Chapter 3: Multipath Propagation

1. Foundations: Random Processes

Why characterize signals?

Wireless signals encounter obstacles causing path loss, shadowing, and fading. We use statistical models to predict behavior.

Path Loss: Power reduction over distance.
Shadowing: Obstruction losses (Large-scale).
Multipath Fading: Interference from multiple paths.

Wide Sense Stationarity (WSS)

Crucial for simplifying mathematical models.

Condition 1: Mean

\mathcal{M}_X(t) = E[X(t)] = \text{Constant}

Condition 2: Autocorrelation

Depends only on time difference ($\tau$).

R_{xx}(t, t+\tau) = E[x(t) x(t + \tau)] = R_{xx}(\tau)

2. Classification of Signal Variation

Signal Variation

Large-Scale Effects (>100m)

Outdoor Pathloss Models

• Okumura-Hata Model
• Break Point Model

Propagation Mechanics

Reflection

Diffraction

Scattering

Shadowing

Statistical Modeling (Log-Normal)

Small-Scale Effects (10m-20m)

Multipath Fading

NLOS (No Line of Sight) Rayleigh Fading

LOS (Line of Sight) Ricean Fading

Time Dispersion

No Time Dispersion Flat Fading

With Time Dispersion Freq-Selective

3. Shadowing & Outage Probability

Log-Normal Shadowing

The received power in dB follows a Gaussian (Normal) distribution.

P_r(d) [\text{dB}] = \overline{P_r}(d) - X_{\sigma}

Where $X_\sigma \sim \mathcal{N}(0, \sigma^2)$

Outage Probability & Coverage

Outage Probability

Prob. that received power < $P_{min}$.

P_{outage} = Q\left(\frac{\overline{P_r}(d) - P_{min}}{\sigma}\right) = Q\left(\frac{\beta}{\sigma}\right)

Acceptance Probability (Coverage)

Prob. of good signal quality.

P_{accept} = 1 - P_{outage} = Q\left(-\frac{\beta}{\sigma}\right)

Example Calculation

Margin ($\beta$): 10 dB
Sigma ($\sigma$): 10 dB
Acceptable Signal: 84%
Since $Q(1) \approx 0.16$, Coverage = $1 - 0.16$

Common Q-Function Values
x	0	1	2	3
Q(x)	0.5	0.159	0.023	0.001

4. Geometric Effect: Two-Ray Model

$g_1, \tau_1$

$g_2, \tau_2$

$\tilde{s}(t)$

$d$

$b-d$

Interference Pattern

The received signal is a sum of the direct path and the reflected path (which has a 180° phase shift).

Transfer Function

\tilde{H}(f) = g_1 e^{-j2\pi f\tau_1} - g_2 e^{-j2\pi f\tau_2}

Coherence Bandwidth

The frequency range where the channel is "flat".

\Delta f = \frac{1}{\Delta \tau} = \frac{c}{2(b-d)}