3.2.5 Finding the optimal weight vector

After optimizing the antenna element positions by the above steps, we introduce convex optimization to obtain the optimal weight vector which can further improve the performance of the array beampattern synthesized by the sparse weight vector:

$$\begin{aligned} \text{Find } \; & \; \mathbf{w}\_{opt} \\ \text{Minimize } \; & \| F(\theta) - F\_d(\theta) \| \_{\infty}, \theta \in [-\mathbf{90}^\circ, \mathbf{90}^\circ] \end{aligned} \tag{12}$$

The optimal sparse weight vector wopt can be obtained from Eq. (12) readily.

#### 3.3 Computer simulations and discussion

The objective is to design an array with the desired beampattern for given the array physical size, as shown in Figure 1, where region j j θ ≤ θ<sup>s</sup> belongs to the mainlobe and region j j <sup>θ</sup> <sup>≥</sup> <sup>θ</sup><sup>s</sup> corresponds to the sidelobe. We set <sup>θ</sup><sup>s</sup> <sup>¼</sup> <sup>2</sup>:3<sup>∘</sup> , and the angle grid for the search area �180<sup>∘</sup> ; <sup>180</sup><sup>∘</sup> ½ � is 2<sup>∘</sup> , that is, we take a "dense set" of �180<sup>∘</sup> ; <sup>180</sup><sup>∘</sup> ½ � with the angles sampled at 2<sup>∘</sup> from �180<sup>∘</sup> to 180<sup>∘</sup> (Figure 3).

To show the performance of our beampattern synthesis, we will consider two cases, same element number array and same beampattern performance, since all formulated problems in Eqs. (6), (10), (11), and (12) are convex, so we adopt the optimization toolbox to solve the formulated problems.

Figure 3. Desired beampattern.

3.3.1 Same element number array with array orientation diversity

DOI: http://dx.doi.org/10.5772/intechopen.88881

(named Array 1, Array 2, Array 3, Array 4, with orientation �10<sup>∘</sup>

Element positions and excitation amplitudes in a 19-element two-array orientation.

Element positions and excitation amplitudes in a 19-element three-array orientation.

ingly bigger than that of our proposed method.

Figure 6.

Figure 7.

45

In this section, we analyzed the influence of the array orientation diversity on the beampattern synthesis by simulation results. We initialize four virtual ULAs

Convex Optimization and Array Orientation Diversity-Based Sparse Array Beampattern Synthesis

respectively) with each subarray aperture of 25λ owning a uniform interspacing λ=8. Besides, we initialize <sup>Q</sup> as a unit matrix and choose <sup>δ</sup> <sup>¼</sup> <sup>10</sup>�<sup>4</sup> and <sup>p</sup> <sup>¼</sup> 2 in our simulations. Figure 4 shows a 19-element beampattern synthesis performance in four cases with one-, two-, three-, and four-array orientations. From Figure 4, we can see that our proposed method and BCS algorithm can improve performance with increasing array orientation diversity (from 1 to 4); the optimal antenna positions and the corresponding excitation amplitudes of the four cases are displayed in Figures 5–8, respectively. Note that for the four cases of Figures 5–8, the required normalized radiated energies of BCS approach [17] are correspond-

, 0<sup>∘</sup> , 10<sup>∘</sup> , 20<sup>∘</sup> ,

#### Figure 4.

A 19-element array performance obtained by BCS inversion algorithm [17] and our method with increasing array orientation diversity. (a) 1 array orientation, (b) 2 array orientations, (c) 3 array orientations, and (d) 4 array orientation.

Figure 5. Element positions and excitation amplitudes in a 19-element one-array orientation.

Convex Optimization and Array Orientation Diversity-Based Sparse Array Beampattern Synthesis DOI: http://dx.doi.org/10.5772/intechopen.88881
