3.1 Problem formulation

The antenna positions and the corresponding weights of the two methods in [3, 17] are listed in Tables 2 and 3, respectively. Compared with the method in [3], we can see from Table 1 and Figure 2 that our proposed algorithm saves six elements without reducing the array performance and the our minimum inter-element spacing of the non-ULA is 0.375λ larger than that of the method of eliminating the elements pair by pair [3]. Compared with the reference method in [17], our proposed method offers an economization of 10 elements as well as 2.3 dB performance improvement, and the minimum inter-element spacing of the sparse array designed by our approach is 0.75λ larger than that of the reference array [17]. We also emphasize that the reference array [17] has 4.5λ larger array aperture than that of

The proposed APS algorithm based on convex optimization and reweighted

Element indices Position (λ) Weight value Element indices Position (λ) Weight value 1,25 12.0 0.2100 8,18 4.5 0.1924 2,24 8.5 0.2605 9,17 3.5 0.2296 3,23 8.0 0.2276 10,16 2.0 0.2282 4,22 7.5 0.2554 11,15 1.5 0.0876 5,21 7.0 0.2103 12,14 0.5 0.1143 6,20 6.0 0.200 13 0 0.2084

Element indices Position (λ) Weight value 1,2 1.375, 0.500 0.0876, 0.1178 3,4 0.375, 0.500 0.0532, 0.1025 5,6 1.250, 1.375 0.0497, 0.1844 7,8 2.125, 4.375 0.1895, 0.1086 9,10 4.500, 5.250 0.0778, 0.2679 11,12 6.125, 7.000 0.2440, 0.1098 13,14 7.125, 8.125 0.0643, 0.2400 15,16 8.875, 10.125 0.1953, 0.2249 17,18 11.000, 11.750 0.2297, 0.0720 19,20 12.000, 12.750 0.1480, 0.1810 21,22 13.750, 13.875 0.0761, 0.0554 23,24 14.875, 15.750 0.0840, 0.1833 25,26 16.625, 16.750 0.1860, 0.0516 27,28 17.375, 19.625 0.1625, 0.0745 29 24.125 0.0317

l1-norm minimization is proven to be effective in reducing array elements, suppressing the sidelobe, and reducing the aperture. This simple and effective design method can be extended to solving the 2D array synthesis problem.

our array.

Advances in Array Optimization

Table 2.

Table 3.

40

7,19 5.0 0.2037

Element positions and weights obtained in a 25-element array [3].

Element positions and weights obtained by the BCS inversion algorithm [17].

We assume that transmit signals and the array are coplanar, so the antenna array synthesis problem can be described as follows:

$$\min(DM) \quad \text{s.t.} \left\{ \min\_{\{R\_{\tilde{w}}, d\_{\tilde{w}}\}\_{a=1,...,D}} ||F\_d(\theta) - F(\theta)||\_{l\_2} \right\} \le \xi \tag{6}$$

where <sup>F</sup>ð Þ¼ <sup>θ</sup> <sup>P</sup><sup>D</sup> α¼1 P<sup>M</sup> <sup>i</sup> <sup>R</sup>αiejkdα<sup>i</sup> cosð Þ <sup>θ</sup>�θα , Fdð Þ<sup>θ</sup> is the desired radiation pattern, <sup>M</sup> is the number of identical antenna elements in each linear array, Rα<sup>i</sup> is the excitation coefficient of the ith element located at dα<sup>i</sup> in the αth array, k is the wavenumber in the free space, and D array orientations θαð Þ α ¼ 1; …; D . The objective of the problem is to synthesize the desired radiation pattern Fdð Þθ with the minimum number of elements under a small tolerance error ξ. For one linear array at orientation θα to the incident plane wave from the bearing θ, the array factor is given by

$$F\_a(\theta) = \sum\_{i}^{M} R\_{ai} \mathcal{e}^{jkd\_{ai}\cos\left(\theta - \theta\_a\right)}\tag{7}$$

Suppose that all the antenna elements in each array orientation θαð Þ α ¼ 1; …; D are symmetrically distributed within a range of �ds to ds along the array orientation θα, respectively, the combination pattern of all the linear orientation arrays can be written as

$$F(\theta) = \sum\_{a=1}^{D} F\_a(\theta) \tag{8}$$

In order to solve Eqs. (7) and (8), we can assume that all the antenna elements are equally spaced from �ds to ds with a small inter-element spacing Δd. Although it is supposed that there is one element at each position, not each antenna element is necessarily radiating waves or excited with current. All the antenna elements can be in two states: "on" states (when the element is in the supposed position or has an excitation) and "off" state (when there is no element in the supposed position or without an excitation). Through discretization, Eq. (8) can be written in a matrix form:

$$[F(\theta)]\_{h\times \mathbf{1}} = [H]\_{h\times n}[r]\_{n\times \mathbf{1}} \tag{9}$$

where h is the number of sampled antenna radiation pattern, <sup>n</sup> <sup>¼</sup> <sup>D</sup> <sup>2</sup>ds Δd � �, the sensing radiation pattern at different angles is contained in vector F ¼ ½ � Fð Þ θ<sup>1</sup> Fð Þ θ<sup>2</sup> ⋯ Fð Þ θ<sup>h</sup> <sup>T</sup>, overcomplete dictionary <sup>H</sup> is an <sup>h</sup> � <sup>n</sup> matrix whose ð Þ <sup>i</sup>; <sup>l</sup> th element is <sup>H</sup>il <sup>¼</sup> ejkd<sup>α</sup><sup>i</sup> cosð Þ <sup>θ</sup>l�θα , l <sup>∈</sup> ð Þ <sup>α</sup> � <sup>1</sup> <sup>n</sup> <sup>D</sup> <sup>þ</sup> <sup>1</sup>; <sup>n</sup> <sup>D</sup> <sup>α</sup> � � for <sup>α</sup> <sup>∈</sup>f g <sup>1</sup>; …; <sup>D</sup> , and h ≪ n. r is an excitation vector, R<sup>α</sup><sup>i</sup> ¼ 0 means the antenna in the lth position of the αth array is absent from the supposed position, and the solution of sparse excitation vector r can be casted as the following convex optimization problem:

$$\begin{array}{ll}\min & \|r\|\_1\\ \text{subject to} & \|\mathbf{F}\cdot\mathbf{H}r\|\_\infty \leq \xi \end{array} \tag{10}$$

3.2.4 Creating the nonuniform arrays

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

3.2.5 Finding the optimal weight vector

number is attained.

create D sparse linear arrays with different orientations.

Find wopt

optimization toolbox to solve the formulated problems.

3.3 Computer simulations and discussion

angle grid for the search area �180<sup>∘</sup>

�180<sup>∘</sup>

Figure 3.

43

Desired beampattern.

After obtaining the sparse weight vector w<sup>s</sup> from step (4), the antenna elements corresponding to nonzero-valued indices of the sparse weight vector are retained to

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

Repeat steps (2, 3, and 4) until the synthesized array beampattern performance is satisfactory or the specified maximum number of iterations or minimum antenna

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:

; <sup>90</sup><sup>∘</sup> ½ � (12)

, that is, we take a "dense set" of

, and the

Minimize k k <sup>F</sup>ð Þ� <sup>θ</sup> Fdð Þ<sup>θ</sup> <sup>∞</sup>, <sup>θ</sup> <sup>∈</sup> �90<sup>∘</sup>

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

The objective is to design an array with the desired beampattern for given the

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

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>

; <sup>180</sup><sup>∘</sup> ½ � is 2<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).

In Eq. (10) the smallest number of nonzero elements in the excitation vector r can be obtained readily by using existing software package, such as CVX [31].
