# Differential Ultra-Wideband Microwave Imaging: Principle Application Challenges
## 2 Signal Model for Small Time-Variant Scattering Objects
### Time-Variant Objects in Multi-Path Environment
A more serious effect comes from the jitter, which leads to the bias term \(\dot{b}_{S}(t)\,\varphi_{j}\) and the random term \(\dot{b}_{S}(t)\Delta_{\underline{T}}\). The bias term is independent of the observation time and thus will be less critical for the target detection (note that \(\varphi_{i}\) is independent of \(t\) and \(T\)). The opposite is valid for the random term. \(\Delta_{\underline{T}}\) is random in \(t\) and \(T\). Furthermore, its random effect on \(c(t,T)\) is weighted by the first derivative \(\dot{b}_{S}(t)\) of the very strong signal scattered from the static objects. Under strong multipath conditions, these signals are typically spread over the whole radar range. Figure 11B gives an example of signal spreading even under simple conditions. To get a better impression of the process of dying out, the data are logarithmically scaled by keeping the signal sign. The scaling function is given by \(b_{\text{log}}=\text{sign}(b)\cdot(\max[20\text{lg}(b/b_{\max}),-D]+D)\). \(D\) [dB] is the dynamic range over which the data are depicted. The strength of the background is often 2-3 orders of magnitude larger than the target reflections and with increasing bandwidth of the radar, the signal derivation leads to an additional emphasis of their influence. Hence, jitter may seriously affect the detection performance of a time-variable target under strong multipath conditions (especially if it is at the same range as a strong target--compare Figure 10--person, and file cabinet). UWB radar experiments for motion detection under (nearly) free space conditions are therefore less trustworthy. So far, we have assumed that the clutter objects are static. In a situation as depicted in Figure 10B, this cannot be strictly presumed, because a living organism can never suppress completely the motion of its limbs. Hence, by referring to the signal model in Equation (30), we also have to take into account a minor time variance of the signal component \(b_{s}(t)\to b_{s}(t,T)\). Under the condition that all motion effects are small (i.e., time-delay modulations are smaller than the rise time of the sounding signal), we can follow the approach introduced with respect to Equation (19). That is, we pick out data samples at different propagation timepoints \(t_{k}\) and observe their magnitude dependent on the observation time \(b(t_{k},T)\) or \(c(t_{k},T)\). The modulation of the majority of data samples will follow the global motion of the scenario under testing conditions (e.g., the arm), and only few data samples will also contain a  
Figure 11: Illustration of multipath influence onto radar data: (**A**) Radar data of a moving person after background removal. The horizontal signal trace represents the third path illustrated in Figure 10: antenna to person, forward scattering of person to the back wall and back to the antenna. (**B**) Radar data after background removal gained from moved sheet metal. The experiment was done in an ordinary laboratory space with a 12 th order M-sequence device (total length of pulse response 307 ns). (**C**) Measurement done with 8 th order M-sequence device showing a too short unambiguity range (length of pulse response 19 ns; see also Section 3.3). modulation caused by the target of interest. The goal is to separate both modulations and to extract the target. Under the assumption that both modulations are independent or orthogonal, respectively, this can be done by principal component analysis (PCA), which exploits singular value decomposition (SVD) [40]. Figure 12 gives an example. Assuming, we want to measure the artery motion in the arm. It is approximated in our example by the sinusoidal modulation \(\chi(T)\) (Figure 12A). Due to the vital motion of the body, we get an additional, irregular modulation \(\zeta(T)\), which is even stronger than the signal of interest. Both modulations overlap in the radar signal, but they are connected with slightly different roundtrip times.