A Parameterized Pattern-Error Objective for Large-Scale Phase-Only Array Pattern Design

A hybrid global-local optimization approach to phase-only array-pattern synthesis

“…That formulation required significant hand-tuning of parameters to achieve convergence, largely due to a poor initialization choice. The modified algorithm described in this paper incorporates the moregeneral objective function from [15] along with a penaltyfunction approach to enforce the phase-only constraint. An optional deep sidelobe region constraint is enforced either by introduction of an additional penalty term or by restricting the step at each iteration to the subspace orthogonal to a linearization of the sidelobe constraints.…”

“…In particular, larger p emphasizes peak approximation errors (thus providing more of an equiripple error) while smaller p emphasizes the average errors. This weighted pattern-error metric is the same as that used in [15], with the exception that here it is a function of the complex array weights rather than of the phase of the weights. This in turn necessitates the constraints (2b) to enforce the fixed magnitude of the complex weights which were structurally imposed in [15].…”

“…This weighted pattern-error metric is the same as that used in [15], with the exception that here it is a function of the complex array weights rather than of the phase of the weights. This in turn necessitates the constraints (2b) to enforce the fixed magnitude of the complex weights which were structurally imposed in [15]. The left side of constraint (2c) has the same form as the objective but with a different weighting function W and potentially different p and q.…”

“…The subscripts "o" and "c" on (W, p, q) indicate objective and constraint, respectively. Compared to the unconstrained optimization of [15], here the extra constraint (2c) allows the error in a specific region (a sidelobe region, say) to be explicitly bounded without tweaking a common weighting function. The degrees of freedom remaining after the constraints are met are applied to minimizing the error in another region (the mainlobe, say) resulting in a Pareto-efficient design.…”

“…We can employ the FFT-based approach detailed in [15], which while restricting the element locations {d n } to a lattice provides considerable computational savings when the weighting function W is nonzero on a significant number of points in the sum.…”