E. Chiprout and M. S. Nakhla. Asymptotic Waveform Evaluation and Moment Matching for Interconnect Analysis. Kluwer Academic Publishers, Boston, MA, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to construct Pad e approximants in the area of control in the early 1970s. Extensions of these techniques to multiple interpolation points followed [42, 43, 44] Of more recent interest, circa 1990, is a class of explicit moment matching methods known as asymptotic waveform evaluation (AWE) [45, 46]. Although the AWE methods themselves vary little in basic concept from the earlier control implementations, the AWE techniques are applied for interconnect model reduction in the area of circuits. The methods received attention for their ability to reduce RC interconnect models involving tens of ....

....the AWE techniques are applied for interconnect model reduction in the area of circuits. The methods received attention for their ability to reduce RC interconnect models involving tens of thousands of variables. A multipoint version of AWE, complex frequency hopping (CFH) is available as well [46]. Unfortunately, all of these explicit moment matching methods are known to exhibit numerical instabilities, particularly as the dimension of the reduced order model M grows. The source of these difficulties was pointed out in [47] and in the independent work of [48] Both efforts point out that ....

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E. Chiprout and M. S. Nakhla, Asymptotic Waveform Evaluation and Moment Matching for Interconnect Analysis. Boston, MA: Kluwer Academic Publishers, 1994.

....construct Pad e approximants in the area of control in the early 1970s. 11] Extensions of these techniques to multiple interpolation points followed . 12, 13, 14] Of more recent interest, circa 1990, is a class of explicit moment matching methods known as asymptotic waveform evaluation (AWE) [15, 16] Although the AWE methods themselves vary little in basic concept from the earlier control implementations, the AWE techniques are applied to interconnect model reduction in the area of circuits. The methods received attention for their ability to reduce RC interconnect models involving tens of ....

....the AWE techniques are applied to interconnect model reduction in the area of circuits. The methods received attention for their ability to reduce RC interconnect models involving tens of thousands of variables. A multipoint version of AWE, complex frequency hopping (CFH) is available as well. [16] Unfortunately, all of these explicit moment matching methods are known to exhibit numerical instabilities, particularly as the dimension of the reducedorder model M grows. The source of these difficulties was pointed out in [17] and in the independent work of [18] The reader is referred to ....

E. Chiprout and M. S. Nakhla. Asymptotic Waveform Evaluation and Moment Matching for Interconnect Analysis. Kluwer Academic Publish14 ers, Boston, MA, 1994.

....current interpolation point, A6) evaluate the residual function. A7) choose new interpolation point where residual function is maximized. A8) Increment m. not a new one, it was mentioned, for example, that others monitor the change in the eigenvalues of the models as the iterations progress [4, 6]. What is new in algorithm 1 is the manner in which a distinction is made between true convergence and stagnated convergence between consecutive models. If ffl m is small for several consecutive values of m, is the modeling error truly small (ffl m (s) 0) or has convergence temporarily ....

....has received a large amount of attention as an approach for speeding the analysis of high frequency circuits. A well publicized algorithm known as Asymptotic Waveform Evaluation (AWE) implements interpolation about a single point [15] although extensions of AWE to multiple points are available [4]. The AWE type algorithm for rational interpolation is a numerically poor one though compared with the Lanczos based algorithms [9, 6] In this section we apply our adaptive rational Lanczos algorithm (algorithm 2) to a relatively well known AWE test problem, see [4, 6] This problem arises from a ....

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E. Chiprout and M. S. Nakhla. Asymptotic Waveform Evaluation and Moment Matching for Interconnect Analysis. Kluwer Academic Publishers, Boston, MA, 1994.

....it improved AWE by exploiting the treelike structure of most interconnect circuits. However, in RICE, transmission lines are modeled as a large number of RLGC sections, which is neither exact nor efficient in computation. Moment computation models of transmission lines have been presented in [23, 24, 26]. In [23] the moment models are formed either by recursively solving second order differential equations or by computing matrix exponentials (even for a single line) which is not very efficient in computation. In [24,30] a comprehensive treatment of the problem of interconnect analysis using ....

....Moment computation models of transmission lines have been presented in [23, 24, 26] In [23] the moment models are formed either by recursively solving second order differential equations or by computing matrix exponentials (even for a single line) which is not very efficient in computation. In [24,30], a comprehensive treatment of the problem of interconnect analysis using complex frequency hopping is given. The work is based on the transmission matrix (ABCD matrix) of transmission lines. A multipoint moment matching technique is introduced and an algorithm for generating higher order moments ....

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E.Chiprout and M.S.Nakhla, "Asymptotic waveform evaluation and moment matching for interconnect analysis," Chap. 3. Kluwer Academic Publishers, 1994.

.... 7 7 7 5 : 15) The poles p j of H q in (9) are then obtained as the roots of the equation a q oe q a q Gamma1 oe q Gamma1 Delta Delta Delta a 1 oe 1 = 0: 16) Finally, the constant k 1 and the residues k j in (9) are computed by solving another linear system of order q, see, e.g. [10] for details. 6 Peter Feldmann and Roland W. Freund 10 0 10 2 10 4 10 6 10 8 10 10 250 200 150 100 50 0 50 Frequency (Hz) Exact AWE, 2 iter. AWE, 5 iter. AWE, 8 iter. Figure 1: Results for simulation of voltage gain with AWE As q is increased, one would expect more and more ....

....fim 2q Gamma1 fi fi fi 1 A 1= 2q Gamma1) 18) The first choice in (18) corresponds to scaling the matrix A in (11) to have Euclidean norm 1. The second choice, proposed in [2] is such that the two first scaled moments, m 0 and 2 m 1 , have the same magnitude. The third choice, suggested in [10], is such that, after scaling, the first and the last computed moment, m 0 and 2q Gamma1 3 m 2q Gamma1 , have the same magnitude. While scaling reduces the ill conditioning somewhat, the scaled moment matrices M (l) q = j k Gamma2 l m j k Gamma2 ] j;k=1;2; q ; l = 1; 2; 3; 19) for ....

E. Chiprout and M.S. Nakhla. Asymptotic Waveform Evaluation and Moment Matching for Interconnect Analysis. Kluwer Academic Publishers, Norwell, MA, 1994.

....large linear circuits. This was first demonstrated by Pillage and Rohrer [6] with their asymptotic 6 Roland W. Freund and Peter Feldmann waveform evaluation (AWE) approach that is based on the computation of the Pad e approximant H q ; for a detailed description of AWE, we refer the reader to [7, 8] and the references given there. In AWE, the Pad e approximant H q is directly generated from the moments (13) Unfortunately, this procedure is inherently numerically unstable, as we illustrated in [1] In fact, this is the reason why, in practice, AWE can be used only for fairly moderate values ....

E. Chiprout and M.S. Nakhla. Asymptotic Waveform Evaluation and Moment Matching for Interconnect Analysis. Kluwer Academic Publishers, Norwell, MA, 1994.

....the problem has yet to converge guarantees that new information is added. That is, one places information into the projection matrices by repeated changes of oe (k) rather than repeated multiplications. This approach is similar in spirit to the complex frequency hopping improvements of AWE [20]. As in CFH, a successful RP implementation tends to require factors of (A Gamma sE) at numerous points. Unlike CFH, the projection technique of the RP algorithm generates a single reduced order model that is a rational interpolant. A projection approach may also enable the use of techniques ....

E. Chiprout and M. S. Nakhla, Asymptotic Waveform Evaluation and Moment Matching for Interconnect Analysis. Boston, MA: Kluwer Academic Publishers, 1994.

....This paper explores the use of Lanczos techniques for the reduced order modeling of large scale dynamical systems. A need for such reduced order models arises in various areas of engineering such as the control of large flexible space structures [5] and the simulation of high speed circuits [6]. The system to be modeled is typically defined via a set of state space equations E x(t) Ax(t) bu(t) and y(t) c T x(t) du(t) 1) where for simplicity, the direct coupling term, d, will be assumed to be zero. As this paper will restrict itself to single input single output (SISO) ....

....becoming an acceptable global approximation of the original system. To overcome this difficulty, several papers in the areas of control and circuits explore the use of a multi point Pad e approximant (denoted a rational interpolant in the systems literature) for approximating (1) see for example [6, 16, 18, 32]) In rational interpolation [1] multi point Pad e [2] a reduced order model is constructed whose transfer function g(s) interpolates the value and subsequent derivatives of g(s) at multiple frequencies foe 1 ; oe 2 ; oe g. Each interpolation point is selected to identify the ....

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E. Chiprout and M. S. Nakhla, Asymptotic Waveform Evaluation and Moment Matching for Interconnect Analysis, Boston, MA: Kluwer Academic Publishers, 1994.

....of dominant poles. Efficient methods of this type can be obtained by using Pad e approximations of the frequency response. This was first demonstrated by Pillage and Rohrer [6] with their asymptotic waveform evaluation (AWE) approach; for a detailed description of AWE, we refer the reader to [7, 8] and the references given there. The basic idea of Pad e approximation of the frequency response (7) is as follows. Let s 0 2 C be an arbitrary, but fixed expansion point such that the matrix G s 0 C is nonsingular. Here, G and C are the matrices from the small signal linear system (5) We now ....

E. Chiprout and M. S. Nakhla, Asymptotic Waveform Evaluation and Moment Matching for Interconnect Analysis. Norwell, MA: Kluwer Academic Publishers, 1994.

....This paper explores the use of Lanczos techniques for the reduced order modeling of large scale dynamical systems. A need for such reduced order models arises in various areas of engineering such as the control of large flexible space structures [5] and the simulation of high speed circuits [6]. The system to be modeled is typically defined via a set of state space equations E x(t) Ax(t) bu(t) and y(t) c T x(t) du(t) 1) where for simplicity, the direct coupling term, d, will be assumed to be zero. As this paper will restrict itself to single input single output (SISO) ....

....becoming an acceptable global approximation of the original system. To overcome this difficulty, several papers in the areas of control and circuits explore the use of a multi point Pad e approximant (denoted a rational interpolant in the systems literature) for approximating (1) see for example [6, 15, 17, 30]) In rational interpolation [1] multi point Pad e [2] a reduced order model is constructed whose transfer function g(s) interpolates the value and subsequent derivatives of g(s) at multiple frequencies foe 1 ; oe 2 ; oe g. Each interpolation point is selected to identify the ....

[Article contains additional citation context not shown here]

E. Chiprout and M. S. Nakhla, Asymptotic Waveform Evaluation and Moment Matching for Interconnect Analysis, Boston, MA: Kluwer Academic Publishers, 1994.

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E. Chiprout and M. S. Nakhla. Asymptotic Waveform Evaluation and Moment Matching for Interconnect Analysis. Kluwer Academic Publishers, Boston, MA, 1994.

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E.Chiprout and M.S.Nakhla, Asymptotic Waveform Evaluation and Moment Matching for Interconnect Analysis. Norwell,MA:Kluwer Academic Publishers, 1994

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Eli Chiprout and Michel S. Nakhla, Asymptotic Waveform Evaluation and Moment Matching for Interconnect Analysis, Kluwer Academic Publishers, 1994.

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E. Chiprout and M.S. Nakhla "Asymptotic Waveform Evaluation and Moment Matching for Interconnect Analysis", Kluwer Academic Publishers, 1994 152

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E. Chiprout and M. S. Nakhla, Asymptotic Waveform Evaluation and Moment Matching for Interconnect Analysis. Boston, MA: Kluwer, 1994.

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