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Page 1 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin One, Two or Many Frequencies: Synchrosqueezing and Multicomponent Signal Analysis Steve McLaughlin Heriot-Watt, School of Engineering & Physical Sciences with thanks to Patrick Flandrin, Mike Davies, Yannis Kopsinis, Sylvain Meignen, Thomas Oberlin

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Page 2 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin

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Page 3 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin

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Page 4 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Structure Introduction Signals, Time and Frequency Empirical Mode Decomposition SynchroSqueezing

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Page 5 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin What are signals? A signal is a time (or space) varying quantity that can carry information. The concept is broad, and hard to define precisely. (Wikipedia)

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Page 6 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin What are signals made of? The Frequency viewpoint (Fourier): Signals can be built from the sum of harmonic functions (sine waves) Joseph Fourier signal Fourier coefficients Harmonic functions

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Page 7 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin What is a Signal? W. Tecumseh Fitch, Calls out of chaos: the adaptive significance of nonlinear phenomena in mammalian vocal production, Animal Behaviour, 2002

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Page 8 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin What is a Signal Frequency modulation (FM) implies the instantaneous frequency varies. This contrasts with amplitude modulation, in which it is the amplitude which is varied while its frequency remains constant. Many signals can be modeled as a sum of AM and FM components.

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Page 9 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Empirical Mode Decomposition Basic idea is very simple: signal = fast oscillation + slow oscillation (& iteration) - Separation fast vs slow data driven - Local analysis based on neighbouring extrema - Oscillation rather than frequency

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Page 10 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Empirical Mode Decomposition Identify local maxima and local minima Deduce an upper envelope and a lower envelope by interpolation (cubic splines) –subtract the mean envelope from the signal –iterate until mean envelope = 0" (sifting) –subtract the obtained mode from the signal –iterate on the residual

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Page 11 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Empirical Mode Decomposition: Sifting

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Page 12 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin EMD - Summary Locality The method operates at the scale of one oscillation Adaptive The decomposition is fully data-driven Multi-resolution The iterative process explores sequentially the natural scales of a signal Oscillations of any type No assumption on the nature of oscillations (e.g., harmonic) 1 nonlinear oscillation = 1 mode No analytic definition The decomposition is only defined as the output of an algorithm analysis and evaluation ?

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Page 13 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin What is frequency? Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency. The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency. (Wikipedia)

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Page 14 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Fatima Miranda Macaque Amolops tormotus What is frequency?frequency

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Page 15 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Hilbert Spectrum Can we produce TF-representation plots based on the IF, IA estimates? Hilbert Spectrum: IA and IF are combined in order to form a spectrogram-like plot, H(t,f ). Specifically, for any time instant t, H(t,f )=α(t) if IF(t) equals to f and zero at the rest of the frequencies. In practice, a Time – Frequency grid is adopted in order to construct H(t,f ).

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Page 16 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin On Instantaneous Frequency (Monocomponent signals) In the case of monocomponent signals, IF coincides with the actual frequency of the signal.

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Page 17 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Instantaneous Frequency Multicomponent signals IF is a non-symmetric oscillation exhibiting spikes that point toward the component with the larger amplitude. The strength of the IF oscillations (that is the height of the spikes) depends on the relative amplitudes of the components. Smoothing,, forces IF to lock at the component with the higher amplitude

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Page 18 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Instantaneous Frequency Multicomponent signals The further apart in frequency the components are, the larger the amplitude of oscillations become. The frequency of oscillations is relative to the frequency difference.

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Page 19 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Beating When a signal is composed of two components with close instantaneous frequencies, EMD exhibits a beating phenomenon [Rilling and Flandrin (2008)]. More precisely, if a signal is composed of two harmonics, i.e., f(t) = cos(2πt)+a cos(2πξ 0 t), Rilling and Flandrin [2008] show an interesting zone in the a vs. ξ plane (amplitude vs. frequency plane) where EMD misidentifies the sum of two components as only a single component; the precise shape of this zone depends on the value ξ 0. They also quantified this phenomenon and termed it beating, because EMD identified the two harmonics as a single oscillating (i.e., beating) signal.

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Page 20 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Time Frequency Reassignment and SSQ Time-Frequency Reassignment originates from a study of the STFT, which smears the energy of the superimposed Instantaneous Frequencies around their center frequencies in the spectrogram. TFR analysis reassigns these energies to sharpen the spectrogram Synchrosqueezing is a special case of reassignment methods [Auger and Flandrin (1995); Chassande–Mottin et al. (2003, 1997)]. –SSQ reallocates the coefficients resulting from a continuous wavelet transform based on the frequency information, to get a concentrated picture over the time-frequency plane, from which the instantaneous frequencies are then extracted –It is adaptive to the signal; –Reconstruction of the signal from the reallocated coefficients is feasible –Solid Theoretical foundations

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Page 21 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin SynchroSqueezing Idea from Daubechies & Maes 1996 –Concentrate (squeeze) wavelet coefficients, at fixed times, on the basis of local frequency estimation (synchronised) Guarantees a sharply localised representation Allows for reconstruction of identified modes Offers a mathematical tractable alternative to EMD

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Page 22 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Synchrosqueezing – Reconstruction Approach We are interested in the retrieval of the components f k of a multicomponent signal f defined by: The problem can be viewed from two perspectives. The first consists of computing an approximation of the modes, either by using EMD or via wavelet projections. The second uses the SST to reallocate the WT before proceeding with multicomponent retrieval

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Page 23 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Underlying theory First recall some results first proposed by Daubechies et al For a signal defined as a superposition of intrinsic-mode-type functions (IMT):

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Page 24 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Underlying theory Where d is a separation condition.

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Page 25 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Theorem

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Page 26 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Theorem - Conditions

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Page 27 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Theorem – A Remark Component retrieval using the SST is thus first based on the computation of the synchrosqueezing operator and then on its integration in the vicinity of curves defined by..

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Page 28 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Retrieval of Multiple Components (RCM): Practical Implemetation (Brevdo et al 2011) The strategy developed by Brevdo et al for the RCM is to proceed on a component by component basis. The idea is to find a curve,in the time-frequency plane such that it maximizes the energy which forces the modes to be smooth through a total variation term penalization and is obtained by computing the following quantity:

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Page 29 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin SST Parameters - WT of Pure Harmonic Signals When dealing with a signal made of pure harmonics The separation of the components using the WT is equivalent to the support of the WT of each component being disjoint that is for all k>1 : Finally, using the definition of the separation condition introduced in Definition 2, when there is no interference between the components and

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Page 30 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin SST Parameters - WT of an ε-IMT So, when is high, a low approximation error requires that must be small. One way to ensure this is to impose that has a fast decay or equivalently, that be large. Let f(t) be a mode as defined in Definition 2, then the WT of f is now approximated by: the approximation error being controlled by (C 1 (f)B 1 +C 2 (f)B 2 +(C 3 (f)B 3 ), (see Brevdo et al 2011), where C i, 1i3, does not depend on the wavelet choice, while

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Page 31 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin SST Parameters - WT of a Superposition of ε-IMT For a multi-component signal as defined in Definition 2 we wish to have the following approximation: So, following the study on a single component, a low approximation error requires that be large enough. Then, we also want each component to be well separated from the other components (which is compulsory if we are to use the WT for mode retrieval), that is for each (a,t) the sum above should be reduced to a single term which is true provided that: Taking into account the separation condition (see Definition 2), we can deduce that the above inequality is true as soon as

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Page 32 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin

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Page 33 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Related Work It is worth noting that the behavior of the SST was studied in detail by Wu et al 2011 for two tone signals. However, no conclusions can be drawn from the latter regarding modulated signals because the modulation considerably alters the wavelet representation of such signals and consequently the results given by the SST. A very similar study was carried out by Mallat, ([14, p. 102]), but using a mother wavelet which was symmetric and compactly supported in the time domain. In such a case, the second order terms in the approximation would be negligible only if for all k: provided

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Page 34 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Observation The size of the frequency support of the mother wavelet plays a fundamental role when dealing with multi-component signals; adjusting this parameter enables us to tune between good separation and good localization of the modes. In the sequel we will use the previous remark regarding the size of the frequency support of the mother wavelet to build a new RCM algorithm. This second constraint when applied to an -IMT implies that Φ k is such that,provided that, which is true when,or equivalently. If we assume that both A k and are bounded below, the quality of the approximation is better if is small when is large. Since one must also have ( where is the frequency bandwidth of the wavelet) to ensure separation of a different component, cannot be taken arbitrarily small. In contrast to the previous approach, the trade-off parameter between separation and localization of the components is now.

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Page 35 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin A New RCM Code at: The algorithm aims at identifying the ridge associated with each component from the magnitude of the WT and then reconstructing the components by using the information in the vicinity of the different ridges. It is based on three distinct steps: - first determine adaptively the number of modes - compute an optimal wavelet threshold - use threshold to retrieve the modes.

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Page 36 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Automatic detection of the number of modes Since, for a given time t, the scales of interest in the SST are those where is large enough, we aim at detecting the modes using the following sets: The natural expectation is that given t and when γ is appropriately chosen, each element will be associated with a single mode as defined above). Note: The normalization of the wavelet transform we use is necessary to give sense to the above set. We compute an estimate of the number of modes at time t is: The number of modes is defined as:

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Page 37 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Wavelet Threshold Once the number of modes is determined, we first consider the sets: and define T 0 as the times t associated with non empty S 0 (t). A threshold for the WT and t in T 0 is then computed as follows:

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Page 38 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin The Algorithm for Retrieving the Modes We consider for each t in T 0 the set I nf-k+1 (t), which enables us to retrieve the mode k by applying the following approximation formula: Note that the index of I in the sum comes from the fact that I k for small k corresponds to a high frequency component and that in Definition (2) the components are arranged in increasing frequency order. This reconstruction procedure is fully automatic and avoids the need to narrow down the curve searching in the time-scale space before proceeding with the mode retrieval.

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Page 39 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Example of a signal made of non-trivial intrinsic mode type signals

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Page 40 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Modulus of the corresponding Wavelet Transform

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Page 41 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Reconstruction using a bump wavelet

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Page 42 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Reconstruction as before but using alternative reconstruction formulae

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Page 43 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin

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Page 44 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Non Uniform Sampling To adaptively choose non-uniform samples to minimize the interpolation error is a thorny issue. A method inspired by the EMD consists of selecting the extrema of the signal, such that they depend locally on the strength of the oscillations. Another possibility would be to choose equi-spaced sample points over the whole signal duration. Methods 1. Consider non-uniform sample points for the location of the extrema of the high frequency mode given by the RCM proposed and then piecewise cubic Hermite interpolation of f at these points. 2. Follow the same framework as the above, but the sample points being equi-spaced with the same average sampling rate.

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Page 45 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin

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Page 46 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Further Work Strong modulation Non-uniform sampling Signals with a piecewise regular TF representation

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Page 47 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin Thank You

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Page 48 of 48 One, Two or Many Frequencies:Synchrosqueezing SIERRA - Lyon, 13 th March 2013 Steve McLaughlin References E. Brevdo, N. S. Fuckar, G. Thakur, and H.-T. Wu, The synchrosqueezing algorithm for time-varying spectral analysis: Robust properties and new plaeoclimate applications, 2011, arXiv preprint ID: E. Chassande-Mottin, I. Daubechies, F. Auger, and P. Flandrin, Differential reassignment, IEEE Signal Process. Lett., vol. 4, no. 10, pp. 293–294, I. Daubechies, J. Lu, and H.-L. Wu, Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool, Appl. Computat. Harmon. Anal., vol. 20, no. 2, pp. 243– 261, H.-T. Wu, P. Flandrin, and I. Daubechies, One or two frequencies? The synchrosqueezing answer, Adv. Adapt. Data Anal., vol. 3, no. 1–2, pp. 29–39, S. Mallat, A Wavelet Tour on Signal Processing. NewYork:Academic,1998. G. Rilling, P. Flandrin, and P. Goncalves, On empirical mode decomposition and its algorithms, in Proc. IEEE-EURASIP Workshop on Nonlin. Signal and Image Process. (NSIP), Grado (I), Jun, 2003.

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