Phase-based homogeneous order separation for improving Volterra series identification Damien Bouvier 1 , Thomas Hélie 1 , David Roze 1 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
Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results y 1 ( t ) Linear subsystem y 2 ( t ) Quadratic subsystem y ( t ) u ( t ) y ( t ) u ( t ) + NL system ≡ y 3 ( t ) Cubic subsystem . . . WNSIB - April 2018 Phase-based homogeneous order separation for improving Volterra series identification 2 / 19
Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results y 1 ( t ) Linear subsystem y 2 ( t ) Quadratic subsystem y ( t ) u ( t ) y ( t ) u ( t ) + NL system ≡ y 3 ( t ) Cubic subsystem . . . y u System { h 1 , h 2 , · · · , h N } Identification Direct identification y 1 Identification h 1 { u } { y } Order . . . . System . . separation h N Identification y N Identification on separated orders WNSIB - April 2018 Phase-based homogeneous order separation for improving Volterra series identification 2 / 19
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 WNSIB - April 2018 Phase-based homogeneous order separation for improving Volterra series identification 3 / 19
Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results Recalls on Volterra series 1 + ∞ � n � � y ( t ) = h n ( τ 1 , . . . , τ n ) u ( t − τ i ) d τ i � �� � R n n =1 i =1 + Volterra kernels 1 Wilson 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
Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results Recalls on Volterra series 1 + ∞ � n � � y ( t ) = h n ( τ 1 , . . . , τ n ) u ( t − τ i ) d τ i � �� � R n n =1 i =1 + Volterra kernels + ∞ � = V n [ u , . . . , u ]( t ) n =1 Properties of operator V n : Symmetry (considering h n symmetric) � � V n [ u 1 , . . . , u n ] = V n , ∀ permutations π u π (1) , . . . , u π ( n ) Multilinearity V n [ u 1 , . . . , λ u k + µ v , . . . , u n ] = λ V n [ u 1 , . . . , u k , . . . u n ] + µ V n [ u 1 , . . . , v , . . . , u n ] Homogeneity � � V n [ α u 1 , . . . , α u n ] = α n V n u 1 , . . . , u n 1 Wilson 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
Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results Order separation using amplitude gains 2 y 1 y 2 z ( t ) = � . . . α N � α α 2 α u ( t ) ( t ) S . . . y N 2 Stephen 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
Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results Order separation using amplitude gains 2 α 2 . . . α N y 1 α 1 u z 1 α 1 1 1 α 2 . . . α N α 2 u z 2 y 2 α 2 2 2 ( t ) ( t ) = ( t ) . S . . . . . ... . . . . . . . . . . . . α 2 . . . α N α K u z K y N α K K K Vandermonde matrix Advantages and disadvantages ✔ Easy implementation ✖ Bad conditioning of Vandermonde matrix when N is large � sensibility to noise ✖ Difficulties in choosing the gains α k : α k > 1 � potential saturation of the system α k < 1 � higher-orders hidden in noise 2 Stephen 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
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 6 / 19
Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results Hypothesis Input signal u ( t ) ∈ C WNSIB - April 2018 Phase-based homogeneous order separation for improving Volterra series identification 7 / 19
Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results Hypothesis Input signal u ( t ) ∈ C y 1 u z 1 1 1 . . . 1 w 2 . . . 1 wu z 2 w y 2 ( t ) ( t ) = ( t ) . S . . . . . . . . . . . . . . . . . w N − 1 u w N − 1 w 2 N − 2 z N . . . 1 y N Discrete Fourier with w = e j 2 π/ N Transform (DFT) matrix of order N WNSIB - April 2018 Phase-based homogeneous order separation for improving Volterra series identification 7 / 19
Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results Hypothesis Input signal u ( t ) ∈ C y 1 u z 1 1 1 . . . 1 w 2 . . . 1 wu z 2 w y 2 ( t ) ( t ) = ( t ) . S . . . . . . . . . . . . . . . . . w N − 1 u w N − 1 w 2 N − 2 z N . . . 1 y N Discrete Fourier with w = e j 2 π/ N Transform (DFT) matrix of order N Im w 1 α 1 α 2 α 3 w 3 Re w 2 WNSIB - April 2018 Phase-based homogeneous order separation for improving Volterra series identification 7 / 19
Introduction Volterra series and order separation Phase-based homogeneous order separation Evaluation and results Hypothesis Input signal u ( t ) ∈ C y 1 u z 1 1 1 . . . 1 w 2 . . . 1 wu z 2 w y 2 ( t ) ( t ) = ( t ) . S . . . . . . . . . . . . . . . . . w N − 1 u w N − 1 w 2 N − 2 z N . . . 1 y N Discrete Fourier with w = e j 2 π/ N Transform (DFT) matrix of 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 WNSIB - April 2018 Phase-based homogeneous order separation for improving Volterra series identification 7 / 19
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 = e j 2 π/ N � w u + w − 1 u � y 1 = V 1 WNSIB - April 2018 Phase-based homogeneous order separation for improving Volterra series identification 8 / 19
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 = e j 2 π/ N Notation: intermodulation term � � V n , q ( t ) = V n u , . . . , u , u , . . . , u ( t ) � �� � � �� � n − q times q times � � w − 1 V 1 , 1 y 1 = + w V 1 , 0 WNSIB - April 2018 Phase-based homogeneous order separation for improving Volterra series identification 8 / 19
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 = e j 2 π/ N Notation: intermodulation term � � V n , q ( t ) = V n u , . . . , u , u , . . . , u ( t ) � �� � � �� � n − q times q times � � w − 1 V 1 , 1 y 1 = + w V 1 , 0 � � � w 2 V 2 , 0 w − 2 V 2 , 2 y 2 = + 2 V 2 , 1 + WNSIB - April 2018 Phase-based homogeneous order separation for improving Volterra series identification 8 / 19
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