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Second-Order Time-Reassigned Synchrosqueezing Transform: Application to Draupner Wave Analysis

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Second-Order Time-Reassigned Synchrosqueezing Transform: Application to Draupner Wave Analysis Second-order Time-Reassigned Synchrosqueezing Transform: Application to Draupner Wave Analysis Dominique Fourer and Franc¸ois Auger July 23, 2019 Abstract This paper addresses the problem of efficiently jointly representing a non- stationary multicomponent signal in time and frequency. We introduce a novel en- hancement of the time-reassigned synchrosqueezing method designed to compute sharpened and reversible representations of impulsive or strongly modulated sig- nals. After establishing theoretical relations of the new proposed method with our previous results, we illustrate in numerical experiments the improvement brought by our proposal when applied on both synthetic and real-world signals. Our ex- periments deal with an analysis of the Draupner wave record for which we provide pioneered time-frequency analysis results. 1 Introduction Time-frequency and time-scale analysis [1, 2, 3, 4] aim at developing efficient and innovative methods to deal with non-stationary multicomponent signals. Among the common approaches, the Short-Time Fourier Transform (STFT) and the Continuous Wavelet Transform (CWT) [5] are the simplest linear transforms which have been in- tensively applied in various applications such as audio [6], biomedical [7], seismic or radar. Unfortunately, these tools are limited by the Heisenberg-Gabor uncertainty prin- ciple. As a consequence, the resulting representations are blurred with a poor energy concentration and require a trade-off between the accuracy of the time or frequency arXiv:1907.09125v1 [eess.SP] 16 Jul 2019 localization. Another approach, the reassignment method [8, 9] was introduced as a mathematically elegant and efficient solution to improve the readability of a time- frequency representation (TFR). The inconvenience is that reassignment provides non- invertible TFRs which limits its interest to analysis or modeling applications. More recently, synchrosqueezing [10, 11] was introduced as a variant of the reas- signment technique due to its capability to provide sharpen and reversible TFRs. This reconstruction capability make this method continuously gaining interest since it paves the way of an infinite number of synchrosqueezing-based applications such as noise removal [12], signal components extraction or separation [10, 13, 14, 4]. 1 Nowadays, efforts are made to efficiently compute the synchrosqueezed version of several linear transforms such as STFT, CWT or S-transform [15, 16] and to im- prove the localization of strongly modulated signals using enhanced instantaneous fre- quency estimators [17, 18, 19]. To deal with impulses and strongly modulated signals, a new variant of the synchrosqueezing was introduced and called time-reassigned syn- chrosqueezing method [20]. However, this method cannot efficiently deal with mixed- content signals containing both impulsive and periodic components. In the present paper, we propose to introduce a novel transform called the second- order horizontal synchrosqueezing aiming to improve the energy localization and the readability of the time-reassigned synchrosqueezing while remaining reversible. To this end, we use an enhanced group-delay estimator which can be mathematically re- lated to our previous results [18]. This paper is organized as follows. In Section 2, the proper definitions of the con- sidered transforms with their properties are presented. In Section 3, we introduce our new second-order time-reassigned synchrosqueezing transform. Section 4 presents nu- merical experiments involving both synthetic and real-world signals. Finally, future work directions are given in Section 5. 2 Time-reassigned synchrosqueezing in a nutshell 2.1 Definitions and properties We define the STFT of a signal x as a function of time t and frequency ! computed using a differentiable analysis window h as: Z h ∗ −j!τ Fx (t; !) = x(τ)h(t − τ) e dτ (1) R where j2=−1 is the imaginary unit and z∗ is the complex conjugate of z.A TFR also h 2 called spectrogram is defined as jFx (t; !)j . Thus, the marginalization over time of h Fx (t; !) leads to: Z ZZ h ∗ −j!τ Fx (t; !) dt = h(t − τ) x(τ) e dtdτ (2) 2 R ZZR = h(u)∗x(τ) e−j!τ dudτ (3) 2 Z R Z = h(u)∗du x(τ) e−j!τ dτ (4) R R ∗ = Fh(0) Fx(!) (5) R −j!t with Fx(!) = x(t) e dt the Fourier transform of signal x. Now, from Eq. (5) R one can compute the Fourier Transform of x as: 1 Z F (!) = F h(t; !) dt (6) x F (0)∗ x h R 2 and the following signal reconstruction formula can be deduced after applying the Fourier inversion formula: ZZ 1 h j!t x(t) = ∗ Fx (τ; !) e dτd!: (7) 2πF (0) 2 h R 2.2 Reassignment To improve the readability of a TFR, reassignment moves the signal energy according to: (t; !) 7! (t^x(t;!); !^x(t;!)), where t^x(t; !) is a group-delay estimator and !^x(t; !) is an instantaneous frequency estimator [9]. Both time-frequency reassignment operators t^and !^ can be computed as follows in the STFT case [21, 16]: T h Fx (t;!) ^ (t;!) ~ (t;!) ~ (t;!) tx = Re tx ; with tx =t − h (8) Fx (t;!) Dh Fx (t;!) (t;!) (t;!) (t;!) !^x = Im (~!x ) ; with! ~x =j! + h (9) Fx (t;!) dh where T h(t) =th(t) and Dh(t) = dt (t) are modified versions of the analysis window h. h Finally, a reassigned spectrogram can be computed as RFx(t; !) = ZZ h 2 ^ jFx (τ; Ω)j δ t − tx(τ; Ω) δ (! − !^x(τ; Ω)) dτdΩ: (10) 2 R The resulting reassigned spectrogram RFx(t; !) is a sharpened but non-reversible TFR due to the loss of the phase information. 2.3 Time-reassigned synchrosqueezed STFT To overcome the problem of non reversibility, synchrosqueezing proposes to move the signal transform instead of its energy, to preserve the phase information of the original transform. Hence, time-reassigned synchrosqueezed STFT can be defined as [20]: Z h h ^ Sx (t; !) = Fx (τ; !)δ t − tx(τ; !) dτ (11) R where t^x(t; !) corresponds to the time reassignment operator which is classically com- puted using Eq. (8). The marginalization over time of the resulting transform leads to: Z ZZ h h ^ Sx (t; !)dt = Fx (τ; !)δ t − tx(τ; !) dtdτ (12) 2 R Z R h ∗ = Fx (τ; !)dτ = Fh(0) Fx(!): (13) R 3 Hence, an exact signal reconstruction from its synchrosqueezed STFT can be de- duced from Eq. (13) as: ZZ 1 h j!t x(t) = ∗ Sx (τ; !) e dτd!: (14) 2πF (0) 2 h R SNR=25.00 dB, L=8.00 SNR=25.00 dB, L=8.00 SNR=25.00 dB, L=8.00 0.45 0.45 0.45 0.4 0.4 0.4 0.35 0.35 0.35 0.3 0.3 0.3 0.25 0.25 0.25 0.2 0.2 0.2 0.15 0.15 0.15 0.1 0.1 0.1 normalized frequency normalized frequency normalized frequency 0.05 0.05 0.05 0 0 0 50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500 time samples time samples time samples (a) spectrogram (b) synchrosqueezing (c) second-order vertical syn- chrosqueezing SNR=25.00 dB, L=8.00 SNR=25.00 dB, L=8.00 SNR=25.00 dB, L=8.00 0.45 0.45 0.45 0.4 0.4 0.4 0.35 0.35 0.35 0.3 0.3 0.3 0.25 0.25 0.25 0.2 0.2 0.2 0.15 0.15 0.15 0.1 0.1 0.1 normalized frequency normalized frequency normalized frequency 0.05 0.05 0.05 0 0 0 50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500 time samples time samples time samples (d) reassigned spectrogram (e) time-reassigned synchrosqueez- (f) second-order horizontal syn- ing chrosqueezing Figure 1: Comparisons of the resulting TFRs of a synthetic multicomponent signal. The TFRs obtained using the synchrosqueezing methods (b),(c), (e) and (f) correpond to their squared modulus. 3 Second-order horizontal synchrosqueezing 3.1 Enhanced group-delay estimation Let’s consider a linear chirp signal model expressed as [18]: x(t) = eλx(t)+jφx(t) (15) t2 with λ (t) = l + µ t + ν (16) x x x x 2 t2 and φ (t) = ' + ! t + α (17) x x x x 2 4 where λx(t) and φx(t) respectively stand for the log-amplitude and phase and with qx = νx + jαx and px = µx + j!x. For such a signal, it can be shown [18] that px =! ~x(t;!) − qx t~x(t;!), and therefore: !x = Im(~!x(t;!) − qx t~x(t;!)) =! ^x(t;!) − Im(qx t~x(t;!)) (18) h The proposed second-order horizontal synchrosqueezing consists in moving Fx (t;!) (2) from the point (t;!) to the point (tx ;!) located on the instantaneous frequency curve, _ (2) dφx (2) (2) i.e. such that φ(tx )= dt (tx ) = !x + αxtx = !. This leads to: (2) ! − !x ^ ! − !^x(t;!) νx ~ tx = = tx(t;!) + + Im(tx(t;!)) (19) αx αx αx which can be estimated by: ( !−!^x(t;!)+Im(^qx(t;!) t~x(t;!)) ifα ^x(t;!) 6=0 ^(2) α^x(t;!) tx (t;!) = (20) t^x(t; !) otherwise where q^x(t;!) =ν ^x(t;!) + jα^x(t;!) is an unbiased estimator of qx. This expression can be compared to the second-order group delay estimator introduced by Oberlin et al. [17] ( !−!^x(t;!) t^x(t; !) + ifα ^x(t;!) 6=0 ^(2b) α^x(t;!) tx (t;!) = : (21) t^x(t; !) otherwise (2b) 2 ^ d λx It can be shown using Eq.(19) that estimator tx is biased when νx = dt2 (t) 6= 0.
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