Signals, sactterers, and statistics

Abstract

In the first part of this paper we consider an ensemble of complex numbers (a set of signals in time, the field scattered by different configurations of scatterers, etc.) and obtain relations among various average functions that may be defined for such an ensemble. In particular, we consider the average total intensity, the coherent intensity, and the incoherent intensity (absolute squared functions of the ensemble); the coherent phase, average phase and average-square phase; the variances of the real and imaginary components of the ensemble and their covariance (or equivalently, the second moments of any phase-quadrature components of the field), as well as the higher moments. The second moments are represented in terms of the incoherent intensity and the real and imaginary parts of an "asymmetry function" <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</tex> ; if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P=0</tex> , then the variances are equal and the covariance is zero. In the second part of the paper, we briefly sketch the formalism that leads to scattering function representations for the intensities and coherent phase, and then develop the corresponding scattering representation of the new function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</tex> . We illustrate the development by explicit results for "gas-like" random distributions of large tenuous scatteres. Thus we obtain direct relations between the various statistical functions mentioned above and the fundamental parameters of the scattering problem. Since the parameters enter differently in the various averages, and since these functions can be measured simultaneously and independently, the results facilitate inverting measured data.