# Differential Ultra-Wideband Microwave Imaging: Principle Application Challenges
## 2 Signal Model for Small Time-Variant Scattering Objects
### Time-Variant Objects in Multi-Path Environment
By considering both noise terms separately, Equation (29) has to be modified to the following:  
\[b(t,T)=b_{S}\bigg{(}t+\Delta_{\overline{\tau}_{j}}\bigg{)}+b_{T}\bigg{(}t+\Delta_ {\overline{\tau}_{j}},T\bigg{)}+b_{ST}\bigg{(}t+\Delta_{\overline{\tau}_{j}},T \bigg{)}+\underline{u}(t,T)\,. \tag{30}\]  
The propagation delay is omitted in the signal components for the sake of a shorter notation. Due to the ergodicity of the additive noise, the randomness of the sampling procedure does not influence its statistical properties, where the jitter \(\Delta_{\overline{\tau}_{j}}\) is omitted in the noise term \(\underline{u}(t,T)\). In order to suppress the static paths, one needs to know \(b_{S}(t)\). It can either be determined from measurements where the target is still absent or it is estimated by averaging over the captured data. In both cases, one performs an integration in observation time in order to reduce the noise influence. So, we can approximately write the following:  
\[\overline{b_{S}(t)}\approx\frac{1}{T_{0}}\underset{T_{0}}{\int}b(t,T)\text{d} T\approx b_{S}(t)*p_{\Delta\tau}(t). \tag{31}\]  
Equation (31) assumes that the variable target parts are canceled out due to a sufficiently long integration. Furthermore, it shows that the time shape of the static signal is slightly modified by \"a low pass filter\" whose IRF is given by the PDF \(p_{\Delta\tau}(t)\) of the jitter. However, this becomes only remarkable if the standard deviation \(\varphi_{j}\) of the jitter is on the order of the rise time of the signals. The additive noise in Equation (31) is omitted, because it is largely suppressed by the averaging. In practical implementation, the integration in Equation (31) may be performed over the whole captured data set or even by slighting averaging or low pass filtering the observation time. The signal \(\overline{b_{S}(t)}\) is often assigned as background. By subtracting \(\overline{b_{S}(t)}\) from the measured data, the result after some manipulation is as follows:  
\[ \begin{array}{ll}c(t,T)&=b(t,T)-\overline{b_{S}(t)}\\ &\approx b_{S}(t)\,*(\delta(t)-p_{\Delta\tau}(t))+\dot{b}_{S}(t)\Delta_{ \overline{\tau}_{j}}+b_{T}(t,T)+\dot{b}_{T}(t,T)\Delta_{\overline{\tau}_{j}} +b_{ST}(t,T)+\dot{b}_{ST}(t,T)\Delta_{\overline{\tau}_{j}}+\underline{u}\\ &\approx\dot{b}_{S}(t)\,\varphi_{j}+\dot{b}_{S}(t)\Delta_{\overline{\tau}_{j} }+b_{T}(t,T)+b_{ST}(t,T)+\underline{u}\,.\end{array}  \tag{32}\]  
The second line in Equation (32) comes from a Taylor series expansion of Equation (30) and the assumption of a not significantly large jitter, so higher terms of the series may be neglected. Further on, the third line ignores the jitter-affected time-variant signals, because they are of very low magnitude, and \(p_{\Delta\tau}(t)\) is approximated by the first element of its Taylor series. As we can observe from Equation (32), we get after background removal the wanted signal \(b_{T}(t,T)\), but it is still bothered by other components. One of them is the multipath component \(b_{ST}(t,T)\). It is illustrated in Figure 11. Figure 11A is based on the same scenario as already discussed in Figure 6, but now it refers to the completely captured data set. Because the person was walking in a room, it created a shadow on the wall opposite to the antenna, which may have the same strength as the wanted signal. However, the signal from the shadow has a larger propagation time compared to the direct target reflection. This gives us the opportunity to single out the signal from the shadow and the direct target reflection. This will work better with a larger bandwidth of radar, because due to the better range resolution, targets close to the wall may be separated from their shadows more easily. Then, we still have the additive noise \(\underline{u}\) in Equation (32) affecting the detection performance. This noise has been treated in many previous papers and will hence not be considered here in detail.