A statistical theory of reverberation and similar first-order scattered fields--II: Moments, spectra and special distributions

Abstract

In Part I a general, first-order scattering model for reverberation has been constructed, from which the characteristic aperture and medium operators have been obtained for general geometries, signals, and arrays. Here, in Part II, the second-order statistics (intensities, covariances, and spectra) of such reverberation processes are determined in detail. In addition to the fluctuation, or purely random component of the received scatter process, an average or purely deterministic component often exists and may be significant, particularly for scattering from surfaces. Since the medium is dispersive (and time varying), reverberation is a non-stationary process (as is well known). Except for purely sinusoidal signals, these processes are (approximately) stationary only for signals of short duration (i.e., at fixed ranges), and only then can be said to possess an intensity spectrum in the usual sense. In this latter instance the familiar reverberation laws (intensity <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sim</tex> input signal, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sim</tex> range <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^{-3}</tex> or range <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^{-2}</tex> , respectively, for surface or volume scatter) hold for the fluctuation component, while different laws (intensity <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sim</tex> range <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^{-2}</tex> , or range <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^{0}</tex> ) govern the deterministic components. With the composite process no such simple behavior usually appears, unless the geometry insures the dominance of a surface or volume effect. Moreover, in the general case these laws are only approximations, which may often be poor ones. Particular attention is also given to narrowband signals. A variety of expressions for the covariances of the slowly-varying components of the received reverberation is developed in detail, for both surface and volume scatter. These results are very simply combined (additively in the means, variances, etc.) to yield analogous results for the complete, or composite reverberation process. Conditions for normality are given and a number of new results for the probability densities of such nonstationary, narrowband Gaussian processes are obtained. The first-order statistics of the envelope and phase of the received scattered return are also derived, for general narrowband signals, extending the earlier work of Rayleigh, Rice, Hoyt, Nakagami, Middleton, and Beckmann. Many special cases for arbitrary illuminating signals are considered, including an approach to complex (distributed) targets, uniform beams, multiple receivers, transmitter and receiver at the same location, large and small doppler, etc., where the general formulation (as in Part I) is broadband, given in terms of frequency-selective apertures, time-varying dynamic cross sections, and arbitrary input signal waveforms. A number of second-order statistical properties of the medium response is defined and evaluated, which are needed for the combined theoretical and experimental exploration of the medium itself. It is found that the present model of the dispersive reverberation channel cannot be fully described by the statistical structures postulated earlier (and successfully) in radar astronomy: a considerably more involved second-order statistic is required (and obtained) here. To what extent it may provide an adequate description of reality remains to be studied. Part II concludes with a short summary of the principal results and assumptions. Many special cases for arbitrary illuminating signals are considered, including an approach to complex (distributed) targets, uniform beams, multiple receivers, transmitter and receiver at the same location, large and small doppler, etc., where the general formulation (as in Part I) is broadband, given in terms of frequency-selective apertures, time-varying dynamic cross sections, and arbitrary input signal waveforms. A number of second-order statistical properties of the medium response is defined and evaluated,which are needed for the combined theoretical and experimental exploration of the medium itself. It is found that the present model of the dispersive reverberation channel cannot be fully described by the statistical structures postulated earlier (and successfully) in radar astronomy: a considerably more involved second-order statistic is required (and obtained) here. To what extent it may provide an adequate description of reality remains to be studied. Part II concludes with a short summary of the principal results and assumptions.