Guaranteed passive parameterized macromodeling by using Sylvester state-space realizations

A novel state-space realization for parameterized macromodeling is proposed in this paper. A judicious choice of the state-space realization is required in order to account for the assumed smoothness of the state-space matrices with respect to the design parameters. This technique is used in combination with suitable interpolation schemes to interpolate a set of state-space matrices, and hence the poles and residues indirectly, in order to build accurate parameterized macromodels. The key points of the novel state-space realizations are the choice of a proper pivot matrix and a well-conditioned solution of a Sylvester equation. Stability and passivity are guaranteed by construction over the design space of interest. Pertinent numerical examples validate the proposed Sylvester realization for parameterized macromodeling

Order bound for the realization of a combination of positive filters

In a problem on the realization of digital ¯lters, initiated by Gersho and Gopinath [8], we extend and complete a remarkable result of Benvenuti, Farina and Anderson [4] on decomposing the transfer function t(z) of an arbitrary linear, asymptotically stable, discrete, time-invariant SISO system as a di®erence t(z) = t1(z) ¡ t2(z) of two positive, asymptotically stable linear systems. We give an easy-to-compute algorithm to handle the general problem, in particular, also the case of transfer functions t(z) with multiple poles, which was left open in [4]. One of the appearing positive, asymptotically stable systems is always 1-dimensional, while the other has dimension depending on the order and, in the case of nonreal poles, also on the location of the poles of t(z). The appearing dimension is seen to be minimal in some cases and it can always be calculated before carrying out the realization

An efficient algorithm for positive realizations

We observe that successive applications of known results from the theory of positive systems lead to an {\it efficient general algorithm} for positive realizations of transfer functions. We give two examples to illustrate the algorithm, one of which complements an earlier result of \cite{large}. Finally, we improve a lower-bound of \cite{mn2} to indicate that the algorithm is indeed efficient in general

A Unifying Framework for Finite Wordlength Realizations.

A general framework for the analysis of the finite wordlength (FWL) effects of linear time-invariant digital filter implementations is proposed. By means of a special implicit system description, all realization forms can be described. An algebraic characterization of the equivalent classes is provided, which enables a search for realizations that minimize the FWL effects to be made. Two suitable FWL coefficient sensitivity measures are proposed for use within the framework, these being a transfer function sensitivity measure and a pole sensitivity measure. An illustrative example is presented

Circulant and skew-circulant matrices as new normal-form realization of IIR digital filters

Normal-form fixed-point state-space realization of IIR (infinite-impulse response) filters are known to be free from both overflow oscillations and roundoff limit cycles, provided magnitude truncation arithmetic is used together with two's-complement overflow features. Two normal-form realizations are derived that utilize circulant and skew-circulant matrices as their state transition matrices. The advantage of these realizations is that the A-matrix has only N (rather than N2) distinct elements and is amenable to efficient memory-oriented implementation. The problem of scaling the internal signals in these structures is addressed, and it is shown that an approximate solution can be obtained through a numerical optimization method. Several numerical examples are included

On a New Characterization of Linear Passive Systems

Characterization of linear passive system

Minimal positive realizations of transfer functions with nonnegative multiple poles

This note concerns a particular case of the minimality problem in positive system theory. A standard result in linear system theory states that any nth-order rational transfer function of a discrete time-invariant linear single-input-single-output (SISO) system admits a realization of order n. In some applications, however, one is restricted to realizations with nonnegative entries (i.e., a positive system), and it is known that this restriction may force the order N of realizations to be strictly larger than n. A general solution to the minimality problem (i.e., determining the smallest possible value of N) is not known. In this note, we consider the case of transfer functions with nonnegative multiple poles, and give sufficient conditions for the existence of positive realizations of order N = n. With the help of our results we also give an improvement of an existing result in positive system theory

Minimal symmetric Darlington synthesis

We consider the symmetric Darlington synthesis of a p x p rational symmetric Schur function S with the constraint that the extension is of size 2p x 2p. Under the assumption that S is strictly contractive in at least one point of the imaginary axis, we determine the minimal McMillan degree of the extension. In particular, we show that it is generically given by the number of zeros of odd multiplicity of I-SS*. A constructive characterization of all such extensions is provided in terms of a symmetric realization of S and of the outer spectral factor of I-SS*. The authors's motivation for the problem stems from Surface Acoustic Wave filters where physical constraints on the electro-acoustic scattering matrix naturally raise this mathematical issue

Positive decomposition of transfer functions with multiple poles

We present new results on decomposing the transfer function t(z) of a linear, asymptotically stable, discrete-time SISO system as a difference t(z) = t(1)(z) - t(2)(z) of two positive linear systems. We extend the results of [4] to a class of transfer functions t(z) with multiple poles. One of the appearing positive systems is always 1-dimensional, while the other has dimension corresponding to the location and order of the poles of t(z). Recently, in [11], a universal approach was found, providing a decomposition for any asymptotically stable t(z). Our approach here gives lower dimensions than [11] in certain cases but, unfortunately, at present it can only be applied to a relatively small class of transfer functions, and it does not yield a general algorithm