From Born Approximation to the Discrete Sensing Matrix

The target region $\Omega$ is discretized onto a uniform grid of $Q$ voxels (pixels in 2D), with centers $\{\mathbf{p}_{q} : q = 1, \ldots, Q\}$ and voxel volume $\Delta V = \text{Vol}(\tilde{\Omega})/Q$. The discretized reflectivity vector is

$$\mathbf{c} = \left\{\frac{\text{Vol}(\tilde{\Omega})}{Q} \, c(\mathbf{p}_{q}) : q = 1, \ldots, Q \right\} \in \mathbb{C}^Q,$$

where $\tilde{\Omega} = \Omega - \mathbf{p}_{0}$ is the target region translated to the origin.

Consider a system with $M$ transmitters $\mathcal{T} = \{\mathbf{s}_{1}, \ldots, \mathbf{s}_{M}\}$, $N$ receivers $\mathcal{R} = \{\mathbf{r}_{1}, \ldots, \mathbf{r}_{N}\}$, and $K$ frequency subcarriers $\mathcal{F} = \{f_1, \ldots, f_K\}$. The scene is discretized into $Q$ voxels.

Define the transmitter steering vector and receiver steering vector for subcarrier $k$ and voxel $q$:

$$\mathbf{a}_{k,q} = \left\{\frac{\sqrt{G^{\text{tx}}_{i}} \, e^{-j2\pi(f_0 + f_k)\tau_i} \, e^{-j\boldsymbol{\kappa}_{i,k}^\mathsf{T} \mathbf{p}_{q}}}{d(\mathbf{s}_{i}, \mathbf{p}_{0})} : i = 1, \ldots, M\right\},$$

$$\mathbf{b}_{k,q} = \left\{\frac{\sqrt{G^{\text{rx}}_{j}} \, e^{-j2\pi(f_0 + f_k)\hat{\tau}_j} \, e^{-j\hat{\boldsymbol{\kappa}}_{j,k}^\mathsf{T} \mathbf{p}_{q}}}{d(\mathbf{p}_{0}, \mathbf{r}_{j})} : j = 1, \ldots, N\right\},$$

where $\tau_i = d(\mathbf{s}_{i}, \mathbf{p}_{0})/\text{c}$ and $\hat{\tau}_j = d(\mathbf{p}_{0}, \mathbf{r}_{j})/\text{c}$.

The sensing matrix $\mathbf{A} \in \mathbb{C}^{MNK \times Q}$ is constructed by stacking:

$$\mathbf{A} = [\mathbf{A}_{1}; \mathbf{A}_{2}; \ldots; \mathbf{A}_{K}], \qquad \mathbf{A}_{k} = [\mathbf{a}_{k,1} \otimes \mathbf{b}_{k,1}, \ldots, \mathbf{a}_{k,Q} \otimes \mathbf{b}_{k,Q}].$$

The observation vector $\mathbf{y} \in \mathbb{C}^{MNK}$ satisfies

$$\boxed{\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}},$$

where $\mathbf{w} \sim \mathcal{CN}(\mathbf{0}, \sigma^2\mathbf{I})$ is AWGN.

The standard baseline image formation is backpropagation (BP):

$$\hat{\mathbf{c}}^{\text{BP}} = \mathbf{A}^{H} \mathbf{D}^{-1} \mathbf{y},$$

where $\mathbf{D}$ is a diagonal matrix that compensates for the distance-dependent power gains:

$$\mathbf{D} = \mathbf{I}_K \otimes \text{diag}\!\left(\frac{G^{\text{tx}}_{i}}{d(\mathbf{s}_{i}, \mathbf{p}_{0})^2} : i = 1, \ldots, M\right) \otimes \text{diag}\!\left(\frac{G^{\text{rx}}_{j}}{d(\mathbf{p}_{0}, \mathbf{r}_{j})^2} : j = 1, \ldots, N\right).$$

Backpropagation coherently sums all measurements after rephasing and compensating for the known propagation losses. It is the adjoint (matched filter) applied to the data.

To operate the model $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$ at a well-defined SNR, Caire normalizes the reflectivity so that

$$\max_q \left(\frac{\text{Vol}(\tilde{\Omega})}{Q} |c(\mathbf{p}_{q})|\right)^2 = 1.$$

With this convention, the receiver SNR at each OFDM subcarrier for the $(i,j)$ Tx-Rx pair is

$$\text{SNR}_{i,j,k} = \frac{G^{\text{tx}}_{i} G^{\text{rx}}_{j}}{d(\mathbf{s}_{i}, \mathbf{p}_{0})^2 \, d(\mathbf{p}_{0}, \mathbf{r}_{j})^2} \cdot \frac{\mathcal{E}_k}{N_0},$$

where $\mathcal{E}_k$ is the transmit energy per symbol on subcarrier $k$ and $N_0$ is the noise PSD. For a target minimum SNR $\rho_{\text{dB}}$, the required transmit power is

$$\left(\frac{\mathcal{E}}{N_0}\right)_{\!\text{dB}} \geq \rho_{\text{dB}} + \max_{i,j}\bigl\{20\log_{10} d(\mathbf{s}_{i}, \mathbf{p}_{0}) + 20\log_{10} d(\mathbf{p}_{0}, \mathbf{r}_{j}) - (G^{\text{tx}}_{i})_{\text{dB}} - (G^{\text{rx}}_{j})_{\text{dB}}\bigr\}.$$