Prerequisites & Notation

Prerequisites for This Chapter

This chapter applies sparse recovery algorithms to the RF imaging model $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$ . The algorithms themselves (ISTA, FISTA, ADMM) were derived in RFI Ch 04 and Telecom Ch 03; here we focus on RF-imaging-specific applications, parameter selection strategies, and extensions including group sparsity, total variation, greedy methods, and gridless super-resolution.

Proximal gradient methods (ISTA, FISTA)(Review RFI Ch 04 / Telecom Ch 03)
Self-check: Can you write the FISTA update for the composite objective $f + h$ ?
ADMM derivation and convergence(Review RFI Ch 04 / Telecom Ch 03)
Self-check: Can you derive the ADMM updates for $\min f(\mathbf{x}) + g(\mathbf{z})$ s.t. $\mathbf{x} = \mathbf{z}$ ?
RIP and LASSO recovery guarantees(Review RFI Ch 11 / FSI Ch 13)
Self-check: Can you state the RIP-based bound for LASSO estimation error?
RF sensing matrix Kronecker structure(Review RFI Ch 08)
Self-check: Can you explain how Kronecker structure accelerates $\mathbf{A}\mathbf{c}$ and $\mathbf{A}^{H}\mathbf{y}$ ?
Matched filter baseline(Review RFI Ch 13)
Self-check: Why does the matched filter $\mathbf{A}^{H}\mathbf{y}$ produce sidelobe artifacts?

Notation and Conventions

Notation used throughout this chapter. All algorithms operate on the linear imaging model $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$ .

Symbol	Meaning	Introduced
$\mathbf{c}$	Discretized reflectivity vector (scene to recover)
$\mathbf{y}$	Measurement vector
$\mathbf{A}$	Sensing / measurement matrix
$\lambda$	Regularization parameter
$\mathbf{w}$	Additive noise vector
$\sigma^2$	Noise variance
$\mathcal{S}_{\lambda}(\cdot)$	Soft-thresholding operator at level $\lambda$
$L_f$	Lipschitz constant of $\nabla f$ , equal to $\sigma_{\max}^2(\mathbf{A})$
$\rho$	ADMM penalty parameter
$\nabla_D$	Discrete gradient operator (finite differences)
$\\|\cdot\\|_{\text{TV}}$	Total variation semi-norm
$\\|\mathbf{X}\\|_{2,1}$	Mixed $\ell_{2,1}$ norm (group / row sparsity)
$\\|\cdot\\|_{\mathcal{A}}$	Atomic norm
$\mathcal{H}_s(\cdot)$	Hard-thresholding operator keeping $s$ largest entries

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