APPROXIMATE SYMBOLIC POLE EXPRESSIONS - VALIDITY N THE DESIGN SPACE F. Constantinescu, M. Nitescu, . V. Marin, D. Marin "Politenica" UniversityBucharest, Romania Abstract Some approximate symbolic pole expressions of a fourth order circuit are computd with a relative error of 10- 3 in a design oint. It ollows that these expressions are valid or a large enough domain in the design space - at least adecadeforeach parameter. I. INTRODUCTION Some symbolic analysis programs are able to ind approximate symbolic expressions of the poles and zeros using the pole splitting technique, analytical formulas [1,2] or the roots of a polynomial up to the fourth degree, and symbolic polynomial deflation [2.3,6]. The siting approach using device parameter elimination, determinant simpliication and factorization, cancellation of factors between numerator and denominator is used in [4] or the computation of symbolic pole expressions. All these procedures are heuristic approaches i.e. they don't necessarily lead to symbolic pole/zero expressions or an arbitrary circuit. In f 5] a characteistic polynomial whose order is reduced to I or 2 is computed by approximating the matrix pencil (A,B) where the circuit equations are written as (A-sB)xO. A detailed reie of these techniques is given in [7]. In [13] the time const ant matrix is used or finding approximate pole/eroexpressions. Anoher approach allowing the determination of the approximate symbolic expressions for all poles/zeros of an arbitrary circuit is presented in [ 8]. TI1is procedure uses the LR algorithm [9] converging to the eigenvalues or a circuit matrix (the state matrix in the case of pole computation). By this way are obtained rational expressions in terms of circuit parameers for real eigenvalues and expresions using at most one square root or complex eigenalue. Even though the convergence is ensured, the computational complexity of the LR algorithm may limit the usefulness of t his netlmd [10]. To oercome this difficult a procedure or computat i on of the symbolic e xpression s or a part of the eigenvalues (belonging to a ''cluster'' in the cunplcx plan e ) only has been developed [I I]. By this way the computational complexity is reduced, the LR algl•rithm working 011 a redced matrix corresponding tll t: "non-�eparahlc" eigenvalues (belonging to the .. cluster") only. Formulae obtai1Kd with this method arc \al id in a vici n ity or the design poi nt. ll1e userutness cif po le/zero expressions is given by 1i,·1r do main of validity i n the design space. n this paper an evaluation of the symbolic ormulae validity in the design space i done. n Section I the pole/ero computation algori is presented. Section III contains an example. A discussion and some conclusions and a uture wm plan are presented in Section V. II. POL/ZERO CPUTATION he main stepsof our algorithmarc: I. Derivaion of the ate matx in symbolic fon 2. Computation of numerical eigenvalues correspondingtonominalvalofcircuitparameters. 3. Cluster identificaon 3.1. Amaximal nuber of minimum module polesare eliminated; 3.2. A maximal numer of maximum module polesareeliminated; 3.3. he remaining ples orm the "cluster". 4. Ordering of tze state matx so that la 1 d�la22\� .. -�lann\. 5. LR terations e reduced state matrix A=[ai1J eing given the LR itetions arc: A=L1R1, R1L1 =A1 , A1= L2R2. R2L2=A2, and o on ( a sequence of decompositions and muliplications). In the LR decomposition L is a lowe riangular matrix with unitarydiagonal and R isanuppertriangularmatri. The program goes out rom the LR iteration loop after the step p if the smallest modle eingenvaluc satisies anerrorcondition (to he deined in the ollowing). f ,, is real the LR iteration wi II e continued with an (11- l) • (1-I) matrix A* obtained nn· AP by removing the last row and the last column. In this case ),, = a,,,, . If ),,, is comple A* is obtained by removi n g the lat to ros and columns. In this cae A,, and ),, 1 are the eigenvalues ofthe 2 • 2 loweright -hand hlock of A1,. Considering the numerical eigenalue c,' a "exact", therelative error of the symbolic eigenvalues, is gi vcn le; - e; 1 by: where ci isa numerical value given by Jc, *I the expression of s, if the .:i:uit paameters have nominal values. Taking into account an imposed relative error E the program :ties out nm the LR iteration loop if: - a1111 - Cn '"i ---< E in the case a real s,, le" *I - - 43 ISBN 973-85072-5-1