# Effects of Movement for High Time-Bandwidths in Batched Pulse Compression Range-Doppler Radar
## 2 Challenges of Pulse Compression with High Time-Bandwidth Products
### Evolution of Radar Waveforms and Reasons for High Instantaneous Time-Bandwidth Products  
In order to cope with different radar measurement tasks, the processed time-bandwidth parameters of the waveform and the modulation design can be pushed. Since it is strongly coupled to waveform design, this part shall not be addressed in detail. In general, large time-bandwidth products come along with high resolution parameters and large possible coherent integration gains. Both can be desired for several reasons. A high range resolution and accuracy can be achieved with a high bandwidth (\(B_{\text{occ}}\) +) and it is beneficial for an improved positioning and localisation capability, especially in multistatic configurations. The maximal achievable range resolution in a stationary channel with propagation speed \(c\) and without parametric super-resolution can in general be approximated by  
\[\delta r_{\text{res}}\geq\frac{c}{2\,B_{\text{occ}}}=\frac{c}{2\,(f_{\text{u}} -f_{\text{l}})}. \tag{5}\]  
In order to support velocity based tracking and particular aspects of object related classification, a better velocity or micro-Doppler resolution (\(T_{\text{cov}}\) +) can be desired. The  
Figure 3: Batched range-Doppler processing principle to build a fast-time (ft) and slow-time (st) domain and to speed-up the radar processing of a CP-OFDM waveform with a block duration \(T_{\text{b}}=T_{\text{u}}\).  
smallest resolution in the analysed Doppler frequency \(\delta f_{\text{d}}\) can be estimated [13] directly from the processed CPI duration as  
\[\delta f_{\text{d}}\geq\frac{1}{T_{\text{cov}}}. \tag{6}\]  
The third ambition can be a large instant dynamic range from the highest to the lowest signal level whereby the waveforms may have a thumbtack or pin head shaped auto-correlation property with a plateau depending on the signal structure [10; 13]. The desired coherent integration gain from relation (3) is usually defined as the integration time-effective bandwidth product to  
\[G_{\text{int,desired}}=T_{\text{int}}\,B_{\text{eff}}. \tag{7}\]  
This will enable a processing approximately down to \(T_{\text{int}}\,B_{\text{eff}}\) below the noise level. A high instantaneous effective bandwidth can thereby pose particular challenges in the radar signal processing chain with respect to linearity, beamforming, hardware stability and delay realisation. With respect to relatively high desired integration gains, the target movement becomes an increasingly considerable factor (Section 2.5). The processing might then need to consider the Doppler influence in more detail (Section 3) and treat it in its actual scaling relation (Section 6.3). In particular for high instantaneous bandwidth waveforms like OFDM the movement impact can be quite strong and it is often underestimated. This is important in for example a passive radar view of coherent multi-channel systems that include 'neighbour channels' in the scenery in their processing. If the increase is high and the Doppler effect is not treated appropriately, an uncompensated coherent range-Doppler processing will lead to the bandwidth paradox (Section 5.4). Despite more spectrally distributed power was available for an increased \(B_{\text{eff}}\), an integration would worsen the result if the stationary assumption (Section 2.5) is violated. Similar effects can be observed for increased covered timespans \(T_{\text{cov}}\). These particular aspects of pulse compression radar and a closer look on the target returns shall be addressed in the following sections.