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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 )

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... The code-efficiency can be expressed in terms of the code-rate R. The asymptotic coding gain GD is now defined as GD = Rdb(coded) db(uncoded) . This equality indicates the saving in required energy-per-bit to noise-power-density ratio of the coded scheme relative to the uncoded case [44]. ...

Baseband modulation codes are widely applied in such diverse fields as digital line transmission [21, 28, 18, 116], digital optical transmission [111, 16], and digital magnetic and optical storage [ 105, 100, 106]. They act to translate the source data sequence d
n
into a sequence a
k
that is transmitted across the channel (Fig. 4.1; see also Chapter 3). The principal goal is to enable the receiver to produce reliable decisions \({\hat d_n}\) about d
n
. Code design should, therefore, account for the characteristics of both channel and receiver. (Because modulation coding is the only type of coding that we consider in this chapter, we shall usually say ‘coding’ where we mean ‘modulation coding’.)

Quad-phase has desirable features for use in digital magnetic recording. The authors propose two novel equalisation schemes that seem attractive at high information densities. One scheme estimates the even and odd data bits in reverse order, while the other uses two equalisers to estimate even and odd bits separately.

For a digital recording system with a binary modulation encoder
and a linearly dispersive channel with additive noise, the author
determines optimum mean-square performance of the linear,
partial-response, and decision-feedback equalizers. The analysis
revolves around the power spectral density A(Ω) of the code and
the folded signal-to-noise ratio X (Ω) of the channel. The
latter function is analyzed for stylized optical and magnetic recording
channels. For all equalizers it is shown that the effect of coding is
similar to increasing X (Ω) by an additive portion
1/A(Ω). This favors depressions of A(Ω) at frequencies where
X (Ω) is poor. However, for A(Ω) to have any
depressions, the signaling rate must be increased with respect to
uncoded storage, and this inevitably degrades X (Ω). At
high information densities the degradation is often too large for coding
to be rewarding. Examples serve to illustrate these results

In digital storage systems, receivers for rate 1/2 modulation
codes are usually oversampled by a factor of two with respect to the
data stream that they attempt to reconstruct. It is shown that
oversampling may be avoided by using partial-response techniques to
detect, instead of the encoded binary signal, a decimated ternary one,
from which the original data can be recovered by means of a simple
decoder. A method is described to find all such decoders for a given
rate 1/2 code. Examples treated are FM, MFM, Miller-squared, (2,7), 3PM,
and quad-phase. The mean-square performances of these reception schemes
are analyzed and compared to those of their predecessors for a
Lorentzian channel. The relative merits are found to depend quite
heavily on the information density, favoring some of the studied schemes
at high densities

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

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

For a data transmission system with intersymbol interference and
noise in which signaling occurs by means of nonoverlapping rectangular
pulses, a relation between the equivalent discrete-time models for
uncoded and coded transmission is derived. It applies to binary
modulation codes with rate R =1/ N , where N is
a positive integer. Examples suggest that these models are often
affected by coding in a manner that is incompatible with a commonly
adopted definition of coding gain

Multi level sequences with a spectral null of order M at frequency
f , meaning that the power spectral density, and its first 2M-1
derivatives vanish at f , are characterized by finite-state
transition diagrams (FSTDs) whose edge labels satisfy bounds on the
variation of the Mth-order running digital sum (RDS). Necessary and
sufficient conditions for FSTDs with higher order null constraints at DC
and at an arbitrary submultiple of the symbol frequency are derived.
Analytical results are given concerning the performance of codes
satisfying an M th-order RDS constraint on partial-response
channels. Specific code designs for quaternary channel inputs are
presented. The Euclidean distance properties of this new class of codes,
aside from their spectral-shaping properties, are demonstrated

The sound from a Compact Disc system encoded into data bits and modulated into channel bits is sent along the 'transmission channel' consisting of write laser - master disk - user disk - optical pick-up. The maximum information density on the disk is determined by the diameter d of the laser light spot on the disk and the 'number of data bits per light spot'. The effect of making d smaller is to greatly reduce the manufacturing tolerances for the player and the disk. The compromise adopted is d approximately equals 1 mu m, giving very small tolerances for objective disk tilt, disk thickness and defocusing.

An analysis of constant envelope digital partial response continuous Phase modulation (CPM) systems is reported. Coherent detection is assumed and the channel is Gaussian. The receiver observes the received signal over more than one symbol interval to make use of the correlative properties of the transmitted signal. The Systems are M -ary, and baseband pulse shaping over several symbol intervals is considered. An optimum receiver based on the Viterbi algorithm is presented. Constant envelope digital modulation schemes with excellent spectral tail properties are given. The spectra have extremely low sidelobes. It is concluded that partial response CPM systems have spectrum compaction properties. Furthermore, at equal or even smaller bandwidth than minimum shift keying (MSK), a considerable gain in transmitter power can be obtained. This gain increases with M . Receiver and transmitter configurations are presented.

A digital magnetic recording system is viewed in this paper as a linear system that inherently includes a correlative level encoder. This encoder can be regarded as a linear finite-state machine like a convolutional encoder. The maximum likelihood decoding method recently devised by Viterbi to decode convolutional codes is then applied to digital magnetic recording systems. The decoding algorithm and its implementation are discussed in detail. Expressions for the decoding error probability are obtained and confirmed by computer simulations. It is shown that a significant improvement in the performance with respect to other methods is achievable by the maximum likelihood decoding method. For example, under the Gaussian noise assumption the proposed technique can reduce raw error rates in the 10<sup>−3</sup> to 10<sup>−4</sup> range by a factor of 50 to 300. These results indicate that the maximum likelihood decoding method gains as much as 2.5 dB in signal-to-noise ratio over the conventional bit-by-bit detection method.

Calderbank, Heegard, and Ozarow [1] have suggested a method of designing codes for channels with intersymbol interference, such as the magnetic recording channel. These codes are designed to exploit intersymbol interference. The standard method is to minimize intersymbol interference by constraining the input to the channel using run-length limited sequences. Calderbank, Heegard, and Ozarow considered an idealized model of an intersymbol interference channel that leads to the problem of designing codes for a partial response channel with transfer function (1 - D^{N}) /2 , where the channel inputs are constrained to be pm 1 . This problem is considered here. Channel inputs are generated using a nontrivial coset of a binary convolutional code. The coset is chosen to limit the zero-run length of the output of the channel and so maintain clock synchronization. The minimum squared Euclidean distance between outputs corresponding to distinct inputs is bounded below by the free distance of a second convolutional code which we call the magnitude code. An interesting feature of the analysis is that magnitude codes that are catastrophic may perform better than those that are noncatastrophic.

Modems for digital communication often adopt the so-called correlative level coding or the partial-response signaling, which attains a desired spectral shaping by introducing controlled intersymbol interference terms. In this paper, a correlative level encoder is treated as a linear finite-state machine and an application of the maximum-likelihood decoding (MLD) algorithm, which was originally proposed by Viterbi in decoding convolutional codes, is discussed. Asymptotic expressions for the probability of decoding error are obtained for a class of correlative level coding systems, and the results are confirmed by computer simulations. It is shown that a substantial performance gain is attainable by this probabilistic decoding method.

This book is written for the design engineer who must build the coding and decoding equipment and for the communication system engineer who must incorporate this equipment into a system. It is also suitable as a senior-level or first-year graduate text for an introductory one-semester course in coding theory. Fundamental concepts of coding are discussed along with group codes, taking into account basic principles, practical constraints, performance computations, coding bounds, generalized parity check codes, polynomial codes, and important classes of group codes. Other topics explored are related to simple nonalgebraic decoding techniques for group codes, soft decision decoding of block codes, algebraic techniques for multiple error correction, the convolutional code structure and Viterbi decoding, syndrome decoding techniques, and sequential decoding techniques. System applications are also considered, giving attention to concatenated codes, coding for the white Gaussian noise channel, interleaver structures for coded systems, and coding for burst noise channels.

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.

The application of Viterbi detection to Class IV partial response on a magnetic recording channel is discussed. The theoretical performance is analyzed in the presence of noise correlation and quantizing errors. A simplified algorithm is described which was implemented on a transverse-scan magnetic tape recorder operating at a linear density of 1800 bits/mm and a transfer rate of 120 Mbits/s. Following an initial 5 bit analog-to-digital conversion of the ternary waveform, the Viterbi detector is implemented entirely in high-speed digital emitter-coupled logic. An advantage of approximately 1.5 dB in effective signal-to-noise ratio is gained over a simple level detector, a result which agrees well with the theoretical development.

The continuous phase modulation (CPM) signaling scheme has gained interest in recent years because of its attractive spectral properties. Data symbol pulse shaping has previously been studied with regard to spectra, for binary data and modulation index 0.5. In this paper these results have been extended to the M -ary case, where the pulse shaping is over a one symbol interval, the so-called full response systems. Results are given for modulation indexes of practical interest, concerning both performance and spectrum. Comparisons are made with minimum shift keying (MSK) and systems have been found which are significantly better in E_{b}/N_{0} for a large signal-to-noise ratio (SNR) without expanded bandwidth. Schemes with the same bit error probability as MSK but with considerably smaller bandwidth have also been found. Significant improvement in both power and bandwidth are obtained by increasing the number of levels M from 2 to 4.

A new method for the transmission of intelligence by means of a signal having certain correlation properties has been evolved. The theoretical and practical aspects of this concept are presented. An important advantage of these techniques is that for a fixed performance criteria, considerably higher speeds are possible compared to the presently known methods. In addition, the implementation is simple and straightforward. An unusual property of these techniques is the capability of error detection without the introduction of redundancy into the original data. Finally, expressions for spectral distributions and error performance as well as methods for practical implementation, including the errordetection process, are presented.

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.

This paper describes a new digital recording code format, Quadra-Phase (QP), designed for high density applications. QP is well suited for use on bandpass channels, such as a tape recorder channel, because its frequency response spectra matches the recording channel. This code is described and compared with other common PCM codes. The encoding and decoding processes are presented, and eye-patterns and spectra of QP and other codes are shown. QP achieves the same packing performance as NRZ, without the need for very low frequency channel response.

A maximum-likelihood sequence estimator for a digital pulse-amplitude-modulated sequence in the presence of finite intersymbol interference and white Gaussian noise is developed, The structure comprises a sampled linear filter, called a whitened matched filter, and a recursive nonlinear processor, called the Viterbi algorithm. The outputs of the whitened matched filter, sampled once for each input symbol, are shown to form a set of sufficient statistics for estimation of the input sequence, a fact that makes obvious some earlier results on optimum linear processors. The Viterbi algorithm is easier to implement than earlier optimum nonlinear processors and its performance can be straightforwardly and accurately estimated. It is shown that performance (by whatever criterion) is effectively as good as could be attained by any receiver structure and in many cases is as good as if intersymbol interference were absent. Finally, a simplified but effectively optimum algorithm suitable for the most popular partial-response schemes is described.

This review of magnetic recording systems concentrates on developments in two particular areas: the video-tape recorder and the rigid disk drive. The consumer VCR represents the state of the art for analog in-contact recording, while the rigid disk drive exemplifies digital recording with rapid access. Various aspects of the mechanical configurations, track-following techniques, heads and media, and signal processing are discussed. More briefly the paper describes other digital systems on flexible media.

Continuous phase modulation-Part 11: Partial response signaling QP, an improved code for high density digital recording

- T Aulin
- N Rydbeck
- C E Sundberg

T. Aulin, N. Rydbeck, and C. E. Sundberg, " Continuous phase modulation-Part 11: Partial response signaling, " ZEEE Truns. Corn-mun., vol. COM-29, pp. 210-225, Mar. 1981. J. A. Bixby and R. A. Ketcham, " QP, an improved code for high density digital recording, " IEEE Trans. Magn., vol. MAG-15, pp. pp. 1557-1569, NOV. 1986. 1465-1467, NOV. 1979.