# Differential Ultra-Wideband Microwave Imaging: Principle Application Challenges
## 2 Signal Model for Small Time-Variant Scattering Objects
### Invariant Object in Free Space and Its Localization
Based on the known wave speed, the propagation distance may be estimated, so a quadric surface (sphere, ellipsoid, elliptic hyperboloid) can be calculated on which the target is localized. By repeating this for different antenna positions, the intersections of all quadric surfaces finally gives the actual target position. In what follows, the localization of a single target shall be illustrated. For the sake of demonstration, we restrict ourselves to mono-static measurements \(b_{ii}(t)\) solely, which provide us the target range via roundtrip time measurement \(r_{iq}=\tau_{iq}c/2\) (refer to Figure 3). Corresponding to the measurement setup in Figure 2, one finds:  
\[r_{iq}^{2}=\big{(}\mathbf{r}_{q}-\mathbf{r}_{i}\big{)}^{T}\big{(}\mathbf{r}_{ q}-\mathbf{r}_{i}\big{)}={r_{q}}^{2}+r_{i}^{2}-2\mathbf{r}_{i}^{T}\mathbf{r}_{q}, \tag{7}\]  
where \(\mathbf{r}_{i}\) and \(\mathbf{r}_{q}\) represent \([3,1]\) position vectors. In order to remove \(r_{q}^{2}\) from the equation, the results from two different antennas are subtracted. For an antenna array of \(K\geq 4\) antennas, this leads to a set of \(P=K!/(2(K-2)! )\geq 6\) combinations, which are arranged in following manner:  
\[\left[ \begin{array}{c}(\mathbf{r}_{1}-\mathbf{r}_{2})^{T}\\ (\mathbf{r}_{1}-\mathbf{r}_{3})^{T}\\ \vdots\\ (\mathbf{r}_{i}-\mathbf{r}_{j})^{T}\\ \vdots\end{array} \right]\mathbf{r}_{q}=\frac{1}{2}\left[ \begin{array}{c}(r_ {1}^{2}-r_{2}^{2})-\left(r_{1q}^{2}-r_{2q}^{2}\right)\\ (r_{1}^{2}-r_{3}^{2})-\left(r_{1q}^{2}-r_{3q}^{2}\right)\\ \vdots\\ \left(r_{i}^{2}-r_{j}^{2}\right)-\left(r_{iq}^{2}-r_{jq}^{2}\right)\\ \vdots\end{array} \right]=\mathbf{R}\,\mathbf{r}_{q}=\frac{1}{2}\mathbf{Q}. \tag{8}\]  
Here, \(\mathbf{R}\) represents a \([P,3]\) matrix of the known antenna position vectors, \(\mathbf{r}_{q}\) is the \([3,1]\) column vector of the wanted target position, and \(\mathbf{Q}\) is a \([P,1]\) column vector built from known antenna distances and the measured target distances. The minimum least square solution of the overdetermined Equation (8) for the target position is given by the following:  
\[\mathbf{r}_{q}=\frac{1}{2}\Big{(}\mathbf{R}^{T}\mathbf{R}\Big{)}^{-1}\mathbf{ R}^{T}\mathbf{Q}. \tag{9}\]  
This type of solution is optimal if the measurement errors of the involved quantities are Gaussian distributed. The squaring of \(r_{iq}\) in Equation (8) leads, however, to a non-Gaussian error distribution, and Equation (9) will then provide a biased solution. Figure 4: Intensity plot of a two-dimensional (2D)-scenario with two point scatterers of equal reflectivity. (**A**) Delay-and-Sum approach. (**B**) Delay-and-Multiply method. In order to avoid the bias, a maximum likelihood approach can be applied, which aims to maximize the probability density function (PDF) in Equation (10) with respect to the target position \(\mathbf{r}_{q}\):  
\[p(\mathbf{r}_{p}|\mathbf{D})=\tfrac{\mathrm{e}^{-\frac{1}{2}(\mathbf{D}_{0}- \mathbf{D})^{T}\Sigma^{-1}(\mathbf{D}_{0}-\mathbf{D})}}{\sqrt{\det(2\pi\Sigma)} }\Rightarrow\mathbf{r}_{q}=\underset{\mathbf{r}_{p}}{\operatorname{argmax}} \{p(\mathbf{r}_{p}|\mathbf{D})\}=\underset{\mathbf{r}_{p}}{\operatorname{ argmin}}\Big{\{}(\mathbf{D}_{p}-\mathbf{D})^{T}\Sigma^{-1}(\mathbf{D}_{p}- \mathbf{D})\Big{\}}. \tag{10}\]  
Here again, one subdivides the volume to be observed in voxels assigned by their position \(\mathbf{r}_{p}\). The distances of a voxel to all antenna positions are summarized in the \([1,K]\) column vector \(\mathbf{D}_{v}=\left[ \begin{array}{cccc}|\mathbf{r}_{1}-\mathbf{r}_{v}|&| \mathbf{r}_{2}-\mathbf{r}_{v}|&\cdots&|\mathbf{r}_{K}-\mathbf{r}_{v}|&\end{array}  \right]^{T}\). The actual distances between the antennas and the target gained from roundtrip time measurements, we have arranged in the \([1,K]\) column vector \(\mathbf{D}=\left[ \begin{array}{cccc}r_{1q}&r_{2q}&\cdots&r_{Kq}\end{array}