A gas of one-dimensional Bose particles interacting via a repulsive delta-function potential has been solved exactly. All the eigenfunctions can be found explicitly and the energies are given by the solutions of a transcendental equation. The problem has one nontrivial coupling constant, γ. When γ is small, Bogoliubov’s perturbation theory is seen to be valid. In this paper, we explicitly calculate the ground-state energy as a function of γ and show that it is analytic for all γ, except γ=0. In Part II, we discuss the excitation spectrum and show that it is most convenient to regard it as a double spectrum—not one as is ordinarily supposed.

Exact analysis of an interacting bose gas. i. the general solution and the ground state

Abyssal recipes II: energetics of tidal and wind mixing

Hitherto communication theory was based on two alternative methods of signal analysis. One is the description of the signal as a function of time; the other is Fourier analysis. Both are idealizations, as the first method operates with sharply defined instants of time, the second with infinite wave-trains of rigorously defined frequencies. But our everyday experiences?especially our auditory sensations?insist on a description in terms of both time and frequency. In the present paper this point of view is developed in quantitative language. Signals are represented in two dimensions, with time and frequency as co-ordinates. Such two-dimensional representations can be called ?information diagrams,? as areas in them are proportional to the number of independent data which they can convey. This is a consequence of the fact that the frequency of a signal which is not of infinite duration can be defined only with a certain inaccuracy, which is inversely proportional to the duration, and vice versa. This ?uncertainty relation? suggests a new method of description, intermediate between the two extremes of time analysis and spectral analysis. There are certain ?elementary signals? which occupy the smallest possible area in the information diagram. They are harmonic oscillations modulated by a ?probability pulse.? Each elementary signal can be considered as conveying exactly one datum, or one ?quantum of information.? Any signal can be expanded in terms of these by a process which includes time analysis and Fourier analysis as extreme cases. These new methods of analysis, which involve some of the mathematical apparatus of quantum theory, are illustrated by application to some problems of transmission theory, such as direct generation of single sidebands, signals transmitted in minimum time through limited frequency channels, frequency modulation and time-division multiplex telephony.

Theory of communication. Part 1: The analysis of information

The measurement of power spectra is a problem of steadily increasing importance which appears to some to be primarily a problem in statistical estimation. Others may see it as a problem of instrumentation, recording and analysis which vitally involves the ideas of transmission theory. Actually, ideas and techniques from both fields are needed. When they are combined, they provide a basis for developing the insight necessary (i) to plan both the acquisition of adequate data and sound procedures for its reduction to meaningful estimates and (ii) to interpret these estimates correctly and usefully. This account attempts to provide and relate the necessary ideas and techniques in reasonable detail — Part I of this article appeared in the January, 1958 issue of THE BELL SYSTEM TECHNICAL JOURNAL.

The Measurement Of Power Spectra: From The Point Of View Of Communications Engineering

Signals which are quantized both in time of occurrence and in magnitude are in fact quite old in the communications art. Printing telegraph is an outstanding example. Here, time is divided into equal divisions, and the number of magnitudes to be distinguished in any one interval is usually no more than two, corresponding to the closed or open positions of a sending switch. It is only in recent years, however, that the development of high speed electronic devices has progressed sufficiently to enable quantizing techniques to be applied to rapidly changing signals such as produced by speech, music, or television. Quantizing of time, or time division, has found application as a means of multiplexing telephone channels.1 The method consists of connecting the different channels to the line in sequence by fast moving switches synchronized at the transmitting and receiving ends. In this way a transmission medium capable of handling a much wider band of frequencies than required for one telephone channel can be used simultaneously by a group of channels without mutual interference. The plan is the same as that used in multiplex telegraphy. The difference is that ordinary rotating machinery suffices at the relatively low speeds employed by the latter, while the high speeds needed for time division multiplex telephony can be realized only by practically inertialess electron streams. Also the widths of frequency band required for multiplex telephony are enormously greater than needed for the telegraph, and in fact have become technically feasible only with the development of wide-band radio and cable transmission systems. As far as any one channel is concerned the result is the same as in telegraphy, namely that signals are received at discrete or quantized times. In the limiting case when many channels are sent the speech voltage from one channel is practically constant during the brief switch closure and, in effect, we can send only one magnitude for each contact or quantum of time. The more familiar word “sampling” will be used here interchangeably with the rather formidable term “quantizing of time”.

Spectra of quantized signals

Fourier integrals for practical applications

Note: The electric wave-filter, an invention of Dr. Campbell, is one of the most important of present day circuit developments, being indispensable in many branches of electrical communication. It makes possible the separation of a broad band of frequencies into narrow bands in any desired manner, and as will be gathered from the present article, it effects the separation much more sharply than do tuned circuits. As the communication art develops, the need will arise to transmit a growing number of telephone and telegraph messages on a given pair of line wires and a growing number of radio messages through the ether, and the filter will prove increasingly useful in coping with this situation. The filter stands beside the vacuum tube as one of the two devices making carrier telegraphy and telephony practicable, being used in standard carrier equipment to separate the various carrier frequencies. It is a part of every telephone repeater set, cutting out and preventing the amplification of extreme line frequencies for which the line is not accurately balanced by its balancing network. It is being applied to certain types of composited lines for the separation of the d.c. Morse channels from the telephone channel. It is finding many applications to radio of which multiplex radio is an illustration. The filter is also being put to numerous uses in the research laboratory. The present paper is the first of a series on the electric wave-filter to be contributed to the Technical Journal by various authors. Being an introductory paper the author has chosen to discuss his subject from a physical rather than mathematical point of view, the fundamental characteristics of filters being deduced by purely physical reasoning and the derivation of formulas being left to a mathematical appendix. — Editor.

Physical theory of the electric wave-filter

Ideal telephone substation and repeater circuits are shown to present output and input requirements which can be met by a type of circuit containing four resistances, each of which has maximum output; maximum output is found to involve biconjugacy, i.e., the four resista. nces fall into two pairs without active electrical connection between the two resistances of either pair; all circuits meeting these requirements with the minimum number of elements are enumerated; the necessary formulas for proportioning these circuits and also circuits having superflous elements are derived. The discussion of the properties of these four-resistance circuits, to which a consideration of three-resistance eire-uits is added, is a contribution to the important problem of the properties of the most general circuit containing ideal transformers.

Maximum Output Networks for Telephone Substation and Repeater Circuits

IN many importa nt practical operations the constant probability of an event happening in a single trial is extremely small but the number of trials is so large that the event may actually occur a sufficient number of times to become a matter of importance The curves of Figs 1 and 2 show the probability P of such an event happening at least c times in a number of trials for which the average number of occurrences is a The probability range shown is from 0000001 to O999999 and the average extends from 0 to 15 in Fig 1 and to 200 in Fig 2 An open scale is obtained at both ends, even when the probability approaches to within one part in a million of the limits 0 and 1, by employing an ordinate scale corresponding to the normal probability integral

http://ieeexplore.ieee.org/iel7/6731005/6767986/06767990.pdf

Probability curves showing poisson's exponential summation

The growing practical importance of transients and other non-periodic phenomena makes it desirable to simplify the application of the Fourier integral in particular problems of this kind and to extend the range of problems which can be solved in closed form by this method. Unless the physicist or technician is in a position to evaluate the definite integrals which occur, by mechanical means, he is usually entirely dependent upon the results obtained by the professional mathematician. To facilitate the use of the known closed form evaluations of Fourier integrals many of them have been compiled and tabulated in Table I. They are presented, however, not as definite integrals but as paired functions, one function being the coefficient for the cisoidal oscillation (or complex exponential) and the other function the reciprocally related coefficient for the unit impulse. This arrangement simplifies the table and promises to be most convenient in practical applications, since it is the coefficients of which immediate use is made, just as in the case of the Fourier series. Applications of the tabulated coefficient pairs to 85 transient problems are given, together with all necessary details, in Table II.

George A. Campbell

Papers

Fourier integrals for practical applications

Physical theory of the electric wave-filter

Maximum Output Networks for Telephone Substation and Repeater Circuits

Probability curves showing poisson's exponential summation

The practical application of the Fourier integral