# Effects of Movement for High Time-Bandwidths in Batched Pulse Compression Range-Doppler Radar
## 1 Introduction  
The capability of a radar to track a target depends strongly on its ability to detect moving targets with good signal level margins. More energy retrieved from an object usually comes along with a better detection performance. The range-Doppler analysis enables hereby the simultaneous determination of range and target velocity that is effectively measurable from the radar node. This information is then used to localise the target and to estimate further movements to establish a track. Despite most objects to be detected are actually moving, most radar processing schemes assume inherently a constant effective velocity and that the target movement can be fully addressed by a single Doppler frequency displacement. As long as this is a valid model, this principle will enhance the radar detection and tracking performance against moving objects. This assumption is obviously limited by range migration and a closer analysis turns out that in fact the time-bandwidth product is crucial to define its boundary [1]. The deviation from this 'narrowband' model becomes more relevant the faster an object moves in relation to the propagation speed \(c\) and the higher the desired coherent integration gain becomes. One implication is that the achieved resolutions will be increased against stationary objects in case of a high time-bandwidth product. If this is not properly considered, it will however become more uncertain against fast moving objects and the occurring range and Doppler frequency spread will affect the overall detection performance. This will already be the case for a purely linear movement. A resolved Doppler frequency difference within a wideinstantaneously processed signal bandwidth may then cause a quite severe bandwidth-dependent impact that it is no longer an extension over some range cells. The principle challenge of Doppler dispersion--which describes the bandwidth-dependent Doppler influence and addresses thereby the range migration--has been studied extensively in the sonar domain in the 1950s and 1960s. It gave the motivation due to a comparably low sound propagation speed \(c_{\mathrm{s}}\lll c_{\mathrm{em}}\) for integration gains as early as 30 dB, see e.g., [2; 3]. Even though the electromagnetic wave propagation speed \(c=c_{\mathrm{em}}\) is much higher, the same challenge will arise similarly in the case of much faster velocities or in the case of higher time-bandwidth products even for comparable speeds. The task of proper compensation has first been addressed in radar in the SAR domain for ground moving target indication. This application was focused to compensate for cross-range blurring in the Doppler domain with a bandwidth-dependent interpolation approach. Thereby, the Keystone transformation has been introduced to compensate a batched/symbol-wise processing of slow ground moving targets with e.g., up to 14 m/s using a processed sample count around \(2^{15}=45\) dB. This has been published in 1997 [4] or, respectively, in 1999 [5]. By the same authors, the application to coherent integration has been published [6] in 2007 that is one method for compensation approaches. High coherent integration gain detection is nowadays a particular challenge of passive radar applications. The inherently bistatic configuration typically suffers from low effectively radiated powers towards the upper hemisphere in which common aerial targets are located, comparatively low receive antenna gains and strong direct path interference (DPI) levels directly received from the transmitter(s). This is worse in the case of continuous broadcast emissions with a high instantaneous signal bandwidth \(B_{\mathrm{eff}}\). Even with shielding towards the transmitter, the dynamic signal level range can easily exceed 80 dB above the averaged reflected target levels. Some exemplary narrowband performance estimations for passive bistatic radar systems are given in [7] et al.