High-resolution frequency-wavenumber spectrum analysis
01 Aug 1969-Vol. 57, Iss: 8, pp 1408-1418
TL;DR: In this article, a high-resolution frequency-wavenumber power spectral density estimation method was proposed, which employs a wavenumber window whose shape changes and is a function of the wave height at which an estimate is obtained.
Abstract: The output of an array of sansors is considered to be a homogeneous random field. In this case there is a spectral representation for this field, similar to that for stationary random processes, which consists of a superposition of traveling waves. The frequency-wavenumber power spectral density provides the mean-square value for the amplitudes of these waves and is of considerable importance in the analysis of propagating waves by means of an array of sensors. The conventional method of frequency-wavenumber power spectral density estimation uses a fixed-wavenumber window and its resolution is determined essentially by the beam pattern of the array of sensors. A high-resolution method of estimation is introduced which employs a wavenumber window whose shape changes and is a function of the wavenumber at which an estimate is obtained. It is shown that the wavenumber resolution of this method is considerably better than that of the conventional method. Application of these results is given to seismic data obtained from the large aperture seismic array located in eastern Montana. In addition, the application of the high-resolution method to other areas, such as radar, sonar, and radio astronomy, is indicated.
Citations
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TL;DR: In this article, a description of the multiple signal classification (MUSIC) algorithm, which provides asymptotically unbiased estimates of 1) number of incident wavefronts present; 2) directions of arrival (DOA) (or emitter locations); 3) strengths and cross correlations among the incident waveforms; 4) noise/interference strength.
Abstract: Processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions. The general problem considers sensors with arbitrary locations and arbitrary directional characteristics (gain/phase/polarization) in a noise/interference environment of arbitrary covariance matrix. This report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution. A description is given of the multiple signal classification (MUSIC) algorithm, which provides asymptotically unbiased estimates of 1) number of incident wavefronts present; 2) directions of arrival (DOA) (or emitter locations); 3) strengths and cross correlations among the incident waveforms; 4) noise/interference strength. Examples and comparisons with methods based on maximum likelihood (ML) and maximum entropy (ME), as well as conventional beamforming are included. An example of its use as a multiple frequency estimator operating on time series is included.
12,446 citations
TL;DR: The article consists of background material and of the basic problem formulation, and introduces spectral-based algorithmic solutions to the signal parameter estimation problem and contrast these suboptimal solutions to parametric methods.
Abstract: The quintessential goal of sensor array signal processing is the estimation of parameters by fusing temporal and spatial information, captured via sampling a wavefield with a set of judiciously placed antenna sensors. The wavefield is assumed to be generated by a finite number of emitters, and contains information about signal parameters characterizing the emitters. A review of the area of array processing is given. The focus is on parameter estimation methods, and many relevant problems are only briefly mentioned. We emphasize the relatively more recent subspace-based methods in relation to beamforming. The article consists of background material and of the basic problem formulation. Then we introduce spectral-based algorithmic solutions to the signal parameter estimation problem. We contrast these suboptimal solutions to parametric methods. Techniques derived from maximum likelihood principles as well as geometric arguments are covered. Later, a number of more specialized research topics are briefly reviewed. Then, we look at a number of real-world problems for which sensor array processing methods have been applied. We also include an example with real experimental data involving closely spaced emitters and highly correlated signals, as well as a manufacturing application example.
4,410 citations
01 Sep 1982
TL;DR: In this article, a local eigenexpansion is proposed to estimate the spectrum of a stationary time series from a finite sample of the process, which is equivalent to using the weishted average of a series of direct-spectrum estimates based on orthogonal data windows to treat both bias and smoothing problems.
Abstract: In the choice of an estimator for the spectrum of a stationary time series from a finite sample of the process, the problems of bias control and consistency, or "smoothing," are dominant. In this paper we present a new method based on a "local" eigenexpansion to estimate the spectrum in terms of the solution of an integral equation. Computationally this method is equivalent to using the weishted average of a series of direct-spectrum estimates based on orthogonal data windows (discrete prolate spheroidal sequences) to treat both the bias and smoothing problems. Some of the attractive features of this estimate are: there are no arbitrary windows; it is a small sample theory; it is consistent; it provides an analysis-of-variance test for line components; and it has high resolution. We also show relations of this estimate to maximum-likelihood estimates, show that the estimation capacity of the estimate is high, and show applications to coherence and polyspectrum estimates.
3,921 citations
01 Nov 1981
TL;DR: In this paper, a summary of many of the new techniques developed in the last two decades for spectrum analysis of discrete time series is presented, including classical periodogram, classical Blackman-Tukey, autoregressive (maximum entropy), moving average, autotegressive-moving average, maximum likelihood, Prony, and Pisarenko methods.
Abstract: A summary of many of the new techniques developed in the last two decades for spectrum analysis of discrete time series is presented in this tutorial. An examination of the underlying time series model assumed by each technique serves as the common basis for understanding the differences among the various spectrum analysis approaches. Techniques discussed include the classical periodogram, classical Blackman-Tukey, autoregressive (maximum entropy), moving average, autotegressive-moving average, maximum likelihood, Prony, and Pisarenko methods. A summary table in the text provides a concise overview for all methods, including key references and appropriate equations for computation of each spectral estimate.
2,941 citations
Book•
01 Jan 2005
TL;DR: 1. Basic Concepts. 2. Nonparametric Methods. 3. Parametric Methods for Rational Spectra.
Abstract: 1. Basic Concepts. 2. Nonparametric Methods. 3. Parametric Methods for Rational Spectra. 4. Parametric Methods for Line Spectra. 5. Filter Bank Methods. 6. Spatial Methods. Appendix A: Linear Algebra and Matrix Analysis Tools. Appendix B: Cramer-Rao Bound Tools. Appendix C: Model Order Selection Tools. Appendix D: Answers to Selected Exercises. Bibliography. References Grouped by Subject. Subject Index.
2,620 citations
Cites methods from "High-resolution frequency-wavenumbe..."
...4 [Capon 1969; Lacoss 1971]....
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...This is the basic idea behind the Capon method, which is an FBA procedure based on a data–dependent bandpass filter [Capon 1969; Lacoss 1971]....
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References
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576 citations
01 Jan 1967
TL;DR: The present work considers the problem of designing a linear filter which combines the outputs of the 25 sensors in a subarray so as to suppress the noise without distorting the signal, or event, and finds that the signal-to-noise improvement given by the frequency- domain procedure is within 2 dB of the gain obtained with the time-domain procedure.
Abstract: The experimental Large Aperture Seismic Array (LASA) represents an attempt to improve the capability to monitor underground nuclear weapons tests and small earthquakes by making a large extrapolation in the existing art of building arrays of spaced and interconnected seismic transducers. The LASA is roughly equivalent to 21 separate subarrays, each consisting of 25 sensors, spread over an aperture of 200 km. The present work considers the problem of designing a linear filter which combines the outputs of the 25 sensors in a subarray so as to suppress the noise without distorting the signal, or event. This filter provides a minimum-variance unbiased estimate of the signal which is the same as the maximum-likelihood estimate of the signal if the noise is a multidimensional Gaussian process. An extensive discussion of the theory of multidimensional maximum-likelihood processing is given. A computer program implementation of the maximum-likelihood filter is presented which employs the cross-correlation matrix of noise measured just prior to the arrival of the event. This time-domain synthesis procedure requires relatively large amounts of computer time to synthesize the filter, and is quite sensitive to the assumption of noise stationarity. The asymptotic theory of maximum-likelihood filtering is also given. An asymptotically optimum frequency-domain synthesis technique is given for two-sided multidimensional filters. This procedure is well suited to machine computation and has the advantage with respect to the time-domain procedure of requiring about 10 times less computation time. A description of a computer program implementation of the frequency-domain synthesis method is given which employs the spectral matrix of the noise estimated just before the arrival of the event. The experimental results obtained by processing several events recorded at the LASA are presented, as well as a comparison of the performance of the frequency-domain method relative to the time-domain synthesis technique. It is found that the signal-to-noise improvement given by the frequency-domain procedure is within 2 dB of the gain obtained with the time-domain procedure, and that the frequency-domain method is relatively insensitive to the assumption of noise stationarity.
261 citations
TL;DR: Frequency-wave number spectra of microseisms were obtained by use of a set of short-period and long-period seismometers at LASA (Large Aperture Seismic Array, Montana) from the phase velocity and direction of body waves, source areas were determined, coinciding with low-pressure regions on the weather map.
Abstract: Frequency-wave number spectra of microseisms were obtained by use of a set of short-period and long-period seismometers at LASA (Large Aperture Seismic Array, Montana). At times of relatively high microseismic activity short-period (shorter than 5 seconds) microseisms consist of both body waves and higher-mode surface waves. From the phase velocity and direction of body waves, source areas were determined, coinciding with low-pressure regions on the weather map. At longer periods, microseisms consist of fundamental- mode Rayleigh and Love waves, the former being dominant. Most microseismic energy arrives at LASA from the northeast and the west.
107 citations
TL;DR: The interpretation of seismograms and spectra influenced by multiple seismic events is discussed in this paper, where two cases of twin earthquakes are analyzed and the significant features of interference are demonstrated.
Abstract: Two or more dispersed wave trains each with constant amplitude will interfere giving a resultant
wave train which is amplitude modulated, if the individual waves have their principal
energies in a common frequency band and if the trains arrive with time separations small
compared to their total length. The dispersive characteristics of the trains need not be the
same. If the component trains are of comparable magnitude, the modulation due to interference
becomes significant and a "beat" phenomenon occurs. Multiple trains of dispersed
seismic surface waves may occur because of a temporal and/or spatial distribution at the
source or because of multipath propagation. Each of these causal mechanisms influences the
amplitude and phase spectra of the resultant wave train; derived properties such as phase
velocities and amplitude ratios are also influenced. In the case of multipath propagation,
wavelength dependent time delays may occur. Two cases of twin earthquakes are analyzed,
and the significant features of interference are demonstrated. In one case, estimates are obtained
for the amplitude ratio and time delay of the second shock with respect to the first.
The interpretation of seismograms and spectra influenced by multiple events is discussed.
80 citations