3.2.2 Finding the sparse weight vector

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 synthesized pattern and the desired pattern:

$$\begin{aligned} \text{Minimize } & \|\mathsf{Q}\boldsymbol{\nu}\|\_{1} \\ \text{Subject to } & \|\boldsymbol{F}(\boldsymbol{\theta}) - \boldsymbol{F}\_{d}(\boldsymbol{\theta})\|\_{\infty} \leq \zeta \text{, } \forall \boldsymbol{\theta} \in [-180^{\circ}, 180^{\circ}] \end{aligned} \tag{11}$$

where ζ is the fitting error between the synthesized pattern and the desired one. 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 be retained. Thus the sparse weight vector w<sup>s</sup> is obtained.
