Ahslmef-The stability and approximation properties of transfer func- tions of generalized Bessel polynomials (GBP) are investigated. Sufficient conditions are ...
325
IEEE TRANSACTIONSON CIRCUITSAND SYSTEMS,VOL. CAS-24, NO. 6, JUNE 1977
Transfer Functions of Generalized Bessel Polynomials JOSk R. MARTINEZ,
Ahslmef-The stability and approximation properties of transfer functions of generalized Bessel polynomials (GBP) are investigated. Sufficient conditions are established for the GBP to be Hmwitz. It is shown that the Pad& approximants of e-’ are related to the GBP. An infinite subset of stable Pad6 functions useful for approximating a constant time delay is defined and its approximation properties examined. The low-pass Pa& functions are compared with aa approximating function suggested by Budak. Basic limitations of Budak’s approximation are derived.
MEMBER, IEEE
parameter IX. When (Y= fi=2 the GBP reduce to the classical Bessel polynomials of circuit theory [5]. III.
HURWITZ
PROPERTY
A necessary condition for B,(s,a,j3) to be Hurwitz is that its coefficients be positive. This implies that (Y> 1- n and p >O, as can be seen from (1). This was essentially I. INTRODUCTION the condition given in [l]. It can be readily shown that the OME RECENT work has suggestedthe possibility of condition is not sufficient to guarantee the Hurwitz propusing ratios of generalized Bessel polynomials erty [6]. Theorem 1 establishes a sufficient condition for (GBP) to approximate the ideal delay function e-’ [l]. However, the question of the stabiiity of such rational B,(s,a,fi) to be Hurwitz. It is recalled that the Hurwitz functions was left unresolved since only a necessary,but character of the special case B, (s, 2,p), /? > 0, was estabnot sufficient, condition for the stability of these poly- lished long ago by Starch [5]. nomials was given. The purpose of this note is to discuss several useful Theorem 1 B, (s,(Y,~), n > 1, is Hurwitz for cx> 0, /3 >O. properties of the GBP and associated rational functions. In particular, we establish a sufficient condition for the The complete proof of Theorem 1 is found in [6]. GBP to be Hurwitz. We also show that nonminimum That the conditions of Theorem 1 are not necessaryis phase rational functions of GBP yield simultaneous maxi- immediately seen by referring to B,(s,cw,/S). Additional mally flat approximations of a certain order of both information on the zeros of the GBP for (Y
