RF Theory › RF Propagation

RF Propagation Models

How RF signals travel from transmitter to receiver — and how much power is lost. From the Friis free-space model to the 3GPP 5G NR models used in real network planning, with annotated plots and worked examples.

01 — Free Space

Friis Free-Space Model

Friis Transmission Equation
Pr(dBm) = Pt(dBm) + Gt(dBi) + Gr(dBi) − FSPL(dB)

FSPL(dB) = 32.44 + 20·log₁₀(f[MHz]) + 20·log₁₀(d[km])
Or: 92.44 + 20·log₁₀(f[GHz]) + 20·log₁₀(d[km])
Freqd=100 md=1 kmd=10 km
433 MHz (IoT)65.6 dB85.6 dB105.6 dB
2.4 GHz (WiFi)80.0 dB100.0 dB120.0 dB
3.5 GHz (5G NR)83.3 dB103.3 dB123.3 dB
28 GHz (5G mmWave)101.4 dB121.4 dB141.4 dB
77 GHz (radar)110.2 dB130.2 dB150.2 dB
02 — Two-Ray Model

Two-Ray Ground Reflection

Beyond the breakpoint distance d_BP, the reflected and direct paths nearly cancel — path loss increases as d⁴ (40 dB/decade) instead of d² (20 dB/decade).

Two-Ray Breakpoint
d < dBP: PL ≈ FSPL (20 dB/decade)
d > dBP: PL ≈ 40·log₁₀(d) − 20·log₁₀(ht·hr) (40 dB/decade)

dBP = 4·ht·hr/λ
Example: h_t=30 m, h_r=1.5 m, f=3.5 GHz → d_BP = 2.09 km
03 — Empirical Model

Log-Distance + Shadowing

Log-Distance Path Loss
PL(d) = PL(d₀) + 10n·log₁₀(d/d₀) + Xσ   [dB]
Xσ = Gaussian shadowing (std dev σ dB)

Path Loss Exponent n

Environmentnσ (dB)
Free space2.00
LOS indoor1.6–1.83–5
NLOS indoor3.0–4.06–10
Urban macro LOS2.1–2.54–8
Urban macro NLOS3.0–4.06–10
Dense urban NLOS4.0–5.08–12
Shadowing margin: For 90% coverage with σ=8 dB (dense urban) → 1.28×8 = 10.2 dB margin needed. Often the dominant link budget term.
04 — Okumura-Hata

Okumura-Hata Model

Hata Urban (150–1500 MHz)
PL = 69.55 + 26.16·log(f_c) − 13.82·log(h_te) − a(h_re) + (44.9−6.55·log(h_te))·log(d)
Suburban: PL_sub = PL_urban − 2·[log(f_c/28)]² − 5.4
Valid: 150–1500 MHz, 1–20 km, h_te = 30–200 m
Example — f=900 MHz, h_te=50 m, h_re=1.5 m, d=2 km, large city

PL_urban = 69.55+26.16×2.954−13.82×1.699+10.17 = 133.5 dB

FSPL at same params: 97.5 dB → Excess urban loss: 36 dB

05 — 3GPP 5G NR

3GPP 5G NR Path Loss Models

UMi Street Canyon LOS (TR 38.901)
PL = 32.4 + 21·log₁₀(d3D) + 20·log₁₀(fc)   [dB]   (f_c in GHz, d in m)
ScenarioPL at 100 m, 3.5 GHzEffective n
UMi LOS85.3 dB2.1
UMi NLOS104.4 dB3.53
UMa LOS82.9 dB2.2
UMa NLOS102.6 dB3.9
InH LOS77.9 dB1.73
InH NLOS106.7 dB3.83
06 — Multipath Fading

Multipath Fading

Rayleigh Fade Margin
For 99% coverage (Rayleigh): fade margin = 20 dB
For 99.9% coverage: fade margin = 30 dB
With 2-antenna MRC diversity: 99% coverage → only 10 dB margin needed
Diversity is far more efficient than margin: 99% → 99.9% requires +10 dB margin (×10 TX power). With 2-antenna receive diversity, only +3 dB needed.
07 — Atmospheric

Rain & Atmospheric Attenuation

FrequencyRain 25 mm/hrRain 50 mm/hrO₂
3.5 GHz0.03 dB/km0.06 dB/km<0.01 dB/km
10 GHz0.5 dB/km1.2 dB/km0.01 dB/km
28 GHz4.5 dB/km8 dB/km0.1 dB/km
60 GHz6 dB/km12 dB/km15 dB/km (O₂ peak)
77 GHz8 dB/km14 dB/km0.4 dB/km
60 GHz limited to short-range by oxygen absorption (15 dB/km). At Ku/Ka-band for satellite backhaul, use ITU-R P.838 rain attenuation models — especially critical for tropical regions with heavy monsoon rainfall.
08 — Comparison

Model Comparison Plot

Figure 1 — Path loss vs distance at 3.5 GHz. Free-space (n=2), two-ray beyond breakpoint (n=4), UMi LOS, UMa NLOS, log-distance NLOS (n=3.5). Divergence between models grows with distance.
09 — Design Rules

Propagation Thumb Rules

#RuleValue
1Every doubling of distance in free space+6.02 dB path loss
2Every doubling of frequency in free space+6.02 dB path loss
35G 28 GHz vs 3.5 GHz path loss difference18 dB (= 20·log₁₀(28/3.5))
490% urban coverage shadowing margin (σ=8 dB)10.2 dB
5Rayleigh fade margin for 99% coverage20 dB
6Concrete wall penetration loss12–15 dB at 3.5 GHz
760 GHz oxygen absorption~15 dB/km — limits range to ~200 m
8Urban path loss exponent NLOSn = 3.5 to 4.0
FSPL Calculator → Link Budget → Link Budget Theory → OFDM & 5G NR →

RF Propagation Models — From Free Space to 5G NR

Path loss between transmitter and receiver determines link margin, cell radius, required antenna gain, and coverage probability. The Friis equation gives the correct answer only in free space. For urban cellular networks, the Okumura-Hata model (below 1.5 GHz) or the 3GPP 38.901 models (above 1 GHz) provide empirically validated accuracy. Choosing the wrong model can result in 20–30 dB path loss errors, directly causing incorrect cell radius sizing.

Why Rain Attenuation Matters for 5G Backhaul

At Ku-band (12–18 GHz) and Ka-band (26–40 GHz) used for satellite backhaul and 5G fronthaul, rain attenuation is significant. In tropical regions with heavy monsoon rainfall — India, Southeast Asia, equatorial Africa — rain rates can exceed 50 mm/hour, producing 8+ dB/km at 28 GHz. Link budgets for mmWave backhaul must include ITU-R P.838 rain margin, typically 10–25 dB for 99.99% availability targets.

Two-Ray Model and the Breakpoint Distance

In terrestrial links, signals arrive via both a direct LOS path and a ground-reflected path. Beyond the breakpoint distance d_BP = 4·h_t·h_r/λ, these paths nearly cancel — path loss increases as d⁴ instead of d². For a 3.5 GHz macro cell with 30 m BS height and 1.5 m UE height, d_BP ≈ 2.1 km. Urban network planners must account for this transition when estimating coverage at cell edges.