# Differential Ultra-Wideband Microwave Imaging: Principle Application Challenges
## 2 Signal Model for Small Time-Variant Scattering Objects
### Time-Variant Objects and Their Emphasis from Noise  
Restricting ourselves only to the multi-static transmission matrix, its components may be expressed as follows in the case of a time-variant scenario (compare also Equation (2)):  
\[S_{\Lambda,ji}(t,T)=\frac{1}{r_{iq}(T)\,r_{jq}(T)}\Theta(t)*\Lambda(t,T)*\delta \bigg{(}t-\bigg{(}\frac{r_{iq}(T)\,+r_{jq}(T)}{c}\bigg{)}\bigg{)}. \tag{11}\]  
Here, \(T\) symbolizes the observation (slow) time. Assuming antennas at fixed positions, the time variance can only affect the target reflectivity \(\Lambda(t,T)\) and the target ranges \(r_{iq}(T)\), \(r_{jq}(T)\). Observation time-independent components in (11) are summarized by \(\Theta(t)=T_{j}(t)*R_{i}(t)*\delta(t-2\tau_{q})\). In order to capture the time variance of the scenario, it is measured in typically regular time intervals \(\Delta T\), and the measurements are displayed as functions of the observation time \(T\) (i.e., \(b(t,T)=S(t,T)*a(t)\)). The related representation is called a radargram, as exemplified in Figure 6A for a walking person, which moves toward the radar antennas and back. From these measurements, the roundtrip time \(\tau_{2}\) has to be estimated. This is not a task of an unambiguous result, because the time shape of the back scattered signal permanently changes (refer Figure 6C). As a consequence, the precision of the target position does not only depend on the range resolution of the UWB radar, but also on the variability of the target response, which is a matter of geometric shape variations during walking. In this paper, as a kind of compromise, the energetic center \(\tau_{2}\) is used to estimate the target distance as follows:  
\[\tau_{2}^{2}(T)=\frac{\int t^{2}(t,T)\mathrm{d}t}{\int p^{2}(t,T)\mathrm{d}t} \tag{12}\]  
\[r_{q}(T)=\frac{\tau_{2}(T)\,c}{2}.\]  
By removing the range influence from the radar data via Equation (13) below, one actually may observe the variability of the backscattering while walking (Figure 6C). \[\widehat{b}\left(t,T\right)=r_{q}^{2}b(t+\tau_{2}(T),T). \tag{13}\]  
In order to image the target motion in space, one can follow one of the approaches discussed in Section 2.1 for every timepoint of the observation time. In many applications, the targets are small, weakly reflecting and moving only a little. This issue, we will address below. A small-sized target leads to Rayleigh-scattering, which provokes a twofold differentiation of the incident field. Hence, in the case of a time variance, such a target may only vary the strength of the backscattering caused by variations of its permittivity, permeability, or volume. This, we will express by:  
\[\Lambda(t,T)=\left(\Lambda_{0}+\Delta\Lambda(T)\right)\frac{\mathrm{d}^{2}}{ \mathrm{d}t^{2}}\cdots \tag{14}\]  
Here, \(\Lambda_{0}\) represents the average reflection strength of the target, and \(\Delta\Lambda\) symbolizes its variable part. If the target finally also moves a little by \(\Delta r\), the receiver signals can be modeled by (15). Without loss of generality, we only refer to the mono-static components here. Joining the observation time-independent part \(\Theta(t)\) with the twofold derivative in Equation (14) \(\Phi(t)=\mathrm{d}^{2}(\Theta(t)*a(t))/\mathrm{d}t^{2}\) and insertion of Equation (14) in Equation (11), the receiving signal becomes as follows:  
\[ \begin{split} b_{ii}(t,T)&=\frac{\Lambda_{0}+ \Delta\Lambda(T)}{\left(r_{iq,0}+\Delta r_{iq}(T)\right)^{2}}\Phi(t)*\delta \left(t-\frac{2\left(r_{iq,0}+\Delta r_{iq}(T)\right)}{c}\right)+\frac{\
u}{ i\bar{i}}(t,T)\\ &=\frac{\Lambda_{0}+\Delta\Lambda(T)}{\left(r_{iq,0}+\Delta r_{iq }(T)\right)^{2}}\Phi\left(\xi-\frac{2\Delta r_{iq}(T)}{c}\right)+\frac{\
u}{ i\bar{i}}(t,T)\,.\end{split}  \tag{15}\]  
To shorten the notation, we include the delay term in the argument of \(\Phi\) and substitute \(\xi=t-2r_{iq,0}/c\), where \(r_{iq,0}=r_{iq}(T=0)\) is the range of the target at the beginning of the observation time.