J. O. Coleman

Uniform circular array pattern synthesis using second-order cone programming
W. Mark Dorsey, Jeffrey O. Coleman, and William R. Pickles
Here we formulate second-order cone programs (SOCPs) for synthesizing complex weights for far-field directional (single-point mainbeam) patterns for narrowband arrays. These formulations, while constructed here with the uniform circular array (UCA) in mind, are actually quite general in that they control the arbitrary-pol sidelobe level (SLL) and co-pol SNR loss relative to ideal by minimizing either while upper-bounding the other. The SLL can be addressed in either an L-infinity sense or an L1 sense, and elements are assumed characterized by individual embedded complex patterns, modeled or measured, and so need not be identical. Conformal arrays are the obvious application, but we leave that for others and here instead apply the SOCPs to uniform circular arrays of directional elements. Design examples assume an antipodal Vivaldi element design with an embedded element pattern obtained through simulation using appropriate unit-cell boundary conditions. Rotation and translation of that simulated pattern provides embedded element patterns for all elements of the circular array.
authors' preprint (1.3Mb)
at IEEE Xplore
In the preprint the plots use color, making it easier reading than the official IEEE/IET version.
This appears in the first paragraph of Section 2.3 [preprint: II-C]:

In this paper, we show two SOCP formulations for synthesizing narrowband directional patterns from a phased array. The first method maximizes SNR while upper bounding SLL, and the second method lower bounds SNR while minimizing peak SLL.

Rather than "In this paper..." it should have been "In 2.3.1 [preprint: 1)] below..." Further, to the end of the above, as modified, it would have been appropriate to add "In 2.3.2 [preprint: 2)], peak SLL is replaced with an approximation of the L1 norm of the array's sidelobe response."
In Section III [preprint: 3]:

"Synthesized patterns are shown in Fig. 3(a)..." should refer to Fig. 3, and "results obtained using SOCP (4)" should refer to SOCP (4) with the point bounds removed.

In the Fig 3 caption, "with SLL minimization formulation" should be removed.

"The pattern in Fig. 4 was synthesized... using SOCP (4)" should refer to SOCP (4) with \delta replaced with \delta_s to apply different SLL bounds in different sidelobe regions.

"The sidelobes for \phi > 0 ... \phi < 0 " should refer to 8 deg rather than 0. The same is true for the Fig 4 caption.

In the sentence before "Section IV Conclusion," the remark "(shown in Fig. 2)" should really be "(magnitudes shown in Fig. 2)."

In the Conclusion, "straightward" of course should be "straightforward."
Published in IET Microwaves, Antennas, and Propagation, June 2015, vol. 9, no. 8, pp. 723-727.
July 2015. Clarifications added December 2020.