# Effects of Movement for High Time-Bandwidths in Batched Pulse Compression Range-Doppler Radar
## 3 Doppler Effect Modelled in Its Scaling Relation  
### Exemplary Processed Targets from Measurement Campaign Datasets  
In the following, target responses from actual passive radar (PR) datasets are shown. The examples were intentionally processed with a 'too high' time-bandwidth product that exceeds the boundary defined in Equation (9) to highlight the involved effects. Thereby, an orthogonal frequency-division multiplexing (OFDM) waveform obtained from a single channel DVB-T in the UHF frequency range was used. This digital broadcast emission inherits both aspects of a high desired integration gain with an integration time-effective bandwidth product according to Equation (7). It provides a continuous illumination and it has a high instantaneous effective bandwidth of \(B_{\text{eff}}\approx B_{\text{occ}}=7.61\,\text{MHz}\). The waveform also features good auto-correlation properties so that the 8k mode symbol duration \(T_{\text{u}}=896\,\text{\mu s}\) has been used as block duration \(T_{\text{b}}\) of the batched processing scheme. The illustrated bistatic sampling is at 33 m per cell. All intensity levels in the bistatic range-Doppler diagrams are plot on a logarithmic scale.  
The target return shown in Figure 5 had been accelerated and then moved with an approximately constant bistatic velocity of about 270 km/h in the processed interval. It has intentionally been processed with a covered timespan \(T_{\text{cov}}\) times effective bandwidth \(B_{\text{eff}}\) equivalent to 68 dB. Thereby, the return becomes extended by \(\Delta f_{\text{D}}=4.5\,\text{Hz}\) in Doppler (4 cells) and by 200 m in bistatic range (6 cells). The effective velocity corresponds to a bistatic range walk of 134.40 m (4 cells) within the covered timespan \(T_{\text{cov}}=896\,\text{ms}\). The two additional range cells could fit to a resolved target extension with an assumed length of 30 m (\(\leq\)2 cells) but the extension over \(\Delta f_{\text{D}}\) is contradicting to its linear movement. The energy becomes spread over 20 cells (13 dB), which is much more than just \(6\,(=4+2)\) bins if a pure extension in range was considered. This'rectangular spread' can be explained by the Doppler frequency difference \(\Delta f_{\text{D}}\) that is resolved within the signal bandwidth. This particular effect of a linear movement will be analysed in the next Section 3.2. Another implication of the extended and fluctuating target return is well visible in the lower right of Figure 5, another track head detached from the actual target at 19.6 km that was likely caused by the sudden motion change and the spread influence.  
Figure 5: Target tracked with an approximately constant effective bistatic velocity \(v\approx 270\,\text{km/h}\) and \(T_{\text{cov}}=896\,\text{ms}\) with \(T_{\text{int}}=717\,\text{ms}\), spread fluctuations led to a detached track visible on the right side.  
An additional example of a strong return response from a close-by target is shown in Figure 5(a). It has departed with a constant bistatic velocity within an even more extended duration of \(T_{\rm cov}=2.24\,\)s. Contrary, in Figure 5(b) a small acceleration is visible at the start of the integration. The target return remained hereby in both cases almost uniform in intensity over the covered \(T_{\rm cov}\) and occupied \(B_{\rm occ}\) at its close bistatic range since it did not fluctuate and it was quite strong not to be overlapped by noise.