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NORM: Compact Model Order Reduction of Weakly Nonlinear Systems

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Title: 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

3
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

5
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

6
Outline
  • Motivation
  • Background
  • Previous work on nonlinear MOR
  • Proposed algorithm -- NORM
  • Numerical results
  • Conclusions and future work

7
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,
8
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

9
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

10
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

11
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

12
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

13
Matrix-Form of Nonlinear Transfer Functions
  • 3rd order

14
Moments of Nonlinear Transfer Functions
  • Moments of linear transfer functions
  • Apply moment matching of these nonlinear transfer
    fcts via projection

15
Moments of Nonlinear Transfer Functions
  • Interaction between moments of transfer functions
    of different orders

16
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

17
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

18
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

19
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

20
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

21
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

22
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

23
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

24
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

25
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

26
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
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