Quadratic Time-frequency Distributions Research Articles

Nonorthogonal Joint Diagonalization/Zero Diagonalization for Source Separation Based on Time-Frequency Distributions

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This paper deals with the blind separation of instantaneous mixtures of source signals using time-frequency distributions (TFDs). We propose iterative algorithms to perform the nonorthogonal zero diagonalization and/or joint diagonalization of given sets of matrices. As an application, we show that the source separation can be realized by applying one of these algorithms to a set of spatial quadratic TFD matrices corresponding only to the so-called cross-source terms and/or to the so-called autosource terms. The determination of the above matrices to be jointly decomposed requires first an automatic selection procedure of useful time-frequency points. Regarding this last point, we also propose a new selection procedure and a modification of an existing one and provide a comparison with other existing ones. The nonorthogonal joint diagonalization and/or zero diagonalization algorithm's main advantage is to not require (in the blind source separation context) a prewhitening stage, which allows them to work even with a class of correlated signals and provides generally improved separation performance. Finally, an analytical example and computer simulations are provided in order to illustrate the effectiveness of the proposed approach and to compare it with classical ones. </para>

Underdetermined Blind Separation of Nondisjoint Sources in the Time-Frequency Domain

This paper considers the blind separation of nonstationary sources in the underdetermined case, when there are more sources than sensors. A general framework for this problem is to work on sources that are sparse in some signal representation domain. Recently, two methods have been proposed with respect to the time-frequency (TF) domain. The first uses quadratic time-frequency distributions (TFDs) and a clustering approach, and the second uses a linear TFD. Both of these methods assume that the sources are disjoint in the TF domain; i.e., there is, at most, one source present at a point in the TF domain. In this paper, we relax this assumption by allowing the sources to be TF-nondisjoint to a certain extent. In particular, the number of sources present at a point is strictly less than the number of sensors. The separation can still be achieved due to subspace projection that allows us to identify the sources present and to estimate their corresponding TFD values. In particular, we propose two subspace-based algorithms for TF-nondisjoint sources: one uses quadratic TFDs and the other a linear TFD. Another contribution of this paper is a new estimation procedure for the mixing matrix. Finally, then numerical performance of the proposed methods are provided highlighting their performance gain compared to existing ones

A proposed High-Resolution Time-Frequency Distribution for the Analysis of Multicomponent and Speech Signals

Abstract— In this paper, we propose a novel time-frequencydistribution (TFD) for the analysis of multi-component signals. In particular, we use synthetic as well as real-life speech signals toprove the superiority of the proposed TFD in comparison to someexisting ones. In the comparison, we consider the cross-termssuppression and the high energy concentration of the signal around its instantaneous frequency (IF). Keywords — Cohen’s Class, Multicomponent signal, Separable Kernel, Speech signal, Time- frequency resolution. I. I NTRODUCTION HE spectrogram, a smoothed version of the well-knownWigner-Ville distribution (WVD), has been widely usedin speech applications [1], [2], [3], [4]. The spectrogram,which is in general a cross-terms free time-frequencydistribution (TFD), suffers from the undesirable trade-offbetween the time concentration and the frequencyconcentration. To address the problem of cross-termssuppression, while keeping a high time-frequency resolution,other TFDs have been proposed. Among these, one can citethe smoothed pseudo WVD (SPWVD) [9], the Zhao-Atlas-Marks distribution (ZAMD) [5] and the B-distribution (BD)[6], just to name a few. In this paper, we present a new distribution for the analysis of multicomponent signals. This distribution, inspired from the Butterworth kernel quadraticTFD [8], has the ability of suppressing the cross-terms while keeping a high-resolution in the time-frequency plane. Toassess the performance of this proposed distribution, we alsopropose to compare it to some existing ones known for theircross-terms suppression property. In the comparisonexamples, we use synthetic as well as real-life data from aspeech application. The comparison results show the highperformance of the proposed TFD in dealing with non-stationary multicomponent signals.The paper is organized as follows: In Section 2, a theoretical aspect of some TFR interest is presented. A proposed hightime-frequency resolution quadratic TFD is introduced inSection 3. In Section 4, simulations and comparison examplesas well as a discussion are presented. Section 5 concludes thepaper.II. THEORETICAL BACKGROUND FOR QUADRATICTFD

Instantaneous frequency estimation based on directionally smoothed pseudo-Wigner-Ville distribution bank

In the cases of low signal-to-noise ratios (SNRs), estimating the instantaneous frequency (IF) curves of signals of interest is an interesting topic with many practical applications. Most of the existing methods are based on quadratic time-frequency (TF) distributions, which, however, yield a number of outliers in the cases of low SNRs. In this paper, we construct a new family of TF distributions, namely, the joint distributions, to estimate the IF curves in order to reduce outliers in the cases of low SNRs. The construction of the joint distributions is based on the definition of the directionally smoothed pseudo-Wigner-Ville distributions (DSPWVD) and pointwise adaptive weight averaging of a bank of DSPWVDs with different directions. The segments of the IF curve whose directions are close to that of the DSPWVD can be highlighted by each DSPWVD and the entire IF curve will be enhanced by the joint TF distribution. Simulation results show that the IF estimator based on the joint distributions outperforms that using quadratic TF distributions and the adaptive optimal adaptive kernel distribution in the cases of low SNRs.

Controlled multifunctional processing of causal signals on the basis of nonlinear (quadratic and cubic) time-frequency distributions

General laws of the controlled multifunctional processing based on nonlinear (quadratic and cubic) time-frequency distributions of finite duration arbitrarily shaped causal signals that do not coincide in time and that have different carrier frequencies are presented. These distributions are compared to the time-frequency representations used in the signal analysis based on the Wigner distribution, the uncertainty function, and other quadratic distributions. Examples of the real-time realization of controlled multifunctional processing of causal signals that is performed on the basis of quadratic and cubic distributions are given. These examples include direct and inverse Fourier transforms, convolution, spectral analysis with varied time and frequency scales, delay, compression, time-domain inversion, and other functions.

The T-class of time–frequency distributions: Time-only kernels with amplitude estimation

This paper presents a class of time–frequency distributions (TFDs) characterized by time-lag kernels which are functions of time only. If the parameters of the time-only kernels are properly chosen, their corresponding TFDs, the T-distributions, are more efficient than their two-dimensional counterparts in terms of cross-terms suppression while keeping a high-energy concentration (resolution) around the IF law of non-stationary signals. The proposed class is a subclass of Cohen's Class of quadratic TFDs. We have shown that separable time-lag kernels should be lag-independent (or time-only) for best resolution. In addition, non-parametric amplitude estimation is possible directly from the T-distributions in case of FM signals, a property that is not verified by other TFDs. Two examples of the T-distributions are given and their performance is compared to other TFDs with numerical examples using synthetic and real-life signals.

Reply to “Comments on `A High-Resolution Quadratic Time-Frequency Distribution forMulticomponent Signals Analysis'”

The author replies to the paper by Hussain (IEEE Trans. Signal Processing, vol. 52, no.1, p.287-90, 2004) which commented on the paper by Barkat and Boashash (IEEE Trans. Signal Processing, vol.49, p.2232-9, 2001). The points raised by Hussain are considered one at a time.

Volterra series representation of time-frequency distributions

This paper addresses a Volterra series representation of bilinear (or quadratic) time-frequency distributions that belong to Cohen's class, whereby the analogy of the bilinear class with a second-order double Volterra series is utilized. In addition, a different viewpoint for the bilinear kernel and a complementary interpretation concerning the quadratic time-frequency distributions are provided.

Resolution measure criteria for the objective assessment of the performance of quadratic time-frequency distributions

This paper presents the essential elements for developing objective methods of assessment of the performance of time-frequency signal analysis techniques. We define a measure for assessing the resolution performance of time-frequency distributions (TFDs) in separating closely spaced components in the time-frequency domain. The measure takes into account key attributes of TFDs, such as components mainlobes and sidelobes and cross-terms. The introduction of this measure allows to quantify the quality of TFDs instead of relying solely on visual inspection of their plots. The method of assessment of performance of TFDs also allows the improvement of methodologies for designing high-resolution quadratic TFDs for time-frequency analysis of multicomponent signals. Different TFDs, including the modified B distribution, are optimized using this methodology. Examples of a performance comparison of quadratic TFDs in resolving closely spaced components in the time-frequency domain, using the proposed resolution measure, are provided.

Identification of Nonlinear Systems Through Time-frequency Filtering Technique

In the previous paper, a class of nonlinear system is mapped to a so-called skeleton linear model (SLM) based on the joint time-frequency analysis method. Behavior of the nonlinear system may be indicated quantitatively by the variance of the coefficients of SLM versus its response. Using this model we propose an identification method for nonlinear systems based on nonstationary vibration data in this paper. The key technique in the identification procedure is a time-frequency filtering method by which solution of the SLM is extracted from the response data of the corresponding nonlinear system. Two time-frequency filtering methods are discussed here. One is based on the quadratic time-frequency distribution and its inverse transform, the other is based on the quadratic time-frequency distribution and the wavelet transform. Both numerical examples and an experimental application are given to illustrate the validity of the technique.

Bilinear signal synthesis in array processing

Multiple source signals impinging on an antenna array can be separated by time-frequency synthesis techniques. Averaging of the time-frequency distributions (TFDs) of the data across the array permits the spatial signatures of sources to play a fundamental role in improving the synthesis performance. Array averaging introduces a weighting function in the time-frequency domain that decreases the noise levels, reduces the interactions of the source signals, and mitigates the crossterms. This is achieved independent of the temporal characteristics of the source signals and without causing any smearing of the signal terms. The weighting function may take noninteger values, which are determined by the communication channel, the source positions, and their angular separations. Unlike the recently devised blind source separation methods using spatial TFDs, the proposed method does not require whitening or retrieval of the source directional matrix. The paper evaluates the proposed method in terms of performance and computations relative to the existing source separation techniques based on quadratic TFDs.

Performance of quadratic time~frequency distributions as instantaneous frequency estimators

General performance analysis of the shift covariant class of quadratic time-frequency distributions (TFDs) as instantaneous frequency (IF) estimators, for an arbitrary frequency-modulated (FM) signal, is presented. Expressions for the estimation bias and variance are derived. This class of distributions behaves as an unbiased estimator in the case of monocomponent signals with a linear IF. However, when the IF is not a linear function of time, then the estimate is biased. Cases of white stationary and white nonstationary additive noises are considered. The well-known results for the Wigner distribution (WD) and linear FM signal, and the spectrogram of signals whose IF may be considered as a constant within the lag window, are presented as special cases. In addition, we have derived the variance expression for the spectrogram of a linear FM signal that is quite simple but highly signal dependent. This signal is considered in the cases of other commonly used distributions, such as the Born-Jordan and the Choi-Williams distributions. It has been shown that the reduced interference distributions outperform the WD but only in the case when the IF is constant or its variations are small. Analysis is extended to the IF estimation of signal components in the case of multicomponent signals. All theoretical results are statistically confirmed.

Adaptive instantaneous frequency estimation of multicomponent FM signals using quadratic time-frequency distributions

An adaptive approach to the estimation of the instantaneous frequency (IF) of nonstationary mono- and multicomponent FM signals with additive Gaussian noise is presented. The IF estimation is based on the fact that quadratic time-frequency distributions (TFDs) have maxima around the IF law of the signal. It is shown that the bias and variance of the IF estimate are functions of the lag window length. If there is a bias-variance tradeoff, then the optimal window length for this tradeoff depends on the unknown IF law. Hence, an adaptive algorithm with a time-varying and data-driven window length is needed. The adaptive algorithm can utilize any quadratic TFD that satisfies the following three conditions: First, the IF estimation variance given by the chosen distribution should be a continuously decreasing function of the window length, whereas the bias should be continuously increasing so that the algorithm will converge at the optimal window length for the bias-variance tradeoff, second, the time-lag kernel filter of the chosen distribution should not perform narrowband filtering in the lag direction in order to not interfere with the adaptive window in that direction; third, the distribution should perform effective cross-terms reduction while keeping high resolution in order to be efficient for multicomponent signals. A quadratic distribution with high resolution, effective cross-terms reduction and no lag filtering is proposed. The algorithm estimates multiple IF laws by using a tracking algorithm for the signal components and utilizing the property that the proposed distribution enables nonparametric component amplitude estimation. An extension of the proposed TFD consisting of the use of time-only kernels for adaptive IF estimation is also proposed.

Scale-invariant nonlinear digital filters

Many signal processing applications involve procedures with simple, known dependences on positive rescalings of the input data; examples include correlation and spectral analysis, quadratic time-frequency distributions, and coherence analysis. Often, system performance can be improved with pre- and/or post-processing procedures, and one of the advantages of linear procedures (e.g., smoothing and sharpening filters) is their scale-invariance (x/sub k//spl rarr/y/sub k/ implies /spl lambda/x/sub k//spl rarr//spl lambda/y/sub k/). There are, however, important cases where linear processing is inadequate, motivating interest in nonlinear digital filters. This paper considers the general problem of designing nonlinear filters that exhibit the following scaling behavior: x/sub k//spl rarr/y/sub k/ implies /spl lambda/x/sub k//spl rarr//spl lambda//sup /spl nu//y/sub k/ for some /spl nu/>0, with particular emphasis on the case v=1. Results are presented for two general design approaches. The first is the top-down design of these filters, in which a relatively weak structural constraint is imposed (e.g., membership in the nonlinear FIR class), and a complete characterization is sought for all filters satisfying the scaling criterion for some fixed /spl nu/. The second approach is the bottom-up design of filters satisfying specified scaling behavior by interconnecting simpler filter structures with known scaling behavior. Examples are presented to illustrate both the simplicity and the utility of these design approaches.

Time-frequency and time-varying analysis for assessing the dynamic responses of cardiovascular control.

Time-frequency or time-variant methods have been extensively applied in the study of the heart-rate variability (HRV) signal. In fact, the frequency content of HRV signal has a strong correlation with the control system assessing heart rate. In particular, the power related to the low-frequency (LF) and high-frequency (HF) components have been demonstrated to correlate to the action of sympathetic and parasympathetic branches of the autonomic nervous system. However, the analysis is restricted to stationary conditions, unless time-frequency methods are employed for detecting dynamic changes that may occur during physiological and pathological conditions. This article reviews the most diffused tools for time-frequency analysis, starting from linear decomposition of the signal (including short-time Fourier transform and wavelet and wavelet packet decomposition), to quadratic time-frequency distributions (including Wigner-Ville transform and Cohen's class of distributions), and finally to adaptive or time-variant autoregressive (AR) models, in both the mono- and bivariate forms. In the past few years, these approaches have been applied in several studies related to cardiovascular responses during nonstationary pathophysiological events. Among them, we will recall and discuss myocardial ischemia (spontaneous or induced), drug infusion, rest-tilt maneuver and syncope, neurophysiological, and sleep investigations.

A high-resolution quadratic time-frequency distribution for multicomponent signals analysis

The paper introduces a new kernel for the design of a high resolution time-frequency distribution (TFD). We show that this distribution can solve problems that the Wigner-Ville distribution (WVD) or the spectrogram cannot. In particular, the proposed distribution can resolve two close signals in the time-frequency domain that the two other distributions cannot. Moreover, we show that the proposed distribution is more accurate than the WVD and the spectrogram in the estimation of the instantaneous frequency of a stepped FM signal embedded in additive Gaussian noise. Synthetic and real data collected from real-world applications are shown to validate the proposed distribution.

Influence of high noise on the instantaneous frequency estimation using quadratic time-frequency distributions

Analysis of time frequency (TF) distributions, as the instantaneous frequency (IF) estimators for low noise, has been previously carried out. In this letter, we extend the analysis to high noise. This noise causes a specific error, which can dominate over all other studied errors. The crucial parameter is the ratio of auto-term (AT) magnitude and distribution standard deviation.

Shift covariant time-frequency distributions of discrete signals

Many commonly used time-frequency distributions are members of the Cohen (1989) class. This class is defined for continuous signals, and since time-frequency distributions in the Cohen class are quadratic, the formulation for discrete signals is not straightforward. The Cohen class can be derived as the class of all quadratic time-frequency distributions that are covariant to time shifts and frequency shifts. We extend this method to three types of discrete signals to derive what we call the discrete Cohen classes. The properties of the discrete Cohen classes differ from those of the original Cohen class. To illustrate these properties, we also provide explicit relationships between the classical Wigner distribution and the discrete Cohen classes.

Performance analysis of instantaneous frequency-based interference excision techniques in spread spectrum communications

Jammers characterized by their instantaneous frequencies can be effectively mitigated in direct sequence spread spectrum communications by using open-loop adaptive excision filters. The primary requirement for these filters is that they must possess a notch in tune with the jammer instantaneous frequency (IF) to annihilate the interference power at every time sample. The interference time-varying frequency can be obtained using existing IF estimators, including quadratic time-frequency distribution methods. Without focusing on any specific estimator, we develop expressions for the receiver performance using a general class of multiple-zero FIR excision filters and show the dependence of the bit error rate (BER) on the filter order and its group delay. The effect of inaccuracies in the jammer instantaneous frequency information on the receiver performance is considered and evaluated as a function of the filter notch bandwidth. The latter is defined by the filter zero multiplicity, which is shown to be an important factor in the analysis of the correlator signal-to-noise ratio (SNR).

A new quadratic time-frequency distribution and a comparative study of several popular quadratic time-frequency distributions

A new quadratic time-frequency distribution (TFD) with a compound kernel is proposed and a comparative study of several popular quadratic TFD is carried out. It is shown that the new TFD with compound kernel has stronger ability than the exponential distribution (ED) and the cone-shaped kernel distribution (CKD) in reducing cross terms, meanwhile almost not decreasing the time-frequency resolution of ED or CKD.