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Abstract

The servo position information of magnetic tape or disk recorders is often recorded as low-frequency components usually called pilot tracking tones. Binary codes giving rise to a spectral null at an arbitrary frequency are used to provide space for the allocation of auxiliary pilot tones. Here, encoding methods are treated in which binary data are mapped into constrained binary sequences for shaping the spectrum. The rate and power spectral density function of memoryless codes that exhibit spectral nulls are computed. The relationship between the code redundancy and spectral notch width is quantified with a parameter called the sum variance. It is found that twice the product of the spectral notch width times the sum variance is approximately unity
... The cardinality of the (3, 3)-Gray code is equal to 27. We determine the z -subset of (r , q) Gray code with kq = 18 elements, leading to z ∈ [4,21] and z 1 = 4. Table 3.11 shows the steps in obtaining the codeword, c. The (3, 3)-Gray code sequence for each z can be veried against Table A.5 in Appendix A. ...
... • Determine z 1 = 4. Therefore, the z -subset of (3, 3)-Gray code contains sequences with z ∈ [4,21]. ...
... (4) shows the set of all balanced sequences of length 4, listed in the lexicographic order, |N (, N (6) presents the set of all balanced sequences of length 6. |N (6)| = 6 3 = 20. ...
Thesis
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Our research deals with the encoding and decoding of balanced sequences using Gray codes. Given that any non-binary sequence, can always be balanced through certain algorithms, we show that the encoding and decoding of a balanced sequence can be performed through a simple and efficient method where the prefix is a Gray code. Our balancing scheme makes use of a generalization of Knuth's balancing algorithm, per- formed on the overall sequence length which includes the information sequence as well as the designed prefix. Our proposed method was firstly applied to certain information source lengths and then generalized for any length. We conclude with a detailed complexity and redundancy analysis for our balancing algorithm.
... For codewords of length M consisting of N interleaved subwords of length z, the cardinality of the codebook C b (M, N ) for the case where N is a prime number is presented by the following formula [20], ...
... The equation (20) is presented by the graph in Fig. 9, where the index-permutation symbols are presented Each grouping in (20) is represented by a subgraph as depicted in Fig. 9. The Hamming distance matrix for all possible index-permutation sequences is presented in (21), where "0" represents the Hamming distance between same sequences or sequences with non connected symbols. ...
Article
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A New graph distance concept introduced for certain coding techniques helped in their design and analysis as in the case of distance-preserving mappings and spectral shaping codes. A graph theoretic construction, mapping binary sequences to permutation sequences and inspired from the k -cube graph has reached the upper bound on the sum of the distances for certain values of the length of the permutation sequence. The new introduced distance concept in the k -cube graph helped better understanding and analyzing for the first time the concept of distance-reducing mappings. A combination of distance and the index-permutation graph concepts helped uncover and verify certain properties of spectral null codes, which were previously difficult to analyze.
... (11) We consider here the spectral properties of full-set block codes, that is, S 2 denotes the set of all possible words, x, that satisfy condition (5). Finding an expression of the spectral properties of a full-set S 2 for large values of n is an open problem as the computation requires the evaluation of (1) for each x ∈ S 2 [12]. ...
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.
... (11) We consider here the spectral properties of full-set block codes, that is, S 2 denotes the set of all possible words, x, that satisfy condition (5). Finding an expression of the spectral properties of a full-set S 2 for large values of n is an open problem as the computation requires the evaluation of (1) for each x ∈ S 2 [12]. ...
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.
... Balanced codes have a lot of applications especially in fiber optics, as well as magnetic and optical storage systems [1]. For optical systems, errors occur in the process of recording data, this is due to the low frequencies of interaction between data written on the disc and the servo systems which follow the track. ...
Conference Paper
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Balancing sequences over a non-binary alphabet is considered, where the algebraic sum of the components (also known as the weight) is equal to some specific value. Various schemes based on Knuth’s simple binary balancing algorithm have been proposed. However, these have mostly assumed that the prefix describing the balancing point in the algorithm can easily be encoded. In this paper we show how non-binary Gray codes can be used to generate these prefixes. Together with a non-binary balancing algorithm, this forms a complete balancing system with straightforward and efficient encoding/decoding.
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