Chapter Summary

Chapter Summary

Key Points

• 1.

  Master scipy.stats for distribution work. Every distribution provides a unified API: pdf, cdf, rvs, fit, ppf. Use scipy.stats for standard distributions and the Cholesky decomposition for generating correlated Gaussian vectors: $\mathbf{x} = \mathbf{L}\mathbf{z}$ where $\mathbf{R} = \mathbf{L}\mathbf{L}^T$. For complex Gaussians, divide by $\sqrt{2}$ to get unit variance.

• 2.

  Always report confidence intervals. A bare BER number is scientifically meaningless. The CI width scales as $1/\sqrt{N}$: halving the CI requires 4x the samples. Use Clopper-Pearson exact intervals for small error counts and bootstrap for non-standard statistics. Run at least 100 errors per SNR point.

• 3.

  Vectorize and validate your Monte Carlo simulations. Process all bits as NumPy arrays, never in Python loops (100-1000x speedup). Use np.random.default_rng() with SeedSequence.spawn() for reproducible parallel simulations. Always validate against a known theoretical result (BPSK AWGN) before simulating complex systems.

• 4.

  Generate channels with correct statistics. Rayleigh = $\mathcal{CN}(0,1)$; Ricean adds a LOS component scaled by $\sqrt{K/(K+1)}$; Kronecker MIMO uses $\mathbf{H} = \mathbf{L}_r \mathbf{H}_w \mathbf{L}_t^T$. Always verify envelope distribution, mean power, spatial correlation, and temporal autocorrelation ($J_0$ for Jakes).

• 5.

  Use variance reduction for low BER. For BER below $10^{-4}$, importance sampling with exponential tilting provides exponential speedup. Antithetic variates give a free 2x improvement. Control variates work when a correlated quantity with known mean is available. Monitor effective sample size to detect weight degeneracy.