A family of synchrosqueezing transforms for multicomponent signal analysis

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A family of synchrosqueezing transforms for

multicomponent signal analysis

Duong-Hung Pham

To cite this version:

Duong-Hung Pham. A family of synchrosqueezing transforms for multicomponent signal analysis.

IEEE-EURASIP Summer School, Signal Processing meets Deep Learning, Sep 2017, Capri, Italy.

2017. �hal-03128357�

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A FAMILY OF SYNCHROSQUEEZING TRANSFORMS

FOR MULTICOMPONENT SIGNALS ANALYSIS

Duong-Hung PHAM - Joint work with Sylvain MEIGNEN

Jean Kuntzmann Laboratory, University of Grenoble-Alpes and CNRS, France. (Email:[email protected] and [email protected])

Introduction

• Time-frequency analysis of multicomponent signals (MCS). • MCS in real life:

• TheSynchroSqueezed Transform(SST) [1] has two purposes:

– sharpenthe time-frequency (TF) representation given by Short-Time Fourier Transform (STFT)

– reconstructautomatically the modes making up the signal.

• The goal of this research: put forward ageneralization of SSTusing a new local estimate of instantaneous frequency (IF) =⇒ achieve a highly concentrated TF representation for a larger class of MCSs + reconstruct their modes with a high accuracy.

Multicomponent signal (MCS)

• Superposition of AM - FM modes: f (t) =

K

P

k=1

fk(t) with fk(t) = Ak(t)ei2πφk(t), for K ∈ N,

Ak(t) > 0, φ ′ k(t) > 0 and φ ′ k+1(t) > φ ′ k(t) for ∀t.

• Hypothesis: all fks are well separated in

fre-quency, i.e. |φ′ k+1(t) − φ

′

k(t)| ≥ 2∆ for ∀t.

STFT

• Fourier transform (FT) of a signal f ∈ L1_(R):

ˆ f (η) =R

Rf (t)e −_i2πηt

dt.

• A signal f ∈ L1_(R)_{and a window g ∈ S(R):}

Vfg(t, η) = Z R f (τ )g∗ (τ − t)e−2iπη(τ −t) dτ.

Reassignment methods

• Reassignment operators: – Local group delay (GD):

ˆ τf(t, η) = t − 1 2π∂ηarg(V g f(t, η)) .

– Local instantaneous frequency: ˆ ηf(t, η) = 1 2π∂targ(V g f(t, η)) • Standard reassignment (RM): – Obliquemapping: (t, η) 7→ (ˆτf, ˆηf). – Operator: Rg f(t, ω) = Z Z R2 |Vg f(τ, η)| 2 × δ (ω − ˆωf(τ, η)) δ (t − ˆτf(τ, η)) dηdτ.

– Ideal TF representation of linear chirps. – Non reconstruction. • SST: – Verticalmapping: (t, η) 7→ (t, ˆηf). – Operator: Tg f(t, ω) = 1 g∗₍₀₎ Z∞ 0 Vg f(t, η)δ (ω − ˆωf(t, η)) dη.

– Ideal TF representation of pure waves. – Reconstruction.

Toward to high-order SST (SSTN)

• Let f ∈ L2_(R),_{frequency modulation operators}_q_˜[p,N ] η,f of φ

(p)_{(t)/(p − 1)! for p = 2, 3, 4 and N = 4 are:}

˜ q[4,4]η,f = G4 Vt0...6 g f , V t0...3 g′ f , ˜ q[3,4]η,f = G3 Vt0...4 g f , V t0...2 g′ f − ˜qη,f[4,4]G3,4 Vt0...5 g f , ˜ q[2,4]η,f = G2 Vt0...2g f , V t0...1g′ f − ˜qη,f[3,4]G2,3 Vt0...3g f − ˜q[4,4]η,fG2,4 Vt0...4g f , where Gp Vt 0...m g f , V t0...n g′ f

is a function of Vftlg for l = 0, . . . , m and V tl_g′ f for l = 0, . . . , n while Gp,j Vt 0...m g f

is associated with coefficient ˜q[j,N ]η,f in the computation of ˜q [p,N ]

η,f for p 6= j.

• IF estimate of order 4is: ˜ ω[4]η,f(t, η) = ˜ωf(t, η) + ˜q[2,4]η,f (t, η) (−x2,1(t, η)) + ˜q [3,4] η,f (t, η) (−x3,1(t, η)) + ˜q [4,4] η,f (t, η) (−x4,1(t, η)) .

– Exact IF estimate for a polynomial chirp of order 4. – ˜ω[2]η,fis obtained by neglecting ˜q

[3,4] η,f and ˜q

[4,4] η,f .

• Synchrosqueezing operator of order N (SSTN) is: TN,fg (t, ω) = 1 g∗₍₀₎ Z∞ 0 Vfg(t, η)δ ω − ˆω[N ]η,f(t, η) dη.

Numerical results

Conclusion

• SSTN: a powerful tool for analysis of MCS con-taining very strongly modulated AM-FM modes. • Combination of a sharp representation (like re-assignment) and a reconstruction (like classical ridge analysis)

• An interesting application on gravitational-wave signal.

Current and future works

• Theoretical analysis of SSTN when applied to noisy signals and when the type of noise is non Gaussian.

• Extension to 2 or 3 dimensions (with monogenic SST).

• More applications for real-life signals (detection, monitoring, etc.).

References

[1] G. Thakur and H.-T. Wu, “Synchrosqueezing-based recovery of instantaneous frequency from nonuniform samples.” SIAM J. Math. Analysis, vol. 43, no. 5, pp. 2078–2095, 2011.

[2] D.-H. Pham and S. Meignen, “High-order syn-chrosqueezing transform for multicomponent signals analysis - with an application to gravitational-wave signal,” IEEE Transactions on Signal Processing, vol. 65, no. 12, pp. 3168–3178, June 2017.

Acknowledgements

The authors acknowledge the support of the French Agence Nationale de la Recherche (ANR) under refer-ence ANR-13- BS03-0002-01 (ASTRES)

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