Free Space vs. Real-World Propagation

Free-space path loss (FSPL) describes signal attenuation in an unobstructed vacuum. Real-world environments impose additional losses from reflection, diffraction, scattering, absorption, and multipath interference. The excess path loss beyond FSPL is termed the propagation model correction factor, and it varies with frequency, antenna height, terrain, clutter, and building density. All empirical propagation models attempt to quantify this excess loss as a statistical function of measurable parameters.

Okumura-Hata Model

The Okumura-Hata model is the most widely used empirical model for macro-cell mobile radio planning in the 150–1500 MHz range. Hata curve-fitted Okumura's original measurement data into closed-form expressions:

Urban median path loss:

L_u (dB) = 69.55 + 26.16·log(f) − 13.82·log(h_te) − a(h_re) + (44.9 − 6.55·log(h_te))·log(d)

where f is frequency in MHz (150–1500), h_te is base station antenna height in metres, h_re is mobile antenna height, d is distance in km, and a(h_re) is an antenna height correction factor. For small and medium cities: a(h_re) = (1.1·log(f) − 0.7)·h_re − (1.56·log(f) − 0.8).

Suburban correction: subtract (2·[log(f/28)]² + 5.4) dB from L_u.

Open rural correction: subtract (4.78·[log(f)]² − 18.33·log(f) + 40.94) dB.

These corrections recognise that clutter in suburban and rural environments is progressively less than in dense urban cores, yielding lower path loss. Typical prediction errors are ±8–12 dB (one standard deviation) for the environments in which the model was calibrated.

COST 231 Extension

The COST 231 project extended Hata's model to the 1500–2000 MHz range (PCS and early 3G bands), producing the COST 231-Hata model. It introduces a metropolitan area correction factor C_m: 0 dB for medium-sized cities and suburbs, 3 dB for metropolitan centres. This model is standard in GSM 1800 and UMTS 2100 planning. Beyond 2 GHz, the model accuracy degrades and ray-tracing or measurement-based models are preferred.

Longley-Rice (ITM) Model

The Irregular Terrain Model (ITM), developed by Rice, Longley, Norton, and Barsis at NTIA/ITS, is the primary propagation model used by the FCC and international regulators for point-to-point and broadcast services across 20 MHz to 20 GHz. Unlike Hata, it is a physics-based statistical model that ingests actual terrain profiles (SRTM 3-arc-second or better data), antenna heights, frequency, polarisation, and required time/location reliability percentages. It computes path loss accounting for diffraction over terrain obstacles using a refined knife-edge method and Bullington equivalent obstructions. The FCC uses ITM in its OET Bulletin 69 tools for FM and TV coverage prediction. NTIA provides the reference C implementation (itm.cpp) and the SPLAT! open-source radio propagation tool wraps ITM for point-to-point analysis.

Indoor Propagation Models

One-slope model: The simplest indoor model: L = L_0 + 10·n·log(d), where L_0 is path loss at 1 m (typically 30–38 dB at 900 MHz) and n is the distance-loss exponent (1.6–3.5 depending on environment). Open-plan offices yield n ≈ 2; corridors can exhibit waveguide effects with n < 2.

Multi-wall model (MWM): More accurate for office buildings, it adds penetration loss for each wall and floor traversed: L = L_0 + 20·log(d) + Σ(n_w · L_w) + Σ(n_f · L_f). Typical wall losses: drywall 3–5 dB; concrete block 10–15 dB; reinforced concrete 15–25 dB. Floor penetration loss: 10–20 dB per floor for concrete decks.

ITU-R P.1238: The international standard indoor model covering 900 MHz to 100 GHz. It uses a power-distance exponent N and a floor attenuation factor (FAF): L = 20·log(f) + N·log(d) + L_f(n) − 28. It is well-calibrated for modern 2.4 GHz and 5 GHz Wi-Fi planning.

Multipath Fading: Rayleigh vs. Rician

When multiple reflected, diffracted, and scattered copies of a signal arrive at a receiver with random amplitudes and phases, the envelope of the received signal follows a statistical distribution. Rayleigh fading occurs when there is no dominant line-of-sight (LOS) component; the signal envelope PDF is: f(r) = (r/σ²)·exp(−r²/2σ²). This is the worst case — the average fade depth at the 1% probability level is approximately 20 dB below the median. Rician fading applies when a strong LOS component exists alongside scattered components; the K-factor (ratio of LOS power to scattered power) characterises how close the statistics are to Rayleigh (K=0) or AWGN (K→∞). Typical K values: urban mobile 0–3 dB; microcell LOS 6–12 dB; fixed point-to-point links with clear Fresnel zone clearance 15–25 dB.

Shadow Fading and Log-Normal Shadowing

Large-scale (slow) fading due to buildings, hills, and vegetation blocking the path is modelled as log-normal: path loss in dB is normally distributed around the median predicted value with standard deviation σ_s. Typical σ_s values: macrocells 6–10 dB; microcells 4–6 dB; indoor 3–5 dB. Network planners use this to compute coverage probability. For a required probability of coverage P_cov at the cell edge, a shadowing margin M_s = σ_s · Q⁻¹(P_cov) must be added to the link budget. For P_cov = 90% and σ_s = 8 dB: M_s = 8 × 1.28 = 10.2 dB.

Diffraction: Knife-Edge Model

When a radio wave encounters a sharp obstacle (ridge, building rooftop), energy is diffracted into the shadow region. The knife-edge diffraction loss is a function of the Fresnel-Kirchhoff diffraction parameter ν:

ν = h · √(2(d₁+d₂) / λ·d₁·d₂)

where h is obstacle height above the direct ray (positive for obstruction). Diffraction loss L_d ≈ 6.02 + 9.11ν + 1.27ν² dB for ν between 0 and 2.4. Negative ν (obstacle below the ray) yields gain relative to free space, consistent with the first Fresnel zone concept. To maintain < 6 dB diffraction loss, 0.6 of the first Fresnel zone radius must be unobstructed at all points along the path.

Practical Frequency Selection

Lower frequencies propagate farther and diffract better over terrain but suffer more interference and require larger antennas. Higher frequencies enable compact antennas and wide bandwidth but attenuate more rapidly. Engineering guidance by band:

  • 400–470 MHz (UHF low): Excellent rural coverage, good building penetration, primary public safety LMR band. Path loss exponent n ≈ 3.5 in urban, 2.5 in rural.
  • 700–800 MHz: FirstNet Band 14 and public safety. Best balance of coverage and capacity. Circa 8–12 dB improvement over 1900 MHz for in-building penetration.
  • 900 MHz ISM: Unlicensed IoT (LoRa, Zigbee). Good non-LOS coverage; competes with licensed 900 MHz SMR.
  • 2.4 GHz: Wi-Fi 802.11b/g/n/ax. Significant wall losses (6–14 dB per interior wall). Dense AP deployment required in large buildings.

As a rule of thumb, each doubling of frequency adds approximately 6 dB to the free-space path loss, and real-world excess loss increases faster due to higher material absorption above ~1 GHz.