Fading Channels

Fading Channels: Multipath Propagation and Signal Variation

Fading channels arise from multipath propagation where signals reach receivers via multiple paths with different delays and attenuations. Understanding fading is critical for wireless system design.

Multipath Propagation

Signal reaches receiver via multiple paths:

1. Direct line-of-sight
2. Ground reflection
3. Diffraction over obstacles
4. Scattering from vegetation/buildings

Received signal: y(t) = Σ_k a_k(t) × e^(-jφ_k(t)) × x(t - τ_k)

• a_k(t) = time-varying amplitude of k-th path
• τ_k = delay of k-th path
• φ_k(t) = phase shift of k-th path

Small-Scale vs. Large-Scale Fading

Large-scale fading:

• Caused by path loss and shadowing
• Varies over 30-100 m distance scale
• Autocorrelation distance: 40-100 m
• Slow variation: predictable, suitable for power control

Small-scale fading:

• Caused by constructive/destructive interference
• Varies over distances comparable to wavelength (λ/2 ≈ 17 cm at 900 MHz)
• Rapid variation: difficult to predict
• Can create deep nulls (20-30 dB fades)

Delay Spread

τ_rms = √(E[τ²] - (E[τ])²)

• RMS delay spread: 0.1-10 μs typical
• Urban: 1-10 μs (severe multipath)
• Rural: 0.1-1 μs (light multipath)
• Indoor: 50-200 ns (light multipath)

Impact on data rate: If symbol period T_s < 5×τ_rms, intersymbol interference severe Maximum data rate: R_max ≈ 0.2/τ_rms

Coherence Bandwidth

B_c ≈ 1/(5×τ_rms)

• Frequency range where channel is approximately flat
• f separated by > B_c experience independent fading
• If signal BW < B_c: flat fading (easier)
• If signal BW > B_c: frequency-selective (requires equalization)

Doppler Spread

Maximum Doppler shift: f_d = f_c × v/c

• v = velocity (m/s)
• c = speed of light
• f_c = carrier frequency

At 900 MHz, 100 km/h: f_d ≈ 83 Hz

Rayleigh Fading Distribution

Amplitude: a ~ Rayleigh(σ) PDF: p(a) = (a/σ²) × exp(-a²/(2σ²))

Probability signal drops below threshold: P(a < A) = 1 - exp(-A²/(2σ²))

Rician Fading Distribution

K-factor = line-of-sight power / scattered power

• K → ∞: Gaussian (free space)
• K >> 1: Near-Gaussian with occasional fades
• K ≈ 1: Significant fading
• K ≈ 0: Rayleigh fading

Interview Q&A

Q1: What is the difference between small-scale and large-scale fading? A: Large-scale fading (path loss + shadowing) varies over 30-100 m, predictable via models, used for power control and coverage planning. Small-scale fading (multipath interference) varies over half-wavelengths (~17 cm at 900 MHz), rapid and difficult to predict, causes deep fades creating error floors if not mitigated by diversity or equalization.

Q2: Explain delay spread and its impact on data rate. A: Delay spread τ_rms is RMS spread of multipath arrival times. If symbol period T_s < 5×τ_rms, previous symbols' echoes overlap current symbol (severe ISI). Maximum data rate approximately 0.2/τ_rms. Urban areas (1-10 μs): max rate 20-200 kbps unequalized. With equalization, can approach Shannon limit.

Q3: What is coherence bandwidth and why does it determine signal processing architecture? A: Coherence bandwidth B_c ≈ 1/(5×τ_rms) is frequency range where channel fades are correlated. If signal BW < B_c: entire signal fades together (frequency-flat fading, single gain variation). If signal BW > B_c: different frequencies fade independently (frequency-selective fading, requires subcarrier equalization). OFDM subcarrier spacing > 1/B_c ensures frequency-flat fading per subcarrier.

Q4: Calculate Doppler spread for a 2 GHz system with 100 km/h mobility. A: f_d = f_c × v/c = 2×10^9 × (100/3.6)/3×10^8 ≈ 185 Hz. Coherence time ≈ 1/(5×f_d) ≈ 1 ms. If symbol period = 10 μs, ~100 symbols pass in coherence time (channel appears stationary). If symbol period = 100 μs, ~10 symbols (channel rapidly varying). This determines equalization/channel estimation strategy.

Q5: Explain Rayleigh fading envelope statistics and fade depth interpretation. A: Rayleigh distribution models amplitude of signal with only scattered components (no LOS). P(a < 0.1×RMS) ≈ 1%, but outage probability much higher. For 10% outage (90% availability): A = 0.34×σ. This means typical signal ±3 dB, but 10% of time < -10 dB (deep fade). Link margin must account for these probabilities; fade margin 10-15 dB typical for 90-99% availability.

Q6: How does K-factor affect channel severity and what K-value indicates different scenarios? A: K-factor = LOS power / scattered power. K >> 10: Nearly free space (rare deep fades). K ≈ 3: Typical outdoor urban (moderate fading). K ≈ 0.5: Severe fading. K ≈ 0: Pure Rayleigh (worst). Higher K means more predictable channel (less deep fades), but blocked LOS creates discontinuity. Mobile receiver experiences K variation: K high in line-of-sight, K ≈ 0 in deep urban canyon.