# Differential Ultra-Wideband Microwave Imaging: Principle Application Challenges
## 2 Signal Model for Small Time-Variant Scattering Objects
### Time-Variant Objects in Multi-Path Environment  
The experimental situation is roughly illustrated in Figure 10 by two examples. One of them is with mostly free space propagation, real static objects (wall, furniture), and a target, which moves over distances much larger than the radar range resolution. The second example deals with minor  
Figure 9: Cross-energy matrix for two independent measurements of a single moving target. motion detection (much smaller than the radar range resolution) and a wave propagation in a lossy environment, which is not very stable, because a person is not able to keep limbs truly motionless. We first restrict ourselves to a stable propagation environment, as depicted in Figure 10A. If threefold and higher order reflections are omitted, we can identify four different types of propagation paths. The first type 1 refers to all paths, which only involve static objects. The second 2 is linked with the object of interest, and the third 3 contains a twofold scattering, one by the target and one by a static object. The forth transmission path 4 symbolizes the transmission behavior of the wall, which affects the antenna signal (e.g., by multiple wall reflections). Merging all this together, the received signal for a single antenna arrangement may be modeled in simplified form as:  
\[ \begin{split} b(t,T)&=T(t)*\Xi(t)*(\Lambda_{S}(t- \tau_{S})+\Lambda_{T}(t-\tau_{T}(T),T)+\Lambda_{S}(t-\tau_{ST})*\Lambda_{T}(t,T) )\\ &*\Xi(t)*R(t)*a(t)+\underline{v}(t,T)\\ &=b_{S}(t-\tau_{S})+b_{T}(t-\tau_{T}(T),T)+b_{ST}(t-\tau_{ST}(T), T)+\underline{v}(t,T)\,.\end{split}  \tag{29}\]  
Here, we neither respect any angular dependency of the impulse response function nor the range influence (i.e., wave spreading and attenuation) of the signal magnitude or the polarization of the electric fields. Without loss in general, we only involve one static and one time-variable target. The different symbols stand for antenna impulse responses \(T(t)\), \(R(t)\), wall transmission \(\Xi(t)\), scattering behavior of the static object \(\Lambda_{S}(t)\), and scattering of time-variable object \(\Lambda_{T}(t,T)\). The symbol \(\tau\) indicates the related path propagation time. In Equation (2), the path delay was respected by convolution with Dirac functions. Here, it is part of the arguments of the function to shorten the notation. In the bottom line of Equation (29), the different components of the transmission paths are merged into different functions. \(b_{S}(t)\) symbolizes all transmission paths (including multiple reflections), which do not change in observation time. This part represents the strongest component of the measured signal. It is often orders of magnitude larger than the other components. \(b_{T}(t,T)\) refers to the time-variable target, in which we are actually interested, and \(b_{ST}(t,T)\) represents multipath components, which involve the time-variant target. \(b_{S}(t)\) and \(b_{ST}(t,T)\) are often assigned as clutter. They must be removed from the measured signal. Before we do that, the noise term in Equation (29) must be accounted for more seriously, because this will be important for the estimation of the clutter reduction. As already mentioned in Equation (17), the randomness of the measurement is caused by amplitude noise (additive noise) \(\underline{n}\sim N\big{(}0,\sigma_{n}^{2}\big{)}\) and \"time\" noise (i.e., sampling jitter) \(\Delta\underline{\tau}_{j}\sim N\big{(}0,\varphi_{j}^{2}\big{)}\). In well-designed receiver electronics, both can be  
Figure 10: Illustration of time-variant multipath scenario. (**A**) Through-wall radar to detect humans behind walls; (**B**) Medical microwave imaging to detect artery pulsation (e.g., in the upper arm of a human). assumed to be Gaussian distributed, white, independent, and ergodic.