# Effects of Movement for High Time-Bandwidths in Batched Pulse Compression Range-Doppler Radar
## 2 Challenges of Pulse Compression with High Time-Bandwidth Products
### Batched Processing Scheme Implementation for Range-Doppler Analysis  
The common implementation of the range-Doppler analysis inherits the pulse compression principle. This fundamental batched processing approach has been illustrated in Figure 3 for a cyclic prefix orthogonal frequency-division multiplexing (CP-OFDM) waveform. It uses short blocks with a duration \(T_{\text{b}}\) that are cut from the received stream (i.e., pulses). Theses blocks are then piecewise correlated with a reference pulse over the fast-time (ft) domain to obtain the measured range \(r_{\text{bi}}\). This step is followed by a Fourier transform over \(n\) processed blocks \(c_{i}\) in the slow-time (st) domain. The Doppler frequency \(f_{\text{d}}\) is thereby evaluated over the timespan \(T_{\text{cov}}\) at each individual range bin. Because the pulse compression is performed over the fast-time pulse duration that represents the range, it is also called range compression. This approach will in the case of a CP-OFDM waveform further profit from signal orthogonality but the block duration has to match the symbol duration \(T_{\text{b}}=T_{\text{u}}\)[9; 10]. Instead of a direct cross-ambiguity function evaluation [13], this approach performs a short-term correlation over the short blocks plus an additional Doppler frequency analysis. Common software-defined radars will further work on digital represented and discretely sampled bins of the actual received signal. The quadrature sampling rate \(f_{\text{s}}\) has then to be  
\[f_{\text{s}}\geq B_{\text{eff}}. \tag{4}\]  
Obviously, a waveform with a high instantaneous effective bandwidth \(B_{\text{eff}}\) will then require a higher quadrature sampling rate \(f_{\text{s}}\) as expressed in Equation (4). This gives a bandwidth implication that is mainly determined by the implementation architecture and will be important later. This batched approach provides a convenient way for fast time (ft) slow-time (st) domain range-Doppler analysis. The scheme can thereby make use of efficiently implementable fast Fourier transform (FFT) algorithms to achieve a cost function of \(O(n\log n)\) for arbitrary \(n\)[16]. Hereby, also the range compression can be performed in the frequency domain [8] as it has been illustrated in Figure 3. The scheme benefits from a significant increase in processing speed [14; 17] and it is similarly used for FMCW pulse compression radar [13] with chirped waveforms--like the one shown in Figure 2--but the evaluated timespan \(T_{\text{cov}}\) will be in this case slightly decreased by the ramp duration \(T_{\text{u}}\). The typical dechirp-on-receive approach also turns out to be a closely approximated matched filter processing [18]. It inherently assumes a piecewise stationary fixed target range and a constant effective velocity over the timespan \(T_{\text{cov}}\). The target  
Figure 2: Occupied bandwidth \(B_{\text{occ}}\approx f_{\text{u}}-f_{\text{l}}\) vs. effective instantaneous bandwidth \(B_{\text{eff}}\) of an exemplary LFM waveform that is measurable before chirping or after dechirp-on-receive operation.  
movement is partly addressed in this processing scheme. An approximation is hereby a constant phase over the block duration \(T_{\text{b}}\) that is introduced by the analysed Doppler frequency \(f_{\text{d}}\)[9] and, respectively, a stationary channel within \(B_{\text{occ}}\) e.g., during a chirp. If all the approximations are valid to consider, the inherent matched filter design will follow the stationary energy relation of Equation (3). This will be evaluated in the discretised and ambiguously processed range \(r_{\text{bi}}\) and Doppler frequency \(f_{\text{d}}\) domain. Its particular implementation specific movement implications will be discussed in Section 4.