Short-time Fourier Transform Coefficients Research Articles

Statistical Synchrosqueezing Transform and Its Application to Seismic Thin Interbed Analysis

Synchrosqueezing transform (SST) benefits from an instantaneous frequency (IF) estimator in the time-frequency domain, providing an energy-concentrated time-frequency representation to describe the time-varying frequency of seismic signals. To enhance the concentration performance of SST, this paper theoretically proposes a statistical SST (SSST) by constructing a spectrum-weighted IF estimator in the short-time Fourier transform (STFT) domain. In this SSST, a linear chirp signal is introduced to better capture its chirp rate, and then the quadratic STFT spectra are weighted by a window-related weighting function. On this basis, two equations related to the IF and chirp rate are constructed to derive the spectrum-weighted IF estimator. Finally, the STFT coefficients are squeezed to the estimated IF trajectories by a frequency fixed-point iterative algorithm, thereby providing a more concentrated time-frequency representation for multi-component signals than existing advanced methods, while enabling to retrieve each component. Two synthetic examples and one field seismic data on thin interbeds are utilized to demonstrate the effectiveness of the proposed SSST and show its ability to highlight the time-varying frequency features of seismic signals, which is a promising seismic data analysis tool, such as characterizing thin interbed thickness variations.

Automatic seizure detection based on Gray Level Co-occurrence Matrix of STFT imaged-EEG

Epilepsy is a complicated neurological disorder, that features recurrent seizures. During a seizure, the repetitive action potentials lead to a high-frequency burst, and a hyper synchronization happens among the activities of a population of cortical neurons concurrently. In this study, we propose a repetition-based seizure detection method, using Gray Level Co-occurrence Matrix (GLCM) and Electroencephalogram (EEG) signals. In this method, imaged-EEGs are made from the Short Time Fourier Transform (STFT) coefficients of two sequential epochs. The GLCMs of these images are calculated and one feature named diagGLCM is extracted from each co-occurrence matrix. This feature demonstrates the repetition of the STFT coefficient of a frequency band in two sequential epochs, which are used for a classification task between ictal (during a seizure) and inter-ictal (between seizures) time segments. Using one frequency band’s STFT repetition, a patient-specific seizure detection algorithm is developed with an accuracy of 99.56%, a sensitivity of 99.52%, and specificity of 99.62%. This accurate classification shows the efficiency of the extracted feature which can be a biomarker for the ictal repetitive action potentials.

A Novel Approach For Identification Of Exon Locations In DNA Sequences Using GLC Window

The application of signal processing techniques for identification of exons in Deoxyribonucleic acid (DNA) sequence is a challenging task. The objective of this paper is to introduce a combinational window approach for locating exons in DNA sequence. In contrast to the traditional single window function for evaluation of short time Fourier transform (STFT), this work proposes a novel method for evaluating STFT coefficients using a combinational window function comprising of Gaussian, Lanczos and Chebyshev (GLC) windows. The chosen combinational window GLC has the highest relative side lobe attenuation values compared to other window functions introduced by various researchers. The proposed algorithm incorporates GLC window function for evaluating STFT coefficients and in the design of FIR bandpass filter. Simulation results revealed its effectiveness in improving the evaluation parameters like Sensitivity, Specificity, Accuracy, Area under curve (AUC), Discrimination Measure (DM). Furthermore, the proposed algorithm has been applied successfully to some universal benchmark datasets like C. elegans, Homosapiens, etc., The proposed method has shown to be an efficient approach for the prediction of protein coding regions compared to other existing methods. All the simulations are done using the MATLAB 2016a.

Time-varying fault feature extraction of rolling bearing via time–frequency sparsity

The joint time–frequency (TF) distribution is a critical method of describing the instantaneous frequency that changes with time. To eliminate the errors caused by strong modulation and noise interference in the process of time-varying fault feature extraction, this paper proposes a novel approach called second-order time–frequency sparse representation (SOTFSR), which is based on convex optimization in the domain of second-order short-time Fourier transform (SOSTFT) where the TF feature manifests itself as a relative sparsity. According to the second-order local estimation of the phase function, SOSTFT can provide a sparse TF coefficient in the short-time Fourier transform (STFT) domain. To obtain the optimal TF coefficient matrix from noisy observations, it is innovatively formulated as a typical convex optimization problem. Subsequently, a multivariate generalized minimax concave penalty is employed to maintain the convexity of the least-squares cost function to be minimized. The aim of the proposed SOTFSR is to obtain the optimal STFT coefficient in the TF domain for extraction of time-varying features and for perfect signal reconstruction. To verify the superiority of the proposed method, we collect the multi-component simulation signals and the signals under variable speed from a rolling bearing with an inner ring fault. The experimental results show that the proposed method can effectively extract the time-varying fault characteristics.

Joint Optimization of Deep Neural Network-Based Dereverberation and Beamforming for Sound Event Detection in Multi-Channel Environments.

In this paper, we propose joint optimization of deep neural network (DNN)-supported dereverberation and beamforming for the convolutional recurrent neural network (CRNN)-based sound event detection (SED) in multi-channel environments. First, the short-time Fourier transform (STFT) coefficients are calculated from multi-channel audio signals under the noisy and reverberant environments, which are then enhanced by the DNN-supported weighted prediction error (WPE) dereverberation with the estimated masks. Next, the STFT coefficients of the dereverberated multi-channel audio signals are conveyed to the DNN-supported minimum variance distortionless response (MVDR) beamformer in which DNN-supported MVDR beamforming is carried out with the source and noise masks estimated by the DNN. As a result, the single-channel enhanced STFT coefficients are shown at the output and tossed to the CRNN-based SED system, and then, the three modules are jointly trained by the single loss function designed for SED. Furthermore, to ease the difficulty of training a deep learning model for SED caused by the imbalance in the amount of data for each class, the focal loss is used as a loss function. Experimental results show that joint training of DNN-supported dereverberation and beamforming with the SED model under the supervision of focal loss significantly improves the performance under the noisy and reverberant environments.

Probabilistic model of speech with high spectral resolution using maximum-likelihood estimation

A probabilistic model for the distribution of narrowband short-time Fourier transform (STFT) coefficients, having frequency resolution of 5 Hz and below, is proposed for speech signals. An important application is to model speech to depict the perceptual stability of human listening capability which allows humans to perceive speech reliably under a wide range of acoustic conditions. Representation of speech with high spectral resolution finds applications in the design of digital hearing aids and cochlear implants for people with hearing disability. While speech is generally considered as a non-stationary signal over segment lengths longer than 20–30 ms, the perceptual stability of human auditory system motivates the need for an invariant representation of speech over long segment lengths. Computer modelling shows that STFT coefficients of speech with high-frequency resolution fit reasonably accurately to Laplace distribution (LD). Parameters of the corresponding LD are estimated using maximum-likelihood estimation. Cramer–Rao bound for the estimated parameters and root mean square error for the fitted distribution confirm the validity of the fitted distribution.

Expectation‐maximisation for speech source separation using convolutive transfer function

This paper addresses the problem of under-determinded speech source separation from multichannel microphone singals, i.e. the convolutive mixtures of multiple sources. The time-domain signals are first transformed to the short-time Fourier transform (STFT) domain. To represent the room filters in the STFT domain, instead of the widely-used narrowband assumption, we propose to use a more accurate model, i.e. the convolutive transfer function (CTF). At each frequency band, the CTF coefficients of the mixing filters and the STFT coefficients of the sources are jointly estimated by maximizing the likelihood of the microphone signals, which is resolved by an Expectation-Maximization (EM) algorithm. Experiments show that the proposed method provides very satisfactory performance under highly reverberant environments.

Feature Selection Based on Binary Tree Growth Algorithm for the Classification of Myoelectric Signals

Electromyography (EMG) has been widely used in rehabilitation and myoelectric prosthetic applications. However, a recent increment in the number of EMG features has led to a high dimensional feature vector. This in turn will degrade the classification performance and increase the complexity of the recognition system. In this paper, we have proposed two new feature selection methods based on a tree growth algorithm (TGA) for EMG signals classification. In the first approach, two transfer functions are implemented to convert the continuous TGA into a binary version. For the second approach, the swap, crossover, and mutation operators are introduced in a modified binary tree growth algorithm for enhancing the exploitation and exploration behaviors. In this study, short time Fourier transform (STFT) is employed to transform the EMG signals into time-frequency representation. The features are then extracted from the STFT coefficient and form a feature vector. Afterward, the proposed feature selection methods are applied to evaluate the best feature subset from a large available feature set. The experimental results show the superiority of MBTGA not only in terms of feature reduction, but also the classification performance.

A New Competitive Binary Grey Wolf Optimizer to Solve the Feature Selection Problem in EMG Signals Classification

Features extracted from the electromyography (EMG) signal normally consist of irrelevant and redundant features. Conventionally, feature selection is an effective way to evaluate the most informative features, which contributes to performance enhancement and feature reduction. Therefore, this article proposes a new competitive binary grey wolf optimizer (CBGWO) to solve the feature selection problem in EMG signals classification. Initially, short-time Fourier transform (STFT) transforms the EMG signal into time-frequency representation. Ten time-frequency features are extracted from the STFT coefficient. Then, the proposed method is used to evaluate the optimal feature subset from the original feature set. To evaluate the effectiveness of proposed method, CBGWO is compared with binary grey wolf optimization (BGWO1 and BGWO2), binary particle swarm optimization (BPSO), and genetic algorithm (GA). The experimental results show the superiority of CBGWO not only in classification performance, but also feature reduction. In addition, CBGWO has a very low computational cost, which is more suitable for real world application.

A Mixing Matrix Estimation Algorithm for the Time-Delayed Mixing Model of the Underdetermined Blind Source Separation Problem

Considering the time-delayed mixing model of the underdetermined blind source separation problem, we propose a novel mixing matrix estimation algorithm in this paper. First, we introduce the short-time Fourier transform (STFT) to transform the mixed signals from the time domain to the time–frequency domain. Second, a neoteric transformation matrix is addressed to construct the linear clustering property of STFT coefficients. Then, a preeminent detection algorithm is raised to identify the single source points. After eliminating the low-energy points and outliers in the time–frequency domain, a potential function of clustering approach is put forward to cluster the single source points and obtain the clustering centers. Finally, the mixing matrix can be estimated through the derivation and calculation. The experimental results validate that the proposed algorithm not only accurately estimates the mixing matrix for the time-delayed mixing model of the underdetermined blind source separation problem but also has certain universality for different array structures. Therefore, both the effectiveness and superiority of the proposed algorithm have been verified.

A Pitch-Synchronous Simultaneous Detection-Estimation Framework for Speech Enhancement

Speech enhancement methods formulated in the short-time Fourier transform (STFT) domain vary in the statistical assumptions made on the STFT coefficients, in the optimization criteria applied or in the models of the signal components. Recently, approaches relying on a stochastic-deterministic speech model have been proposed. The deterministic part of the signal corresponds to harmonically related sinusoids, often used to represent voiced speech. The stochastic part models signal components that are not captured by the deterministic components. In this paper, we consider this scenario under a new perspective yielding three main contributions. First, a pitch-synchronous signal representation is considered and shown to be advantageous for the estimation of the harmonic model parameters. Second, we model the harmonic amplitudes in voiced speech as random variables with frequency bin dependent Gamma distributions. Finally, distinct estimators for the different models of voiced speech, unvoiced speech, and speech absence are derived. To select from the arising estimates, we take into account the mutual impact of detection and estimation by proposing a binary decision framework that is derived from a Bayesian risk function. The resulting pitch-synchronous stochastic-deterministic estimator outperforms several benchmark methods in terms of speech intelligibility and perceived quality predicted by instrumental measures for various noise types and different signal-to-noise ratios.

Underdetermined blind separation of overlapped speech mixtures in time-frequency domain with estimated number of sources

Noise suppression and the estimation of the number of sources are two practical issues in applications of underdetermined blind source separation (UBSS). This paper proposes a noise-robust instantaneous UBSS algorithm for highly overlapped speech sources in the short-time Fourier transform (STFT) domain. The proposed algorithm firstly estimates the unknown complex-valued mixing matrix and the number of sources, which are then used to compute the STFT coefficients of corresponding sources at each auto-source time-frequency (TF) point. After that, the original sources are recovered by the inverse STFT. To mitigate the noise effect on the detection of auto-source TF points, we propose a method to effectively detect the auto-term location of the sources by using the principal component analysis (PCA) of the STFTs of noisy mixtures. The PCA-based detection method can achieve similar UBSS outcome as some filtering-based methods. More importantly, an efficient method to estimate the mixing matrix is proposed based on subspace projection and clustering approaches. The number of sources is obtained by counting the number of the resultant clusters. Evaluations have been carried out by using the speech corpus NOIZEUS and the experimental results have shown improved robustness and efficiency of the proposed algorithm.

Fast computation of the spectral correlation

Although the Spectral Correlation is one of the most versatile spectral tools to analyze cyclostationary signals (i.e. signals comprising hidden periodicities or repetitive patterns), its use in condition monitoring has so far been hindered by its high computational cost. The Cyclic Modulation Spectrum (the Fourier transform of the spectrogram) stands as a much faster alternative, yet it suffers from the uncertainty principle and is thus limited to detect relatively slow periodic modulations. This paper fixes the situation by proposing a new fast estimator of the spectral correlation, the Fast Spectral Correlation, based on the short-time Fourier transform (STFT). It proceeds from the property that, for a cyclostationary signal, the STFT evidences periodic flows of energy in and across its frequency bins. The Fourier transform of the interactions of the STFT coefficients then returns a quantity which scans the Spectral Correlation along its cyclic frequency axis. The gain in computational cost as compared to the conventional estimator is like the ratio of the signal length to the STFT window length and can therefore be considerable. The validity of the proposed estimator is demonstrated on non trivial vibration signals (very weak bearing signatures and speed varying cases) and its computational advantage is used to compute a new quantity, the Enhanced Envelope Spectrum.

Speech dereverberation using weighted prediction error with correlated inter-frame speech components

In this paper, we propose a new dereverberation approach based on the weighted prediction error (WPE) method implemented in the short-time Fourier transform (STFT) domain. Our main contribution is to model the temporal correlation of the STFT coefficients across analysis frames, referred to as inter-frame correlation (IFC), and exploit it in the dereverberation process. Since accurate modeling of the IFC is not tractable, we consider an approximate model wherein only a finite number of consecutive speech frames are considered correlated. It is shown that, given an estimate of the IFC matrix, the proposed approach results in a convex quadratic optimization problem with respect to the reverberation prediction weights, and a closed-form solution can be accordingly derived. Furthermore, an efficient method for the estimation of the underlying IFC matrix is developed based on the extension of a recently proposed speech variance estimator. We evaluate the performance of our approach incorporating the estimated IFC matrix and compare it to the original and several variants of the WPE method. The results reveal lower residual reverberation and higher overall quality of the enhanced speech when the proposed method is employed.

On the STFT Inversion Redundancy

The inversion of the short-time Fourier transform (STFT) calculated with common windows is revisited. It has been shown that, using a commonly windowed STFT, a full signal reconstruction may be achieved from the same number of the STFT coefficients as the number of signal samples. Architectures for the redundant and the nonredundant STFT realization are presented. Inversion stability is checked through a noise analysis in the reconstruction process.

Recent Developments in Speech Enhancement in the Short-Time Fourier Transform Domain

In this paper, we present an overview on the topic of noise reduction in the short-time Fourier transform (STFT) domain. First, we briefly review the conventional literature in the single- and multichannel cases separately. In the single-channel scenario, we focus on the spectral subtractive methods, Wiener filter based methods, speech amplitude estimators and estimators of the complex STFT coefficients. In the multi-channel scenario, we investigate in short a selection of key beamforming approaches as well as conventional post-filtering methods. Next, a detailed survey of the most recent advances in the STFT-based noise reduction methods is provided. This includes STSA estimators with super-Gaussian priors, noise power spectral density (PSD ) estimation, estimation methods in the modulation domain, estimation of spectral phase and noise PSD matrix estimation for multi-channel applications. Finally, we summarize the presented material and draw important conclusions on each of the investigated topics.

Direction-of-arrival estimation of passive acoustic sources in reverberant environments based on the Householder transformation.

This paper presents an approach to the direction-of-arrival (DOA) estimation problem in acoustic environments using microphone arrays. It works in the short-time Fourier transform (STFT) domain. It first transforms the noisy speech signals received at the array into the STFT domain. A Householder transformation is then constructed and applied to the multichannel STFT coefficients in each subband. This transformation converts the multichannel STFT coefficients into two components: one is a single coefficient that is dominated by the signal of interest and the other consists of the M - 1 coefficient that is dominated by noise (or even consists of noise-only if there is no reverberation), where M is the number of sensors. A cost function is then formed from the outputs of the Householder transformation and the DOA information can subsequently be obtained by searching the extremum value of this cost function in the angle range between 0° and 180°. Simulation results are provided to illustrate the performance of this approach.

A minimum variance distortionless response filter based on the bifrequency spectrum for single-channel noise reduction

This paper deals with the problem of single-channel noise reduction in the short-time Fourier transform (STFT) domain. Many algorithms have been developed to solve this important problem, most of which generally assume that the STFT coefficients in different frequency bands are uncorrelated, so the noise reduction is achieved by applying a gain function to the STFT of the noisy speech in each frequency band. However, this assumption is not accurate and the STFT coefficients of speech signals between neighboring frequency bands are correlated in practice due to the use of small lengths of the fast Fourier transform (FFT) and overlap add/save techniques in implementation. This paper formulates the noise reduction problem by taking into account the interband correlation using the so-called bifrequency spectrum. Based on this formulation, a single-channel minimum variance distortionless response (MVDR) filter is derived, which is shown to be able to significantly improve the signal-to-noise ratio (SNR) and meanwhile maintain the desired speech not much distorted. Simulations are presented to justify the claimed merits of the developed MVDR filter.

Markov random field in speech enhancement: Application for tonal languages

This paper proposed speech enhancement algorithm based on Markov random field (MRF) model for Thai, a tonal language. Firstly, a noisy speech signal is transformed using the short time Fourier transform (STFT). In so doing, noise is removed and speech is preserved, especially harmonics information as f0 patterns are relevant perceptual cues for lexical tones. The voice activity detector is used to classify each STFT time frame into voiced and unvoiced. Harmonics information is retrieved from each voiced time frame, where four neighborhoods of the analyzed STFT coefficients include its adjacent time frames (left, right) and nearest harmonics (top, bottom). For the unvoiced, four adjacent coefficients (left, right, top, and bottom) are used. A two-state MRF model is used to classify STFT coefficients into speech and noise. Those with speech state are retained, while the rest is set to zero. The enhanced speech is estimated by the inverse STFT. Results from quality evaluation test on four sets of Thai rhyming words corrupted by white noise at SNR levels of 0, 5, and 10 dB showed that the proposed algorithm significantly improved SNR of noisy speeches compared with spectral subtraction (1.3 dB on average) and Wiener filtering (1.9 dB on average).

Consistent Wiener Filtering for Audio Source Separation

Wiener filtering is one of the most ubiquitous tools in signal processing, in particular for signal denoising and source separation. In the context of audio, it is typically applied in the time-frequency domain by means of the short-time Fourier transform (STFT). Such processing does generally not take into account the relationship between STFT coefficients in different time-frequency bins due to the redundancy of the STFT, which we refer to as consistency. We propose to enforce this relationship in the design of the Wiener filter, either as a hard constraint or as a soft penalty. We derive two conjugate gradient algorithms for the computation of the filter coefficients and show improved audio source separation performance compared to the classical Wiener filter both in oracle and in blind conditions.