Standard wireless models are great at modelling unbostructed line-of-sight connections. However, they fail to accurately model the network states at peak noise, in complicated urban environments both indoors and outdoors. In addition, these models live entirely in the proprietary world of the IEEE website. Implementing these models in Python would go a long way to modelling the bigger problem. This is the first step. Since some quantities required for these models are unknown, a gridsearch algorithm will be used to optimize these algorithms.
Since these models require different sets of parameters, what features our dataset needs is an open question. The first method will be using a k-random-forest algorithm to see what features are relevant to the signal strength between two nodes.
The next method will take the most useful features and use them to build a linear (or maybe quadratic) regressor that predicts signal to noise ratio between two nodes.
Then, each of these models (the set of 802.11 standard models and my from-scratch regressor) will be compared using a Xi-Square test. Differences between my 'simplified' model and the more complex 802.11 models will be compared and analyzed with respect to the cost of finding the additional features.
Machine Learning Tools can signinficantly reduce the cost of measurement.
In the simplest scenario, we can model the free space path loss. This function of received power in terms of distance is known as Friis' Law:
$P_{Rx}(d)=P_{Tx}G_{Tx}G_{Rx}(\frac{\lambda}{4*\pi*d})^2A_{Rx}$
Where $P$ is the power, $G$ is the gain, and $A$ is the area of the receiving antenna, $d$ is distance between the transmitters and $lambda$ is the frequency. This equation only applies for systems that are separated by at least one Rayleigh distance defined by
$d_{R}=\frac{2{L_a}^2}{\lambda}$
This is known as the far field. When dealing with link budgets, it is best to use a logarithmic scale because signal levels will vary across many orders of magnitude.
$P_{Rx}(d)=P_{Tx}G_{Tx}G_{Rx}20\log(\frac{\lambda}{4*\pi*d})^2A_{Rx}$
if and only if the powers and gains are consistently in dB or dBm.
### Kirhhoff Theory
In the same way that Rayleigh distance defines the breaking point of Friis system, Rayleigh roughness can be thought of as the
#### Perturbation Theory
The next model, log-normal shadowing can be thought of as an extension of the Friis model with the added inclusion of a random variable. If the receiver is in the far field of the receiver (where $d>d_R$), $PL(d_0)$ is the path loss measured at a distance $d_0$ from teh transmitter, then the path loss when moving from distance $d0$ to $d$ is given by the equation
$PL{d_0\rightarrow d}(dB) PL(d_0)+10n\log_{10}\frac{d}{d_0}$
$n$ is the path loss exponent, given by the table below.
| Environment | Path Loss Exponent min | Path Loss Exponent Max |
|---|---|---|
| Free Space | 2 | 2 |
| Urban area cellular radio | 2.7 | 3.5 |
| Shadowed urban cellular radio | 3 | 5 |
| Inside-LoS | 1.6 | 1.8 |
| Obstructed in building | 4 | 6 |
| Obstructed in Factory | 2 | 3 |
The next model builds on the Log-Normal shadowing model by including a random variable, $chi$. This law can be expressed as:
$PL_{d_0\rightarrow d}(dB)=PL(d_0)+10n\log_{10}\frac{d}{d_0}+\chi$
Where $\chi$ is a zero-mean Gaussian distributed random variable. This variable is only used when there is a shadowing effect. Equivalently, $X=0$ when no shadowing effect is present.
| Model | Required Features | ||||||
|---|---|---|---|---|---|---|---|
| Friis | Receive Power | Transmit Power | Transmitter Gain | Receiver Gain | distance | frequency |