# Effects of Movement for High Time-Bandwidths in Batched Pulse Compression Range-Doppler Radar
## 3 Doppler Effect Modelled in Its Scaling Relation
### Effect of a Constant Effective Velocity on an Instantaneous Broadband Waveform Processing  
Matched filter based processing concepts are independent of the signal bandwidth only for stationary considerations as shown in (3) before. Thereby it can be regarded an optimal filter in the presence of preferable white noise [15]. This consideration is limited for moving objects due to the fact that these do not always result in temporary stationary signals. In order to explain the counterintuitive finding of an additional extent over the Doppler frequency and the surface spread, the Doppler effect has to be analysed more in-depth. In a non-stationary situation, it becomes necessary to model the Doppler effect in its original scaling [22; 23] relations. This is an actual dilation or compression of a scattered signal [1; 3]. It has hereby to be taken into account that the Doppler effect occurs for radar a second time at the scattering point in contrast to an emission from a moving object. Since the classical processing in range-Doppler follows the Fourier relations in the frequency domain, the laws of Fourier transform may be used. This consideration follows the time scaling property of the Fourier transform (FT) so that it is equal to frequency compression or dilation of the received signal \(x(t)\),  
\[x(st)\circ\raisebox{-1.72 pt}{\includegraphics[width=1.72 pt]{figures/figures  
The scaling relation in Equation (13) is then the non-relativistic case for electromagnetic waves with an effective velocity \(v\) equal to an object observed from its heading that moves with \(v\) (i.e., \(\cos\varphi=0\)[23]) as long as the actual \(|v|\ll c\) so that just longitudinal Doppler scaling is treated. The scale factor \(s\) is in the order of \(1\pm 10^{-8}\) to \(1\pm 1.5\times 10^{-6}\) in case of effective velocities between \(5\,\mathrm{km/h}\) and \(800\,\mathrm{km/h}\) for electromagnetic waves with a propagation speed \(c\) in vacuum (\(c=c_{\mathrm{em}}=c_{0}\)) or, respectively, air. Due to these small scale factors, the change in peak intensity caused by motion [22], which is considered as \(1/|s|\) in Equation (12), can usually be neglected. The Doppler frequency displacement \(f_{\mathrm{D}}\) of only a single RF carrier scaled at \(f_{\mathrm{c}}\) will relate to the narrowband approximation of a single Doppler frequency shift by  
\[f_{\mathrm{D}}(v,f_{\mathrm{c}})=\left(1-s\right)f_{\mathrm{c}}\approx\frac{2 \,v}{c}\,f_{\mathrm{c}}. \tag{14}\]  
For all sub-carriers \(f_{\mathrm{c}}\in[f_{\mathrm{v}},f_{\mathrm{u}}]\) within \(B_{\mathrm{occ}}\), this leads to a Doppler frequency spread  
\[\Delta f_{\mathrm{D}}=|s-1|\left(f_{\mathrm{u}}-f_{\mathrm{f}}\right)\approx \frac{2\,|v(t)|}{c}\,\frac{\left(f_{\mathrm{u}}-f_{\mathrm{f}}\right)}{B_{ \mathrm{occ}}}\,. \tag{15}\]  
The bandwidth-dependent scaling relation shown in Equation (15) becomes resolved for a fine analysed frequency resolution \(\delta f_{\mathrm{d}}\) and a tiny range cell spacing \(\delta r_{\mathrm{bi}}\) of the range-Doppler processing. An object that moves linearly from \(r_{1}\) to \(r_{2}\) will then result in a type of uniform rectangular spread. Particularly, if a broadband waveform--similar to Figure 1--is considered that occupies a wide bandwidth \(B_{\mathrm{eff}}\approx B_{\mathrm{occ}}\) instantaneously, the Doppler frequency spread \(\Delta f_{\mathrm{D}}\) will appear at every passed range cell. This can simply be explained by the fact that the Doppler frequency difference between the lower \(f_{\mathrm{f}}\) and the upper frequency carrier \(f_{\mathrm{u}}\) (Figure 1) becomes resolved. Figure 7 contains an illustration of this effect for a point scatterer return that extends in a uniform spread for a high time-bandwidth product. The covered timespan \(T_{\mathrm{cov}}\) will thereby determine the extent in range and the occupied bandwidth the amount of \(\Delta f_{\mathrm{D}}\). It is noteworthy that this counter-intuitive spread over Doppler frequency is not another effect.