3.2.3 Updating the weight matrix

After obtaining the original weight vector <sup>w</sup> <sup>¼</sup> ½ � <sup>w</sup>1; <sup>w</sup>2; … <sup>T</sup> from step (2), the weight matrix <sup>Q</sup> is updated according to <sup>Q</sup> <sup>¼</sup> diag ð Þ j j <sup>w</sup><sup>1</sup> <sup>þ</sup> <sup>δ</sup> �<sup>p</sup> , wð Þ j j <sup>2</sup> <sup>þ</sup> <sup>δ</sup> �<sup>p</sup> ð Þ ½ � , … in each iteration; usually, p is an integer greater than 1, while it was demonstrated experimentally that p ¼ 2 is a better choice for our APS problem. To ensure regular update Q especially for zero-valued components in w, we bring in the parameter δ>0 which should be set slightly smaller than the expected nonzero magnitudes of w. Reweighted l<sup>1</sup> minimization can improve the signal reconstruction performance. Convex Optimization and Array Orientation Diversity-Based Sparse Array Beampattern Synthesis DOI: http://dx.doi.org/10.5772/intechopen.88881

## 3.2.4 Creating the nonuniform arrays

αth array is absent from the supposed position, and the solution of sparse excitation

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].

In this subsection, the new solution of Eq. (6) can be summarized as follows:

To place more antenna elements than those of a conventional array with the same array size, we first create D virtual linear orientation arrays with much smaller interspacing λ=16 (in general, the inter-element spacing of the conventional ULA is λ=2). Using the reweighted l1-norm minimization in the following step, we set a

Let Fð Þθ be a synthesized beampattern by using a weight vector, and the weight vector can be obtained by solving the following weighted l1-norm minimization convex problem which is to try to minimize the peak value of the error between the

where ζ is the fitting error between the synthesized pattern and the desired one.

After obtaining the original weight vector <sup>w</sup> <sup>¼</sup> ½ � <sup>w</sup>1; <sup>w</sup>2; … <sup>T</sup> from step (2), the

in each iteration; usually, p is an integer greater than 1, while it was demonstrated experimentally that p ¼ 2 is a better choice for our APS problem. To ensure regular update Q especially for zero-valued components in w, we bring in the parameter δ>0 which should be set slightly smaller than the expected nonzero magnitudes of w. Reweighted l<sup>1</sup> minimization can improve the signal reconstruction performance.

Subject to k k <sup>F</sup>ð Þ� <sup>θ</sup> Fdð Þ<sup>θ</sup> <sup>∞</sup> <sup>≤</sup>ζ, <sup>∀</sup><sup>θ</sup> <sup>∈</sup> �180<sup>∘</sup>

Minimizing k k Qw <sup>1</sup> makes the vector Qw sparse, which is useful to create D nonuniformly spaced linear orientation arrays. Here, let the weight vector w ¼ ½ � <sup>w</sup>1; <sup>w</sup>2; … <sup>T</sup> obtained from Eq. (11) be the original weight vector for convenience. The weighted l1-norm minimization will make some weights of the original weight vector be very small, so they can be adjusted to zero without significantly reducing the array performance. That is, if the absolute value of an element from the original weight vector is smaller than a threshold which is set according to the array performance requirement, the element will be assigned zero; otherwise, the element will

subject to k k <sup>F</sup>‐H<sup>r</sup> <sup>∞</sup> <sup>≤</sup><sup>ξ</sup> (10)

; <sup>180</sup><sup>∘</sup> ½ � (11)

, wð Þ j j <sup>2</sup> <sup>þ</sup> <sup>δ</sup> �<sup>p</sup> ð Þ ½ � , …

vector r can be casted as the following convex optimization problem:

min k kr <sup>1</sup>

3.2 The proposed algorithm

Advances in Array Optimization

3.2.2 Finding the sparse weight vector

3.2.3 Updating the weight matrix

42

synthesized pattern and the desired pattern:

Minimize k k Qw <sup>1</sup>

be retained. Thus the sparse weight vector w<sup>s</sup> is obtained.

weight matrix <sup>Q</sup> is updated according to <sup>Q</sup> <sup>¼</sup> diag ð Þ j j <sup>w</sup><sup>1</sup> <sup>þ</sup> <sup>δ</sup> �<sup>p</sup>

3.2.1 Initializing a virtual array and a weight matrix

DM � DM dimension weight matrix Q as a unit matrix.

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 create D sparse linear arrays with different orientations.

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