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A direct procedure based on using a variant partial-fraction expansion (VPFE) is presented for finding a canonical minimal realization of cascade-parallel structure for 2-D transfer functions having a separable denominator. A similarity transforming matrix that transforms a 2-D separable-denominator state-space model from a phase-variable canonical form to the VPFE canonical form is presented. A numerical example is provided to illustrate the procedure. >

A practical algorithm for computing the partial fraction expansion of a general rational function without derivative evaluations is given. In addition, it is shown how to reduce the roundoff error in its implementation. The procedure is rather straightforward, accurate, efficient, and easy to program. >

This paper considers the multiplicative and additive minimum-phase decompositions of a two-dimensional (2-D) rational power density spectrum (commonly known as the spectral factorization and partial fraction expansion problems). It is shown that both of these decompositions generally involve filters which are ratios of 2-D minimum-phase filters, with finite support in one variable, and infinite support in the other variable. It is shown that a sufficient condition for a 2-D autoregressive (AR) spectrum to have a rational additive decomposition is for the minimum-phase whitening filter to have finite reflection coefficient support.

A method is presented for computing a minimal J-form realization (A, B, C) of a transfer function matrix from its partial fraction expansion (PFE). The decoupling inherent in the Jordan form enables the computational algorithm for the general multipole case to be developed by considering the case of a transfer function matrix having one pole of multiplicity alpha . After developing matrix equations relating the column of C and rows of B to the fraction expansion coefficient, a single sweep method which utilizes orthogonal matrices generated sequentially by singular value decomposition (SVD) is developed for determining the column of C, rows of B and the structure of the J-form for A. At the ith state in the sequence, the matrix requiring SVD is composed from PFE coefficient matrices and matrices generated by previous SVDs. Other computational aspects of the proposed method are presented and discussed. >

Several techniques are presented to obtain ladder and parallel structures for block filters based on continued and block partial fraction expansions of a block filter transfer function matrix. These techniques are useful where a block length that is a greater than the order of the filter is not required. Some of these structures are canonic with respect to both vector delays and matrix multipliers and very attractive in multiplier range standpoint. Illustrative examples are provided.

A new O(N^2) algorithm for partial fraction expansion with multiple poles is presented, which is simpler and somewhat faster than the known methods. A short and general WATFIV program is included.

A novel method is presented to determine the SmithMacmillan form of a rational m \times n matrix R(p) from Laurent expansions in its poles and zeros. Based on that method, a numerically stable algorithm is deduced, which uses only a minimal number of terms of the Laurent expansion, hence providing a shortcut with respect to cumbersome and unstable procedures based on elementary transformations with unimodular matrices. The method can be viewed as a generalization of Kublanovkaya's algorithm for the complete solution of the eigenstructre problem for \lambda I - A . From a system's point of view it provides a handy and numerically stable way to determine the degree of a zero of a transfer function and unifies a number of results from multivariable realization and invertibility theory. The paper presents a systematic treatment of the relation between the eigen-information of a transfer function and the information contained in partial fraction or Laurent expansions. Although a number of results are known, they are presented in a systematic way which considerably simplifies the total picture and introduces in a natural way a number of novel techniques.

This paper gives the outline of a new method for the calculation of electromagnetic transients on transmission lines above homogeneous ground. The basic idea is that line transfer functions which normally are calculated in terms of hyperbolic functions of \omega are now expressed in terms of s = \sigma + j \omega . This is possible due to the existence of analytical expressions of line impedances and admittances in terms of j \omega , or s in the new approach. It is then possible to find the poles of the transfer function by the secant method and then the corresponding residues. The partial fraction approximation of the transfer functions is incomplete and a correction may be used to account for the missing poles. The calculations are performed numerically only for samples of s needed for the calculation of poles and residues. The end result is a set of state equations with \upsilon and i as inputs/outputs at the switch terminals. Inputs and outputs are interchanged as the switch poles open or close. The integration can be numerical or closed form for sinusoidal inputs (generator voltages) and the state variables carry the past information from one stage of switching to the next. Essential parts of the procedures are illustrated by numerical results.

The realization of a digital transfer function using a variation of partial fraction is accomplished. By this method, the obtained network features a new cascade network structure. Further, this paper proves relationships between coefficients of the digital transfer function and the values of the network elements.

Brugia's work on the noniterative computation of high order derivatives of rational functions with application to multiple-pole fraction expansion is simplified by the use of operators. The formulas for the derivatives are given in a recursive scheme and finally the Laurent expansion is developed and applied to the partial fraction expansion of improper rational functions.

A new method is suggested for solving the polynomial equation A(s)X(s)+B(s)Y(s)=C(s) where A (s), B (s), C (s) are given polynomials and X (s), Y (s) are to be determined. The method is then used for the formulation of the partial-fraction expansion of transfer functions, and for computing the greatest common divisor of polynomials

An analysis is presented of the class of rational functions in two variables realized by two-dimensional state-space models satisfying the commutativity assumption A/sub 1/A/sub 2/=A/sub 2/A/sub 1/. A complete characterization of these transfer functions is given in terms of the existence of a suitable partial fraction expansion. It is shown that, under the commutativity assumption, minimal realizations are unique modulo algebraic equivalence, and the minimality is equivalent to local observability and reachability. >

Application of the impulse invariance technique to z -transformation of certain Laplace transfer functions can be difficult if the function is not reduced to a constituent partial fraction form. This may be overcome by use of the z -form method, originally proposed by Boxer and Thaler [1]. However, the technique often requires a substantial amount of algebraic manipulation. Q -matrix methods are widely employed for such applications involving the bilinear transformation and substantially reduce the required amount of manipulation. This paper presents a general method for the formation and application of a z -form Q -matrix.

In this note it is shown that the 2-D composition \frac{A(z_1,z_2)A(z_{1}^{-1},z_{2}^{-1})} {B(z_1,z_2)B(z_{1}^{-1},z_{2}^{-1})} = \frac{P(z_1,z_2)}{B(z_1,z_2)} + \frac{P(z_{1}^{-1},z_{2}^{-1})}{B(z_{1}^{-1},z_{2}^{-1})} contrary to the 1-D case, does not exist in general. Conditions for the existence of such a decomposition are obtained and applied to two examples.

The minimal roundoff noise in fixed point digital filters is determined by a certain determinant, generally denoted by det (KW)^{1/2} . This determinant may be expressed by the poles and zeros of the filter transfer function H(z). This paper presents a simple and direct derivation of this expression, based upon an expansion of H(z) in partial fractions of the first order.

Based on a variation of partial fraction expansion of the digital transfer function, an algorithm, more efficient than that suggested by Ishii [1], for calculating the values of network elements is suggested.

It is shown that there is a possibility, at least in some cases, for the decomposition given in [1] to exist for N > 1 , as it is when N = 1 , where N is the degree of the 2-D polynomials A(z_{1}, z_{2}),B(z_{1}, z_{2}) and P(z_{1}, z_{2}) considered in [1].