NORM: Compact Model Order Reduction of Weakly Nonlinear Systems

1
NORM Compact Model Order Reduction of Weakly
Nonlinear Systems

• Peng Li and Larry Pileggi
• Carnegie Mellon University
• June 4, DAC 2003

2
Motivation

• Non-digital blocks are often the design
  bottleneck for mixed-signal SoCs

• Analog and RF flows lack the modeling continuity
  that facilitates digital design

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Motivation

• Compact sub-block macromodels are the key to
  whole-system verification
• Back annotation of such models facilitates
  system-level what-if analysis

Specifications
System-level Design
(Time-Varying) Weakly Nonlinear Reduced Order
Models
Modeling Gap
Analog Design
Circuit-level Design/Synthesis

• Important to model the weak nonlinearities for
  analog and RF
• IIP3, THD, gain compression,

Layout
Validation
4
Proposed Work

• Can we build efficient analog macromodels to
  capture linear behavior time variation
  distortion ?

• Challenge is to provide sufficient modeling
  accuracy while controlling model complexity

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Challenges

• Nonlinear model order reduction challenges
• System description is more complex less
  friendly to work with
• Weakly nonlinear effects dramatically complicate
  model reduction
• Model order reduction becomes a high-dimensional
  problem

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Outline

• Motivation
• Background
• Previous work on nonlinear MOR
• Proposed algorithm -- NORM
• Numerical results
• Conclusions and future work

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Volterra Series

• Volterra Series to describe weakly nonlinear
  systems

output
input
Applicable to a broad class of circuits weakly
nonlinear amps, switching mixers, and
switch-capacitor circuits,
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Previous Work

• Recent nonlinear reduced order modeling
• Projection Volterra-based
• Extension of LTI MOR Roychowdhury TCAS99
  Phillips CICC00
• Growth of the reduced order model size
• Bilinearization Phillips DAC00
• Significant increase of the problem size in a
  bilinear form
• Trajectory Piecewise-Linear
• Rewienski ICCAD01
• Training-input dependent
• Symbolic Modeling
• Wambacq DATE00
• Volterra-based, used for high-level symbolic
  model generation

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Projection-Based Prior Work

• Model a weakly nonlinear system as a set of
  linear networks using Volterra
• Reduce each linear network using projection
  Roychowdhury TCAS99 Phillips CICC00

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Projection-Based Prior Work

• Reduce system matrices using projection

qxn
qxq3
nxn3
n3xq3

• Issues and Limitations
• How well is the overall nonlinear system behavior
  matched?
• How can the reduced-model-order size be
  optimized?
• Reduction size can be extremely large the
  problem for nonlinear MOR

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Proposed Approach NORM

• Consider MOR from a nonlinear system perspective
• Use (nonlinear) transfer functions as MOR
  criterion
• Consider (nonlinear) Padé approximation
  explicitly

Weakly Nonlinear

• Requires matrix-form nonlinear transfer functions

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Matrix-Form of Nonlinear Transfer Functions

• Computation of nonlinear transfer functions
• Accomplished in a recursive procedure nonlinear
  current method
• Requires a more formal matrix description for MOR

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Matrix-Form of Nonlinear Transfer Functions

• 3rd order

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Moments of Nonlinear Transfer Functions

• Moments of linear transfer functions

• Apply moment matching of these nonlinear transfer
  fcts via projection

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Moments of Nonlinear Transfer Functions

• Interaction between moments of transfer functions
  of different orders

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Moment Matching of Nonlinear Transfer Fcts.

• Dependency of moment matching between different
  nonlinear TFs
• Constraint on the moment matching orders
• Explicitly enforced in NORM
• Not necessary for prior projection approaches --
  suggests optimal strategies

• Decompose moment spaces into a set of minimum
  Krylov subspaces
• Optimal model size

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Multi-Point Expansion

• Single-point vs. Multi-point

H2
f2
f1

• Less economical to match higher order moments
• Multi-point expansions further improve the model
  compactness
• Zero-th order multi-point expansions
• Dimension of Krylov subspaces number of
  moments matched
• Additional cost can be minimized by matrix/vector
  reuse

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NORM Summary

• Formulate matrix-form nonlinear transfer
  functions
• Perform nonlinear Padé approximation
• Explicitly consider the moment matching of
  nonlinear TFs
• Fully capture interactions between transfer
  functions of different orders
• Moment spaces are further decomposed into a set
  of minimum Krylov subspaces
• Optimal model size
• Multi-point expansions can further improve model
  compactness
• Unique to nonlinear model order reduction

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Model Size Comparison

• Worst-case R.O.M. size for single-input
  multiple-output nonlinear systems
• k-th order model in H1 and H1 H2

output
input

• Prior work Roychowdhury TCAS99 Phillips
  CICC00
• NORM-mp equivalent multi-point NORM matching
  same number of moments

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Example A Double-Balanced Mixer

• Characterized using time-varying Volterra series
  w.r.t. RF input based on 2403 time-sampled
  circuit variables
• Each nonlinearity is modeled as a third-order
  polynomial about the time varying operating point
  due to large-signal LO
• Test frequency range 100MHz-1.5GHz

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Example A Double-Balanced Mixer

• The harmonic of the time-varying H3(t,f1,f2,f0)
  specifying the translated IM3, f0 -900MHz

• Original H3 Size 2403

• Prior work Size 60

• NORM-sp Size 19

• NORM-mp Size 14

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Example A Double-Balanced Mixer

• Harmonic-balance simulation as a function of RF
  frequency and amplitude
• Simulation speedups
• Prior method (60-order) lt5x
• NORM-sp(19-order) 350x380x
• NORM-mp(14-order) 730x840x

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Example A Subharmonic Direct-Conversion Mixer

• Characterized using time-varying Volterra series
  based on 4130 time-sampled circuit variables
• Each nonlinearity is modeled as a third-order
  polynomial about the varying operating point due
  to the 6-phase 800MHz LO
• 2 transistor size mismatch is introduced
• Second-order distortions become important
• Test frequency range 2.2GHz-2.6GHz

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Example A Subharmonic Direct-Conversion Mixer
The DC component of the time-varying H2(t,f1,f2)
specifying IM2 at the base band

• Original H2 Size 4130

• Prior work Size 122

• NORM-sp Size 34

• NORM-mp Size 22

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System-Level Simulation

• Reduced order models can be used in
  time/frequency domain nonlinear analysis or
  Volterra type simulation

• Now working on efficient techniques for whole
  receiver modeling

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Conclusions and Future Work

• Developed an approach for direct moment matching
  of nonlinear transfer functions NORM
• NORM controls reduced order model complexity via
  careful moment analysis of the nonlinear transfer
  functions
• Minimum Krylov subspaces
• Multi-point expansions
• The resulting reduced-models can be efficiently
  incorporated into system-level simulation
  environment
• Future direction
• Hybrid approach projection the circuit
  internal structure