Showing papers on "Time–frequency analysis published in 1997"

Optimal filtering in fractional Fourier domains

Abstract: For time-invariant degradation models and stationary signals and noise, the classical Fourier domain Wiener filter, which can be implemented in O(N log N) time, gives the minimum mean-square-error estimate of the original undistorted signal. For time-varying degradations and nonstationary processes, however, the optimal linear estimate requires O(N/sup 2/) time for implementation. We consider filtering in fractional Fourier domains, which enables significant reduction of the error compared with ordinary Fourier domain filtering for certain types of degradation and noise (especially of chirped nature), while requiring only O(N log N) implementation time. Thus, improved performance is achieved at no additional cost. Expressions for the optimal filter functions in fractional domains are derived, and several illustrative examples are given in which significant reduction of the error (by a factor of 50) is obtained.

Fractional wavelet transform.

Abstract: The wavelet transform, which has had a growing importance in signal and image processing, has been generalized by association with both the wavelet transform and the fractional Fourier transform. Possible implementations of the new transformation are in image compression, image transmission, transient signal processing, etc. Computer simulations demonstrate the abilities of the novel transform. Optical implementation of this transform is briefly discussed.

Comments on the interpretation of instantaneous frequency

Abstract: Instantaneous frequency, taken as the derivative of the phase of the signal, is interpreted in the time-frequency literature as the average frequency of the signal at each time. We point out some difficulties with this interpretation, and show that for a generic two-component AM-FM signal, the interpretation holds only when the components are of equal strength. We conclude that instantaneous frequency and the average frequency at each time are generally two different quantities. One possible interpretation of the difference between these two quantities is suggested.

Parameter estimation of spread spectrum frequency-hopping signals using time-frequency distributions

Abstract: In this paper we propose a method for estimating the parameters of frequency-hopping (FH) signals embedded in noise. The method is based on two main steps: (i) representation of the signal in the time-frequency domain; (ii) integration in the time-frequency domain along paths expressed in a parametric form depending on the parametric law characterizing the signal instantaneous frequency. The method does not make any assumption about the alphabet of hopping frequencies, the duration of each hop, or the synchronization. The performance of the method is given in terms of estimation variances.

A four-parameter atomic decomposition of chirplets

Abstract: A new four-parameter atomic decomposition of chirplets is developed for compact representation of signals with chirp components. The four-parameter atom is obtained by scaling the Gaussian function, and then applying the fractional Fourier transform (FRFT), time-shift and frequency-shift operators to the scaled Gaussian. The decomposition is realized by extending the matching pursuit algorithm to four parameters. For this purpose, the four-parameter space is discretized to obtain a dense subset in the Hilbert space. Also, a related time-frequency distribution is developed for clear visualization of the signal components. The decomposition provides a more compact and precise representation of chirp components as compared to the three-parameter ones.

An architecture for the realization of a system for time-frequency signal analysis

Abstract: An architecture of the system for time-frequency signal analysis is presented. This system is based on the S-method, whose special cases are two the most important distributions: the spectrogram and the Wigner distribution. Systems with constant and signal-dependent window widths are presented.

The Continuous Wavelet Transform and Variable Resolution Time-Frequency Analysis

Abstract: W avelet transforms have recently emerged as a mathematical tool for multiresolution decomposition of signals. They have potential applications in many areas of signal processing that require variable time‐frequency localization. The continuous wavelet transform is presented here, and its frequency resolution is derived analytically and shown to depend exclusively on one parameter that should be carefully selected in constructing a variable resolution time‐frequency distribution for a given signal. Several examples of application to synthetic and real data are shown.

Multi-Dimensional Signal Processin Using an Algebraically Extended Signal Representation

Abstract: Many concepts that are used in multi-dimensional signal processing are derived from one-dimensional signal processing. As a consequence, they are only suited to multi-dimensional signals which are intrinsically one-dimensional. We claim that this restriction is due to the restricted algebraic frame used in signal processing, especially to the use of the complex numbers in the frequency domain. We propose a generalization of the two-dimensional Fourier transform which yields a quaternionic signal representation. We call this transform quaternionic Fourier transform (QFT). Based on the QFT, we generalize the conceptions of the analytic signal, Gabor filters, instantaneous and local phase to two dimensions in a novel way which is intrinsically two-dimensional. Experimental results are presented.

A novel FSK demodulation method using short-time DFT analysis for LEO satellite communication systems

Abstract: This paper proposes a novel frequency-shift keying (FSK) demodulation method using short-time discrete Fourier transform (ST-DFT) analysis for low-Earth-orbit (LEO) satellite communication systems. The ST-DFT-based FSK demodulation method is simple and robust to a large and time-variant frequency offset because it expands the received signal in a time-frequency plane and demodulates it only by searching the instantaneous spectral peaks with no complicated carrier-recovery circuit. Two kinds of demodulation strategies are proposed: a bit-by-bit demodulation algorithm and an efficient demodulation-algorithm frequency-sequence estimation (FSE) based on the Viterbi algorithm. In addition, in order to carry out an accurate ST-DFT window synchronization, a simple DFT-based ST-DFT window-synchronization method is proposed.

An introduction to discrete finite frames

Abstract: The frame concept was first introduced by Duffin and Schaeffer (1952), and it is widely used today to describe the behavior of vectors for signal representation. The Gabor (1946) expansion and wavelet transform are two special well-known cases. The goal of this article is to describe the frame theory and introduce a simple tutorial method to find discrete finite frame operators and their frame bounds. An easily implementable method for finding the discrete finite frame and subframe operators has been presented by Kaiser (1994). We introduce the method of Kaiser to compute the discrete finite frame operator. Using subframe operators, the biorthogonal basis and projection vectors in a subspace can be easily calculated. Gabor and wavelet analysis are two popular tools for signal processing, and they can reveal time-frequency distribution for a nonstationary signal. Both schemes can be regarded as signal decompositions onto a set of basis functions, and their basis functions are derived from a single prototype function through simple operations. Therefore, the basis functions used in Gabor and wavelet analysis can be regarded as special frames. For completeness we also make some simple introductions on the results of special frames such as discrete Gabor and wavelet analysis.

An adaptive, on-line, statistical method for bearing fault detection using stator current

Abstract: It is well-known that motor current is a nonstationary signal whose properties vary with respect to the time varying operating conditions of the motor. As a result Fourier analysis makes it difficult to recognize fault conditions from the normal operating conditions of the motor. Time-frequency analysis, on the other hand, unambiguously represents the motor current which makes signal properties related to fault detection more evident in the transform domain. In this paper, we present an adaptive, statistical, time-frequency method for the detection of bearing faults. Due to the time varying normal operating conditions of the motor and the effect of motor geometry on the current, we employ a training base approach in which the algorithm is trained to recognize the normal operating conditions of the motor before the actual testing starts. The experimental results from our study suggests that the proposed method provides a powerful, and a general approach to the motor current based fault detection.

Special purpose hardware for time frequency analysis

Abstract: A special purpose hardware system for time-frequency signal analysis is presented. This system is based on the S-method which has the significant advantage of using the short time Fourier transform as an intermediate step in its implementation. The hardware designed for a fixed point arithmetic is well-structured and suitable for VLSI implementation. An example including an error analysis is also provided.

Generalized instantaneous parameters and window matching in the time-frequency plane

Abstract: The concept of instantaneous parameters, which has previously been associated exclusively with 1-D measures like the instantaneous frequency and the group delay, are extended to the 2-D time-frequency plane. Such generalized instantaneous parameters are associated with the short-time Fourier transform. They may also be interpreted as local moments of certain time-frequency distributions. It is shown that these measures enable local signal behavior to be characterized in the time-frequency plane for nonstationary deterministic signals. The usefulness of the generalized instantaneous parameters is demonstrated in their application to optimal selection of windows for spectrograms. This is achieved through window matching in the time-frequency plane. An algorithm is provided that illustrates the performance of this window matching. Results based on simulated and real data are presented.

Time-frequency analysis of the first heart sound. Part 2: An appropriate time-frequency representation technique.

Abstract: A simulated first heart sound (S1) signal is used to determine the best technique for analysing physiological S1 from the following five time-frequency representations (TFR): the spectrogram, time-varying autoregressive modelling, binomial reduced interference distribution, Bessel distribution and cone-kernel distribution (CKD). To provide information on the time and frequency resolutions of each TFR technique, the instantaneous frequency and the -3 dB bandwidth as functions of time were computed for each simulated component of the S1. The performance index for selecting the best technique was based on the relative error and the correlation coefficient of the instantaneous frequency function between the theoretical distribution and the computed TFR. This index served to select the best technique. The sensitivity of each technique to noise and to small variations of the signal parameters was also evaluated. The results of the comparative study show that, although important limitations were found for all five TFRs tested, the CKD appears to be the best technique for the time-frequency analysis of multicomponent signals such as the simulated S1.

An accurate error analysis model for fast Fourier transform

Abstract: An error propagation model is proposed for the in-place decimation-in-time version of the radix-2 FFT algorithm. With the model, an accurate error expression and error variance for the computation of FFT are derived. This correspondence deals with fixed-point and block floating-point arithmetic. Simulation results agree closely with the theoretical predicted ones. We find that some roundoff errors at different stages correlate with each other. The density of correlations is closely associated with the round-off approach used in butterfly calculations.

Global frequency modulation laws extraction from the Gabor transform of a signal: a first study of the interacting components case

Abstract: Under some asymptotic conditions, modulated trajectories in the time-frequency plane can be directly extracted from the coefficients of the Gabor (1946) transform (or wavelets) of a signal. For simple modulated signals, it has been shown that these trajectories allow us to accurately characterize the frequency-modulation (FM) laws of the analyzed signals. For more modulated signals (for instance musical sounds), some interactions between the trajectories may occur and complex time-frequency structures appear. These interactions and the time-frequency diagrams thus obtained are mathematically and qualitatively described. In particular, some connections with the diagrams of the phase of the transform and with acoustical interaction phenomena (like beats phenomena) are discussed.

S-class of time-frequency distributions

Abstract: A new general class of distributions (S-class of distributions) for time–frequency signal analysis is proposed. This class is derived by generalising recently defined S-distribution. It is possible to define the S-counterpart distribution for each known distribution from the Cohen class, such that some of the performances may be improved. This class of distributions may be treated as a variant of the author's L-class of distributions, but it may satisfy unbiased energy conditions, time marginal as well as the frequency marginal in the case of asymptotic signals. A method for the realisation of the S-distribution which will be, in the case of multicomponent signals, equal to the sum of S-distributions of each component separately, is presented. Theory is illustrated by examples.

Time-frequency plane Wiener filtering of the high-resolution ECG: background and time-frequency representations

Abstract: This paper introduces the concept of a posteriori Wiener filtering (APWF), performed in the time-frequency plane. The objective is to improve the signal-to-noise ratio (SNR) of the ensemble-averaged high-resolution electrocardiogram (HRECG). APWF was developed to address the problem of a limited ensemble size for estimating ensemble-averaged evoked potentials. For the HRECG, the authors identify the major challenge as adapting the time-frequency structure of the filter to that of low-level cardiac signals. Technical limitations and the characteristics of HRECG signals make time-frequency analysis of the ensemble average problematic. Normal and abnormal signal components are difficult to distinguish due to low time-frequency energy concentration and limited spectrotemporal resolution. However, considering the entire ensemble of repetitive ECG records, signal and noise components are separable in the time-frequency plane. This forms the basis of the new time-frequency plane Wiener (TFPW) filter, applicable to any ensemble averaging problem involving repetitive deterministic signals mixed with uncorrelated noise.

Fractional Gabor transform.

Abstract: A fractional Gabor transform (FRGT) is proposed. This new transform is a generalization of the conventional Gabor transform (GT) based on the Fourier transform to the windowed fractional Fourier transform (FRFT). The FRGT provides analyses of signals in both the real space and the FRFT frequency domain simultaneously. The space-FRFT frequency pattern can be rotated as the fractional order changes. The FRGT has an additional freedom, compared with the conventional GT, i.e., the transform order. The FRGT may offer a useful tool for guiding optimal filter design in the FRFT domain in signal processing.

Time-Frequency and Ambiguity Function Approaches in Structural Identification

Abstract: In the identification and control of some structures it is important to be able to resort to special techniques that may exploit environmental excitation in normal serviceability conditions. Hence, preliminary signal processing must be designed to extract significant data, even in the presence of unknown, weak, or transient excitations. In the processing and representation of nonstationary dynamic signals, great importance has been taken on in the literature by the transforms in the joint time-frequency domain. In this context, a method based on the time-frequency representation and some properties of the ambiguity function is proposed to identify the natural frequencies, draw the modal shapes, and detect nonlinearity in such testing conditions. To this end, the performances of the new Choi-Williams exponential kernel are compared with those of the Wigner-Ville transform. Finally techniques based on the instantaneous cross-correlation function are proposed to improve the performance of the identification method, and an application to a real bridge is discussed.

Wave-oriented signal processing of dispersive time-domain scattering data

Abstract: Phase-space data processing is receiving increased attention because or its potential for furnishing new discriminants relating to classification and identification of targets and other scattering environments. Primary emphasis has been on time-frequency processing because of its impact on transient, especially wideband, short-pulse excitations. Here, we investigate the windowed Fourier transform, the wavelet transform, and model based superresolution algorithms within the context of a fully quantified and calibrated test problem investigated by us previously: two-dimensional (2-D) short-pulse plane wave scattering by a finite periodic array of perfectly conducting coplanar flat strips. Because the forward problem has been fully calibrated and parametrized, some quantitative measures can be assigned with respect to the tradeoffs of these time-frequency algorithms, yielding tentative performance assessments of the tested processing algorithms.

Modelling of natural sounds by time–frequency and wavelet representations

Abstract: Sound modelling is an important part of the analysis–synthesis process since it combines sound processing and algorithmic synthesis within the same formalism. Its aim is to make sound simulators by synthesis methods based on signal models or physical models, the parameters of which are directly extracted from the analysis of natural sounds. In this article the successive steps for making such systems are described. These are numerical synthesis and sound generation methods, analysis of natural sounds, particularly time–frequency and time–scale (wavelet) representations, extraction of pertinent parameters, and the determination of the correspondence between these parameters and those corresponding to the synthesis models. Additive synthesis, nonlinear synthesis, and waveguide synthesis are discussed.

Shift and scale invariant detection

Abstract: Different signal realizations generated from a given source may not appear the same. Time shifts, frequency shifts, and scales are among the signal variations commonly encountered. Time-frequency distributions (TFDs) covariant to time and frequency shifts and scale changes reflect these variations in a predictable manner. Based on such TFDs, representations invariant to these signal distortions are possible. Presented here are two approaches for discriminating between signal classes where within class translation and scale variation occur. The first method uses an auto-correlation followed by a scale transform to achieve the invariances. The second method treats the TFD as a two-dimensional probability density function and applies a transformation that removes the mean and variance to provide the shift and scale invariance. Each method employs discrimination mechanisms to yield powerful results.

A new digital filter for phasor computation. I. Theory

Abstract: A new pair of orthogonal filters for phasor computation is presented. They have excellent time-frequency characteristics for fault location and measurement. Their impulse responses are obtained applying the inverse Fourier transform to a single-lobe function with a strong stopband. From this process, a new window emerges. Without side lobes, it overcomes the temporal barriers imposed by the rectangular window, implicit in digital Fourier filtering. Its length is not restricted to a multiple of one cycle, and it can be adjusted to cover totally the available samples of the fault, extracting the fundamental component with growing precision. On the other hand, its sampling frequency can be reduced to twice the fundamental frequency. In particular, at this minimum sampling frequency, the digital cosine filter rejects the even harmonics and the aperiodic component.