Noise Corruption of Empirical Mode Decomposition and Its Effect on Instantaneous Frequency

Huang's Empirical Mode Decomposition (EMD) is an algorithm for analyzing nonstationary data that provides a localized time-frequency representation by decomposing the data into adaptively defined modes. EMD can be used to estimate a signal's instantaneous frequency (IF) but suffers from poor performance in the presence of noise. To produce a meaningful IF, each mode of the decomposition must be nearly monochromatic, a condition that is not guaranteed by the algorithm and fails to be met when the signal is corrupted by noise. In this work, the extraction of modes containing both signal and noise is identified as the cause of poor IF estimation. The specific mechanism by which such "transition" modes are extracted is detailed and builds on the observation of Flandrin and Goncalves that EMD acts in a filter bank manner when analyzing pure noise. The mechanism is shown to be dependent on spectral leak between modes and the phase of the underlying signal. These ideas are developed through the use of simple signals and are tested on a synthetic seismic waveform.Comment: 28 pages, 19 figures. High quality color figures available on Daniel Kaslovsky's website: http://amath.colorado.edu/student/kaslovsk

2-D Prony-Huang Transform: A New Tool for 2-D Spectral Analysis

This work proposes an extension of the 1-D Hilbert Huang transform for the analysis of images. The proposed method consists in (i) adaptively decomposing an image into oscillating parts called intrinsic mode functions (IMFs) using a mode decomposition procedure, and (ii) providing a local spectral analysis of the obtained IMFs in order to get the local amplitudes, frequencies, and orientations. For the decomposition step, we propose two robust 2-D mode decompositions based on non-smooth convex optimization: a "Genuine 2-D" approach, that constrains the local extrema of the IMFs, and a "Pseudo 2-D" approach, which constrains separately the extrema of lines, columns, and diagonals. The spectral analysis step is based on Prony annihilation property that is applied on small square patches of the IMFs. The resulting 2-D Prony-Huang transform is validated on simulated and real data.Comment: 24 pages, 7 figure

Spectral analysis for nonstationary audio

A new approach for the analysis of nonstationary signals is proposed, with a focus on audio applications. Following earlier contributions, nonstationarity is modeled via stationarity-breaking operators acting on Gaussian stationary random signals. The focus is on time warping and amplitude modulation, and an approximate maximum-likelihood approach based on suitable approximations in the wavelet transform domain is developed. This paper provides theoretical analysis of the approximations, and introduces JEFAS, a corresponding estimation algorithm. The latter is tested and validated on synthetic as well as real audio signal.Comment: IEEE/ACM Transactions on Audio, Speech and Language Processing, Institute of Electrical and Electronics Engineers, In pres

Instantaneous PSD Estimation for Speech Enhancement based on Generalized Principal Components

Power spectral density (PSD) estimates of various microphone signal components are essential to many speech enhancement procedures. As speech is highly non-nonstationary, performance improvements may be gained by maintaining time-variations in PSD estimates. In this paper, we propose an instantaneous PSD estimation approach based on generalized principal components. Similarly to other eigenspace-based PSD estimation approaches, we rely on recursive averaging in order to obtain a microphone signal correlation matrix estimate to be decomposed. However, instead of estimating the PSDs directly from the temporally smooth generalized eigenvalues of this matrix, yielding temporally smooth PSD estimates, we propose to estimate the PSDs from newly defined instantaneous generalized eigenvalues, yielding instantaneous PSD estimates. The instantaneous generalized eigenvalues are defined from the generalized principal components, i.e. a generalized eigenvector-based transform of the microphone signals. We further show that the smooth generalized eigenvalues can be understood as a recursive average of the instantaneous generalized eigenvalues. Simulation results comparing the multi-channel Wiener filter (MWF) with smooth and instantaneous PSD estimates indicate better speech enhancement performance for the latter. A MATLAB implementation is available online

Novel Fourier Quadrature Transforms and Analytic Signal Representations for Nonlinear and Non-stationary Time Series Analysis

The Hilbert transform (HT) and associated Gabor analytic signal (GAS) representation are well-known and widely used mathematical formulations for modeling and analysis of signals in various applications. In this study, like the HT, to obtain quadrature component of a signal, we propose the novel discrete Fourier cosine quadrature transforms (FCQTs) and discrete Fourier sine quadrature transforms (FSQTs), designated as Fourier quadrature transforms (FQTs). Using these FQTs, we propose sixteen Fourier-Singh analytic signal (FSAS) representations with following properties: (1) real part of eight FSAS representations is the original signal and imaginary part is the FCQT of the real part, (2) imaginary part of eight FSAS representations is the original signal and real part is the FSQT of the real part, (3) like the GAS, Fourier spectrum of the all FSAS representations has only positive frequencies, however unlike the GAS, the real and imaginary parts of the proposed FSAS representations are not orthogonal to each other. The Fourier decomposition method (FDM) is an adaptive data analysis approach to decompose a signal into a set of small number of Fourier intrinsic band functions which are AM-FM components. This study also proposes a new formulation of the FDM using the discrete cosine transform (DCT) with the GAS and FSAS representations, and demonstrate its efficacy for improved time-frequency-energy representation and analysis of nonlinear and non-stationary time series.Comment: 22 pages, 13 figure

The analysis of nonstationary vibration data

The general methodology for the analysis of arbitrary nonstationary random data is reviewed. A specific parametric model, called the product model, that has applications to space vehicle launch vibration data analysis is discussed. Illustrations are given using the nonstationary launch vibration data measured on the Space Shuttle orbiter vehicle