# A Novel High-Precision Imaging Radar for Quality Inspection of Building Insulation Layers
## 3 Radar System Design
### Radar Signal Selection  
The range resolution \(\Delta\)R is determined by the time-domain pulse width W and the wave velocity in the medium v, with the expression as follows,  
\[\Delta\text{R}=\frac{\text{Wv}}{2}, \tag{1}\]  
where the time-domain pulse width is related to its frequency bandwidth,  
\[\text{W}=\frac{1}{\text{B}}, \tag{2}\]  
Here, we take the \(-10\) dB bandwidth as B.  
The wave velocity v in a medium depends on the velocity c of the electromagnetic wave in a vacuum, being \(\text{c}=3.0\times 10^{8}\text{m/s}\), and the relative dielectric constant \(\varepsilon_{r}\) of the medium,  
\[\text{v}=\frac{\text{c}}{\sqrt{\varepsilon_{r}}}, \tag{3}\]  
We can derive that,  
\[\text{B}=\frac{\text{c}}{2\sqrt{\varepsilon_{r}}\Delta\text{R}}, \tag{4}\]  
To achieve the resolution of the insulation board (\(\varepsilon_{r}=1\)-\(3\), \(\Delta\)R = 50 mm) of the insulation layer, the bandwidth of the radar should not be less than 3000 MHz with \(\varepsilon_{r}=1\).  
According to the radar's detection depth D, range resolution \(\Delta\)R, and the average relative dielectric constant \(\bar{\varepsilon}_{r}\) of the insulated panel and its surface mortar, the center frequency \(f_{c}\) of the radar-transmitted pulse can be selected as follows [31],  
\[\frac{75}{\Delta R\sqrt{\bar{\varepsilon}_{r}}}<f_{c}<\frac{1200\sqrt{\bar{ \varepsilon}_{r}-1}}{D} \tag{5}\]  
The unit of \(f_{c}\) is megahertz (MHz). Insulated panels are made of foam covered with a thin layer of mortar and they have an average relative dielectric constant \(\bar{\varepsilon}_{r}\) of 2 to 4.  
Take the maximum value on the left side of the equation, which means the denominator should take the minimum value. Take the minimum value on the right side of the equation, which also means \(\bar{\varepsilon}_{r}\) should take the minimum value. Range resolution \(\Delta\)R is substituted into the calculation with a value of 10 mm and the average relative dielectric constant of 2. Finally, we obtain \(5304\text{MHz}<f_{c}<10,000\text{MHz}\).  
According to the Fresnel principle, assuming that the target buried depth is D, the lateral resolution \(\Delta\)L can be calculated according to the following formula,  
\[\Delta\text{L}=\sqrt{\frac{\Delta\text{D}}{2}}=\sqrt{\frac{\text{Dc}}{2\sqrt{ \bar{\varepsilon}_{r}}f_{c}}} \tag{6}\]  
The smaller \(\bar{\varepsilon}_{r}\) is, the bigger \(f_{c}\) is. When D = 120 mm, \(\bar{\varepsilon}_{r}=2\), we need \(f_{c}\geq 5091\text{MHz}\) to achieve \(\Delta\text{L}=50\text{mm}\).  
Considering the design cost and the miniaturization of the radar system, the X2 impulse radar transceiver from Novelda is selected for the radar. The flexible transceiver makes X2 the perfect choice for implementing high-accuracy, high-resolution sensing systems with low power consumption [32]. Table 2 shows the TX parameters when choosing the parameter PGSelect = 9, 10.  
Based on various technical parameters of the radar and the parameters of the X2 impulse radar transceiver, it is determined that the center frequency is 8800 MHz and the operating bandwidth is 3100 MHz for settings of PGSelect = 10. Output pulse frequency spectra (at PRF = 100 MHz) are given in Figure 4.