# Effects of Movement for High Time-Bandwidths in Batched Pulse Compression Range-Doppler Radar
## 6 Discussion of Findings and Consideration for Processing Approaches
### Constrains of Actual Non-Uniform and Non-Rectangular Target Returns  
A time-scale analysis approach with an inherent constant scale factor \(s\) (Section 6.2) considers a pure uniform rectangular return spread, regardless of the chosen implementation technique (Section 6.3). This is the continuation of an assumed constant effective velocity within the whole processing interval and thereby it considers a purely linearly moving single point scatterer as illustrated in Figure 7 (Section 3.2). In the case of a confined dominant point scatterer and an idealised propagation channel with a transfer function \(|H_{\text{trg}}(f)|=const\), this might be a plausible model (Figure 6). The downside is that any deviation from this assumed pure uniform rectangular shape relates to a lower achievable coherent integration gain. Any deviation from a uniform spread will then lead to a suboptimal refocusing that includes parts with less energy and more integrated noise - a kind of 'collapsing loss' [13] may occur. As it has been shown by examples of an actual target response, a practical extended target return might fluctuate even within an occupied bandwidth of just 8 MHz in excess of 300 ms (Section 5.5). The observable in-band fluctuations raise the question how to properly process and define a kind of wideband or high time-bandwidth product related radar cross section (RCS). The classical'reflected power' to 'incident power density' model [13] and the Swerling cases are typically used for confined 'narrowband' and mostly monostatic target returns. The target model of extended targets [11] consists contrarily of one or more scattering regions with individually associated complex amplitudes. If a high bandwidth provides a range resolution that is smaller than the target dimension, these scattering regions on a target will become resolved. This has to be addressed in further considerations but the totally distinct time-bandwidth product sets as expressed in Table 1 (Section 2.5) for the same fixed boundary cannot be treated equally. Furthermore, the bistatic geometry might lead to another bistatic RCS definition because the radar return will change for varying aspect angles [11; 20]. A joint processing [29; 47] applied to extended target returns might however regain some of its lost visibility. It is a subtle but noteworthy difference that very weak extended target responses close to the detection threshold might also look similar to an in-band fluctuating return but this is simply caused by the overlap of present noise. A detailed CEM object and propagation environment modelling might be useful to give more insight to highly fluctuating returns because these are likely caused by a combination of actual target extension, aspect angle changes and multipath propagation. Last but not least, the effective acceleration of the target return will give an instantaneous Doppler relation (Section 3.3) so that the scale factor \(s(t)\) will not anymore be constant within the CPI duration. A time-dependent instantaneous effective velocity yields from Equation (13) with the bistatic range \(R_{\text{bi}}\) (Section 2.4)  
\[v_{\text{est}}(\tau)=\frac{1}{2}\left.\frac{\partial\,R_{\text{bi}}(t)}{ \partial\,t}\right|_{t=\tau}\approx c\left.\frac{1-s(t)}{1+s(t)}\,\right|_{t= \tau}. \tag{31}\]  
Even a slight acceleration would then not only cause less energy--due to parts cut from the rectangular return--but also pure noise being integrated in regions that are no longer inside the otherwise assumed rectangular shaped extension. This is illustrated in Figure 15 for the example of an actual target return. An uncompensated processing will obviously be much worse for stronger acceleration (Section 5.6). This motion influence could however be addressed if the target return acceleration was partly predictable like the change of the effectively measurable velocity in a bistatic radar geometry (Section 2.4).