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## Über dieses Buch

Non-Gaussian Signal Processing is a child of a technological push. It is evident that we are moving from an era of simple signal processing with relatively primitive electronic cir­ cuits to one in which digital processing systems, in a combined hardware-software configura.­ tion, are quite capable of implementing advanced mathematical and statistical procedures. Moreover, as these processing techniques become more sophisticated and powerful, the sharper resolution of the resulting system brings into question the classic distributional assumptions of Gaussianity for both noise and signal processes. This in turn opens the door to a fundamental reexamination of structure and inference methods for non-Gaussian sto­ chastic processes together with the application of such processes as models in the context of filtering, estimation, detection and signal extraction. Based on the premise that such a fun­ damental reexamination was timely, in 1981 the Office of Naval Research initiated a research effort in Non-Gaussian Signal Processing under the Selected Research Opportunities Program.

## Inhaltsverzeichnis

### 1. Bispectral Characterization of Ocean Acoustic Time Series: Nonlinearity and Non-Gaussianity

Abstract
Previous research into the Gaussianity of ocean acoustical time series has examined univariate marginal densities. In this paper we present research which examines this issue from a time series point of view. Even series which previously passed univariate tests for normality are shown to be non-Gaussian time series. Additionally, these time series are shown to be nonlinear time series, so that such acoustical series must be modeled in a nonlinear fashion.
Patrick L. Brockett, Melvin Hinich, Gary R. Wilson

### 2. Class a Modeling of Ocean Acoustic Noise Processes

Abstract
Previous work has shown that some ocean acoustic noise processes can be represented as Class A noise. Likelihood ratio and threshold detectors have been developed to detect signals in the presence of Class A noise. The performance of these detectors is significantly affected by the accuracy with which the parameters of the Class A noise can be estimated. This paper presents two methods of estimating the Class A parameters, a minimum distance method and a maximum likelihood method. These methods are compared to a previously developed method using estimates of the moments of the noise process and are generally found to be superior estimators.
Dennis R. Powell, Gary R. Wilson

### 3. Statistical Characteristics of Ocean Acoustic Noise Processes

Abstract
A statistical analysis is given of ambient noise data from several ocean acoustic environments. Included in the analysis are statistical tests for homogeneity and randomness, statistical tests for normality, sample autocorrelation functions, and kernel density estimates of the instantaneous amplitude fluctuations. The test results indicate that a randomness hypothesis may be rejected when Nyquist rate sampling is employed. A randomization procedure is applied to the data in order to create ensembles which pass the tests for randomness and homogeneity. Analysis of these ensembles indicates that a stationary Gaussian assumption is not justified for some ocean environments. The largest deviations from normality occur in the tail regions of the density function and are often attributable to non-stationary characteristics of the data.
Fredrick W. Machell, Clark S. Penrod, Glen E. Ellis

### 4. Conditionally Linear and Non-Gaussian Processes

Abstract
A new theory and corresponding methodology is evolving for certain classes of nonlinear non-Gaussian signal processing. This research considers particular statistical model structures such as conditionally linear or bilinear. Some non-Gaussian distributions, which arise in underwater acoustical signal processing, are included, as are others. The results are based on rigorous developments related to, but not limited to, bilinear and conditionally Gaussian processes. Parameter or state estimation (e.g., acoustic source location) are studied as well as preliminary results in information transmission and coding.
R. R. Mohler, W. J. Kolodziej

### 5. A Graphical Tool for Distribution and Correlation Analysis of Multiple Time Series

Abstract
This paper proposes the use of the parallel coordinate representation for the representation of multivariable statistical data, particularly for data arising in the time series context. We discuss the statistical interpretation of a variety of structures in the parallel coordinate diagrams including features which indicate correlation and clustering. One application is to graphically assess the finite dimensional distribution structure of a time series. A second application is to assessing structure of multichannel time series. An example of this latter application is given. It is shown that the parallel coordinate representation can be exploited as a graphical tool for using beam forming for short segments of ocean acoustic data.
Edward J. Wegman, Chris Shull

### 6. Comments on Structure and Estimation for NonGaussian Linear Processes

Abstract
A great deal of the research in time series analysis has been based on insights derived from the structure of Gaussian stationary processes. Suppose (X n), n = …,-1,0,1,… is a Gaussian stationary sequence with mean zero and spectral distribution function F(λ).
M. Rosenblatt

### 7. Harmonizable Signal Extraction, Filtering and Sampling

Abstract
The purpose of this paper is to describe some aspects of analysis on a class of non-stationary and non Gaussian processes dealing with linear filtering, signal extraction from observed data, and sampling the process. The class to be considered consists of harmonizable processes which uses some suitably generalized spectral methods of the classical theory. Let us elaborate these statements.
M. M. Rao

### 8. Fisher Consistency of Am-Estimates of the Autoregression Parameter Using Hard Rejection Filter Cleaners

Abstract
An AM estimate $$\hat(\phi)$$ of the AR(1) parameter φ is a solution of the M-estimate equation $$\sum\limits_1^n {\hat x_{t - 1} } \psi \left( {[y_t - \hat \phi \hat x_{t - 1} ]/s_t } \right) = 0$$ where $$\hat {x}_t-1$$, t=0,2,…,satisfies the robust filter recursion $$\hat {x}_t = \hat {\phi} \hat {x}_t-1 + s_t \psi \ast ([y_t - \hat {\phi} \hat {x}_t-1]/s_t)$$, and s t is a data dependent scale which satisfies an auxiliary recursion. The AM-estimate may be viewed as a special kind of bounded-influence regression which provides robustness toward contamination models of the type y t = (1 - z t) x t + z t w t where z t is a 0–1 process, w t is a contamination process and x t is an AR(1) process with parameter φ. While AM-estimates have considerable heuristic appeal, and cope with time series outliers quite well, they are not in general Fisher consistent. In this paper, we show that under mild conditions, $$\hat(\phi)$$ is Fisher consistent when [mathtype] is of hard-rejection type.
R. D. Martin, V. J. Yohai

### 9. Bayes Least Squares Linear Regression Is Asymptotically Full Bayes: Estimation of Spectral Densities

Abstract
The Bayes least squares linear method of estimating regression functions using orthogonal expansions may be implemented by making first an orthogonal transformation that yields uncorrelated variables. The theorem presented here gives conditions sufficient for joint asymptotic normality of the transformed variables. Thus when the coefficients are independent and normal according to the prior, one expects that in the presence of much data they will be (approximately) independent normal according to the posterior. The method is illustrated using a suggestion of Wahba (1980) to transform the problem of estimating a spectral density into a regression problem.
H. D. Brunk

### 10. Signal Detection for Spherically Exchangeable (Se) Stochastic Processes

Abstract
Consider signal detection in the following context. The background pure noise (PN) is produced by nature’s choosing a parameter value and then generating a time series according to a (stochastic process) law with that parameter value. This process may be repeated N times. If there is a signal present, what is generated is a noise-plus-signal (N + S), time series. The statistical properties of the latter-type of time series are different from those of the former.
C. B. Bell

### 11. Contributions to Non-Gaussian Signal Processing

Abstract
Data from the 1980 and 1982 Arctic experiments and from in air helicopter-radiated noise measurements are presented and discussed as part of the physical evidence for the existence of non-Gaussian signals and noises. After the data are presented, theoretical considerations are given for frequency domain kurtosis estimation. Statistical models are introduced and discussed which support and correspond to the measurements made on the real data in the frequency domain. It was discovered from frequency domain measurements that at times Arctic under-ice noise consists of narrowband non-Gaussian interference components. These components can severely degrade the performance of sonar systems. Therefore, a method was developed that effectively removes narrowband interference by implementing a non-linearity in the frequency domain.
Roger F. Dwyer

### 12. Detection of Signals in the Presence of Strong, Signal-Like Interference and Impulse Noise

Abstract
We assume that the noise which interferes with signal detection can be considered to be a mixture of non-stationary, high-amplitude, non-Gaussian components plus a low amplitude Gaussian stationary component. Such a model appears to be widely applicable. The methodology that we propose for signal detection is to identify, categorize, model, and remove the non-Gaussian components in a piece-wise fashion based on their ease of separability from the background Gaussian noise and weak signals. This approach to modeling and processing complicated and non-stationary data is similar to that of experimental physicists going back at least to the time of Newton and perhaps most clearly articulated by Eugene Wigner in his Nobel Prize lecture [1]. Liu and Nolte [2] and Claus, Kadota, and Romain [3] have shown that when the noise is Gaussian and consists of a sum of a strong highly coherent component and a weak component of independent noise samples, then estimation and subtraction of the coherent noise component is nearly optimal. The application of special data smoothers and data cleaners by Martin and Thomson [4] for obtaining robust spectral estimates when the data is contaminated by outliers has provided another motivation for our use of adaptive differential quantization for robust detection.
D. W. Tufts, I. P. Kirsteins, P. F. Swaszek, A. J. Efron, C. D. Melissinos

### 13. On NonGaussian Signal Detection and Channel Capacity

Abstract
This paper contains a discussion of some recent research in signal detection and communications. No proofs are included; the emphasis is on motivation and results. Some precise definitions and results are contained in the Appendix. For a more detailed discussion and additional results, reference is made to [8,9,10] for the signal detection problems, and to [7,8] for the channel capacity problems. The present paper is couched in terms of problems in underwater acoustics; as will be seen, the models and results are of general applicability.
Charles R. Baker

### 14. Detection in a Non-Gaussian Environment: Weak and Fading Narrowband Signals

Abstract
Procedures for the detection of both weak and narrowband signals in non–Gaussian noise environments are discussed. For the weak signal case, nonlinear processors based on the Middleton Class A noise model and the mixture representation are developed. Significant processing gains are achievable with some rather simple procedures. A variant of the mixture model leads to a nonstationary detector, called the “switched detector.” Experiments with this detector on ambient arctic and shrimp noises show processing gains of 1.4 to 4.1 dB, respectively.
Signals with a moderate signal-to-noise ratio in non-Gaussian noise are modeled as fading narrowband signals. A new processor is developed which combines a robust estimator (for the fading signal) with a robust detection procedure. The robust estimator-detector prserves the structure of the quadrature (envelope) matched filter and is shown to be asymptotically optimal for a wide range of decision rules and several common target models encountered in sonar and radar. Various degrees of robustness are achieved, depending on the assumed availability for noise reference samples.
Stuart C. Schwartz, John B. Thomas

### 15. Energy Detection in the Ocean Acoustic Environment

Abstract
The performance of the energy detector is evaluated using ambient noise data from several ocean acoustic environments. Estimates of the false alarm probability are presented as a function of the detection threshold for each environment. Estimated values for the corresponding minimum detectable signal-to-noise ratio (MDS) are also given for an artifically generated white Gaussian signal. The results presented here indicate that non-Gaussian noise statistics can have a significant impact on the relationship between the false alarm probability and the detection threshold. This threshold adjustment results in a serious degradation of energy detector performance in terms of the MDS for some non-Gaussian noise environments.
Fredrick W. Machell, Clark S. Penrod
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