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Phase-based homogeneous order separation for improving Volterra series identification Damien Bouvier 1 , Thomas Hlie 1 , David Roze 1 1Project-team S3: Systems, Signals and Sound (http://s3.ircam.fr/) Science and Technology of Music and Sound

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Phase-based homogeneous order separation for improving Volterra series identification

Damien Bouvier1, Thomas Hélie1, David Roze1

1Project-team S3: Systems, Signals and Sound (http://s3.ircam.fr/) Science and Technology of Music and Sound UMR 9912 Ircam-CNRS-UPMC

12 April 2018

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

NL system u(t) y(t) ≡ Linear subsystem y1(t) Quadratic subsystem y2(t) Cubic subsystem y3(t) . . . u(t) + y(t)

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

NL system u(t) y(t) ≡ Linear subsystem y1(t) Quadratic subsystem y2(t) Cubic subsystem y3(t) . . . u(t) + y(t) System u Identification y {h1, h2, · · · , hN}

Direct identification

System {u} Order separation {y} Identification y1 h1 Identification yN hN . . . . . .

Identification on separated orders

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Summary

Introduction Volterra series and order separation

  • Recalls on Volterra series
  • Order separation using amplitude gains

Phase-based homogeneous order separation

  • Theoretical method for complex-valued signals
  • Extension for real-valued signals

Evaluation and results

  • Evaluation of order separation on a simulated system
  • Application to the Silverbox benchmark

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Recalls on Volterra series1

y(t) =

+∞

  • n=1
  • Rn

+

hn(τ1, . . . , τn)

  • Volterra kernels

n

  • i=1

u(t − τi)dτi

1Wilson J. Rugh. Nonlinear system theory. Johns Hopkins University Press Baltimore, 1981. WNSIB - April 2018 Phase-based homogeneous order separation for improving Volterra series identification 4 / 19

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Recalls on Volterra series1

y(t) =

+∞

  • n=1
  • Rn

+

hn(τ1, . . . , τn)

  • Volterra kernels

n

  • i=1

u(t − τi)dτi =

+∞

  • n=1

Vn[u, . . . , u](t) Properties of operator Vn: Symmetry (considering hn symmetric) Vn[u1, . . . , un] = Vn

  • uπ(1), . . . , uπ(n)
  • , ∀ permutations π

Multilinearity Vn[u1, . . . , λuk + µv, . . . , un] = λVn[u1, . . . , uk, . . . un] + µVn[u1, . . . , v, . . . , un] Homogeneity Vn[αu1, . . . , αun] = αnVn

  • u1, . . . , un
  • 1Wilson J. Rugh. Nonlinear system theory. Johns Hopkins University Press Baltimore, 1981.

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Order separation using amplitude gains2

S αu(t) z(t) = α α2 . . . αN

   

y1 y2 . . . yN

   (t)

2Stephen P. Boyd, Y. S. Tang, and Leon O. Chua. “Measuring volterra kernels”. In: Circuits and Systems,

IEEE Transactions on 30.8 (1983), pp. 571–577.

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Order separation using amplitude gains2

S

   

α1u α2u . . . αKu

   (t)    

z1 z2 . . . zK

   (t) =    

α1 α2

1

. . . αN

1

α2 α2

2

. . . αN

2

. . . . . . ... . . . αK α2

K

. . . αN

K

       

y1 y2 . . . yN

   (t)

Vandermonde matrix

Advantages and disadvantages

✔ Easy implementation ✖ Bad conditioning of Vandermonde matrix when N is large sensibility to noise ✖ Difficulties in choosing the gains αk: αk > 1 potential saturation of the system αk < 1 higher-orders hidden in noise

2Stephen P. Boyd, Y. S. Tang, and Leon O. Chua. “Measuring volterra kernels”. In: Circuits and Systems,

IEEE Transactions on 30.8 (1983), pp. 571–577.

WNSIB - April 2018 Phase-based homogeneous order separation for improving Volterra series identification 5 / 19

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Introduction Volterra series and order separation

  • Recalls on Volterra series
  • Order separation using amplitude gains

Phase-based homogeneous order separation

  • Theoretical method for complex-valued signals
  • Extension for real-valued signals

Evaluation and results

  • Evaluation of order separation on a simulated system
  • Application to the Silverbox benchmark

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Hypothesis

Input signal u(t) ∈ C

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Hypothesis

Input signal u(t) ∈ C S

   

u wu . . . wN−1u

   (t)

with w = ej2π/N

   

z1 z2 . . . zN

   (t) =    

1 1 . . . 1 w w2 . . . 1 . . . . . . . . . wN−1 w2N−2 . . . 1

       

y1 y2 . . . yN

   (t)

Discrete Fourier Transform (DFT) matrix

  • f order N

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Hypothesis

Input signal u(t) ∈ C S

   

u wu . . . wN−1u

   (t)

with w = ej2π/N

   

z1 z2 . . . zN

   (t) =    

1 1 . . . 1 w w2 . . . 1 . . . . . . . . . wN−1 w2N−2 . . . 1

       

y1 y2 . . . yN

   (t)

Discrete Fourier Transform (DFT) matrix

  • f order N

Re Im

α1 α2 α3 w1 w2 w3

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Hypothesis

Input signal u(t) ∈ C S

   

u wu . . . wN−1u

   (t)

with w = ej2π/N

   

z1 z2 . . . zN

   (t) =    

1 1 . . . 1 w w2 . . . 1 . . . . . . . . . wN−1 w2N−2 . . . 1

       

y1 y2 . . . yN

   (t)

Discrete Fourier Transform (DFT) matrix

  • f order N

Advantages and disadvantages

✔ Optimal conditioning ✔ Reduces measurement noise by a factor √ N (supposing Gaussian white noise) ✔ Predictable behaviour if wrong truncation order N ✖ Need complex signals as input & output theoretical method

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Approach for real-valued signals

Choice of input signal: Re[wu(t)], with w = ej2π/N y1 = V1

  • w u + w−1u

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Approach for real-valued signals

Choice of input signal: Re[wu(t)], with w = ej2π/N Notation: intermodulation term Vn,q(t) = Vn

  • u, . . . , u

n−q times

, u, . . . , u

q times

  • (t)

y1 = w V1,0

  • +

w−1V1,1

  • WNSIB - April 2018

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Approach for real-valued signals

Choice of input signal: Re[wu(t)], with w = ej2π/N Notation: intermodulation term Vn,q(t) = Vn

  • u, . . . , u

n−q times

, u, . . . , u

q times

  • (t)

y1 = w V1,0

  • +

w−1V1,1

  • y2 =

w2 V2,0

  • +

2V2,1

  • +

w−2V2,2

  • WNSIB - April 2018

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Approach for real-valued signals

Choice of input signal: Re[wu(t)], with w = ej2π/N Notation: intermodulation term Vn,q(t) = Vn

  • u, . . . , u

n−q times

, u, . . . , u

q times

  • (t)

y1 = w V1,0

  • +

w−1V1,1

  • y2 =

w2 V2,0

  • +

2V2,1

  • +

w−2V2,2

  • y3 =

w3 V3,0

  • +

3w V3,1

  • +

3w−1V3,2

  • +

w−3V3,3

  • WNSIB - April 2018

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

S

   

Re[u] Re[wu] . . . Re w2Nu

   (t)

with w = ej2π/(2N+1)

   

z1 z2 . . . z2N+1

   (t) =    

1 1 . . . 1 w w2 . . . 1 . . . . . . . . . w2N w4N . . . 1

            

2V2,1 V1,0 + 3V3,1 V2,0 V3,0 V3,3 V2,2 V1,1 + 3V3,2

        

(t) Discrete Fourier Transform (DFT) matrix

  • f order 2N + 1

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Input signal: Re[(w1 + w2)u(t)] w1 w2

  • 3
  • 2
  • 1

1 2 3 | | | | | | |

  • 3
  • 2
  • 1

1 2 3 − − − − − − − V3,3 V3,3

  • V3,2

V3,3

  • V1,1 + V3,2
  • V3,1

V1,1 + V3,2

  • V3,3
  • V1,0 + V3,1
  • V3,0

V3,2

  • V1,0 + V3,1
  • V3,0

V3,1

  • V3,0

V3,0

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Create input collection u Measure

  • utputs
  • uk1,k2 = Re

(wk1 + wk2)u Invert 2D-DFT

  • zk1,k2
  • Separate

combinatorics Regroup terms {Vn,q} {yn}

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Create input collection u Measure

  • utputs
  • uk1,k2 = Re

(wk1 + wk2)u Invert 2D-DFT

  • zk1,k2
  • Separate

combinatorics Regroup terms {Vn,q} {yn} Separation method Amplitude-based Phase-based Signal type real-valued complex-valued real-valued Method parameters Gains αk None None Number of measurements N N (2N+1)(N+1) Conditioning Bad Optimal Good Noise reduction ✔ ✔ Order rejection ✔

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Introduction Volterra series and order separation

  • Recalls on Volterra series
  • Order separation using amplitude gains

Phase-based homogeneous order separation

  • Theoretical method for complex-valued signals
  • Extension for real-valued signals

Evaluation and results

  • Evaluation of order separation on a simulated system
  • Application to the Silverbox benchmark

WNSIB - April 2018 Phase-based homogeneous order separation for improving Volterra series identification 12 / 19

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Order separation evaluation test Test description

Simulated system:

  • Kernel with memory length of 5 samples
  • Truncation order N = 9
  • In total, 2001 parameters to estimate

Tested separation methods:

  • Phase-based, with 190 test signals
  • Amplitude-based, with 9 test signals
  • Amplitude-based, with 190 test signals

Input signal: Gaussian white noise Kernel identification: Least-Squares method (time-domain)

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Order separation error

1 2 3 4 5 6 7 8 9 −80 −60 −40 −20 20 Order n NRMSE (dB)

  • Amp. method
  • Amp. method (2)

Phase method Noise level

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Kernel identification error

1 2 3 4 5 6 7 8 9 −100 −80 −60 −40 −20 20 40 Order n NRMSE (dB) Direct True orders

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Kernel identification error

1 2 3 4 5 6 7 8 9 −100 −80 −60 −40 −20 20 40 Order n NRMSE (dB) Direct True orders

  • Amp. method
  • Amp. method (2)

Phase method

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Silverbox benchmark Test description

  • Simulation of the Silverbox model
  • Separate orders of the test data using the phase-based method
  • Suppose truncation order N = 9 (≈ 10 hours of measurement needed)

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Silverbox benchmark: spectra of separated orders

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Silverbox benchmark: spectra of separated orders

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Conclusion Homogeneous order separation for identification

✔ Modular approach: any identification method can be used ✔ Applicable to any polynomial series expansion

  • Volterra series
  • Block systems with polynomial nonlinearities (Wiener, Hammerstein,

Wiener-Hammerstein)

  • Polynomial NARMAX
  • Polynomial Nonlinear State-Space (PNLSS)

✖ Need a great number of measurements (i.e. (2N + 1)(N + 1) ≡ O(N2)) Open-source Python toolbox available at https://github.com/d-bouvier/pyvi.

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Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results

Conclusion Homogeneous order separation for identification

✔ Modular approach: any identification method can be used ✔ Applicable to any polynomial series expansion

  • Volterra series
  • Block systems with polynomial nonlinearities (Wiener, Hammerstein,

Wiener-Hammerstein)

  • Polynomial NARMAX
  • Polynomial Nonlinear State-Space (PNLSS)

✖ Need a great number of measurements (i.e. (2N + 1)(N + 1) ≡ O(N2)) Open-source Python toolbox available at https://github.com/d-bouvier/pyvi.

Thanks for your attention!

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Writing PNLSS output as sum of homogeneous order

˙ x(t) = Ax(t) + Bu(t) +

  • (p,q)∈N2

2≤p+q

Mpq

  • x(t), . . . , x(t)
  • p

, u(t), . . . , u(t)

  • q
  • (n=1)

(n=2) (n=3) (p+q =2) (p+q =3)

u(t) x1(t) x1(t) x2(t) x2(t) x3(t) x(t) B W W W M2,0 M1,1 M0,2 M2,0 M2,0 M1,1 M3,0 M2,1 M1,2 M0,3

With W(s) = [sI − A]−1

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