Prerequisites & Notation

Before You Begin

This chapter builds the central theoretical framework of the book. It assumes familiarity with the Born approximation (Ch 05) and connects wave physics to the linear algebra of imaging. The reader should be comfortable with Fourier transforms, array signal processing, and basic electromagnetic scattering.

Born approximation and the linearized scattering integral(Review ch05)
Self-check: Can you write the scattered field as a linear functional of the contrast function under the Born approximation?
Green's functions in free space (2D and 3D)(Review ch05)
Self-check: Can you write the free-space Green's function in 3D and explain its physical meaning?
Fourier transforms (continuous and discrete)(Review ch04)
Self-check: Can you state the Fourier transform pair and the Poisson summation formula?
Array steering vectors and beamforming(Review ch07)
Self-check: Can you write the steering vector for a UPA and explain matched-filter beamforming?
OFDM signal model and subcarrier structure(Review ch14)
Self-check: Can you describe how OFDM decomposes a wideband channel into narrowband subchannels?
Linear algebra: Kronecker products and SVD(Review ch01)
Self-check: Can you compute a Kronecker product and state what the SVD reveals about a matrix?

Notation for This Chapter

Symbols used throughout this chapter. Many are introduced in Caire's "On the Illumination and Sensing Model for RF Imaging" and carried throughout the rest of the book.

Symbol	Meaning	Introduced
$\mathbf{s}_{i}$	Transmitter $i$ position	s01
$\mathbf{r}_{j}$	Receiver $j$ position	s01
$\mathbf{p}$	Target / voxel position; $\mathbf{p}_{0}$ = reference center of target region	s01
$\Omega$	Target region in space	s01
$c(\mathbf{p})$	Complex reflectivity function	s01
$\mathbf{c}$	Discretized reflectivity vector ( $Q \times 1$ )	s02
$\mathbf{A}$	Sensing / measurement matrix ( $MNK \times Q$ )	s02
$\mathbf{y}$	Observation vector: $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$	s02
$\kappa$	Wavenumber: $\kappa = 2\pi/\lambda = 2\pi(f_0 + f)/\text{c}$	s01
$\boldsymbol{\kappa}_s$	Transmitter wavenumber vector	s03
$\boldsymbol{\kappa}_r$	Receiver wavenumber vector	s03
$\kappa_{\mathbf{s},\mathbf{r}}$	Combined Tx-Rx wavenumber: $\kappa_{\mathbf{s},\mathbf{r}} = \boldsymbol{\kappa}_s + \boldsymbol{\kappa}_r$	s03
$\tilde{c}(\boldsymbol{\kappa})$	Spatial Fourier transform of the reflectivity	s04
$G^{\text{tx}}, G^{\text{rx}}$	Transmit and receive antenna gains	s02
$\tau_{i,j}$	Round-trip propagation delay for Tx $i$ , Rx $j$	s02
$\hat{\mathbf{c}}^{\text{BP}}$	Backpropagation (matched-filter) image estimate	s02
$Q$	Number of voxels (pixels) in the discretized scene	s02
$M, N, K$	Number of transmitters, receivers, and frequency subcarriers	s02

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