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     

Analysis and synthesis of multicomponent signals using positivetime-frequency distributions

A new approach to the analysis and reconstruction of multicomponent nonstationary signals from their time-frequency distribution (TFD) is presented. Specifically, we consider a TFD based on the recently introduced minimum cross entropy principle (MCE). This positive TFD is cross-terms free and, hence, has an advantage over the family of bilinear distributions. Based on the MCE-TFD, a new algorithm for reconstructing the phase and amplitude parameters of each component of the signal is developed. To evaluate the accuracy of the algorithm. Monte Carlo simulations are presented and compared with the corresponding Cramer-Rao bound. It is shown that the new algorithm is superior to presently available methods in both efficiency and performance. It is concluded that together with the MCE-TFD representation, the proposed approach provides a powerful tool for analysis of nonstationary multicomponent signals embedded in additive Gaussian noise     

Adaptive instantaneous frequency estimation of multicomponent FMsignals 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     

基于科恩分布的地球物理信号的时频分析

在简要讨论几种科恩类广义双线性时频分布的基础上,探讨双线性时频分布的算法实现与仿真流程,进而模拟出地磁脉动信号的几种时频分布,同时还对Pc脉动的几种双线性时频分布进行了比较,给出了相应的交叉项抑制和参数选取的方法。结果表明,在核函数参数选取适当的情况下,广义双线性时频表示是分析地球物理信号的有力工具。     

The time-frequency distributions of nonstationary signals based ona Bessel kernel

A kernel based on the first kind Bessel function of order one is proposed to compute the time-frequency distributions of nonstationary signals. This kernel can suppress the cross terms of the distribution effectively. It is shown that the Bessel distribution (the time-frequency distribution using Bessel kernel) meets most of the desirable properties with high time-frequency resolution. A numerical alias-free implementation of the distribution is presented. Examples of applications in time-frequency analysis of the heart's sound and Doppler blood flow signals are given to show that the Bessel distribution can be easily adapted to two very different signals for cardiovascular signal processing. By controlling a kernel parameter, this distribution can be used to compute the time-frequency representations of transient deterministic and random signals. The study confirms the potentials of the proposed distribution in nonstationary signal analysis     

The running time-frequency distributions

This paper introduces the running kernels that yield recursive structures for time-frequency distributions (TFDs). The running kernels offer important properties not possessed by the commonly used block distribution kernels. The introduced kernels allow an invariance in computations with respect to the extent of the kernel in the time or the lag variable. However, contrary to the wide class of block kernels that satisfy the desired timefrequency (t-f) properties, most recursive (running) time-frequency distributions (RTFDs) violate the marginal and the support properties. This paper considers both the direct and the indirect types of recursion and presents examples for illustration.This research was supported in part by the US Air Force, grant no. AFOSR F49620-93-C0063 and a grant from the Office of Research and Sponsored Projects at Villanova University.     

On generalized-marginal time-frequency distributions

We introduce a family of time-frequency (TF) distributions with generalized marginals, i.e., beyond the time-domain and the frequency-domain marginals, in the sense that the projections of a TF distribution along one or more angles are equal to the magnitude squared of the fractional Fourier transforms of the signal. We present a necessary and sufficient condition for a TF distribution in Cohen's class to satisfy generalized marginals. We then modify the existing well-known TF distributions in Cohen's class, such as Choi-Williams (1989) and Page distributions, so that the modified ones have generalized marginals. Numerical examples are presented to show that the proposed TF distributions have the advantages of both Wigner-Ville and other quadratic TF distributions, which only have the conventional marginals. Moreover, they also indicate that the generalized-marginal TF distributions with proper marginals are more robust than the Wigner-Ville and the Choi-Williams distributions when signals contain additive noise     

Kernel decomposition of time-frequency distributions

Bilinear time-frequency distributions (TFDs) offer improved time-frequency resolution over linear representations, but suffer from difficult interpretation, higher implementation cost, and the lack of associated low-cost signal synthesis algorithms. In the paper, the authors introduce some new tools for the interpretation and quantitative comparison of high-resolution TFDs. These tools are used in related work to define low-cost high-resolution TFDs and to define linear, low-cost signal synthesis algorithms associated with high-resolution TFDs. First, each real-valued TFD is associated with a self-adjoint linear operator ψ. The spectral representation of ψ expresses the TFD as a weighted sum of spectrograms (SPs). It is shown that the SP decomposition and Weyl correspondence do not yield useful interpretations for high-resolution TFDs due to the fact that ψ is not positive     

Principal decomposition of time-frequency distributions

The authors present a novel time-frequency analysis technique which uses principal components analysis to map any given time-frequency distribution (TFD) of a signal into a set of three 1-D principal decomposition functions. These three functions may then be considered to be `separable' components of a time-frequency function which they refer to as the principal approximation function for the original TFD. They show how principal decomposition analysis is useful for the enhancement and frequency-tracking of nonstationary harmonic signals     

Alias-free generalized discrete-time time-frequency distributions

A definition of generalized discrete-time time-frequency distribution that utilizes all of the outer product terms from a data sequence, so that one can avoid aliasing, is introduced. The new approach provides (1) proper implementation of the discrete-time spectrogram, (2) correct evaluation of the instantaneous frequency of the underlying continuous-time signal, and (3) correct frequency marginal. The formulation provides a unified framework for implementing members of Cohen's class, which was formulated in the continuous-time domain. Some requirements for the discrete-time kernel in the new approach are discussed in association with desirable distribution properties. Some experimental results are provided to illustrate the features of the proposed method     

Virtues and vices of quartic time-frequency distributions

We present results concerning three different types of quartic (fourth order) time-frequency distributions (TFDs). First, we present new results on the previously introduced local ambiguity function and show that it provides more reliable estimates of instantaneous chirp rate than the Wigner distribution. Second, we introduce the class of quartic, shift-covariant, time-frequency distributions and investigate distributions that localize quadratic chirps. Finally, we present a shift covariant distribution of time and chirp rate     

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.     

Localized Radon-Wigner transform and generalized-marginal time-frequency distributions

This paper introduces the localized Radon transform (LRT) into time-frequency distributions and presents the localized Radon-Wigner transform (LRWT). The definition of LRWT and a fast algorithm is derived, the properties of LRWT and its relationship with Radon-Wigner transform, Wigner distribution (WD), ambiguity function (AF), and generalized-marginal time-frequency distributions are analyzed.     

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.     

An analysis of some time-frequency and time-scale distributions

This paper presents an analysis of the representation of instantaneous frequency and group delay using time-frequency transforms or distributions of energy density domain. The time-frequency distributions which ideally represent the instantaneous frequency or group delay (itfd) are defined. Closeness to the itfd is chosen as a criterion for comparison of various commonly used distributions. It is shown that the Wigner distribution is the best among them, with respect to this criterion. The wavelet and scaled forms of the Wigner distribution are defined and analyzed. In the second part of the paper we extended the analysis to the multicomponent signals and cross terms effects. On the basis of that analysis, an efficient method, derived from the analysis of the Wigner distribution defined in the frequency domain, is proposed. This method provides some substantial advantages over the Wigner distribution. The theory is illustrated on numerical examples.     

Positive time-frequency distributions based on joint marginalconstraints

The authors study the formulation of members of the Cohen-Posch (1985) class of positive time-frequency energy distributions. Minimization of cross-entropy measures with respect to different priors and the case of no prior or maximum entropy were considered. They conclude that, in general, the information provided by the classical marginal constraints is very limited, and thus, the final distribution heavily depends on the prior distribution. To overcome this limitation, joint time and frequency marginals are derived based on a “direction invariance” criterion on the time-frequency plane that are directly related to the fractional Fourier transform     

Spatial polarimetric time-frequency distributions for direction-of-arrival estimations

Time-frequency distributions (TFDs) are traditionally applied to a single antenna receiver with a single polarization. Recently, spatial time-frequency distributions (STFDs) have been developed for receivers with multiple single-polarized antennas and successfully applied for direction-of-arrival (DOA) estimation of nonstationary signals. In this paper, we consider dual-polarized antenna arrays and extend the STFD to utilize the source polarization properties. The spatial polarimetric time-frequency distributions (SPTFDs) are introduced as a platform for processing polarized nonstationary signals, which are received by an array of dual-polarized double-feed antennas. This paper deals with narrow-band far-field point sources that lie in the plane of the receiver array. The source signals are decomposed into two orthogonal polarization components, such as vertical and horizontal. The ability to incorporate signal polarization empowers the STFDs with an additional degree of freedom, leading to improved signal and noise subspace estimates for direction finding. The polarimetric time-frequency MUSIC (PTF-MUSIC) method for DOA estimation based on the SPTFD platform is developed and shown to outperform the time-frequency, polarimetric, and conventional MUSIC techniques, when applied separately.     

A NEW QUADRATIC TIME-FREQUENCY DISTRIBUTIONAND A COMPARATIVE STUDY OF SEVERAL POPULARQUADRATIC 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.     

Highly concentrated time-frequency distributions: pseudo quantumsignal representation

Distributions that are highly concentrated in the time-frequency plane are presented. Since the idea for these distributions originated from the Wigner representation in the quantum mechanics, a review of this representation is done in the first part of the paper. Abstracting the physical sense of the quantum mechanics representation, we defined the “pseudo quantum” signal representation. On the basis of a signal, the “pseudo wave function” with the corresponding “pseudo particle” having the “pseudo momentum” _p=ℏ_fω is generated. By varying the value of ℏ_f, one is in a position to influence the concentration of the “pseudo quantum” (time-“pseudo momentum”) signal's presentation while keeping its most important time-frequency properties invariant. From this reflection, an efficient distribution for the time-frequency (time-“pseudo momentum”) signal analysis is obtained. This distribution produces as high a concentration in the time-frequency (time-“pseudo momentum”) plane as the L-Wigner distribution; however, it may satisfy the marginal properties. The theory is illustrated with examples     

Diversity and channel estimation using time-varying signals and time-frequency techniques

We propose the use of time-varying (TV) signaling in modulation schemes to provide multiuser detection and multipath diversity in TV wireless channels. Specifically, we design an orthogonal linear chirp modulation scheme that is based on assigning different users with optimally designed parameters in order to reduce multiple-access interference. We also derive conditions on the parameters of the modulation signals to achieve multipath diversity. Furthermore, we propose the use of TV pilot signals with nonlinear instantaneous frequency and matched time-frequency (TF) techniques to estimate fast-fading channels with unknown state information. The proposed algorithm simplifies to the estimation of the parameters of multiple linear chirps, which we perform using the modified matching pursuit decomposition. We compare our estimation method with the use of pilot signals with linear instantaneous frequency, which we implement using the reassigned spectrogram. The proposed modulation scheme is applied to a frequency-hopped code-division multiple-access system for which we demonstrate improved performance when compared with frequency-shift-keying (FSK) modulation due to the designed multipath diversity and low multiple-access interference. Our simulations also demonstrate the increased estimation performance when pilot signals with nonlinear structures are used instead of linear structured ones to estimate TV channel parameters.