J.2 Combining the sub-models
Figure J.2.1 shows the structure of the overall model. The double outlining of the boxes and the double-width lines in the diagram represent the fact that what flows through the diagram is not a signal or power, but a whole statistical distribution of power/loss. Specifically it is the inverse cumulative distribution function (ICDF) of the models. This specifies the distribution of basic transmission loss values L as a function of time percentage p. The model combiners are represented by the circles. The letter in a combiner specifies the correlation property of the combiner: “C” means fully correlated, “E” means mutually exclusive and “U” means uncorrelated. The “S” combiners are scalar combiners where one of the quantities being combined is a simple number (typically a median value) rather than a full distribution.
The details of how the sub-model predictions are combined are given explicitly in §§ 4 and 5. But for information, the basic formulae used for combining two distributions, assuming the four types of correlation properties used, are given here.
In the following equations L represents a basic transmission loss of one of the four parallel end‑to‑end propagation models. A represents the attenuation relative to free space produced by one of the additive models. We write L(p) or A(p) for the value of an ICDF at a time percentage p. Suffixes in1, in2 and out are used in the obvious way for ICDFs and scal for a scalar input.
Combining two ICDFs that are fully correlated (a “C” combiner) is simply a matter of adding powers or losses at time percentage p. The way this is done depends on whether the models to be combined are expressed in terms of two basic transmission losses, or a basic transmission loss and an attenuation relative to free space:
dB (J.2.1a)
dB (J.2.1b)
Combining an ICDF and a constant value (an “S” combiner), for example a single median value of attenuation, is also simple. The output ICDF is just the input ICDF “shifted” along the power/loss axis by the value of the input scalar quantity:
dB (J.2.2)
Note that “C” and “S” combinations can be done “point by point”, that is the output value at p% of the time only depends on the p% values of the input models, and does not require the full distributions.
Combining mutually exclusive mechanisms (an “E” combiner) is more difficult to implement computationally, but is conceptually simple. The time percentages of the two input ICDFs are added at each value of loss:
dB (J.2.3)
This requires an iterative procedure that uses the full distributions of the input quantities. This method is used for combining the clear-air and precipitation mechanisms.
Surprisingly perhaps, combining two ICDFs that are uncorrelated (a “U” combiner) is the most difficult. Indeed numerical techniques, such as Monte-Carlo, are required to do this properly. When WRPM is used for Monte-Carlo simulations, the structure of the WRPM model allows the statistics to be correctly modelled in a fairly straightforward way. This is described in § 5.3.
However, it is recognized that WRPM will often be used in circumstances that do not justify the computational complexity of a Monte-Carlo simulation. In that case a simple ansatz is applied to allow the full-model basic transmission loss to be calculated at a single value of time percentage. The principle is to select the strongest signal, or equivalently the lowest value of basic transmission loss, from the two (or more) signal paths at each time percentage p. A “blend” function can be used to eliminate the slope discontinuities that a simple picking of the minimum value would entail. The following method is used in § 5.2:
dB (J.2.4)
Although this looks very similar to the method of equation (J.2.1a) and has the advantage that the combination can be done on a “point by point” basis, “U” and “C” combiners are statistically very different. Maintaining the logical separation here makes it easier for an implementer of the model to apply numerical methods to give a more statistically correct result than the simple analytical approach of equation (J.2.4).
Note that equations (J.2.1a) and (J.2.4) could encounter a numerical problem if the sub-model basic transmission losses are very large. Numerical limitations could cause the argument of the log function to be zero. This is avoided by using the mathematically equivalent formulation of these equations given in § 5. This factors out the basic transmission loss of the dominant sub-model and adds to it a correction that takes account of the other sub-models.
FIGURE J.2.1
Sub-model combination diagram
