Chapter Summary

Chapter 16 Summary

Key Points

• 1.

  AirComp model (§16.1): users transmit $x_k = b_k s_k$ synchronously over a Gaussian MAC; receiver observes $\sum_k h_k b_k s_k + \mathbf{w}$; magnitude alignment $b_k h_k = \eta$ gives $\hat{y} = \sum_k s_k + \mathbf{w}/\eta$ — aggregate in one channel use.

• 2.

  Zero-forcing MSE (§16.2): $\mathsf{MSE}^{\star} = \sigma^2/\min_k \gamma_k$ where $\gamma_k = |h_k|^2 P_k / \sigma_s^2$; the weakest user sets the MSE. Threshold scheduling traces the Pareto frontier between MSE and user count.

• 3.

  Nomographic functions (§16.3): $f = \psi(\sum_k \varphi_k(s_k))$ — means, weighted sums, geometric means, moments, smooth-max. Kolmogorov-Arnold guarantees every continuous aggregate has a nomographic representation. Non-linear $\psi$ introduces Jensen bias.

• 4.

  Native privacy (§16.4): MAC superposition hides individual $s_j$; per-user MI bound is $O(1/n)$ under single-antenna honest-but-curious server. AirComp amplifies Gaussian dither by $\sqrt{n}$ — communication-efficient DP.

• 5.

  Misalignment floor: carrier-phase error $\phi_{\max}$ produces SNR-independent MSE floor $\sigma_s^2(1 - \mathrm{sinc}^2(\phi_{\max}))$. Budget $\phi_{\max} \leq 17°$ for $3$-dB floor.