The ASTRES Toolbox for Mode Extraction of Non-Stationary Multicomponent Signals

Transcription

1 The ASTES Toolbo for Mode Etraction of Non-Stationary Multicomponent Signals Dominique Fourer, Jinane Harmouche, Jérémy Schmitt, Thomas Oberlin, Sylvain Meignen, François Auger and Patrick Flandrin Abstract In this paper, we introduce the ASTES toolbo which offers a set of Matlab functions for non-stationary multicomponent signal processing. The main purposes of this proposal is to offer efficient tools for analysis, synthesis and transformation of any signal made of physically meaningful components e.g. sinusoid, trend or noise. The proposed techniques contain some recent and new contributions, which are now unified and theoretically strengthened. They can provide efficient time-frequency or time-scale representations and they allow elementary components etraction. Usage and description of each method are then detailed and numerically illustrated. I. INTODUCTION eal world signals are often non-stationary and made of several components. They require advanced techniques to be efficiently processed. Unfortunately, the Short-Time Fourier Transform STFT and the Continuous Wavelet Transform CWT which belong among the most usual approaches, are limited due to a poor energy localization in the time-frequency plane []. Two solutions, the reassignment and the synchrosqueezing methods [], [3], [4], can improve the readability of a time-frequency representation TF. More particularly, the synchrosqueezing can provide sharpened and invertible TFs allowing many applications like noise removal and signal decomposition into modes [3], [5], [6]. In addition, the Empirical Mode Decomposition EMD [7] and the Singular Spectrum Analysis SSA [8], which belong to data-driven methods, allow unsupervised signal decomposition into a sum of physically meaningful components e.g. periodic functions, trend or noise. Several recent developments were proposed to enhance these techniques [9] and to enable a fully automatic version of SSA, in the same flavor as EMD. This paper proposes a collection of Matlab functions related to the new methods developed in the Analysis, Synthesis, Transformation by eassignment, EMD and Synchrosqueezing ASTES research project. This new toolbo cf. Fig., designed for automatic signal mode etraction, can be viewed as an etension of the previously proposed Time-Frequency ToolBo TFTB. It contains implementations of some recent contributions related to reassignment, synchrosqueezing [], [], [6], [], EMD [3], [4], and SSA [9] methods. This paper is organized as follows. In Section II, proper definitions This research was supported by the French AN ASTES project AN- 3-BS toolbo TFs STFT CWT S transform reassignment synchrosqueezing ridge detection Data driven methods EMD SSA Mode etraction Fig.. ASTES toolbo content description freely available online. of the considered transforms and their TFs are presented. In Section III, reassigned and synchrosqueezed versions of each transform are described with two ridge detection methods. Section IV describes data-driven methods, EMD and SSA, with their corresponding new developments. Finally, the proposed methods are illustrated by numerical eperiments on real world signal in Section V, before concluding the paper in Section VI. II. TIME-FEQUENCY EPESENTATIONS A. Short-time Fourier transform STFT Let Ft,ω h denote the STFT of signal, using a differentiable analysis window h, defined as Ft,ω h = uht u e jωu du, z being the comple conjugate of z. Thus, the spectrogram can be computed by Ft,ω h cf.tfrstft,tfrgab. Eq. admits the following synthesis formula when ht, allowing to recover signal with a delay t cf. rectfrgab t t = ht + Ft,ωe h jωt t dω π. An efficient recursive implementation suitable for real-time computation, is proposed in [], []. It uses a specific causal one-sided analysis window which can be epressed as h k t= tk T k k! e t/t Ut, with k, where Ut is the Heaviside step function and T a time spread parameter cf. recursive_stft and stft_rec. B. Continuous wavelet transform CWT The CWT of a signal is defined for an admissible mother wavelet function Ψ as [] W Ψ t,s = + τ t τψ dτ 3 s s

2 If we define the scale as s = ω ω, ω being an arbitrary frequency, Eq. 3 can now be epressed as a time-frequency transform as CW t,ω = ω + ω τψ τ t dτ. 4 ω The scalogram can be computed by W t,s or CW t,ω. If we use the Morlet wavelet [], epressed as Ψt = π /4 T e t T e jωt, we finally obtain MW t,ω = + ω T t τ τe ω ω T e jωτ t dτ 5 π cf. tfrscalo and MW. The synthesis formula of Eq. 3 is given by cf. recmw t = W Ψ t,s s 3/ ds, with C ψ = F Ψ ω dω C ψ ω 6 where F Ψ ω denotes the Fourier transform of Ψt. C. Stockwell transform S-transform The S-transform [5] can be defined for a given zero-mean signal as cf. tfrst ST t,ω = πω T + t τ τe ω ω T e jωτ dτ 7 where T is the width of the Gaussian analysis window when ω = ω. The corresponding TF, called Stockwellogram, is provided by ST t,ω.the S-transform can be viewed as a STFT using a frequency-varying window width, ST h t,ω = ω + τh ω ω t τ e jωτ dτ 8 wherehis a Gaussian window defined asht = πt e t T. The S-transform can also be related to the Morlet wavelet transform by ST t,ω = πω T e jωt MW t,ω, 9 which shows that the S-transform can be viewed as a normalized phase-shifted Morlet wavelet transform. Thus, substituting MW in Eq. 6 leads to the S-transform synthesis formula cf. rectfrst [6] t = C h ω T jωt dω ST t,ωe, where C h ω T is a proportionality factor defined by C h ω T = F h ξ ω dξ ξ. III. EASSIGNMENT AND SYNCHOSQUEEZING eassignment and synchrosqueezing are sharpening techniques designed to improve TFs. The main difference between these methods is the reconstruction capability of synchrosqueezing and a quite poorer time-frequency localization compared to the reassignment, which is not invertible. A. eassignment For each transform, the TF values are moved according to the map t,ω ˆtt,ω,ˆωt,ω, where ˆtt,ω = e tt,ω and ˆωt,ω = Im ωt,ω resp. ŝt,ω = Im st,ω, are the reassignment operators. STFT: eassignment operators can be computed as [] tt,ω = t FT h t,ω FDh F h, ωt,ω = jω + t,ω t,ω F h, t,ω with T ht = tht and Dht = dh dt t. Thus, the reassigned spectrogram can be computed as cf. tfrrsp and recursive_rsp F t,ω = Fτ,Ω h δ t ˆtτ,Ω δω ˆωτ,ΩdτdΩ. 3 A recent etension of the reassignment process, based on the Levenberg-Marquardt algorithm, allows to adjust the energy localization in the time-frequency plane through a damping parameter µ [7]. The new reassignment operators can be computed as ˆt µ t,ω ˆω µ t,ω t = t h h ω t,ω+µi t,ω 4, t ht,ω= h with h t,ω= t ˆtt,ω ω ˆωt,ω t t,ω h ω t,ω where I is the identity matri. As a result, the Levenberg-Marquardt reassigned spectrogram can be obtained by replacing ˆt,ˆω by ˆt µ,ˆω µ in Eq. 3 cf. tfrlmrgab and recursive_lmrsp. CWT: Since, t WΨ t,s = s WDΨ t, s, the CWT reassignment operators can be epressed by [], [], [8] tt,s = t+s WT Ψ t, s W Ψ t,s, 5 st,s = ω ωt,s = sω W Ψ t,s W DΨ t,s. 6 Then, the reassigned scalogram can be computed as cf. rmw W t,s = W Ψ τ,s δ t ˆtτ,s δs ŝτ,s s ŝτ,s dτds. 7 3 S-transform: eassignment operators can be computed as for the STFT cf. proofs in 8, [6]. Thus, both classical and Levenberg-Marquardt reassigned Stockwellograms are obtained through Eqs. and 3, by replacing Ft,ω h by ST h t,ω, F T h by ST T h and F Dh and tfrlmrst. Thus we have ST t,ω = by ST Dh cf. tfrrst.m ST τ,ω δ t ˆtτ,Ω δω ˆωτ,Ωdτ dω 8 B. Synchrosqueezing Each synchrosqueezed transform can be deduced from the simplified synthesis formula given by Eqs, 6 and, respectively for the STFT, the CWT and the S-transform.

3 STFT: The synchrosqueezed STFT can be defined for any t such that ht cf. tfrsgab, recursive_sstft SF h t,ω = Ft,ω h e jω t t δω ˆωt,ω dω, 9 Its squared modulus provides a sharpened TF, and it can be inverted by ˆt t = ht SF h t,ω dω π, eplacing ˆω by ˆω µ computed by Eq. 4 allows to make synchrosqueezing adjustable as for the Levenberg-Marquardt reassignment cf. tfrlmsgab, recursive_lmsstft. CWT: The synchrosqueezed CWT can be defined as [3] cf. smw SW t,s = W t,s s 3/ δs ŝt,s ds, and can be inverted by cf. recsmw ˆt = SW t,sds, C Ψ 3 S-transform: The synchrosqueezed S-transform can be defined as [6] cf. tfrsst, tfrlmsst SST t,ω= ST t,ω e jω t δω ˆωt,ω dω ω, 3 and can be inverted by cf. rectfrsst ˆt = SST t,ω dω C h ω T, 4 C. Second-order synchrosqueezing Second-order synchrosqueezing was first proposed by Oberlin et al. in [] to improve TFs of strongly modulated chirps signal. Assuming a signal model epressed as t = ate jφt, with Φt = ϕ +ω t+α t, 5 where at and Φt stand for the time-varying amplitude and phase. Second-order synchrosqueezing proposes to use local modulation estimation to improve the resulting TF. Oblique synchrosqueezing only proposed for the STFT cf. tfrosgab can be viewed as a time-frequency reassignment with a phase correction term [], OSF t,ω = Fτ,ω h e jω ˆα t τt τ δt ˆtτ,ω δω ˆωτ,ω dτdω. 6 Vertical synchrosqueezing uses an improved estimation of the instantaneous frequency instead of the classical reassignment operator, thanks to the chirp rate estimation ˆα t,ω = Im q t,ω see [] for a further investigation. The improved frequency estimator can be computed as { ˆω ˆωt,ω+ ˆα t,ωt ˆtt,ω if ˆα t,ω < t,ω = ˆωt, ω otherwise, 7 For amplitude modulated signals, it is shown in [] that 7 is biased, and can be improved using a slightly different estimator epressed as when q t,ω < ˆω 3 t,ω = Im ωt,ω+ q t,ωt tt,ω. 8 An estimation of q t,ω can be computed for each transform as follows: STFT: In [], we proposed several local modulation estimators, valid for any differentiable analysis window, n, which can be used in Eq. 7 or in 8 to obtain a vertical synchrosqueezed STFT transform cf. tfrvsgab, recursive_vstft e F Dn h ˆα Kn t,ω = Im F T Dn h q tn q ωn t,ω = FD n h F Dn h t,ω = FT n Dh t,ωf Dn h t,ω t,ωf Dn h t,ω, 9 t,ωf h h t,ω FDn t,ωf Dh t,ω, t,ωf T h t,ω FT Dn h t,ωf h t,ω + n F T n h F T n h F T h F h F T n h F T n h F h 3 F Dh, 3 where t,ω was omitted in Eq. 3 for the sake of clarity. In [], these estimators were implemented and compared in terms of accuracy for n =. CWT: Using the local modulation estimator q t,s = ω t t,s t t t,s proposed for the CWT [8], we obtain cf. vsmw q t,s = s W D Ψ t,sw Ψ t,s WDΨ t,s W DΨ t,sw T Ψ t,s W T DΨ t,sw Ψ 3 t,s. 3 S-transform: A new local modulation estimator can also be derived to compute the vertical synchrosqueezed S- transform [6] cf. tfrvsst, q t,ω = ST D h t,ωst t,ω ST Dh t,ω ST Dh t,ωstt h Dh t,ω STT t,ωst t,ω D. idge detection for mode etraction. 33 Let us consider a multicomponent signal denoted by t = I i= it. Then, each mode i t = a i te jφit can be etracted from its synchrosqueezed transform, by restricting the integration area to the vicinity of the ridge, denoted Ω i t, in each reconstruction formula given by Eqs., and 4. The ridge Ω i t can be directly estimated from a TF using a ridge detector as: Brevdo et al. method: This technique [9] aims at finding the best frequency curve Ωt in the TF S, which maimizes the energy with a smoothness constraint through a total variation penalization term epressed as cf. ridge_detect_brvmask ˆΩ=argma S t,ωt dt λ dω Ω dt t dt, 34 where λ controls the importance of the smoothness constraint. For multi-component etraction i.e. I >, this method can be iterated after setting S to zero in the vicinity of the previously detected ridge.

4 Delaunay triangulation method: Frandrin proposes in [6] a new way to disentangle the different components in the time-frequency plane through a simplified representation provided by the Delaunay triangulation attached to the spectrogram zeros. For a particular Gaussian analysis window gt = π /4 e t, the zeros z n = ω n + jt n can be determined from the STFT, through Bargmann transform as: Ft,ω g = e z 4 F z, with F z = Az,ττdτ, and Az,τ = π /4 e τ jτz+z It admits a Weierstrass-Hadamard factorization such as [6]: F z z e z zn + z zn. 36 z n n= Then, the Delaunay triangulation is performed over zeros z n and the domains attached to signal components are related to outlier edges cf. spz_delaunay_dom4a. IV. DATA-DIVEN METHODS The ASTES toolbo also includes several implementations of the well known EMD method [7], which was reformulated and etended for two dimensional signals [3], [4]. Due to paper size limitation, we chose to focus on more recent developments allowing an unsupervised usage of SSA. A. Singular spectrum analysis SSA [8] can epand a signal in a sum of periodic components, trends and noise. It considers a finite length time series e.g. a sampled signal s = {s n,n =...N} and a user parameter L. Its algorithm cf. ssa is described by: Construction from of the trajectory Hankel matri X of size L K, with K = N L+. Singular Value Decomposition SVD of X, to obtain X = UΣV T = X i, with X i = σ i u i v T i, 37 i= σ i being the singular values of X and u i resp. v i the column vectors of U resp. V. 3 Elementary components reconstruction from each X i by anti-diagonal averaging through i n = n L n X i m,n m+ for n<l m= L X i m,n m+ for L n K m= N n+ m=n K+ L X i m,n m+ for K + n N. 4 Unsupervised mode etraction through elementary components grouping, based on Hierarchical Clustering HC [9] cf. ssa_decomp. The main idea of this algorithm [] is to look for the two nearest classes and to merge them to build a new class. This operation is iterated until the desired number of classes i.e. I is reached. To compare two classes C i and C j containing several time series, the dissimilarity is defined as the minimal distance between two distinct time series of each class DC i,c j = min C i,y C j d,y, d,y = <,y> y, Thus, each etracted mode is obtained by summing the elementary components included into the resulting classes C i. V. NUMEICAL SIMULATIONS A. Time-frequency representations Fig. compares the TFs of the gravitational wave signal GW594 [], resulting from a binary black hole collision. These TFs are computed using respectively, the Gabor transform STFT using a Gaussian analysis window, the Morlet Wavelet transform and the S-transform, with their respective first- and second-order synchrosqueezed versions. The econstruction Quality Factor QF given by QF, ˆ = ln ˆ is indicated above each invertible TF. B. Empirical mode decomposition For this eperiment cf. Fig. 3, we consider a real-world multi-component audio signal recorded from a cello. This signal is analyzed using the synchrosqueezed STFT combined with the Brevdo et al. method described in Section III-D, the EMD method and the proposed SSA method [9]. Synchrosqueezing leads to better separation results compared to other methods which are more likely to create interference. VI. CONCLUSION AND FUTUE WOKS The ASTES toolbo was introduced as a collection of Matlab functions for processing non-stationary and multicomponent signals. This toolbo which is freely available online, unifies into the same framework several recent techniques developed into the ASTES project. Some capabilities of these methods designed for efficient TFs computation and mode etraction, were also illustrated by application eamples on real world signals. Future works consist in theoretically strengthening these tools, and in proposing new applications. EFEENCES [] P. Flandrin, Time-Frequency/Time-Scale analysis. Acad. Press, 998. [] F. Auger and P. Flandrin, Improving the readability of time-frequency and time-scale representations by the reassignment method, IEEE Trans. Signal Process., vol. 43, no. 5, pp , May 995. [3] I. Daubechies, J. Lu, and H.-T. Wu, Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool, Applied and Computational Harmonic Analysis, vol. 3, no., pp. 43 6,. [4] F. Auger, P. Flandrin, Y. Lin, S. McLaughlin, S. Meignen, T. Oberlin, and H. Wu, TF reassignment and synchrosqueezing: An overview, IEEE Signal Process. Mag., vol. 3, no. 6, pp. 3 4, Nov. 3. [5] S. Meignen, T. Oberlin, and S. McLaughlin, A new algorithm for multicomponent signals analysis based on synchrosqueezing: with an application to signal sampling and denoising, IEEE Trans. Signal Process., vol. 6, no., pp , Nov.. [6] P. 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5 Gabor spectrogram, T=.3 Morlet scalogram, ft= squ. mod. synch. STFT, T=.3, QF=66.3 db squ. mod. synch. MW, ft=., QF=6.8 db squ. mod. synch. ST, ft=., QF=35.37 db squ. mod. vertical synch. STFT T=.3, QF=44.89 db squ. mod. vertical synch. MW, ft=., QF=6.8 db squ. mod. vertical synch. ST, ft=., QF=3.5 db Stockwellogram, ft= Fig.. Comparison between the Gabor STFT, Morlet CWT and the Stockell transform applied on the Livingston GW594 signal [] a synchrosqueezed STFT SSA etracted components EMD etracted components synchrosqueezing etracted components b EMD c SSA Fig. 3. Comparison of the etracted modes resulting from a real-world audio signal, using synchrosqueezed STFT, EMD and the proposed SSA method. [8] J. Elsner and A. Tsonis, Singular Spectrum Analysis, A New Tool in Time Series Analysis. Plenum Press, 996. [9] J. Harmouche, D. Fourer, F. Auger, P. Flandrin, and P. Borgnat, One or two components? the singular spectrum analysis answers, in Structured Low-ank Approimation SLA 5, Grenoble, France, Jun. 5. [] D. Fourer, F. Auger, and P. Flandrin, ecursive versions of the Levenberg-Marquardt reassigned spectrogram and of the synchrosqueezed STFT, in Proc. IEEE ICASSP, Mar. 6, pp [] T. Oberlin, S. Meignen, and V. Perrier, Second-order synchrosqueezing transform or invertible reassignment? Towards ideal time-frequency representations, IEEE Trans. Signal Process., vol. 63, no. 5, pp , Mar. 5. [] D. Fourer, F. Auger, K. Czarnecki, S. Meignen, and P. Flandrin, Chirp rate and instantaneous frequency estimation: application to recursive vertical synchrosqueezing, IEEE Signal Process. Lett., 7. [3] T. Oberlin, S. Meignen, and V. Perrier, An alternative formulation for the empirical mode decomposition, IEEE Trans. Signal Process., vol. 6, no. 5, pp ,. [4] J. Schmitt, N. Pustelnik, P. Borgnat, P. Flandrin, and L. Condat, D prony-huang transform: A new tool for D spectral analysis, IEEE Trans. Image Process., vol. 3, no., pp , 4. [5]. Stockwell, L. Mansinha, and. Lowe, Localization of the comple spectrum: the S-transform, IEEE Trans. Signal Process., vol. 44, no. 4, pp. 998, Apr [6] D. Fourer, F. Auger, and J. Hu, eassigning and synchrosqueezing the Stockwell transform: Complementary proofs, Tech. ep., Dec. 5. [Online]. Available: [7] F. Auger, E. Chassande-Mottin, and P. Flandrin, Making reassignment adjustable: the Levenberg-Marquardt approach, in Proc. IEEE ICASSP, Mar., pp [8] T. Oberlin and S. Meignen, The second-order wavelet synchrosqueezing transform, in Proc. IEEE ICASSP, Mar. 7, p. to appear. [9] E. Brevdo, N. S. Fuckar, G. Thakur, and H.-T. Wu, The synchrosqueezing algorithm: a robust analysis tool for signals with time-varying spectrum, Ariv preprint,. [] J. H. Ward, Hierarchical grouping to optimize an objective function, JASA, vol. 58, pp , 963. [] P. Flandrin, A note on the time-frequency analysis of GW594, E cole normale supe rieure de Lyon, Tech. ep., Sep. 6. [Online]. Available: