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Showing papers in "IEEE Transactions on Circuits and Systems I-regular Papers in 1959"


Journal ArticleDOI
TL;DR: The relation of the theory developed in this paper to Huffman's description of linear sequence transducers in terms of the D operator is discussed, as well as unsolved problems and directions for further generalization.
Abstract: Analysis and synthesis techniques for a class of sequential discrete-state networks are discussed. These networks, made up of arbitrary interconnections of unit-delay elements (or of trigger flip-flops), modulo-p adders, and scalar multipliers (modulo \alpha , prime p ), are of importance in unconventional radar and communication systems, in automatic error-correction circuits, and in the control circuits of digital computers. In addition, these networks are of theoretical significance to the study of more general sequential networks. The basic problem with which this paper is concerned is that of finding economical realizations of such networks for prescribed autonomous (excitation-free) behavior. To this end, an analytical-algebraic model is described which permits the investigation of the relation between network logical structure and state-sequential behavior. This relation is studied in detail for nonsingular networks (those with purely cyclic behavior). Among the results of this investigation is the establishment of relations between the state diagram of the network and a characteristic polynomial derived from its logical structure, An operation of multiplication of state diagrams is shown to correspond to multiplication of the corresponding polynomials. A criterion is established for the realizability of prescribed cyclic behavior by means of linear autonomous sequential networks. An effective procedure for the economical realization of such networks is described, and it is shown that linear feedback shift registers constitute a canonical class of realizations. Examples are given of the realization procedure. The problem of synthesis with only one-cycle length specified is also discussed. A partial solution is obtained to this "don't care" problem. Some special families of feedback shift registers are investigated in detail, and the state-diagram structures are obtained for an arbitrary number of stages and an arbitrary (prime) modulus. Mathematical appendixes are included which summarize the pertinent results in Galois field theory and in the factorization of cyclotomic polynomials into irreducible factors over a modular field. The relation of the theory developed in this paper to Huffman's description of linear sequence transducers in terms of the D operator is discussed, as well as unsolved problems and directions for further generalization.

301 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that linearity and passivity imply causality in the most general linear, passive, time-invariant n -port (e.g., networks which may be both distributed and non-reciprocal) from an axiomatic point of view, and a completely rigorous theory was constructed by the systematic use of theorems of Bochner and Wiener.
Abstract: 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 (and as such need not be single-valued). Under very weak conditions on the domain of \Phi , it is shown that linearity and passivity imply causality. In every case, \Phi_a , the n -port corresponding to \phi augmented by n series resistors is always causal ( \Phi is the "augmented network," Fig. 2). Under the further assumptions that the domain of \Phi_a is dense in Hilbert space and \phi is time-invariant, it is proved that \Phi possesses a frequency response and defines an n \times n matrix S(z) (the scattering matrix) of a complex variable z = \omega + i\beta with the following properties: 1) S(z) is analytic in Im z > 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.

170 citations


Journal ArticleDOI
C. Coates1
TL;DR: In this article, the authors derived topological expressions for a non-reference vertex of a flow graph by a linear combination of the weights of the reference vertices of the flow graph and showed that these results are similar to those given for the Signal-Flow-Graph.
Abstract: A weighted, oriented topological structure, denoted by G and called a flow graph, is associated with a set of m equations in n variables, denoted by KX = 0 , such that K is a connection matrix and X a vertex weight matrix of the associated graph. This same set of equations can be written as A_{-v:}^{-} C(A+)'X = 0 where A_{-}^{-v:} and A^{+} are negative and positive incidence matrices and where C and X are respectively branch and vertex weight matrices of the graph. By familiar algebraic procedures, an expression for the weight x_p , of a nonreference vertex of G is obtained as a linear combination of the weights of the reference vertices (vertices with zero negative order) and can be written as x_p = \Sigma_{j=1}^{s} \zeta p{\dot}r_{j}x_{r_j} . To these algebraic results there correspond topological expressions in terms of subgraphs of G for the coefficients, \zeta P{\dot}r_j . A similar correspondence is obtained between the topological operation of deleting a vertex from the flow graph and the algebraic operation of eliminating a variable from the set of equations. These results are derived from the algebraic equations written in terms of the incidence and weight matrices of the graph. They are similar to those given for the familiar Signal-Flow-Graph, although they are more convient to use, since the topological properties of the flow graph depend only upon the algebraic properties of the set of equations. A flow graph can be drawn directly from an electric network diagram, and the flow-graph properties, used to obtain a solution of the network equations. Examples of this for two types of feedback networks are shown.

144 citations


Journal ArticleDOI
J. Bertram1, P. Sarachik1
TL;DR: In this article, the stability of non-linear and linear systems with random time varying parameters has been studied and sufficient conditions for stability in the mean square are obtained by an extension of the Lyapunov's Second Method to stochastic problems.
Abstract: This paper is concerned with the stability, in a stochastic sense, of circuits or systems described by ordinary differential equations with randomly time varying parameters. Sufficient conditions for stability in the mean square are obtained by an extension of "Lyapunov's Second Method" to stochastic problems. The general result while appliable to non-linear as well as linear systems, presents formidable computational difficulties except for a few special cases which are tabulated. The linear case with certain assumptions concerning the statistical independence of parameter variation is carried out in detail.

100 citations


Journal ArticleDOI
TL;DR: In this article, a class of finite-state machines, called information-lossless Canonical circuits, were introduced, which can transform 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.
Abstract: 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

97 citations


Journal ArticleDOI
TL;DR: Methods are described for determining the number of symmetry classes for functions of n variables, and for ascertaining whether or not two functions belong to the same class.
Abstract: Two Boolean functions which differ only by permutation and complementation of their n input variables belong to the same symmetry class. Methods are described for determining the number of symmetry classes for functions of n variables, and for ascertaining whether or not two functions belong to the same class. This classification is achieved via a complete set of invariants, characteristic of the class, and easily computable from any function in it. The invariants also provide information concerning the size and symmetry properties of the class. Analogous techniques apply to other symmetry classifications of Boolean functions, and to more general categories of discrete mappings.

93 citations


Journal ArticleDOI
TL;DR: In this paper, the effects of arbitrarily located stray delays in such circuits are analyzed, and it is shown that proper operation can be assured regardless of the presence of stray delays and without the introduction of delay elements by the designer.
Abstract: 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.

73 citations



Journal ArticleDOI
TL;DR: In this paper, the problem of decomposing a polynomial with real coefficients into the differences of two polynomials with negative real zeros such that the latter polynominals have coefficients which are as small as possible is addressed.
Abstract: Negative-impedance conversion methods of active RC synthesis lead to networks which are highly sensitive to active and passive parameters. In the design of such networks there is some freedom available, which may be used to minimize the sensitivity of the structure to both the active and passive parameters. In most cases, the problem is that of decomposing a polynomial with real coefficients into the differences of two polynomials with negative real zeros such that the latter two polynominals have coefficients which are as small as possible. The optimum design is presented.

60 citations


Journal ArticleDOI
TL;DR: The paper deals with matrices K which may be decomposed in a congruence ADA \prime where A is a rectangular unimodular- and D a diagonal- matrix with constant, positive and real diagonal elements.
Abstract: In this paper some properties of unimodular (or E -) and paramount (or M -) matrices are discussed. The paper deals with matrices K which may be decomposed in a congruence ADA \prime where A is a rectangular unimodular- and D a diagonal- matrix with constant, positive and real diagonal elements. It is shown that such a decomposition, if at all possible, is essentially unique and a direct algebraic procedure is given which results either in finding the pair of matrices A and D or in a proof that such decomposition is impossible. Since the admittance matrices of n -ports described on pure resistance networks (or RLC networks for positive, real values of the complex frequency) with n + 1 nodes, or dually the impedance matrices of n -ports inscribed into R -networks with exactly n independent links belong to the Class of ADA \prime matrices the paper defines a method of decomposition of such matrices into the product ADA . The synthesis of the corresponding n -port may then be realized by known methods.

52 citations


Journal ArticleDOI
TL;DR: In this article, it was proved that the modified Bott-Duffin (or the Reza-Pantell-Fialkow-Gerst) realization using seven elements is rigorously minimal in number of elements.
Abstract: The purpose of this paper is to obtain rigorously minimal realizations of the biquadratic minimum positive real function without the use of transformers. For this purpose a few theorems are proved about the structure of the network realizing a minimum pr function. This is followed by an exhaustive search of networks in increasing order of number of elements. It is proved that the modified Bott-Duffin (or the Reza-Pantell-Fialkow-Gerst) realization using seven elements is rigorously minimal in number of elements, except for the special cases Z(0) = 4 Z(\infty) and Z(\infty) = 4 Z(0) . These two special cases have five element realizations.

Journal ArticleDOI
TL;DR: In this paper, a class of RC driving-point immittances characterized by nearly constant argument over an extended frequency range is considered, where the poles of admission are geometrically spaced along the negative real frequency axis, and consequently the elements of the network can be thought of as "spaced".
Abstract: This paper deals with a class of RC driving-point immittances characterized by nearly constant argument over an extended frequency range. These arguments may have an average value between the limits zero and \pi/2 radians. Networks having near constant argument are of importance in shaping the phase character of the forward gain in feedback systems. These networks have arguments that oscillate about a mean value and the nature of this oscillation is discussed. The poles of admittance are geometrically spaced along the negative-real frequency axis, and consequently the elements of the network' can be thought of as "spaced." The immittance functions and argument oscillations for the 22-1/2° , 45° and 67- 1/2° cases as a function of spacing are fully discussed. An application to a feedback amplifier design is given.

Journal ArticleDOI
TL;DR: In this paper, a transform table similar to that used in Laplace transform theory is developed and applied to network problems, which can be described as: performing the integral operation, applying the table of operations on known transform pairs, and deriving the Hankel transform from the Laplace Transform.
Abstract: Integral transform techniques for solving linear integro-differential equations can provide insight and flexibility in solving physical problems, especially network problems. The type of differential equation which describes the physical system will dictate the transform that should be applied to simplify the solution and this paper deals with two transforms, namely, the Mellin transform and the Hankel transform. The Laplace transform can be used to solve linear constant coefficient differential equations or networks which are represented by this type of equation. A familiarity with this transform is assumed and is not covered in this paper. Mellin transforms may be applied to networks which yield the Euler-Cauchy differential equation. This transform will simplify the solution of such an equation. A transform table, similar to that type used in Laplace transform theory, is developed and applied to network problems. Hankel transforms may be applied to networks which yield the Bessel differential equation or variations of this equation. Unlike the Laplace and Mellin transforms, the Hankel transform is symmetric and the transformed variable is a real, rather than a complex variable. A transform table of both operations and functions is developed anti applied to network problems as before. Three methods can be used to establish the table of transform pairs. They can be described as: performing the integral operation, applying the table of operations on known transform pairs, and deriving the Hankel transform from the Laplace transform. With both transforms, the applications are made to problems in analysis, instrumentation, and synthesis.

Journal ArticleDOI
TL;DR: In this article, the exact conversion functions for series modulators with simple poles and simple zeros were derived for networks with one periodically operated switch, where the switch is alternately open and closed during equally long time intervals.
Abstract: The exact conversion functions are calculated for networks containing one periodically operated switch, using familiar pole-zero and Fourier methods of analysis. It is first assumed that the switch is alternately open and closed during equally long time intervals. Circuits whose driving-point impedance Z(p) seen from the switch has neither pole nor zero at infinity are treated in detail. The analysis is then extended in order to allow for impedances Z(p) having either a pole or a zero at p = \infty . Complete results are also given for circuits whose switch is alternately open during time intervals of duration, T_1 , and closed during intervals of duration, T_2 eq T_1 . The general analysis is applied to a series modulator and the realization of a given function of frequency as conversion function of such a modulator is investigated. Throughout this paper, the impedance Z(p) is assumed to have only simple poles and simple zeros.

Journal ArticleDOI
TL;DR: A search for optimum group codes using the IBM 704 computer is described, and a number of counter-examples to typical conjectures on binary group codes are listed.
Abstract: This paper describes a search for optimum group codes using the IBM 704 computer. Some theory of the relation. ship between equivalent codes used in narrowing the search is described, and the method of searching is outlined. The newly found optimum codes are listed, along with a number of counter-examples to typical conjectures on binary group codes.

Journal ArticleDOI
TL;DR: The optimum polynomials of even degrees are presented in general forms, from which the optimum filters with monotonic response are derived.
Abstract: Recently, Papoulis has developed a new class of filters, which has the maximum cutoff rate under the condition of a monotonically-decreasing response. These new filters are based on the optimum monotonically-increasing polynomials of odd degrees. In this paper, the optimum polynomials of even degrees are presented in general forms, from which the optimum filters with monotonic response are derived. Characteristics of these filters are illustrated by several examples which include frequency response, pole locations and the ladder realizations.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the number of degrees of freedom of a RLC network can be determined from the energy-storing elements and the topology of the network.
Abstract: It is shown here that the number of degrees of freedom, or what is equivalent-the number of natural frequencies-of any RLC network can readily be determined from the number of energy-storing elements and the topology of the network. The effect of loss (resistance) in altering the number of degrees of freedom is explained.

Journal ArticleDOI
TL;DR: In this article, a linear sequential circuit is defined in terms of the modular field GF(p) and vectors and matrices defined thereover, and the behavior of the circuit is described in a finite state space of k dimensions and p^k states.
Abstract: Sequential circuits comprising 1) modulo- p ( p = prime) summers, 2) amplifiers whose gains are integers , and 3) unit delays are considered in this paper which constitutes an extension of earlier work by Huffman. Such circuits are characterized in terms of the modular field GF(p) and vectors and matrices defined thereover. A summary of the properties of GF(p) is given. A linear sequential circuit is defined in terms of \overrightarrow{y}(n) = C \overrightarrow{s}(n) + D \overrightarrow{x}(n) \overrightarrow{s}(n + 1) = A \overrightarrow{s}(n) + B \overrightarrow{x}(n) where A, B, C, and D are k \times k matrices defined over GF(p) . The latter equations constitute a canonical representation of any circuit comprising the above listed components. It is shown that circuits of this type meet the usual additivity criterion of linear systems. The behavior of the circuit is described in a finite state space of k dimensions and p^k states. The autonomous circuit (A, B, C, D = constant and \overrightarrow{x}(n) \equiv \overrightarrow{O} , all n ) is characterized by the matrix A . If A is nonsingular all initial states are either finite equilibrium points or lie in periodic sequences of length T_{\tau} \leq T_{\max} = p^{k} - 1 . If the minimum polynomial of A_{\tau} has distinct roots, T_{\tau} , divides r(p - 1) . If A is singular, there are some singular initial states to which the circuit cannot return in the absence of excitation. The use of Z transforms for linear modular sequential circuits is demonstrated. Inputs and outputs are represented by their "transforms" and the circuit by its "transfer function." The transform of the output is the product of the transfer function and the transform of the input. Several illustrative examples are included.

Journal ArticleDOI
TL;DR: The well-known Routh's criterion uses a very efficient computational method that has been found to reduce greatly calculational labor and chances of error in a number of other important applications to circuit theory.
Abstract: The well-known Routh's criterion uses a very efficient computational method, or algorithm, that has been found to reduce greatly calculational labor and chances of error in a number of other important applications to circuit theory. Among these applications are finding common factors of polynomials, computing Sturm's functions, synthesizing RC, RL, or LC ladder networks by means of continued-fraction expansions, determining RC, RL, or LC realizability of a given immittance function, and analysis of ladder networks. Methods of handling the first two problems, both in normal and special cases, are given and illustrated.

Journal ArticleDOI
TL;DR: The theory of random linear systems is extended to systems containing one or more non-independent parameters under the assumption that the parameter processes and the solution process have very widely separated spectra.
Abstract: The theory of random linear systems is extended to systems containing one or more non-independent parameters under the assumption that the parameter processes and the solution process have very widely separated spectra. It is shown that the second product moment of the solution satisfies a linear integral equation which can be solved in closed form in some important special cases. The mean square stability theory of equations containing one purely random coefficient initiated by Samuels .nd Eringen is developed further and extended to systems containing one narrow-band random parameter. Specific mean sjuare stability criteria are worked out for an RLC circuit with capacity variations that are a narrow-band stochastic function.

Journal ArticleDOI
Sidney Darlington1
TL;DR: This paper may be regarded as an extension, to nonstationary systems, of methods applied to stationary systems by Bode and Shannon, using primarily circuit theory concepts.
Abstract: Nonstationary signal and noise statistics are assumed, such that ensembles with the same covariances can be generated by passing white noise through finite networks of linear, time-variable, positive elements. Linear least-squares smoothing and prediction operations are to be found. This paper may be regarded as an extension, to nonstationary systems, of methods applied to stationary systems by Bode and Shannon, using primarily circuit theory concepts. Analogous results are obtained by examining analogous operations in frequency domain, and differential equations terms.

Journal ArticleDOI
TL;DR: In this paper, it is shown how another set of functions, which are just the Jacobi polynomials whose argument is an exponential, may be used instead of the Laguerre functions.
Abstract: When problems having to do with transients are solved by the Laplace transform or equivalent methods, one may be left with the necessity of solving a rather complicated equation in the transform variable. This may be avoided, in many cases, by getting the solution in the form of a series. Laguerre functions have had some use for that purpose. It is shown here how another set of functions, which are just the Jacobi polynomials whose argument is an exponential, may be used instead. The use of this latter set permits a rather elegant means of evaluating the coefficients in the expansion to be used. In an appendix, ways of applying the mathematical techniques used are investigated. These involve the complex "Faltung" theorem, for investigating questions of orthogonality and orthonormality in general.

Journal ArticleDOI
TL;DR: The theory of irredundant networks is significant for the design of redundant networks, i.e., networks generating functions with a certain protection against temporary branch errors.
Abstract: Certain Boolean functions can be generated by irredundant branch-networks, i.e., with networks with only one branch for each variable (literal) of the function. A simple solution (based on graph theory) is given to the realizability problem for irredundant 2- and n -terminal networks. The theory of irredundant networks is significant for the design of redundant networks, i.e., networks generating functions with a certain protection against temporary branch errors. A few examples on redundant networks are given and the method of design is compared with coding theory.

Journal ArticleDOI
W. Cadden1
TL;DR: Three types of sequential circuits are defined, two of which are synchronous and one of which is asynchronous, and transformation procedures are given for transforming a state table of one type into state tables of the other types.
Abstract: Three types of sequential circuits are defined, two of which are synchronous and one of which is asynchronous. The concept of equivalent sequential circuits as discussed by Huffman, Mealy, and Moore is extended to circuits of different types. Transformation procedures are given for transforming a state table of one type into state tables of the other types. One of these transformations can also be used to introduce unit delay between corresponding inputs and outputs for a synchronous circuit. The transformation methods allow a comparison of circuits, or state tables of different types to be made for a given sequential circuit problem. A few general conclusions are drawn about the different types of sequential circuits.

Journal ArticleDOI
TL;DR: It is shown that, provided the impedance seen by the time-varying element becomes capacitive at very high frequencies, the complete solution can be found within an arbitrary amount of accuracy.
Abstract: This paper presents a method of steady-state analysis of a linear network, of arbitrary degree of complexity, containing a single periodically-varying element. The proposed method makes full use of circuit theoretical ideas, such as impedance matching and tearing apart, and of iteration techniques which are particularly suitable for automatic computation. The proposed method has the additional feature of leading to the amplitude and phase of all sidebands and of giving a bound on the error if the iterations are stopped at any particular point. More precisely, it is shown that, provided the impedance seen by the time-varying element becomes capacitive at very high frequencies, the complete solution can be found within an arbitrary amount of accuracy.

Journal ArticleDOI
TL;DR: In this article, lossless, reciprocal transformations of two-state and non-reciprocal two-ports are discussed and the minimum loss of such networks for various applications are shown to depend upon a single characteristic number which is invariant under the transformations studied.
Abstract: Lossless, reciprocal transformations of two-state and nonreciprocal two-ports are discussed. (Two-state networks are networks whose parameters may be switched simultaneously between two different values.) The minimum loss of such networks for various applications are shown to depend upon a single characteristic number which is invariant under the transformations studied. When a two-state or nonreciprocal material is used to realize such a network, there is an upper limit for the value of the characteristic number, and this is defined as a figure of merit for the material.

Journal ArticleDOI
TL;DR: The treatment is expository, but introduces the unifying framework of graph theory for these various considerations, including hoff's Laws, mesh and node equations, and matrix tree theorem.
Abstract: In this discussion, we mention the following topics concerning electric networks in graph theoretic terms: Kirch. hoff's Laws, mesh and node equations, and matrix tree theorem; flow problems and Menger's theorem; boolean functions and enumeration and synthesis problems; information theory and Markov chains; cut sets and incidence matrices; the "crummy relay" results of Moore and Shannon; and the treatment using electrical concepts of the dissections of rectangles into squares by Brooks, Smith, Stone, and Tuttle. The treatment is expository, but introduces the unifying framework of graph theory for these various considerations.

Journal ArticleDOI
TL;DR: The matrices introduced are closely related to the relation matrices of the calculus of relations and provide a formal tool for discussing state diagrams and several of the well-known theorems on state diagrams are consequences of properties of transition matrices, which remain invariant under matrix multiplication.
Abstract: In this paper a matrix technique is introduced for the analysis of state diagrams of synchronous sequential machines. The matrices introduced are closely related to the relation matrices of the calculus of relations and provide a formal tool for discussing state diagrams. It is shown that several of the well-known theorems on state diagrams are consequences of properties of transition matrices, which remain invariant under matrix multiplication. A reduction procedure for state diagrams, based on transition matrices, which is similar to Moore's technique, is given. A method of extending the results to asynchronous machines is also included.


Journal ArticleDOI
TL;DR: The method recognizes that construction of a tree (and hence the pertinent graph) from a given matrix can be done by inspection once the pattern of its growth has been established, and may be regarded as a constructive test for fulfillment of necessary and sufficient conditions.
Abstract: The method recognizes that construction of a tree (and hence the pertinent graph) from a given matrix can be done by inspection once the pattern of its growth has been established. To this end it is only necessary that we have a mechanism, applicable to a given cut-set matrix, which sorts out those rows that correspond to the outermost twigs or tips of the tree, for we can then form an abbridged cut-set matrix corresponding to what is left of the total graph after the tree tips with their uniquely attached links are pruned away. This remainder again has tips which can be found and eliminated in the same way. Continuation thus reveals the desired growth pattern. Since the method cannot fail to yield a graph if its existence is compatible with the structure of the given matrix, it may be regarded as a constructive test for fulfillment of necessary and sufficient conditions.