Quadratic Time-frequency Research Articles

Pilot Decontamination over Time Frequency and Space Domains in Multi-Cell Massive MIMO System

In this article, we show that Pilot contamination problem can be seen as a source separation problem using time, frequency, and space domains. Our method capitalizes on a nonunitary joint diagonalization of spatial quadratic time-frequency (STFD) signal representation to identify the desired and interfering users. We first compute the noise subspace from the STFD matrices selected appropriately. Secondly, we use the noise subspace obtained to estimate the Elevation (El) and the Azimuth (Az) angles for which the MUSIC cost function is maximized. Numerical simulations are provided to illustrate the effectiveness and the behavior of the proposed approach.

Time-Frequency Represetation of Radar Signals Using Doppler-Lag Block Searching Wigner-Ville Distribution

Radar signals are time-varying signals where the signal parameters change over time. For these signals, Quadratic Time-Frequency Distribution (QTFD) offers advantages over classical spectrum estimation in terms of frequency and time resolution but it suffers heavily from cross-terms. In generating accurate Time-Frequency Representation (TFR), a kernel function must be able to suppress cross-terms while maintaining auto-terms energy especially in a non-cooperative environment where the parameters of the actual signal are unknown. Thus, a new signal-dependent QTFD is proposed that adaptively estimates the kernel parameters for a wide class of radar signals. The adaptive procedure, Doppler-Lag Block Searching (DLBS) kernel estimation was developed to serve this purpose. Accurate TFRs produced for all simulated radar signals with Instantaneous Frequency (IF) estimation performance are verified using Monte Carlo simulation meeting the requirements of the Cramer-Rao Lower Bound (CRLB) at SNR > 6 dB.

EEG-Based Emotion Recognition Using Quadratic Time-Frequency Distribution.

Accurate recognition and understating of human emotions is an essential skill that can improve the collaboration between humans and machines. In this vein, electroencephalogram (EEG)-based emotion recognition is considered an active research field with challenging issues regarding the analyses of the nonstationary EEG signals and the extraction of salient features that can be used to achieve accurate emotion recognition. In this paper, an EEG-based emotion recognition approach with a novel time-frequency feature extraction technique is presented. In particular, a quadratic time-frequency distribution (QTFD) is employed to construct a high resolution time-frequency representation of the EEG signals and capture the spectral variations of the EEG signals over time. To reduce the dimensionality of the constructed QTFD-based representation, a set of 13 time- and frequency-domain features is extended to the joint time-frequency-domain and employed to quantify the QTFD-based time-frequency representation of the EEG signals. Moreover, to describe different emotion classes, we have utilized the 2D arousal-valence plane to develop four emotion labeling schemes of the EEG signals, such that each emotion labeling scheme defines a set of emotion classes. The extracted time-frequency features are used to construct a set of subject-specific support vector machine classifiers to classify the EEG signals of each subject into the different emotion classes that are defined using each of the four emotion labeling schemes. The performance of the proposed approach is evaluated using a publicly available EEG dataset, namely the DEAPdataset. Moreover, we design three performance evaluation analyses, namely the channel-based analysis, feature-based analysis and neutral class exclusion analysis, to quantify the effects of utilizing different groups of EEG channels that cover various regions in the brain, reducing the dimensionality of the extracted time-frequency features and excluding the EEG signals that correspond to the neutral class, on the capability of the proposed approach to discriminate between different emotion classes. The results reported in the current study demonstrate the efficacy of the proposed QTFD-based approach in recognizing different emotion classes. In particular, the average classification accuracies obtained in differentiating between the various emotion classes defined using each of the four emotion labeling schemes are within the range of –. Moreover, the emotion classification accuracies achieved by our proposed approach are higher than the results reported in several existing state-of-the-art EEG-based emotion recognition studies.

Refining the ambiguity domain characteristics of non-stationary signals for improved time–frequency analysis: Test case of multidirectional and multicomponent piecewise LFM and HFM signals

This paper aims at providing a more accurate description of the ambiguity domain characteristics of a piecewise multicomponent non-stationary signals with focus on piece-wise linear frequency modulated (LFM) (PW-LFM) signal and a mixed LFM and hyperbolic FM (HFM). The main motivation comes from the observed PW-LFM nature of several real life signals. It is essential that the characteristics of these types of signals be taken into account for the design of high resolution Time–Frequency Distributions (TFDs) and therefore to improve the application and interpretation of time–frequency signal analysis and processing. In this paper the ambiguity function (AF) of a general PW-LFM signal is derived exactly and then analyzed to deduce important properties. The precise location and behavior of both auto-terms and cross-terms of a general piecewise LFM signal can be deduced from its AF. Numerical simulations using different types of test signals confirm the analytical derivations. An extension of the PW-LFM test signal is also presented by using HFM signals. For such signals, the exact analytical expression of auto-terms is given as well as the expression of cross-terms between HFM and LFM. The results obtained can be used in future studies to design more advanced quadratic time–frequency distributions (QTFDs) that exhibit improved properties in terms of resolution and accuracy.

Automatic Detection and Classification of Audio Events for Road Surveillance Applications.

This work investigates the problem of detecting hazardous events on roads by designing an audio surveillance system that automatically detects perilous situations such as car crashes and tire skidding. In recent years, research has shown several visual surveillance systems that have been proposed for road monitoring to detect accidents with an aim to improve safety procedures in emergency cases. However, the visual information alone cannot detect certain events such as car crashes and tire skidding, especially under adverse and visually cluttered weather conditions such as snowfall, rain, and fog. Consequently, the incorporation of microphones and audio event detectors based on audio processing can significantly enhance the detection accuracy of such surveillance systems. This paper proposes to combine time-domain, frequency-domain, and joint time-frequency features extracted from a class of quadratic time-frequency distributions (QTFDs) to detect events on roads through audio analysis and processing. Experiments were carried out using a publicly available dataset. The experimental results conform the effectiveness of the proposed approach for detecting hazardous events on roads as demonstrated by 7% improvement of accuracy rate when compared against methods that use individual temporal and spectral features.

A structure for representation of localized time-frequency distributions in presence of noise

This paper considers multi-component non-stationary signals contaminated with additive white Gaussian noise and suggests a structure to remove high noise from quadratic time-frequency distribution (QTFD) of these signals while localizing auto-terms. Proposed structure includes three steps. It first reduces some of the noise power by filtering the distribution in ambiguity domain and thus enhances signal-to-noise ratio. In the second step, the regions corresponding to the auto-terms are determined and by presenting a time-varying threshold, the noise out of auto-terms is eliminated in each time interval. Finally, using reassignment principle, the auto-terms are concentrated to local centroids of their energy. The typical reassignment method localizes both auto-terms and noise components while the proposed procedure relocates only the auto-terms to their original positions. Simulation results illustrate the performance of the proposed structure as compared to more standard state-of-the-art QTFDs.

Time–frequency analysis in infant cry classification using quadratic time frequency distributions

This paper presents a new investigation of time–frequency (t–f) based signal processing approach using quadratic time–frequency distributions (QTFDs) namely spectrogram (SPEC), Wigner–Ville distribution (WVD), Smoothed–Wigner Ville distribution (SWVD), Choi–William distribution (CWD) and modified B-distribution (MBD) for classification of infant cry signals. t–f approaches have proved as an efficient approach for applications involving the non stationary signals. In feature extraction, a cluster of t–f based features were extracted by extending the time-domain and frequency-domain features to the joint t–f domain from the generated t–f representation. Conventional features such as mel-frequency cepstral coefficients (MFCCs) and linear prediction coefficients (LPCs) were also extracted in order to compare the effectiveness of the t–f methods. The efficacy of the extracted feature vectors was validated using probabilistic neural network (PNN) and general regression neural network (GRNN). The proposed methodology was implemented to classify different sets of binary classification problems of infant cry signals from different native. The best empirical result of above 90% was reported and revealed the good potential of t–f methods in the context of infant cry classification.

On the Symplectic Covariance and Interferences of Time-Frequency Distributions

We study the covariance property of quadratic time-frequency distributions with respect to the action of the extended symplectic group. We show how covariance is related to and in fact in competition with the possibility of damping the interferences which arise due to the quadratic nature of the distributions. We also show that the well-known fully covariance property of the Wigner distribution in fact characterizes it (up to a constant factor) among the quadratic distributions $L^{2}(\mathbb{R}^{n})\rightarrow C_{0}({ \mathbb{R}^{2n}})$. A similar characterization for the closely related Weyl transform is given as well. The results are illustrated by several numerical experiments for the Wigner and Born--Jordan distributions of the sum of four Gaussian functions in the so-called diamond configuration.

Discrete Time-Frequency Signal Analysis and Processing Techniques for Non-Stationary Signals

This paper presents the methodology, properties and processing of the time-frequency techniques for non-stationary signals, which are frequently used in biomedical, communication and image processing fields. Two classes of time-frequency analysis techniques are chosen for this study. One is short-time Fourier Transform (STFT) technique from linear time-frequency analysis and the other is the Wigner-Ville Distribution (WVD) from Quadratic time-frequency analysis technique. Algorithms for both these techniques are developed and implemented on non-stationary signals for spectrum analysis. The results of this study revealed that the WVD and its classes are most suitable for time-frequency analysis.

A robust parameter estimation of FHSS signals using time–frequency analysis in a non-cooperative environment

Frequency hopping spread spectrum (FHSS) signals are widely implemented in both modern civilian and military applications. They are robust to channel impairments because of their low probability of interception. For applications that require the interception of FHSS signals, the signal parameters such as the hopping frequencies, hopping duration and hopping sequence should be accurately measured. In this paper, an accurate FHSS signal parameter estimation method is proposed based on quadratic time–frequency distributions (QTFDs). The extended modified B-distribution (EMBD) and the adaptive smoothed Wigner–Ville (SWWVD) are used which have the properties of high time–frequency resolution. The adaptive SWWVD requires no prior knowledge of the signal parameters since the kernel parameters are estimated from the signal characteristics and are compared to the EMBD which operates at the optimal kernel parameters. The proposed instantaneous frequency (IF) estimate method is compared to the time–frequency (TF) moments method and benchmarked with the Cramer–Rao lower bounds (CRLBs). The computational complexity of the IF estimation method is reduced by a factor of five compared to the TF moments method. Furthermore, the results show that the IF estimation method outperforms moments method where the mean-squared error (MSE) of the hopping frequencies estimate meets at minimum SNR of −3 dB and the hopping duration estimate MSE meets the CRLB at SNR of 0 dB.

Quadratic Time-Frequency Transforms-Based Brillouin Optical Time-Domain Reflectometry

Linear and quadratic Time-Frequency (T-F) transforms are proposed for the signal processing of the Brillouin optical time-domain reflectometry, to improve the system performance, in terms of the transition responsivity and frequency resolution. Various T-F transforms, including linear T-F transform (short time Fourier transform) and quadratic transforms (Choi–Williams, Zhao-Atlas-Marks, smoothed pseudo Wigner–Ville, S-method, and adaptive spectrogram) are applied to the experimental backscattered time-domain spontaneous Brillouin signals. Multiple T-F approaches can be jointly applied to efficiently improve the time and frequency resolutions simultaneously. Results show that the SWPV transform provides the best transition responsivity and frequency resolution simultaneously among the six T-F transforms because of its smoothing operations on frequency and time axis for reducing the cross-term interference.

Improving DOA Estimation Algorithms Using High-Resolution Quadratic Time-Frequency Distributions

This paper addresses the problem of direction of arrival (DOA) estimation and blind source separation (BSS) for nonstationary signals in the underdetermined case. These two problems are strongly related to the mixing matrix estimation problem. To deal with the nonstationary characteristics of signals, this study uses high-resolution quadratic time-frequency distributions (TFDs) to reduce cross-terms while keeping a good resolution for the construction of spatial TFDs. The main contributions of this paper are two-fold. First, the formulation of a statistical test for the noise thresholding step improves robustness and avoids the use of empirical parameters; this test performs multisource selection of the time-frequency points where the signal of interest is present. Second, an algorithm based on image processing methods performs an auto-source selection for mixing matrix estimation. The results on simulated signals demonstrate an improvement of 10 dB in terms of normalized mean square error for BSS and 7% in terms of relative error for DOA estimation over standard methods.

Time-Frequency Analysis for GNSSs: From interference mitigation to system monitoring

In this article, we discuss the important role of time-frequency (TF) signal representations in enhancing global navigation satellite system (GNSS) receiver performance. Both linear transforms and quadratic TF distributions (QTFDs) are considered. We review recent advances of antijam techniques that exploit the distinction in the TF signatures between the desired and undesired signals, enabling effective jammer excision with minimum distortion to the navigation signals. The characterization of jammers by their instantaneous frequencies (IFs) lends itself to sparse structures in the TF domain and, as such, invites compressive sensing (CS) and sparse reconstruction to aid in joint-variable domain jamming localization and suppression. Furthermore, the integration of the spatial domain with the TFDs, through the use of multiantenna receivers, permits the applications of space-time processing for effective jamming mitigation. The article also describes the fundamental role of TFDs in monitoring the performance of new GNSSs, including satellite clocks and ionospheric scintillation data. Real GNSS data collected in the presence of jamming are used to demonstrate the effectiveness of TF-based antijam approaches

Sparsity-Aware Adaptive Directional Time–Frequency Distribution for Source Localization

Multi-component characteristics and missing data samples introduce artifacts and cross-terms in quadratic time–frequency distributions, thus affecting their readability. In this study, we propose a new time–frequency method that employs directional smoothing and compressive sensing to reduce cross-terms and mitigate artifacts associated with missing samples. The efficacy of the proposed time–frequency distribution for solving real-life problems is illustrated by employing it to estimate direction of arrival of sparsely sampled sources in under-determined scenario. Numerical results show that the proposed method is superior to other state-of-the-art methods both in terms of obtaining clear time–frequency representation and accurately estimating direction of arrival.

An Improved Design of High-Resolution Quadratic Time–Frequency Distributions for the Analysis of Nonstationary Multicomponent Signals Using Directional Compact Kernels

This paper presents a new advanced methodology for designing high resolution time–frequency distributions (TFDs) of multicomponent nonstationary signals that can be approximated using piece-wise linear frequency modulated (PW-LFM) signals. Most previous kernel design methods assumed that signals auto-terms are mostly centered around the origin of the $(\nu,\tau)$ ambiguity domain while signal cross-terms are mostly away from the origin. This study uses a multicomponent test signal for which each component is modeled as a PW-LFM signal; it finds that the above assumption is a very rough approximation of the location of the auto-terms energy and cross-terms energy in the ambiguity domain and it is only valid for signals that are well separated in the $(t,f)$ domain. A refined investigation led to improved specifications for separating cross-terms from auto-terms in the $(\nu,\tau)$ ambiguity domain. The resulting approach first represents the signal in the ambiguity domain, and then applies a multidirectional signal dependent compact kernel that accounts for the direction of the auto-terms energy. The resulting multidirectional distribution (MDD) approach proves to be more effective than classical methods like extended modified B distribution, S-method, or compact kernel distribution in terms of auto-terms resolution and cross-terms suppression. Results on simulated and real data validate the improved performance of the MDD, showing up to 8% gain as compared to more standard state-of-the-art TFDs.

SSP-Based UBI Algorithms for Uniform Linear Array

The underdetermined blind identification (UBI) for uniform linear array (ULA) with complex-valued mixing matrix can be processed effectively by the algorithms based on single-source-point (SSP) detection. In this paper, we propose two novel SSP-based methods of UBI in the time–frequency (TF) domain for ULA. One method, called UBI based on linear TF transform (UBI-LT), presents a new SSP detection criterion based on short-time Fourier transform, which modifies IME-RSSP (proposed by Li) by exploiting the phase information of mixture. The other method proposes a new SSP detection criterion based on a cross-term suppression quadratic TF distribution called UBI based on modified quadratic TF distribution (UBI-MQD), which can be seen as an improved version of the SSP-based algorithm proposed by Su. After performing these SSP detection criteria, two methods employ the peak detection and a clustering algorithm to estimate the complex-valued mixing matrix. Two methods have their own advantages and can be chosen by robust systems or high-performance systems. Numerical simulation results show that (1) the proposed methods have better performance than the existing methods with the same means of TF analysis (linear TF transform or quadratic TF distribution), and (2) UBI-LT is more robust than UBI-MQD even on the condition that the source number is large and the signal-to-noise (SNR) is low, while UBI-MQD has higher performance than UBI-LT when the source number is small and the SNR is high.

Wave physics of the frequency difference autoproduct: Bilinear time-frequency analysis

Frequency difference beamforming (Abadi et al., 2012, JASA, 132, 3018-3029) and matched field processing (Worthmann et al., 2015, JASA, 138, 3549-3562) are array signal processing techniques that determine acoustic ray-path directions and source locations, respectively, from sparse array recordings by shifting the signal processing to a lower, out-of-band, difference-frequency bandwidth. This frequency down-shift is accomplished through the use of the frequency difference autoproduct (Δf-AP), a quadratic product of complex frequency-domain acoustic field amplitudes at two different frequencies. In this presentation, the physical and mathematical underpinnings of this quadratic product are discussed. Specifically, in the branch of applied mathematics termed bilinear time-frequency analysis, this Δf-AP is actually one of four quadratic time-frequency functions, each inter-related through forward and inverse Fourier transforms. The other three functions are the time-domain autoproduct, the Wigner distribution, and the ambiguity function. These four functions and several of their interesting properties are reviewed, including their relationship to the Wiener-Khintchine Theorem, and the spatial and spectral coherence and correlation functions. Additionally, the behavior of these functions when evaluated from simulated acoustic recordings from ideal single-path and multi-path acoustic environments is shown and compared. [Sponsored by ONR and NSF.]

An automatic fast optimization of Quadratic Time-frequency Distribution using the hybrid genetic algorithm

This paper presents a novel framework for a fully automatic optimization of Quadratic Time-frequency Distributions (QTFDs). This ‘black box’ approach automatically adjusts the QTFD kernel parameters by using a hybrid genetic algorithm (HGA). This results in an optimal use of QTFDs suitable for non-specialist users without requiring any additional input except for the signal itself. This optimization problem has been formulated as the minimization of the cost function of a modified energy concentration measure. The efficiency of the proposed method has been demonstrated by representing selected non-stationary signals in the time-frequency domain and testing robustness under different SNR conditions by estimating the instantaneous frequency. A fast implementation of QTFD optimization reduces computation time significantly; e.g., the computation time of a real world bat signal of 400 samples reduces to 3.5885±0.3942s from its standard implementation (53.0910±1.445s).

How useful are nonlinear methods for EEG and MEG data analysis?

This paper presents a novel framework for a fully automatic optimization of Quadratic Time-frequency Distributions (QTFDs). This ‘black box’ approach automatically adjusts the QTFD kernel parameters by using a hybrid genetic algorithm (HGA). This results in an optimal use of QTFDs suitable for non-specialist users without requiring any additional input except for the signal itself. This optimization problem has been formulated as the minimization of the cost function of a modified energy concentration measure. The efficiency of the proposed method has been demonstrated by representing selected non-stationary signals in the time-frequency domain and testing robustness under different SNR conditions by estimating the instantaneous frequency. A fast implementation of QTFD optimization reduces computation time significantly; e.g., the computation time of a real world bat signal of 400 samples reduces to 3.5885±0.3942 s from its standard implementation (53.0910±1.445 s).

A multivariate time-frequency approach for tracking QT variability changes unrelated to heart rate variability.

The beat-to-beat variability of the QT interval (QTV) is a marker of ventricular repolarization (VR) dynamics and it has been suggested as an index of sympathetic ventricular outflow and cardiac instability. However, QTV is also affected by RR (or heart rate) variability (RRV), and QTV due to RRV may reduce QTV specificity as a VR marker. Therefore, it would be desirable to separate QTV due to VR dynamics from QTV due to RRV. To do that, previous work has mainly focused on heart rate corrections or time-invariant autoregressive models. This paper describes a novel framework that extends classical multiple inputs/single output theory to the time-frequency (TF) domain to quantify QTV and RRV interactions. Quadratic TF distributions and TF coherence function are utilized to separate QTV into two partial (conditioned) spectra representing QTV related and unrelated to RRV, and to provide an estimates of intrinsic VR dynamics. In a simulation study, a time-varying ARMA model was used to generate signals representing realistic RRV and VR dynamics with controlled instantaneous frequencies and powers. The results demonstrated that the proposed methodology is able to accurately track changes in VR dynamics, with a correlation between theoretical and estimated patterns higher than 0.88. Data from healthy volunteers undergoing a tilt table test were analyzed and representative examples are discussed. Results show that the QTV unrelated to RRV dynamics quickly increased during orthostatic challenge.