Emmanuel Bacry

Audio Denoising by Time-Frequency Block Thresholding

Removing noise from audio signals requires a nondiagonal processing of time-frequency coefficients to avoid producing ldquomusical noise.rdquo State of the art algorithms perform a parameterized filtering of spectrogram coefficients with empirically fixed parameters. A block thresholding estimation procedure is introduced, which adjusts all parameters adaptively to signal property by minimizing a Stein estimation of the risk. Numerical experiments demonstrate the performance and robustness of this procedure through objective and subjective evaluations.

Analysis of sound signals with high resolution matching pursuit

Sound recordings include transients and sustained parts. Their analysis with a basis expansion is not rich enough to represent efficiently all such components. Pursuit algorithms choose the decomposition vectors depending upon the signal properties. The dictionary among which these vectors are selected is much larger than a basis. Matching pursuit is fast to compute, but can provide coarse representations. Basis pursuit gives a better representation but is very expensive in terms of calculation time. This paper develops a high resolution matching pursuit: it is a fast, high time-resolution, time-frequency analysis algorithm, that makes it likely to be used far musical applications.

Audio Signal Denoising with Complex Wavelets and Adaptive Block Attenuation

We investigate a new audio denoising algorithm. Complex wavelets protect phase of signals and are thus preferred in audio signal processing to real wavelets. The block attenuation eliminates the residual noise artifacts in reconstructed signals and provides a good approximation of the attenuation with oracle. A connection between the block attenuation and the decision-directed a priori SNR estimator of Ephraim and Malah is studied. Finally we introduce an adaptive block technique based on the dyadic CART algorithm. The experiments show that not only the proposed method does eliminate the residual noise artifacts, but it also preserves transients of signals better than short-time Fourier based methods do.

Linear processes in high dimensions: Phase space and critical properties

In this work we investigate the generic properties of a stochastic linear model in the regime of high dimensionality. We consider in particular the vector autoregressive (VAR) model and the multivariate Hawkes process. We analyze both deterministic and random versions of these models, showing the existence of a stable phase and an unstable phase. We find that along the transition region separating the two regimes the correlations of the process decay slowly, and we characterize the conditions under which these slow correlations are expected to become power laws. We check our findings with numerical simulations showing remarkable agreement with our predictions. We finally argue that real systems with a strong degree of self-interaction are naturally characterized by this type of slow relaxation of the correlations.

Random cascade model in the limit of infinite integral scale as the exponential of a non-stationary

In this paper we propose a new model for volatility fluctuations in financial time series. This model relies on a nonstationary Gaussian process that exhibits aging behavior. It turns out that its properties, over any finite time interval, are very close to continuous cascade models. These latter models are indeed well known to reproduce faithfully the main stylized facts of financial time series. However, it involves a large-scale parameter (the so-called integral scale where the cascade is initiated) that is hard to interpret in finance. Moreover, the empirical value of the integral scale is in general deeply correlated to the overall length of the sample. This feature is precisely predicted by our model, which, as illustrated by various examples from daily stock index data, quantitatively reproduces the empirical observations.

1/f

The Wavelet Transform Modulus Maxima method is used to analyze the fractal scaling properties of DNA sequences. This method, based on the definition of partition functions which use the values of the wavelet transform at its modulus maxima, allows to determine accurately the singularity spectrum of a given singular signal. By considering analyzing wavelets that make the wavelet transform microscope blind to `patches of different nucleotide compositions which are ubiquitous to genomic sequences, we demonstrate and quantify the existence of long-range correlations in the noncoding regions. The fluctuations around the patchy landscapes of the DNA walks reconstructed from both the noncoding and coding regions are found to have Gaussian statistics. Whereas the fluctuations from the former behave like fractional brownian motions, those of the latter cannot be distinguished from uncorrelated random brownian walks.

noise. Application to volatility fluctuations in stock markets

Pattern formation in systems far from equilibrium is a subject of considerable current interest1–6. Recently, much effort has been directed towards the study of fractal growth phenomena in physical, chemical and biological systems7,8. Unfortunately, the understanding of phenomena like viscous fingering in Hele-Shaw cells9 and electrochemical deposition10 is hampered by the mathematical complexity of the problem. Highly ramified structures are generally produced in the zero surface tension limit. In this limit, both processes are equivalent to a Stefan problem11 : a diffusion problem for the pressure or the electrochemical potential, with boundary values specified on the moving interface, whose local velocity is in turn determined by the normal gradient of the Laplace field. This highly nonlinear problem is not readily amenable even to modern numerical simulations. When solving the Stefan problem by direct means, the interface develops unphysical cusps in a finite time9. One is thus led to introduce some short-distance cutoff which in some sense mimics surface tension12,13. Thus far no computer simulations of the equations of motion achieve the necessary size to make definite conclusions about the deterministic character of the fractal patterns observed in the experiments1–6.