# Effects of Movement for High Time-Bandwidths in Batched Pulse Compression Range-Doppler Radar
## 5 Influence of Linear Movement within the Signal Bandwidth due to Doppler
### Target Response Coherency over \(T_{\rm cov}\) and \(B_{\rm occ}\) from Measurement Datasets  
An actual extended target return might also differ [11] from a pure uniform return of one dominant confined scatterer as it was the case in the previous examples in Figure 6a,b. While distributed multipath delayed trails behind the target have been observed [30], an extended return can vary as well within the occupied bandwidth over the CPI duration. In the case of range walk and a high instantaneous bandwidth, the target returns become resolved in the range-Doppler plot. This enables a detailed analysis within the covered timespan \(T_{\rm cov}\) and over the whole instantaneously occupied bandwidth.  
In order to illustrate these target response fluctuations, a strong target return has been processed with fine resolution parameters of \(T_{\rm cov}\,B_{\rm eff}=72\,\)dB. It was evaluated at \(f_{\rm c}=690\,\)MHz within \(B_{\rm occ}\approx B_{\rm eff}=7.61\,\)MHz. The target approached with an almost constant effective velocity during the processed intervals. The result is shown in Figure 11 and the overlapped response \(896\,\upmu\)s later is shown in Figure 12. It thereby clearly deviates from the uniform spreads of target returns that were shown earlier.  
Figure 11: Fluctuating approaching target return moving with a constant \(v\approx 225\,\)km/h and integrated in \(B_{\rm eff}=7.61\,\)MHz over \(T_{\rm int}=1.79\,\)s during a coherent processing interval of \(T_{\rm cov}=2.24\,\)s.  
Figure 10: Extension loss in range-Doppler for multiple time-bandwidth products and linear velocities. (**a**) Expressed with \(f_{\rm s}\approx B_{\rm eff}\) for multiple \(T_{\rm cov}\,B_{\rm eff}\) sets. (**b**) Expressed for constant effective velocities.  
By using Equations (14) and (17), the Doppler frequency difference \(\Delta f_{\rm D}=f_{\rm D}(f_{\rm u})-f_{\rm D}(f_{\rm l})\) can shown to be  
\[f_{\rm D}(f_{\rm u}=f_{\rm c}+B_{\rm occ}/2) =f_{\rm D}(f_{\rm c})\ +f_{\rm D}(B_{\rm occ}/2) \tag{24}\] \[f_{\rm D}(f_{\rm l}=f_{\rm c}-B_{\rm occ}/2) =\underbrace{f_{\rm D}(f_{\rm c})}_{\rm Gross\ f_{\rm D}}\underbrace {-f_{\rm D}(B_{\rm occ}/2)}_{\rm independent\ of\ f_{\rm c}}. \tag{25}\]  
The bar marked with \(\Delta f_{\rm D}\) in Figures 11 and 12 shows the return resolved within the CPI duration \(T_{\rm cov}\) from the lower to the upper bandwidth in Doppler frequency over slow-time in steps of  
\[\delta f_{\rm step}=c\ /\ (2\ v_{\rm eff}\ T_{\rm cov})=1.1\,\rm MHz. \tag{26}\]  
Thus, in the case \(B_{\rm eff}\approx B_{\rm occ}\), this resolves the target return instantaneously at \(f_{\rm c}\) within the full spectrum of the occupied bandwidth \(B_{\rm occ}\) over fast-time in a time spacing of  
\[\delta t_{\rm step}=\ c\ /\ (2\ v_{\rm eff}\ f_{\rm s})\ =263\,\rm ms. \tag{27}\]  
It can hereby be seen that the target response during its approach with a constant effective velocity might vary strongly over time and even within 8 MHz over the bandwidth. If the described aspect is ignored, this will lead to a severe impact on classical schemes. Due to missing parts compared to a purely uniform rectangular response (Figure 6a), this would lead to a specific uncertainty for the posteriori determination of position and speed in an uncompensated range-Doppler processing scheme (Section 5.2). The resolved target fluctuations--likely caused by multipath and aspect angle changes [11; 20] or an altered effective antenna gain towards the target--highlight that it is not always optimal to re-focus the whole response within an assumed rectangular spread. The fluctuations could be interpreted similar to 'narrowband' Swerling cases [13] if the whole frequency dependent return was simply summed up. However, their original statistical modelling usually follows a slightly different approach.