# 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
In order to approach realistic conditions, we have also added receiver noise now, which is modeled by the random process \(\
u_{i\bar{i}}(t,T)\). Developing Equation (15) in a Taylor series and omitting higher terms and cross-terms yields the following (using \(\dot{\Phi}=\mathrm{d}\Phi/\mathrm{d}t\)):  
\[ \begin{split} b_{ii}(t,T)&\approx\frac{\Lambda_{0} }{r_{iq,0}}\left[\underbrace{\Phi(\xi)}_{\mathrm{A}}+\underbrace{\left(\frac{ \Delta\Lambda(T)}{\Lambda_{0}}-\frac{\Delta r_{iq}(T)}{r_{iq,0}}\right)}_{ \mathrm{B}}\Phi(\xi)+\underbrace{\frac{2}{c}\dot{\Phi}(\xi)\Delta r_{iq}(T)}_{ \mathrm{C}}+\right.\\ &\left.+\underbrace{\frac{2}{c^{2}}\ddot{\Phi}(\xi)\big{(}\Delta r _{iq}(T)\big{)}^{2}}_{\mathrm{D}}+\cdots\right]+\
u_{i\bar{i}}(t,T)\,.\end{split}  \tag{16}\]  
As seen from Equation (16), the radar response \(b_{ii}(t,T)\) is composed from several components. Term A represents the observation time-independent part (i.e., the response of the target in a static state). Term B is an amplitude modulation of the channel response, where the second term \(\Delta r_{iq}/r_{iq,0}\) is usually negligible compared to the effect in terms C and D. Terms C and D are caused from time-delay modulations. The modulation term C is dominant at the signal edges (large first derivative) of the received signals. It is proportional to the target motion \(\Delta r_{iq}(T)\). The modulation term D mainly appears in the region of peak values of the receiving signal, because there the magnitude of the second derivative is maximum. This term is usually of minor interest, because it is less sensitive than term C and it provides only the squared target motion. Figure 7 illustrates both terms for a weak sinusoidal target motion. Figure 7A picks out the signal section, which is located around the target reflection (compare Figure 3 for the complete signal). In practice, the received signal is sampled. Hence, we  
Figure 6: Data of a walking person measured from a mono-static radar. (**A**) radargram; (**B**) target range; and (**C**) time variance of target backscattering. know it only at discrete timepoints. Five such sampling points \(t_{1}\cdots t_{5}\) are selected, and their variation in observation time is emphasized. Two points at \(t_{1}\), \(t_{5}\) are placed on a falling signal edge, one point at \(t_{3}\) is located at the rising edge, and two points at \(t_{2}\), \(t_{4}\) are to be found close to signal peaks. Figure 7B plots the observation time variation of the different samples. One can observe the following:  
* the largest modulation is provided by the sample located at the steepest part of the received signal;
* the modulations at rising and falling edges are inverted; and
* the modulation at the signal peaks has double frequency (due to the squaring) and is quite weak. For motionless targets, where only its reflectivity \(\Delta\Lambda\) is time variant, the modulation term B is responsible. It provides the largest contribution at the signal peaks. In the case of properly designed measurement receivers, which are typically based on sub-sampling, the receiver noise \(\underset{ij}{\
u}(t,T)\) of the different measurements may be considered as Gaussian distributed with a white spectrum in both \(t\) and \(T\), as well as uncorrelated between different response functions and measurement channels. Moreover, as will be seen in the next section, the strength of the noise will depend on the received signal itself, and its variance becomes dependent on propagation time:  
\[\text{var}\{\
u_{ii}(t,T)\}=\sigma_{v}^{2}(t)=\sigma_{n}^{2}+\left(\dot{b}_{ii }(t)\right)^{2}\varphi_{j}^{2}, \tag{17}\]  
(i.e., the noise increases at steep signal edges). Here, \(\sigma_{n}^{2}\) characterizes the variance of the additive random effect as thermal and quantization noise, while \(\varphi_{j}^{2}\) refers to the variance of the sampling jitter, which can be considered as \"time noise\".