# Effects of Movement for High Time-Bandwidths in Batched Pulse Compression Range-Doppler Radar
## 2 Challenges of Pulse Compression with High Time-Bandwidth Products  
### Stationary Pulse Compression Principle  
Modern radars are typically very agile and can often make use of bandwidths which enable resolutions smaller or equal to the target dimensions. By adapting pulse and radiation parameters, different radar scenarios can be addressed individually [13]. In order to quantify the impact of the time-bandwidth product, a definition of these two operands is necessary. This enables a clear differentiation between the covered and effective timespan, respective bandwidths and sampling frequency relations. In this section, the stationary aspects of a range-Doppler analysis shall be recapitulated shortly before the target movement induced effects are introduced. A classical batched range-Doppler processing scheme performs thereby an inherent pulse compression by correlation similar to pulse-Doppler radars [14]. This principle is used to reduce the peak power of short pulses while a similar range resolution capability and covered range can be achieved [13].  
During the pulse compression, the received PEP power \(P_{\text{rx}}(t)\) gets integrated within the coherent processing interval (CPI). The actual integration timespan \(T_{\text{int}}\) will depend on the waveform and it can be smaller or equal to the CPI duration \(T_{\text{cov}}\) as it is shown for a pulsed waveform in Figure 1. This gives a relation of  
\[T_{\text{int}}\leq T_{\text{cov}}. \tag{1}\]  
The received energy \(E=\int_{0}^{T_{\text{int}}}P_{\text{rx}}(t)\,\text{d}t\) will hereby be collected over the timespan \(T_{\text{int}}\). In the case of additive white noise with a frequency independent noise power per unit bandwidth \(N_{0}\), the maximal target return signal-to-noise ratio (SNR) [13; 15] will only be worsen by the receiver noise factor \(F\) to  
\[SNR\leq\frac{E}{N_{0}\,F}. \tag{2}\]  
In the case of AWGN and low ambiguities, the inequation in (2) becomes maximal. It can however be preprocessed by pre-whitening for coloured noise [15]. Due to the inherent integration stage, actual energy levels are considered \((\text{Ws}=\text{Joule})\). A better SNR will obviously lead to an improved detection capability in the presence of noise. If the averaged collected energy is approximated by \(E\approx P_{\text{rx,avg}}\,T_{\text{int}}\) and (2) is formulated with noise energy \(N\approx N_{0}\,B_{\text{noise}}\) / \(B_{\text{eff}}\), it will result in the fundamental relation  
\[SNR\leq\frac{P_{\text{rx,avg}}}{N_{0}\,F\,B_{\text{noise}}}\underbrace{T_{ \text{int}}\,B_{\text{eff}}}_{=\,\text{G}_{\text{int,desired}}}. \tag{3}\]  
The involved bandwidth definitions of the waveform may however differ. The occupied bandwidth \(B_{\text{occ}}\) will describe the maximum range resolution in a stationary channel. The effective instantaneous bandwidth \(B_{\text{eff}}\) contributes to the integration gain against noise  
Figure 1: Spectrogram of an emitted pulsed waveform with its \(T\)-\(B\) characteristics, RF carrier \(f_{\text{c}}\), lower \(f_{\text{l}}\), upper \(f_{\text{u}}\), baseband intermediate \(f_{\text{if}}\) and quadrature sampling frequency \(f_{\text{s}}\).  
and it sets the dynamic range extent below the averaged noise level. The example of a chirped waveform has been illustrated in Figure 2. It has a comparatively low \(B_{\text{eff}}\) but it measures a large bandwidth \(B_{\text{occ}}=f_{\text{u}}-f_{\text{l}}\) over the pulse duration \(T_{\text{u}}\). This relation yields \(B_{\text{eff}}\leq B_{\text{occ}}\). Further, the bandwidth dependency in (3) will vanish if \(B_{\text{noise}}\approx B_{\text{eff}}\) but only in the case that a stationary channel and target response can be considered. This becomes particularly important in a non-stationary channel with moving targets (Section 2.5).