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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

Content uploaded by Kees Schouhamer Immink

Author content

All content in this area was uploaded by Kees Schouhamer Immink on Apr 12, 2019

Content may be subject to copyright.

... 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. ...

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. ...

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]. ...

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]. ...

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. ...

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.

We investigate a new approach for designing spectral shaping block codes with a target spectrum, H_t(f), that has been specified at a plurality of frequencies. We analyze the probability density function of the spectral power density function of uncoded n-symbol bipolar code words. We present estimates of the redundancy and the spectrum of spectral shaping codes with specified target spectral densities H_t(f_i) at frequencies f_i. Constructions of low-redundancy codes with suppressed low-frequency content are presented that compare favorably with conventional dc-balanced codes currently used in data transmission and data storage devices with applications in consumer electronics.

An efficient concatenation of error correction codes with constrained codes is proposed in this paper. Generally, constrained codes are designed to match specified channels, whereas error correction coding schemes are designed to correct the channel errors. They both play important roles to ensure the integrity of data in data storage systems. In this study, we first investigate the design of k constrained codes combined with a low-density parity-check (LDPC) code, and then we extend the idea to the design of DC-free k constrained LDPC codes. Simulation results show that the proposed designs achieve an improved bit error rate (BER) performance, compared to prior art schemes. Especially, the proposed design for the DC-free k constrained codes not only fully eliminates the effect of error propagation in a reverse configuration, but also achieves significant DC suppression.

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.

Hybrid overlay/underlay mechanism is a smart way for secondary users to access spectrum in cognitive radio (CR). In this way, secondary users coexist with primary users in the same frequency band. For all we know, in the RF end, multicarrier modulation has been the major technique to realize this kind of spectrum access mechanism. From the perspective of system baseband, we proposed a promising system to help secondary users accessing spectrum in hybrid overlay/underlay mechanism, called composite sequence cognitive radio (CS-CR). In this system, we design a composite sequence, which is compounded by spectral null codes and m sequence. With our design composite sequence, secondary users can transmit low power over a wide frequency band and have spectral nulls in the band where primary users have been used. Bit error performance of primary and secondary users in AWGN channel is simulated in our system. The simulation results indicate that secondary users can guarantee the quality of service (QoS) of primary users, and also suppress the interference from primary users at the same time.

In digital transmission it is sometimes desirable for the channel stream to have low power near zero frequency. Suppression of the low-frequency components is achieved by constraining the unbalance of the transmitted positive and negative pulses. Rate and spectral properties of unbalance constrained codes with binary symbols based on simple bi-mode coding schemes are calculated.

Key factors for a compact optical recording system for filing applications (as demonstrated in 1978 at Philips Research Laboratories) are semiconductor lasers and pregrooved disks with sensitive stable materials. Three aspects of recent optical systems will be treated in more detail.

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.

The engineering design of a head-positioning system for an interchangeable-medium disk file is considered. Emphasis is placed upon three specific functions within the positioning system: (1) encoding and demodulation of information from the dedicated servo surface, (2) compensation and dynamics of the track-following control system, and (3) implementation of control electronics for a quasi-time-optimal, track-accessing control system. The examples used are taken from the IBM 3340 Disk Storage Facility.

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.

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.

This paper provides a tutorial introduction to recording codes for magnetic disk storage devices and a review of progress in code construction algorithms. Topics covered include: a brief description of typical magnetic recording channels; motivation for use of recording codes; methods of selecting codes to maximize data density and reliability; and techniques for code design and implementation.

A continuous servo system that shares the surface area of the medium with data on the same track, to improve the accuracy of servo-track following, uses only conventional media and a single head. A long-gap head records low-frequency servo signals deeply into the medium coating, and a short-gap head then erases the surface portion to stabilize the servo amplitude. Short-gap read-write heads then write and read high-frequency data signals without interference. Using frequency-separation filters and an ac-biased pulse-write method, an inductive head reads servos while simultaneously writing data. Servo-signal amplitude increases during data-write operations. Servo signals, which do not degrade after stabilization, provide information needed for track following and for various timing and control functions. Data reliability is unimpaired.

Current head-positioning servo methods require a spatial separation between the data and servo. This separation limits the accuracy with which a head-positioning servo system can follow a data track. The buried servo techniques described here allow servo information to be recorded in the same medium area as that used for data, a single head is used for both data and servo. These techniques provide high-bandwidth continuous head-position information with a potential for a high track-density servo system. The servo pattern consists of two single-frequency signals recorded deeply into the magnetic medium below the area used for data. The servo signals are detected by synchronous detection techniques and provide a continuous head-position error signal. The servo system uses this head-position error signal to follow a given data track and to seek new data tracks. Buried servo was implemented on a flexible disk drive. The track-following system had performance advantages when compared to a sector servo scheme implemented on a similar device.