Precise Algorithms for Pole-Zero Analysis in …

1 Precise Algorithms for Pole-Zero Analysis inElectronic circuit DesignJosef Dobe s , Dalibor Biolek , Jan M chal , David Cern y , and Libor Sl ama Czech Technical University in Prague, Department of Radio Engineering, Technick a 2, 16627 Praha 6, Czech Republic Brno University of Technology, Department of Microelectronics, Udoln 53, 60200 Brno, Czech RepublicAbstract The Pole-Zero Analysis is generally known to be verysensitive to the numerical precision of the computer arithmetics. In thepaper, various methods are suggested for solving that problem. First,an optimal pivoting strategy of the algorithm that reduces the generaleigenvalue problem to the standard one is presented for both full-and sparse-matrix procedures.

2 The algorithm increases the precisionof the semisymbolic Analysis , especially for the large-scale radio-frequency circuits. A novel technique is also incorporated recognizingmultiple poles or zeros, which are often computed inaccurately bystandard Algorithms . A new type of this procedure called secondaryroot polishing is described in the paper. The accuracy is furthermoreincreased using longer numerical data. First, the long double precisionis utilized. Further, a novel application of a suitable multiple-precisionarithmetic library is suggested. Finally, using the longer numericaldata to eliminate possible imprecision of the multiple eigenvaluesis evaluated. The algorithm is demonstrated in both low- and high-frequency domains.

3 In the low-frequency domain, necessity of usingthe longer numerical data is demonstrated by a power operationalamplifier with poles and zeros located in both hertz and gigahertzranges, which are often computed inaccurately by the standard algo-rithms. In the high-frequency domain, the algorithm is demonstratedby estimating the frequency of a distributed microwave oscillator, andby estimating the bandwidth of a distributed microwave Eigenvalues and eigenfunctions, poles and zeros, res-onant or large-scale electronic circuits, sparse matrices, variable-length arithmetic, cascade filters, transmission lines, power amplifiers,distributed microwave amplifiers, distributed microwave INTRODUCTIONTHE Pole-Zero Analysis belongs indisputably to themost important parts of the design of electronic , the Analysis is known to be very sensitive to thenumerical precision of algebraic operations during the pro-cess [1] [6].

4 Consequently, many of the theoretically exactmethods fail, especially for the large-scale radio-frequency andmicrowave major types of improvements to these methods areproposed here: the first one consists in a meticulous algorithm designregarding the choice of pivots during the reduction to thestandard eigenvalue problem, the second one suggests the primary or secondary rootpolishing for a substantial improvement of the results ofboth reduction and QR algorithm, the third one is based on using longer numerical datatypes together with more Precise arithmetic for the sparse-matrix reduction either as fully utilizing the givenhardware capabilities or by applying a suitable multiple-precision software arithmetic library,This paper has been supported by the Grant Agency of the Czech Republic,grant No102/08 1.

5 Final position of the nonzero matrix elements after the reduction. ThematricesP11andQ22are diagonal, the matrixQ11is of arbitrary structure. and the fourth one suggests the longer numerical datato be used for the QR algorithm to eliminate expectedimprecision of the multiple eigenvalues (the imprecisionof unequal eigenvalues with the same absolute value isresolved using the shift of algorithm s origin).II. PRINCIPLE OF THEREDUCTIONALGORITHMA system of linear circuit equations (or equations that arelinearized around an operating point) can be written by meansof the Laplace transformation(sP+Q)X=Y,(1)wheresmarks the Laplace operator,P/Qare the matrices as-sociated with the dynamic/static parts of the model derivatives,respectively,Xis the vector of the Laplace representation ofcircuit variables, andYis the vector of external poles of all the transfer functions and the zeros of atransfer function jthcircuit variable divided byithexternalsource can be computed by solving the equationsdet(sP+Q)= 0for poles,det(sP(0j)+Q(ij))= 0for zeros,(2)where the matricesP(0j)andQ(ij)

6 Arise from the originalones the first by zeroingjthcolumn, and the second byreplacingjthcolumn byYwith all its elements zeroed withthe exception of the one corresponding the general eigenvalue problems defined by (2) ismore difficult than solving the standard ones. Therefore, asystematic reduction is applied during the transformation of(2) to the standard form, which is shown in Fig. 1 it is analternative of the method in [7]. After the transformation, theINTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSINGI ssue 3, Volume 5, 2011228determinant can be computed in a classical way:det(sP+Q)=( 1)nexchn i=m+1Q22iidet(P11P 111(sP11+Q11))=( 1)nexchm i=1P11iin i=m+1Q22iidet(s1+P 111Q11),(3)wherenexchis the total number of row and column exchangesduring the reduction, and1is the unity matrix.

7 The operationsthat transform the matrixsP+Qto the form shown in Fig. 1are certain modifications of the Gauss elimination only exception occurs when the matrixP11contains anondiagonal element that is not reducible by the diagonalelements of this matrix. In such cases, it is necessary tomultiply a row from the lower part of the matrix by thesoperator, which is equivalent to moving a row of theQ22matrix left. The nondiagonal element of theP11matrix canthen be reduced by means of the transferred row. Note that thetwo products in the equation (3) could be extremely big forthe large-scale RF circuits and, therefore, only their signs andlogarithms may be stored in the computer that for a general efficiency of the reduction algorithm(3), utilizing matrix sparsities is necessary.

8 Various methodsof exploiting the sparsity of matrices are described in [7] [9].The final step for determining the poles or zeros of the trans-fer function is naturally the computation of the eigenvalues ofthe matrix ( , solving the standard eigenvalue problem, forwhich the QR algorithm with a shift of origin is mostly used)Q = P 111Q11.(4)III. ENHANCING THEACCURACY OF THEREDUCTIONA. Optimal Pivoting for the Reduction AlgorithmsThe reduction process is very difficult from the point of viewof numerical precision, especially in the case of the large-scalesystems. The matrices often contain elements of very differentmagnitudes. Therefore, afull pivotingshould be used for thechoice of thekthkey element:Pkk:= maxk6i6nk6j6n|Pij|, k= 1.

9 , n.(5)However, the key element determined in the above stated wayis regarded to be zero if it is too small in comparison with themaximum element of thekthcolumn:if|Pkk|6 eigenmax16i<k|Pik|, thenPkk:= 0.(6) eigenis an important parameter of the algorithm. An inap-propriately large value of the parameter causes ignoring some(real, in fact) poles or zeros, inappropriately small value causescomputing superfluous (spurious) poles or key elements determined in the upper and lower parts ofthe matrices are used for the reduction of remaining elementsof the matrices if they are not too small:if|Pi j |6 roundmaxk6i6nk6j6n|Pij|, thenPi j := 0,k= 1, .. , n 1, i >k j >k,(7a) 2. Cascade of two 2nd-order building blocks. The application ofsecondary root polishing is necessary for Precise |Qi j |6 roundmaxm<i6km<j6k|Qij|, thenQi j := 0,k=n.

10 , m+ 1, m < i 6k m < j 6k. (7b) roundis a second parameter of the algorithm. It prevents thereduction of tiny elements that can arise due to roundoff reduction algorithm should be implemented to use thesparsity of matricesPandQ. These matrices are sparseenough for not too complicated tasks. However, the applicationof the full pivoting is then difficult from the programming pointof view. Therefore, onlypartial pivotingmust be used here thekthkey element is chosen from the rest of thekthcolumnof a reduced matrix:Pkk:= maxk6i6n|Pik|, k= 1, .. , n.(8)However, the key element determined in the above stated wayis regarded to be zero if it is too small in comparison with themaximum element of thekthrow:if|Pkk|6 maxk<j6n|Pkj|, thenPkk.