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Chapter 7 Reading on Moment Calculation

Chapter 7 Reading on Moment Calculation. Time Moments of Impulse Response h ( t ). Definition of moments. i -th moment. Note that m 1 = Elmore delay when h ( t ) is monotone voltage response of impulse input. Pade Approximation .

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Chapter 7 Reading on Moment Calculation

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  1. Chapter 7 Reading onMoment Calculation

  2. Time Moments of Impulse Response h(t) • Definition of moments i-th moment • Note that m1 = Elmore delay when h(t) is monotone voltage response of impulse input

  3. Pade Approximation • H(s) can be modeled by Pade approximation of type (p/q): • where q < p << N • Or modeled by q-th Pade approximation (q << N): • Formulate 2q constraints by matching 2q moments to compute ki’s & pi’s

  4. (ii) General Moment Matching Technique • Basic idea: match the moments m-(2q-r), …, m-1, m0, m1, …, mr-1 • When r = 2q-1: (i) initial condition matches, i.e.

  5. Compute Residues & Poles EQ1 match first 2q-1 moments

  6. Basic Steps for Moment Matching Step 1: Compute 2q moments m-1, m0, m1, …, m(2q-2) of H(s) Step 2: Solve 2q non-linear equations of EQ1 to get Step 3: Get approximate waveform Step 4: Increase q and repeat 1-4, if necessary, for better accuracy

  7. Components of Moment Matching Model • Moment computation • Iterative DC analysis on transformed equivalent DC circuit • Recursive computation based on tree traversal • Incremental moment computation • Moment matching methods • Asymptotic Waveform Evaluation (AWE) [Pillage-Rohrer, TCAD’90] • 2-pole method [Horowitz, 1984] [Gao-Zhou, ISCAS’93]...

  8. Basis of Moment Computation by DC Analysis • Applicable to general RLC networks • Used in Asymptotic Waveform Evaluation (AWE) [Pillage-Rohrer, TCAD’90] • Represent a lumped, linear, time-invariant circuit by a system of first-order differential equations: where xrepresents circuit variables (currents and voltages) G represents memoryless elements (resistors) C represents memory elements (capacitors and inductors) bu represents excitations from independent sources y is the output of interest

  9. Transfer Function • Assume zero initial conditions and perform Laplace transform: where X, U, Y denote Laplace transform of x, u, y, respectively • Transfer function: • Let s = s0 + s, where s0 is an arbitrary, but fixed expansion point such that G+s0C is non-singular

  10. Taylor Expansion and Moments • Expansion of H(s) about s= 0: • Recursive moment computation:

  11. Taylor Expansion and Moments (Cont’d) • Expansion of H(s) around • Recursive moment computation:

  12. When s0 = 0, equivalent to DC analysis: • shorting inductors (0V) and opening capacitors (0A) • compute currents through inductors and voltages across capacitors as moments Inductor Voltage source Capacitor Current source Interpretation of Moment Computation • Compute: • Convert:

  13. When s0 = 0, equivalent to DC analysis: • voltage sources of inductor L= LmL, current sources of capacitor C = CmC • external excitations = 0 • compute currents through inductors and voltages across capacitors as moments Inductor Voltage source Capacitor Current source Interpretation of Moment Computation (Cont’d) • Compute: • Convert:

  14. When s0 = 0, equivalent to DC analysis: • moments as currents through inductors and voltages across capacitors • external excitations = 0 • compute voltage sources of inductors and current sources of capacitors Inductor Voltage source Capacitor Current source Interpretation of Moment Computation (Cont’d) • Compute: • Convert:

  15. Moment Computation by DC Analysis • Perform DC analysis to compute the (i+1)-th order moments voltage across Cj => the (i+1)-th order moment of Cj current across Lj => the (i+1)-th order moment of Lj • DC analysis: • modified nodal analysis (used in original AWE ) • sparse-tableau • …… • Time complexity to compute moments up to the p-th order: p time complexity of DC analysis

  16. Advantage and Disadvantage ofMoment Computation by DC Analysis • Recursive computation of vectors uk is efficient since the matrix (G+s0C) is LU-factored exactly once • Computation of uk corresponds to vector iteration with matrix A • Converges to an eigenvector corresponding to an eigenvalue of A with largest absolute value

  17. Moment Computation for RLC Trees • Most interconnects are RLC trees • Exploit special structure of G and C matrices to compute moments • Two basic approaches: • Recursive moment computation [Yu-Kuh, TVLSI’95] • Incremental moment computation [Cong-Koh, ICCAD’97] • Both papers used a slightly different definition of moment • We continue to use the same definition as before

  18. To source Basis of Moment Computation for RLC Trees • Wire segment modeled as lumped RLC element • Definition:

  19. Transfer Function for RLC Trees • Let • Apply KCL at node k: • Voltage drop from root to node i: • Transfer function:

  20. Recursive Formula for Moments • Transfer function: • Expanding Hi(s), Hk(s) into Taylor series: • Define p-th order weighted capacitance of Ck: • Recursive formula for moments:

  21. Recursive Formula for Moments (Cont’d) • Define p-th order weighted capacitance in subtree Tk • Can show that: • Equivalently: • Similarity with definition of Elmore delay

  22. Compute all p-th order weighted capacitance • Compute the p-th order weighted capacitance in subtree Tk • Top-down tree traversal phase; O(n) for n nodes • Compute the moment at each capacitor node k Recursive Moment Computation for RLC Trees[Yu-Kuh, TVLSI’95] • Initialize (-1)-th and 0-th order moments • Compute moments from order 1 to order p successively • Bottom-up tree traversal phase: O(n) for n nodes • Time complexity to compute moments up to the • p-th order = O(np)

  23. Incremental Moment Computation[Cong-Koh, ICCAD’97] • Iterative tree traversal approaches such as [Kuh-Yu, TVLSI’95] assume a static tree topology • More suitable for interconnect analysis • Not suitable for tree topology optimization • After each modification of tree topology, need to recompute all p-th order moments • Incremental updates of sink moments • More suitable for tree optimization algorithm that construct topology in a bottom-up fashion • As we modify the tree topology, update the transfer function Hi(s) for sink si incrementally

  24. u v w Basis for Bottom-Up Moment Computation • Assume sink w is originally in subtree Tv • Merge subtree Tv with another subtree and the new tree is rooted at node u • Consider the computation of moments for sink w • Definition:

  25. u u v w Bottom-Up Moment Computation • As we merge subtrees, compute new transfer function Hu~v(s) and weighted capacitance recursively: • Maintain transfer function Hv~w(s) for sink w in subtree Tv, and moment-weighted capacitance of subtree: • New transfer function for node w • New moment-weighted capacitance of Tu:

  26. Complexity Analysis ofIncremental Moment Computation • Assume topology has n nodes • Complexity due to each edge: O(p2) • Total time complexity: O(np2) • If there are k nodes of interest, time complexity = O(knp2)

  27. Moment Matching by AWE[Pillage-Rohrer, TCAD’90] • Recall the transfer function obtained from a linear circuit • Let s = s0 + s, where s0 is an arbitrary, but fixed expansion point such that G+s0C is non-singular • When matrix A is diagonalizable

  28. q-th Pade Approximation • Pade approximation of type (p/q): • q-th Pade approximation (q << N): • Equivalent to finding a reduced-order matrix AR such that eigenvalues lj of AR are reciprocals of the approximating poles pj for the original system

  29. Asymptotic Waveform Evaluation • Recall EQ1: • Let

  30. Asymptotic Waveform Evaluation (Cont’d) where • Rewrite EQ1: • Solving for k: • Let Need to compute all the poles first

  31. Structure of Matrix AR • Matrix: has characteristic equation: • Therefore, AR could be a matrix of the above structure • Note that: Characteristic equation becomes the denominator of Hq(s):

  32. Solving for Matrix AR • Consider multiplications of ARon ml produces produces

  33. Solving for Matrix AR (Cont’d) • After q multiplications of ARon ml produces • Equating m’ with m:

  34. Summary of AWE Step 1: Compute 2q moments, choice of q depends on accuracy requirement; in practice, q  5 is frequently used Step 2: Solve a system of linear equations by Gaussian elimination to get aj Step 3: Solve the characteristic equation of AR to determine the approximate poles pj Step 4: Solve for residues kj

  35. Numerical Limitations of AWE • Due to recursive computation of moments • Converges to an eigenvector corresponding to an eigenvalue of matrix A with largest absolute value • Moment matrix used in AWE becomes rapidly ill-conditioned • Increasing number of poles does not improve accuracy • Unable to estimate the accuracy of the approximating model • Remedial techniques are sometimes heuristic, hard to apply automatically, and may be computationally expensive

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