2.1.4.1 The electronic multiple null steering method

Here in this subsection, the solution proposed in Section 2.1.2 is further extended to include which elements are to be optimized and to what extent. The small number of the selected elements should have an impact on the controllable nulls. Therefore, we examine three different selection strategies. The first strategy selects the most effective elements that are located on the extremes of the array; while in the second and third strategies, the selections consider the elements that are located at the center of the array or randomly chosen from the whole array elements [21]. The idea of the second strategy was employed in a sidelobe adaptive canceller system [27]. By adapting few elements at the center of the array, the system is capable to produce multiple nulls toward a number of interfering signals [27]. Experience with these three selection strategies showed that the first strategy provides best performance for interference suppression [21]. To apply the first strategy, first, consider an array of an even number of elements 2N, with uniform amplitude excitations and mechanically fixed locations with uniform inter-element spacing d, symmetrically positioned about the origin (i.e., N elements are placed on each side of the origin). Assuming a subset of only 2M elements (out of the 2Nelement array) is optimized to generate the required nulls at unwanted directions (i.e., M outer elements on each end of the array). The remaining 2N-2M elements are kept unchanged, i.e., having uniform amplitude and equal-phase excitations. The overall far-field pattern due to the 2N-2M uniformly excited array elements and the 2M adaptive array elements can be written as [21]:

$$\text{AF}(\mathbf{u}) = 2 \underbrace{\sum\_{n=1}^{(\mathbf{N}-\mathbf{M})} \cos\left[\frac{(2\mathbf{n}-\mathbf{1})}{2}kdu\right]}\_{\text{2N-2M uniform array}} + \underbrace{\sum\_{n=\mathbf{N}-\mathbf{M}+1}^{\text{N}} \left\{\mathbf{A}\_{m}\mathbf{e}^{j\left(\frac{(2m-1)}{2}kdu + P\_{m}\right)} + \mathbf{A}\_{m}\mathbf{e}^{-j\left(\frac{(2m-1)}{2}kdu + P\_{m}\right)}\right\}}\_{\text{2M elements array}},\tag{2}$$

where k ¼ 2π=λ. From (2), it can be noted that the amplitudes (Amr and Aml for right and left subset elements) and phases (Pmr and Pml for right and left subset elements) of 2M adjustable elements array can be considered either symmetric or nonsymmetric (i.e., Aml 6¼ Amr and Pml ¼6 Pmr) like the earlier method. Also note that the 2M adjustable elements are selected from the extremes of the array and they play an important role in generating the required nulls. The structure of the interior 2N-2M uniformly excited array elements with adjustable amplitude and phase excitations of the outer M elements on each side of the array is shown in Figure 9 [21].
