# Effects of Movement for High Time-Bandwidths in Batched Pulse Compression Range-Doppler Radar
## 4 Implication of a Linear Target Movement in Batched Range-Doppler Processing  
In the following analysis, the implications on a batched range-Doppler processed target response are illustrated in more detail. It is thereby important to consider the involved influences individually to create a good basis for potential improvements. The first important aspect is that the interpretation of the Doppler effect can be separated for an RF carrier frequency \(f_{\rm c}\gg B_{\rm occ}\). This helps to split the instantaneous Gross Doppler implications mainly related to the processing scheme from to the ones of the bandwidth-dependent part by the movement during the processed interval. \[f_{\rm D}(f)=\underbrace{f_{\rm D}(f_{\rm c})}_{\rm Gross\ Doppler}+\underbrace{f_{\rm D }(f-f_{\rm c})}_{\rm Bandwidth\ dependent} \tag{17}\]  
By Equation (17), the implication of the Doppler effect can be differentiated between physically caused movement impacts and these that are related to the processing scheme. A batched implementation as shown in Section 2.2 may  
1. introduce a frequency sensitivity [14; 24] due to the Gross Doppler displacement \(f_{\rm D}(f_{\rm c})\propto v\). This is caused by its intra-pulse phase approximation [9] over the block duration \(T_{\rm b}=T_{\rm u}\). If a cyclic prefix waveform with guard duration \(\alpha\) is considered, this will lead to a sensitivity degradation \[Loss_{\rm PC}\big{(}T_{\rm b},f_{\rm D}(f_{\rm c}),\tau,\alpha\big{)}=\left( \frac{1}{\left(1-\frac{\mu}{T_{\rm b}}\right)\,{\rm sinc}(\pi\,f_{\rm D}\,(T_ {\rm b}-u))}\right)^{2}.\] (18)  
The relation (18) is thereby only range (\(\tau\)) dependent with \(u=\max(0,\,\tau-\alpha)\) outside of the guard duration \(\alpha\)[25]. Its derivation is given in Appendix A. In addition, only ambiguously measurable parameters are determined by a batched processing. This will impact subsequent processing stages like the tracking. In particular multi-carrier waveforms with long symbol durations \(T_{\rm u}\) and a low Doppler tolerance might result in a limited unambiguous Doppler processing from the radar perspective [26]. The unambiguity window in the analysed frequency domain \(f_{\rm d}\) is \(f_{\rm D,u}=\pm\,0.5/\left(T_{\rm u}+\alpha\right)\) as illustrated in Figure 8. This gives an unambiguously measurable velocity span of \(v_{\rm D,u}=\pm\,c/\left(4\,(T_{\rm u}+\alpha)\,f_{\rm c}\right)\). If the resulting fold-over effect was not considered, it would have a severe impact on tracking because the wrong radial direction and speed would be determined. For chirped signals the ambiguity impact is often worsen by a strong range-Doppler coupling of the waveform [13]. The directly unambiguously measurable range is \(R_{\rm num}\leq c\,\left(T_{\rm u}+\alpha\right)/2\). This influence can be resolved by pulse staggering or similar modifications to the transmit waveform. The target displacement in the discretised digital analysis domain of a software defined radar implementation will further cause  
* a temporary impact on a moving scatterer's maximum \(SNR\) that is inherent from the DSP architecture. This leads to an DFT scalloping [27] and range straddling [13] loss whenever a point scatterer is time-dependently displaced over multiple discretised range (ft) or Doppler (st) cells. A single point can be split over up to four processed range-Doppler cells as illustrated in Figure 9. This gets in particular worse for quadrature sampled signals, confined and point scatterer like target returns and rectangular high effective instantaneous bandwidth waveforms. The range straddling impact will increase for high ratios \(f_{\mathrm{s}}^{-1}\,B_{\mathrm{occ}}\) up to \(3.9\,\mathrm{dB}\). The DFT scalloping depends on the relation of signal displacement \(f_{\mathrm{D}}\) to the frequency bins and the number of processed samples. The scalloping loss can reach another \(3.9\,\mathrm{dB}\) for quadrature sampling. Despite this effect is in principle widely known in digital signal processing, it is rarely considered in radar processing schemes [28; 29].