James L. Massey

Optimum Frame Synchronization

This paper considers the optimum method for locating a sync word periodically imbedded in binary data and received over the additive white Gaussian noise channel. It is shown that the optimum rule is to select the location that maximizes the sum of the correlation and a correction term. Simulations are reported that show approximately a 3-dB improvement at interesting signal-to-noise ratios compared to a pure correlation rule. Extensions are given to the phase-shift keyed (PSK) sync case where the detector output has a binary ambiguity and to the case of Gaussian data.

Codes, automata, and continuous systems: Explicit interconnections

Close relationships are established between convolutional codes and zero-state automata and between cyclic codes and zero-input automata. Furthermore, techniques of automata theory and continuous system theory are used to elaborate on the coding problem; and approaches from coding and automata are used to establish and interpret typical structural conditions in continuous systems. The investigation incorporates basic coding concepts into the currently emerging common basis for automata and continuous systems, and it gives explicit examples of the resulting benefits accruing to each of these areas from the others.

Error-correcting codes in computer arithmetic.

This chapter is intended to summarize the most important results which have been obtained in the theory of coding for the correction and detection of errors in computer arithmetic. The rapid growth in the size and speed of digital computers has placed stringent reliability demands on the arithmetic unit. Attempts to satisfy these demands have generally followed one of three directions: (1) Attempts to improve the reliability of the components used in the construction of the arithmetic unit, (2) attempts to improve reliability by incorporating hardware redundancy so that the result of a computation is unaffected by the failure of one or more of the replicated units which form the arithmetic unit, or so that the failure of one or more of the replicated units can be detected and the faulty units replaced, and (3) attempts to incorporate redundancy into the numbers themselves which are being processed so that erroneous results can be corrected or detected. This third approach, which is the subject of this chapter, tacitly assumes that it is possible to build the “decoder” which corrects or detects erroneous results much more reliably than the arithmetic unit which it monitors, so that the decoder can be considered error-free for practical purposes.

Error Bounds for Tree Codes, Trellis Codes, and Convolutional Codes with Encoding and Decoding Procedures

One of the anomalies of coding theory has been that while block parity-check codes form the subject for the overwhelming majority of theoretical studies, convolutional codes have been used in the majority of practical applications of “error-correcting codes.” There are many reasons for this, not the least being the elegant algebraic characterizations that have been formulated for block codes. But while we may for aesthetic reasons prefer to speculate about linear block codes rather than convolutional codes, it seems to me that an information-theorist can no longer be inculpably ignorant of non-block codes. It is the purpose of these lectures to provide a reasonably complete and self-contained treatment of non-block codes for a reader having some general familiarity with block codes.

Note on Finite-Memory Sequential Machines

One of the design problems of digital computers is in code conversion from analog form to its equivalent digital form, and vice versa. In data logging, reduction, and process control applications of digital computers, input data are frequently in analog or shaft position form. Conventionally weighted binary numbers are not used directly for analog readings because at certain positions these may be ambiguously expressed. Gray suggested a code in this application which eliminates the ambiguity of natural binary numbers. The basic rule in Gray code generation is to allow only one bit change between adjacent numbers.1 The Gray code, unfortunately, is difficult to use in computation. Hence, it is customary to perform code conversion before attempting computer arithmetic operations.

Advances in Threshold Decoding

Publisher Summary This chapter presents the recent advances in threshold decoding. In a linear code, whether of the block type or of the convolutional type, the redundant bits in the encoded set are each formed as a modulo-2 summation of selected information bits. Threshold decoding is a simple solution to the decoding problem based on a special subset of parity checks. Threshold decoding has two common forms: (1) majority decoding and (2) a posteriori probability (APP) decoding. The importance of APP decoding is that it permits use of the statistical information regarding the received bits available at the receiver in contrast with purely algebraic decoding techniques. In many systems such as those signaling over the Gaussian white noise channel, this results in an unacceptable degradation of performance. It seems highly significant that most of the new codes found for implementation by threshold decoders have their origins in number theory rather than in algebra. Further use of number theory in this way promises to provide additional classes of codes suitable for threshold decoding.

Comments on "Inversion of multivariable linear systems"

The purpose is to give a simple algebraic proof to the necessary part of the criterion of functional reproducibility of multivariable linear systems given by Silverman and to incorporate a slight correction to his criterion. Furthermore, the criteria of the functional reproducibility of Silverman and Sain and Massey [1] are shown to be equivalent.

Inverse problems in coding, automata, and continuous systems

The investigation presents explicit interconnections between inverse problems in the theories of convolutional codes, automata, and continuous time linear dynamical systems. In a code-generating, or transfer function, matrix framework, necessary and sufficient conditions are given for a feedforward linear sequential circuit to have a feedforward inverse, either instantaneous or with delay. In the corresponding state-oriented realization, techniques which have been applied to construct inverses for continuous time linear dynamical systems are used to outline the construction of inverse automata. Finally, the feedforward inverse results are applied to continuous time systems. Relationships between the results are discussed.

A modified inverse for linear dynamical systems

In earlier work, the authors have treated the questions of existence and construction of linear time invariant dynamical inverses for linear time invariant dynamical systems. For systems having inverses, these procedures provide an inverse system whose output is a uniformly delayed or uniformly integrated version of the original system input. Recognizing that some applications may require each component of the original system input to be recovered with individually minimum delay or integration, the authors present here alternative procedures to be used in such cases.