# 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
\begin{array}{ll}\mathds{E}\big{\{}C_{x_{0}^{\zeta}}(k)\big{\}}=x_{0}(t_{0 })\sum_{n=1}^{N}\chi(n\Delta T)\zeta((n-k)\Delta T)\\ \text{var}\big{\{}C_{x_{0}^{\zeta}}(k)\big{\}}=\sigma_{v}^{2}(t_{0})\sum_{n=1 }^{N}\xi^{2}(n\Delta T)=N\sigma_{v}^{2}(t_{0})\zeta_{rms}^{2}\,.\end{array}  \tag{21}\]  
In the case of a perfect match between modulation and reference function \(\zeta(T)=\chi(T-\tau_{0})\), the signal-to-noise ratio at a given time sample \(t_{0}\) yields the following:  
\[SNR(t_{0})=\frac{\text{max}\Big{(}\mathds{E}^{2}\big{\{}C_{x_{0}^{\zeta}}(k) \big{\}}\Big{)}}{\text{var}\big{\{}C_{x_{0}^{\zeta}}(k)\big{\}}}=\frac{\big{(} x_{0}(t_{0})N\zeta_{rms}^{2}\big{)}^{2}}{N\sigma_{v}^{2}(t_{0})\zeta_{rms}^{2}}=N \zeta_{rms}^{2}\frac{x_{0}^{2}(t_{0})}{\sigma_{v}^{2}(t_{0})}. \tag{22}\]  
As seen from Equation (22), the detection performance can be arbitrarily increased by extending the integration time \(T_{0}\) (i.e., increasing \(N\)). However, for vital data detection, the modulation is not stable over time. Hence, \(T_{0}\) should not be selected too long in order to avoid de-correlation [38]. In the simplest case of sinusoidal modulation, the correlation function (20) is implemented by a Fourier-transform in observation time direction (\(\phi\)--\"observation time\" frequency) as follows:  
\[|\underline{B}_{ii}(t,\phi)|=\int\limits_{T_{0}}b_{ii}(t,T)\mathrm{e}^{- \beta T\phi T}\mathrm{d}T. \tag{23}\]  
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Figure 8 depicts an example. Obviously it is hardly possible to identify the moving target in the original radar data if it is too noisy. However, already the correlation over the short duration \(T_{0}\) makes the target visible, and further enlargement of the integration time increasingly suppresses the noise. In many cases, however, the target modulation is not predictable in advance. Hence, the question arise from where to take the reference modulation in Equation (20). One possible option is to illuminate the target from a second antenna at a slightly different position. The resulting radar data is \(b_{jj}(t,T)\). According to Equation (19), we again pick out data samples captured at \(t_{1}\) and call the related signal \(y(T)\) as follows:  
\[y(T)=b_{jj}(t_{1},T)-\overline{b_{jj}(t_{1})}=y_{0}(t_{1})\,\chi(T)+\underset {y}{\
u}.\] (24)  
Because the antennas observe the same object, the modulation function \(\chi(t)\) is identical in \(x(t)\) and \(y(t)\). Taking the cross energy from both signals, we get the following:  
\[E_{yx}(t_{0},t_{1}) = \sum_{n=1}^{N}y(n\Delta T)\,x(n\Delta T)=\sum_{n=1}^{N}\left(y_{0 }(t_{1})\chi(n\Delta T)+\underset{y}{\
u}\right)\left(x_{0}(t_{0})\,\chi(n \Delta T)+\underset{x}{\
u}\right)\] \[\mathrm{E}\{E_{yx}(t_{0},t_{1})\} = x_{0}(t_{0})\,y_{0}(t_{1})\sum_{n=1}^{N}\chi^{2}(n\Delta T)=Nx_{0 }(t_{0})\,y_{0}(t_{1})\chi^{2}_{rms}\] (25) \[\mathrm{var}\{E_{yx}(t_{0},t_{1})\} = \sum_{n=1}^{N}\mathrm{var}\bigg{\{}y_{0}\mathcal{N}_{x}+x_{0} \,\underset{y}{\
u}+\underset{y}{\  
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which gives a result of SNR only twice as poor as in the ideal case for Equation (22).  
Because the optimum sampling positions \(t_{0}\), \(t_{1}\) are not known prior, one has to run through all possible combinations leading to the cross-energy matrix, as illustrated in Figure 9. To calculate the \([M,M]\) cross-energy matrix \(\mathbf{E}_{ij}\), we assume that the radar data are given by two \([M,N]\) matrices \(\mathbf{B}_{ii}\) and \(\mathbf{B}_{jj}\) (\(M\) = number of samples in propagation time; \(N\) = number of impulse responses measured during the observation interval \(T_{0}\)). In a first step, the DC-value of every row is removed from both matrices leading to \(\widetilde{\mathbf{B}}_{ii}\) and \(\widetilde{\mathbf{B}}_{jj}\). Finally, from this, the cross-energy is calculated from the following:  
\[\mathbf{E}_{ij}=\widetilde{\mathbf{B}}_{ii}\left(\widetilde{\mathbf{B}}_{jj} \right)^{T}. \tag{27}\]  
The maximum position of that matrix gives the roundtrip times from both antennas.