are intended for use on channels where the input alphabet size is limited. These channels arise in the context of digital data recording and certain data transmission applications. The codes-called matched-spectral-null codes -satisfy the property that the frequencies at which the code power spectral density vanishes correspond precisely to the frequencies at which the channel transfer function is zero. It is shown that matched-spectral-null sequences provide a distance gain on the order of 3 dB and higher for a broad class of partial-response channels, including many of those of primary interest in practical applications. The embodiment of the matched-spectral-null coded partial-response system incorporates a sliding-block code and a Viterbi detector based upon a reduced-complexity trellis structure, both derived from canonical diagrams that characterize spectral-null sequences. The detectors are shown to achieve the same asymptotic average performance as maximum-likelihood sequence-detectors, and the sliding-block codes exclude quasicatastrophic trellis sequences in order to reduce the required path memory length and improve "worst-case" detector performance. Several examples are described in detail.

Let T(n,k) denote the set of all words of length n over the alphabet {+1, -1}, having a kth order spectral-null at zero frequency. A subset of T(n,k) is a spectral-null code of length n and order k. Upper and lower bounds on the cardinality of T(n,k) are derived. In particular we prove that (k - 1) log2 (n/k) less-than-or-equal-to n - log2 \T(n,k)\ less-than-or-equal-to O(2k log2 n) for infinitely many values of n. On the other hand, we show that T(n,k) is empty unless n is divisible by 2m, where m = left-perpendicularlog2 kright-perpendicular + 1. Furthermore, bounds on the minimum Hamming distance d of T(n,k) are provided, showing that 2k less-than-or-equal-to d less-than-or-equal-to k(k - 1) + 2 for infinitely many n. We also investigate the minimum number of sign changes in a word x is-an-element-of T(n,k) and provide an equivalent definition of T(n,k) in terms of the positions of these sign changes. An efficient algorithm for encoding arbitrary information sequences into a second-order spectral-null code of redundancy 3 log2 n + O(log log n) is presented. Furthermore, we prove that the first nonzero moment of any word in T(n,k) is divisible by k! and then show how to construct a word with a spectral null of order k whose first nonzero moment is any even multiple of k!. This leads to an encoding scheme for spectral-null codes of length n and any fixed order k, with rate approaching unity as n --> infinity.

It is proved that codes devised by J. Berger (see Inf., and
Control. vol.4, p.68-73, (1961)) that detect random bursts of length
b and simultaneously detect any number of unidirectional errors
( b -bED/AUED codes) are optimal. The construction is modified in
two ways: (1) to obtain b -bED/ s -UED codes in which a
random burst of length b and up to s unidirectional
errors are detected, and (2) to obtain b -bED/ s -UbED
codes in which a random burst of length b and a unidirectional
burst of length s are detected. A lower bound on the number of
check bits for such codes is obtained. The lower bounds indicate that
the proposed b -bED/ s -UED codes are close to optimal
and the b -bED/ s -UbED codes are asymptotically optimal

A coding technique for improving the reliability of digital
transmission over noisy partial-response channels with characteristics
(± D <sup>m</sup>), m =1, 2, where the channel
input symbols are constrained to be ±1, is presented. In
particular, the application of a traditional modulation code as an inner
code of a concentrated coding scheme in which the outer code is designed
for maximum (free) Hamming distance is considered. A performance
comparison is made between the concentrated scheme and a coding
technique presented by Wolf and G. Ungerboeck (see ibid., vol. COM-34,
p.765-773, Aug. 1986) for the dicode channel with transfer function (1-
D )

A method is presented for designing binary channel codes in such a way that both the power spectral density function and its low-order derivatives vanish at zero frequency. The performance of the new codes is compared with that of channel codes designed with a constraint on the unbalance Of the number of transmitted positive and negative pulses. Some remarks are made on the error-correcting capabilities of these codes.

In honor of the twenty-fifth anniversary of Huffman coding, four new results about Huffman codes are presented. The first result shows that a binary prefix condition code is a Huffman code iff the intermediate and terminal nodes in the code tree can be listed by nonincreasing probability so that each node in the list is adjacent to its sibling. The second result upper bounds the redundancy (expected length minus entropy) of a binary Huffman code by P_{1}+ log_{2}[2(log_{2}e)/e]=P_{1}+0.086 , where P_{1} is the probability of the most likely source letter. The third result shows that one can always leave a codeword of length two unused and still have a redundancy of at most one. The fourth result is a simple algorithm for adapting a Huffman code to slowly varying esthnates of the source probabilities. In essence, one maintains a running count of uses of each node in the code tree and lists the nodes in order of these counts. Whenever the occurrence of a message increases a node count above the count of the next node in the list, the nodes, with their attached subtrees, are interchanged.

A coset of a convolutional code may be used to generate a zero-run
length limited trellis code for a 1-D partial-response channel. The free
squared Euclidean distance, d<sub>free</sub><sup>2</sup>, at the channel
output is lower bounded by the free Hamming distance of the
convolutional code. The lower bound suggests the use of a convolutional
code with maximal free Hamming distance, d<sub>max</sub>(R,N), for given
rate R and number of decoder states N. In this paper we present cosets
of convolutional codes that generate trellis codes with d<sub>free</sub>
<sup>2</sup>>d<sub>max</sub>(R,N) for rates 1/5⩽R⩽7/9 and (d
<sub>free</sub><sup>2</sup>=d<sub>max</sub>(R,N) for
R=13/16,29/32,61/64, The tabulated convolutional codes with R⩽7/9
were not optimized for Hamming distance. Instead, a computer search was
used to determine cosets of convolutional codes that exploit the memory
of the 1-D channel to increase d<sub>free</sub><sup>2</sup> at the
channel output. The search was limited by only considering cosets with
certain structural properties. The R⩾13/16 codes were obtained using
a new construction technique for convolutional codes with free Hamming
distance 4. Newly developed bounds on the maximum zero-run lengths of
cosets were used to ensure a short maximum run length at the 1-D channel
output

A bound on the probability that the length of any source code will
be shorter than the self information by γ bits is easily obtained
using a Chebyshev-type argument. From this bound, one can establish the
competitive optimality of the self information and of the Shannon-Fano
code (up to one bit). In general, however, the Huffman code cannot be
examined using this technique. Nevertheless, in the present work, the
competitive optimality (up to one bit) of the Huffman code for general
sources is also established using a different technique

Let X be a discrete random variable drawn according to a
probability mass function p ( x ), and suppose
p ( x ), is dyadic, i.e., log(1/ p ( x ))
is an integer for each x . It is shown that the binary code
length assignment l ( x )=log(1/ p ( x ))
dominates any other uniquely decodable assignment
l '( x ) in expected length in the sense that
El ( X )< El '( X ), indicating
optimality in long run performance (which is well known), and
competitively dominates l '( x ), in the sense that Pr{
l ( X )< l '( X )}>Pr{ l (
X )> l '( X )}, which indicates l is
also optimal in the short run. In general, if p is not dyadic
then l =[log 1/ p ] dominates l '+1 in expected
length and competitivity dominates l '+1, where l ' is
any other uniquely decodable code

Systematic codes capable of detecting burst unidrectional errors of length up to 2<sup>r−1</sup>using r check bits where r ≥ 3 are presented. Moreover, b-adjacent unidirectional error-detecting codes using [log 2 (b + 1)] check bits are also described. These codes are shown to be optimal or near optimal. The encoding/decoding and the totally self- checking checker design methods for these codes are also given.

In this paper we present some basic theory on unidirectional error correcting/detecting codes. We define symmetric, asymmetric, and unidirectional error classes and proceed to derive the necessary and sufficient conditions for a binary code to be unidirectional error correcting/detecting.

Families of systematic unidirectional burst-detecting codes are
presented. When the number of information bits is large enough, the
codes can detect longer bursts than previously known codes. Encoding and
decoding procedures are indicated

Higer-order spectral-null codes-Constructions and bounds Engle-wood Cliffs Theory of unidirectional error cor-rectingldetecting codes Burst unidirectional error detecting codes

- R M Roth
- P H Siegel
- A B Vardy
- T R N Bose
- B Rao
- Bose

R. M. Roth, P. H. Siegel, and A. Vardy, " Higer-order spectral-null codes-Constructions and bounds, " IEEE Trans. Inform. Theory, vol. 40, no. 6, pp. 1826-1840, Nov. 1994. U. A. S. Immink, Coding Techniques for Digital Recorders. Engle-wood Cliffs, NJ: Prentice-Hall, 1991. B. Bose and T. R. N. Rao, " Theory of unidirectional error cor-rectingldetecting codes, " IEEE Trans. Comput., vol. C-31, no. 6, pp. 521-530, June 1982. B. Bose, " Burst unidirectional error detecting codes, " ZEEE Trans. Comput., vol. '2-35, no. 4, pp. 350-353, Apr. 1986. M. Blaum, " Systematic unidirectional burst detecting codes, " IEEE Trans. Comput., vol. 37, no. 4, pp. 453457, Apr. 1988.

- G Owen

G. Owen, Game Theory, 2nd ed. New York: Academic Press, 1982.
IEEETRANSACTIONS
ONINFORMATIONTHEORY,VOL.41,NO.6,NOVEMBER
1995

Matched spectral-null codes for partialresponse channels

- R Karabed
- P H Siegel

R. Karabed and P. H. Siegel, "Matched spectral-null codes for partialresponse channels," IEEE Trans. Inform. Theory, vol. 37, no. 3, pp.
818-855, May 1991.

Higer-order spectral-null codes-Constructions and bounds

- M Roth
- P H Siegel
- A Vardy

M. Roth, P. H. Siegel, and A. Vardy, "Higer-order spectral-null
codes-Constructions and bounds," IEEE Trans. Inform. Theory, vol.
40, no. 6, pp. 1826-1840, Nov. 1994.