Synthesis of low-sensitivity orthogonal digital filters

A novel method is proposed for the synthesis of low-sensitivity digital filters meeting any prescribed transfer function. The method is based on extracting an orthogonal matrix from the filter state matrix, resulting in structures that need n Givens rotations and at most n+1 multipliers. Thus the proposed realization is canonic in the sense that it has the same degrees of freedom as the original transfer function. It is also shown that when the filter transfer function is a reciprocal reactant bounded function, it can be decomposed into allpass functions that need only Givens rotations for their realization. As the basic module in the realization is the Givens rotation, the CORDIC computation algorithm can be applied directly. This means considerable savings in computation time and complexity. It also results in structures that are less sensitive to the effects of finite word length. Illustrative examples, including the design and synthesis of linear phase selective filters, are given to show the extremely low sensitivity with respect to finite word length of the resulting realizations when compared with other methods. © 1997 by John Wiley & Sons, Ltd.     

Realization of n-port resistance matrices

The problem of realization of a given real symmetric matrix R of order n as an open-circuit resistance matrix of some resistive n-port network having nullity n is considered in this paper. The solution presented is in the form of a realization technique which implies the necessary and sufficient conditions for this type of realization. Derived from fundamental topological considerations in the analysis of resistive n-port networks, the realization technique presented here is found to be superior to the existing ones.     

A new formulation and solution method for the maximum loading point in electrical power systems

This paper proposes a new efficient formulation and solution method for a maximum loading point or saddle node bifurcation point in electrical power systems. This point, corresponding to a tip of the P(Q)-V curve, is characterized by singularity of the load flow Jacobian. The proposed formulation is of dimension n + 1, instead of 2n + 1 in the standard formulation, for n-dimensional load flow equations. The proposed method uses a 1-dimensional singularity condition, obtained from a reduction of the standard n + 1-dimensional singularity conditions. For this reduction, one of the diagonal elements of the load flow Jacobian is selected. We also propose an index for this selection to make the proposed method reliable. The solution for the proposed formulation can effectively be obtained based on the Newton-Raphson method with sparse matrix techniques. The computational performance of the proposed method is demonstrated on 6, 14, and 118 bus test systems. © 1997 Scripta Technica, Inc. Electr Eng Jpn, 121(1): 17–25, 1997     

On the realization of gyrators by nullors and resistors

It is proved that for the realization of a gyrator by nullors and resistors, at least two nullors and at least two resistors are needed. A generalization is the result that any n-port network possessing both an impedance and an admittance matrix and consisting of nullors and resistors must contain at least n resistors. It is conjectured that a realization of a gyrator with only two nullors requires four resistors and a realization of a gyrator with only two resistors requires three nullors.     

Synthesis of n-port active rc networks with a minimum number of capacitors

A new and practical synthesis procedure is presented for the realization of an arbitrary n × n matrix of real rational functions of the complex frequency variable as the short-circuit admittance matrix of a transformerless active RC n-port network. The realization requires a theoretically minimum number of capacitors p, where p is the degree of the matrix, and no more than (n + p + 1) grounded finite-gain phase-inverting voltage-controlled voltage sources. All the capacitors and ports are grounded. The freedom implicit in the synthesis procedure allows the inclusion of constraints on the passive element values. Furthermore, in special cases the realization is achieved with a reduced number of conductances and voltage-controlled voltage sources. The synthesis procedure is simple to apply and can readily be implemented on a digital computer. Several examples are given.     

Realization of a general all‐pole current transfer function by using CBTA

In this article, a general all‐pole current transfer function synthesis procedure using current backward transconductance amplifiers (CBTAs) is proposed. The proposed configuration uses n current backward transconductance amplifiers and n grounded capacitors as the only type of passive elements. The circuit is eligible to realize any all‐pole transfer characteristics with a given strictly Hurwitz (stable) denominator polynomial. Further, it is straightforward to find the values of the passive elements from the coefficients of this polynomial by using the Routh–Hurwitz algorithm as in the realization of a two‐element kind passive network synthesis. In this sense and as far as the author's knowledge, it is the only active structure that can be synthesized like a passive two‐element kind Cauer circuit. The simulations that are performed using PSPICE exhibit satisfactory results coherent with the theory. Copyright © 2011 John Wiley & Sons, Ltd.     

Cascaded lattice realization of digital filters

A new algorithm is proposed to realize a digital transfer function. The final realization is in the form of cascaded lattice two-pairs with the right-most two-pair being constrained such that one of its output terminals is connected directly to an input terminal. The complete realization uses m delays, 2m + 1 multipliers and 2m two-input adders where m is the order of the transfer function. Several special cases are included, along with an example illustrating the general algorithm.     

Short-circuit admittance matrix synthesis with rc common-ground networks and a minimum number of grounded finite-gain phase-inverting voltage-controlled voltage sources

A new procedure for the synthesis of active RC networks when grounded finite-gain phase-inverting voltage-controlled voltage sources serve as active elements is developed. It is proved that an arbitrary n × n matrix of real rational functions of the complex frequency variable can be realized as the short-circuit admittance matrix of a grounded transformerless active RC n-port network containing (n+1) grounded finite-gain phase-inverting voltage-controlled voltage sources (VCVSs). In general all the (n+1) grounded VCVSs are necessary. The structure proposed to prove a general theorem is later simplified for the realization of a restricted but important class of real rational matrices to obtain considerable savings in the computation volume and in the number of passive components used for the realization of the network. Examples are given to illustrate presented synthesis procedures.     

Interpolation with positive real functions of several variables

The interpolation problem for n-dimensional positive real functions is formulated in two different settings: with interpolation knots either contained on the distinguished boundary of the Cartesian product of open right half planes or in this n-dimensional ‘open right half plane’. A complete solution of the first problem is given. Necessary conditions of existence of a solution of the second problem are proved and the well known Nevanlinna-Pick algorithm is generalized so that construction of interpolating n-dimensional positive real functions becomes possible at least in case when the interpolation knots are real vectors. Examples illustrate the suggested methods, show the complexity and significance of the problem and methods under discussion. Interpolation in special classes of functions has for some time been an important problem in mathematics. General schemes of solution in linear functional spaces with orthogonality are well known. For classes of functions which do not satisfy the axioms of a linear space special methods have been studied in detail, because they are closely connected to other important parts of mathematics such as approximation theory, numerical analysis and the theory of moments. Considerable efforts have been put into interpolation in the complex domain with functions analytic in the unit disc and with positive real functions. the significance of interpolation procedures in the class of one-variable positive real functions in network theory, e.g. in broadband matching and approximation problems has been recognized in References 1-3. an excellent survey paper has recently been devoted to these problems;⁴ in this paper some new ideas for the use of interpolation in system theory were outlined. the growing interest in multivariable network theory calls for generalizations of interpolation procedures to multivariable cases. Many of the one-variable applications can easily be extended to the multivariable case and it is hoped that in passive network theory and in problems of n-dimensional digital systems the solution of the interpolation problem will contribute to the difficult synthesis problems in the multivariable theory. In this paper conditions for the existence of multivariable positive real interpolating functions will be investigated and some procedures for the construction of such functions will be described.     

Optimal design of lattice wave digital fractional‐order Butterworth filter

This letter presents the design of digital fractional‐order Butterworth filter (DFOBF) of order (n+α) , where n is an integer, and α ∈ (0,1) , from the perspective of optimal realization. The magnitude–frequency characteristic of the DFOBF is optimally modeled using the computationally efficient lattice wave digital filters (LWDFs). Design examples for the third‐ and fifth‐order LWDF‐DFOBFs with various values of n, α_, and cut‐off frequencies are presented.     

Biquadratic n-dimensional impedances

A discussion of n-dimensional quadratic Hurwitz polynomials and biquadratic positive real functions is given. The formulated necessary and sufficient conditions are easily verifiable and make the discussion of the influence of coefficient changes possible. Some more general results and examples are included.     

Solving piecewise-linear equations for resistive networks

Nonlinear resistive networks can be characterized by the equation f(x) = y where f (x) is a continuous piecewise-linear mapping of Rⁿ into itself. The n-dimensional Euclidean space is divided into a finite number of regions, and, in each region say region R_m, we can express f by J ^(m) x + w ^(m) where J ^(m) is a constant n × n Jacobian matrix and w ^(m) is a constant n-vector. In this paper we obtain the following results: If all the Jacobian determinants in the unbounded regions have the same sign, the equation f(x)= y has at least one solution and an algorithm is developed, which obtains one or more solutions in a finite number of steps. The work represents a generalization of early work by Fujisawa, Kuh and Ohtsuki and relaxes the condition imposed on the function. For example, in the bounded regions, the Jacobian matrices can be singular and the sign of Jacobian determinants can be arbitrary.     

New theorems in the synthesis of passive multiplexers

Coupling networks that divide the frequency spectrum into separate frequency bands and also provide desired filtering are known as multiplexers. In this paper we present four new theorems concerning the synthesis-theoretic structure of lumped passive multiplexers constructed as lossless reciprocal n-ports. In particular we prove in Theorem 1 that any set of n – 1 admissible power gains, complementary or otherwise, can be realized by a series interconnection of n–1 appropriately chosen non-Foster positive-real functions. In Theorems 2–4 we establish various necessary and sufficient conditions for the realization of bounded-real gains s₂₁(p),…, s_n1(p) given both in magnitude and phase. These conditions reveal that the general problem is one of Nevalinna-Pick interpolation. Lastly, five worked examples are used to illustrate all facets of the theory.     

Linear bipolar OTAs employing exponential‐law circuits

This paper presents design of linear bipolar OTAs, which are composed of two function blocks; one is an exponential‐law circuit and the other is a core cell. Multi‐tanh cells are employed as the core cell. This kind of OTA has lower power dissipation relatively to the conventional multi‐tanh cell. According as the order of the multi‐tanh core cell becomes higher, the number of circuit realization for the core cell increases. For example, we have two OTAs for the core circuit of an emitter‐coupled pair and four OTAs for the doublet core cell. Thus, we consider the generalized OTAs for an arbitrary order n of the core cell and obtain a formula to give the realization number of the linear OTAs for n. According to the formula, there must be eight OTAs in the case of n=3. All of the eight OTAs are examined. Analysis and simulation results show that the OTAs have advantage in their characteristics, such as linear input range, power dissipation, noise, and frequency response. Copyright 2004 John Wiley & Sons, Ltd.     

On the number of conductances required for realizing Y and K matrices

Upper bounds are established on the number of conductances required for realizing a real symmetric matrix Y as the short-circuit conductance matrix of a resistive n-port network containing no negative conductances, and for the realization of a real matrix K as the potential factor matrix of a similar network without negative conductances. These results are the consequence of the properties of the modified cut-set matrix of an n-port and a theorem in the theory of linear programming.     

Digital architectures realizing piecewise‐linear multivariate functions: Two FPGA implementations

Digital architectures for the circuit realization of multivariate piecewise‐linear (PWL) functions are reviewed and compared. The output of the circuits is a digital word representing the value of the PWL function at the n‐dimensional input. In particular, we propose two architectures with different levels of parallelism/complexity. PWL functions with n = 3 inputs are implemented on an FPGA and experimental results are shown. The accuracy in the representation of PWL functions is tested through three benchmark examples, two concerning three‐variate static functions and one concerning a dynamical control system defined by a bi‐variate PWL function. Copyright © 2009 John Wiley & Sons, Ltd.     

A class of nonrealizable paramount matrices

Although paramountcy¹ is a necessary and sufficient condition for the realizability of a 3 by 3 matrix as either an open-circuit impedance matrix or a short-circuit admittance matrix of a resistive 3-port,² Weinberg's 4 by 4 example^3,5 shows that paramountcy, although necessary for open-circuit impedance realization, is not sufficient. Here we exhibit a class of 4 by 4 completely irreducible paramount matrices, including Weinberg's example, no member of which is realizable as an open-circuit impedance matrix. This is accomplished by examining those topological constraints inherited by a realizing network when algebraic conditions are imposed on the paramount matrix which the network realizes. Our development deals with an n-port network, for arbitrary n, and also provides new criteria guaranteeing nonrealizability of an irreducible¹ paramount matrix.     

Transfer matrix realization using RC:GIC Networks

A general synthesis procedure for realizing any stable voltage transfer matrix using current-conversion type generalized impedance converter (GIC)s is outlined. Each row of the transfer matrix is realized separately as an M-input, single-output grounded RC:GIC network where M is the number of columns in the transfer matrix. The realization procedure is illustrated by an example. Using this approach, a second-order filter has been obtained capable of realizing simultaneously highpass, band-pass and low-pass transfer functions and using fewer passive components and operational amplifiers than the popular state-variable realizations. The various sensitivities of the second-order realizations are low and comparable to the state-variable realizations. A similar method can be developed for current transfer matrix invoking duality and thus using the voltage-conversion type GICs.     

The jaumann structure in wave-digital filters

This paper deals with an extension of the wave-digital filter structures introduced by Fettweis. A new structure is introduced, viz, the digital translation of the Jaumann structure. The Jaumann structure is a well-known canonical realization of analogue, symmetric, grounded lossless two-port networks. With this structure, transfer functions with complex transmission zeros can be obtained in a wave-digital realization, which is not possible by means of a wave-digital ladder filter. Two examples are given, one of which has been implemented using TTL digital integrated circuits showing excellent properties.