# 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  
The generic situation is depicted in Figure 2. The antennas and target are referred to as points in space assigned by their position vectors \(\mathbf{r}_{i}\) and \(\mathbf{r}_{q}\). We call these points radiation and scattering centers, respectively. The interaction of the involved objects--the antennas and target--with the electric field, we describe by impulse response functions (IRF) \(T_{i}(t)\), \(R_{i}(t)\) and \(\Gamma_{i}(t)\), \(\Lambda(t)\). Herein, \(t\) refers to the propagation time (often also called fast time). \(T_{i}(t)\) and \(R_{i}(t)\) are the transmission IRFs of antenna \(i\) if it works in transmitter and receiver mode, respectively. Both are linked via the reciprocity relation \(T_{i}(t)=\frac{1}{2\pi c}\frac{\mathrm{d}}{\mathrm{d}t}R_{i}(t)\), where \(c\) is the speed of light in the propagation medium. \(\Gamma_{i}(t)\) and \(\Lambda(t)\) represent reflection IRFs of the antenna feeding and the target. For an introduction into the concept of impulse response functions see [1]. Any angular dependencies and polarimetric issues of \(T_{i},\ R_{i}\) and \(\Lambda\), as well as the aperture reflection of the antenna, we will omit here for the sake of brevity. For such a scenario, the individual response functions in (1) can be expressed by the time domain Friis-formula and radar equation by (\(\delta(t)\)--Dirac delta function):\[S_{ii}(t) = \Gamma_{i}(t)*\delta(t-2\tau_{a})+\frac{1}{\tau_{ii}}T_{i}(t)*\Lambda (t)*R_{i}(t)*\delta\big{(}t-2(\tau_{iq}+\tau_{a})\big{)}\] \[S_{ji}(t) = \frac{1}{\tau_{ji}}T_{j}(t)*R_{i}(t)*\delta\big{(}t-\big{(}\tau_{ji }+2\tau_{a}\big{)}\big{)}+\frac{1}{\tau_{ii}\tau_{ji}}T_{j}(t)*\Lambda(t)*R_{i}(t )*\delta\big{(}t-\big{(}\tau_{ij}+\tau_{ji}+2\tau_{a}\big{)}\big{)} \tag{2}\] \[\tau_{iq} = \frac{\tau_{ij}}{c};r_{iq}=|\tau_{q}-\tau_{i}|;\tau_{ji}=\frac{ \tau_{ij}}{c};r_{ij}=|\tau_{i}-\tau_{j}|.\]  
Herein, the left term of \(S_{ii}\) represents the feed point reflections of the antenna, and the right term is a mono-static radar equation, while \(S_{ji}\) is derived from the Friis-formula (left term; often called crosstalk) and a bi-static radar equation (right term). Multiple reflections between antenna-antenna and antennas-target have been omitted in Equation (2). The delay times \(\tau_{a}\), \(\tau_{ji}\), \(\tau_{iq}\) refer to the propagation delay between the measurement plan and the radiation center of the antennas, the propagation time between two antennas, and the propagation time between the antenna and the target, respectively. Following (2), the response matrix \(\mathbf{S}\) can be decomposed into two parts as follows:  
\[\mathbf{S}=\mathbf{S}_{0}+\mathbf{S}_{\Lambda}, \tag{3}\]  
where \(\mathbf{S}_{0}\) only involves antenna effects (i.e., feed point reflection and crosstalk) and \(\mathbf{S}_{\Lambda}\) is called the multi-static response matrix, which covers all transmission paths including the scattering object. Figure 3 illustrates typical signals for an idealized scenario with electrically short antennas and a point scatterer. Note that the antenna IRF in transmission and receiving mode is simply the derivative \(T_{i}\sim\frac{\mathrm{d}}{\mathrm{d}t}\cdots\) and a delta function \(R_{j}(t)\sim\delta(t)\), respectively. The IRF of a point scatterer leads to a second derivative \(\Lambda(t)\sim\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\cdots\)[1]. The signal \(b_{ji}\) refers to the received signal of antenna \(j\), if antenna \(i\) is stimulated by the pulse. For the reconstruction of the target locations (i.e., the image of the scenario) from the measurements \(b_{ji}\), one has several options. _Method 1_: The volume to be observed is subdivided into a grid of voxels. In order to get the intensity value \(I_{p}\) of the voxel located at position \(\mathbf{r}_{v}\), one superimposes the signal components received by all antennas with propagation times corresponding to the related voxel-antenna distance. Note, for that purpose, the wave speed is supposed to be known.