# Differential Ultra-Wideband Microwave Imaging: Principle Application Challenges
## 3 Device Requirements for Differential Imaging
### Random Effects
\begin{array}{rl}\left.\begin{array}{rl}\mathrm{ var}\{c_{lm}(t)\}\end{array}\right|_{\mathrm{max}}&=\frac{1}{N_{m}}\left( \sigma_{n}^{2}+\overline{\overline{\dot{m}_{b}^{2}}}\,\varphi_{j}^{2}\right)= \frac{1}{N_{m}}\left(\sigma_{n}^{2}+\,\varphi_{j}^{2}\int|j2\pi fM_{b}(f)|^{2} \mathrm{d}f\right)\\ &\approx\frac{1}{N_{m}}\left(\sigma_{n}^{2}+\,\frac{1}{3}\left(\pi V_{0}f_{c} \varphi_{j}\right)^{2}\right).\end{array}  \tag{41}\]  
In order to stay in the digital domain, we characterize the overall sensitivity of the receiver (i.e., ADC resolution and track-and-hold noise) by its effective number of bits \(ENOB=\log_{2}(2V/\sigma_{n})\) (\(2\,V\)--ADC full scale range) so that we get from Equation (41) equality between quantization/thermal noise and jitter induced noise under the following condition:  
Figure 14: Impact of sampling jitter on data capturing. (**A**) Illustration of the effect at a single sampling point \(t_{0}\). (**B**) Measurement example. (**C**) Probability density functions of a rising signal edge. \[\varphi_{j,0}\approx\frac{\sqrt{3}}{\pi\,f_{c}2^{ENOB-1}}\,. \tag{42}\]  
This is an important design rule for M-sequence devices in order to balance between resolution, technical effort, and data rate. Assuming an operational frequency band \(0\cdots 5\,\mathrm{GHz}\) (\(f_{c}=10\,\mathrm{GHz}\)) and \(ENOB=10\,\mathrm{bit}\), the standard deviation of the jitter should not exceed \(\varphi_{j}\leq 100\,\mathrm{fs}\). This is a challenging task. Therefore, the timing concept of the M-sequence radar is specifically designed for low jitter generation (see [21]). The time reference is provided by a low-phase-noise single-tone microwave source. Other timing-critical components, such as LFSR, binary divider, and track-and-hold circuit are monolithically integrated in a high frequency SiGe-technology providing low-noise signals of steep trigger edges. Figure 15 illustrates the noise behavior of a commercial M-sequence device [43]. Figure 15A shows the pure additive noise \(\underline{\
u}\), because no signal was fed into the receiver (blue spectrum). Figure 15B,C are showing the spectrum of the observation time noise for the maximum input signal (red spectra). Figure 15B refers to a voltage sample placed at a horizontal signal part, and Figure 15C depicts the fluctuations at the steepest signal edge. The noise level in Figure 15A,B are nearly identical. Hence, from Equation (42), the sampling jitter of the device is sufficiently small, so it does not degrade the noise performance of the device. In Figure 15C, the noise level for frequencies above \(\phi\geq 0.6\,\mathrm{Hz}\) is about the same as in the two previous cases, which also confirms Equations (41) and (42). For frequencies below \(\phi<0.6\,\mathrm{Hz}\), the noise increases inversely with frequency. The actual reason for this behavior has still to be investigated. Because the measurement was done in a non-temperature-controlled room, one reason could be minor temperature variations, which affect the propagation time of the cable connecting transmitter and receiver. Thanks to the noise suppression of correlation (see Equation (40)), the remaining temporal fluctuation of an M-sequence device is typically in the range of few femtoseconds. Temperature fluctuations in the range of only one-hundredth of a degree lead already to runtime changes on RF-cables that exceed the measurement uncertainty of the device. Finally, it should be pointed to an effect that leads to an erroneous roundtrip time measurement due to noise [44]. Assuming the roundtrip time is determined by the first crossing of a threshold. The situation is illustrated in Figure 16. The rising edge of a signal is affected by random noise, whose standard deviation is indicated by error bars. A related presentation is depicted in Figure 14C showing the PDFs \(p_{b(i\Delta t)}(V)\) of the voltage samples collected at different sampling points \(i\Delta t\). Figure 15: Spectral noise power of 12 th order M-sequence device m: explore. (**A**) receiver input open.