Chapter Summary

Chapter Summary

Key Points

• 1.

  Use the right path loss model for your scenario. Free-space for satellites, log-distance for general analysis ($n = 2$-$5$), and 3GPP TR 38.901 for standards-compliant 5G simulations. Shadow fading adds $\mathcal{N}(0, \sigma_{\text{SF}}^2)$ dB fluctuations with typical $\sigma = 4$-$8$ dB.

• 2.

  Generate fading with correct statistics. Rayleigh: h = (randn + 1j*randn)/sqrt(2) for unit power. Ricean: add a LOS component scaled by $\sqrt{K/(K+1)}$. Always verify np.mean(|h|^2) = 1. Use the Jakes spectral shaping for time-correlated fading.

• 3.

  MIMO channels need spatial correlation. Use the Kronecker model $\mathbf{H} = \mathbf{L}_r\mathbf{H}_w\mathbf{L}_t^T$ with exponential correlation matrices. Correlation reduces capacity: keep antenna spacing $\ge \lambda/2$ for low correlation.

• 4.

  Fading fundamentally changes BER behavior. Rayleigh fading degrades BPSK BER from $Q(\sqrt{2E_b/N_0})$ to $\approx 1/(4\bar\gamma)$ at high SNR: a catastrophic loss. Diversity order $L$ restores the slope to $1/\bar\gamma^L$.

• 5.

  Visualize channel properties for insight. Plot the PDP, Doppler spectrum, and frequency response. Compute $\tau_{\text{rms}}$ and $B_c$ to guide OFDM design: the CP must exceed $\tau_{\max}$ and each subcarrier bandwidth must be less than $B_c$.