2. Nonuniform array pattern synthesis using reweighted l1-norm minimization

## 2.1 Problem formulation

Consider a narrowband linear array with M isotropic antennas located at x1, …, xM ∈R<sup>2</sup> . Assume that a harmonic plane wave with wavelength λ propagates across the array with incident direction θ. The M signal outputs si are converted to the baseband, weighted by the weights wi, and summed. Then the array response can represented as

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

$$G(\theta) = \sum\_{i=1}^{M} w\_i \exp\left(j2\pi\mathbf{x}\_i \sin\theta/\lambda\right) = \mathfrak{w}^T \mathfrak{a}(\theta) \tag{1}$$

where ϕ<sup>i</sup> ¼ 2πxi sin θ=λ is the phase delay due to propagation, complex weight vector w ¼ ½ � w1; …; wM <sup>T</sup> <sup>∈</sup>CM, and the steering vector <sup>a</sup>ð Þ<sup>θ</sup> .

Let Gdð Þθ be the desired array response at the direction θ. The APS problem is to find the complex weight vector <sup>w</sup> such that <sup>G</sup>ð Þ¼ <sup>θ</sup> Gdð Þ<sup>θ</sup> for all <sup>θ</sup> <sup>∈</sup> �90<sup>∘</sup> ; <sup>90</sup><sup>∘</sup> ½ �. For the array described above, how well Gð Þθ approximates Gdð Þθ can be measured by using the peak error across θ, i.e.,

$$\min\_{\omega} \max\_{\theta \in \Theta} |G(\theta) - G\_d(\theta)| \tag{2}$$

where <sup>Θ</sup> <sup>∈</sup> �90<sup>∘</sup> ; <sup>90</sup><sup>∘</sup> ½ � is a dense set of arrival angles that we are of interest. The goal of the proposed algorithm is to find both optimal antenna locations and corresponding weights that approach the desired array pattern as well as possible.
