>In the study of the stability of a system, it is never completely satisfactory to know only that an equilibrium state is asymptotically stable. As a practical matter, it is necessary to have some idea of the size of the perturbations the system can undergo and still return to the equilibrium state. It is never possible to do this by examining only the linear approximation. The effect of the nonlinearities must be taken into account. Liapunov's second method provides a means of doing this. Mathematical theorems underlying methods for determining the region of asymptotic stability are given, and the methods are illustrated by a number of examples.

Quantization is an operation that takes place when a physical quantity is represented numerically. It is the assignment of an integral value to a physical quantity corresponding to the nearest number of units contained in it. Quantization is like "sampling-in-amplitude," which should be distinguished from the usual "sampling-in-time." The probability density distribution of the signal is sampled in this case, rather than the signal itself. Quantized signals take on only discrete levels and have probability densities consisting of uniformly-spaced impulses. If the quantization is fine enough so that a Nyquist-sampling restriction upon the probability density is satisfied, statistics are recoverable from the grouped statistics in a way similar to the recovery of a signal from its samples. When statistics are recoverable, the nature of quantization "noise" is understood. As a matter of fact, it is known to be uniformly distributed between plus and minus half a unit, and it is entirely random (first order) even though the signal may be of a higher-order process, provided that a multidimensional Nyquist restriction on the high-order distribution density is satisfied. This simple picture of quantization noise permits an understanding of round-off error and its propagation in numerical solution, and of the effects of analog-to-digital conversion in closed-loop control systems. Application is possible when the grain size is almost two standard deviations. Here the dynamic range of a variable covers about three quantization boxes.

In this paper the most general linear, passive, time-invariant n -port (e.g., networks which may be both distributed and non-reciprocal) is studied from an axiomatic point of view, and a completely rigorous theory is constructed by the systematic use of theorems of Bochner and Wiener. An n -port \Phi is defined to be an operator in H_n , the space of all n -vectors whose components are measurable functions of a real variable t, (- \infty 0 ; 2) Q(z) = I_n - S^{\ast}(z)S(z) is the matrix of a non-negative quadratic form for all z in the strict upper half-plane and almost all \omega . Conversely, it is also established that any such matrix represents the scattering description of a linear, passive, time-invariant n -port \Phi such that the domain of \Phi_a contains all of Hilbert space. Such matrices are termed "bounded real scattering matrices" and are a generalization of the familiar positive-real immittance matrices. When \Phi and \Phi^{-1} are single-valued, it is possible to define two auxiliary positive-real matrices Y(z) and Z(z) , the admittance and impedance matrices of \Phi , respectively, which either exist for all z in Im z > 0 and almost all \omega or nowhere. The necessary and sufficient conditions for an m>n \times n matrix A_{n}(z) to represent either the scattering or immittance description of a linear, passive, time-invariant n -port \Phi are derived in terms of the real frequency behavior of A_{n}(\omega) . Necessary and sufficient conditions for \Phi_a to admit the representation i(t) = \int_{-\infty}^{\infty} dW_{n}(\tau)e(t - {\tau}) for all integrable e(t) in its domain are given in terms of S(z) . The last section concludes with a discussion concerning the nature of the singularities of S(z) and the possible extension of the theory to active networks.

The concept of network-element-value solvability is introduced and its importance in the establishment of methods of automatic trouble-shooting of electronic equipment is pointed out. First, a set of definitions is given which enables an objective discussion to be made of network solvability of arbitrary passive, linear, lumped parameter networks with respect to a restricted set of external terminals (available and partly available). Next, a relation is obtained connecting the number of available and partly available terminals of a network with the number of admittance functions determining the measurable behavior of the network. Theorems A, B and C then give solvability conditions for purely resistive networks. It is shown that the theory developed can be extended to include networks with internal energy sources (Theorem D). A general necessary condition for network-element-value solvability is then obtained (Theorems E and F). Finally, examples are given showing applications and limitations of the theorems obtained.

This paper presents a new result generalizing Richards' theorem. It is shown that this result leads to a complete, simple and unified theory of cascade synthesis which yields the types A, B, Brune, C and D sections in a direct and natural manner. The element values of the various sections are obtained in closed form in terms of three or six indexes. Thus the extraction cycle is performed once and for all for the whole class of positive-real functions. Several problems are worked out in detail and a chart is constructed to facilitate the computations. The formulas are easily programmed on a digital computer.

This paper describes a design theory for a class of strip-line filters on an insertion loss basis. The important part of the paper is the equivalent transformations showing the close correspondence between lumped and coupled-line distributed parameter circuits. An introduction is provided for clarification. Line type, low-pass ladder, high-pass ladder, and band-pass ladder type filters are realized in coupled strip-lines. Their physical configurations are depicted. Finally a development is given of characteristic-immittance formulas.

The physical processes of HF loss are discussed. Two cases in transient analysis for coaxial cables are presented; the first considers skin effect in plated conductors while the second is an analysis based upon an attenuation approximation of the form (frequency) ^m , 0 < m < 1 . A graphical transient analysis technique is described which allows one to easily analyze cables whose losses are due to a combination of physical processes. Generalized curves based upon the (frequency) ^m approximation are presented by which the transient response of a cable can be rapidly evaluated for purposes of engineering design.

Both the loop and node methods of network analysis produce a system of second-order differential equations. A method of analysis is proposed which produces a set of first-order differential equations. With this method, the network equations obtained can be expressed in the form F + dy/dt = Ay , where F and y are column matrices and A is a square matrix. The variables, y , are currents through inductances and voltages across capacitances; the forcing functions. F are proportional to voltage and current sources. The elements of A are inductances, capacitances, and resistances, or combinations thereof. Characteristic roots (natural frequencies) of the network are identical with the eigenvalues of the A matrix.

This paper treats the problem of finding the steadystate currents and voltage drops in an electrical network of twoterminal elements, each of which has the property that its current-vs-voltage-drop graph, or "characteristic," is a curve going upward and to the right. (Thus, "tunnel diodes" are excluded, but nonlinear resistances, current and voltage sources, rectifiers, etc. are permitted.) The construction methods are specifically designed for digital computation techniques (either automatic or manual). The principal tools are: 1) the application of theorems from graph theory ("network-topology"), and 2) quantization of the variables (permitting them to take on only a discrete set of values).

An important class of finite-state machines transforms input sequences of digits into output sequences in a way such that, after an experiment of any finite length on the machine, its input sequences may be deduced from a knowledge of the corresponding output sequence, its initial and final states, and the set of specifications for the transformations by which the machine produces output sequences from input sequences. These machines are called "information-lossless." Canonical circuit forms are shown into which any information-lossless machine may be synthesized. The existence of inverses for these circuits is investigated; and circuits for their realization are derived.

A method of estimation and approximate precompensation for the effects of element losses in resistance-terminated LC circuits is presented. Some new theorems are utilized for the special case of ladder networks. The process is valid for a nonuniform distribution of losses; there is, however, a limit to the amount of loss that can be compensated for. The procedure is based on a simple expression giving the firstorder increment of the complex transfer function in terms of quantities normally available from the lossless design. A formula is derived for the lossy response of the network; this has a form which brings out the individual effects of the different parasitic elements in producing distortion, and shows the effects of Q-tolerances of each lossy element. The predistorted transmission function is also given explicitly. The method'is not restricted to treating the effects of dissipation. It is also applicable to other parasitic phenomena (stray capacitances, element tolerances, etc.). The formulas become especially simple and useful for ladder networks. In the case of equal loss factors in elements of the same kind, they present an alternative to Darlington's design equations. Examples are given to illustrate the relative simplicity and accuracy of the method for practical network synthesis.

A distributed RC circuit analogous to a continuously tapped transmission line can be made to have a rational short-circuit transfer admittance and one rational short-circuit driving-point admittance. A subcircuit of the same structure has a rational open-circuit transfer impedance and one rational open-circuit driving-point impedance. Hence, rational transfer functions may be obtained while considering either generator impedance or load impedance. The functions have poles only along the negative real axis. Although the number of poles is arbitrary, only two may be chosen with complete freedom in a single unit. The realization of a transfer function with many poles may require a number of units either connected in parallel or isolated. Even for a structure with only two poles, there is a considerable saving in the number of interconnections over the lumped circuit. The loss and total capacity required is often the same order of magnitude as those for lumped circuits.

When a signal is approximated by a finite set of component signals which span a subspace \Phi , the least-square approximation may be interpreted geometrically in signal space as the projection of the true signal vector upon this finite dimensional subspace. In case the component signals are one-sided exponentials, the projection operators may be realized by simple physical filters following Kautz' procedure for constructing orthogonalized exponentials. The purpose of this paper is to describe the 'present-instant' error, the 'complementary' signal and the 'complementary' filter which are useful concepts in approximating a signal by one-sided exponential components. An attempt is made to interpret directly in the time domain Kautz' procedure using the 'present-instant' error concept. Some important properties of the 'complementary' signals which prove to be of value in simplifying the process of error energy evaluation and synthesis of the approximating signal, are derived and discussed. In particular, it is found that the 'complementary' filter for a given finite exponential basis is simply an all-pass rational transmittance having zeros in the frequency domain which match the exponents of the basis. This familiar all-pass filter indeed represents an orthogonal transformation which preserves the energy of the signal under transformation. A signal to be approximated is transformed by this filter into the 'complementary' signal which can be separated in time domain into two parts, namely a 'complementary' approximating signal and a 'complementary' error signal. The actual approximating signal and error signal may be recovered independently from them by making certain physical filtering operations.

A communication net consists of branches representing communication channels with the weight of each branch being a positive real number which represents the capacity of transferring information through the branch (called a "branch capacity"). The terminal capacity between the vertices i and j of a communication net is the capacity of transferring information between the vertices i and j by considering the net as a whole. To indicate the terminal capacities between all possible pairs of vertices in a net, a terminal capacity matrix is defined. Then the necessary and sufficient conditions for a terminal capacity matrix are given. To represent the structure of a communication net, a branch capacity matrix is defined. Then the synthesis of a communication net from a given terminal capacity matrix is to obtain a branch capacity matrix from the given terminal capacity matrix.

This paper is concerned with asynchronous, sequential switching circuits in which the variables are represented by voltage levels, not by pulses. The effects of arbitrarily located stray delays in such circuits are analyzed, and it is shown that, for a certain class of functions, proper operation can be assured regardless of the presence of stray delays and without the introduction of delay elements by the designer. All other functions require at least one delay element in their circuit realizations to insure against hazards. In the latter case it is shown that a single delay element is always sufficient. The price that must be paid for minimizing the number of delay elements is that of greater circuit complexity.

The identification problem involves the determination of the identity of a black box from the observation of its responses to a set of input signals. In this paper attention is focused on the identification of zero-memory multipoles and two-poles of class n_1 . The test signals are sine waves of different amplitudes and frequencies, and the measured quanity is the describing function of the device. In the case of two-poles of class n_1 , it is found that the describing function is related to the characteristic function by an integral equation of second order which can be solved explicitly by the use of the Fourier-Hankel transformation.

The theoretical background exists for the design of 90° phase-difference networks. Continued interest in the design of such wide-band networks for single-sideband and other applications indicates the need for methods to ease the computational difficulties inherent in network design involving elliptic functions. Concise design data are given in the form of equations, curves and tables which simplify synthesis of LC or RC all-pass networks by eliminating the above computational difficulties. The normalized design curves presented cover a range of bandwidths of 2000 to 1 and permit the designer to write the response function of all-pass phase-difference networks with over-all complexity of up to 12 real pole-zero pairs.

The article is devoted to the theory of weak resonance action on self-oscillating systems. The theory is based on the method of secondary-simplification of "shortened" equations for amplitudes and phases of the self-oscillating process. A series of new results is obtained. These results are taken as a basis for the creation of new radiotechnical devices.

The topological implications of irreducibility of the admittance or impedance matrix of an n-port network are studied. Special attention is given to the cases when in (n - 1) rows of such a matrix of order n the main diagonal elements are equal to the absolute values of some off-diagonal elements. It is shown that the conditions of realizability of a Y or Z matrix of this type reduce to the known conditions of realizability by means of a network with (n + 1) vertices or exactly n independent circuits. Some examples show realizable Z matrices which cannot be realized as Y matrices and vice versa. Other examples give nonrealizable, paramount Y and Z matrices showing that paramountcy is not a sufficient condition for realizability.