Prerequisites & Notation

Before You Begin

Chapter 16 opens Part V. AirComp exploits the wireless multiple-access channel's built-in superposition property to compute the sum (or more general nomographic function) of users' values in a single channel use — bypassing the per-user digital upload that dominates secure aggregation (Chapter 10) and wireless FL (Chapter 17). Prerequisites span basic wireless communication (Gaussian MAC, fading, power control) and Part III's federated learning aggregation problem.

Gaussian multiple-access channel (MAC)(Review ch14)
Self-check: Write the signal model for $n$ users transmitting over a Gaussian MAC; identify what the receiver observes.
Channel inversion / power control(Review ch06)
Self-check: Given per-user channel gain $h_k$ , what transmit scaling equalizes received magnitudes at the access point?
Aggregation goal in FL (§9.1, §10.1)(Review ch09)
Self-check: Why does the server need only $\sum_k \mathbf{g}_k$ — not individual $\mathbf{g}_k$ ?
Minimum mean-squared error estimation(Review ch05)
Self-check: State the MMSE estimator for a Gaussian signal in AWGN.
Basic complex Gaussian $\mathcal{CN}$ notation(Review ch10)
Self-check: Recall: $\mathcal{CN}(0, \sigma^2)$ has real and imaginary parts i.i.d. $\mathcal{N}(0, \sigma^2/2)$ .

Notation for This Chapter

AirComp notation draws from wireless ( $h_k$ , noise, power) and from FL (gradient $\mathbf{g}_k$ ). The central quantity is the aggregation MSE.

Symbol	Meaning	Introduced
$n$	Number of users (transmitters) sharing the MAC	s01
$x_k \in \mathbb{C}$	Symbol transmitted by user $k$ (pre-processed from source value)	s01
$h_k \in \mathbb{C}$	Complex channel gain from user $k$ to the access point	s01
$b_k \in \mathbb{C}$	Transmit scaling (power-control coefficient) at user $k$	s02
$\mathbf{w}$	Receiver additive noise, $\mathcal{CN}(0, \sigma^2)$	s01
$P_k$	Per-user transmit power budget, $\mathbb{E}[\|b_k x_k\|^2] \leq P_k$	s02
$\eta$	Common receive amplitude target, $b_k h_k = \eta$	s02
$\hat{y}$	Receiver estimate of $\sum_k x_k$	s01
$\mathsf{MSE}$	Aggregation mean-squared error, $\mathbb{E}[\|\hat{y} - \sum_k x_k\|^2]$	s02
$f(x_1, \ldots, x_n)$	Target nomographic function (§16.3)	s03
$\text{SNR}_{k}$	Per-user receive SNR, $\|h_k\|^2 P_k / \sigma^2$	s02

← Ch 15 The AirComp Model