## 4 Sampling the Aperture
### Grating Lobes  
The effect of interfering sidelobes due to a discontinuous phase between sub-apertures is different to the effect of grating lobes. If the sensor spacing is larger than \(\lambda\), the peak of a grating lobe already occurs within the region of propagating signals. The problem occurs due to the length of the receiver array between the transmitter positions building in turn a pseudo transmitter array. The proposed concept of sub-apertures implies separated transmitter positions with a spacing \(L/2\) causing already grating lobes even within a smaller visible region. Although most energy is suppressed by the receiving beampattern, this effect needs to be considered. Figure 6(a) shows the overlay of a transmitter/receiver beam pattern according to the concept of a uniformly weighted linear array (11) (see [28]).  
The combined beam pattern is separated into the transmitter pattern having four different positions plus the full receiver pattern (\(N_{sf}=384\)) of the physical length of all sub-apertures. The combination  
Figure 6: Phase relation of all sensors at a depth of 10 m from a point-scatterer (placed at 10 m). The y-axis shows the relation along the lateral range of four array lengths. The x-axis shows an SAS configuration of adjacent arrays.  
Figure 7: Theoretical beam pattern of uniform linear arrays. Comparison of grating lobes using different distances between sub-apertures. Parameters are according to the example with four array positions (Section 3).  
for two different lengths of overlapping arrays is shown in Figure 6(b). The overlap of 50% achieves a sufficient suppression since the grating lobes lie within the Nulls of the receiver pattern. This fact is due to the ratio of positions for grating lobes and zeros of uniform linear arrays. Grating lobes (\(u_{gr}\)) and zeros (\(u_{null}\)) occur corresponding to (10). Sticking to an array overlap of 50%, the transmitter sensor distance \(D_{trans}\) can be expressed as \(\frac{N_{o}}{2D_{rec}}\) with \(D_{rec}\) as the receiver spacing. This leads to a ratio of \(u_{null}\) and \(u_{gr}\), which equals \(\frac{2m_{1}}{m_{2}}\). Hence, multiples by a factor 2 of grating lobes fall automatically into zeros of the receiver beampattern.  
As positions of 50% overlap cannot always be acquired in reality, the concept of null-steering can be applied in a straightforward way. Figure 8 shows an example with significant grating lobes of up to 20 dB. Null-steering approximates a desired pattern subject to a set of null constraints. The matrix in brackets (14) is the projection matrix orthogonal to the constraint subspace, which results in the optimal weighting coefficients (see [28]). The matrix \(C\) (13) defines the null-steering positions given in \(u_{n}\). The coefficients \(W_{o}\) are applied on the receiver data segments where the overlapping regions are processed with the same coefficients. The derivation is given in [28], using the same notation \(u\) defined in the range from \(-1\) to \(+1\), which corresponds to a coverage of 180\({}^{\circ}\). The plots in Figures 7 and 8 are calculated for a depth of 10 m and range of 2 m according to the experiments.  
\[u_{gr}=m_{1}\frac{\lambda}{D_{trans}}\,\quad u_{null}=m_{2}\frac{\lambda}{N_{sf}D _{rec}} \tag{10}\]  
\[B_{u}(u)=\frac{\sin(\frac{\pi Nd}{\lambda}u)}{\sin(\frac{\pi d}{\lambda}u)} \quad-1<u<1 \tag{11}\]  
\[W_{d}=\frac{1}{N}\quad n=0,...,N-1; \tag{12}\]  
\[C=[e^{-i\pi nu_{n}(1)}e^{-i\pi nu_{n}(2)}\...\ ] \tag{13}\]  
\[W_{o}=W_{d}(I_{N}-C[C^{H}C]^{-1}C^{H}) \tag{14}\]