Prerequisites & Notation

Before You Begin

This chapter is the communications-specific culmination of Chapters 13 and 14 on compressed sensing and sparse recovery. We assume familiarity with the basic $\ell_0$ / $\ell_1$ formulation, RIP, and at least one recovery algorithm (LASSO, OMP, or AMP). On the communications side, we use pilot-based channel estimation from Chapter 10 and the matched-filter analysis of Chapter 2.

Sparse signal model, RIP, $\ell_1$ recovery guarantees (Ch 13)(Review ch13)
Self-check: Given a $k$ -sparse signal and Gaussian sensing matrix, can you state the scaling of the number of measurements needed for stable recovery?
Recovery algorithms: LASSO, OMP, AMP (Ch 14)(Review ch14)
Self-check: Can you write down one iteration of OMP and explain its stopping condition?
Pilot-based channel estimation (Ch 10)(Review ch10)
Self-check: Can you derive the LS channel estimate from $\mathbf{y} = \boldsymbol{\Phi}\mathbf{h} + \mathbf{w}$ and its MSE?
Matched-filter detection, ROC (Ch 2)(Review ch02)
Self-check: Can you plot $P_d$ vs $P_{fa}$ for a known-signal detector?
Uniform linear array and steering vectors
Self-check: Can you write the steering vector of a ULA with half-wavelength spacing for angle $\theta$ ?

Notation for This Chapter

Symbols used throughout Chapter 15. Since pilot matrices play a central role, we follow the communications convention of denoting them by $\boldsymbol{\Phi}$ .

Symbol	Meaning	Introduced
$\boldsymbol{\Phi} \in \mathbb{C}^{M \times L}$	Pilot matrix / sensing matrix (rows = pilot observations, columns = delay taps or angular bins)	s01
$\mathbf{h} \in \mathbb{C}^{L}$	Sparse channel vector in delay or angular domain ( $s$ -sparse with $s \ll L$ )	s01
$s$	Sparsity level: number of nonzero entries of $\mathbf{h}$ or $\mathbf{x}$	s01
$M$	Number of measurements (pilot symbols, snapshots, or pilot length)	s01
$K_{\text{total}}$ , $K_a$	Total number of users and number of active users in massive access	s02
$\mathbf{a}(\theta)$	Array steering vector at angle $\theta$	s03
$\\|\mathbf{x}\\|_{2,1}$	Group $\ell_{2,1}$ -norm: $\sum_{g} \\|\mathbf{x}_g\\|_2$ with $g$ indexing groups	s04
$\\|\mathbf{x}\\|_\mathcal{A}$	Atomic norm induced by a set $\mathcal{A}$ of atoms	s03

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