# Effects of Movement for High Time-Bandwidths in Batched Pulse Compression Range-Doppler Radar
## 6 Discussion of Findings and Consideration for Processing Approaches
### Common Time-Scale Analysis Implementation Techniques for Batched Schemes
A DVB-T2 waveform can for example incorporate up to 27,841 active sub-carriers in one 32k OFDM symbol duration of \(T_{\text{u}}=3.5\) ms in 8 MHz channels [25; 40]. Depending on the interpolation method, this can lead to a significant amount of added latency. In order to speed up the computation, several publications focused on the implementation of efficient interpolation methods to reduce the added complexity and the introduced latency [41; 42; 43]. The processing effort could also be decreased by skipping e.g. every second fast-time sub-carriers to reduce the total number of carriers. Hereby their energy would be lost and the unambiguous range--which is in this case typically not a problem--is halved but the constant velocity spread effect is still properly addressed. Since this transform is commonly applied after pulse compression on the ft/st pulse compressed matrix [41; 42], an unambiguous Doppler limited processing has to identify the relevant unambiguous range-Doppler window separately. This is typically required if the block duration \(T_{\text{b}}\) is determined by a given waveform (Section 2.2) that leads to the fold-over effect in Doppler as it has been illustrated in Figure 8. One method to address the next unambiguous window after Equation (29) is to multiply the term \(\exp(-\text{i}2\pi\ell_{\text{{dover}}}(f_{\text{{c}}}/(f_{\text{{H}}}+f_{ \text{{c}}})\,m)\) to the std-domain [36; 44]. For this, the foldover or ambiguity number [6; 36]\(\
eq 0\) needs a heuristic approach for likely values or a proper identification method like [26]. This is in general necessary to obtain the correct velocity estimate from a detection and to establish a target track. Further, the Gross Doppler sensitivity loss of a batched implementation (Section 4) will not be addressed by this multiplication if the pulses had already been compressed. Another despite similar method of this normalisation class is to compensate the KT based frequency relation of Equation (29) in the time domain as a dispersive time-domain correction. This tries to address the problem over all the sub-carriers in a dispersive relation over the ft-bandwidth \(B_{\text{occ}}\) that makes it possible to implement this compensation in the time-domain as a dispersive FIR filter. It can then be moved to arbitrary positions in the processing chain and therefore also before the pulse compression. This method has been described in 2019 as time-domain keystone transform in [45]. Since different kinds of dispersive implementations are possible for speed-up, this way is promising to address the bandwidth-dependent Doppler implication (Section 5) by this time-scale approach. Solely in the case of a narrowband waveform, it is a valid technique to multiply an extra single phasor by a complex exponential function to correct a narrowband chirp related to the Stolt interpolation [46]. This can address the bandwidth influence over the occupied bandwidth \(B_{\text{occ}}\) as it is used in SAR processing to address the relative platform motion and it is easier but it will only work for an instantaneous narrowband chirped waveform with a low effective bandwidth \(B_{\text{eff}}\) - as shown in Figure 2--because it may address the bandwidth-dependent Doppler during the chirp duration \(T_{\text{u}}\) sequentially. The extra phasor can be related by its range profile [18]. It works similar to a single carrier signal \(x_{\text{sc}}(t)=\text{e}^{-\text{i}2\pi f_{\text{{c}}}(t)\,t}\) scaled by Equation (12) to \(x_{\text{sc}}(s\,t)=1/|s|\,\left[\,x_{\text{sc}}(t)\,\right]\,\text{e}^{-\text {i}2\pi f_{\text{{D}}}(f_{\text{{c}}})\,t}\) so that it is displaced by the Fourier shift theorem due to its piecewise Doppler frequency  
\[f_{\text{{D}}}(f_{\text{{c}}})|_{f_{\text{{c}}}=f_{\text{{c}}}(t)}\approx(1- s)\,f_{\text{{c}}}(t) \tag{30}\]  
which is approximately \(f_{\text{{D}}}(t)\approx f_{\text{{c}}}(t)\,(2\,v_{\text{eff}})/c\) from Equation (14).