Prerequisites & Notation

Prerequisites

This chapter builds on the unified forward model developed in Chapter 6 and uses linear algebra tools from Telecom Ch 01 and optimization concepts from Telecom Ch 03.

Unified forward model: $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$ (Review ch06)
Self-check: Can you write out the sensing matrix $\mathbf{A}$ entry-by-entry from the Born approximation?
Kronecker product definition and properties ( $\otimes$ )(Review ch01)
Self-check: Can you compute $(\mathbf{B} \otimes \mathbf{C})\text{vec}(\mathbf{X}) = \text{vec}(\mathbf{C}\mathbf{X}\mathbf{B}^T)$ ?
Singular value decomposition (SVD)(Review ch01)
Self-check: Can you state the SVD of a rectangular matrix and its relationship to eigenvalues of $\mathbf{A}^{H}\mathbf{A}$ ?
Condition number and ill-posedness(Review ch02)
Self-check: What does $\kappa(\mathbf{A}) \gg 1$ imply about reconstruction stability?
Iterative optimization algorithms (gradient descent, proximal methods)(Review ch03)
Self-check: What is the step-size constraint for gradient descent on $\|\mathbf{A}\mathbf{c} - \mathbf{y}\|^2$ ?
Wavenumber-domain analysis and k-space coverage(Review ch06)
Self-check: How does each Tx-Rx-frequency triple map to a point in k-space?

Chapter Notation

Key symbols introduced or used throughout this chapter. Symbols from earlier chapters retain their meaning; new symbols specific to the Kronecker analysis are marked below.

Symbol	Meaning	Introduced
$\mathbf{A}$	Sensing matrix, $\mathbb{C}^{M \times N}$
$\mathbf{A}_{\text{Tx}}$	Transmit spatial factor of the sensing matrix
$\mathbf{A}_{\text{Rx}}$	Receive spatial factor of the sensing matrix
$\mathbf{A}_{f}$	Frequency (range) factor of the sensing matrix
$\mathbf{c}$	Discretized reflectivity vector, $\mathbb{C}^N$
$\mathbf{y}$	Observation vector, $\mathbb{C}^M$
$\mathbf{w}$	AWGN noise vector
$\kappa(\mathbf{A})$	Condition number $\sigma_{\max}/\sigma_{\min}$
$\mu(\mathbf{A})$	Mutual coherence of $\mathbf{A}$
$\mathbf{G} = \mathbf{A}^{H}\mathbf{A}$	Gram matrix (point-spread function in operator form)
$N_t$	Number of transmit antennas
$N_r$	Number of receive antennas
$N_f$	Number of frequency bins (subcarriers)
$\kappa$	Wavenumber $2\pi/\lambda$
$\otimes$	Kronecker product
$\text{vec}(\cdot)$	Vectorization operator (column-stacking)

← Ch 6 Kronecker Product Structure of the Sensing Matrix