Analysis of local singularities If the time-frequency representations are useful to bring to the fore the progression with time of the frequency of a signal, the time-scale representations are more adapted to the analysis of irregular structures and singularities, or of signals presenting self-similarities (such as fractional Brownian motion, [GF92]). We give in the following such an example with the analysis of local singularities, thanks to the scalogram and the Unterberger distribution. The local regularity of a signal can be characterized by its Holder (or Lipschitz or scaling) exponent: for a signal $x(t)$ which is uniformly Holder $H$, there exists a constant $C$ such that $$\begin{eqnarray*} \vert x(s)-x(t)\vert\ \ \leq\ \ C\ \vert s-t\vert^H,\ \ 0<H<1. \end{eqnarray*}$$ $H$ then represents the exponent of regularity of the signal. If we consider the wavelet transform $T_x(t,a;\Psi)$ of this signal, with an analyzing wavelet $\Psi$ such that $t\ \Psi(t)$ is absolutely integrable, then one can show that $$\begin{eqnarray*} \vert T_x(t,a;\Psi)\vert&\leq& C\ \vert a\vert^{H+1/2}\ \int_{... ...t\Psi(t)\vert\ dt\\ &=& O(\vert a\vert^{H+1/2})\ \ \forall\ t, \end{eqnarray*}$$ or, in terms of scalogram and behavior when $a$ tends to 0, $$\begin{eqnarray*} E\left[\vert T_x(t,a;\Psi)\vert^2\right] \ \sim\ \vert a\vert^{2H+1},\ \ a\rightarrow 0. \end{eqnarray*}$$ where $E[.]$ refers to the expectation. This means that the regularity of the signal can be recovered from the behavior of its scalogram at small scales, and it is possible to show that the reciprocal is true. Since they are time-dependent in nature, the wavelet-based techniques also allow an estimation of the local regularity of a signal. In some sense, time-scale methods offer in this respect a framework similar to the one provided by time-frequency analysis for tracking the time evolution of spectral features. Indeed, if we now have, at a given time $t_0$, $$\vert x(t_0+\tau)-x(t_0)\vert\ \leq\ C\ \vert\tau\vert^{H(t_0)},\ \ 0<H(t_0)<1,$$ (5.2) then we can establish the inequality $$\begin{eqnarray*} \vert T_x(t,a;\Psi)\vert&\leq& C\ \vert a\vert^{H(t_0)+1/2}\ \... ... &=& O(\vert a\vert^{H(t_0)+1/2} + \vert t-t_0\vert^{H(t_0)}). \end{eqnarray*}$$ We then obtain an image of the signal's regularity at the small scales of its wavelet transform (or scalogram), but accompanied with a time localization. The reciprocal is also true, which means that an appropriate decrease of the wavelet (scalogram) coefficients in a cone-shaped region of the time-frequency plane allows one to estimate the local regularity of a signal. If we further impose to condition (5.2) that the signal presents an asymptotic spectral decrease, $$\begin{eqnarray*} X(\nu) \ \sim\ \vert\nu\vert^{-(1+2H(t_0))}\ e^{j2\pi \nu t_0}\ \ \mbox{ for }\ \ \vert\nu\vert\rightarrow\infty, \end{eqnarray*}$$ then we have the following approximation for the active Unterberger distribution: $$\begin{eqnarray*} U_x(t,a) \ \sim\ \vert a\vert^{2(1+H(t_0))}\ \delta(t-t_0),\ \ a\rightarrow 0. \end{eqnarray*}$$ Thus, the Unterberger distribution follows a law along scales which gives access to the strength of the singularity ($H$), and along time to the localization of this singularity. The file holder.m estimates the Holder exponent of any signal from an affine time-frequency representation of it. o Example For instance, we consider a 64-points Lipschitz singularity (see anasing.m) of strength $H=0$, centered at $t_0=32$, >> sig=anasing(64); and we analyze it with the scalogram (Morlet wavelet with half-length = 4, see fig. 5.7), >> [tfr,t,f]=tfrscalo(sig,1:64,4,0.01,0.5,256,1); Figure 5.7: Scalogram of a Lipschitz singularity at time $t=32$, of strength $H=0$ The time-localization of the singularity can be clearly estimated from the scalogram distribution at small scales : >> H=holber(tfr,f,1,256,32) ------> H=-0.0381 If we now consider a singularity of strength H=-0.5 (see fig. 5.8), >> sig=anasing(64,32,-0.5); >> [tfr,t,f]=tfrscalo(sig,1:64,4,0.01,0.5,256,1); Figure 5.8: Scalogram of a Lipschitz singularity at time $t=32$, of strength $H=-0.5$ we notice the different behavior of the scalogram along scales, whose decrease is characteristic of the strength $H$. The estimation of the Holder exponent at $t=32$ gives : >> H=holber(tfr,f,1,256,32) ------> H=-0.5107 which is close to 0.5. The same conclusions can be observed from the active Unterberger distribution.
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