Prerequisites

This chapter bridges machine learning and wireless communications. If any item below feels unfamiliar, revisit the linked material before proceeding.

Linear algebra: least squares, SVD, matrix gradients(Review ch01)
Self-check: Can you compute the gradient of a quadratic cost $f(\mathbf{x}) = \|\mathbf{A}\mathbf{x} - \mathbf{b}\|^2$ and write the closed-form minimiser $\hat{\mathbf{x}} = (\mathbf{A}^T\mathbf{A})^{-1}\mathbf{A}^T\mathbf{b}$ ?
Probability: Gaussian distributions, maximum likelihood, KL divergence(Review ch02)
Self-check: Can you write the log-likelihood of a Gaussian observation model $\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{n}$ with $\mathbf{n} \sim \mathcal{CN}(\mathbf{0}, \sigma^2\mathbf{I})$ and derive the ML estimate of $\mathbf{x}$ ?
Digital modulation: constellations, BER, signal space geometry(Review ch03)
Self-check: Can you draw the constellation for QPSK and 16-QAM, compute the average symbol energy, and explain how minimum-distance detection works?
Estimation and detection theory: MMSE, MAP, Cramer-Rao bound(Review ch09)
Self-check: Can you state the MMSE estimator $\hat{\mathbf{x}}_{\text{MMSE}} = \mathbb{E}[\mathbf{x}|\mathbf{y}]$ , explain the bias-variance trade-off, and describe when the linear MMSE equals the full MMSE?

Symbols introduced in this chapter. See also the NGlobal Notation Table master table in the front matter.

Symbol	Meaning	Introduced
$f_\theta(\cdot)$	Parametric function (neural network) with learnable parameters $\theta$	s01
$\mathcal{L}(\theta)$	Loss function to be minimised during training	s01
$\hat{\mathbf{H}}_{\mathrm{NN}}$	Neural-network channel estimate	s01
$\mathcal{S}_\theta$	Soft-thresholding operator with learnable threshold $\theta$	s02
$\mathbf{W}^{(l)}, \boldsymbol{\theta}^{(l)}$	Weight matrix and threshold at layer $l$ of an unfolded network	s02
$\pi(a\|s)$	Policy mapping state $s$ to action $a$ in reinforcement learning	s03
$Q(s,a)$	State-action value function (Q-function)	s03
$r_t$	Reward at time step $t$	s03
$\gamma$	Discount factor in the cumulative reward objective	s03
$\mathbf{w}_{\text{global}}$	Global model parameters in federated learning	s04
$C$	Number of clients (base stations) in federated learning	s04
$E$	Number of local SGD epochs per communication round	s04