A explain why this 2d lter design problem is convex b

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Unformatted text preview: shading coefficients, and will be the design variables in the problem. For a given set of weights, the combined output G(θ) is a function of the angle of arrival θ of the plane wave. The design problem is to select weights wi that achieve a desired directional pattern G(θ). We now describe a basic weight design problem. We require unit gain in a target direction θtar , i.e., G(θtar ) = 1. We want |G(θ)| small for |θ − θtar | ≥ ∆, where 2∆ is our beamwidth. To do this, we can minimize max |G(θ)|, tar |θ −θ |≥∆ where the maximum is over all θ ∈ [−π, π ] with |θ − θtar | ≥ ∆. This number is called the sidelobe level for the array; our goal is to minimize the sidelobe level. If we achieve a small sidelobe level, then the array is relatively insensitive to signals arriving from directions more than ∆ away from the target direction. This results in the optimization problem minimize max|θ−θtar |≥∆ |G(θ)| subject to G(θtar ) = 1, with w ∈ Cn as variables. The objective function can be approximated by discretizing the angle of arrival with (say) N values (s...
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