# Differential Ultra-Wideband Microwave Imaging: Principle Application Challenges
## 2 Signal Model for Small Time-Variant Scattering Objects
### Invariant Object in Free Space and Its Localization
In a mathematically generalized form, inspired from the p-norm, this may be expressed in different ways, as for example by (\(p=0,1,2,3,\cdots\)):  
\[I_{1,p}(\mathbf{r}_{v}) = \sqrt{\int\limits_{-\infty}^{\infty}w(t)\sum\limits_{i=1}^{K} \sum\limits_{j=1}^{K}\left(h_{ji}(t)*b_{ji}(t+\tau_{v})\right)^{p}}\bigg{|} \mathrm{d}t \tag{4}\] \[I_{2,p}(\mathbf{r}_{v}) = \sqrt{\int\limits_{-\infty}^{\infty}w(t)\sum\limits_{i=1}^{K} \sum\limits_{j=1}^{K}\left|h_{ji}(t)*b_{ji}(t+\tau_{v})\right|^{p}}\mathrm{d}t\] \[\tau_{v} = \frac{|\mathbf{r}_{v}-\mathbf{r}_{v}|+|\mathbf{r}_{v}-\mathbf{r} _{v}|}{2\tau_{a}}+2\tau_{a}.\]  
Figure 2: Arbitrary distributed antenna array with single target in free space. If the voxel position coincides with the target position \(\mathbf{r}_{v}=\mathbf{r}_{q}\), the signals are coherently superimposed, leading to a large intensity value. In the opposite case (i.e., \(\mathbf{r}_{v}\
eq\mathbf{r}_{q}\)), the signals are incoherently added, so the voxel intensity tends to small values. For the suppression of measurement errors, sidelobes, or other image defects, it may be meaningful to modify every signal \(b_{ji}(t)\) by an individual weighting function \(h_{ji}(t)\) before summing (see e.g., [11; 12; 13; 14; 15] for examples). Furthermore, in Equation (4), \(w(t)\) represents a gating function whose duration is on the order of the width of the stimulus pulse. This ensures that only the desired signal sections are added up. In Equation (4), for \(p\), typically 1 or 2 are also assigned as a delay-and-sum approach, which is widely used in the literature. Figure 4A illustrates an intensity plot based on the delay-and-sum approach for a simple two-dimensional (2D)-scenario with a five-element linear antenna array. Obviously, the method provokes many sidelobes, which limit the contrast of the image. To improve the contrast, the number of antennas has to be increased or other methods of signal superposition have to be applied. Equations (5) and (6) depict two examples. The first method, we call the delay-and-multiply approach. A related result is demonstrated in Figure 4B. Obviously, the contrast is dramatically increased, because the sidelobes have disappeared. Nevertheless, the method needs some care, because an unwanted zero in the data, which actually should be superimposed, will \"destroy\" the related target. \[I_{3}(\mathbf{r}_{v})=\int\limits_{-\infty}^{\infty}w(t)\prod_{i}\prod_{j} \bigl{|}h_{ji}(t)*b_{ji}(t+\tau_{v})\bigr{|}\mathrm{d}t\,. \tag{5}\]  
Finally, the routine Equation (6) superimposes a set of cross-correlation functions--also assigned as delay-multiply-and-sum--determined from the captured response signals [16; 17; 18; 19; 20]. \[ \begin{array}{l}I_{4}(\mathbf{r}_{v})=\left|\begin{array}{l}\int\limits_{- \infty}^{\infty}w(t)\quad\sum\limits_{\begin{array}{l}k=1\\ i=1\end{array}}^{K}\quad\sum\limits_{\begin{array}{l}k=1\\ j=1\end{array}}^{K}\quad\sum\limits_{\begin{array}{l}k=1\\ i\
eq k,l\end{array}}^{K}h_{jit}(t)*b_{ji}\Bigl{(}t+\tau_{v}^{(ji)}\Bigr{)}*b_ {lk}\Bigl{(}-t-\tau_{v}^{(lk)}\Bigr{)}\mathrm{d}t\\ \tau_{v}^{(ji)}=\frac{|\mathbf{r}_{i}-\mathbf{r}_{v}|+|\mathbf{r}_{i}-\mathbf{ r}_{v}|}{\tau_{v}}+2\tau_{u}\tau_{v}^{(lk)}=\frac{|\mathbf{r}_{i}-\mathbf{r}_{v}|+| \mathbf{r}_{i}-\mathbf{r}_{v}|}{\tau_{v}}+2\tau_{u}.\end{array}\right|\\ \end{array}  \tag{6}\]  
Figure 3: Typical signals for an idealized scenario with electrically small antennas and a point scatterer. Antenna \(i\) is stimulated with a Gaussian pulse, while all other antennas only act as receiver \(a_{j}=0\); \(j\
eq i\). Method 2: Instead of superimposing the measured signals as demonstrated in Equations (4)-(6), one may also try to numerically solve the problem of target localization. If we consider a single measurement \(b_{ji}(t)\) (compare Figure 3) or even a cross-correlation function between signals \(b_{ji}(t)\) and \(b_{lk}(t)\), we can identify the propagation time of arrival (ToA) and the propagation time difference of arrival (TDoA), respectively, between the involved antennas and the target.