18.3.1 Backgroud

The free space model and the two-ray model predict the received power as a deterministic function of distance. They both represent the communication range as an ideal circle. In reality, the received power at certain distance is a random variable due to multipath propagation effects, which is also known as fading effects. In fact, the above two models predicts the mean received power at distance $d$. A more general and widely-used model is called the shadowing model [29].

The shadowing model consists of two parts. The first one is known as path loss model, which also predicts the mean received power at distance $d$, denoted by $\overline{P_r(d)}$. It uses a close-in distance $d_0$ as a reference. $\overline{P_r(d)}$ is computed relative to $P_r(d_0)$ as follows.

$$\frac{P_r(d_0)}{\overline{P_r(d)}} = {\left( \frac{d}{d_0} \right)}^\beta$$ (18.4)

$\beta$ is called the path loss exponent, and is usually empirically determined by field measurement. From Eqn. (18.1) we know that $\beta = 2$ for free space propagation. Table 18.1 gives some typical values of $\beta$. Larger values correspond to more obstructions and hence faster decrease in average received power as distance becomes larger. $P_r(d_0)$ can be computed from Eqn. (18.1).

Table 18.1: Some typical values of path loss exponent $\beta$

Environment		$\beta$
Outdoor	Free space	2
	Shadowed urban area	2.7 to 5
In building	Line-of-sight	1.6 to 1.8
	Obstructed	4 to 6

Table 18.2: Some typical values of shadowing deviation $\sigma _{dB}$

Environment	$\sigma _{dB}$ (dB)
Outdoor	4 to 12
Office, hard partition	7
Office, soft partition	9.6
Factory, line-of-sight	3 to 6
Factory, obstructed	6.8

The path loss is usually measured in dB. So from Eqn. (18.4) we have

$${\left[ \frac{\overline{P_r(d)}}{P_r(d_0)} \right]}_{dB} = -10 \beta \log \left( \frac{d}{d_0} \right)$$ (18.5)

The second part of the shadowing model reflects the variation of the received power at certain distance. It is a log-normal random variable, that is, it is of Gaussian distribution if measured in dB. The overall shadowing model is represented by

$${\left[ \frac{P_r(d)}{P_r(d_0)} \right]}_{dB} = -10 \beta \log \left( \frac{d}{d_0} \right) + X_{dB}$$ (18.6)

where $X_{dB}$ is a Gaussian random variable with zero mean and standard deviation $\sigma _{dB}$. $\sigma _{dB}$ is called the shadowing deviation, and is also obtained by measurement. Table  18.2 shows some typical values of $\sigma _{dB}$. Eqn. (18.6) is also known as a log-normal shadowing model.

The shadowing model extends the ideal circle model to a richer statistic model: nodes can only probabilistically communicate when near the edge of the communication range.