Propagation, LOS/NLOS and Empirical Models

Tip

Exam Focus

Write free-space path loss $L_{fs}=32.44+20\log_{10}d_{km}+20\log_{10}f_{MHz}$, contrast the two-ray $d^{-4}$ law, state the Okumura and Hata model ranges and the Hata urban equation, and classify fading (fast/slow, flat/selective). For reuse: $N=i^2+ij+j^2$, $D/R=\sqrt{3N}$. FSPL, Doppler and reuse-distance numericals are all high-yield.

Path Loss and Propagation Mechanisms

Caution

Key Formula — Free-space path loss (Friis)

$$\frac{P_r}{P_t}=G_tG_r\!\left(\frac{\lambda}{4\pi d}\right)^{\!2},\qquad L_{fs}(dB)=32.44+20\log_{10}d_{km}+20\log_{10}f_{MHz}.$$

Radio energy reaches the receiver by several mechanisms; a non-line-of-sight (NLOS) link relies on them when the direct line-of-sight (LOS) path is blocked.

Multipath propagation: direct LOS plus ground/building reflection, rooftop diffraction and scattering; the building casts a shadow (NLOS) region and the moving mobile experiences Doppler.

Mechanism	Description
Reflection	From surfaces large compared with $\lambda$
Diffraction	Bending around edges; gives some shadow-region coverage
Scattering	From rough surfaces / small objects; spreads energy widely
Shadowing	Slow log-normal variation behind obstacles

Propagation mechanisms.

Note

Key Point — Two-ray ground-reflection model

Beyond the crossover distance the direct and ground-reflected rays give

$$P_r\approx P_tG_tG_r\frac{h_t^2h_r^2}{d^4},$$

so power falls as $d^{-4}$ (path-loss exponent $n_p=4$) rather than the free-space $d^{-2}$.

Multipath Fading and Doppler

Multipath components add constructively or destructively, causing fading.

Fading	Meaning
Fast	Rapid variation from multipath + motion
Slow	Slow variation from shadowing
Flat	Signal bandwidth $\ll$ coherence bandwidth; one gain
Frequency-selective	Delay spread resolves paths $\Rightarrow$ ISI

Fading classification.

Caution

Key Formula — Doppler shift

$$f_d=\frac{v}{\lambda}\cos\phi,\qquad |f_d|_{\max}=\frac{v}{\lambda}.$$

Empirical Path-Loss Models: Okumura and Hata

Free-space loss is optimistic for cluttered cellular routes, so measurement-based models are used in the 150–1500 MHz band.

Note

Key Point — Okumura model

Fully empirical: a median free-space loss plus a measured median attenuation $A_{mu}(f,d)$ and correction factors for base/mobile antenna height, land-use (urban/suburban/open) and terrain. It is accurate but read from curves rather than a closed form.

[!NOTE]

Key Point — Hata model (urban), a closed-form fit to Okumura

$$L_{u}=69.55+26.16\log_{10}f-13.82\log_{10}h_b-a(h_m) +(44.9-6.55\log_{10}h_b)\log_{10}d,$$

with $f$ in MHz ($150$–$1500$), $h_b$ ($30$–$200$ m), $h_m$ ($1$–$10$ m), $d$ in km ($1$–$20$). For a small/medium city $a(h_m)=(1.1\log_{10}f-0.7)h_m-(1.56\log_{10}f-0.8)$. Suburban and open-area versions subtract correction terms.

Frequency Reuse

Frequency reuse assigns the full channel set once per cluster of $N$ cells, then repeats it in clusters far enough away to keep interference acceptable. Reuse factor $=1/N$.

Caution

Key Formula — Cluster size and reuse distance

$$N=i^2+ij+j^2,\qquad D=R\sqrt{3N},\qquad Q=\frac{D}{R}=\sqrt{3N},$$

$i,j$ non-negative integers; common $N=3,4,7,12$.

Seven-cell cluster ($N=7$): cell radius $R$ and the reuse distance $D=R\sqrt{3N}$ to the nearest co-channel cell using frequency group A.

Note

Key Point — Co-channel and adjacent-channel interference

Co-channel interference (CCI) is unwanted energy from cells reusing the same channels; it cannot be filtered out (same frequency). Reduce it by larger $N/D$, sectoring, directional antennas/beamforming and power control. A useful edge estimate is $\frac{C}{I}\approx\frac{(3N)^{n_p/2}}{i_0}$ ($i_0\approx6$ first-tier interferers). Adjacent-channel interference leaks from nearby channels; reduce it with guard bands, filtering and careful assignment.

Capacity Improvement

Technique	Idea	Cost
Cell splitting	Split a congested cell into smaller cells	More sites, more handovers
Sectoring	Directional antennas ($3\times120^\circ$/$6\times60^\circ$) cut interferers	More antennas/handovers
Microcell zone	Zones share one base station	Reduces handoff burden

Capacity-improvement methods.

Worked Examples

Note

Example — Cluster size and reuse distance

$i=2$, $j=1$, cell radius $R=2\,km$.

$$N=2^2+2\cdot1+1^2=7,\qquad D=R\sqrt{3N}=2\sqrt{21}\approx\boxed{9.17\,km}.$$

[!NOTE]

Example — Free-space path loss and Doppler

$f=900\,MHz$, $d=2\,km$, mobile at $72\,km/h=20\,m/s$.

$$L_{fs}=32.44+20\log_{10}2+20\log_{10}900\approx\boxed{97.55\,dB},$$$$\lambda=\tfrac{c}{f}=0.333\,m,\quad f_d=\tfrac{v}{\lambda}=\tfrac{20}{0.333}\approx\boxed{60\,Hz}.$$