Ferkans — Interactive Telecom Tutor

Definition:
Two-Ray Ground-Reflection Model

Over a flat, reflecting ground plane, the received signal is the sum of a direct ray and a ground-reflected ray. For antenna heights $h_t$ and $h_r$ and distance $d$ :

$P_r = P_t G_t G_r \left(\frac{\lambda}{4\pi d}\right)^2 \left|1 + \Gamma\,e^{-j\Delta\phi}\right|^2$

where $\Gamma \approx -1$ (grazing incidence) and the phase difference is

$\Delta\phi = \frac{2\pi}{\lambda}\left(\sqrt{d^2 + (h_t + h_r)^2} - \sqrt{d^2 + (h_t - h_r)^2}\right).$

For $d \gg h_t, h_r$ (far field):

$P_r \approx P_t G_t G_r \frac{h_t^2 h_r^2}{d^4}.$

The path-loss exponent changes from 2 (free space) to 4 beyond the breakpoint distance:

$d_c = \frac{4 h_t h_r}{\lambda}.$

,

Theorem: Two-Ray Model — Near and Far Regimes

The two-ray model exhibits two distinct regimes:

Near field ( $d < d_c$ ): constructive and destructive interference cause oscillations around free-space path loss. Average exponent $\approx 2$ .
Far field ( $d > d_c$ ): the direct and reflected rays nearly cancel, giving $P_r \propto d^{-4}$ (exponent 4). The received power becomes independent of wavelength:

$P_r = P_t G_t G_r \frac{h_t^2 h_r^2}{d^4} \qquad (d \gg d_c).$

Beyond the breakpoint, the ground reflection nearly cancels the direct ray. The residual power comes from the small path-length difference between the two rays, which scales as $h_t h_r / d$ . Squaring gives the $d^{-4}$ dependence.

Proof

Far-field approximation

For $d \gg h_t + h_r$ , the path-length difference is $\Delta \ell \approx 2h_t h_r / d$ (geometric approximation).

The phase difference is $\Delta\phi = 2\pi\Delta\ell/\lambda = 4\pi h_t h_r / (\lambda d)$ .

For $d \gg d_c$ , $\Delta\phi \ll 1$ , so $|1 + \Gamma e^{-j\Delta\phi}| = |1 - e^{-j\Delta\phi}| \approx \Delta\phi = 4\pi h_t h_r / (\lambda d)$ .

Substituting into the Friis equation:

$P_r = P_t G_t G_r \left(\frac{\lambda}{4\pi d}\right)^2 \left(\frac{4\pi h_t h_r}{\lambda d}\right)^2 = P_t G_t G_r \frac{h_t^2 h_r^2}{d^4}$ . $\blacksquare$

Free-Space vs. Two-Ray Path Loss

Compare free-space path loss ( $\propto d^{-2}$ ) with the two-ray model. Adjust antenna heights to see how the breakpoint distance $d_c = 4h_t h_r/\lambda$ shifts.

Parameters

TX antenna height

h_t

(m)30

RX antenna height

h_r

(m)1.5

Frequency (MHz)900

Example: Breakpoint Distance for a Cellular System

A cellular base station has $h_t = 30$ m. A mobile user has $h_r = 1.5$ m. The carrier frequency is 1800 MHz. Find the breakpoint distance and the path loss at $d = 1$ km.

Solution

Wavelength and breakpoint

$\lambda = 3 \times 10^8 / 1.8 \times 10^9 = 0.167$ m.

$d_c = 4 h_t h_r / \lambda = 4 \times 30 \times 1.5 / 0.167 = 1078$ m $\approx 1.1$ km.

Path loss at 1 km

Since $d = 1000$ m $< d_c$ , we are in the oscillation region near the free-space regime.

$PL_{\text{fs}} = 20\log_{10}(4\pi \times 1000 / 0.167) = 97.5$ dB.

The two-ray model predicts oscillations around this value. At exactly $d = d_c$ , the path loss transitions to the $d^4$ regime. $\blacksquare$

Two-Ray Ground-Reflection Model Geometry — Geometry of the two-ray model. The direct ray (solid) and ground- reflected ray (dashed) combine at the receiver. The path-length difference $\Delta\ell \approx 2h_t h_r / d$ for $d \gg h_t, h_r$ . Beyond the breakpoint $d_c = 4h_t h_r / \lambda$ , the two rays nearly cancel, producing $P_r \propto d^{-4}$ .

Two-Ray Interference Pattern

Watch how the received power oscillates between constructive and destructive interference as distance increases, then transitions to the smooth

d^{-4}

regime beyond the breakpoint.

The oscillation pattern arises from the phase difference between direct and reflected rays. Beyond

d_c

, the envelope settles to

d^{-4}

.

Definition:
Log-Distance Path-Loss Model

A general model that captures the average path loss as a function of distance:

$PL(d) = PL(d_0) + 10n\log_{10}\!\left(\frac{d}{d_0}\right) \quad \text{dB}$

where:

$d_0$ is a reference distance (typically 1 m or 100 m)
$PL(d_0)$ is the path loss at $d_0$ (often set to free-space)
$n$ is the path-loss exponent (PLE)

Environment	Typical $n$
Free space	2
Urban cellular	2.7–3.5
Suburban	3–5
Indoor (same floor)	1.6–3.3
Indoor (through floors)	4–6
Dense urban (mmWave)	2–4

Path-Loss Model Comparison

Compare Friis (free-space), two-ray, and log-distance models on a single plot. Adjust the path-loss exponent $n$ to see how different environments affect received power vs. distance.

Parameters

Path-loss exponent

n

3.5

TX height

h_t

(m)30

RX height

h_r

(m)1.5

Frequency (MHz)900

Why This Matters: Path-Loss Exponent in Network Planning

The path-loss exponent $n$ is a key input to cellular network planning. A higher $n$ means faster signal decay, which reduces inter-cell interference but also reduces coverage. In dense urban deployments, $n$ is typically 3–4, leading to cell radii of 100–500 m. In rural areas, $n \approx 2.5$ allows cells to cover several kilometres. 5G millimeter-wave cells have $n \approx 2$ – $3$ but suffer high penetration loss, limiting cells to tens of metres indoors.

Common Mistake: Path-Loss Exponent Is NOT Universal

Mistake:

Using $n = 2$ (free-space) for all environments, or a single value of $n$ for all distances.

Correction:

The path-loss exponent depends on the environment, frequency, antenna heights, and distance range. Even within one environment, $n$ may differ in LOS vs. NLOS conditions. Always use measurement- calibrated values or standardised models (Section 5.4) for engineering calculations.

Quick Check

In the two-ray model, what is the path-loss exponent beyond the breakpoint distance?

2

3

4

6

Correction:

4

Correct. Beyond the breakpoint $d_c = 4h_t h_r/\lambda$ , the direct and reflected rays nearly cancel, giving $P_r \propto d^{-4}$ .

Quick Check

How does the breakpoint distance $d_c = 4h_t h_r/\lambda$ change when the frequency doubles?

It halves

It doubles

It stays the same

It quadruples

Correction:

It doubles

Correct. $d_c = 4h_t h_r / \lambda = 4h_t h_r f / c$ . Doubling $f$ doubles $d_c$ .

Two-Ray Model

A propagation model that accounts for the direct ray and a single ground-reflected ray. Predicts $P_r \propto d^{-4}$ beyond the breakpoint distance.

Breakpoint Distance

$d_c = 4h_t h_r / \lambda$ . The distance beyond which the two-ray model transitions from the $d^{-2}$ to the $d^{-4}$ regime.

Path-Loss Exponent

The exponent $n$ in $PL \propto d^n$ . Free space: $n=2$ . Typical urban: $n = 3$ – $4$ . Dense indoor: $n = 4$ – $6$ .

Path-Loss Models

Definition: Two-Ray Ground-Reflection Model

Theorem: Two-Ray Model — Near and Far Regimes

Far-field approximation

Free-Space vs. Two-Ray Path Loss

Parameters

Example: Breakpoint Distance for a Cellular System

Wavelength and breakpoint

Path loss at 1 km

Two-Ray Ground-Reflection Model Geometry

Two-Ray Interference Pattern

Definition: Log-Distance Path-Loss Model

Path-Loss Model Comparison

Parameters

Why This Matters: Path-Loss Exponent in Network Planning

Common Mistake: Path-Loss Exponent Is NOT Universal

Quick Check

Quick Check

Two-Ray Model

Breakpoint Distance

Path-Loss Exponent

Definition:
Two-Ray Ground-Reflection Model

Definition:
Log-Distance Path-Loss Model