Envelope Probability Density Function Research Articles

On the PDF of the sum of random vectors

There are various cases in physics and engineering sciences (especially communications) where one requires the envelope probability density function (PDF) of the sum of several random sinusoidal signals. According to the correspondence between a random sinusoidal signal and a random vector, the sum of random vectors can be considered as an abstract mathematical model for the above sum. Now it is desired to obtain the PDF of the length of the resulting vector. Considering the common and reasonable assumption of uniform distributions for the angles of the vectors, many researchers have obtained the PDF of the length of the resulting vector only for special cases. However in this paper, the PDF is obtained for the most general case in which the lengths of vectors are arbitrary dependent random variables. This PDF is in the form of a definite integral, which may be inappropriate for analytic manipulations and numerical computations. So an appropriate infinite Laguerre expansion is also derived. Finally, the results are applied to solve a typical example in computing the scattering cross section of random scatterers.

Implications of nonfractal seafloor stochasticity on acoustical scattering from the Mid-Atlantic Ridge

Acoustical and bathymetric data were collected near the Mid-Atlantic Ridge as part of the Acoustical Reverberation Special Research Program (ARSRP) in 1993. The wideband time-domain envelope statistics of backscatter from prominent bathymetric features exhibit a non-Rayleigh character which previous researchers have described as event-like. The envelope probability density functions (pdf’s) show enhanced tails at high levels which can be a source of active sonar clutter. It is proposed that the origin of such clutter is single-scale, nonfractal roughness. High-resolution bathymetry reveals a power spectral density (PSD) with power-law form. Wavelet analyses reveal a single-scale character to the roughness. In order to test the hypothesis, multiscale and single-scale realizations of the measured PSDs are used in numerical simulations of time-domain backscatter. At the range and azimuthal resolutions in the experiment, the envelope pdfs for multiscale surfaces are Rayleigh, while the single-scale surfaces lead to enhanced tails as observed in the data. The conclusions drawn are that along with an interface’s rms height, correlation length, and power spectral shape, one should also be concerned with its scale structure because it plays an important role in the physics of wave interaction at random interfaces.

Non-Gaussian random vector identification using spherically invariant random processes

With the modeling of non-Gaussian radar clutter in mind, elegant and tractable techniques are presented for characterizing the probability density function (PDF) of a correlated non-Gaussian radar vector. The need for a library of multivariable correlated non-Gaussian PDFs in order to characterize various clutter scenarios is discussed. Specifically,. the theory of spherically invariant random processes (SIRPs) is examined in detail. Approaches based on the marginal envelope PDF and the marginal characteristic function have been used to obtain several multivariate non-Gaussian PDFs. An important result providing the PDF of the quadratic form of a spherically invariant random vector (SIRV) is presented. This result enables the problem of distributed identification of a SIRV to be addressed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The response envelope probability density function of a Duffing oscillator with random narrow-band excitation

A Duffing oscillator excited by narrow-band random noise can exhibit multi-valued behaviour, with random jumps between two pseudo-stable states. In this paper an approximate form of the probability density function or the envelope of the response is obtained. State equations for the response envelope with the random excitation envelope as input are obtained by stochastic averaging. The joint probability density function of response envelope and input envelope can be found approximately for certain ranges of the system parameters. The response envelope probability density function shows two maxima in the cases where the response is multi-valued. Numerical simulation confirms the results.

Evaluation and verification of bottom acoustic reverberation statistics predicted by the point scattering model

The point scattering model offers a parametrization of the reverberation envelope probability density function (pdf) in terms of the average number of scatterers contributing to the return and the presence of a coherent component in the received process. Computer simulations were used to verify model predictions and to evaluate their usefulness in the context of sea floor classification. Central to the verification strategy was the successful solution of the inverse problem based on synthetic reverberation data. The average number of scatterers was determined from estimates of the kurtosis of the instantaneous reverberation pdf. The magnitude of a coherent component embedded in the scattered return was recovered from envelope histograms with the assistance of the Kolmogorov goodness‐of‐fit test. Following the verification study, the model was perturbed by introducing clustered and ordered distributions of scatterers in addition to the standard Poisson. The initial results indicate that the reverberation envelope pdf differs significantly for the three scatterer distributions. The clustered distribution led to a rapid increase in kurtosis, while the ordered distribution displayed evidence of intermittent coherent scattering. The usefulness of this parametrization was further tested with real reverberation data representing several distinct seafloor regimes. [Research funded by ONR.]

Methods for Estimation of Statistical Properties of Envelopes of Ultrasonic Echoes from Myocardium

Several investigators have characterized various forms of heart disease from the statistical properties of envelopes of ultrasonic echos from myocardium. In particular, the mean-to-standard deviation ratio (MSR), skewness, and kurtosis of the envelope probability density function have been used for the detection of myocardial ischemia, infarction, reperfusion, and hypertrophy. In this paper, the effects of phenomena other than tissue acoustic properties upon estimates of statistical parameters are investigated. These include system characteristics (center frequency, bandwidth, beam width, etc.), sample volume dimensions, and tissue velocity. In myocardium, relatively small amounts of tissue are available for interrogation. It is shown that, under these limited data acquisition conditions, substantial systematic biases in the estimates of statistical parameters may occur. Analytic forms for errors in the envelope variance estimate are derived. Estimation of the envelope mean, variance, MSR, skewness, and kurtosis is investigated experimentally, using a commercial medical ultrasound scanner and a tissue-mimicking phantom.

Sonar estimates of seafloor microroughness

This paper presents an analysis of the effects of microroughness of the ocean bottom on a sonar signal. The results best apply to features where the roughness amplitude is less than one-quarter of an acoustic wavelength such as with ripples, beds of rocks, and nodules. The shape of the probability density function (PDF) of the echo envelope is examined in terms of the rms roughness and a new parameter, the correlation area of the bottom. The area is equal to the product of the x and y correlation distances along the floor. The PDF is shown to be extremely sensitive to small changes in the roughness. Furthermore by determining the rms roughness from standard coherent reflection measurements, the correlation area may be extracted directly from the PDF. Thus both vertical as well as lateral information is obtainable from sonar data. The technique can be used to discriminate between different types of bottoms that may have the same roughness but different correlation areas, for example a floor with ripples versus one with rocks or nodules. The analysis combines (1) a general statistical model employing the Rice PDF [S. O. Rice, in Selected Papers on Noise and Stochastic Processes, edited by N. Wax (Dover, New York, 1954), pp. 133–294], and (2) a theory originated by Eckart [C. Eckart, J. Acoust. Soc. Am. 25, 566–570 (1953)]. The analysis is applied to sonar data collected from the continental shelf near Cape Hatteras, North Carolina. The results are consistent with the known characteristics of the area.

Sonar estimates of sea floor microroughness

This paper presents an analysis of the effects of microroughness of the ocean bottom on a sonar signal. The results best apply to features where the roughness amplitude is less than one‐quarter of an acoustic wavelength such as with ripples, beds of rocks, and nodules. The shape of the probability density function (PDF) of the echo envelope is examined in terms of the rms roughness and a new parameter, the correlation area of the bottom. The area is equal to the product of the x and y correlation distances along the floor. The PDF is shown to be extremely sensitive to small changes in the roughness. Furthermore by determining the rms roughness from standard coherent reflection measurements, the correlation area may be extracted directly from the PDF. Thus both vertical as well as lateral information is obtainable from sonar data. The technique can be used to discriminate between different types of bottoms that may have the same roughness but different correlation areas, for example a floor with ripples ve...

Envelope probability density function of the sum of signal, noise and interference

A new closed-form expression of the envelope probability density function for the sum of signal, Gaussian noise and c.w. interference is found.