# Effects of Movement for High Time-Bandwidths in Batched Pulse Compression Range-Doppler Radar
## 2 Challenges of Pulse Compression with High Time-Bandwidth Products
### Limits Introduced by the Bistatic Velocity and the Time-Bandwidth Product
Hereby, all sets have a maximum integration gain from Equation (7)  
\[G_{\mathrm{int}}\leq T_{\mathrm{int}}\,B_{\mathrm{eff}}\leq TB_{\mathrm{max}}=58.8 \,\mathrm{dB}. \tag{11}\]  
Related to the values expressed in Table 1, the integration time will roughly be limited to \(50\,\mathrm{ms}\) before range walk occurs if a maximal range resolution down to \(10\,\mathrm{m}\) is achieved. However, the pairs cannot be treated and compared independently of the channel and target response. In particular, if the resolution gets smaller than the target dimension, it will likely introduce bandwidth-dependent and RCS modelling related effects so that each set might perform differently. This effect can even be present within the \(8\,\mathrm{MHz}\) bandwidth of a single DVB-T channel as it will be shown in Section 5.5 for actual target return measurement examples. The coherent processing of a 'high resolution mode' has been further discussed in e.g., [11; 20]. The consideration of a fixed (resolution based) integration limit from Equation (10) seems at first plausible but it might be required to exceed this limit set on the maximum time-bandwidth product. This can be desired either because not all objects are moving with maximum speed, a finer ground resolution is desired or simply a higher coherent integration gain is needed for the detection of small radar cross sections or to achieve an increased coverage. Thereby it is important that a difference of solely \(6\,\mathrm{dB}\) relates to a factor of \(\sqrt{2}\) in covered monostatic range. This difference can even be a factor of up to 2 in one-sided bistatic range because the inequation \(R_{1}^{2}\,R_{2}^{2}\leq(R_{1}+R_{2})^{4}\) is valid in the oval of Cassini power relation [19]. This raises the question of improving this limitation and which effects, loss expectations and further considerations need to be kept in mind for finer resolutions or faster moving objects. If more than \(59\,\mathrm{dB}\) as related to Table 1 shall be achieved in the case of \(v_{\mathrm{eff}}=700\,\mathrm{km}/\mathrm{h}\), then a loss will have to be considered without a properly compensated processing. This will happen equally if the bandwidth or time is increased or e.g., a processing fixed to \(60\,\mathrm{dB}\) is used. In particular bistatic passive radar applications often require an even higher coherent integration gain to enable a reliable target detection at bistatic ranges of multiple tenfold kilometres. Since most targets are moving, this boundary becomes highly relevant for effective velocities larger than \(200\) to \(300\,\mathrm{km}/\mathrm{h}\) and desired integration gains that exceed \(65\,\mathrm{dB}\). If the target changes its bistatic range within the duration of the CPI significantly and this violates Equation (9), the range walk effect will occur but it will then not only cover some range cells for common effective bandwidths \(B_{\mathrm{eff}}\) of broadcast emissions (Section 3) although it is primarily considered as range migration in standard radar definitions [21]. An actual extended target can further introduce a distribution over several range cells [11] for a high resolution waveform if the target dimension is larger than the resolved bistatic range cell area (\(\propto f_{\mathrm{s}}^{-1}\)). The main implication is that a point scatterer return will not be fixed to one range cell and further a Doppler spread may occur even for a constant effective velocity so that a target track might be lost. This is shown in the next Section 3 and its impact will be quantified in Sections 4 and 5. \begin{table}
\begin{tabular}{c c c c c} \hline
**Time** & **Bandwidth** & **Max. Resolution** & **Frequency** & \(T\mathbf{B}_{\mathrm{max}}\) \\
**\(T_{\mathrm{cov}}\)/s** & \(B_{\mathrm{occ}}\)/MHz** & \(\delta\mathbf{r_{\mathrm{bg}}}\)/m & \(\delta\mathbf{f_{\mathrm{d}}}\)/Hz & \(\mathrm{dB}\) \\ \hline
1 & 0.77 & 194.8 & 1 & 58.8 \\
0.5 & 1.50 & 100.0 & 2 & 58.