## 4 Sampling the Aperture  
A constraint for the processing of sub-apertures is similar to the sampling of any signal. A continuous sampled signal along the processed aperture is necessary to avoid spectral artifacts. In [31], a similar problem of nonuniformly synthetic aperture positions for SAR nondestructive evaluation imaging is investigated. There, an approach is introduced, which allows sampling from non-uniform positions using a non-uniform fast FT (NUFFT) for the Stolt interpolation. Note that a gap within the aperture can be seen as a multiplication of a rectangular window with subsequent sub-apertures (single scan-data) in the spatial domain.  
The example from Section 3 representing a point-spread function from multiple sub-apertures is now repeated, but with an information gap between four neighboring antenna positions. A corresponding beampattern compared to the ideal beampattern in Figure 3(b) shows therefore considerably higher sidelobes. Figure 5 shows a comparison of beampattern without a continuous phase along the sub-apertures. Figure 4(a) shows the case with arrays spatially directly connected. Nevertheless, the result is a significant interfering sidelobe. Figure 4(b) shows another invalid configuration with a spatial gap of 20% of the array length between every sub-aperture. The sidelobes in Figure 4(b) are stronger, which can be explained by the physical interrupt between the neighboring arrays.  
As shown by the beampattern in Figure 4(a), a straighforward linkage of arrays for synthetic aperture processing is not possible. The reason for the discontinuities within the received data from all sub-apertures are the different sender positions from which each single scan is recorded. Figure 6 shows a detailed phase response for all four arrays. The phase is displayed for a point-scatterer (_PS_) at a fixed depth and calculated at this depth \(z_{PS}\), \(S_{bf}(s,z_{PS})\). The x-axis displays all sensors of all subsequent arrays where the y-axis displays this phase along the lateral range for \(z_{PS}\). The normalized representation shows four apertures with spacing \(\frac{\lambda}{2}\) and 128 sensors. The lateral range along the y-axis is the same length as the four array apertures on the x-axis. According to Equation (5), the array manifold vector is summed along all sensors for an image point reconstruction. Figure 5(a) shows interruptions along the data of the sub-apertures. In contrast, Figure 5(b) shows a continuous phase along all sub-apertures. The 50% overlap between neighboring arrays keeps the position from one transmitter (center) \(S1\) and the outer receiver element \(R2_{o}\) from a neighboring array at the same position. The consequence is the same run-time to any spatial position \(\tau_{S1}=\tau_{R2o}\). The same holds for the second transmitter (center) \(S2\) and outer receiver element \(R1_{o}\) from the first array \(\tau_{S2}=\tau_{R1o}\). Hence, an overlap of 50% guarantees the same phase at the transition from one sub-aperture to the other, which yields an ideal reconstruction, as shown in Figure 2(a). This explains the erroneous sidelobes in  
Figure 5: Comparison of image reconstruction (beampattern from point scatterer) using different distances between array sub-apertures.  
the beampattern Figure 4(a). A practical implementation to obtain single scans at appropriate positions is recording a dense set and selecting the suitable positions after image registration.