2.2.5 Optimizing the sparse weight vector

After obtaining the array antenna positions by steps (B, C, D), the optimal weight vector wopt is further obtained by solving following convex optimization problem, which is to improve the performance of the array beampattern synthesized by the sparse weight vector:

$$\begin{aligned} \text{Find } \; & \; \boldsymbol{w}\_{opt} \\ \text{Minimize } & \; \| \boldsymbol{G}(\boldsymbol{\theta}) - \boldsymbol{G}\_d(\boldsymbol{\theta}) \|\_{\infty}, \; \boldsymbol{\theta} \in [-\boldsymbol{\Psi} \mathbf{0}^\circ, \mathbf{\mathcal{M}}^\circ] \end{aligned} \tag{5}$$

Figure 2.

Figure 1.

The desired beampattern.

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

Table 1.

39

"our 19-element array beampattern."

Our element positions and weights in a 19-element antenna array.

A "25-element array beampattern obtained by [3] and a 29-element array beampattern obtained by [17]" vs.

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

Element indices Position (λ) Weight value Element indices Position (λ) Weight value 1,19 10.500 0.2289 6,14 3.875 0.2677 2,18 8.625 0.2583 7,13 2.875 0.2195 3,17 7.6250 0.2207 8,12 1.875 0.1813 4,16 6.750 0.2904 9,11 1.000 0.2347 5,15 4.750 0.1567 10 0 0.2427

#### 2.3 Computer simulation and discussions

Given the array aperture, the objective is to design an array with the beampattern as shown in Figure 1, where region j j θ ≤θ<sup>s</sup> corresponds 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>:5<sup>∘</sup> , and the angle grid of the interval �90<sup>∘</sup> ; <sup>90</sup><sup>∘</sup> ½ � is 1<sup>∘</sup> . We design a virtual ULA with the array aperture of 25:5λ having a uniform inter-element spacing of λ=8.

The beampattern of Figure 2 is obtained by using our approach for a 19-element array, and the optimal beampattern exhibits the maximum sidelobe of �15.46 dB. The optimal antenna positions and the corresponding weights are displayed in Table 1. The designs proposed in [3, 17] describe a 25-element and a 29-element non-ULA with the approximate desired array pattern shown in Figure 2, respectively. The 25-element array beampattern described in [3] by eliminating the elements pair by pair has a maximum sidelobe �13.75 dB, while the maximum sidelobe of 29-element array beampattern obtained by the BCS algorithm [17] is �13.165 dB. Convex Optimization and Array Orientation Diversity-Based Sparse Array Beampattern Synthesis DOI: http://dx.doi.org/10.5772/intechopen.88881

Figure 1. The desired beampattern.

APS performance and convergence rate. Because more elements of the original weight vector will be pruned if the threshold value η is increased, which make us probably cannot find the optimal array element positions of the array, correspondingly the array synthesis performance is not optimal for a given array element number. Conversely, if the threshold value is decreased, less elements of the original weight vector will be pruned in each iteration, which increases the algorithm complexity. So we should make a good balance between APS performance and

After obtaining the original weight vector <sup>w</sup> <sup>¼</sup> ½ � <sup>w</sup>1; <sup>w</sup>2; … <sup>T</sup>, the weight matrix <sup>Q</sup>

greater than 1; it was demonstrated experimentally that p ¼ 2 is a better choice for our APS problem). To ensure that the weight matrix is effectively updated when a zero-valued component in w, we introduce a parameter δ>0. It is empirically demonstrated that δ should be set slightly smaller than the expected nonzero mag-

The sparse weight vector w<sup>s</sup> is obtained by pruning the original weight vector, and then the antenna elements corresponding to nonzero-valued indices of w<sup>s</sup> are

The above steps (A, B, C) are repeated until the final synthesized array performance is satisfactory or the specified maximum number of iterations is attained.

After obtaining the array antenna positions by steps (B, C, D), the optimal weight vector wopt is further obtained by solving following convex optimization problem, which is to improve the performance of the array beampattern synthe-

Minimize k k <sup>G</sup>ð Þ� <sup>θ</sup> Gdð Þ<sup>θ</sup> <sup>∞</sup>, <sup>θ</sup> <sup>∈</sup> �90<sup>∘</sup>

Given the array aperture, the objective is to design an array with the beampattern as shown in Figure 1, where region j j θ ≤θ<sup>s</sup> corresponds 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>:5<sup>∘</sup>

The beampattern of Figure 2 is obtained by using our approach for a 19-element array, and the optimal beampattern exhibits the maximum sidelobe of �15.46 dB. The optimal antenna positions and the corresponding weights are displayed in Table 1. The designs proposed in [3, 17] describe a 25-element and a 29-element non-ULA with the approximate desired array pattern shown in Figure 2, respectively. The 25-element array beampattern described in [3] by eliminating the elements pair by pair has a maximum sidelobe �13.75 dB, while the maximum sidelobe of 29-element array beampattern obtained by the BCS algorithm [17] is �13.165 dB.

; <sup>90</sup><sup>∘</sup> ½ � is 1<sup>∘</sup>

aperture of 25:5λ having a uniform inter-element spacing of λ=8.

; <sup>90</sup><sup>∘</sup> ½ � (5)

. We design a virtual ULA with the array

, and the

; ð Þ j j <sup>w</sup><sup>2</sup> <sup>þ</sup> <sup>δ</sup> �<sup>p</sup> ð Þ ½ � ; … (usually, <sup>p</sup> is an integer

convergence rate when setting the value of η.

2.2.3 Updating the weight matrix

Advances in Array Optimization

nitudes of w.

is updated as <sup>Q</sup> <sup>¼</sup> diag wð Þ j j <sup>1</sup> <sup>þ</sup> <sup>δ</sup> �<sup>p</sup>

2.2.4 Forming the nonuniform array

2.2.5 Optimizing the sparse weight vector

sized by the sparse weight vector:

angle grid of the interval �90<sup>∘</sup>

38

Find wopt

2.3 Computer simulation and discussions

retained to form a nonuniform array with fewer elements.

#### Figure 2.

A "25-element array beampattern obtained by [3] and a 29-element array beampattern obtained by [17]" vs. "our 19-element array beampattern."


#### Table 1.

Our element positions and weights in a 19-element antenna array.

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 our array.

3. Beampattern synthesis using reweighted l1-norm minimization and

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

To address left-right radiation pattern ambiguity problem, we allow exploitation

We assume that transmit signals and the array are coplanar, so the antenna array

k k Fdð Þ� θ Fð Þθ <sup>l</sup><sup>2</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>

9 >>=

>>;

jkdα<sup>i</sup> cosð Þ <sup>θ</sup>�θα (7)

Fαð Þθ (8)

Δd � �, the

½ � <sup>F</sup>ð Þ<sup>θ</sup> <sup>h</sup>�<sup>1</sup> <sup>¼</sup> ½ � <sup>H</sup> <sup>h</sup>�<sup>n</sup>½ �<sup>r</sup> <sup>n</sup>�<sup>1</sup> (9)

<sup>T</sup>, overcomplete dictionary <sup>H</sup> is an <sup>h</sup> � <sup>n</sup> matrix whose

<sup>D</sup> <sup>α</sup> � � for <sup>α</sup> <sup>∈</sup>f g <sup>1</sup>; …; <sup>D</sup> , and

<sup>D</sup> <sup>þ</sup> <sup>1</sup>; <sup>n</sup>

≤ξ (6)

<sup>R</sup>α<sup>i</sup> f g ;dα<sup>i</sup> <sup>α</sup> <sup>¼</sup> <sup>1</sup>, :…,D i ¼ 1, :…,M

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

array orientation diversity

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

3.1 Problem formulation

where <sup>F</sup>ð Þ¼ <sup>θ</sup> <sup>P</sup><sup>D</sup>

written as

form:

41

F ¼ ½ � Fð Þ θ<sup>1</sup> Fð Þ θ<sup>2</sup> ⋯ Fð Þ θ<sup>h</sup>

of the array orientation diversity in the CS framework.

minð Þ DM s:t: min

8 >><

>>:

the incident plane wave from the bearing θ, the array factor is given by

M

Rαie

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

α¼1

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

i

<sup>F</sup>ð Þ¼ <sup>θ</sup> <sup>X</sup> D

where h is the number of sampled antenna radiation pattern, <sup>n</sup> <sup>¼</sup> <sup>D</sup> <sup>2</sup>ds

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

sensing radiation pattern at different angles is contained in vector

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>F</sup>αð Þ¼ <sup>θ</sup> <sup>X</sup>

synthesis problem can be described as follows:

α¼1 P<sup>M</sup>

The proposed APS algorithm based on convex optimization and reweighted 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.


Table 2.

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


Table 3. Element positions and weights obtained by the BCS inversion algorithm [17]. Convex Optimization and Array Orientation Diversity-Based Sparse Array Beampattern Synthesis DOI: http://dx.doi.org/10.5772/intechopen.88881
