Showing papers on "Time–frequency analysis published in 1995"

Time-Frequency Analysis

Abstract: 1. The Time and Frequency Description of Signals. 2. Instantaneous Frequency and the Complex Signal. 3. The Uncertainty Principle. 4. Densities and Characteristic Functions. 5. The Need for Time-Frequency Analysis. 6. Time-Frequency Distributions: Fundamental Ideas. 7. The Short-Time Fourier Transform. 8. The Wigner Distribution. 9. General Approach and the Kernel Method. 10. Characteristic Function Operator Method. 11. Kernel Design for Reduced Interference. 12. Some Distributions. 13. Further Developments. 14. Positive Distributions Satisfying the Marginals. 15. The Representation of Signals. 16. Density of a Single Variable. 17. Joint Representations for Arbitrary Variables. 18. Scale. 19. Joint Scale Representations. Bibliography. Index.

Time-frequency analysis: theory and applications

Abstract: Featuring traditional coverage as well as new research results that, until now, have been scattered throughout the professional literature, this book brings together in simple language the basic ideas and methods that...

Analysis of multicomponent LFM signals by a combined Wigner-Hough transform

Abstract: The aim of the paper is the performance evaluation of a method for the analysis of mono- or multicomponent linear-frequency modulation (LFM) signals, based on the Hough transform of the Wigner-Ville distribution of the signals. A closed form expression is found for the signal-to-noise ratio and the parameter estimation accuracy. The overall method, as any nonlinear method, exhibits a threshold effect. Nevertheless, it is shown to be asymptotically efficient and offers a good rejection capability of the cross terms. >

An analysis of instantaneous frequency representation using time-frequency distributions-generalized Wigner distribution

Abstract: This paper presents an analysis of the representation of instantaneous frequency using time-frequency distributions of energy density domain. Similarity to the "ideal" instantaneous frequency presentation is chosen as a criterion for comparison of various distributions. Although all the commonly used distributions suffer from the artifacts along frequency axis, it is shown that the Wigner distribution is the best among them, with respect to this criterion. The generalization of Wigner distribution-LWD-is introduced to decrease the artifacts. The properties of the LWD are analyzed. It is shown that, at the expense of an insignificant increase in computation time, much better results are obtained. The theory is illustrated by a numerical example with the frequency modulated signals. >

Instantaneous frequency estimation using an adaptive short-time Fourier transform

Abstract: Instantaneous frequency estimation is important in many applications, including FM demodulation. Time-frequency-based methods have been widely studied for IF estimation, particularly for low SNR, where they prove to be the most effective methods. We propose a new adaptive short-time Fourier transform algorithm, with chirping windows which is tailored for near-optimal time-frequency-based IF estimation. Extensive performance evaluations with both synthetic and real data demonstrate modest (1-2 dB) improvements in the threshold SNR over the best fixed-window STFTs. The computational cost of the new method is about ten times that of a fixed-window STFT.

Instantaneous frequency and time-frequency distributions

Abstract: It is generally stated that the conditional mean frequency of a time-frequency distribution (TFD) should equal the instantaneous frequency of the signal. The commonly accepted definition of instantaneous frequency as the derivative of the phase of the analytic signal sometimes leads to curious results. Although it is commonly held that positivity of the TFD and satisfaction of the so-called "instantaneous frequency constraint" are generally incompatible, we show that one can always find a complex signal, the real part of which is the given signal, for which the derivative of the phase is consistent with the marginals and positivity. Furthermore, for the cases considered, the derivative of the phase of this signal, which by design equals the conditional mean frequency of a positive TFD, is a reasonable, readily interpretable choice for instantaneous frequency.

Discrete Gabor structures and optimal representations

Abstract: The idea of Gabor's (1946) signal expansion is to represent a signal in terms of a discrete set of time-shifted and frequency modulated signals that are localized in the time-frequency (or phase) space. We present detailed descriptions of the block and banded structures for the Gabor matrices. Based on the explicit descriptions of the sparsity of such matrices, we can establish the sparse form of the Gabor matrix and obtain the dual Gabor atom (mother wavelet), the inverse of the Gabor frame operator, and carry out the discrete finite Gabor transform in a very efficient way. Some explicit sufficient and also necessary conditions are derived for a Gabor atom g to generate a Gabor frame with respect to a TF-lattice (a, b). >

Fractional Fourier transform used for a lens-design problem.

Abstract: The fractional Fourier transform has been used in optics so far for wave propagation and for signal processing. Now we show that this new transform can also be helpful for lens design, especially for specifying a lens cascade.

On the local frequency, group shift, and cross-terms in some multidimensional time-frequency distributions: a method for multidimensional time-frequency analysis

Abstract: Presents an analysis of the representation of local frequency and group shift using multidimensional time-frequency distributions. In the second part of the correspondence, the authors extend 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 multidimensional Wigner distribution defined in the frequency domain, is proposed. This method provides some substantial advantages over the Wigner distribution: the well known cross-terms effects are reduced or completely removed; the oversampling of signals is shown to be unnecessary; and the computation time can be significantly reduced, as well. The theory is illustrated by a two-dimensional numerical example. >

Equivalence of DFT filter banks and Gabor expansions

Abstract: Recently connections between the wavelet transform and filter banks have been established. We show that similar relations exist between the Gabor expansion and DFT filter banks. We introduce the "z-Zak transform" by suitably extending the discrete-time Zak transform and show its equivalence to the polyphase representation. A systematic discussion of parallels between DFT filter banks and Weyl-Heisenberg frames (Gabor expansion theory) is then given. Among other results, it is shown that tight Weyl-Heisenberg frames correspond to paraunitary DFT filter banks. The wavelet transform, filter banks, and multiresolution signal analysis have recently been unified within a single theory.'6 This led to new results and deeper insights in both areas. In this paper, we show that an important linear time-frequency representation known as the Gabor expansion79 and the computationally efficient DFT filter banks'°'4'6 can be unified in a similar manner.15 We show that the theory of Weyl-Heisenberg frames ( WHFs) ,16-18 which is a fundamental concept in Gabor expansion theory, allows to establish known and new results on DFT filter banks. We also extend the Zak transfoin,'92' a transformation particularly useful for the Gabor expansion, to the complex plane (z-plane), and we show that the resulting "z-Zak transform" is equivalent to the polyphase representation used in filter bank theory.22"°'6"2"3

Analysis and processing of shaft angular velocity signals in rotating machinery for diagnostic applications

Abstract: The paper presents the application of some modern signal processing methods to the analysis of angular velocity signals in a rotating machine for diagnostic purposes. The signal processing techniques considered in this paper include: classical non-parametric spectral analysis; principal component analysis; joint time-frequency analysis; the discrete wavelet transform; and change detection algorithm based on residual generation. These algorithms are employed to process shaft angular velocity data measured from an internal combustion engine, with the intent of detecting engine misfire. The results of these analyses show that these algorithms have potential for on-board diagnostic application in passenger and commercial vehicles, and more generally for failure detection of other classes of rotating machines.

Linear vs. bilinear time-frequency methods for interference mitigation in direct sequence spread spectrum communication systems

Abstract: Time frequency signal representations have been examined for interference mitigation in spread spectrum (SS) communication systems. We implement a direct sequence spread spectrum (DS-SS) system with additive jamming and utilize two distinct time-frequency (t-f) methods to mitigate the interference, namely the shift-covariant class of time frequency distributions (TFD) and the Gabor (1946) transform. The TFD is a bilinear transform and is shown to improve the receiver characteristics in a highly nonstationary jamming environment. The Gabor transform is a linear method that has real-time computational advantages yet has a limited class of jammers against which it performs well. A common feature in these two excision methods is that the t-f points are analyzed with the same time resolutions and frequency resolutions. The performance of both methods under several jamming scenarios is discussed and computational considerations are addressed.

Adaptive SVD based AR model order determination for time frequency analysis of Doppler ultrasound signals

Abstract: The short-time Fourier transform provides a picture of the spectral components temporal location in time-varying signals, but its performance is limited by the intrinsic trade-off between time and frequency resolutions. In the present study, this problem is addressed using a spectral estimator based on a combination of the autoregressive (AR) modeling technique and a new automatic model order selection method. The order estimation is achieved by means of the singular value decomposition (SVD) of an appropriate data matrix in conjunction with a new criterion (dynamic mean evaluation, DME). The latter is used to decide which singular values correspond to the signal and which to the noise subspaces, avoiding an a priori threshold definition, thus giving the variable AR model order on consecutive short-time segments. Combination of the AR high frequency resolution capabilities and the SVD plus DME robustness and simplicity make the overall method reliable in many practical applications, mainly in the analysis of time-varying signals corrupted by noise. The proposed procedure has been applied to benchmark as well as to Doppler signal analysis. Some examples are reported confirming the above-mentioned properties.

A Capon's time-octave representation application in room acoustics

Abstract: In time-frequency analysis, Capon's estimator has proven its efficiency in precise applications. In a context where a time-octave representation is also necessary, the authors propose a new method combining both Capon's estimator and a time-octave representation. The main objective is to obtain legibility in the time frequency plane using a variable frequency resolution with a fixed time resolution. This fixed time resolution is possible owing to the good resolution properties of Capon's estimator compared to the Fourier transform. This choice leads to a particular repartition of basic cells in the time-frequency plane that seems more adapted to a physical interpretation in the application presented. Nevertheless, a parallel with the wavelet transform is displayed: the constructed wavelet is adapted to the signal at each octave or at each fraction of octave. The proposed method is presented both in continuous and discrete formulations. Its structure is studied and a simplification is proposed when precise hypotheses are verified. Simulations and comparisons with classical representations (spectrogram, scalogram) are discussed. The contribution of each method, essentially in the duality of time-frequency and time-scale, are shown up in relation to the analyzed signal. Lastly, the proposed method and classical ones are applied on rear signals issued from room acoustics where the aim is the time-frequency characterization of concert halls from impulse responses. >

More results on orthogonal wavelets with optimum time-frequency resolution

Abstract: Signal decomposition techniques are an important tool for analyzing nonstationary signals. The proper selection of time-frequency basis functions for the decomposition is essential to a variety of signal processing applications. The discrete wavelet transform (DWT) is increasingly being used for signal analysis, but not until recently has attention been paid to the time-frequency resolution property of wavelets. This paper describes additional results on our procedure to design wavelets with better time-frequency resolution. In particular, our optimal duration-bandwidth product wavelets (ODBW) have better duration-bandwidth product, as a function of wavelet-defining filter length N, than Daubechies' minimum phase and least- asymmetric wavelets, and Dorize and Villemoes' optimum wavelets over the range N equals 8 to 64. Some examples and comparisons with these traditional wavelets are presented.© (1995) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Block-circulant Gabor-matrix structure and discrete Gabor transforms

Abstract: We develop the block-circulant structure of Gabor matrices, and establish that Gabor matrices are unitarily block-diagonalizable simultaneously. It opens a new way of implementing the discrete Gabor transforms. For the most interesting cases, if the product ab of the lattice constants divides the signal length N (in particular, in the critical-sampling cases), we prove that the Gabor operators are simultaneously unitarily equivalent to non-negative pointwise multiplication operators. This leads to fast computations of the inverse of the Gabor operator and the square root of the inverse of the Gabor operator, as well as the dual Gabor wavelet and the tight Gabor wavelet. Gabor syntheses turn out to be simple, and we can also easily predetermine the stability of Gabor reconstructions.

Comparison of three running spectral analysis methods for electrogastrographic signals.

Abstract: IN THE human stomach, there is electrical activity which controls the contractions of the stomach. The gastric myoelectrical activity is composed of the slow wave (or electrical control activity) and spikes (or electrical response activity) (SARNA, 1975). Electrogastrography (EGG), i.e. recording gastric electrical activity by placing cutaneous electrodes on the abdomen over the stomach, provides a non-invasive method of studying gastric electrical behaviour (ALVAREZ, 1922; CHEN and MCCALLUM, 1991a). Previous studies have shown that cutaneous electrodes are able to pick up the rhythm of the gastric slow wave but not that of spikes (SMOUT eta/ . , 1980). The dominant frequency of the EGG represents the frequency of the gastric slow wave. Spikes are reflected in the EGG as an increase in amplitude, i.e. relative amplitude change of the EGG reflects spikes or contractions of the stomach. Unlike other surface electrophysilogical recordings such as the ECG and EEG, the EGG has a low signal-to-noise ratio. It Contains noise, such as respiratory and motion artifacts and possible myoelectrical activity from other organs (C~',~ and MCCALLt/M, 1991a). As a result, direct visual interpretation of the EGG time signal is almost impossible. Quantitative analysis of the EGG has relied on spectral methods, including fast Fourier transform, autoregressive modelling and smoothed power spectral analysis (Ln'~KENS, and DATARDtNA, 1978; et al., 1981; STERN et al., 1987; CrtEN, 1992). In particular, running spectral analysis (or time-frequency representation) is widely used for both qualitative and quantitative analyses of the EGG. Three running spectral analysis methods have been introduced or developed for EGG analysis. These include the shorttime Fourier transform (STFT) NAWAB and Qt/ATmm, 1988), the adaptive spectral analysis (CI-IEN et a t , 1990) and the exponential distribution (ED) (Crtol and WILLIAMS, 1989). The first method introduced into the area was the short-time Fourier analysis (VAN DER SCdEE and GRASHUIS, 1987). It was called running spectral analysis by some investigators. In this work, however, running spectral analysis is used as a general term. The adaptive spectral analysis was developed (CHIN et al.,

Phase-space localization and discrete representations of wave fields

Abstract: The high-frequency behavior of wave fields in free space is characterized by localization in phase space, exemplified by the description of geometrical optics in which distinct wave vectors are associated with the rays passing through each spatial point. I explore the manifestation of phase-space localization in discrete representations of the wave field, in particular the discrete Fourier transform (DFT) and the Gabor representation. A number of auxiliary concepts, such as spectral truncation, the Lagrange manifold, and the Landau–Pollak (LP) theorem, are described and exploited in the process of understanding the behavior of high-frequency fields, and it is shown that the Lagrange manifold is the source in phase space of the dominant contributions to these discrete representations in a number of specially selected examples. The LP theorem specifies the number of discrete degrees of freedom required for a given field to be approximated to a prescribed accuracy, and the theorem controls both the number of samples for DFT implementation and the number of Gabor coefficients required. The behavior of the Gabor coefficients away from the Lagrange manifold is studied for a Fresnel wave (quadratic phase variation on a line), and it is shown that these coefficients decay exponentially, signifying the localization of the Gabor representation. The number of operations required for computing the Gabor representation is compared with the fast-Fourier-transform (FFT) implementation of the DFT, with the LP dimension used as the common cardinality of the two representations, the result being that the FFT is asymptotically more efficient than the Gabor representation even after one allows for the localization of the latter. However, the Gabor representation can be used in circumstances in which the FFT is inapplicable, when the explicit localization is a significant advantage.

Wavelet Analysis of Dispersive Stress Waves

Abstract: The time-frequency analysis of dispersive stress waves is reviewed. It is shown that the wavelet transform using the Gabor wavelet effectively decomposes the strain response into its time-frequency components, and the peaks of the time-frequency distribution indicate the arrival times of waves. The flexural waves induced in a beam by lateral impact are considered and it is shown that the wavelet transform enables us to identify the dispersion relation of the group velocity, and to estimate the impact location. In addition, the potential of using the wavelet transform for more detailed nondestructive evaluation of material damage is shown.

Evolutionary spectral analysis and the generalized Gabor expansion

Abstract: We present a connection between the discrete Gabor expansion and the evolutionary spectral theory. Including a scale parameter in the Gabor expansion, we obtain a new representation for deterministic signals that is analogous to the Wold-Cramer decomposition for non-stationary processes. The energy distribution resulting from the expansion is easily calculated from the Gabor coefficients. By choosing Gaussian windows and appropriate scales, the expansion can represent narrow-band and wide-band signals, as well as their combination. As an application, we consider the masking of signals in the time-frequency space and provide an approximate implementation using the new Gabor expansion. Examples illustrating the time-frequency analysis and the masking are given.

Gabor-type matrices and discrete huge Gabor transforms

Abstract: A Gabor family is obtained from a Gabor atom (or Gabor window, or basic building block) by time-frequency shifts along some discrete TF-lattice. Such a family is usually not orthogonal. Therefore the determination of appropriate coefficients in order to obtain a series representation of a given signal in terms of this family has been considered a computational intensive task for a long time. We introduce a class of matrices, called Gabor-type matrices and show that the product of two Gabor-type matrices is again a Gabor-type matrix of the same type. The key point for applications is based on the observation that the multiplication of Gabor-type matrices can be replaced by some special "multiplication" of associated small block matrices. We propose an efficient algorithm, which we call the block-multiplication, and which makes explicit use of the sparsity of those Gabor-type matrices. As an interesting consequence, we show that Gabor operators corresponding to Gabor triples (g/sub k/,a,b) commute for arbitrary signals g/sub k/ (k=1,2) provided that ab divides the signal length.

The discrete-time phase derivative as a definition of discrete instantaneous frequency and its relation to discrete time-frequency distributions

Abstract: We compare two definitions of the instantaneous frequency of a discrete signal: the two-point symmetric phase difference and the discrete-time phase derivative. The phase derivative definition avoids the pitfalls associated with the two-point definition and is equal to the first moment w.r.t. frequency of a properly defined, alias-free time-frequency distribution of the signal, which is consistent with the continuous-time case. >

Application of Choi—Williams Reduced Interference Time Frequency Distribution to Machinery Diagnostics

Abstract: This article discusses time frequency analysis of machinery diagnostic vibration signals. The short time Fourier transform, the Wigner, and the Choi–Williams distributions are explained and illustrated with test cases. Examples of Choi—Williams analyses of machinery vibration signals are presented. The analyses detect discontinuities in the signals and their timing, amplitude and frequency modulation, and the presence of different components in a vibration signal.

Fast and parametric algorithm for discrete Gabor expansions and the role of various dual windows

Abstract: We present a fast and parametric algorithm for discrete Gabor expansions (DGE). The fast algorithm allows us to obtain any different pseudo-dual window within a fraction of a second based on the (usual) dual window. We discuss important roles of pseudo-duals in general, and in particular for DGEs. An example of using pseudo-duals to achieve the dimension invariance for DGEs is presented. A new multi-Gabor expansion scheme is also defined and characterized. Various other issues on DGEs are also discussed.© (1995) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Structured autoregressive instantaneous phase and frequency estimation

Abstract: A new approach to estimate the phase and amplitude signal parameters of a quite general class of complex valued signals is presented. The proposed algorithm can estimate the signal parameters of a sum of complex signals, the amplitudes may be time varying and the phase functions are modelled by some continuous functions a/sub l/(t). The data can be evenly or unevenly sampled in time. The signal parameter estimates minimizes a loss function based on the prediction errors of a new, time dependent, structured autoregressive filter. The instantaneous phase and frequency is easily obtained from the estimated signal parameters. The structured AR filter is a model based time-frequency representation.

ウェーブレット変換による分散性応力波の時間-周波数解析 : 群速度の同定と超音波材料評価への応用

Abstract: The wavelet transform is applied to time-frequency analysis of dispersive stress waves. The Gabor function is adopted as the analyzing wavelet. The magnitude of the wavelet transform of wave data takes its maximum value at the time when the stress wave reaches the observation point with its group velocity at each frequency. An experiment on the flexural wave in a beam shows that the dispersion relation for the group velocity can be accurately identified by the wavelet transform of measured data. In addition, the application of the wavelet transform to ultrasonic testing of a polymer alloy shows that changes in velocity and attenuation coefficient due to mechanical damaging can be evaluated at each frequency. These results suggests that the wavelet transform has potential ability to present more detailed evaluation of material damages.

Mechanical Signature Analysis Using Time-Frequency Signal Processing /

Abstract: Signature analysis consists of the extraction of information from measured signal patterns. The work presented in this paper illustrates the use of time-frequency (TF) analysis methods for the purpose of mechanical signature analysis. Mechanical signature analysis is a mature and developed field; however, TF analysis methods are relatively new to the field of mechanical signal processing, having mostly been developed in the present decade, and have not yet been applied to their full potential in this field of engineering applications. Some of the ongoing efforts are briefly reviewed in this paper. One important application of TF mechanical signature analysis is the diagnosis of faults in mechanical systems. In this paper we illustrate how the use of joint TF signal representations can result in tangible benefits when analyzing signatures generated by transient phenomena in mechanical systems, such as might be caused by faults or otherwise abnormal operation. This paper also explores signal detection concepts in the joint TF domain and presents their application to the detection of internal combustion engine knock.

Reconstruction of signals from irregular samples of its short-time Fourier transform

Abstract: The short-time Fourier transform (STFT) leads to a highly redundant linear time-frequency signal representation. In order to remove this redundancy it is usual to sample the STFT on a rectangular grid. For such regular sampling the basic features of the reconstruction problem are well understood. In this paper, we consider the problem of reconstructing a signal from irregular samples of its STFT. It may happen that certain samples of the STFT from a regular grid are lost or that the STFT has been purposely sampled in an irregular way. We investigate that problem using Weyl-Heisenberg frames, which are generated from a single atom by time- frequency-shifts (along the sampling set). We compare various iterative methods and present typical numerical experiments. Whereas standard frame iterations are doing not very well it turns out that for many reasons the conjugate gradient algorithm behaves best, most often even better than one might expect from the observations made for general frame operators.