## 2 Materials and Methods  
### Speckle Whitening of Complex SAR Data  
#### 2.1.1 SAR System: An Overview  
SAR images are computed images synthesized by processing radar returns captured from a moving platform [9]. A train of pulses, also known as chirps, is transmitted along a leaning direction, referred to as the slant range, orthogonal to the track of the platform, which constitutes the conjugate azimuth direction. The SAR system receives and stores the complex returns (amplitude and phase) of the objects encountered in the cross-track direction, which are located, thanks to the delays of their responses, through Doppler processing [29]. In-phase and quadrature components of the complex envelope of the received signal are sampled at a frequency at least twice the bandwidth of the chirps, quantized, and stored; they constitute the two-dimensional (2D) array of raw data.  
The 2D array of raw data, which exhibits a grainy, noise-like appearance and a mean that slowly varies in space, has to be focused by the second segment of the SAR system,called the SAR processor, which is usually located on the ground because of its high processing power requirements. SAR processing consists of a 2D deconvolution, typically separable, of the shape of the band-limited pulse [30] along the slant-range direction (range focusing), and of the so-called azimuth chirp, caused by the varying Doppler shift within the antenna beamwidth in the azimuth direction, which develops along the track of the platform and whose shape is squeezed with the range (azimuth focusing) [9; 29]. Fourier-domain deconvolution is fast and accurate but cannot be used for drones, whose trajectory is unpredictable [31].  
The range and azimuth deconvolutions are intended to emulate (a) a compressed range pulse, which is exactly band-limited and thus has a perfectly rectangular Fourier transform; and (b) a synthetic antenna beam with a narrow main lobe in the azimuth direction. Unfortunately, after the 2D deconvolution, the range chirp becomes a sinc function, that is, the inverse Fourier transform of a rect function; the sinc function has tails that decay slowly along the range. Similarly, in the azimuth direction, after the azimuth chirp has been deconvolved, the same issue occurs. The undesirable consequence of the deconvolution process is that a bright point target becomes surrounded by a dashed cross, not exactly square unless the squint effect [29] is corrected, caused by the 2D Gibbs effect.  
The on-ground SAR processor usually employs frequency windows, typically Hamming or Kaiser windows, to avoid the dispersion of point targets. The drawback is that fully developed speckle, which, in principle, should be spatially uncorrelated if the raw data are not oversampled, becomes autocorrelated and presents a grainy appearance. Speckle stems from the coherent combination of radar returns from multiple elementary scatterers at the receiver. If the cell contains few dominant scattering elements, speckle is said to be partially developed. This happens in textures typically found in urban areas [32]. The autocorrelation function of speckle may impact the subsequent analysis/processing of complex SAR data.  
Figure 1 shows the end-to-end flowchart of the system, including the (optional) spatial decorrelation stage. The cascade of onboard SAR sensors and the on-ground SAR processor make up the complete SAR system. In principle, the SAR system should yield the product of radar reflectivity and complex speckle that is found at its input. In practice, the SAR processor may include tapering windows along both the range and azimuth directions; hence, the output may become autocorrelated. However, it can be decorrelated by applying the inverse filter of the tapering window.