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This research offers an additional approach to the increased interest in information theoretic techniques utilized in the Theory of Communications, Electrical Engineering and Signal Processing disciplines for extracting nonlinear behavior in dynamical systems. This new approach was, in part, motivated by a diligent effort to create a man-made system that mimics the sound generation of a cicada. This insect has tremendous sound production capacity for its size. For the Okanagana and Magicicada species studied in this research, these cicadae ranged in size from five to six centimeters and produce sounds that are heard several hundred meters away. The evolution of this new signal processing algorithm from this bio-inspired research is explained in this article. This investigation initially examined the cicada hypothesized nonlinear system, by employing a number of numerical techniques in which to identify nonlinearity in a measurement times series. One such technique, the Nuttall modified-Volterra approach would serve as the validation and verification process for confirming that the inherent artificiality introduced by converting the sound production system of the biologic system to a man-made device did not corrupt the inherent dynamics of the cicada mating call. The technical advantage gained from quantification of the expansion kernels using the Nuttall approach, is the creation of more characterization clues by extending beyond the linear kernel response. This unique method is based on an extension of earlier developments of Vito Volterra and Norbert Wiener. The new Nuttall-Wiener-Volterra (NWV) method identifies the existence of nonlinearity in a measurement time series and determines the power distribution of individual nonlinear components. Moreover, the NWV method, unlike other methods that are likely less computationally efficient due to the Curse of Dimensionality (COD), significantly reduces the computational workload, thereby making characterizations of nonlinear systems with memory at higher orders possible. The nonlinear system kernel responses reveal identification and characterization of linear and nonlinear dynamics contained within the system under investigation. Thus, the nonlinear kernel responses computed for the cicada exposed a critical development for the NWV technique, namely that in order to obtain meaningful NWV kernel responses (i.e., to have physically and mathematically sound computational results), there are restrictive requirements for the system input excitation to be (a) band-limited, (b) white Gaussian and (c) zero mean. By studying the anatomical structures in the cicada sound production system and developing the wave propagation and finite element (FE) models this effort also then attempted an approach to confirm the accuracy of these models by employing the NWV nonlinear (and linear) analysis method.

This research has developed a mathematical model for characterizing nonlinear systems utilizing the earlier methods of Vito Volterra (1860-1940)1 and later by Norbert Wiener (1894-1964)2. More recently, Albert Nuttall, working with coauthors, has made significant improvements over the earlier methods. These enhancements include kernel estimation using a least-square approach for obtaining a best fit to the measured system response. These techniques, among others, include: maximally-sparse sampling of the kernels (for first-, second- and third-order), combatting ill-conditioning on its own terms, realizing an exponential decrease of the growth of the number of kernel coefficients, capturing frequency content beyond input excitation (transmit) band, and estimating the effects of noise on the kernel responses. Extensions of this theory are relevant to problems in acoustics.

The present article discusses novel improvements in nonlinear signal processing made by the prime algorithm developer, Dr. Albert H. Nuttall and co-authors, a consortium of research scientists from the Naval Undersea Warfare Center Division, Newport, RI. The algorithm, called the Nuttall-Wiener-Volterra or 'NWV' algorithm is named for its principal contributors [1], [2],[ 3] . The NWV algorithm significantly reduces the computational workload for characterizing nonlinear systems with memory. Following this formulation, two measurement waveforms are required in order to characterize a specified nonlinear system under consideration: (1) an excitation input waveform, x(t) (the transmitted signal); and, (2) a response output waveform, z(t) (the received signal). Given these two measurement waveforms for a given propagation channel, a 'kernel' or 'channel response', h= [h 0 ,h 1 ,h 2 ,h 3 ] between the two measurement points, is computed via a least squares approach that optimizes modeled kernel values by performing a best fit between measured response z(t) and a modeled response y(t). New techniques significantly diminish the exponential growth of the number of computed kernel coefficients at second and third order and alleviate the Curse of Dimensionality (COD) in order to realize practical nonlinear solutions of scientific and engineering interest.

The present article discusses some inroads in nonlinear signal processing made by the prime algorithm developer, Dr. Albert H. Nuttall and co-authors, a consortium of research scientists from the Naval Undersea Warfare Center Division, Newport, RI. The algorithm, called the Nuttall-Wiener-Volterra 'NWV' algorithm is named for its principal contributors [1], [2],[ 3] over many years of developmental research. The NWV algorithm significantly reduces the computational workload for characterizing nonlinear systems with memory. Following this formulation, two measurement waveforms on the system are required in order to characterize a specified nonlinear system under consideration: (1) an excitation input waveform, x(t) (the transmitted signal); and, (2) a response output waveform, z(t) (the received signal). Given these two measurement waveforms for a given propagation channel, a 'kernel' or 'channel response', h= [h 0 ,h 1 ,h 2 ,h 3 ] between the two measurement points, is computed via a least squares approach that optimizes modeled kernel values by performing a best fit between measured response z(t) and a modeled response y(t). New techniques significantly diminish the exponential growth of the number of computed kernel coefficients at second and third order in order to combat and reasonably alleviate the curse of dimensionality.

An analysis of cicada mating calls, measured in field experiments, indicates that the very high levels of acoustic energy radiated by this relatively small insect are mainly attributed to the nonlinear characteristics of the signal. The cicada emits one of the loudest sounds in all of the insect population with a sound production system occupying a physical space typically less than 3 cc. The sounds made by tymbals are amplified by the hollow abdomen, functioning as a tuned resonator, but models of the signal based solely on linear techniques do not fully account for a sound radiation capability that is so disproportionate to the insect's size. The nonlinear behavior of the cicada signal is demonstrated by combining the mutual information and surrogate data techniques; the results obtained indicate decorrelation when the phase-randomized and non-phase-randomized data separate. The Volterra expansion technique is used to fit the nonlinearity in the insect's call. The second-order Volterra estimate provides further evidence that the cicada mating calls are dominated by nonlinear characteristics and also suggests that the medium contributes to the cicada's efficient sound propagation. Application of the same principles has the potential to improve radiated sound levels for sonar applications.

This study investigates laser beam propagation through an atmospheric boundary layer near the ocean surface. Objectives of this research are to ascertain feasibility limits for achieving maximum energy efficiency at extended ranges in the face of atmospheric and other distortions as the laser beam penetrates through transitional (anisotropic) and turbulent (isotropic) boundary layer regimes. Various aspects of turbulence modeling of laser beam propagation near the ocean surface are discussed including: Kolmogorov’s model of atmospheric turbulence, parameterized structure functions (e.g., velocity and temperature gradients, gradients in refractive index) and other important factors affecting near surface propagation such as humidity, aerosols, and wave slap. Various preliminary modeled propagation results are shown, and a new methodology is proposed for improving existing model estimates with new time domain measurement procedures.

A well-known mathematical model for characterizing nonlinear systems was originally proposed by Vito Volterra (1860-1940) and later developed by Norbert Wiener (1894-1964). More recently, Albert Nuttall, working with coauthors, has made substantial improvements to Wiener’s approach, in which kernel estimates are modeled directly rather than computed from Wiener functionals. Other enhancements made include kernel estimation using a least-square approach for obtaining a best fit to the measured system response, and procedural methods to obtain uncorrelated column-wise basis vectors in the design matrix. These techniques, among others, include: maximally-sparse sampling of the kernels (for first-, second- and third-order), combatting ill-conditioning on its own terms, and realizing an exponential decrease of the growth of the number of kernel coefficients.

The theoretical directivity of a single combined acoustic receiver, a device that can measure many quantities of an acoustic field at a collocated point, is presented here. The formulation is developed using a Taylor series expansion of acoustic pressure about the origin of a Cartesian coordinate system. For example, the quantities measured by a second-order combined receiver, denoted a dyadic sensor, are acoustic pressure, the three orthogonal components of acoustic particle velocity, and the nine spatial gradients of the velocity vector. The power series expansion, which can be of any order, is cast into an expression that defines the directivity of a single receiving element. It is shown that a single highly directional dyadic sensor can have a directivity index of up to 9.5 dB. However, there is a price to pay with highly directive sensors; these sensors can be significantly more sensitive to nonacoustic noise sources.

Some of the features unique to beamforming a linear array of acoustic velocity sensors, which are not present with scalar-sensing elements (such as conventional pressure sensors), are described in this paper. Four types of sensors are considered here: a uniaxial motion sensor, which measures acoustic particle velocity along a single axis; a biaxial motion sensor measuring velocity in two orthogonal directions; a triaxial motion sensor that measures all three orthogonal components of the velocity vector; and a sensor, denoted as an acoustic vector sensor, that measures acoustic pressure as well as the complete velocity vector. Comparisons are made of the directivity index for each type of sensor and for linear arrays of sensors. It is shown that uniaxial velocity sensors can have a maximum directivity factor three times greater than an omnidirectional pressure sensor, a gain in directivity index of 4.8 dB. Not surprisingly, this directivity gain is highly dependent on signal arrival direction. Indeed, a un...

Realistic simulations with spatially coherent noise have been run in order to compare the performance of adaptive beamforming (ABF), inverse beamforming (IBF), and conventional beamforming (CBF) for the case of finite-averaging times, where the actual spatial coherence of the acoustic field, or covariance matrix, is not known a priori, but must be estimated. These estimation errors cause large errors in the ABF estimate of the directionality of the acoustic field, partly because ABF is a highly nonlinear algorithm. In addition, it is shown that ABF is fundamentally limited in its suppression capability at very low frequency (VLF), based on the sidelobe level of the conventional beampattern in the direction of the noise interferer [G. L. Mohnkern, "Effects of Errors and Limitations on Interference Suppression," NOSC Technical Document 1478, Naval Ocean Systems Center (1989)]. The simulations include a low-level plane wave signal of interest, a stronger noise plane wave interferer, and spatially random background noise. Both IBF and ABF performed significantly better than CBF, and IBF's performance was slightly better than ABF's performance. The performances of IBF and the ABF algorithm, the minimum variance distortionless response (MVDR) [A. H. Nuttall and D. W. Hyde, "Unified Approach to Optimum and Suboptimum Processing for Arrays," USL Report Number 992, Naval Underwater Systems Center, New London, CT (22 April 1969)] were recently compared independently [J. S. D. Solomon, A. J. Knight, and M. V. Greening, "Sonar Array Signal Processing for Sparse Linear Arrays," Defense Science and Technology Organization (DSTO) Technical Report (June 1999)] using measured data, with the result that IBF outperformed MVDR. This result is significant because MVDR requires orders of magnitude more processing power than IBF or CBF.

A generalized model for characterizing nonlinear systems was originally proposed by Italian mathematician and physicist Vito Volterra (1860-1940). A further development by American mathematician and MIT Professor Norbert Wiener (1894-1964) was published in 1958. After direct involvement with Norbert Wiener publication, Albert H. Nuttall has recently made new inroads along with his coauthors in applying the Wiener-Volterra model. A general description of a nonlinear system to the third order is termed the Nuttall-Wiener-Volterra model (NWV) after its co-founders. In this formulation, two measurement waveforms on the system are required in order to characterize a specified nonlinear system under consideration: an excitation input, x(t) (the transmitted signal) and a response output, z(t) (the received signal). Given these two measurement waveforms for a given system, a kernel response, h= [h0,h1,h2,h3] between the two measurement points, is computed via a least squares approach that optimizes modeled kernel...

Free-wave propagation of an infinite, tensioned string, supported along its length by repeating segments of multiple spring-mass connections, is examined. The segments can consist of an arbitrary number of different support sets and be of any overall length. Periodicity is intrinsic, since the segments repeat; the goal, though, is to examine what effect variations within the segments have on dispersion. The formulation reveals an unexpected amount of complexity for such a simply posed system. Each support set has independent mass, stiffness, and viscous damping, and the sets are allowed to be offset from one another. A free-wave dispersion formula is derived for two sets of supports (Q = 2) and compared to the well-known ideally periodic expression (Q = 1). A means to obtain general dispersion formulas, for any Q, is discussed. It is shown that the systems’ dispersion curves are primarily governed by the material properties of the string and by the location of the supports.

The possibility of estimating the directionality of a stationary homogeneous noise field, directly from the element outputs of a line array, is investigated and found to be feasible for large arrays without encountering ill conditioning. This technique, called inverse beamforming for historical reasons, is applicable to line and planar as well as volumetric arrays, and requires no more than two‐dimensional fast Fourier transforms (FFTs) for its realization. Derivations and results are presented both for a Fourier series method and a Fourier integral method.

This report analyzes the zero-pole locations of an infinite length of tensioned string that has attached periodic supports. The dynamic response of the system is derived for distributed wave number forcing and discrete point forcing acting on the string. These wave number-frequency transfer functions are then written in zero-pole format by a mathematical transformation of their infinite series. Once this is accomplished, the locations of the system's poles and zeros become apparent, and they can be plotted in the wave number frequency plane. It is shown that there are specific regions where an infinite number of poles can exist and specific regions where poles cannot exist. For the system with wave number forcing, the system zeros correspond very closely to the system poles except in the area of the fundamental unsupported string resonance. For the system with point forcing, the zeros can exist in the entire wave number frequency plane except at the fundamental resonance. A numerical example is included, and the different zones of the system are demonstrated.

An innovative noise suppression algorithm, called the coherent onion peeler (COP) (patent detained) is derived by minimizing the error in the coherent subtraction of the data from a single plane wave model. The COP solution "collapses" for all beamforming algorithms to the same solution at the complex hydrophone FFT level. Thus, any beamformer can be applied to the residual FFTs after the COP has been applied. The COP is applied to the strongest interferer first and the residual FFTs represent the acoustic field with this interferer coherently "peeled back" like the outer skin of an onion. This procedure can be repeated any number of times to coherently suppress the noise from all interferers present. A new nonadaptive beamformer is developed that "blends" together the best beam pattern properties of conventional beamforming (CBF) near array design frequency and the generalized fourier integral method (GFIM) at very low frequencies (VLF). In between these frequency extremes, the GFIM-CBF Blend (patent pending) (G-C Blend) is a frequency-dependent weighted average of both individual beamformers. COP and G-C Blend are applied to the broadband noise emitted by the tow ship, which severely degrades the performance of the towed array; COP reduced this broadband noise by over 19 dB in a single pass.

Closed-form relations have been obtained for the covariances of the spectral estimates, as well as the correlation estimates, that result from weighted overlapped FFT processing of a stationary process with Gaussian characteristics. In particular, estimates that result from time-displaced overlapped data segments and different frequency bins have been considered. The effects of wraparound in the time-delay domain of the correlation estimates have also been incorporated. The fundamental parameters of the study are the weighting segment length K, the amount of lag L, and the FFT size N, as well as the input data correlation function C(k) and the particular weighting sequence w(k).

Several possibilities for computing the generalized Q(sub M) (a, b) function for complex arguments a, b are presented and evaluated. Each approach has some limitations, such as loss of significance or complicated steepest paths of descent in an integral evaluation. However, a procedure is derived that will retain significance for all values of the parameters a, b, while avoiding underflow or overflow, and the procedures has been automated. The need for this function arises when the joint probability density function of the M- 1 largest squared envelopes of a matched filter output, plus the sum of the remainder, is considered. In particular, a Bromwich contour integral repeatedly utilizes this function in its evaluation.

A set of N independent random variables ?Xn! with arbitrary probability density functions (pn(X)! are ordered into a new set of dependent random variables, each with a different probability density function. From this new set, the M - I largest random variables are selected. Then, the sum of the remaining N + I - M random variables is computed, giving a total of M dependent random variables. Several statistics have been computed for these M random variables, including their joint probability density function and a quantity called the "combined probability and joint probability density function" of particular selections. Most of the results can be expressed in terms of a single Bromwich contour integral in the moment-generating domain. A saddlepoint approximation and first-order correction term are derived for this contour integral and then applied to several typical examples. Actual numerical calculation of these saddlepoint approximations requires evaluation of several functions that have high-order removable singularities, thereby requiring power series expansions near these removable singularities. Expansions with 13 to 14 decimal-digit accuracy have been derived and programmed for this purpose.

A set of N independent random variables (Xn! with arbitrary probability density functions ?pn(x)! are ordered into a new set of dependent random variables, each with a different probability density function. From this latter set, the M-1 largest random variables are selected. Then, the sum of the remaining N + 1 - M random variables is computed, giving a total of M dependent random variables. Several statistics are computed for these M random variables, including their joint probability density function, a quantity called the "combined probability and joint probability density function" of particular selections, and their distribution. Most of the results can be expressed as a single Bromwich contour integral in the moment-generating domain. This integral is most easily numerically evaluated by locating (approximately) the real saddlepoint of the integrand and passing the contour through that point. Very high accuracy in the joint probability density function evaluations is available by using numerical integration on this latter contour, instead of a saddlepoint approximation.

A method is presented for reducing the dimensionality of the search space when some of the unknown parameters appear linearly in the model fit. After elimination of the linear parameters, the gradient vector and the Hessian matrix of the resultant Hermitian form are derived so that an efficient minimization procedure can be developed in multiple dimensions. A 'destabilizing' term is identified in the Hessian matrix and can be dropped from the calculations if desired. This approach is expected to be more reliable; it also does not require any second-order partial derivatives, leading to fewer computations for finding the minimum in the multidimensional search space.

The evaluation of the joint probability density function from the joint moment generating function involves an M-dimensional inverse laplace transform. The analytic and numerical difficulty of performing this task for large values of M prompts consideration of an approximate technique such as the saddlepoint method. Advantage can be taken of the fact that the joint probability density function is real and positive, to show that the dominant saddlepoint in the original region of analyticity of the joint moment generating function is on the real axes. Furthermore, inside this region of analyticity, the integrand of the inverse Laplace transform has a positive-definite Hessian matrix on these real axes, indicating a single minimum for the saddlepoint location, when it exists. These properties serve to reduce the numerical effort required to locate the dominant M-dimensional saddlepoint. Examples of some statistical problems where this issue is of importance are included.

This report discusses an array processing technique that allows optimal source amplitudes and arrival angles to be determined from the data received on arbitrary arrays. Used with either a one- or two-planewave fit to arbitrarily spaced linear arrays, the method shows superior performance to conventional beamforming when more than one source is present within the angular resolution of the array. With such situations becoming more and more likely as operating frequencies are driven to lower values, in an effort to increase detection ranges, this new approach can be used to coherently subtract strong arrivals from received data, allowing the detection of additional weak arrivals in close angular proximity. In the case where a planewave arrives at an arbitrary two-dimensional array from a moving source, a similar technique yields optimal values of the source amplitudes of the planewave, as well as the starting and ending positions of the moving source. The new processing thus performs the functions of a combined beamformer and tracker in such a way that the total error over the observation interval is minimized.

A set of N independent, identically distributed random variables ?X(sub n)), with common probability density function p(x), are ordered into a new set of dependent random variables ?X'(sub n)), each with a different probability density function. From this latter set, the n1-th largest random variable through the n(sub M-1)-th largest random variable are selected. Then, the sum of the remaining N+1-M random variables is computed, giving a total of M dependent random variables. The joint probability density function of these M random variables is derived in a form involving a single Bromwich contour integral in the moment-generating function domain. The integral is most easily numerically evaluated by locating (approximately) the real saddlepoint of the integrand and passing the contour through this point. Very high accuracy in the probability density function evaluation is available by using numerical integration instead of a saddlepoint approximation.

In this paper, we analyze the statistics of two general classes of statistics. The first class is "M quadratic and linear forms of correlated Gaussian random variables". Examples include both cyclic and non-cyclic autocorrelation function (ACF) estimates of a correlated Gaussian process or the magnitude-squared of the output samples of a filtered Gaussian process. The second class consists of a subset of order statistics together with a remainder term. An example is the largest M - 1 bins of a discrete Fourier transform (DFT) or discrete wavelet transform (DWT), together with the sum of the remaining energies, forming an M-dimensional statistic. Both classes of statistics are useful in classification and detection of signals. In this paper, we solve for the joint probability density functions (PDFs) of both classes. Using the PDF projection method, these results can be used to transform the feature PDFs into the corresponding high-dimensional PDFs of the raw input data.

The determination of the statistics of a set of random variables (RVs) is frequently achieved by assuming the RVs to be joint Gaussian or to be statistically independent of each other. Although this assumption greatly simplifies the analysis, it can lead to very misleading probability measures, especially on the tails of the distributions, where the exact details of the particular RVs can be important This report presents a new method for deriving saddlepoint approximations (SPAS) for a number of very useful statistics of a general set of M dependent RVs, including the joint moments, the joint cumulative distribution function (CDF), the joint exceedance distribution function (EDF), and the joint probability density function (PDF). In particular, application to the maximum RV of a set of M dependent non-Gaussian RVs will be thoroughly investigated for two different examples. Also, the statistics of the range variate (maximum - minimum) are determined for the same two examples. The calculation of these M-dimensional statistics sometimes requires that multiple M-dimensional SPs be determined in order to evaluate an EDF or PDF at a single point in probability space. Also, the storage requirements and/or execution times can rapidly grow unmanageable as the number of dimensions, M, increases. Finally, care must be taken to ensure that the SP is located in the correct region of M-dimensional space; this allowed region of analyticity varies with the particular statistic under investigation.

The detection performance of an or-ing device with pre-averaging and post-averaging has been determined in the form of numerous receiver operating characteristics covering a wide range of input deflections (signal-to-noise ratios). Numerical evaluation of the false alarm probability P(sub f) and detection probability P(sub d) has been conducted for the case of a deterministic signal in the presence of additive Gaussian noise for a wide range of values of K, the amount of pre-averaging before or-ing; N, the number of channels or-ed; and M, the amount of post-averaging after or-ing. A MATLAB program that can be used to extend these results to parameter values outside the range studied here is also provided. The tradeoffs associated with switching from post-averaging to pre-averaging, or vice versa, have been thoroughly investigated and tabulated for a standard operating point (P(sub f) = 1 E-3, P(sub d) = 0.5) and for a high-quality operating point (P(sub f) = 1 E-6, P(sub d) = 0.9). The losses associated with performing too little pre-averaging can be severe, especially for large numbers, N, of or-ed channels.

The detection performance of an or-ing device with pre-averaging and post-averaging has been determined in the form of numerous receiver operating characteristics covering a wide range of input signal-to-noise ratios. Numerical evaluation of the false alarm probability P(sub f) and detection probability P(sub d) has been conducted for the case of a phase-incoherent signal in the presence of additive Gaussian noise for a wide range of values of K, the amount of pre-averaging before or-ing; N, the number of channels or-ed; and M, the amount of post-averaging after or-ing. A MATLAB program that can be used to extend these results to parameter values outside the range studied here is also listed. The tradeoffs associated with switching from post-averaging to pre-averaging, or vice-versa, have been thoroughly investigated and tabulated for a standard operating point (P(sub f) = 1E-3, P(sub d) = 0.5) and for a high-quality operating point (P(sub f) = 1E-6, P(sub d) = 0.9). The losses associated with doing too little pre-averaging can be severe, especially for large numbers, N, of or-ed channels.

The detection performance of an or-ing device with pre-averaging and post-averaging has been determined in the form of numerous receiver operating characteristics covering a wide range of input signal-to-noise ratios. Numerical evaluation of the false alarm probability P(sub f) and detection probability P(sub d) has been conducted for the case of a random Gaussian signal in the presence of additive Gaussian noise, for a wide range of values of K, the amount of pre-averaging before or-ing; N, the number of channels or-ed; and M, the amount of post-averaging after or-ing. Also, a MATLAB program is listed that can be used to extend these results to parameter values outside the range studied here. The tradeoffs associated with switching from post-averaging to pre-averaging, or vice-versa, have been thoroughly investigated and tabulated for a standard operating point P(sub f) = 1 E-3, P(sub d) = 0.5, and for a high-quality operating point P(sub f) = 1 E-6, P(sub d) = 0.9. The losses associated with doing too little pre-averaging can be severe, especially for large numbers, N, of or-ed channels.

The long-term goal of this project is to detect weak random signals that have little or no known structure. Furthermore, this is intended to be accomplished in near-optimum fashion with practical realistic processor forms. The major scientific objective of this investigation is to maximize the processor s signal detection probability while maintaining a specified (low) false alarm probability. Furthermore, the losses of practical processors are to be compared with the fundamental detection limits possible in the noisy environment of interest. This work is supported by ONR Biological Oceanography (Code 322BC).

An efficient, fast, and accurate Fourier transform technique for obtaining small tail probabilities for both the probability density function and the exceedance distribution function, directly from the characteristic function, is derived and demonstrated numerically for several examples. The method is especially useful when analytic or asymptotic expressions for the probabilities are unavailable or unknown. By choosing the shift parameter r close to the highest singularity of the characteristic function in the complex plane, very small values of the tail probabilities of the density function and exceedance distribution function can be realized. The cost in this approach is that the sampling increment must then be taken small, in order to avoid aliasing. Finer sampling necessitates more computer time and effort, but it does not require more storage; rather, prealiasing can be advantageously employed to keep the fast Fourier transform size at reasonable values. The fast Fourier transform size has no effect upon the errors caused by aliasing and truncation; rather, the size merely controls the spacing at which the probability density function and exceedance distribution function outputs are calculated. Tail probabilities in the E-50 range are readily available with a computer limited to 15 significant decimal digits.

A random signal (if present) is located somewhere in a time interval characterized by a total of N search bins, along with uniform noise. The signal is burst-like and occupies a contiguous set of M bins, but the location of the M bins occupied by the signal is unknown. Also, the average signal level S in an occupied bin is arbitrary and unknown. The optimum (likelihood ratio) processor for this scenario is derived and simulated to determine its receiver operating characteristics. Practical approximations to this likelihood ratio processor lead to a class of suboptimum processors, called the linear-and-power (LAP) processors, that have a control parameter mu that can be varied for best signal detection capability. Simulations of various LAP processors reveal that near-optimum performance can be achieved by letting the control parameter mu tend to infinity; the resultant processor, called the Maximum processor, compares the maximum of all possible partial contiguous linear sums of the observations with a fixed threshold. For search size N = 1024, the loss in detectability of the Maximum processor relative to the unrealizable likelihood ratio processor is less than 0.1 dB over the complete range of values of M, the signal burst size.

A signal (if present) is located somewhere in a band of frequencies characterized by a total of N search bins, along with uniform noise of unknown level per bin, N. The signal occupies an arbitrary set of N of these bins, where not only is the extent N unknown, but, in addition, the locations of the particular N bins occupied by the signal (if present) are unknown. Also, the average signal level in an occupied bin, S, is arbitrary and unknown. In order to realize a specified false alarm probability, the power-law processor has been normalized by division with an estimate of the noise level, either from a noise only reference or from the measured data itself. Various combinations of normalizer forms have been investigated quantitatively through their receiver operating characteristics. It has been found that if the number of bins, M, occupied by the signal is small relative to the search size N, the additional signal-to-noise ratio required by the normalizer, in order to maintain the standard operating point, is not significant. However, if M is of the order of N/4 or larger, the degradations begin to become substantial. A partial remedy for the inherent losses caused by an unknown noise level is the use of a noise-only data reference, if available. However, eventually, as M increases and tends to N, the detection situation becomes progressively more difficult, finally becoming impossible. This is not a limit of the normalized power-law processor, but, rather, of the fact that detection of a white signal in white noise of unknown level is a theoretical impossibility.

A signal (if present) is located somewhere in a band of frequencies characterized by a total of N search bins with uniform noise normalized to unit power. The signal occupies an arbitrary set of M of these bins, where not only is extent M unknown, but in addition, the locations of the particular M occupied bins are unknown. Also, the average signal strengths per bin, ?S sub m!for 1

A signal (if present) is located somewhere in a band of frequencies characterized by a total of N search bins. The signal occupies an arbitrary set of M_ of these bins, where not only is M_ unknown, but also, the locations of the particular M_ occupied bins are unknown. Also, the signal strength is unknown. A class of processors, called the power‐law processors, is investigated, in which the available data is raised to the ν‐th power prior to summation over all data values. The receiver operating characteristics have been determined for values of power ν=1, 2, 2.5, 3, ∞ for a wide range of values of M_. These results allow for accurate extraction of required signal‐to‐noise ratios to achieve a specified level of performance, as measured by the false alarm and detection probabilities, Pf and Pd. One of the most surprising and useful results of this study is the discovery that the power‐law processor with ν=2.4 performs near the absolute optimum, even without any knowledge of the number of occupied bins M_ or the signal‐to‐noise ratio.

The detection and false alarm probabilities, for a normalizer operating in a partially-correlated fading medium with an additive unknown noise level, are derived exactly in the form of series expansions which can be quickly and accurately computed. This new work is an extension of (1), where the characteristic function of the decision variable, for a system operating in a known background noise level, was derived exactly and evaluated for a variety of cases. These new results also replace the approximate analysis of the same normalizer system, that was conducted in (2), with an exact analysis here.

The radiated signals from sources are of great importance for passive sonar, which is designed to exploit peculiarities of this form of signal and distinguish it from ambient noise, in which it is normally observed. The radiated signals usually consist of both narrowband and broadband components. In this paper, we describe techniques that exploit the broadband signals for detection and possible classification of the underwater sources.

We compare the modified Wigner distribution functions obtained via the Choi-Williams kernel and its rotation, as well as by the tilted Gaussian kernel. Based on several commonly used examples, we demonstrate that the modified Wigner distribution obtained via the Gaussian kernel can minimize the artifacts more effectively and has the capability of selectively filtering out undesired components.© (1991) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Because of the critical role of non-Gaussian noise processes in modern signal processing, which usually involves nonlinear operations, it is important to examine the effects of the latter on such noise and the extension to added signal inputs. Here, only non-Gaussian (specifically Class A) noise inputs, with an additive Gaussian component, are considered. The 'classical' problems of zero-memory nonlinear (ZMNL) devices serve to illustrate the approach and to provide a variety of useful output statistical quantities, e.g., mean or dc values, mean intensities, covariances, and their associated spectra. Here, Gaussian and non-Gaussian noise fields are introduced, and their respective temporal and spatial outputs are described and numerically evaluated for representative parameters of the noise and the ZMNL devices.

The system of interest here includes the combined nonlinear transformations of normalization and or-ing in the data processing chain. The required threshold settings for specified false alarm probabilities are determined, as well as the required input signal-to-noise ratios for specified detection probabilities. Specializations to normalization or or-ing alone are available by particular choices of parameter values built into the models. When the sizes of the transformations are changed, that is, the amount of normalization and/or or-ing, the thresholds will also have to be changed if the earlier probabilities are to be maintained; these options are included in the first program listed.

A number of new two-dimensional Fourier transforms of combinations of cross Wigner distribution functions, W, of convolution form or correlation form are derived. In addition, similar relations are obtained for combinations of cross complex ambiguity functions, X. Their great generality subsumes most of the already known available properties, such as: the volume constraint of magnitude-squared ambiguity functions; the positivity of the convolution of two Wigner distribution functions; and Moyal's theorem. Keywords: Two-dimensional, Convolution, Correlation, Temporal correlation, Wigner distribution, Ambiguity function, Fourier transform, Spectral correlation.

The performance of several weighted energy detectors of gaussian signals in noise are investigated, both by exact procedures and by five different approximation procedures. In particular, receiver operating characteristics, for false alarm probabilities ranging from 1E-10 to .1 and detection probabilities ranging from .01 to .999, are quantitatively compared. The standard Gaussian approximation is found to be severely deficient and generally optimistic for small false alarm probabilities, while two different fourth-order approximations have excellent capability over the entire range of probabilities considered. A method of avoiding the calculation of the eigenvalues of a covariance matrix, and yet accurately predicting performance of a fading medium, is presented. It requires only sums of products of the covariance elements directly, the precise number depending on the order of the approximation.

For a line array immersed in a stationary homogeneous noise field composed of plane waves of differing speeds and angles of arrival, standard beamforming is not useful for estimating the noise spectrum directionality. However, the two‐dimensional spectrum of the received noise field enables resolution in temporal‐frequency and in spatial‐frequency and can thereby indicate the temporal‐frequency content and the apparent speed along the line array of plane‐wave arrivals. The interrelationships and attributes of the space‐time correlation, the cross‐spectrum, and the two‐dimensional spectrum are presented and interpreted via simple examples. A technique for estimation of the two‐dimensional spectrum for a sparse line array and examples of its use and interpretation are given.

This correspondence presents a brief summary of a generalized framework for power spectral estimation and shows how three previous estimation methods fit into this framework as special cases. Further, this correspondence clarifies some recent comments made on spectral estimation; in particular specific references are given that strongly support the use of overlapped weighted segment averaging for spectral estimation.

The false alarm and detection probabilities of an N‐channel system subject to L+1 level quantization, OR‐ing, and accumulation of M time samples, are evaluated exactly, with no Gaussian assumptions, for arbitrary values of N, L, M, and for a quantizer with arbitrary breakpoint locations. The channel noises are independent but can have arbitrary statistics. The signal occupies one unknown (possibly changing) channel, if present. Two FFTs suffice to sweep out a complete detection probability versus threshold curve. The optimum placement of quantizer breakpoints, for a fixed total number of levels, L+1, is a subtle one and shown to depend on N, M, and the desired level of performance; a simple rule‐of‐thumb is presented which yields near‐optimum capability over the useful range of the operating characteristics.

Several observations are made about a recently proposed semi-infinite programming (SIP) method for computation of linear Chebyshev approximations to complex-valued functions. A particular discretization of the SIP problem is shown to be equivalent to replacing the usual absolute value of a complex number with related estimates, resulting in a class of quasi-norms on the complex number field C \mathbf {C} , and consequently a class of quasi-norms on the space C ( Q ) C(Q) consisting of all continuous functions defined on Q ⊂ C Q \subset {\mathbf {C}} , Q compact. These quasi-norms on C ( Q ) C(Q) are estimates of the L ∞ {L_\infty } norm on C ( Q ) C(Q) and are useful because the best approximation problem in each quasi-norm can be solved by solving (i) an ordinary linear program if Q is finite or (ii) a simplified SIP if Q is not finite.

Recent investigations of fluid‐loaded plates have suggested the possibility of three real roots for the equation governing the free wave number of fluid‐loaded plates. These three real roots are purported to correspond to three separate and distinct free wave numbers for the fluid‐loaded plates, a result at variance with the classical single free wave‐number solution. This paper reexamines the free vibration of the fluid‐loaded, thin plate to clarify this question of multiple roots. The results of this study confirm the classical single free wave‐number result, and show that multiple real roots can only be obtained by violating the radiation condition. [Work supported by CNM.]

The phase distortion of a plane‐wave pulse owing to bottom reflection at incident angles more grazing than the critical angle is considered. Two general methods of investigation, using the time and frequency domains, are given; they employ, respectively, the Hilbert transform of the incident pulse and the complex envelope signal. Applications to two typical SONAR pulses—a carrier frequency amplitude‐modulated by a rectangular pulse and a Gaussian pulse—are given, along with graphs of the results. It is concluded that negligible distortion occurs, there being only a phase shift of the carrier, when the Q of the waveform is moderately larger than unity, the exact value depending on the smoothness of the transmitted pulse.

The acoustic theory for the reflection of a single‐frequency plane wave from a fluid layer lying between two semi‐infinite fluid media, where the layer and the bottom medium are subject to attenuation, is reviewed. Applying the experimental fact that the attenuation is proportional to the first power of frequency to this theoretical model of the ocean bottom, we show that the effect on impingent plane waves may be treated as a parallel linear‐filtering operation with a component transfer function of the basic form H (f) = exp[−2πb|f|+ie sgn (f)]. The parameters b and e depend on the media physical constants, the angle of incidence, and the particular reflected component under consideration. Here, f is frequency in cycles per second, 2πb is attenuation in nepers per cycles per second, e is phase shift in radians, and sgn (f) = 1 for f>0, −1 for f

The characteristic function of spectral estimates obtained via overlapped FFT processing of windowed data is presented for a random process containing a signal tone and Gaussian noise. For the special case of noise- alone, the probability distribution of the estimate is plotted and compared with an approximation utilizing only the first two moments and found to be in excellent agreement in probability over the range (.0001, .9999) for several data windows, overlaps, and time-bandwidth products. This result means that knowledge of the equivalent degrees of freedom of the spectral estimate is adequate for a complete probabilistic description, even when the overlap results in significant statistical dependence of the component FFT outputs.