Statistical distribution of the amplitude and phase of a multiply scattered field

Abstract

Th e probability dist ribu Lion of t he am pli tude and phase o f the s um o f a large num ber of random two-di m ensional vecto rs is derived under t he fo llowing general condit io ns : Both t he ampli tudes a nd th e phases of the component vectors a re random, t he di sLribu tions b eing a rbitrary within the vali dit.\7 of t he Central Lim it Th eorem ; in part icular, the d ist ributions of t he individual veetors need not be identical, the amplitude and phase of each co mponent v ec tor n eed not be indep end e nt a nd the di tribu tion s Ileed not b e symmel ri cal. T he di sLributions form erly d erived by R ayleigh , nice, H oy t, a nd Heckmanll are shown lo be special cases of this di sLribu t ion .

Generalized MGF of Beckmann Fading With Applications to Wireless Communications Performance Analysis

“…In the same way, the parameter q measures the power imbalance between the in-phase (I) and quadrature (Q) NLOS components as in the Hoyt (Nakagamiq) fading model. The parameter r also indicates a power imbalance between the I and Q components, but now for the LOS component 2 . Finally the parameter Ω can be regarded as the average received power Ω = E |v| 2 = E R 2 .…”

“…This topic was originally addressed by Beckmann [2,3] in its more general form by assuming arbitrary mean and variance for the real and imaginary parts of v, i.e., X ∼ N (µ x , σ 2 x ) and Y ∼ N (µ y , σ 2 y ), being X and Y independent 1 . This corresponds to the most accurate way to characterize the scattering of electromagnetic waves from rough surfaces [5], on which the distribution of the received signal envelope R = |v| is that of the modulus of a complex Gaussian RV.…”

Making Aircraft Vortices Visible to Radar by Spraying Water into the Wake

“…Figure 8 shows that the probability density p(|V|) of the modulus |V| of complex voltage is unchanged by the resolution refinement. The probability densities are very well fit by the Rayleigh distribution (Beckmann 1962) (40) having the same mean as the data. The Rayleigh distribution results when the scattering amplitude is the same for all droplets and the phases uniformly distributed.…”

A single-cell spiking model for the origin of grid-cell patterns

“…In this case, the traveled distance α p follows a Rayleigh distribution with density [149] where for A pm uniformly distributed in interval (0, 1). Therefore, the first two moments of α p read …”