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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]. ...
Article
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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]. ...
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 (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 ...
Article
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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. ...
Article
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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We present a comprehensive survey of error-correcting codes for channels corrupted by synchronization errors. We discuss potential applications as well as the obstacles that need to be overcome before such codes can be used in practical systems.
... 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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We will present coding techniques for transmission and storage channels with unknown gain and/or offset. It will be shown that a codebook of length-n q-ary codewords, S, where all codewords in S have equal balance and energy show an intrinsic resistance against unknown gain and/or offset. Generating functions for evaluating the size of S will be presented. We will present an approximate expression for the code redundancy for asymptotically large values of n.
... 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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Optical Orthogonal Codes (OOC) are special classes of codes for applications in code division multi-access (CDMA) transmission using an optical carrier. We show a new relationship between Constant Weight Cyclically Permutable Codes (CPC) and OOC's which can be investigated further. The main result ties together the CPC's and the solution of an algebraic system of congruences. For a particular case, there is a recurrence relationship for the code cardinality. More general cases need to be elaborated in coming research. We discuss also so-called Super OOC's, which are codes with stronger correlation properties, and show several examples of such codes derivable from CPC's designed by applying the method proposed.
... 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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The paper gives a survey of spectrum shaping codes used for digital recording systems. This class of codes belongs to the broader class of modulation codes, which are widely used in recording systems for adjusting the source characteristics to the characteristics of the recording channel. The Shannon noiseless capacities of recording channels are considered, as well as the spectra of maxentropic sequences of M-ary recording constraints. In addition, some practical encoding and decoding schemes are discussed.
... rejection of the DC-component and enhancement in distance property [75]- [80]. ...
Conference Paper
Full-text available
... 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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Full-text available
We will present coding techniques for transmission and storage channels with unknown gain and/or offset. It will be shown that a codebook of length-n q-ary codewords, S, where all codewords in S have equal balance and energy show an intrinsic resistance against unknown gain and/or offset. Generating functions for evaluating the size of S will be presented. We will present an approximate expression for the code redundancy for asymptotically large values of n.
... 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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Let n be an arbitrary integer, let p be a prime factor of n. Denote by ! 1 the p t h primitive unity root, omega(1) : = e 2 pi i/p Define omega(i) : = omega 1(i) for 0 <= i <= p - 1 and B : = {1; omega 1, ... , omega(p-1)}(n) subset of C(n). Denote by K (n; p) the minimum k for which there exist vectors upsilon(1,) ... , upsilon(k) is an element of B such that for any vector omega is an element of B, there is an i, 1 <= i <= k, such that v(i) . omega = 0, where upsilon center dot omega is the usual scalar product of upsilon and omega. Grobner basis methods and linear algebra proof gives the lower bound K ( n; p) = n (p-1). Galvin posed the following problem: Let m = m ( n) denote the minimal integer such that there exists subsets Lambda(1,) ..., Lambda(m) of {1, ... , 4n} with vertical bar Lambda i vertical bar = 2n for each 1 <= i <= n, such that for any subset B subset of [4n] with 2 n elements there is at least one i, 1 <= i <= m, with A(i) boolean AND B having n elements. We obtain here the result m (p) >= p in the case of p > 3 primes.
... 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. ...
Article
Full-text available
Constrained codes are a key component in 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
... 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]. ...
Preprint
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In this paper, two moment balancing schemes, namely a variable index scheme and a fixed index scheme, for either single insertion/deletion error correction or multiple substitution error correction are introduced for coded sequences originally developed for correcting substitution errors only. By judiciously flipping bits of the original substitution error correcting code word, the resulting word is able to correct either a reduced number of substitution errors or a single insertion/deletion error. The number of flips introduced by the two schemes can be kept small compared to the code length. It shows a practical value of applying the schemes to a long substitution error correcting code for a severe channel where substitution errors dominate but insertion/deletion errors can occur with a low probability. The new schemes can be more easily implemented in an existing coding system than any previously published moment balancing templates since no additional parity bits are required which also means the code rate remains same and the existing substitution error correcting decoder requires no changes. Moreover, the work extends the class of Levenshtein codes capable of correcting either single substitution or single insertion/deletion errors to codes capable of correcting either multiple substitution errors or single insertion/deletion error.
... 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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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.
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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.