The Weyl correspondence and time-frequency analysis

Describes the Weyl correspondence and its properties, showing how it gives a “window-independent” definition of time-frequency concentration for use in models in signal detection. The definition of concentration is justified by showing that it gives reasonable answers in certain intuitive cases. The Weyl correspondence expresses a linear transformation as a weighted superposition of time-frequency shifts of the signal, and then authors explain why this is not the same as “transforming” a signal into the time-frequency domain, multiplying by a weight in the transform domain and taking the inverse. The investigation into time-frequency concentration and the Weyl correspondence is justified by a new result. The authors show that convolving the Wigner distribution with a general smoothing function is equivalent to evaluating a weighted sum of spectrograms. This is a new interpretation of the process of smoothing the Wigner distribution to reduce cross-terms. It relates smoothing of the Wigner distribution to the “multiple window” technique pioneered by Thomson (1982)     

The Wigner distribution of a linear signal space

A time-frequency representation of linear signal spaces, called its Wigner distribution (WD), is introduced. Similar to the WD of a signal, the WD of a linear signal space describes the space's energy distribution over the time-frequency plane. It is shown that the WD of a signal space can be defined both in a deterministic and in a stochastic framework, and it can be expressed in a simple way in terms of the space's projection operator and the bases. It is shown to satisfy many interesting properties which are often analogous to corresponding properties of the WD of a signal. The results obtained for some specific signal spaces are found to be intuitively satisfactory. The cross-WD of two signal spaces, a discrete-time WD version, and the extension of the WD definition to arbitrary quadratic signal representation are also discussed     

Comments on “On amplitude and frequency demodulation usingenergy operators”

The authors comment that while testing some of the results obtained by Maragos, Kaiser and Quatieri (see ibid., vol.41, no.4, p.1532-50, 1993) with a discrete FM signal of the form x(n)=Acos(φ(n)) where φ(n)=Ω_cn+Ω_m /Ω_fsin(Ω_fn)+&thetas; they noticed that the error in the instantaneous frequency increased as Ω_f increased. One would hope that frequency tracking of a constant frequency sinusoidal signal would be very accurate. However, this does not agree with one of their proposition as there is an error in its proof     

Integral transforms covariant to unitary operators and theirimplications for joint signal representations

Fundamental to the theory of joint signal representations is the idea of associating a variable, such as time or frequency, with an operator, a concept borrowed from quantum mechanics. Each variable can be associated with a Hermitian operator, or equivalently and consistently, as we show, with a parameterized unitary operator. It is well known that the eigenfunctions of the unitary operator define a signal representation which is invariant to the effect of the unitary operator on the signal, and is hence useful when such changes in the signal are to be ignored. However, for detection or estimation of such changes, a signal representation covariant to them is needed. Using well-known results in functional analysis, we show that there always exists a translationally covariant representation; that is, an application of the operator produces a corresponding translation in the representation. This is a generalization of a recent result in which a transform covariant to dilations is presented. Using Stone's theorem, the “covariant” transform naturally leads to the definition of another, unique, dual parameterized unitary operator. This notion of duality, which we make precise, has important implications for joint distributions of arbitrary variables and their interpretation. In particular, joint distributions of dual variables are structurally equivalent to Cohen's class of time-frequency representations, and our development shows that, for two variables, the Hermitian and unitary operator correspondences can be used consistently and interchangeably if and only if the variables are dual     

A four-parameter atomic decomposition of chirplets

A new four-parameter atomic decomposition of chirplets is developed for compact and precise representation of signals with chirp components. The four-parameter chirplet atom is obtained from the unit Gaussian function by successive applications of scaling, fractional Fourier transform (FRFT), and time-shift and frequency-shift operators. The application of the FRFT operator results in a rotation of the Wigner distribution of the Gaussian in the time-frequency plane by a specified angle. The decomposition is realized by using the matching pursuit algorithm. For this purpose, the four-parameter space is discretized to obtain a small but complete subset in the Hilbert space. A time-frequency distribution (TFD) is developed for clear and readable visualization of the signal components. It is observed that the chirplet decomposition and the related TFD provide more compact and precise representation of signal inner structures compared with the commonly used time-frequency representations     

Measuring time-frequency information content using the Renyientropies

The generalized entropies of Renyi inspire new measures for estimating signal information and complexity in the time-frequency plane. When applied to a time-frequency representation (TFR) from Cohen's class or the affine class, the Renyi entropies conform closely to the notion of complexity that we use when visually inspecting time-frequency images. These measures possess several additional interesting and useful properties, such as accounting and cross-component and transformation invariances, that make them natural for time-frequency analysis. This paper comprises a detailed study of the properties and several potential applications of the Renyi entropies, with emphasis on the mathematical foundations for quadratic TFRs. In particular, for the Wigner distribution, we establish that there exist signals for which the measures are not well defined     

Theoretical analysis of word-level switching activity in thepresence of glitching and correlation

This paper presents a novel analytical approach to compute the switching activity in digital circuits at the word level in the presence of glitching and correlation. The proposed approach makes use of signal statistics such as mean, variance, and autocorrelation. It is shown that the switching activity α_f at the output node f of any arbitrary circuit in the presence of glitching and correlation is computed as α_f=Σ_i=1^S-1α(f _i,i+1)=Σ_i=1^{S- 1}p(f_i+1)(1-p(f_i))(1-ρ(f_i,i+1 )) (1) where ρ(f_i,i+1)=ρ(f_i,i+1)=(E[f_i(Sn)f _i+1(Sn)]- p(f_i)p(f_i+1))/(√(p(f _i)-p(f_i)²)(p(f_i+1)- p(f_i+1²))) (2). S number of time slots in a cycle; ρ(f_i,+1) time-slot autocorrelation coefficient; E[x]=expected value of x; p_x=probability of the signal x being “one”. The switching activity analysis of a signal at the word level is computed by summing the activities of all the individual bits constituting the signal. It is also shown that if the correlation coefficient of the higher order bits of a normally distributed signal x is ρ(x_c), then the bit P₀ where the correlation begins and the correlation coefficient is related hy ρ(x_c)=erfc{(2(P₀-1)-1)/(√2σ_x )} where erfc(x)=complementary error function; σ_x=variance of x. The proposed approach can estimate the switching activity in less than a second which is orders of magnitude faster than simulation-based approaches. Simulation results show that the errors using the proposed approach are about 6.1% on an average and that the approach is well suited even for highly correlated speech and music signals     

无限方差噪声环境下的分数低阶空间时频盲源分离

对脉冲噪声α稳定分布环境下的时频分布进行了研究,改进了适合α稳定分布信号或强脉冲噪声环境的分数低阶时频分布方法,用分数低阶空间时频矩阵代替空间时频矩阵,基于时频盲分离算法提出了一种改进的分数低阶空间时频盲源分离算法,并归纳了算法步骤。通过对FLO-TF-UBSS算法和已有的TF-UBSS算法及MD-BSS算法进行详细比较,仿真结果表明,所提出的FLO-TF-UBSS算法有效的降低了信号的均方误差（MSE）,能较好的对α稳定分布噪声环境下的非平稳信号进行盲分离,并实现了对实际的稳定分布舰船信号的盲提取,性能优于已有TF-UBSS算法和MD-BSS算法,且具有一定的韧性。      

Tomographic time-frequency analysis and its application towardtime-varying filtering and adaptive kernel design for multicomponentlinear-FM signals

Since line integrals through the Wigner spectrum can be calculated by dechirping, calculation of the Wigner spectrum may be viewed as a tomographic reconstruction problem. In the paper, the authors show that all time-frequency transforms of Cohen's class may be achieved by simple changes in backprojection reconstruction filtering. The resolution/cross-term tradeoff that occurs in time-frequency kernel selection is shown to be analogous to the resolution-ringing tradeoff that occurs in computed tomography (CT). “Ideal” reconstruction using a purely differentiating backprojection filter yields the Wigner distribution, whereas low-pass differentiating filters produce cross-term suppressing distributions such as the spectrogram or the Born-Jordan distribution. It is also demonstrated how this analogy can be exploited to “tune” the reconstruction filtering (or time-frequency kernel) to improve the ringing/resolution tradeoff. Some properties of the projection domain, which is also known as the Radon-Wigner transform, are characterized, including the response to signal delays or frequency shifts and projection masking or convolution. Last, time-varying filtering by shift-varying convolution in the Radon-Wigner domain is shown to yield superior results to its analogous Cohen's class adaptive transform (shift-invariant convolution) for the multicomponent, linear-FM signals that are investigated     

An acoustooptic Wigner processor for time-varying signal analysis

The Wigner distribution, or WD, produces a time-frequency signal representation from which the time- and/or frequency-domain properties of a signal can be extracted. To operate at very high data rates, an acoustooptic specialized Wigner distribution processor is proposed.     

基于熵的Gabor变换窗函数宽度自适应选择算法

该文针对Gabor变换中窗函数宽度选择的问题,提出了以提高Gabor表示的聚集性和时频分辨率为目的的窗函数宽度自适应选择算法。提出对香农熵的取值范围进行改进,使其更适合度量时频分布的聚集性,进而根据熵度量实现了与信号非平稳性相适应的最优窗函数宽度选择。仿真结果表明该算法对单分量及多分量信号都能有效地选择最优窗函数宽度,能够获得聚集性好、时频分辨率高的Gabor表示,并具有很好的抗噪性能。     

Elimination of interference component in Wigner-Ville distributionfor the signal with 1/f spectral characteristic

A new technique for interference-term suppression in the Wigner-Ville distribution (WVD) is proposed for signals with a 1/f spectrum shape. The spectral characteristic of the signal is altered by f^α filtering before time-frequency analysis and compensated after analysis. With the utilization of the proposed technique in a smoothed pseudo Wigner-Ville distribution, an excellent suppression of the interference component can be achieved. The authors apply this technique to the heart rate variability signal     

基于数据增强和特征嵌入的自动调制识别

基于端到端的深度学习模型已经被广泛应用于自动调制识别。现有的深度学习方案大多数依赖于丰富的样本分布,而大批量的标记训练集通常很难获得。提出了一种基于数据驱动和选择性核卷积神经网络(Convolutional Neural Network,CNN)的自动调制识别框架。首先开发深度密集生成式对抗网络增强5种调制信号的原始数据集;其次选择平滑伪Wigner-Ville分布作为信号的时频表示,并将注意力模块用于聚焦时频图像分类中的差异区域;最后将真实信号输入轻量级卷积神经网络进行时间相关性提取,并融合信号的时频特征完成分类。实验结果表明,所提算法提高了在低信噪比情况下的识别精度,表现出较强的鲁棒性。     

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     

Time-frequency projection filters and time-frequency signalexpansions

We consider the problems of designing a linear, time-varying filter with a specified “time-frequency (TF) pass region” and of constructing an orthonormal basis for the parsimonious expansion of signals located in a given TF support region. These problems of TF filtering and TF signal expansion are reduced to the problem of designing a “TF subspace”, i.e., a linear signal space comprising all signals located in a given TF legion. Specifically, the TF filter is taken to be the orthogonal projection operator on the TF subspace. We present an optimum design of TF subspaces that is based on the Wigner distribution of a linear signal space and is an extension of the well-known signal synthesis problem. The optimum TF subspace is shown to be an “eigenspace” of the TF region, and some properties of eigenspaces are discussed. The performance of TF projection filters and TF signal expansions is studied both analytically and via computer simulation     

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.     

Frequency hopping modulation recognition based on time-frequency energy spectrum texture feature

For frequency hopping modulation identification,a novel method based on time-frequency energy spectrum texture feature was proposed.Firstly,the time-frequency diagram of the frequency hopping signal was obtained by smoothed pseudo Wigner-Ville distribution,and the background noise of the time-frequency diagram was removed by two-dimensional Wiener filtering to improve the resolution of the time-frequency diagram under low SNR conditions.Then,the connected-domain detection algorithm was used to extract the time-frequency energy spectrum of each hop signal and convert it into a time-frequency gray-scale image.The histogram statistical features and the gray-scale co-occurrence matrix feature were combined to form a 22-dimensional eigenvector.Finally,the feature set was trained,classified and identified by optimized support vector machine classifier.Simulation experiments show that the multi-dimensional feature vector extracted by the algorithm has strong representation ability and avoids the misjudgment caused by the similarity of single features.The average recognition accuracy of the six modulation methods of frequency hopping signals BPSK,QPSK,SDPSK,QASK,64QAM and GMSK is 91.4% under the condition of -4 dB SNR.     

Time-Frequency Analysis Using Local Taylor Series Expansion

We describe an approach to time-frequency analysis based on the local approximation of the signal by a first order Taylor series. We show that the Taylor approximation provides a representation of the signal in terms of its instantaneous frequency and instantaneous bandwidth. This representation can be translated into the frequency domain in a straightforward manner. The key to this approach is the local decomposition of the signal into its components, which is similar to the problem of estimating the parameters of of complex exponentials from observation of their sum. The resulting time-frequency representation (TFR) does not have the time and frequency marginal properties shared by many of the time-frequency distributions presented in the literature, but is additive over the signal components and, by its construction, does not have cross-terms.