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I've seen that, despite some phased array antennas are rectangular, many of them are not. They are octagonal, decagonal and even more.

enter image description here

Why are they built in this way?

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    \$\begingroup\$ From a pure RF and DSP SNR point of view, the 'ideal' shape is circular, for a number of reasons that should be obvious. Octagonal is a good approximation to this, while retaining Cartesian support for addressing the elements and physically building the array - two considerations not to be sniffed at. \$\endgroup\$
    – Neil_UK
    Jan 28 at 18:24
  • \$\begingroup\$ @Neil_UK Why is it the ideal shape? A circular shape makes me think of some reasons of symmetry. But I don't understand why this should be desired. \$\endgroup\$
    – Kinka-Byo
    Jan 28 at 18:33
  • \$\begingroup\$ You ask a lot of the right questions on this site, you appear to be looking in the right direction. Think about the properties that a circle has that other shapes don't have. Perhaps why bubbles are spheres, what controls the shape of a bubble? \$\endgroup\$
    – Neil_UK
    Jan 28 at 19:21
  • \$\begingroup\$ And you should give credit to where you pulled that picture from. \$\endgroup\$
    – SteveSh
    Jan 28 at 20:40
  • \$\begingroup\$ @Kinka-Byo - I guess I don't understand what Neil_UK means by ideal. A circular shape does have some amount of symmetry to it, but I'm not sure why that makes it ideal. Note that I've been working with phased array antenna systems for most of my career (over 50 years) and have never worked with an ideal circular array. \$\endgroup\$
    – SteveSh
    Jan 28 at 22:48

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There are several reasons for those shapes.

First, the cut off/rounded corners make it easier to fit into the nose of an aircraft. This is true of the phased arrays, both active and passive, that are found on the B-1B, F-16 (newer versions), F-22, F-35, etc.

Most square or rectangular arrays are found in ground or ship born applications, where packaging constraints aren't quite as severe. Typical of such an array is the Ground/Air Task Oriented Radar (G/ATOR) for the United States Marine Corps. Picture below is from Defence Blog:

enter image description here

Second, if the array is an AESA (Active Electronically Steered Array), there's a whole bunch of expensive electronics (LNAs, filters, phase shifters, attenuators) behind each antenna element. For cost reasons, you like to keep the number of elements to the minimum needed to provide the aperture gain needed.

Most arrays use amplitude weighting in both the X and Y axis of the antenna in receive in order to control the sidelobe level. This amplitude weighting means that the end elements along both axis, and especially the elements in what would be the square corner of the array are heavily attenuated and so contribute little to the overall performance of the array. So in the case of a receive-only antenna those elements are removed in order to save cost.

For an antenna that's used to transmit (and receive), the name of the game is usually to put as much power out as possible. That is, to maximize the Effective Isotropic Radiated Power, or EIRP. In that application, edge or corner elements are usually left in place to take advantage of the transmit output power of those elements, even if they do not contribute much to receive operation. That is, assuming they're not removed in order to fit the array into the volume constraints.

Added Antenna Below

Here's an array antenna that is similar to some that I have worked with in the past. It is shaped like a parallelogram, and the elements are located parallel to the tilted side.

enter image description here

An antenna shaped like this has certain characteristics related to sidelobes that make it attractive for some applications. Note that the individual radiating elements, denoted by the "X" in the diagram, are not located on a nice Cartesian grid. This makes the computation of the element's phase shifts a bit more difficult, but not overly so.

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