Showing papers in "IEEE Transactions on Circuits and Systems in 1974"

Narrow-band multiple-coupled cavity synthesis

Abstract: A synthesis procedure of identical narrow-band high-Q cavities is presented. The procedure handles multiple couplings in a noncascaded form by synthesizing coupling impedances in one step. Necessary and sufficient conditions for realizability of singly and doubly terminated networks are obtained. An example of a sixth-order elliptic function waveguide filter is included.

Digital all-pass networks

Abstract: A systematic method is outlined to realize an m th-order all-pass digital transfer function using only m multipliers as a cascade of first-order and/or second-order all-pass sections. The realization is based on the multiplier extraction approach in which the n th-order filter section is considered as a digital (n + 1) -pair of which n pairs of input and output terminal variables are constrained by n multipliers. The transfer matrix parameters of the digital (n + 1) -pair, containing only delays and adders, are first identified from which the realization is obtained by inspection. Both canonic and noncanonic realizations are derived. All realizations are then compared with regard to the effect of multiplication roundoff and hardware requirements.

Zeros and poles of matrix transfer functions and their dynamical interpretation

Abstract: The given rational matrix transfer function H(\cdot) is viewed as a network function of a multiport. The no X ni matrix H(s) is factored into D_{l}(S)^{-1} N_{l}(s) = N_{r}(s)D_{r}(s)^{-1} ,where D_{l}(\cdot),N_{l}(\cdot),N_{r}(\cdot) , and D_{r}(\cdot) are polynomial matrices of appropriate size, with D_{l}(\cdot) and N_{i}(\cdot) left coprime and N_{r}(\cdot) and D_{r}(\cdot) right coprime. A zero of H(\cdot) is defined to be a point z where the local rank of N_{l}(\cdot) drops below the normal rank. The theorems make precise the intuitive concept that a multiport blocks the transmission of signals proportional to e^{zt} if and only if z is a zero of H(\cdot) . We show that p is a pole of H(\cdot) if and only if some "singular" input creates a zero-state response of the form re^{pt} , for t > 0 . The order m of the zero z is similarly characterized. Although these results have state-space interpretation, they are derived by purely algebraic techniques, independently of state-space techniques. Consequently, with appropriate modifications, these results apply to the sampled-data case.

Design of digital notch filters

Abstract: It is shown that a second-order digital notch filter is uniquely characterized by two distinct parameters, the notch frequency and the 3-dB rejection bandwidth. As a result, such a filter can be realized using only two multipliers. Methods are outlined to design a notch filter for a prescribed notch frequency and a prescribed 3-dB rejection bandwidth, along with procedures for postdesign adjustment of these parameters. All two-multiplier, canonic and noncanonic, notch filter configurations are developed using the multiplier extraction approach. These networks are then compared with regard to the effect of internal multiplication roundoff errors. Results of computer simulation of the notch filter configurations are also included.

Recursive algorithm for spectral factorization

Abstract: This paper describes a recursive computational algorithm for computing spectral factors of continuous-time and discrete-time power spectrum matrices. The matrices need not be positive definite. Convergence rates of the recursive equation are studied.

A versatile active RC building block with inherent compensation for the finite bandwidth of the amplifier

Abstract: An earlier paper by the authors describes an easily trimmed universal building block for active RC filters which possesses the valuable characteristic that, with suitable design, the Q -value can be made approximately independent of the gain-bandwidth product of the operational amplifiers This makes the filter usable for high frequencies, while at the same time the dependence of the Q -value on temperature variations in the operational amplifiers is drastically reduced Design formulas are presented, as well as comparative measurements which verify the theory The building block is shown to have excellent characteristics both as a universal second-order building block and as a standard block for active ladder synthesis of bandpass filters

Analytical solution of large numbers of mutually coupled nearly sinusoidal oscillators

Abstract: Attempts to produce a mathematical model of the electrical activity in certain sections of the digestive tract in animals have led to complex systems of interconnected nonlinear oscillators. The analytical solution of mutually coupled Van der Pol oscillators using the method of harmonic balance and based on the assumption of nearly sinusoidal oscillations is presented. Algebraic equations are derived from the system dynamic equations, which can be easily solved in the case of identical oscillators to reveal the effect of coupling on the overall system frequency, amplitudes, and phases. A simple hillclimbing method is used for the solution of the algebraic equations for the nonidentical oscillator case. Using this technique, a jump transition is found in the unstable limit cycle of the overall system that can be predicted from a manipulation of the algebraic equations.

Recent advances in monolithic operational amplifier design

Abstract: The state of the art in monolithic operational amplifier design is surveyed. A set of large- and small-signal performance parameters are defined and discussed. Relationships between the important operational amplifier parameters of slew rate, offset voltage, and unity gain bandwidth are demonstrated. The dependence of settling time upon the fine structure of the open-loop frequency response is discussed, and the recent technological and circuit design approaches to the minimization of settling time are reviewed.

Stability of multidimensional digital filters

Abstract: Conditions are obtained for a digital filter in three or more variables to be stable, and computational techniques for checking the stability are examined.

Transfer-function synthesis using a current conveyor

Abstract: This letter demonstrates that a network composed of a current conveyor and RC one-ports is capable of realizing any real rational voltage transfer function.

Chemical reaction networks

Abstract: A set of chemical species "interconnected" by reaction pathways is commonly referred to as a chemical network. We show that chemical networks are mathematically equivalent to a class of multiport networks. Hence, abstract circuit theory can be applied to study chemical systems. Both linear-graph and bond-graph representations of reaction networks are presented. We also indicate how network methods can be employed to treat coupled transport and chemical reaction processes.

Filter analysis by use of pencil of functions: Part II

Abstract: A pair of functions, when linearly combined via a parameter, produces a mathematical entity called a pencil of functions. These pencils are especially interesting when a signal g_{1} (t) is processed by a cascade of simple operators such as first-order filters (FOF's) 1/(s +q_{l}), q_{l} > 0, i = 1,{\cdots}, n, because the pencils formed by pairs of the resulting signal ensemble g_{l} + y_{l}g_{l+1} possess some very useful properties. Most useful of these concerns the linear dependence of the set of pencils thus produced. It is shown in Parts I and II that a necessary condition for a set of pencil of functions to be linearly dependent is a polynomial equation that must be satisfied by their parameters. Applications of the result include linear system identification and rational modeling of the power density spectrum of a random signal. The former of these is discussed in Part I. System dynamics is estimated in closed form requiring no prior estimates. The estimated parameters coincide with true values in the event of noise-free data. Inner products are utilized for computations, and minimum variance corrections are made when the data are noisy.

Explicit topological formulation of Lagrangian and Hamiltonian equations for nonlinear networks

Abstract: This paper presents an explicit topological formulation of the Lagrangian and the Hamiltonian equations for a large class of nonlinear networks. In particular, formulations are given for networks containing both linear and nonlinear controlled sources. Classically, the Lagrangian and Hamiltonian equations are derived from variational techniques; in this paper topological techniques are used for the formulation.

Synthesis of a low-sensitivity multiloop feedback active RC filter

Abstract: A general, low-sensitivity multiloop feedback configuration for realizing higher order transfer functions is presented. A synthesis procedure, which incorporates the desirable building block approach of cascade design, is described. In particular, the paper is concerned with the realization of higher order narrow-band bandpass (BP) functions where either biquad or Tarmy-Ghausi (TG) networks are used as active blocks. A sensitivity expression, dependent explicitly on the individual active block Q's and loop gains, is derived and minimized. The results demonstrate the low-sensitivity performance of the structure which, for some designs is superior to that of the "leapfrog" (LF) configuration, especially over a wide-band of frequencies.

Continued fraction expansion and inversion of the Cauer third form

Abstract: The operation of expanding a rational transfer function into a continued fraction expansion of the Cauer third form and inverting a continued fraction expansion of the Cauer third form to a rational transfer function is extremely time consuming and laborious.

Analytic signals and product theorems for Hilbert transforms

Abstract: Analytic signals are introduced as certain eigenfunctions of the Hilbert transform operator; that is, z(\cdot) is termed "analytic" if and only if \hat{z}(t) = -jz(t) for all t , where \hat{z}(\cdot) is the Hilbert transform of z(\cdot) . Similarly, "dual-analytic" signals are defined as solutions of the homogeneous equation \hat{u} = ju . Using this characterization of analytic signals (shown to be equivalent to the usual definition due to Ville [1]), simple proofs are obtained for all known product theorems of the form \hat{f}g = f\hat{g} , which are useful in the representation and analysis of modulated waveforms. In addition, parallel theorems for the class of dual-analytic and frequency-translated dual-analytic signals are proven.

RC-active networks with reduced sensitivity to amplifier gain bandwidth product

Abstract: A sensitivity analysis for a general second-order multiamplifier RC-active network is described. The conditions for minimized sensitivity to amplifier gain-bandwidth product (GBWP) are established after first formulating the transfer function gain dependence under the assumption of a single-pole representation of the complex amplifier gain. On the basis of a perturbation technique, approximate expressions are presented for the evaluation of the selectivity and frequency sensitivities. By distinguishing a general form of denominator decomposition, the methods available for realization of high Q's with improved sensitivity are explored. Three special cases of the decomposition are identified; two of these, in common with the general case, suggest the use of twin-T null networks with their attendant high passive sensitivities, but the third gives rise to an additional group of circuits 'realizable as double first-order all-pass sections. By comparison with some previous circuits which require matched amplifiers to achieve a low sensitivity, the proposed configurations are shown to be unconditionally insensitive to variations of the GBWP of the amplifiers.

Recursive digital filters having equiripple group delay

Abstract: In a recent paper Thiran has proposed a method for the approximation of constant group delay in an equiripple manner, based on the method of Ulbricht and Piloty. A different, conceptually simpler solution to the same problem is proposed. The solution is based on an algorithm of the pole-shifting type, related to Remez's second algorithm. The choice of initial conditions is discussed and numerical solutions are given.

The bordered triangular matrix and minimum essential sets of a digraph

Abstract: A partitioning strategy of sparse matrices is dealt with. In particular, the problem of transforming a nonsingular matrix by symmetric permutation to an optimal bordered triangular form (BTF) is solved. It is shown that the problem is equivalent to the determination of a minimum essential set of a directed graph. An efficient algorithm is given for finding minimum essential sets of a digraph. The method depends on, as a preliminary step, graph simplication using local information at a vertex. A circuit-generation technique based on vertex elimination is then introduced. The algorithm is illustrated with a complete example. A simple electrical network is used to illustrate the use of the BTF in the sparse tableau approach of network analysis.

Low-sensitivity active filters using the operational amplifier pole

Abstract: The utilization of the operational amplifier pole in the design leads to filters with satisfactory high-frequency performance. The approach reduces the number of capacitors required leading to circuits suitable for integration. Circuit arrangements for a bandpass filter and a second-order Butterworth low-pass filter have been suggested using only resistors and two operational amplifiers. The circuits have low parameter sensitivities.

On optimization of cascade fixed-point digital filters

Abstract: The generation of roundoff accumulation noise in implementing a cascade digital filter can be regarded as a multistage process; therefore, the minimum-noise realization is obtained by dynamic programming. The effect of scaling on the noises is discussed by a numerical example.

l_p approximation of the group delay response of one- and two-dimensional filters

Abstract: An algorithm is developed to design all-pass filters that best approximate in the l_{p} , sense a prescribed group delay response. This algorithm can be used for both the cases of one- and two-dimensional filters. To illustrate the method, several examples for designing one- and two-dimensional filters are solved.

Matrix representations for sorting and the fast Fourier transform

Abstract: The different versions of the fast Fourier transform (FFT) are described here for arbitrary base in terms of the matrix factors of the discrete Fourier transform matrix T_{N} . The Kronecker product notation and the ideal shuffle base r permutation operator form the basis for a unifying theory through which the various versions of the FFT can be viewed. The properties of the ideal shuffle base r permutation operator are used to arrive at FFT versions with such desirable properties as in-place computation or identical geometry from stage to stage. The FFT versions previously described in the literature are derived here. At the same time, algorithms for the sorting of FFT data in digit-reversed order are generated. These are explored and new sorting versions amenable to hardware implementation with sequential memory are presented. As an example of how the unifying theory is used, a number of FFT versions with identical geometry from stage to stage are derived. The hardware necessary for these algorithms is described for the base 4 case with N = 1024 data points.

Synthesis of linear time-varying passive networks

Abstract: A state-space synthesis procedure is given for linear timevarying passive impedance matrices. The synthesis uses only passive components.

Bilinear transformation of polynomials

Abstract: An algorithm for computing the coefficients of a polynomial equation resulting from a bilinear transformation of its variable is described, which compares favorably with the matrix method proposed by Power and improved by Jury and Chan, and which is simpler and easier to program.

A comparison of active, multiple-loop feedback techniques for realizing high order bandpass filters

Abstract: This paper reviews and compares the four multiple-loop feedback techniques used to realize high-order narrow-band filters; namely: follow-the-leader feedback (FLF), primary resonator block (PRB), leapfrog (LF), and modified leapfrog (MLF) techniques. The comparison is based upon the following criteria: design ease and flexibility, sensitivity performance, insensitivity to design errors due to nonideal operational amplifiers, and tunability. In all comparisons the simple but highly sensitive cascade technique is used as a reference. Several examples are cited which indicate comparative trends in sensitivity performance. Finally the computer program used to realize minimum sensitivity FLF networks is described.

General passive networks-Solvability, degeneracies, and order of complexity

Abstract: A topological expansion of the network determinant is developed which makes possible the investigation of some properties of general passive networks modeled by two-terminal elements and gyrators. Necessary and sufficient conditions are obtained for the unique solvability of the network equations. These conditions are expressed in terms of the network graph and certain gyrator-only networks, called G -networks, which are derived from the original network by removing and contracting two-terminal elements. The solvability of G -networks is also shown to be fully related to the degeneracies existing in the state-variable characterization of the network. A procedure is given for determining the actual number of state equations (order of complexity) needed for describing an arbitrarily degenerate passive network. The theoretical results obtained are illustrated with some examples.

A new class of equal-ripple filtering functions with low Q-factors: The MUCROER polynomials

Abstract: A new class of polynomials is defined as a generalization of the polynomials 1 + {\varepsilon}^{2} \cdot C_{m}^{2}(-jp) , where C_{m} indicates the Chebyshev polynomial of degree m ; these polynomials, called multiple critical root equal ripple (MUCROER), are very convneient for the approximation of RC active filter characteristics. Their behavior is equal ripple in the passband but the critical root pair (the one nearest to the j\omega axis) is multiple as in the case of the maximally flat MUCROMAF polynomials; then the polynomial degree is higher but the Q factor of the critical root pair is lower than those of the corresponding Chebyshev polynomial. Thus they allow the realization of RC active filters with more sections but with much lower sensitivity. Some design examples show the remarkable reduction of the critical Q factor.