# Effects of Movement for High Time-Bandwidths in Batched Pulse Compression Range-Doppler Radar
## 2 Challenges of Pulse Compression with High Time-Bandwidth Products
### Limits Introduced by the Bistatic Velocity and the Time-Bandwidth Product  
As it has been pointed out previously, the shown fundamental axioms are based on the assumption of a piecewise stationary channel. This view is obviously limited for moving target detection. Most classical range-Doppler processing schemes are however still based on the piecewise assumption of a single Doppler frequency displacement, a constant effective velocity and a fixed bistatic range. The movement is thereby implicitly treated by just its Gross Doppler frequency displacement \(f_{\mathrm{D}}\) of the RF carrier \(f_{\mathrm{c}}\) and at a single time delay. This is equally valid for the common batched implementation scheme of a range-Doppler analysis that has been described in Section 2.2. The pulse compressed energy retrieved by matched filtering will hereby be independent of the processed bandwidth according to Equation (3) as long as the channel with the target return can be considered stationary (Section 2.1). The common and often called 'narrowband' boundary for classical range-Doppler processing is generically defined to confine a point scatterer movement to one range cell with propagation speed \(c\) as [1]  
\[\frac{2\,v_{\mathrm{eff}}}{c}\ll\frac{1}{T_{\mathrm{cov}}\,B}. \tag{9}\]  
The bandwidth \(B\) in Equation (9) has thereby not been clearly defined yet. It can be either the occupied bandwidth \(B_{\mathrm{occ}}\) or the quadrature sampling rate \(f_{\mathrm{s}}\geq B_{\mathrm{eff}}\) (4) at which the sampled data is represented. More precisely expressed, the constant velocity leads to the two following constrains. First, the analysed frequency resolution \(\delta f_{\mathrm{d}}\approx T_{\mathrm{cov}}^{-1}\) shall be larger than \(\frac{2\,v_{\mathrm{eff}}}{c}\left(f_{\mathrm{u}}-f_{\mathrm{f}}\right)\) to confine the Doppler extent. Second, the bistatic range change of \(v_{\mathrm{eff}}\,T_{\mathrm{cov}}\) shall be smaller \(c\) / \(\left(2\,f_{\mathrm{s}}\right)\) to limit the extent to the range cell spacing \(\delta v_{\mathrm{bi}}\) (\(\propto f_{\mathrm{s}}^{-1}\)). If both conditions are combined, this will give an upper boundary on \(G_{\mathrm{int}}\). The stationary boundary for a linear movement that is valid for \(f_{\mathrm{s}}\geq B_{\mathrm{occ}}\) then yields  
\[G_{\mathrm{int}}\leq T_{\mathrm{int}}\,B_{\mathrm{eff}}\leq T_{\mathrm{cov}} \,B_{\mathrm{occ}}\ \stackrel{{!}}{{\leq}}\frac{c}{2\,v_{\mathrm{eff}}}\leq TB_{ \mathrm{max}}\approx T_{\mathrm{cov}}\,f_{\mathrm{s}}. \tag{10}\]  
The boundary shown in Equation (10) could be slightly relaxed if the sampling rate is within the range \(B_{\mathrm{eff}}\leq f_{\mathrm{s}}\leq B_{\mathrm{occ}}\) from Equation (4) for low \(B_{\mathrm{eff}}\) since the instantaneous Doppler spread is \(\propto B_{\mathrm{eff}}\) and will tend to \(\Delta f_{\mathrm{D}}\) acc. to Equation (15) on every extended range bin if \(B_{\mathrm{eff}}\to B_{\mathrm{occ}}\). For the next considerations, the wide instantaneous bandwidth of a broadcast OFDM waveform will be considered so that the approximation \(B_{\mathrm{occ}}\approx(f_{\mathrm{u}}-f_{\mathrm{f}})\approx B_{\mathrm{eff}} \approx f_{\mathrm{s}}\) is fulfilled. The explicit key parameters computed for a fixed time-bandwidth product with the speed of light in vacuum \(c=c_{\mathrm{em}}\) and a constant effective velocity of \(v_{\mathrm{eff}}=700\,\mathrm{km}/\mathrm{h}=194\,\mathrm{m}/\mathrm{s}\) are shown in Table 1. It contains various sets of timespans and bandwidths with their maximal achievable range resolutions and analysed frequency  
Figure 4: Bistatic setup with a baseline of \(L=50\,\mathrm{km}\) between a transmitter (TX) and receiver (RX) pair with bistatic angle \(\beta\) and a target moving in direction \(\delta\) at a speed of \(|\vec{v}_{\mathrm{trg}}|=800\,\mathrm{km}/\mathrm{h}\) over ground. accuracies for pulse compression.