Chapter Summary

Chapter Summary

Key Points

• 1.

  Build constellations systematically. Use qam_constellation(M) for rectangular QAM and psk_constellation(M) for constant-envelope PSK. Always normalize to unit energy ($E_s = 1$) for consistent SNR calculations. Gray coding ensures $P_b \approx P_s / \log_2 M$ at moderate SNR.

• 2.

  Simulate BER with statistical rigor. Run until at least 100 errors per SNR point. Report Clopper-Pearson confidence intervals. Always validate against a known theoretical result (BPSK AWGN: $P_b = Q(\sqrt{2E_b/N_0})$) before simulating complex systems.

• 3.

  The matched filter is optimal for AWGN. The matched filter $h(t) = s^*(T-t)$ maximizes output SNR to $2E_s/N_0$ by the Cauchy-Schwarz inequality. It is equivalent to a correlator receiver and forms a sufficient statistic for ML detection.

• 4.

  Higher-order modulation trades SNR for throughput. Going from QPSK to 16-QAM doubles the bits per symbol but costs about 3.8 dB in required $E_b/N_0$. Adaptive modulation selects the best trade-off based on current channel quality.

• 5.

  Vectorize everything in NumPy. Process all symbols in parallel using broadcasting: np.abs(rx[:, None] - const[None, :]) computes all distances at once. This gives 100-1000x speedup over Python loops and is essential for large-scale BER simulations.