New Algebraic estimation techniques in signal processing Mamadou - PowerPoint PPT Presentation

New Algebraic estimation techniques in signal processing Mamadou Mboup UFR de Math´ ematiques et Informatique Universit´ e Ren´ e Descartes-Paris 5 Projet ALIEN, INRIA-Futurs Email: mboup@math-info.univ-paris5.fr Algebraic estimation Control Theory, Estimation and Signal Processing, M. Fliess anniversary, IHP mars 30-31, 2006.

Outline General overview 1 Mathematical background 2 Applications 3 Polynomial phase signal Signal analysis and representation Introduction Derivative estimation Signal Denoising - Change points detection Concluding remarks 4 Algebraic estimation Control Theory, Estimation and Signal Processing, M. Fliess anniversary, IHP mars 30-31, 2006.

A simple example - Notations and algorithm n ( t ) � �� � ae − t 2 : σ 2 unknown parameter σ 2 + γ + n 0 ( t ) y ( t ) = � �� � x ( t ) Differential equation : σ 2 ˙ x ( t ) = − tx ( t ) − tγ → σ 2 sx = x ′ + σ 2 x (0) − γ s 2 Elimination of structured perturbations: Π( · ) = d 3 ds 3 s 2 ( · ) ⇒ ( s 3 x (3) + 9 s 2 x ′′ + 18 sx ′ + 6 x ) σ 2 = s 2 x (4) + 6 sx (3) + 6 x ′′ Linear estimator: ( s 3 y (3) + 9 s 2 y ′′ + 18 sy ′ + 6 y ) � σ 2 = s 2 y (4) + 6 sy (3) + 6 y ′′ (Strictly) proper estimator: multiply both sides by s − ν , ν > 2 : � � � s ν − 3 + 9 y ′′ y (3) y ′ σ 2 = y (4) s ν − 2 + 6 y (3) s ν − 1 + 6 y ′′ y s ν − 2 + 18 s ν − 1 + 6 s ν − 1 s ν The estimator is strictly proper for ν � 4 . Algebraic estimation Control Theory, Estimation and Signal Processing, M. Fliess anniversary, IHP mars 30-31, 2006.

A simple example - Notations and algorithm n ( t ) � �� � ae − t 2 : σ 2 unknown parameter σ 2 + γ + n 0 ( t ) y ( t ) = � �� � x ( t ) Differential equation : σ 2 ˙ x ( t ) = − tx ( t ) − tγ → σ 2 sx = x ′ + σ 2 x (0) − γ s 2 Elimination of structured perturbations: Π( · ) = d 3 ds 3 s 2 ( · ) ⇒ ( s 3 x (3) + 9 s 2 x ′′ + 18 sx ′ + 6 x ) σ 2 = s 2 x (4) + 6 sx (3) + 6 x ′′ Linear estimator: ( s 3 y (3) + 9 s 2 y ′′ + 18 sy ′ + 6 y ) � σ 2 = s 2 y (4) + 6 sy (3) + 6 y ′′ (Strictly) proper estimator: multiply both sides by s − ν , ν > 2 : � � � s ν − 3 + 9 y ′′ y (3) y ′ σ 2 = y (4) s ν − 2 + 6 y (3) s ν − 1 + 6 y ′′ y s ν − 2 + 18 s ν − 1 + 6 s ν − 1 s ν The estimator is strictly proper for ν � 4 . Algebraic estimation Control Theory, Estimation and Signal Processing, M. Fliess anniversary, IHP mars 30-31, 2006.

A simple example - Notations and algorithm n ( t ) � �� � ae − t 2 : σ 2 unknown parameter σ 2 + γ + n 0 ( t ) y ( t ) = � �� � x ( t ) Differential equation : σ 2 ˙ x ( t ) = − tx ( t ) − tγ → σ 2 sx = x ′ + σ 2 x (0) − γ s 2 Elimination of structured perturbations: Π( · ) = d 3 ds 3 s 2 ( · ) ⇒ ( s 3 x (3) + 9 s 2 x ′′ + 18 sx ′ + 6 x ) σ 2 = s 2 x (4) + 6 sx (3) + 6 x ′′ Linear estimator: ( s 3 y (3) + 9 s 2 y ′′ + 18 sy ′ + 6 y ) � σ 2 = s 2 y (4) + 6 sy (3) + 6 y ′′ (Strictly) proper estimator: multiply both sides by s − ν , ν > 2 : � � � s ν − 3 + 9 y ′′ y (3) y ′ σ 2 = y (4) s ν − 2 + 6 y (3) s ν − 1 + 6 y ′′ y s ν − 2 + 18 s ν − 1 + 6 s ν − 1 s ν The estimator is strictly proper for ν � 4 . Algebraic estimation Control Theory, Estimation and Signal Processing, M. Fliess anniversary, IHP mars 30-31, 2006.

. . . A simple example - Notations and algorithm Numerical estimate (time domain) � t { [ ν ] 2 τ 2 - 6[ ν ] 1 τ ( t - τ ) + 6( t - τ ) 2 } τ 2 ( t - τ ) ν - 3 y ( τ ) dτ � 0 σ 2 ( t ) = � t { 6( t - τ ) 3 - [ ν ] 3 τ 3 + 9[ ν ] 2 τ 2 ( t - τ ) - 18 τ ( t - τ ) 2 } ( t - τ ) ν - 4 y ( τ ) dτ 0 where [ ν ] i = � i k =1 ( ν − k ) � t � t α − 1 � t 1 y ( k ) ( − 1) k τ k y ( τ ) dτdt 1 · · · dt α − 1 s α �→ · · · 0 0 0 � t ( − 1) k ( t − τ ) α − 1 τ k y ( τ ) dτ = ( α − 1)! 0 The estimation time t may be very small ⇒ fast estimation. The noise effect is attenuated by the iterated integrals Algebraic estimation Control Theory, Estimation and Signal Processing, M. Fliess anniversary, IHP mars 30-31, 2006.

. . . A simple example - Notations and algorithm y ( t ) = − 2 π 2 ( t − d ) e − 2 π 2 ( t − λ ) 2 + n ( t ) , λ : unknown parameter PPM in UWB application: y − y + 2 π 2 t 2 y = ( ˙ y + 4 π 2 ty ) λ − (2 π 2 y ) λ 2 Differential equation: t ˙ � � � λ 2 π 2 y (3) − sy ′′ − 3 y ′ = ([ sy ] ′ − 4 π 2 y ′′ ) λ − (2 π 2 y ) λ 2 � P = Q � λ 2 �� t � Integral equation: y ( t ) = − 2 π 2 ( t − λ ) y ( τ ) dτ + c 0 t + c 1 + c 2 0 ds 3 { s 3 y − 2 π 2 ( s 2 y ′ − sy ) } = { 2 π 2 d 3 d 3 λ = q ds 3 ( s 2 y ) } λ � � p Algebraic estimation Control Theory, Estimation and Signal Processing, M. Fliess anniversary, IHP mars 30-31, 2006.

Simulation: SNR = 23 dB Noise correlation function 3 1.0 0.8 2 0.6 1 0.4 0 0.2 −1 0.0 −2 −0.2 −3 −0.4 −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 0 10 20 30 40 50 60 70 80 90 100 110 Estimations of lambda vs time Zooming 15 0.16 λ = 0.11 0.15 ^ 10 Diff. Eq.: λ 0.14 ^ 2 Diff. eq.: sqrt( ) λ 0.13 5 0.12 Int. eq. 0.11 0 0.10 0.09 −5 0.08 0.07 −10 0.06 −0.5 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 The noise need not be white, Gaussian, etc ... ! Algebraic estimation Control Theory, Estimation and Signal Processing, M. Fliess anniversary, IHP mars 30-31, 2006.

Mathematical background: Differential algebra - operational calculus Differential algebra, Commutative ring/field, equipped with a derivation � a differential ring/field, provides a powerful and elegant mean to exhibit hidden linear structures. In control theory: nonlinear inversion, flatness,... In signal processing: nonlinear inversion (equalizability). Significant breakthrough is expected Algebraic estimation Control Theory, Estimation and Signal Processing, M. Fliess anniversary, IHP mars 30-31, 2006.

Some definitions Differential field extension L/K : Two differential fields K , L K ⊂ L restriction to K of the derivation of L is the derivation of K . L/K is termed differentially algebraic iff, ds , . . . , d n x ∀ x ∈ L, ∃ P, polyn over K | P ( x, dx ds n ) = 0 . Otherwise, L/K is said to be differentially transcendental . Mikusi´ nski’s field of operators: C = ( { f : [0 , + ∞ ) → C , f continuous } , + , ⋆ ) � commutative ring without zero divisors. Denote by M , the quotient field of C ; its elements { f } are called operators . Equip M with d f ds = {− tf } (algebraic derivative) � differential field of operators Algebraic estimation Control Theory, Estimation and Signal Processing, M. Fliess anniversary, IHP mars 30-31, 2006.

Identifiability Let k 0 be a differential field of constants and Θ = ( θ 1 , . . . , θ r ) a set of unknown parameters . λ 1 θ 2 1 θ 2 Let k = k 0 ( Θ ) . Example: 2 ∈ k 0 ( Θ ) , λ i ∈ k 0 λ 2 + λ 3 θ 3 Let K/k ( s ) be a finitely generated differentially algebraic extension A signal is an element of K . Consider a finite collection of signals: x = ( x 1 , . . . , x κ ) The parameters Θ are linearly identifiable with respect to x if, and only if, P Θ = Q P i,j , Q j ∈ span k 0 ( s )[ d ds ] (1 , x ) , i, j = 1 , . . . , r and det( P ) � = 0 . weakly linearly identifiable with respect to x if, and only if, Θ ′ = ( θ ′ 1 , . . . , θ ′ q ′ ) are linearly identifiable, wrt x and i (resp. θ i ) is algebraic over k 0 ( Θ ) (resp. k 0 ( Θ ′ )) each θ ′ projectively linearly identifiable wrt x if, and only if, θ ǫ , . . . , θ ǫ − 1 θ ǫ , θ ǫ +1 θ 1 θ ǫ , . . . , θ r θ ǫ are linearly identifiable, for some θ ǫ � = 0 . Algebraic estimation Control Theory, Estimation and Signal Processing, M. Fliess anniversary, IHP mars 30-31, 2006.

Perturbations A perturbation noise n is either structured : ⇒ ∃ Π ∈ k 0 ( s )[ d ⇐ ds ] , Π � = 0 | Π n = 0 . s ν is annihilated by Π = νs ν − 1 + s ν d Example : n = γ ds non structured : rapid oscillating (high frequency) signal, attenuated by the iterated integrals Non standard analysis description [M. Fliess] Algebraic estimation Control Theory, Estimation and Signal Processing, M. Fliess anniversary, IHP mars 30-31, 2006.

Outline General overview 1 Mathematical background 2 Applications 3 Polynomial phase signal Signal analysis and representation Introduction Derivative estimation Signal Denoising - Change points detection Concluding remarks 4 Algebraic estimation Control Theory, Estimation and Signal Processing, M. Fliess anniversary, IHP mars 30-31, 2006.

Polynomial phase signal Noisy chirp signal : y ( t ) = A sin ϕ ( t ) + n ( t ) , where ϕ ( t ) = ϕ 0 + ϕ 1 t + ϕ 2 t 2 , n ( t ) is the noise. Set x ( t ) = y ( t ) − n ( t ) : noise-free signal. ... ϕ ( t ) 2 ˙ D ϕ ⋆ : x ( t ) + ˙ x ( t ) + 3 ˙ ϕ ( t ) ¨ ϕ ( t ) x ( t ) = 0 which reads in the operational domain as � � 2 s d 2 1 s + s 3 ) + 4 ϕ 2 ( ϕ 2 + ϕ 1 s ) d (2 ϕ 1 ϕ 2 + ϕ 2 ds + 4 ϕ 2 x ds 2 x (0) + x (0) ϕ 2 x (0) s + x (0) s 2 = (¨ 1 ) + ˙ x ( t ) is a differentially rational signal � ϕ ( t ) 3 x ( t ) = 0 D ϕ ⋆ : ϕ ( t )¨ ˙ x ( t ) − ¨ ϕ ( t ) ˙ x ( t ) + ˙ Different estimator for different D ϕ ⋆ Algebraic estimation Control Theory, Estimation and Signal Processing, M. Fliess anniversary, IHP mars 30-31, 2006.