Channel Models

Channel Models: Mathematical Models for Communication Channels

Channel models allow prediction of signal degradation and system performance. This guide covers free space, path loss, shadowing, fading, and composite models.

Free Space Path Loss

Signal power decreases with distance due to spreading:

Path loss (dB) = 20 × log₁₀(d) + 20 × log₁₀(f) + 20 × log₁₀(4π/c) - G_tx - G_rx

Simplified Friis equation: P_r = P_t × G_t × G_r × (λ/(4πd))²

• P_r, P_t = received, transmitted power
• G_t, G_r = transmit, receive gains
• λ = wavelength = c/f
• d = distance
• Exponent = -2 (path loss exponent for free space)

Large-Scale Path Loss Models

Okumura-Hata Model (empirical, 150 MHz - 1.5 GHz): Path loss = 69.55 + 26.16×log₁₀(f) - 13.82×log₁₀(h_tx) - C_h + (44.9 - 6.55×log₁₀(h_tx))×log₁₀(d)

• f = frequency (MHz)
• h_tx = transmitter height (m)
• C_h = terrain correction
• d = distance (km)

Shadowing Model

Slow fading due to obstacles (trees, buildings):

• Log-normal distribution: P_r = P_r0 × 10^(-χ/10)
• χ ~ N(0, σ²), σ = 4-13 dB typical
• Autocorrelation distance: 30-100 m typical

Small-Scale Fading

Rayleigh Fading:

• No line-of-sight component
• Multiple scatterers equally likely
• Envelope ~ Rayleigh distribution
• Phase ~ uniform

Rician Fading:

• Dominant line-of-sight + scattered components
• K-factor = (Line-of-sight power) / (scattered power)
• K >> 1: Nearly free space
• K ≈ 0: Rayleigh fading

Fading Channel Characterization

Delay Spread: τ_rms = time dispersion (1-10 μs urban) Coherence Bandwidth: B_c ≈ 1/(5×τ_rms) (frequency selectivity) Doppler Spread: f_d = v/λ × max(cos(θ)) (mobility) Coherence Time: T_c ≈ 1/(5×f_d) (temporal selectivity)

Composite Model Example

Doppler Effect in Mobile

Received frequency shift: f_r = f_t × (c + v_r)/(c + v_t) Maximum Doppler shift: f_d = f_c × v/c

At 900 MHz, 100 km/h: f_d ≈ 83 Hz

Interview Q&A

Q1: How does free space path loss differ from real-world propagation? A: Free space assumes no obstacles, ideal antenna radiation patterns, line-of-sight. Real world has: buildings (shadowing), terrain, multipath reflections. Empirical models (Okumura-Hata, Cost-231) fit measurements showing path loss exponent 2.5-4 typical (vs. 2 free space), higher in urban (more obstacles).

Q2: Explain log-normal shadowing and its significance. A: Slow fading from obstruction by buildings/terrain follows log-normal distribution with 4-13 dB standard deviation. Creates ~ ±10 dB slow variations over 30-100 m distance. Fade margin of 10-15 dB required for system reliability. Link budget planning requires accounting for shadowing, not just path loss.

Q3: What is K-factor in Rician fading and what does it mean? A: K-factor = line-of-sight power / scattered power ratio. High K (>10): Fading minimal, nearly free space. K ≈ 1: LOS and scattered equal. K ≈ 0: Pure scattering (Rayleigh). K indicates how much LOS dominance affects channel—higher K means more predictable channel (less deep fades), but also higher path loss when LOS blocked.

Q4: Why is coherence bandwidth important for modulation selection? A: Coherence bandwidth B_c ≈ 1/(5×delay_spread) represents frequency range over which channel is nearly flat. If signal bandwidth < B_c, frequency-flat fading (single fade affects all spectrum). If signal bandwidth > B_c, frequency-selective fading (some frequencies fade, others pass). OFDM bandwidth is partitioned into subcarriers with spacing > 1/B_c, ensuring frequency-flat fading per subcarrier (simplifies equalization).

Q5: Explain Doppler spread and coherence time. A: Doppler spread f_d = v/λ represents maximum frequency shift due to mobility. Coherence time T_c ≈ 1/(5×f_d) is duration over which channel appears stationary. If symbol period >> T_c, fading changes within symbol (time-selective fading, severe). If symbol period << T_c, fading constant across symbol (easier to equalize). Channel stationarity time limits how long channel estimate is valid.

Q6: Describe a typical urban wireless channel model combining path loss, shadowing, and fading. A: Composite model: Received power = [Free space path loss] + [Log-normal shadowing] + [Rayleigh small-scale fading]. At 1 km: Free space ≈ -80 dBm, Shadowing ±8 dB (70-90% likely), Fading ±20 dB deep nulls. Total variation: 60-100 dBm range possible. Link budget needs margin for all three, not just average path loss.