Ferkans — Interactive Telecom Tutor

The Random Component of Large-Scale Fading

Path-loss models predict the average received power at a given distance. In reality, two locations at the same distance from a base station can experience vastly different path losses due to buildings, hills, and foliage blocking the signal. This location-dependent variation is shadowing (also called shadow fading or slow fading). Modelling shadowing is essential for computing coverage probability and outage.

Definition:
Log-Normal Shadowing Model

The received power in dB at distance $d$ is modelled as

$P_r(d)\,\text{[dBm]} = P_t\,\text{[dBm]} - PL(d)\,\text{[dB]} + X_\sigma$

where $X_\sigma \sim \mathcal{N}(0, \sigma^2)$ is a Gaussian random variable in dB. Equivalently, the path loss is

$PL(d) = \overline{PL}(d) + X_\sigma = PL(d_0) + 10n\log_{10}(d/d_0) + X_\sigma.$

In linear scale, the received power is log-normally distributed:

$P_r \sim \text{LogNormal}(\mu, \sigma^2)$

Environment	Typical $\sigma$ (dB)
Outdoor urban	6–10
Outdoor suburban	6–8
Indoor	3–14
Factory floor	4–8

Theorem: Outage Probability

The outage probability is the probability that the received power falls below a minimum threshold $P_{\min}$ (receiver sensitivity):

$P_{\text{out}}(d) = P(P_r(d) < P_{\min}) = Q\!\left(\frac{\overline{P_r}(d) - P_{\min}}{\sigma}\right)$

where $\overline{P_r}(d) = P_t - \overline{PL}(d)$ is the mean received power (dBm) and $Q(\cdot)$ is the Gaussian Q-function.

Equivalently, the coverage probability is $1 - P_{\text{out}}(d)$ .

Shadowing adds a random dB offset to the mean path loss. The Q-function measures the probability that this offset pushes the signal below the threshold. Larger $\sigma$ (more shadowing variability) increases the outage probability.

Proof

Derivation

$P_r(d) = \overline{P_r}(d) + X_\sigma$ where $X_\sigma \sim \mathcal{N}(0, \sigma^2)$ .

$P(P_r < P_{\min}) = P(X_\sigma < P_{\min} - \overline{P_r}) = \Phi\!\left(\frac{P_{\min} - \overline{P_r}}{\sigma}\right) = Q\!\left(\frac{\overline{P_r} - P_{\min}}{\sigma}\right)$ . $\blacksquare$

,

Key Takeaway

Shadowing transforms deterministic coverage into a probabilistic problem. To guarantee $p$ % coverage, the link budget must include a fade margin of $Q^{-1}(1 - p/100) \cdot \sigma$ dB. For typical urban shadowing ( $\sigma = 8$ dB) and 95% coverage, this margin is $\approx 13$ dB — a substantial overhead that directly affects cell size and base station density.

Combined Path Loss and Shadowing

See how shadowing adds random variation around the mean path loss. Adjust $\sigma$ to see the impact on outage probability at a given distance.

Parameters

Path-loss exponent

n

3.5

Shadow std dev

\sigma

(dB)8

Frequency (MHz)900

Theorem: Cell Coverage Area with Shadowing

The fraction of a circular cell of radius $R$ that achieves the minimum received power $P_{\min}$ is

$C(R) = \frac{1}{\pi R^2} \int_0^R Q\!\left(\frac{\overline{P_r}(d) - P_{\min}}{\sigma}\right) 2\pi d\,dd$

For the log-distance model, this evaluates to

$C(R) = 1 - e^{2ab + b^2}\,Q(2b + a) + 2\,Q(a)$

(Jakes' formula) where

$a = \frac{\overline{P_r}(R) - P_{\min}}{\sigma}, \qquad b = \frac{10n\log_{10}(e)}{\sigma} = \frac{10n}{(\ln 10)\sigma} \approx \frac{4.343n}{\sigma}.$

Points closer to the base station have higher mean received power and thus higher coverage probability. The integral averages this over the entire cell area. The result depends on the ratio of shadowing spread to the path-loss slope.

Proof

Setup

The coverage probability at distance $d$ is $P_{\text{cov}}(d) = Q\!\left(\frac{\overline{P_r}(d) - P_{\min}}{\sigma}\right)$ .

The areal average is $C(R) = \frac{1}{\pi R^2}\int_0^R P_{\text{cov}}(d)\, 2\pi d\,dd$ .

Substituting the log-distance model and performing the change of variable $u = 10n\log_{10}(d/R)/\sigma$ yields the closed-form expression. $\blacksquare$

Cell Coverage Area

Adjust the path-loss exponent and shadowing standard deviation to see how they affect the fraction of the cell area with adequate coverage.

Parameters

Path-loss exponent

n

3.5

Shadow std dev

\sigma

(dB)8

Link margin at cell edge (dB)10

Definition:
Correlated Shadowing

Shadowing between two links sharing a common endpoint (e.g., a mobile receiving from two base stations) is correlated because the same obstacles may block both paths.

The Gudmundson model describes the spatial correlation of shadowing as exponentially decaying:

$\rho(\Delta d) = e^{-\Delta d / d_{\text{corr}}}$

where $d_{\text{corr}}$ is the decorrelation distance (typically 20–100 m in urban environments, up to several hundred metres in suburban areas).

Correlated shadowing affects handover margins and interference statistics in multi-cell systems.

Example: Fade Margin Design

A cellular system requires 95% coverage at the cell edge (outage probability $\le 5$ %). The shadowing standard deviation is $\sigma = 8$ dB. How much fade margin must be added to the link budget?

Solution

Required Q-function value

$P_{\text{out}} = 0.05 = Q(a)$ $\Rightarrow$ $a = Q^{-1}(0.05) = 1.645$ .

Fade margin

$\text{Fade margin} = a \cdot \sigma = 1.645 \times 8 = 13.2$ dB.

This means the system must be designed so that the mean received power at the cell edge exceeds $P_{\min}$ by at least 13.2 dB to guarantee 95% coverage. $\blacksquare$

Common Mistake: Confusing Shadowing with Small-Scale Fading

Mistake:

Treating shadowing and multipath (small-scale) fading as the same phenomenon.

Correction:

Shadowing (Section 5.5) is caused by large obstacles (buildings, hills) and varies over distances of tens to hundreds of metres — it is a large-scale effect. Small-scale fading (Chapter 6) is caused by constructive/destructive interference of multiple signal copies and varies over distances of half a wavelength ( $\approx$ cm at GHz frequencies). They are modelled independently: shadowing as log-normal, small-scale fading as Rayleigh/Rice.

Quick Check

If $\sigma = 10$ dB and we want 90% coverage probability at the cell edge, what is the required fade margin?

10 dB

12.8 dB

16.5 dB

20 dB

Correction:

12.8 dB

Correct. $Q^{-1}(0.10) = 1.282$ , so fade margin $= 1.282 \times 10 = 12.8$ dB.

Shadow Fading

Random variation of received power (in dB) around the mean path loss, caused by large obstacles. Modelled as $X_\sigma \sim \mathcal{N}(0, \sigma^2)$ (log-normal in linear scale).

Outage Probability

The probability that the received signal falls below the minimum required level: $P_{\text{out}} = Q((\overline{P_r} - P_{\min})/\sigma)$ .

Fade Margin

Extra link budget margin (dB) added to combat shadowing. For $p$ % coverage: margin $= Q^{-1}(1-p/100) \cdot \sigma$ .

Shadowing

The Random Component of Large-Scale Fading

Definition: Log-Normal Shadowing Model

Theorem: Outage Probability

Derivation

Key Takeaway

Combined Path Loss and Shadowing

Parameters

Theorem: Cell Coverage Area with Shadowing

Setup

Cell Coverage Area

Parameters

Definition: Correlated Shadowing

Example: Fade Margin Design

Required Q-function value

Fade margin

Common Mistake: Confusing Shadowing with Small-Scale Fading

Quick Check

Shadow Fading

Outage Probability

Fade Margin

Definition:
Log-Normal Shadowing Model

Definition:
Correlated Shadowing