# Atmospheric and Fog Effects on Ultra-Wide Band Radar Operating at Extremely High Frequencies
## 5 Distance Measurement Accuracy and Resolution  
In the following we demonstrate that the deviation in the detected intermediate frequency causes errors in the absolute range estimation and may also lead to some degradation in the longitudinal resolution. We previously showed that according to Equation (14), the longitudinal range resolution of \(\delta d_{vacuum}=1.5\) cm can be obtained by the wideband FMCW radar. This value is not affected by the distance to the target, nor by free space loss and attenuation of the wave while propagating in the atmosphere. Power loss causes a decrease in the signal-to-noise ratio, reducing the accuracy of the absolute distance measurement. We assume here that the received signal strength is well above the minimum detected signal and there is enough fade margin. This can be achieved by employing directive antennas, compensating for the low power [27; 28]. It was shown in the previous section that in the dispersive atmosphere, the frequency \(f_{IF}(t)\) of the detected signal is deviated from the expected \(f_{m}\), resulting in an error in the estimated range. Under foggy conditions, the distance to target is calculated by:  
\[d_{fog}\left(t\right)=\frac{T_{sweep}c}{2\Delta f}f_{IF}(t) \tag{20}\]  
The difference between the measured value in foggy conditions and the real distance to the target (determined by \(f_{m}\)) is:  
\[\Delta d=d_{fog}\left(t\right)-d=\frac{T_{sweep}c}{2\Delta f}\left[f_{IF}(t)-f _{m}\right] \tag{21}\]  
Using Equation (12) we obtain:  
\[\Delta d=\frac{f_{IF}(t)-f_{m}}{f_{m}}\cdot d \tag{22}\]  
Figure 4 illustrates the resulting error in the distance measurement for the discussed FMCW millimeter wave radar with \(\Delta f~{}=~{}10\) GHz and \(T_{sweep}~{}=~{}1\) ms while detecting an object at a distance of \(d~{}=~{}1\) km under foggy conditions. We choose to demonstrate the effect at frequencies of 94 GHz and 325 GHz, which are being used in many wideband millimeter and sub-millimeter radars. The graphs of Figure 4 show that under foggy conditions, there is a systematic offset \(\Delta d\) in distance measurements, depending on the carrier frequency. This offset leads to an erroneous estimation \(\overline{\Delta d}/d\) of the absolute distance to the target (here \(\overline{\Delta~{}d}\) is the average value of the offset from the real distance \(d\)). The calculated range appears greater than the actual one. Moreover, it can be seen that during the sweep time, the error deviation varies between a minimum value \(\Delta d_{\min}\) to a maximum \(\Delta d_{\max}\), leading to a spread in the longitudinal point response. This instantaneous variation is due to the frequency-dependent dispersion component in Equation (15) and may cause a degradation \(\left(\Delta d_{\max}-\Delta d_{\min}\right)/\delta d\) in resolution in some frequency bands. This error increases with the distance. Table 1 presents calculations of the percentage errors in distance measurements \(\overline{\Delta d}/d\) and resolution degradations \(\left(\Delta d_{\max}-\Delta d_{\min}\right)/\delta d\) for several frequencies in the millimeter and sub-millimeter bands for a target 1 km away. Broadening the frequency sweep \(\Delta f\) (in our case up to 10 GHz) of the chirp, atmospheric conditions will not only affect the amplitude of the received signal  
(due to absorption) but also its phase, leading to a reduction in accuracy of distance estimations. The transmission of wide FMCW signals in atmospheric absorption resonances causes some degradation in resolution, as seen at 60 GHz, 183 GHz and 325 GHz. In imaging radar systems [27], the time-varying range offset would degrade the longitudinal resolution.