Prerequisites & Notation

Prerequisites for This Chapter

This chapter develops the matched filter (backpropagation) estimator and its extensions -- the simplest and most widely used approaches to RF imaging. These methods form the baseline against which all reconstruction algorithms in Parts IV--VI are compared.

The forward model $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$ and the sensing operator(Review ch08)
Self-check: Can you write down the $(m,k)$ entry of $\mathbf{A}$ for a multi-static stepped-frequency radar?
Array signal processing and beamforming(Review ch07)
Self-check: Can you derive the Capon (MVDR) beamformer for a ULA?
k-space coverage and the Ewald sphere(Review ch06)
Self-check: How does each Tx-Rx-frequency triple map to a point in $\mathbf{k}$ -space?
Compressed sensing basics: sparsity model and sensing matrix properties(Review ch12)
Self-check: What does the RIP constant $\delta_s$ measure?

Notation and Conventions

Key symbols used throughout this chapter. All notation follows the conventions established in Ch 08 for the forward model.

Symbol	Meaning	Introduced
$\mathbf{y} \in \mathbb{C}^M$	Measurement vector
$\mathbf{c} \in \mathbb{C}^N$	Discretized reflectivity vector (scene to recover)
$\mathbf{A} \in \mathbb{C}^{M \times N}$	Sensing (forward) matrix
$\hat{\mathbf{c}}^{\text{BP}}$	Backpropagation / matched filter image estimate
$\mathbf{G} = \mathbf{A}^{H} \mathbf{A}$	Gram matrix (discrete PSF)
$\mathbf{w} \sim \mathcal{CN}(0, \sigma^2 \mathbf{I})$	Circularly-symmetric complex Gaussian noise
$\mathbf{s}_{i}$	Position of transmitter $i$
$\mathbf{r}_{j}$	Position of receiver $j$
$\mathbf{p}_{q}$	Position of voxel (pixel) $q$
${\tau_{i,j}}_{i,j}(\mathbf{p})$	Round-trip delay from Tx $i$ to target $\mathbf{p}$ to Rx $j$
$\Delta_r = c/(2B)$	Range resolution
$\Delta_{\text{cr}} = \lambda / (2D_{\text{eff}})$	Cross-range resolution

← Ch 12 The Matched Filter / Backpropagation Estimator