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.     

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.     

A signal-dependent time-frequency representation: optimal kerneldesign

A new time-frequency distribution (TFD) that adapts to each signal and so offers a good performance for a large class of signals is introduced. The design of the signal-dependent TFD is formulated in Cohen's class as an optimization problem and results in a special linear program. Given a signal to be analyzed, the solution to the linear program yields the optimal kernel and, hence, the optimal time-frequency mapping for that signal. A fast algorithm has been developed for solving the linear program, allowing the computation of the signal-dependent TFD with a time complexity on the same order as a fixed-kernel distribution. Besides this computational efficiency, an attractive feature of the optimization-based approach is the ease with which the formulation can be customized to incorporate application-specific knowledge into the design process     

An adaptive optimal-kernel time-frequency representation

Time-frequency representations with fixed windows or kernels figure prominently in many applications, but perform well only for limited classes of signals. Representations with signal-dependent kernels can overcome this limitation. However, while they often perform well, most existing schemes are block-oriented techniques unsuitable for on-line implementation or for tracking signal components with characteristics that change with time. The time-frequency representation developed in the present paper, based on a signal-dependent radially Gaussian kernel that adapts over time, surmounts these difficulties. The method employs a short-time ambiguity function both for kernel optimization and as an intermediate step in computing constant-time slices of the representation. Careful algorithm design provides reasonably efficient computation and allows on-line implementation. Certain enhancements, such as cone-kernel constraints and approximate retention of marginals, are easily incorporated with little additional computation. While somewhat more expensive than fixed kernel representations, this new technique often provides much better performance. Several examples illustrate its behavior on synthetic and real-world signals     

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     

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     

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     

二次时频表示中核函数的优化设计

二次时频分布是分析非平稳信号的有力工具,在具有许多优良特性的同时,存在严重的交叉干扰项。在Wigner-Ville分布及Cohen类时频分布具有固定核函数的基础上,研究了基于信号的核函数优化设计的两种方法,径向高斯核函数和最优相位核函数的设计方法。基于信号的核函数的时频表示可以有效地抑制或转移交叉分量,提高时频表示的可读估计,改善其主要性能。     

Near-field time-frequency localization method using sparse representation

This paper presents a novel near-field source localization method based on the time-frequency sparse model.Firstly,the method converts the time domain data of array output into time-frequency domain by time-frequency transform;then constructs sparse localization model by utilizing the specially selected time-frequency points,and finally the greedy algorithms are chosen to solve the sparse problem to localize the source.When the coherent sources exist,we propose an additional iterative selection procedure to improve the estimation performance.The proposed method is suitable for uncorrelated and coherent sources,moreover,the improved estimation accuracy and the robustness to low signal to noise ratio(SNR) are achieved.Simulations results verify the efficiency of the proposed algorithm.     

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     

A signal-dependent time-frequency representation: fast algorithmfor optimal kernel design

A time-frequency representation based on an optimal, signal-dependent kernel has been previously been proposed in an attempt to overcome one of the primary limitations of bilinear time-frequency distributions: that the best kernel and distribution depend on the signal to be analyzed. The optimization formulation for the signal-dependent kernel results in a linear program with a unique feature: a tree structure that summarizes a set of constraints on the kernel. The authors present a fast algorithm based on sorting to solve a special class of linear programs that includes the problem of interest. For a kernel with Q variables, the running time of the algorithm is O(Q log Q), which is several orders of magnitude less than any other known method for solving this class of linear program. This efficiency enables the computation of the signal-dependent, optimal-kernel time-frequency representation at a cost that is on the same order as a fixed-kernel distribution. An important property of the optimal kernel is that it takes on essentially only the values of 1 and 0     

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     

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     

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.     

Iterative detection for multicarrier transmission employing time-frequency concentrated pulses

We consider multicarrier transmission schemes in which the elementary signal pulses relate to the elements of a Weyl-Heisenberg system, i.e., resulting from a prototype function shifted in time and frequency. The overlapping of the information-bearing signal parts at the output of doubly dispersive channels and the resulting interference are confined by utilizing a prototype function whose energy is concentrated in both time and frequency. We derive a symbol detector which first calculates a sufficient statistic for the unknown data symbols from the linearly combined output signals of a filter bank, and second, performs an iterative maximization of the likelihood function. The presented receiver takes full advantage of the confined pulse overlapping to limit the computational effort. An analysis of the computational complexity and bit-error rate performance of the iterative detection scheme is provided for wide-sense stationary uncorrelated scattering channels.     

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.     

Orthonormal-basis partitioning and time-frequency representation of cardiac rhythm dynamics

Although a number of time-frequency representations have been proposed for the estimation of time-dependent spectra, the time-frequency analysis of multicomponent physiological signals, such as beat-to-beat variations of cardiac rhythm or heart rate variability (HRV), is difficult. We thus propose a simple method for 1) detecting both abrupt and slow changes in the structure of the HRV signal, 2) segmenting the nonstationary signal into the less nonstationary portions, and 3) exposing characteristic patterns of the changes in the time-frequency plane. The method, referred to as orthonormal-basis partitioning and time-frequency representation (OPTR), is validated using simulated signals and actual HRV data. Here we show that OPTR can be applied to long multicomponent ambulatory signals to obtain the signal representation along with its time-varying spectrum.     

Modified Cohen-Lee time-frequency distributions and instantaneousbandwidth of multicomponent signals

Cohen (1989, 1995) has introduced and extensively studied and developed the concept of the instantaneous bandwidth of a signal. Specifically, instantaneous bandwidth is interpreted as the spread in frequency about the instantaneous frequency, which is itself interpreted as the average frequency at each time. This view stems from a joint time-frequency distribution (TFD) analysis of the signal, where instantaneous frequency and instantaneous bandwidth are taken to be the first two conditional spectral moments, respectively, of the distribution. However, the traditional definition of instantaneous frequency, namely, as the derivative of the phase of the signal, is not consistent with this interpretation, and new definitions have therefore been proposed previously. We show that similar problems arise with the Cohen-Lee (1988, 1989) instantaneous bandwidth of a signal and propose a new formulation for the instantaneous bandwidth that is consistent with its interpretation as the conditional standard deviation in frequency of a TFD. We give the kernel constraints for a distribution to yield this new result, which is a modification of the kernel proposed by Cohen and Lee. These new kernel constraints yield a modified Cohen-Lee TFD whose first two conditional moments are interpretable as the average frequency and bandwidth at each time, respectively     

Direction of arrival estimation using adaptive directional time-frequency distributions

Time-frequency distributions (TFDs) allow direction of arrival (DOA) estimation algorithms to be used in scenarios when the total number of sources are more than the number of sensors. The performance of such time–frequency (t–f) based DOA estimation algorithms depends on the resolution of the underlying TFD as a higher resolution TFD leads to better separation of sources in the t–f domain. This paper presents a novel DOA estimation algorithm that uses the adaptive directional t–f distribution (ADTFD) for the analysis of close signal components. The ADTFD optimizes the direction of kernel at each point in the t–f domain to obtain a clear t–f representation, which is then exploited for DOA estimation. Moreover, the proposed methodology can also be applied for DOA estimation of sparse signals. Experimental results indicate that the proposed DOA algorithm based on the ADTFD outperforms other fixed and adaptive kernel based DOA algorithms.     

Interference mitigation in spread spectrum communication systemsusing time-frequency distributions

The capability of the time-frequency distributions (TFDs) to properly represent a single as well as multiple component signals in time and frequency permits the application of a new approach for interference excision in spread spectrum communication systems. The instantaneous frequency (IF) estimate from the TFD is used to construct a finite impulse response filter that reduces the interference power with a minimum possible distortion of the desired signal. The proposed technique is therefore a case of open-loop adaptive filtering. Three- and five-coefficient zero-phase excision filters are considered. Closed-form expressions of the improvement of SNR at the receiver correlator output using the TFD-based adaptive filtering are derived for two extreme cases of time-varying interferers, namely, those of fixed frequency sinusoids and randomly changing instantaneous frequencies. Simulation results including the bit error rates are presented for both swept and frequency hopping jammers