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Binary transmission codes with higher order spectral zeros at zero frequency

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Abstract

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.

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... A codeword, x, is dc 2 -balanced if it satisfies [13,25] n i=1 ...
... The set S 2 is empty if n mod 4 = 0 [13]. Let x ∈ S 2 then its reverse x r = (x n , . . . ...
... Figure 1 shows results of computations for n = 32, 64, and 128. As a comparison we plotted the exact auto-correlation function of a full set of dc 2 -balanced sequences, denoted byρ(i), which was computed using an enumeration technique and generating functions [13]. ...
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We apply the central limit theorem for deriving approximations to the auto-correlation function and power density function (spectrum) of second-order spectral null (dc2-balanced) codes. We show that the auto-correlation function of dc2-balanced codes can be accurately approximated by a cubic function. We show that the difference between the approximated and exact spectrum is less than 0.03 dB for codeword length n=256.
... A codeword, x, is dc 2 -balanced if it satisfies [10], [21] n i=1 ...
... The set S 2 is empty if n mod 4 = 0 [10]. Let x ∈ S 2 then its reverse x r = (x n , . . . ...
... Figure 1 shows results of computations for n = 32, 64, and 128. As a comparison we plotted the exact auto-correlation function of a full set of dc 2 -balanced sequences, denoted byρ(i), which was computed using an enumeration technique and generating functions [10]. ...
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We apply the central limit theorem for deriving approximations to the auto-correlation function and power density function (spectrum) of second-order spectral null (dc2-balanced) codes.We show that the auto-correlation function of dc2-balanced codes can be accurately approximated by a cubic function. We show that the difference between the approximate and exact spectrum is less than 0.04 dB for codeword length n = 256.
... A codeword, x, is dc 2 -balanced if it satisfies [10], [21] ...
... The set S 2 is empty if n mod 4 ̸ = 0 [10]. Let x ∈ S 2 then its reverse x r = (x n , . . . ...
... Figure 1 shows results of computations for n = 32, 64, and 128. As a comparison we plotted the exact auto-correlation function of a full set of dc 2 -balanced sequences, denoted byρ(i), which was computed using an enumeration technique and generating functions [10]. ...
Preprint
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We apply the central limit theorem for deriving approximations to the auto-correlation function and power density function (spectrum) of second-order spectral null (dc^2-balanced) codes. We show that the auto-correlation function of dc^2-balanced codes can be accurately approximated by a cubic function. We compare the approximate auto-correlation function and spectrum with the exact auto-correlation function and spectrum of full set dc^2-balanced codes. We show that the difference between the approximate and exact spectrum is less than 0.04 dB for codeword length n = 256. We compare the spectral performance of dc-balanced versus dc^2-balanced codes in the low-frequency range.
... Higher-order spectral null codes, introduced by Immink [11] and Immink and Beenker [12], exhibit a spectral null at the zero frequency, ω = 0, and also the derivatives of the power density function or spectrum H(ω) at ω = 0 are zero. Higherorder spectral zeros will result in a substantial decrease of the power at low frequencies for a fixed code redundancy as compared with the conventional designs on the bounded digital sum concept. ...
... Higherorder spectral zeros will result in a substantial decrease of the power at low frequencies for a fixed code redundancy as compared with the conventional designs on the bounded digital sum concept. Higher-order spectral null block codes also show attractive distance properties, which have been studied in [12], [13]. A challenging combinatorics problem of estimating the redundancy of codes that satisfy higher-order spectral null constraints have been solved by, for example, Freiman and Litsyn [14], and Roth, Siegel, and Vardy [10]. ...
... There are various ways of implementing higher-order spectral null codes. Immink and Beenker [12] defined a block code comprising a set of K-th order balanced codewords, S K , by ...
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We present an estimate of the power density function (spectrum) of binary K-th order spectral null codes. We work out the auto-correlation model in detail for second-and third-order spectral null codes. We compare the auto-correlation functions and spectra predicted by the model with those generated by full-set K-th order spectral null block codes.
... [2]. dc 2 -Balanced or second-order spectral zero codes exhibit the property that both the power spectrum and its second derivative are zero at zero frequency, resulting in significant suppression of spectral components at low frequencies [3]. Implementation examples of higherorder spectral codes have been presented by Roth et al. [4], Skachek et al. [5], and Yang [6]. ...
... In [3], constrained codes were presented, where the codebook S is chosen such that each codeword x ∈ S satisfies two conditions, namely ...
... dc 2 -Balanced codes have desirable practical features. First, the minimum Hamming distance of S is four [3]. Secondly, both the power spectrum and its second derivative are zero at zero frequency, which is a prerequisite for some channels. ...
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Two constructions of a low-complexity near-optimal detection method of dc2-balanced codes are presented. The methods presented are improvements on Slepian's algorithm for optimal detection of permutation codes.
... For the purpose of this correspondence, let us return to (1). We note here that first the series ix i , which sums all indices i where x i = 1, represents the codeword moment, also more precisely referred to as the first-order moment [4]. By considering the integer codeword moment ix i , before it is reduced modulo m, as is done in (1) and in most investigations, we present some new insight in the binary structure of Levenshtein's codes in Section II. ...
... In [14], we presented the construction of an (n; k) = ( 16; 8) balanced or dc-free block code with d min = 4. Blaum improved on this result by constructing an (n; k) = (16; 9) balanced code in [15], while the bounds in, e.g., [12] indicates the existence of an (n; k) = (16; 10) code. Immink and Beenker [4] proposed an n = 16 dc 2 -constrained code with cardinality 526 for this purpose. ...
... Immink and Beenker [4] described a subclass of balanced codes, sometimes called higher order dc-constrained codes, which suppresses low-frequency components in the code's power spectral density function. Briefly, they investigated codewords x x ...
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Levenshtein proposed a class of single insertion/deletion correcting codes, based on the number-theoretic construction due to Varshamov and Tenengolt’s. We present several interesting results on the binary structure of these codes, and their relation to constrained codes with nulls in the power spectral density function. One surprising result is that the higher order spectral null codes of Immink and Beenker are subcodes of balanced Levenshtein codes. Other spectral null subcodes with similar coding rates, may also be constructed. We furthermore present some coding schemes and spectral shaping markers which alleviate the fundamental restriction on Levenshtein’s codes that the boundaries of each codeword should be known before insertion/deletion correction can be effected.
... The design of codes with low power near the zero frequency was considered for data storage devices as well as communication over metallic cables [95]. ...
... Immink and Beenker [95] studied codewords of K-th-order zero-disparity and proved that the first 2K + 1 derivatives of their power spectral density function are zero at zero frequency. Noting the similarities between (1) and (2), Ferreira et al. [74] pointed out that the codes of Immink and Beenker are subsets of constant-weight Levenshtein codes, thus they can correct one synchronization error. ...
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... and maximize the code size |S| = ∑ i |S wi | by a proper choice of the parameters d c and d p . The above conditions are reminiscent of the conditions imposed on binary codewords that satisfy higher-order spectral constraint [7]. In the next section, we will show that we can find the K subsets that satisfy the constant balance and energy constraint using generating functions [9]. ...
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... The original proof of the distance-enhancing properties of MSN codes was based upon a number-theoretic lower bound on the minimum Hamming distance of zero-disparity codes, due to Immink and Beenker [108]. They proved that the minimum Hamming distance (and, therefore, the minimum Euclidean distance) of a block code over the bipolar alphabet with orderspectral-null at DC grows at least linearly in Specifically, they showed that, for any pair of lengthsequences in the code and This result for block codes can be suitably generalized to any constrained system with orderspectral null at DC. ...
... Spectral-null codes were extensively studied, see e.g. 2,3,4,5,7,13,14,16,21,23,24,27,28,25,26,29] and references therein. The power spectrum of the code words has a wide notch at zero frequency and thus allows e cient rejection of the low-frequency components. ...
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The spectral-null code S(n, k) of kth order and length n is the union of n-tuples with ±1 components, having kth-order spectral-null at zero frequency. We determine the exact asymptotic in n behavior of the size of such codes. In particular, we prove that for n satisfying some divisibility conditions, log<sub>2</sub>|S(n, k)|=n-k <sup>2</sup>/2log<sub>2</sub>n+c<sub>k</sub>+o(1), where c<sub>k</sub> is a constant depending only on k and o(1) tends to zero when n grows. This is an improvement on the earlier known bounds due to Roth, Siegel, and Vardy (see ibid., vol40, p.1826-40, 1994)
... The main point of our discussion is given in Theorem 1 below. The OOC's construction approach in it somehow resembles the higher order constrained codes design [8] for digital recorders [9]. Let p be a prime number and g be a primitive root for p. ...
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... A number of constructions of codes with higher order spectral zeros, based on FSTD's can be found in [4]. Finally, it should be noted that codes that generate K-RDS f constrained sequences improve the error correcting capabilities of recording systems [14]. ...
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... rejection of the DC-component and enhancement in distance property [75]- [80]. ...
... and maximize the code size |S| = i |S wi | by a proper choice of the parameters d c and d p . The above conditions are reminiscent of the conditions imposed on binary codewords that satisfy higher-order spectral constraint [7]. In the next section, we will show that we can find the K subsets that satisfy the constant balance and energy constraint using generating functions [9]. ...
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... component; i.e., in any long block, there will be about as many + 1's as − 1's. For other problems and solutions in the design of simple balanced signal sets, see [2,3,6,10,11]. ...
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... The original proof of the distance-enhancing properties of MSN codes was based upon a number-theoretic lower bound on the minimum Hamming distance of zero-disparity codes, due to Immink and Beenker [108]. They proved that the minimum Hamming distance (and, therefore, the minimum Euclidean distance) of a block code over the bipolar alphabet with order-spectral-null at DC grows at least linearly in Specifically, they showed that, for any pair of lengthsequences in the code and This result for block codes can be suitably generalized to any constrained system with order-spectral null at DC. ...
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... rejection of the DC-component and enhancement in distance property [68]- [73]. ...
... rejection of the DC-component and enhancement in distance property [75]- [80]. ...
... Higher order zero disparity DC free codes require that the code word moments denoted by meet the following conditions [4]: ...
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... The VT construction has been further implemented in single nonbinary insertion/deletion error correcting codes presented by Tenengolts [8] and ST codes presented by Abdel-Ghaffar [10]. In [9], the high-order spectrumnull code construction published in [13] was found to be a subset of a Levenstein code. Helberg and Ferreira [14] presented a class of codes which can correct multiple insertion/deletion errors based on a construction which is a generalization of the Varshamov-Tenengolts construction [10], [11]. ...
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... It follows directly from Descartes Rule of Signs that the minimum squared distance between codewords is at least , where is the minimum distance of the integer alphabet employed (for the bipolar alphabet , this gives a bound of . This simple observation is the starting point for the construction of many codes used in magnetic recording applications; more details can be found in Immink and Beenker [115], Karabed and Siegel [121], Eleftheriou and Cideciyan [63], and the survey paper [152]. The objective in all these papers is to separate signals at the output of a partial-response channel by generating codewords at the input with spectral nulls that are matched to those of the channel. ...
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Kth order zero disparity codes have been considered in several recent papers. In the first part of this paper we remove the zero disparity condition and consider the larger class of codes, Kth order disparity D codes. We establish properties of disparity D codes showing that they have many of the properties of zero disparity codes. We give existence criteria for them, and discuss how new codewords may be formed from ones already known. We then discuss Kth order disparity D codes that have the same number of codewords. We discuss the minimum distance properties of these new codes and present a decoding algorithm for them. In the second part of the paper we look at how the minimum distance of disparity D codes can be improved. We consider subsets of a very specialised subclass, namely first order zero disparity codes over alphabet Aq of size q. These particular subsets have q codewords of length n and minimum Hamming distance n. We show that such a subset exists when q is even and nis a multiple of 4, and also when q is odd and n is even. These subsets have the best error correction capabilities of any subset of q first order zero disparity codewords.
Article
A rarely used definition of the Gosset lattice is revived to construct codes which (unlike other Gosset lattice codes) have a null at DC in their baseband spectrum. A closed-form expression for the code spectrum is derived, and the bandpass spectrum of this code is compared to the spectrum of an E <sub>8</sub> construction of the same code. A fast sequential method of demodulation is described. This two-stage demodulator consists of a simplified algorithm that can declare erasures to provide speed in demodulation and a maximum-likelihood algorithm used to maintain error performance
Conference Paper
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We investigate the use of higher order code word moments to shape the power spectral density of codes. Several codes are presented that combine different spectral shaping techniques. The insertion/deletion and additive error correcting capabilities of these codes are investigated and simple decoding procedures are presented. Finally, we give an example for the use of a higher order spectral zero code
Chapter
Codes \({\mathcal{C}}(m,r)\) of length 2m over {1, -1} are defined as null spaces of certain submatrices of Hadamard matrices. It is shown that the codewords of \({\mathcal{C}}(m,r)\) all have an rth order spectral null at zero frequency. Establishing the connection between \({\mathcal{C}}(m,r)\) and the parity-check matrix of Reed-Muller codes, the minimum distance of \({\mathcal{C}}(m,r)\) is obtained along with upper bounds on the redundancy of \({\mathcal{C}}(m,r)\). An efficient algorithm is presented for encoding unconstrained binary sequences into \({\mathcal{C}}(m,r)\).
Article
Codes C(m, r) of length 2 m over 1, -1 are defined as null spaces of certain submatrices of Hadamard matrices. It is shown that the codewords of C(m, r) all have an rth order spectral null at zero frequency. Establishing the connection between C(m, r) and the parity-check matrix of Reed-Muller codes, the minimum distance of C(m, r) is obtained along with upper bounds on the redundancy of C(m, r). An efficient algorithm is presented for encoding unconstrained binary sequences into C(m, 2).
Book
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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
Book
Error-correcting codes constitute one of the key ingredients in achieving the high degree of reliability required in modern data transmission and storage systems. This book introduces the reader to the theoretical foundations of error-correcting codes, with an emphasis on Reed-Solomon codes and their derivative codes. After reviewing linear codes and finite fields, Ron Roth describes Reed-Solomon codes and various decoding algorithms. Cyclic codes are presented, as are MDS codes, graph codes, and codes in the Lee metric. Concatenated, trellis, and convolutional codes are also discussed in detail.
Book
Full-text available
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
Article
The authors generalise the construction of Doppler-tolerant Golay complementary waveforms by Pezeshki-Calderbank-Moran-Howard to complementary code sets having more than two codes, which they call Doppler-null codes. This is accomplished by exploiting number-theoretic results involving the sum-of-digits function and a generalisation to more than two symbols of the classical two-symbol Thue-Morse sequence. Two approaches are taken to establish higher-order nulls of the composite ambiguity function: one by rewriting it in terms of equal sums of powers (ESP) and the other by factoring it in product form to reveal a higher-order zero, analogous to spectral-null codes. They conclude by describing an application of minimal ESP sets to multiple-input-multiple-output radar.
Article
Let be the set of all words of length N over the m-polar alphabet Φ m ={−(m−1),−(m−3),…,+(m−3),+(m−1)}, having a q-th order spectral-null at zero frequency. Any subset of is a spectral-null code of length N and order q. This paper gives an equivalent formulation of in terms of codes over the m-ary alphabet ℤ m ={0, 1,…, m−1}, derives a recursive expression for the cardinality of , shows combinatorial properties of , gives new simple ways to obtain systematic m-ary q-th order spectral-null codes, and finally, presents new efficient recursive design methods to encode k m-ary information digits into second-order spectral-null codes over ℤ m of length N(k)=n(k)+N(⌈log m n(k)(n(k)−1)/2⌉+1), whose implementing algorithm requires T=O(mk log m k) m-ary digit operations and S=O(k) m-ary digit storing elements.
Article
Constrained codes are a kev component in the digital recording devices that have become ubiquitous in computer data storage and electronic entertainment applications. This paper surveys the theory and practice of constrained coding, tracing the evolution of the subject from its origins in Shannon's classic 1948 paper to present-day applications in high-density digital recorders. Open problems and future research directions are also addressed.
Article
Codes whose spectral density and its lower order derivatives vanish at zero frequency are investigated, and necessary and sufficient conditions for the spectral properties are given. In particular, for the most common balanced alphabets, a complete characterization is furnished for the class of symbol-by-symbol codes with vanishing spectrum and second derivative. The fundamental results are extended to block codes, and suitable criteria for their design are given. Specific examples of binary and ternary block codes are presented; their performance in terms of low power content at low frequencies compares advantageously with that of some conventional codes
Conference Paper
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In this paper we derive a formula for the number of code words of a DC2-balanced codes. This number is expressed as a coefficient of a generating function in two variables. In addition, we establish a lower as well as an upper bound for this number. In particular, we show that the information rate tends to unity when the code length tends to infinity.
Article
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Error correction coding gain can be obtained with dc free magnetic recording codes having a minimum Hamming distance d//m//i//n greater than equivalent to 4. This information theoretic paper establishes lower bounds on the minimum Hamming distance achievable with (2n,k) (ie R equals k/2n) block codes with runlength constrained or dc free sequences, and present a table with the computed guaranteed minimum values of k for (2n,k) dc free block codes with (b equals 0, L-SCRIPT equals 2n minus l,C equals n) sequences, a specified block length 2n and desired minimum Hamming distance d//m//i//n. Encoding and decoding schemes for the synthesized code are also proposed.
Article
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A method is presented for designing binary channel codes in such a way that both the power spectral density function and its second-derivative vanish at zero frequency. Recursion relations are derived to determine the number of codewords that can be used in this coding scheme. A simple algorithm for encoding and decoding codewords is developed. The performance of the new codes is compared with that of classical channel codes designed with a constraint on the unbalance of the number of transmitted positive and negative pulses.
Article
Full-text available
Error correction coding gain can be obtained with dc free magnetic recording codes having a minimum Hamming distance d_{min}geq 4 . In this information theoretic paper we establish lower bounds on the minimum Hamming distance achievable with (2n,k) (ie R = k/2n) block codes with runlength constrained or dc free sequences, and present a table with the computed guaranteed minimum values of k for (2n,k) dc free block codes with (b=0, l=2n-1, C=n) sequences, a specified block length 2n and desired minimum Hamming distance d_{min} . Subsequently, we present the formal synthesis of a (16,8) d_{min}=4 code with sequences which have, in NRZ notation, a minimum runlength of 1 and maximum runlength of 8, while the bound on the maximum instantaneous accumulated charge is C = 5. This accumulated charge returns to zero at the end of each 16 bit code word. Encoding and decoding schemes for the synthesized code are also proposed.
Article
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Article
The stochastic process appearing at the output of a digital encoder is investigated. Based upon the statistics of the code being employed, a systematic procedure is developed by means of which the average power spectral density of the process can be determined. The method is readily programmed on the digital computer, facilitating the calculation of the spectral densities for large numbers of codes. As an example of its use, the procedure is applied in the case of a specific multi-alphabet, multi-level code.
Article
The role of line coding is to convert source data to a digital form resistant to noise in combination with such other impairments as a specific medium may suffer (notably intersymbol interference, digit timing jitter and carrier phase error), while being reasonably economical in the use of bandwidth. This paper discusses the nature and role of various constraints on code words and word sequences, including those commonly used on metallic lines, optical fibres, carrier channels and radio links ; and gives some examples from each of these applications. It should serve both as a general review of the subject and as an introduction to the companion papers on specific topics.
Article
In digital transmission of binary (+1,-1) signals it is desirable that the stream of pulses which constitutes the signal have no dc, that is, that the power spectrum go to zero at zero frequency. It is desirable that, for a given efficiency or entropy, the spectrum rise slowly with increasing frequency. We have obtained the spectrum for selected blocks with equal numbers of plus ones and minus ones. For a given efficiency, this is better than the spectrum obtained by Rice, using the Monte Carlo method, for block encoding using polarity pulses. An algorithm given by Schalwijk should allow simple encoding into selected blocks.
Article
When alphabets of digital symbols are used to represent information for data processing, storage, and transmission, redundancy in the alphabets is traditionally used for the purpose of error compensation. This paper deals with alphabets of redundant codes, both binary and higher level, where the emphasis is on using redundancy to produce code alphabets with unique properties in their frequency spectra that can be exploited in the design of the system in which they are used. In particular, techniques are presented for synthesizing alphabets that produce spectral nulls at frequencies 1/kT, where T is the duration of a word element. Some of the interesting alphabets are a 10-word, 5-bit alphabet with spectrum zero at 1/2T; a 10-word, 6-bit alphabet with spectrum zero at 1/3T; a 36-word, 8-bit alphabet with zero at 1/4T; and a 36-word, 8-bit alphabet with zeros at both 0 and 1/2T.
Article
A practical method is described for encoding an unrestricted binary signal into a form suitable for transmission through a binary regenerated signal path while incurring only a small increase in modulation rate.
Article
In pulse-amplitude modulation (PAM) digital transmission systems line encoding is used for shaping the spectrum of the encoded symbol sequence to suit the frequency characteristics of the transmission channel. In particular, it is often required that the encoded symbol sequence have a zero mean and spectral density vanishing at zero frequency. We show that the finite running digital sum condition is a necessary and sufficient condition for this to occur. The result holds in particular for alphabetic codes, which are the most widely used line codes.
Article
Let {(X_{k}, Y_{k}) }^{ infty}_{k=1} be a sequence of independent drawings of a pair of dependent random variables X, Y . Let us say that X takes values in the finite set cal X . It is desired to encode the sequence {X_{k}} in blocks of length n into a binary stream of rate R , which can in turn be decoded as a sequence { hat{X}_{k} } , where hat{X}_{k} in hat{ cal X} , the reproduction alphabet. The average distortion level is (1/n) sum^{n}_{k=1} E[D(X_{k},hat{X}_{k})] , where D(x,hat{x}) geq 0, x in {cal X}, hat{x} in hat{ cal X} , is a preassigned distortion measure. The special assumption made here is that the decoder has access to the side information {Y_{k}} . In this paper we determine the quantity R ast (d) , defined as the infimum ofrates R such that (with varepsilon > 0 arbitrarily small and with suitably large n )communication is possible in the above setting at an average distortion level (as defined above) not exceeding d + varepsilon . The main result is that R ast (d) = inf [I(X;Z) - I(Y;Z)] , where the infimum is with respect to all auxiliary random variables Z (which take values in a finite set cal Z ) that satisfy: i) Y,Z conditionally independent given X ; ii) there exists a function f: {cal Y} times {cal Z} rightarrow hat{ cal X} , such that E[D(X,f(Y,Z))] leq d . Let R_{X | Y}(d) be the rate-distortion function which results when the encoder as well as the decoder has access to the side information { Y_{k} } . In nearly all cases it is shown that when d > 0 then R ast(d) > R_{X|Y} (d) , so that knowledge of the side information at the encoder permits transmission of the {X_{k}} at a given distortion level using a smaller transmission rate. This is in contrast to the situation treated by Slepian and Wolf [5] where, for arbitrarily accurate reproduction of {X_{k}} , i.e., d = varepsilon for any varepsilon >0 , knowledge of the side information at the- encoder does not allow a reduction of the transmission rate.
Principles of digital line codingBinary code suitable for line transmissionSpectra and efficiency of binary codes without dcRedundant alphabets with desirable frequency propertiesThe spectral density of a coded digital signal
  • K W Cattermole
  • J M Inf
  • J N Griffitha
  • J R Franklin
  • Pierce
PI PI [31 [41 [51 161 [71 181 [91 1101 IllI 1121 K. W. Cattermole, “Principles of digital line coding,” Inf. J. Electron., vol. 55, pp. 3-33, 1983. J. M. Griffitha, “Binary code suitable for line transmission,” Electron. Lat., vol. 5, pp. 79-81, 1969. J. N. Franklin and J. R. Pierce, “Spectra and efficiency of binary codes without dc,” IEEE Trans. Commun., vol. COM-21, pp. 1182-1184, 1972. E. Gorog, “Redundant alphabets with desirable frequency properties,” IBM J. Res. Develop., vol. 12, pp. 234-241, 1968. B. S. Bosik, “The spectral density of a coded digital signal,” Bell Syst
Spectra and efficiency of binary codes without dc Redundant alphabets with desirable frequency properties The spectral density of a coded digital signal Codes for zero spectral density at zero frequency
  • J M J N Griffitha
  • J R Franklin
  • E Pierce
  • B S Gorog
  • G L Bosik
  • Pierobon
J. M. Griffitha, " Binary code suitable for line transmission, " Electron. Lat., vol. 5, pp. 79-81, 1969. J. N. Franklin and J. R. Pierce, " Spectra and efficiency of binary codes without dc, " IEEE Trans. Commun., vol. COM-21, pp. 1182-1184, 1972. E. Gorog, " Redundant alphabets with desirable frequency properties, " IBM J. Res. Develop., vol. 12, pp. 234-241, 1968. B. S. Bosik, " The spectral density of a coded digital signal, " Bell Syst. Tech. J., vol. 51, pp. 921-932, 1972. G. L. Pierobon, " Codes for zero spectral density at zero frequency, " IEEE Trms. Inform. Theory, vol. IT-30, pp. 435-439, 1984. J. Riordan, An Introduction to Combinatorial Analysis. Princeton, NJ: Princeton Univ. Press, 1980.
The rate distortion function of a binary symmetric smrce Principles of digital line coding
  • Englewood Cliffs
  • Nj K J Kerpez
  • K W Cattermole
Englewood Cliffs, NJ: Prentice-Hall, 1971. K. J. Kerpez, " The rate distortion function of a binary symmetric smrce K. W. Cattermole, " Principles of digital line coding, " Inf. J. Electron., vol. 55, pp. 3-33, 1983.
Spectrum shaping with binary DC*-constrained channel codes A. Papoulis, The Fourier Integral and its Applications On the number of codewords of a DC'-balanced code
  • K A S Immink
K. A. S. Immink, " Spectrum shaping with binary DC*-constrained channel codes, " Philips J. Res., vol. 40, pp. 40-53, 1985. A. Papoulis, The Fourier Integral and its Applications. New York: McGraw-Hill, 1962. G. F. M. Beenker and K. A. S. Immink, " On the number of codewords of a DC'-balanced code, " in Proc. 6th Symp. Information Theory m the Benelux, 1985, pp. 133-139.
Redundant alphabets with desirable frequency properties The spectral density of a coded digital signal
  • E B S Gorog
  • Bosik
E. Gorog, " Redundant alphabets with desirable frequency properties, " IBM J. Res. Develop., vol. 12, pp. 234-241, 1968. B. S. Bosik, " The spectral density of a coded digital signal, " Bell Syst. Tech. J., vol. 51, pp. 921-932, 1972.
Binary code suitable for line transmission
  • J M Griffitha
J. M. Griffitha, "Binary code suitable for line transmission," Electron. Lat., vol. 5, pp. 79-81, 1969.