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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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time moments of impulse response h t
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
pade approximation
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
general moment matching technique

(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.

compute residues poles
Compute Residues & Poles

EQ1

match first 2q-1 moments

basic steps for moment matching
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

components of moment matching model
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]...
basis of moment computation by dc analysis
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

transfer function
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
taylor expansion and moments
Taylor Expansion and Moments
  • Expansion of H(s) about s= 0:
  • Recursive moment computation:
taylor expansion and moments cont d
Taylor Expansion and Moments (Cont’d)
  • Expansion of H(s) around
  • Recursive moment computation:
interpretation of moment computation

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:
interpretation of moment computation cont d

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:
interpretation of moment computation cont d14

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:
moment computation by dc analysis
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
advantage and disadvantage of moment computation by dc analysis
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
moment computation for rlc trees
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
basis of moment computation for rlc trees

To source

Basis of Moment Computation for RLC Trees
  • Wire segment modeled as lumped RLC element
  • Definition:
transfer function for rlc trees
Transfer Function for RLC Trees
  • Let
  • Apply KCL at node k:
  • Voltage drop from root to node i:
  • Transfer function:
recursive formula for moments
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:
recursive formula for moments cont d
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
recursive moment computation for rlc trees yu kuh tvlsi 95

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)
incremental moment computation cong koh iccad 97
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
basis for bottom up moment computation

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:
bottom up moment computation

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:
complexity analysis of incremental moment computation
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)
moment matching by awe pillage rohrer tcad 90
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
q th pade approximation
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
asymptotic waveform evaluation cont d
Asymptotic Waveform Evaluation (Cont’d)

where

  • Rewrite EQ1:
  • Solving for k:
  • Let

Need to compute all the poles first

structure of matrix a r
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):

solving for matrix a r
Solving for Matrix AR
  • Consider multiplications of ARon ml

produces

produces

solving for matrix a r cont d
Solving for Matrix AR (Cont’d)
  • After q multiplications of ARon ml

produces

  • Equating m’ with m:
summary of awe
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

numerical limitations of awe
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