# 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  
The main implication of the time-scale approach expressed in Equation (28) is that its implementation requires a modification of the common time-frequency batched processing scheme (Section 2.2). Thereby, the idea is to keep the principle of a batched processing scheme but to compensate the range migration induced spread. The principle implementation methods can be split into two classes depending on their view on the Doppler relation. This is similar to the identical consideration of the scaling in range or in the Doppler extent over the occupied bandwidth \(B_{\text{occ}}\) (Section 3.2). The one class tries thereby to address the Doppler scaling relation with a time-domain based compensation by re-scaling, shifting or resampling. The other class performs a normalisation in the frequency domain to compensate the relative Doppler frequency deviation \(\Delta f_{\text{D}}\) in relation to a reference carrier \(f_{\text{c}}\). After this compensation has been performed, the radar processing chain can further process the received signal with classical steps. The first implementation class with a scaling relation in the time-domain can be understood as time re-scaling of the whole received signal. The common effective aircraft velocities between 5 km/h and 800 km/h cause scale factors of \(1\pm 10^{-8}\) to \(1\pm 1.5\times 10^{-6}\) in radar (Section 3.2). Several methods are possible whereby the most straight forward approach is rational \(L/M\) resampling [1]. Hereby, it turns out that in particular the small factors briefly deviating from 1 are most challenging because these may require large \(L\) and \(M\). Other scaling techniques can be based on different kinds of linear or e.g., polyphase interpolation signal scaling [33] or multi-rate signal processing. Another method is to use a precomputed filter based on a pre-known reference signal [34]. The fractional delays form hereby another set of time-based methods that introduce small time delays before individual compressed pulses to address a particular target motion [35], the actual shift is however implementable in the frequency domain as well. These time-domain re-scaling methods have to be performed either target specific whereby a particular target motion could be addressed or alternatively for multiple velocity spans. Hereby, a limited set of spans is needed because a small deviation \(|\Delta v_{\text{eff}}|<100\) km/h leads to a difference of only a fraction of one to two dB according to Figure 10b. However, this condition tightens for higher time-bandwidth products so that smaller spans had to be used. Alternatively, a bandwidth-dependent view as expressed in Equation (15) forms the second class. The idea is to perform a normalisation of the Doppler frequency deviation \(\Delta f_{\text{D}}\) within the occupied bandwidth by relating the fast-time carriers--that are subject to slightly different Doppler shifts--to that of a fixed reference carrier \(f_{\text{c}}\). A common technique for high instantaneously processed bandwidths is the Keystone transform (KT) [36; 37]. It is based on the ft/st domain of a batched processing scheme (Section 2.2). The KT compensation works as originally described to rescale the slow-time axis \(t_{\text{st}}\) of every active fast-time frequency sub-carrier \(f_{\text{ft}}\in[f_{\text{L}},f_{\text{u}}]\) within \(B_{\text{occ}}\) in the slow-time domain by transforming [5; 6]  
\[t_{\text{st}}(f_{\text{ft}})=\left(\frac{f_{\text{c}}}{f_{\text{ft}}+f_{\text {c}}}\right)t_{\text{ks}}. \tag{29}\]  
The advantage of the Keystone transform is that the whole unambiguous range-Doppler window is compensated for its linear velocity migration [38; 39]. It is hereby almost too accurate because it addresses all resolved velocities in the unambiguous Doppler window. The implementation has to consider that this technique requires an interpolation of every fast-time sub-carrier so that the effort increases proportional to this count.