# Motion Compensation for Long Integration Times and DoA Processing in Passive Radars
## 2 Passive Radar Operating Principle  
The basic geometry of a PR based on satellite IoO is depicted in Figure 1: \(L\) is the baseline or IoO-PR distance; \(R_{T}\) is the IoO-target distance; \(R_{R}\) is the PR-target distance; \(\beta\) is the bistatic angle; and \(\vec{V}_{sat}\) and \(\vec{V}_{tg}\) are the velocity vectors of the satellite and the target, respectively. Since DVB-S satellites have a GEO orbit, they can be considered stationary, which simplifies PR scenario geometry and guarantees the availability of stable illuminating signals.  
Two receiving channels are used: the reference one, which acquires the direct signal from the IoO; and the surveillance one, which captures the echoes from the targets when they are illuminated by the IoO. If the bistatic range origin is located at the PR receiver, the target echo delay is expressed as \(t_{target}=t_{T}+t_{R}-t_{L}\). \(t_{target}\) is calculated from the bistatic range, \(R_{bis}=R_{T}+R_{R}\), and the baseline as \(t_{target}=(R_{bis}-L)/c\), being \(c\) the light speed; associated propagation delays are \(t_{L}=L/c\), \(t_{T}=R_{T}/c\) and \(t_{R}=R_{R}/c\).  
The cross ambiguity function (CAF) is the main coherent processing stage. It performs the matched filter, allows for the estimation of the bistatic range and Doppler shift of the target, and provides an integration gain whose theoretical value equals the product  
Figure 1: Basic geometry of a satellite IoO-based passive radar.  
\(T_{int}\cdot BW\). The acquisition time, \(T_{acq}\), is divided into coherent processing intervals (CPIs) of duration \(T_{int}\) for performing signal processing (Figure 2). The time between CPIs is the pulse repetition frequency (PRI), which can be lower than \(T_{int}\) to process overlapped CPIs, equal to \(T_{int}\) for processing all of the available data, or higher than \(T_{int}\) to obtain a compromise solution between the computational cost and the refresh rate.  
To generate the CAF, delayed and Doppler-shifted copies of the reference signal are correlated with the surveillance one (1). This operation is usually performed at base-band after analog-to-digital conversion. At this point, the reference and surveillance sequences are \(s_{ref}[n]\) and \(s_{surv}[n]\) (1) [2,34].  
\[S^{CAF}[m,p]=\sum_{n=0}^{N-1}S_{ref}^{*}[n-m]\cdot S_{surv}[n]\cdot\exp\left(- j2\pi\frac{p}{N}n\right) \tag{1}\]  
where  
* \(N=T_{int}\cdot f_{s}\), is the number of sample acquired during \(T_{int}\) (s), which defines the duration of the coherent processing interval (CPI), and \(f_{s}\) (Hz) is the sampling frequency;
* \(m=0,\ldots,M-1\) is the time bin associated with a delay \(t_{m}=m/f_{s}\), with \(M\) being the number of range cells for a given instrumented radar coverage and \(t_{m}\in[0,T_{int}]\).
* \(p=-P/2,\ldots,(P/2)-1\) is the Doppler-shift, corresponding to \(f_{D}=f_{s}(p/N)\), with \(P\) being the number of Doppler shift cells for a the Doppler interval of interest defined in \([f_{s}(-(P/2)/N),f_{s}(((P/2)-1)/N)]\).  
For each target, the result of the CAF is the ambiguity function (AF) of the transmitted signal, scaled and shifted to be centered on the target bistatic time delay and Doppler shift. The stationary targets' main contributions appear along the zero Doppler line of the CAF output domain (the Range-Doppler Map, RDM), although, depending on the signal AF, their contributions can spread throughout all of the RDM. This is the case of the DVB signals (terrestrial and satellite), which are B1-type signals.  
So far, defined signal models have a focus on the target. Actually, the reference signal is noisy, and it can be affected by multipaths. The surveillance signal also includes the direct signal from the IoO (direct path interference, DPI) and clutter. These elements are filtered by zero Doppler rejection techniques, so the detection in the zero Doppler line is not considered by focusing this work on moving targets' returns and the reduction of integration gain losses when long integration times are used.