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Deterministic Channel Models — When Physics Suffices

A deterministic channel model computes propagation from first principles. For RF imaging, deterministic models directly construct the sensing matrix $\mathbf{A}$ and are essential for high-fidelity image reconstruction. This section builds up from free-space (single bounce) to multipath and through-wall models.

Definition:
Free-Space (Single-Bounce) Forward Model

Under the Born approximation with far-field conditions, the signal received at Rx $j$ due to Tx $i$ at frequency $f_k$ is (from Chapter 6 and Caire's unified model):

$y(\mathbf{s}_{i}, \mathbf{r}_{j}; f_k) = \frac{\sqrt{G^{\text{tx}}_{i} G^{\text{rx}}_{j}}\,e^{-j\kappa_{k}(d(\mathbf{s}_{i}, \mathbf{p}_{0}) + d(\mathbf{p}_{0}, \mathbf{r}_{j}))}}{d(\mathbf{s}_{i}, \mathbf{p}_{0})\,d(\mathbf{p}_{0}, \mathbf{r}_{j})} \int_{\Omega} c(\tilde{\mathbf{p}})\, e^{-j(\boldsymbol{\kappa}_s + \boldsymbol{\kappa}_r)^T \tilde{\mathbf{p}}}\,d\tilde{\mathbf{p}} + w_{i,j,k},$

where $\tilde{\mathbf{p}} = \mathbf{p} - \mathbf{p}_{0}$ is the position relative to the scene center, and the Tx and Rx wavenumber vectors are:

$\boldsymbol{\kappa}_{s} = \kappa_{k} \frac{\mathbf{p}_{0} - \mathbf{s}_{i}}{d(\mathbf{s}_{i}, \mathbf{p}_{0})}, \quad \boldsymbol{\kappa}_{r} = \kappa_{k} \frac{\mathbf{p}_{0} - \mathbf{r}_{j}}{d(\mathbf{p}_{0}, \mathbf{r}_{j})}.$

After grid discretization, this becomes $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$ with the $(i,j,k)$ -th row of $\mathbf{A}$ encoding the round-trip phase and path loss for each voxel.

Definition:
Ray-Tracing Channel Model

Ray tracing (geometric optics) models propagation as a set of discrete rays, each characterized by departure angle, arrival angle, delay, amplitude, and phase. For a scene with $L$ propagation paths:

$h(\tau, \theta_{\text{tx}}, \theta_{\text{rx}}) = \sum_{l=1}^{L} \alpha_l\,\delta(\tau - \tau_l)\, \delta(\theta_{\text{tx}} - \theta_{\text{tx},l})\, \delta(\theta_{\text{rx}} - \theta_{\text{rx},l}),$

where $\alpha_l$ , $\tau_l$ , $\theta_{\text{tx},l}$ , $\theta_{\text{rx},l}$ are the complex amplitude, delay, departure angle, and arrival angle of the $l$ -th ray.

Each ray corresponds to a propagation mechanism:

Line-of-sight (LOS): Direct path, single-bounce Born model.
Specular reflection: Mirror-like reflection from flat surfaces.
Diffraction: Bending around edges (GTD/UTD models).
Scattering: Rough-surface or volumetric scattering.

For imaging, each ray directly maps to a column (or set of columns) in the extended sensing matrix $\mathbf{A}_{\text{full}}$ .

Theorem: Multipath Extension of the Sensing Matrix

For a scene with $N_{\text{mp}}$ multipath orders (each order corresponding to an additional bounce), the extended forward model is:

$\mathbf{y} = \sum_{p=0}^{N_{\text{mp}}} \Gamma^p\,\mathbf{A}^{(p)}\mathbf{c} + \mathbf{w} = \mathbf{A}_{\text{eff}}\mathbf{c} + \mathbf{w},$

where $\mathbf{A}^{(p)}$ is the sensing matrix for paths with $p$ reflections, $\Gamma$ is the surface reflection coefficient, and:

$\mathbf{A}_{\text{eff}} = \sum_{p=0}^{N_{\text{mp}}} \Gamma^p\,\mathbf{A}^{(p)}.$

If multipath targets have independent reflectivities, the model instead becomes:

$\mathbf{y} = [\mathbf{A}^{(0)} \;|\; \Gamma\mathbf{A}^{(1)} \;|\; \cdots]\, \mathbf{c}_{\text{ext}} + \mathbf{w},$

with $\mathbf{c}_{\text{ext}}$ stacking direct and ghost reflectivities.

Each multipath bounce adds a "ghost" copy of the scene at a reflected position. Including these ghosts in $\mathbf{A}$ prevents them from corrupting the reconstruction of the true scene.

Proof

Single-bounce case

For $p=1$ (one reflection off a flat surface at position $z_w$ with reflection coefficient $\Gamma$ ), the image method places a virtual Tx (or Rx) at the mirror position. The path length for the reflected ray is $d(\mathbf{s}_{i}, \mathbf{p}_{q}^{\text{mirror}}) + d(\mathbf{p}_{q}^{\text{mirror}}, \mathbf{r}_{j})$ , giving a modified steering vector that defines $\mathbf{A}^{(1)}$ .

General $p$-bounce

By induction, the $p$ -th bounce applies $p$ image reflections, each attenuated by $\Gamma$ . The sensing matrix $\mathbf{A}^{(p)}$ uses the $p$ -times-reflected virtual source/receiver positions.

Superposition

Since Maxwell's equations are linear, the total received signal is the superposition of all bounce orders. Collecting terms yields $\mathbf{A}_{\text{eff}} = \sum_p \Gamma^p \mathbf{A}^{(p)}$ . When we need to separate direct from ghost targets, we expand the unknown vector instead, giving the block-column form. $\blacksquare$

Definition:
Through-Wall Propagation Model

For through-wall imaging, the signal traverses one or more walls before reaching the target. Each wall introduces:

Attenuation: The transmission coefficient depends on wall material ( $\varepsilon_r$ ), thickness $d_w$ , frequency, and incidence angle $\theta$ :

$T(\theta, f) = \frac{4\eta_0\eta_w}{(\eta_0 + \eta_w)^2}\, e^{-j\kappa_{w} d_w / \cos\theta_w},$

where $\eta_w = \eta_0/\sqrt{\varepsilon_r}$ and $\theta_w = \arcsin(\sin\theta/\sqrt{\varepsilon_r})$ (Snell's law).
Refraction: The ray bends at each wall interface, shifting the apparent target position.
Dispersion: For wideband signals, the frequency-dependent wall attenuation distorts the pulse shape.

The sensing matrix incorporates these effects:

$[\mathbf{A}_{\text{tw}}]_{(i,j,k),q} = T_i(\theta_{i,q}, f_k)\, T_j(\theta_{j,q}, f_k)\,[\mathbf{A}_{\text{free}}]_{(i,j,k),q},$

where $T_i$ and $T_j$ are the wall transmission coefficients for the Tx-side and Rx-side walls.

Example: Indoor Through-Wall Imaging at 5 GHz

A radar system operates at $f_0 = 5$ GHz with bandwidth $W = 1$ GHz. The system images through a concrete wall ( $\varepsilon_r = 6$ , thickness $d_w = 20$ cm) at normal incidence.

(a) Compute the one-way wall attenuation in dB.

(b) Compute the apparent range shift due to the higher propagation speed inside the wall.

(c) If the wall has a reflection coefficient $|\Gamma| = 0.42$ , what fraction of the incident power is transmitted?

Solution

Wall attenuation

For lossless concrete at normal incidence, the power transmission coefficient is:

$|T|^2 = \frac{4\varepsilon_r}{(\sqrt{\varepsilon_r} + 1)^2} = \frac{4 \times 6}{(\sqrt{6} + 1)^2} = \frac{24}{(2.449 + 1)^2} = \frac{24}{11.90} \approx 2.02.$

Wait -- this exceeds 1, indicating we must use the full Fabry-Perot formula accounting for both interfaces and the phase through the wall. The simplified Fresnel reflection at one interface gives $|\Gamma|^2 = \left(\frac{\sqrt{6}-1}{\sqrt{6}+1}\right)^2 = \left(\frac{1.449}{3.449}\right)^2 \approx 0.176$ . So the single-interface power transmission is $1 - 0.176 = 0.824$ . Two interfaces: $|T_{\text{total}}|^2 \approx 0.824^2 = 0.679$ , i.e., about $-1.7$ dB per one-way traversal.

Range shift

Inside the wall, the propagation speed is $v = c/\sqrt{\varepsilon_r} = c/\sqrt{6}$ . The electrical path length through the wall is $d_w \sqrt{\varepsilon_r} = 0.20 \times \sqrt{6} \approx 0.49$ m, compared to the physical 0.20 m. The apparent range shift is $\Delta R = d_w(\sqrt{\varepsilon_r} - 1) = 0.20 \times 1.449 \approx 0.29$ m. Targets appear shifted by 29 cm deeper than their true position.

Power transmission with given $|\Gamma|$

Example: From Ray Tracing to the Sensing Matrix

Consider a 2D scene with one flat wall at $y = 3$ m (reflection coefficient $\Gamma = 0.7 e^{j\pi}$ ) and one point target at $(2, 2)$ m. A monostatic radar is at the origin, operating at $f_0 = 10$ GHz.

(a) Compute the LOS path delay.

(b) Using the image method, find the ghost target position.

(c) Construct the $2 \times 1$ extended sensing vector (LOS + 1-bounce) for this single measurement.

Solution

LOS path delay

Distance to target: $d = \sqrt{2^2 + 2^2} = 2\sqrt{2} \approx 2.83$ m. Round-trip delay: $\tau = 2d/c = 2 \times 2.83 / (3 \times 10^8) \approx 18.9$ ns. Phase: $\phi = -2\kappa d = -2 \times \frac{2\pi \times 10^{10}}{3 \times 10^8} \times 2.83 \approx -1186$ rad.

Ghost target via image method

The image of the target at $(2, 2)$ reflected in the wall at $y = 3$ is at $(2, 4)$ m. The round-trip distance for the reflected path is $d_{\text{ghost}} = \sqrt{2^2 + 4^2} = \sqrt{20} \approx 4.47$ m.

Extended sensing vector

The $2 \times 1$ sensing vector stacks the LOS and ghost contributions:

$\mathbf{A}_{\text{ext}} = \begin{bmatrix} \frac{e^{-j2\kappa \cdot 2\sqrt{2}}}{(2\sqrt{2})^2} \\[6pt] \Gamma \cdot \frac{e^{-j2\kappa \cdot \sqrt{20}}}{(\sqrt{20})^2} \end{bmatrix}.$

The extended unknown is $\mathbf{c}_{\text{ext}} = [\alpha, \alpha_{\text{ghost}}]^T$ . Without including the ghost column, the reconstruction would show a spurious target at $(2, 4)$ m.

Hierarchy of Deterministic Models

Model	Accuracy	Cost	Use case
Born approximation	Low-medium	$O(MQ)$	Weak scatterers, initial design
Ray tracing (GO)	Medium	$O(L \cdot K)$	Large scenes, urban environments
Physical optics	Medium-high	$O(N_s^2)$	RCS prediction, SAR simulation
FDTD/FEM/MoM	High	$O(N^4)$	Reference, complex materials

For imaging inversion, the model used to construct $\mathbf{A}$ must balance accuracy against computational cost. The Born model with Kronecker structure ([?ch07:s06]) provides the best cost-accuracy tradeoff for most scenarios. Ray tracing is used when multipath must be modeled explicitly.

Modern tools like Sionna RT (NVIDIA) provide differentiable ray tracing, enabling gradient-based optimization of sensing geometries.

Multipath Ghost Targets

Visualize how multipath reflections create ghost targets in the reconstructed image. The left panel shows the true scene with a reflecting wall; the right panel shows the back-projection image with and without multipath modeling in $\mathbf{A}$ .

Parameters

Wall reflection coefficient

|\Gamma|

0.7

Multipath order1

Historical Note: Ray Tracing — From Optics to RF

1962-present

Ray tracing has roots in geometric optics dating to the 17th century (Fermat, Huygens). Its adaptation to radio propagation began with Keller's Geometrical Theory of Diffraction (GTD, 1962), which extended geometric optics to include diffracted rays. Kouyoumjian and Pathak's Uniform Theory of Diffraction (UTD, 1974) resolved the singularities of GTD at shadow boundaries, making ray tracing practical for engineering applications. Today, GPU-accelerated ray tracers like Sionna RT process millions of rays in real time, enabling their use as differentiable forward models for optimization.

Historical Note: Through-Wall Radar Imaging

1990s-present

Through-wall radar imaging emerged from military and law-enforcement needs in the 1990s. Early systems used UWB pulses (0.5-3 GHz) to penetrate concrete and brick walls. The key challenge was not just wall attenuation but the multipath ghosts created by wall reflections. Amin's group at Villanova University developed the image-method framework for multipath mitigation that we present in this section. Modern through-wall systems combine MIMO arrays with compressed sensing to image through multiple walls at once.

Born approximation

The assumption that each scatterer interacts only with the incident field, not with fields scattered by other objects. Valid when $|\chi| \cdot \kappa a \ll 1$ , where $\chi$ is the contrast and $a$ is the object size. Enables the linear forward model $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$ .

Image method

A technique for modeling multipath reflections by placing virtual (image) sources at mirror positions relative to reflecting surfaces. Each image source generates a ghost target in the reconstructed image.

Common Mistake: Ignoring Multipath in Through-Wall Imaging

Mistake:

Using the free-space sensing matrix $\mathbf{A}$ for reconstruction in environments with strong reflecting surfaces (floors, walls).

Correction:

Include multipath paths in the extended sensing matrix $\mathbf{A}_{\text{eff}}$ or use the image method to model ghost targets. Without this, ghost targets appear at incorrect positions and can be mistaken for real targets.

Key Takeaway

Deterministic channel models build $\mathbf{A}$ from physics. The free-space Born model gives the standard linear system. Multipath extends $\mathbf{A}$ with additional columns per bounce order. Through-wall propagation adds frequency-dependent attenuation and apparent range shift. The model hierarchy (Born, ray tracing, PO, full-wave) trades accuracy for computational cost.

Deterministic Channel Models for Imaging