This appendix includes the calculations of the probability density functions and the moments of the total field and power variables. By total variables, we mean the fields or the powers measured by probes or antennas, which simultaneously collects the three Ex, Ey, Ez complex field projections.
The total field variable taking Et as a notation is given by the square root of the sum of six terms made up of all the square complex components of each Ex y zprojection i.e.:
[A3.1]
The normalized variable of the total field is designated for the occasion by the lower case et notation. This variable comes from the extension of the definition introduced in section 3.2.3:
We find in this formula the standard deviation σν, which is attached to the complex components of Ex y z.
Knowing that the Ex y z variables all follow the normal probability distribution,the square amplitude of the normalized total et field will be the χ2 variable with six degrees of freedom:
[A3.3]
We can thus add to this variable, the pdf of χ2, which is expressed by the notation conventions of formulas[3.23] and [3.24].
Hence:
After use of simplified notations and insertion of the numerical value of Γ(3),this equation takes the final form:
The pdf brought back to the normalized variable of the total field et relates to the pe(et) notation.
Under the previous notation, taking the derivate of p6(α) and convert the resultdp in respect of the normalized field et variables, we obtain a new equation in order to determine pe(et).
After insertion of function [A3.5] then immediately gives access to pe(et), i.e.:
Using the change of variable suggested in [A3.2] and after insertion into [A3.6],we reach the pE(Et) function giving the pdf of the total field where the absolute amplitude Et of the total field projections appears.
[A3.8]
We will notice the use of simplified exponents notations in place of (Et)5 and(Et)2.
By using the suitable notation, the mean amplitude of the normalized total field variable will be determined by the calculation of the moment of et, i.e.:
[A3.9]
After insertion of function [A3.7] and solving the integral analytically, we reach the numerical value of . i.e.:
[A3.10]
From the previous equation and from the change of variable [A3.2], we take :
The total power variable taking the symbol pt is the extension of the definition of the power found in equation [3.33] where the term |Ex,y,z|2 is replaced by (Et)2 with the Et variable previously introduced, hence:
[A3.12]
We find in equation [A3.12] the coefficient of physical scale A0.
The pdf of the total power taking the notation p6(pt) comes from algebraic computations linking the variable α, which is defined in [A2.4], to the variable pt:
[A3.13]
Using the derivate form of [A3.5], we reach the sought after relationships:
[A3.14]
After insertion of function [A3.5], we take the pdf of pt:
The establishment of the previous equation suggests adopting as the definition of the total reduced power, the ratio of pt with the p0 parameter defined below:
The probability density function of the total normalized power using the notation p6(ptr), is easily deduced from equation [A3.15] after a few simple transformations:
[A3.17]
We establish the pdf of the total normalized power, i.e.:
[A3.18]
The mean of the total normalized power, taking the symbol , comes from the calculation of the moment of ptr as found below:
[A3.19]
After solving the integral, we obtain:
[A3.20]
By using equation [A3.16], we easily find the mean of the total power:
[A3.21]
We will notice that the obtained numerical value is exactly similar to three times the pmv mean power, which is determined on an electric field component. A comparison of this result to formulas [3.32] and [A3.11] shows that the ratio of the mean powers is indeed different from the ratio of the square roots of the mean fields:
The result shows that the ratio of the moment of the total power pt over the mean power of one field projection pmv is not similar to the square of the ratio of the mean value of the absolute total field Et over the moment of one Ex,y,z projection. Taking the square root of the previous [A3.22] equation then leads to:
[A3.23]