# 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
These properties of the raw (i.e., unfiltered) data for mono- and bi-static channels are summarized in the following (Gaussian distributed; white; uncorrelated):  
\[ \begin{split}\underset{\mu}{\
u}(t,T)&\sim N\big{(} 0,\sigma_{v}^{2}(t)\big{)}\\ \text{cov}\bigg{\{}\underset{\mu}{\
u}(t_{1},T),\
u_{ii}(t_{2},T) \bigg{\}}=\left\{\begin{array}{c}\sigma_{v}^{2}(t);t_{1}=t_{2}=t\\ 0;t_{1}\
eq t_{2}\end{array}\right.;\text{cov}\bigg{\{}\underset{\mu}{\
u}(t,T _{1}),\
u_{ii}(t,T_{2})\bigg{\}}=\left\{\begin{array}{c}\sigma_{v}^{2}(t);T_ {1}=T_{2}\\ 0;T_{1}\
eq T_{2}\end{array}\right.\\ \text{cov}\bigg{\{}\underset{\mu}{\
u}(t);t=k;j=l\\ 0;t\
eq k;j\
eq l\,.\end{split}  \tag{18}\]  
Because in the case of weak targets, the modulation effects will also be quite weak, the major challenge will be to detect the time-variable targets under noise conditions. Many methods are investigated with that goal [29; 30; 31; 32; 33; 34; 35; 36; 37]. Here, for the sake of brevity, we will only consider the most effective method, which is based on a matched filter concept. For that purpose, we pick out the data  
Figure 7: Weakly sinusoidal moving point scatterer. (**A**) Section of the time-variant receiving signal for the noise-free case. The variation of some data samples captured at propagation time \(t_{1}\cdots t_{5}\) is emphasized. (**B**) Time variance of the selected data samples (DC-value ignored). samples captured at an arbitrary time point \(t_{0}\) and call the related signal \(x(T)\). Removing the DC-value \(\overline{b_{ii}(t_{0})}\), \(x(T)\) may be decomposed into an observation time-dependent modulation function \(\chi(T)\) and a propagation time-dependent value \(x_{0}(t_{0})\) that determines the strength of the modulation dependent on the sample position as follows:  
\[ \begin{array}{ll}x(T)&=x_{0}(t_{0})\,\chi(T)+\underset{x}{\
u}=b_{ii}(t_{0}, T)-\overline{b_{ii}(t_{0})}\\ &=\left\{\begin{array}{rl}\frac{\
u}{x}&\text{case 1}\\ b_{ii}(t_{0})\cdot\chi(T)+\underset{x}{\
u}&\text{case 2;}\chi(T)=\Delta\Lambda(T)/ \Lambda_{0}\\ \dot{b}_{ii}(t_{0})\cdot\chi(T)+\underset{x}{\
u}&\text{case 3;}\chi(T)=2 \Delta r_{iq}(T)/c\\ \ddot{b}_{ii}(t_{0})\cdot\chi^{2}(T)+\underset{x}{\
u}&\text{case 4:}\chi(T)= \sqrt{2}\Delta r_{iq}(T)/c\,.\end{array}\right.\end{array}  \tag{19}\]  
In case 1, the selected time sample is located within a flat part of the receiving signal, so we get only noise (\(x_{0}=0\)). In case 2, the time sample is located close to a signal peak, and the time variance is caused from a variation of the target reflectivity. Cases 3 and 4, refer to a weakly moving target if the sampling time is placed on a signal edge (case 3; i.e., one of the blue signals in Figure 7B) or close to a signal peak (case 4; i.e., one of the red signals in Figure 7B). Case 4 is mostly out of interest and will not be considered further. In order to suppress the noise, the deterministic part \(x_{0}(t_{0})\,\chi(T)\) in Equation (19) has to be emphasized. Assuming we know approximately the time shape of the modulation function, except for an unknown time delay \(\zeta(T)\approx\chi(T-\tau_{0})\), this is the case when the test scenario may be externally modulated by the operator or, in the case of heartrate and breathing detection, where one can assume an approximately sinusoidal modulation. From the assumed modulation function and the measurement data, one can establish a summation over the observation time period \(T_{0}=N\,\Delta T\) (\(\Delta T\) = repetition interval between consecutive measurements, see Figure 7; \(N\) = number of repetitions), which finally represents a cross-correlation function as follows:  
\[C_{x_{0}^{\zeta}}(k)=\sum_{n=1}^{N}x(n\Delta T)\zeta(n\Delta T-k\Delta T)= \sum_{n=1}^{N}\bigg{(}x_{0}(t_{0})\chi(n\Delta T)+\underset{x}{\
u}\bigg{)} \zeta(n\Delta T-k\Delta T)\,. \tag{20}\]  
Using Equation (18), its expected value and variance result in the following:  
\[