# Differential Ultra-Wideband Microwave Imaging: Principle Application Challenges
## 3 Device Requirements for Differential Imaging
### Random Effects  
As already mentioned above, UWB devices are subjected to two types of randomness. These are the uncertainties of voltage capturing due to thermal noise and quantization effects symbolized by \(\underline{n}\) and the uncertainties (jitter) of the sampling time \(\Delta_{\underline{\tau}_{j}}\). Assuming a frequency-independent noise power spectral density \(\Phi_{n}\), the power of the additive noise \(P_{n}=\text{var}\Big{\{}\underline{n}\Big{\}}=\sigma_{n}^{2}\approx\Phi_{n} B\) will rise with the bandwidth \(B\). The measures to combat against the impact of this noise are to increase the average power of the sounding signal and to emphasize the deterministic measurement effects against the noise by an appropriate integration over a long observation time (e.g., synchronous averaging; matched filtering--compare Equation (20) and Figures 8 and 9). The impact of sampling jitter is illustrated in Figure 14. Sampling jitter causes data capturing at incorrect and unknown time instances. Because the captured voltage has to be assigned to the pre-defined sampling time \(t_{0}\), the timing uncertainty transforms into amplitude noise dependent on the signal slope. The integral effect of the jitter on the signal quality may be expressed by the signal-to-noise ratio relating the average signal power \(P\) to the jitter induced power \(P_{j}\) as follows:  
\[SNR_{j}=\frac{P}{P_{j}}=\frac{\frac{\frac{1}{\tau_{R}}\int b^{2}(t)\text{d}t}{ \frac{\sigma_{n}^{2}}{\tau_{R}}\int\limits_{\tau_{R}}\hat{b}^{2}(t)\text{d}t} }{\frac{\sigma_{n}^{2}}{\tau_{R}}\int\limits_{\tau_{R}}\hat{b}^{2}(t)\text{d}t }=\frac{\int\limits_{-\infty}^{\infty}\left|\underline{B}(f)\right|^{2}\text{ d}f}{4\pi^{2}\varphi_{j}^{2}\int\limits_{-\infty}^{\infty}f^{2}| \underline{B}(f)|^{2}\text{d}f}\approx\frac{3}{\pi^{2}\varphi_{j}^{2}B^{2}}. \tag{39}\]  
\(\left|\underline{B}(f)\right|^{2}\) represents the power spectrum of the sounding signal. Applying [PERSON]'s theorem and the differentiation rule of Fourier transform and assuming for the sake of simplicity a constant spectrum of the sounding signal within the band limits \(-B/2\cdots B/2\), one finds that the jitter effect is independent of the actual time shape and the power of the sounding signal. This result is confirmed by simulations and measurements in [42]. However, this approach obstructs the view of the actual situation. Jitter-induced noise is non-ergodic. Therefore, its actual dependency on time should be considered for better understanding of its impact. As is obvious from Figure 14, jitter develops its effect mainly on signal edges. Hence, in the case of pulse radar, the whole jitter power is concentrated at the edges of strong signal parts. It remains the question how the output signal \(c_{bm}(t)\) of an M-sequence radar (see Equation (35)) is affected by jitter and noise:As we can observe from Equation (40), the expected value is a bias-free estimation of the wanted correlation function, and the total noise depends on the strength of the received signal due to the jitter. Unlike the pulse radar, the noise level is not focused on signal edges but is evenly distributed throughout the signal (i.e., it is converted into additive noise, which depends on the signal level and bandwidth (due to differentiation)). Additionally, the noise level decays with increasing length \(N_{m}\) of the M-sequence. The maximum noise level appears if there is a direct connection between the receiver and transmitter (i.e., \(b_{0}(t)=m_{b}(t)\)). Assuming again constant spectral power within the stimulation band \(-B/2\cdots B/2\) or \(-f_{c}/2\cdots f_{c}/2\), respectively, we find the following:  
\[\begin{array}{rl}\left.