Basic propagation models and far-field distance

Propagation models predict the average received signal strength at a given distance (large-scale models) or its rapid fluctuations over short distances (small-scale/fading models).

Basic propagation models

1. Free space model (Friis) Used when there is a clear, unobstructed LOS path, e.g. satellite and microwave links.

Pr(d) = Pt·Gt·Gr·λ² / [(4π)²d²L]
PL(dB) = 20 log(4πd/λ)

Power falls as 1/d² (20 dB/decade). Valid only in the far field (d ≥ df).

2. Two-ray ground reflection model Considers the direct path plus a ground-reflected path. For large d:

Pr = Pt·Gt·Gr·ht²·hr² / d⁴

Power falls as 1/d⁴ (40 dB/decade). More accurate than free space for long distances over flat ground.

3. Log-distance path loss model

PL(d) = PL(d0) + 10n log(d/d0)

n = 2 free space, 2.7–3.5 urban, 3–5 shadowed urban.

4. Log-normal shadowing Adds a zero-mean Gaussian random variable Xσ (dB) to log-distance loss to account for obstacles: PL(d) = PL(d0) + 10n log(d/d0) + Xσ.

Empirical outdoor models (Okumura, Hata) and indoor models (partition loss, Ericsson) build on these.

Far field distance

The far field (Fraunhofer region) begins at

df = 2D²/λ,  with df ≫ D and df ≫ λ

Given D = 1 m, f = 900 MHz:

λ  = 3×10⁸ / 900×10⁶ = 0.333 m
df = 2 × 1² / 0.333 = 6 m
Check: 6 m ≫ D (1 m) and 6 m ≫ λ (0.333 m)

Answer: Far field distance df = 6 m.