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Classical modulation theory deals mainly with single-parameter proportional modulation techniques, i.e., with processes such as AM and FM, wherein the amplitude or angle of a sinusoidal carrier wave is varied as a linear function of an information signal. Classical modulation theory has been immensely useful, but it is a restrictive theory because it can deal tractably with only a limited class of possible modulation processes. In this paper we seek to take the first few steps toward a broader and more unified theory of modulation--a theory which will encompass several recently proposed hybrid modulation schemes as well as conventional techniques, which will suggest new modulation processes, and which might lead toward unification of modulation theory and coding theory. The paper is divided into two parts. Part I is devoted to theoretical development as summarized below. The results of Part I are applied in Part II to study modulation processes per se. The paper as a whole is in the nature of a wide ranging survey rather than a rigorously detailed analysis. The most general form of modulated wave exhibits simultaneous phase and envelope fluctuations. In the first half of Part I, equations are developed to describe phase-envelope relationships when the wave is band-limited. These equations prove to be intimately dependent on the real and complex zeros of the wave. Real zeros ("zero crossings") are familiar physical attributes. Complex zeros, however, are physically rather covert and are a concept foreign to many engineers. Both kinds of zeros are easy to interpret mathematically in terms of the factorization of a Fourier series. In the second half of Part I, three simple examples are used to illustrate relationships between zeros, spectra, envelopes, and phase functions. These examples lead to operational procedures for representing signals in terms of zeros and, most importantly, to a viewpoint wherein zeros are regarded as fundamental informational attributes of signals. This viewpoint leads, in Part II, to the concept of modulation as zero manipulation.

Between 1948?when Shannon first proposed his basic theorems on information theory?and the start of the space age, little practical application developed from the lessons of coding theory. This article presents an overview of the Shannon theorem, interesting practical codes, and their application to the space channel. It turns out that a simple encoder in combination with a decoder of modest complexity placed into an uncoded communications system can increase the data rate by a factor of four or more depending on the coding scheme and the allowable error rate. Use of a convolutional code with sequential decoding has proved to be the outstanding scheme for these applications. It appears that, in the future, coding will find a place in most new digital space communication systems.

As in previous summaries, the words ‘information theory’ are used in the wide sense to include such areas as (1) statistical communication theory, (2) detection theory, (3) pattern recognition and adaptive systems, (4) coding theory, and similar examples of the interaction of random processes and communication systems. For classification purposes, progress in information theory will be described under the four headings just listed.

Described in this report are the results of a comprehensive technical survey of all published Soviet literature in coding theory and its applications--over 400 papers and books appearing before March 1967. The purpose of this report is to draw attention to this important collection of technical results, which are not well known in the West, and to summarize the significant contributions. Particular emphasis is placed upon those results that fill gaps in the body of knowledge about coding theory and practice as familiar to non-Soviet The most noteworthy Soviet contributions have occurred in those areas that deal with codes for the noiseless channel, codes that correct asymmetric errors, decoding for cyclic codes, randomcoding bounds on the amount of computation required, and various application criteria--that is, when to use which code, and how well it performs. Other important but isolated results have been reported on the construction of optimal low-rate codes, bounds on nonrandom codes, linear (continuous) coding, codes for checking arithmetic operations, properties of code polynomials, linear transformations of codes, multiple-burst-correcting codes, special synchronization codes, and certain broad generalizations of the conventional coding problem. Little or no significant work has been done on pseudorandom sequences, unit-distance codes (with one exception), the application of codes to the design of redundant computers and memories, the search for good cyclic codes, and the physical realization of sequential decoding algorithms. Section II of this report is directed to the nonspecialist, and describes the status of the field of coding theory in the Soviet Union, summarizes the major technical results, and compares these with corresponding work in the West. Section III discusses in detail for the coding specialist new theoretical results, details of coding procedures, and analytical tools described in the Soviet literature. A complete bibliography is included.

This algorithm implements the Hu-Tucker method of variable length, minimum redundancy alphabetic binary encoding [1]. The symbols of the alphabet are considered to be an ordered forest of n terminal nodes. Two nodes in an ordered forest are said to be tentative-connecting if the sequence of nodes between the two given nodes is either empty or consists entirely of nonterminal nodes.

Among the allegedly determinable traits of the modern mathematician are his passions for generality and precision. Such traits go unnoticed among the fraternity but they sometimes provide cultural shocks for those outside the fold. In this paper both traits are explicitly exhibited.

The work concerns: (1) synthesis of coding and other digital apparatus, (2) further results on the symbol string algebra reported on previously (AD-273 185), and (3) topics concerning digital codes. (Author)

The present paper is partly expository and does not assume any previous knowledge of information theory or coding theory. It is meant to be the first in a series of papers on coding theory for noisy channels, this series replacing the report [12] which was widely circulated. A few of the results were reported in [11], [27] and [28]. In Sections 2 and 3 we present certain refinements and generalizations of known methods of Shannon, Fano, Feinstein and Gallager. A discussion of some other methods may be found in the surveys by Kotz [15] and Wolfowitz [28] such as those of Khintchine [14], McMillan [18] and Wolfowitz [25], [26]. Section 4 contains a number of applications. Most of this section is devoted to a certain memoryless channel with additive noise. Some of the proofs have been collected in Section 5. Finally, Section 6 describes some new results on the relative entropy $H(\mu_1\mid\mu_2),$ of one measure $\mu_1$ relative to another measure $\mu_2$, and its relation with the total variation $\|\mu_1 - \mu_2\|$.

The report reviews progress to date in what is often referred to as probabilistic coding theory. An extensive bibliography is provided.

The earliest studies of reliability assurance by the introduction of redundancy were basically concerned with the notion of simple composition (replication). Composition methods are time-honored and merely entail the complementation of a particular component or module of a system with one or more redundant replicas of the component or module, which either take over the function of the primary object in the event of failure (passive, or “cold,” composition) or operate simultaneously with the primary object (active, or “hot” composition). Essentially, however, the modern theory of reliability assurance through the introduction of redundancy was developed and its fundamental disciplines defined at a later date. This theory had its genesis in von Neumann’s six lectures on probabilistic logic at the California Institute of Technology in 1952; the lectures were subsequently published in the now-classical treatise [269]. Somewhat later Moore and Shannon published their paper [263] on the synthesis of reliability circuits from unreliable relays. With the advent of information theory, particularly the branch thereof known as coding theory, the growing tendency was to regard discrete systems as special kinds of channels, namely computation channels, and to treat them with the reliability improvement techniques used in communication channels. The legitimacy of this approach was investigated by Elias [196], and the practical application of coding to discrete systems of a particular type was explored by Gavrilov [9] and Zakrevskii [35].

The purpose of this paper is to describe a redundancy technique which would require mere duplication to achieve the same failure-masking capabilities as von Neumann's triplication and majority-voting technique. An analysis of the circuit-failure problem is approached from the viewpoint of coding theory with comparisons made between the noisy channel and circuit-failure problems. Some of the difficulties of extrapolating from the former to the latter are discussed, as well as recent attempts to minimize the redundancy 'overhead' by coding over larger numbers of bits. Following a description of the binary erasure channel model, a proposal of a failure-erasure technique based upon it is outlined. The method enables failure-masking at duplicative rather than triplicative costs. There are constraints which this scheme imposes upon the circuit elements, however, and the characteristics of the ideal circuit element and logic signaling are proposed. The paper concludes with a discussion of existing hardware which approximates the desired characteristics.