Lisa A. Pflug

Properties of higher‐order correlations and spectra for bandlimited, deterministic transients

Higher‐order correlations and spectra may be used for detection, time delay estimation, classification, and discrimination of signals. For these applications, a detailed knowledge of their attributes can be highly useful. In this paper, the properties of the bicorrelation and tricorrelation of bandlimited deterministic transients, i.e., energy signals, and their spectra, the bispectrum and trispectrum are studied. Bandlimited transients that contain frequencies down to and including zero and those that have a nonzero lower cutoff frequency are both considered. Using symmetries inherent in the bispectrum of a signal, the entire bispectrum can be mapped from bispectral elements defined in two polygons, one for the unaliased and one for the aliased domain, each of which is one‐twelfth the area of its total domain. The nonredundant unaliased region of the trispectrum is contained in two principal unaliased polyhedra, each replicated 48 times to reproduce the full trispectrum. If there is aliasing in the trisp...

Performance of some sparseness criterion blind deconvolution methods in the presence of noise

A comparison of the spareseness (simplicity) norm criterion blind deconvolution methods of Cabrelli and Wiggins is made in order to ascertain relative performance for underwater acoustic transient source signal estimation, especially in the presence of noise. Both methods perform well at high signal-to-noise ratios, producing source estimates that are significant improvements over the original received signal for classification purposes. At moderate and lower SNRs, the Cabrelli method tends to generate results that are superior to the Wiggins method. This is especially true for a damped sinusoid transient source, for which the Wiggins method fails completely at lower SNRs, while the Cabrelli method can still produce good source estimates.

Detection of Oscillatory and Impulsive Transients Using Higher Order Correlations and Spectra

Higher‐order cross and ordinary correlation detectors are applied to four deterministic transients contaminated by uncorrelated Gaussian noise only. Histograms and moments are used to examine the properties of the signals and their effect on detector performance. Receiver operating characteristic (ROC) curve analysis and limiting signal‐to‐noise ratios for ‘‘good’’ detection provide comparative measures for different detectors. Probability density functions of detection ordinate values of signal‐present and noise‐only correlations are used to explain ROC curve behavior. Using a known source, the cross‐correlation detector performs better than the higher‐order correlation detectors for each transient studied. However, for an unknown narrow pulse source signal, the bicorrelation and tricorrelation detectors outperform the cross‐correlation detector. In contrast, the bicorrelation detector performs very poorly for low‐frequency narrow‐band signals with a small third moment embedded in uncorrelated Gaussian n...

Variability in higher order statistics of measured shallow-water shipping noise

Many underwater acoustic signal processing algorithms are designed for use in stationary and/or Gaussian noise. While these assumptions are often valid for applications in deep water ocean areas, they may not be appropriate for shallow water areas, especially in the presence of local shipping activity. Local shipping also produces spatial correlation in the noise and introduces additional complexity for multichannel processing. In this paper, two 30-minute sets of ambient ocean noise, recorded near the San Diego, California coast, are analyzed for stationarity and Gaussianity using the Kolmogorov-Smirnov test. Since processing algorithms based on higher order statistics often assume Gaussianity, time-dependent fluctuations in the third and fourth order cumulants are also analyzed. The analysis reveals significant variability in the time lengths of stationary periods, and episodic periods of nonGaussianity that last for up to five minutes. Statistical fluctuations appear predominantly in the second and fourth order cumulants rather than the third order cumulant. The shipping noise is also shown to be correlated between pairs of hydrophones with the level of correlation varying over time and the correlation ranging from positive to negative with increasing channel separation.

Sampling requirements and aliasing for higher‐order correlations

While sampling at a Nyquist frequency equal to the highest frequency present in the data (critical sampling) is sufficient to prevent aliasing in both the data and the autocorrelation of a bandlimited energy signal, the sampling requirements for the avoidance of aliasing in higher‐order correlations and spectra are not the same. Also, there is a difference in aliasing effects depending on whether one samples the original continuous‐time signal and calculates the autocorrelation or one samples the continuous‐time autocorrelation. This distinction between sampling procedures must be made for correlations of higher order, as well, for which not only the type of aliasing but also the sampling requirements to prevent aliasing differ. In particular, if one samples the continuous‐time autobicorrelation or autotricorrelation, critical sampling is sufficient to prevent aliasing. In practice, however, it is not usually the continuous‐time autobicorrelation or autotricorrelation that is sampled. Generally, it is the...

Prefiltering for improved correlation detection of bandlimited transient signals

Prefiltering, or limiting the passband of a received signal, can be used to improve ordinary correlation threshold detector performance for an unknown source model. For the known source model, the cross‐correlation detector is equivalent to matched filtering, and intrinsically contains prefiltering. Prefiltering also improves higher‐order correlation threshold detector performance, but often with more advantage than seen in the ordinary correlation detector. This is true for both the unknown and known source models. Geometric interpretations are given to provide insight into the origin of potential higher‐order advantage. Eight energy signals, each with three different Fourier magnitude‐based filters, are used to test the cross‐correlation, bicorrelation, and tricorrelation detectors by Monte Carlo simulation and hypothesis testing. Significant signal‐to‐noise ratio (SNR) gains are evident for both the known and unknown source models with the tricorrelation exhibiting the largest gains. The tricorrelation...

Sampling requirements for nth‐order correlations

As has been derived previously [Pflug, MS thesis, Univ. of New Orleans (1990); Pflug et al., J. Acoust. Soc. Am. 91, 975–988 (1992); Pflug et al., J. Acoust. Soc. Am. 94, 2159–2172 (1993)], the sampling intervals required to prevent nonremovable and removable aliasing in higher‐order correlations calculated from discrete‐time data are Δt3≤1/(3ft) for the bicorrelation and Δt4≤1/(4ft) for the tricorrelation, where ft is the highest or top frequency present in a signal. It is shown here that a general expression for the sampling required to prevent aliasing for each order n of correlation calculated from discrete‐time data is Δtn≤1/(nft). Time‐domain computer calculations of correlation central ordinate values for orders two through eight are consistent with this result. Removable aliasing can be eliminated for data which are sampled such that Δt≤1/(2 ft) by (1) the application of n‐dimensional transform or time‐domain masking filters, (2) one‐dimensional filtering of the transform domain diagonal factor, o...

Prefiltering for higher order advantage

Prefiltering, or limiting the spectral domain of data, improves maximum magnitude correlation peak detectors, both ordinary and higher order, for known and unknown sources. The only exception is the matched filter, which intrinsically contains prefiltering. For the cases studied, prefiltering generally has a higher order advantage, i.e., for higher order and in higher dimensions, it is even more effective than in one dimension for the ordinary correlation. Geometrical considerations can give some insight into this advantage. The tricorrelation detector with prefiltering performs best for all eight tested signals in the unknown source tests, and is the best detector for seven of the eight signals in the known source tests. Prefiltering for higher order correlation detectors involves only one-dimensional filtering and so is computationally efficient.<<ETX>>

Principal domains of the trispectrum, signal bandwidth, and implications for deconvolution

Fourth-order statistics can be useful in many signal processing applications, offering advantages over or supplementing second-order statistical techniques. One reason is that fourth-order statistics can discriminate between non-Gaussian signals and Gaussian noise. Another is that fourth-order statistics contain phase information, whereas second-order statistics do not. In the continuing development of the mathematical properties of fourth-order statistics, several researchers have derived existence conditions and definitions for the unaliased and aliased principal domains of the discrete trispectrum, which is significantly more complex than the power or energy spectrum. The consistencies and inconsistencies of these results are presented and resolved in this paper. The most flexible definitions give four individual principal domains for the discrete trispectrum: two unaliased and two aliased. The most useful combinations are those that combine the two unaliased domains together and the two aliased domains together, which can be done easily from the four individual domains. The relationship between the individual trispectral domains and signal bandwidth is important when using the fourth-order statistic for applications because they have particular properties that can be detrimental to some deconvolution algorithms. The reasons for this, as well as the validity of proposed solutions to this problem, are explained by the trispectral structure and its origins.

Minimum entropy filtering for improving nonstationary sonar signal classification

The passive sonar classification problem can be decomposed into two stages: (l) recovering the source time signature of a transient event from a set of received signals by accounting for environmental distortion effects, and (2) applying a pattern recognition algorithm to the estimated source signature for final classification. The minimum entropy method is studied with regard to its performance in removing multipath distortion from passive transients, to improve the performance of classifiers. It was found that the method often works well if the kurtosis of the associated multipath Greens function is high enough, and that signal stationarity is not required. We also found that, while there are usually a few filter lengths at which the best solutions are obtained with conventional convergence criteria, good solutions exist across a much broader range of filter lengths if the iterations are not allowed to proceed to convergence. That is, kurtosis needs to be increased, but not maximized. In many cases, two or three iterations is sufficient.