for many decades, there has been a general perception in the literature that Fourier methods are not suitable for the analysis of nonlinear and non-stationary data. In this paper, we propose a novel and adaptive Fourier decomposition method (FDM), based on the Fourier theory, and demonstrate its efficacy for the analysis of nonlinear and non-stationary time series. The proposed FDM decomposes any data into a small number of ‘Fourier intrinsic band functions’ (FIBFs). The FDM presents a generalized Fourier expansion with variable amplitudes and variable frequencies of a time series by the Fourier method itself. We propose an idea of zero-phase filter bank-based multivariate FDM (MFDM), for the analysis of multivariate nonlinear and non-stationary time series, using the FDM. We also present an algorithm to obtain cut-off frequencies for MFDM. The proposed MFDM generates a finite number of band-limited multivariate FIBFs (MFIBFs). The MFDM preserves some intrinsic physical properties of the multivariate data, such as scale alignment, trend and instantaneous frequency. The proposed methods provide a time–frequency–energy (TFE) distribution that reveals the intrinsic structure of a data. Numerical computations and simulations have been carried out and comparison is made with the empirical mode decomposition algorithms.

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2016.0871

The Fourier decomposition method for nonlinear and non-stationary time series analysis.

Fourier–Bessel series expansion based empirical wavelet transform for analysis of non-stationary signals

Time–frequency representation (TFR) is useful for non-stationary signal analysis as it provides information about the time-varying frequency components. This study proposes a novel TFR based on the improved eigenvalue decomposition of Hankel matrix and Hilbert transform (IEVDHM–HT). In the proposed method, first the authors decompose non-stationary signals using the IEVDHM with suitably defined criterion for eigenvalue selection, requirement of number of iterations, and new component merging criteria. Furthermore, the HT is applied on extracted components in order to obtain the TFR of non-stationary signals. The performance of proposed TFR has been evaluated on synthetic signals in clean and white noise environment with different signal-to-noise ratios. The proposed method gives good performance in terms of Renyi entropy measure in comparison with other existing methods. Application of the proposed TFR is also shown for the classification of epileptic seizure electroencephalogram (EEG) signals. The least-square support vector machine (LS-SVM) with radial basis function kernel is used for classification of seizure and seizure-free EEG signals obtained from the publicly available database by the University of Bonn, Germany. The proposed method has achieved classification accuracy 100% for the studied EEG database.

https://ietresearch.onlinelibrary.wiley.com/doi/pdf/10.1049/iet-smt.2017.0058

Time–frequency representation using IEVDHM–HT with application to classification of epileptic EEG signals

Cross-terms reduction in the Wigner-Ville distribution using tunable-Q wavelet transform

Epileptic seizure identification using entropy of FBSE based EEG rhythms

The decomposition of multi-component signals finds use in signal analysis. A new iterative approach to decompose a multi-component non-stationary signal into amplitude–frequency modulated (AM–FM) mono-component signals is presented in this paper. The proposed iterative approach is based on repeatedly performing eigenvalue decomposition (EVD) of the Hankel matrix, initially constructed from the samples of the multi-component signal. The components corresponding to significant eigenvalue pairs of the Hankel matrix are extracted. The process of constructing the Hankel matrix, performing EVD of the Hankel matrix and extraction of components corresponding to significant eigenvalue pairs, which we refer to as ‘Iteration’ is repeated till all the extracted components satisfy the defined mono-component signal criteria (MSC). The proposed decomposition approach being adaptive is suitable for analysis of non-stationary signals. The experimental results obtained by decomposing different kinds of synthetic and natural multi-component signals using the proposed iterative approach are presented and compared with the results obtained by empirical mode decomposition (EMD) and singular spectrum analysis (SSA). It is shown that unlike EMD, the ability of the proposed iterative approach to separate constituent mono-component signals is neither affected by the ratio of their mean frequencies nor by their relative amplitudes.

An iterative approach for decomposition of multi-component non-stationary signals based on eigenvalue decomposition of the Hankel matrix

We propose a robust event-based method for estimation of the instantaneous fundamental frequency of a voiced speech signal. The amplitude and frequency modulated (AM-FM) signal model of voiced speech in the low frequency range (LFR) indicates the presence of energy only around its instantaneous fundamental frequency (F0) and its few harmonics. The time-varying F0 component of a voiced speech signal is extracted by a robust algorithm which iteratively performs eigenvalue decomposition (EVD) of the Hankel matrix, initially constructed from samples of the LFR filtered voiced speech signal. The negative cycles of the extracted time-varying F0 component provide a reliable coarse estimate of intervals where glottal closure instants (GCIs) may be present. The negative cycles of the LFR filtered voiced speech signal occurring within these intervals are isolated. There is a sudden decrease in the glottal impedance at GCIs resulting in high signal strength. Therefore, GCIs are detected as local minima in the derivative of the falling edges of the isolated negative cycles of the LFR filtered voiced speech signal, followed by a selection criterion to discard false GCI candidates. The instantaneous F0 is estimated as the inverse of the time interval between two consecutive GCIs. Experiments were performed on the Keele and CSTR speech databases in white and babble noise environments at various levels of degradation to assess the performance of the proposed method. The proposed method substantially reduces the gross F0 estimation errors in comparison to some state of the art methods.

Event-based method for instantaneous fundamental frequency estimation from voiced speech based on eigenvalue decomposition of the Hankel matrix

In this paper, we present a novel method for robust and accurate identification of glottal closure instants (GCIs) from the voiced speech signal. The proposed method employs a new iterative algorithm based on the eigen value decomposition (EVD) of Hankel matrix to extract the time-varying fundamental frequency (F0) component of the voiced speech signal. The extracted F0 component is used to isolated the peak negative cycles of the low frequency range (LFR) filtered voiced speech signal. The GCIs are identified by detecting local minimas in the derivative of falling edges of peak negative cycles of the LFR filtered voiced speech signal which is followed by a selection criterion. The experimental results on speech signals under the white noise environment at various levels of degradation demonstrate that the proposed method outperforms existing methods in terms of accuracy and identification rate.

GCI identification from voiced speech using the eigen value decomposition of Hankel matrix

Epochs present in the voiced speech are defined as time instants of significant excitation of the vocal tract system during the production of speech. Nonstationary na- ture of excitation source and vocal tract system makes accurate identification of epochs a difficult task. Most of the existing methods for epoch detection require prior knowledge of voiced regions and a rough estimation of pitch frequency. In this paper, we propose a novel method that relies on time-order representation (TOR) based on short-time Fourier- Bessel (FB) series expansion which can be employed on entire speech signal to detect epochs without any prior information. The proposed method automatically detects voiced regions in the speech signal by computing the marginal energy density with respect to time in the low frequency range (LFR) from the energy distribution in the time-frequency plane. An estimate of pitch frequency for each detected voiced region is then obtained by computing the marginal energy density with respect to frequency in the LFR from the en- ergy distribution in the time-frequency plane. Epochs are located for each detected voiced region as peaks in the derivative of the low pass filtered (LPF) signal corresponding to falling edges of peak negative cycles in the LPF signal synthesized from TOR coefficients corresponding to LFR. Experimental results obtained by the proposed method on speech signals taken from the CMU-Arctic database are found to be promising. The proposed method detects epochs with high accuracy and reliability.

Time-Order Representation Based Method for Epoch Detection from Speech Signals

In this paper, we present a pseudo Wigner–Ville distribution (PWVD) based novel method for the voiced/non-voiced (V/NV) detection in noisy speech signals. The energy distribution of the speech signal on the time–frequency plane is obtained by computing the PWVD coefficients of the analytic speech signal over the low frequency range (LFR). The marginal energy density with respect to time (MEDT) over the low frequency range (LFR) derived from the energy distribution of the speech signal on the time–frequency plane is used as a feature to provide the instantaneous V/NV detection. The experimental results on speech signals from the CMU-Arctic database under white, babble and vehicular noise environments taken from the NOISEX-92 database at various signal to noise ratio (SNR) are obtained to assess the performance of the proposed method. A significant performance improvement in the V/NV detection accuracy is obtained by the proposed method over the existing methods for the V/NV detection under the white noise and babble/vehicular noise environments, respectively.

Pooja Jain

Papers

An iterative approach for decomposition of multi-component non-stationary signals based on eigenvalue decomposition of the Hankel matrix

Event-based method for instantaneous fundamental frequency estimation from voiced speech based on eigenvalue decomposition of the Hankel matrix

GCI identification from voiced speech using the eigen value decomposition of Hankel matrix

Time-Order Representation Based Method for Epoch Detection from Speech Signals

Marginal energy density over the low frequency range as a feature for voiced/non-voiced detection in noisy speech signals