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Suppose I have a traditional linear antenna array. My element factor follows a cosine shape between 0-180 degrees. The relationship between my theoretical total antenna array pattern and the element factor is illustrated by this graph (screen shot from analog.com, article written by Peter Delos, Bob Broughton and Jon Kraft).

screen shot from analog.com, article written by Peter Delos, Bob Broughton and Jon Kraft

My understanding is that the theoretical array factor does not consider element pattern, and the total pattern is closest to the array factor at the angle where the element factor is at maximum.

If one of the antennas in my antenna array is tilted, is it still possible to achieve maximum total array gain by setting the tilted antenna element at a magical phase value? If one of the antennas is tilted,is the performance of the beam-forming always worse than the ideal condition?

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  • \$\begingroup\$ " is it still possible to achieve maximum total array gain by setting the tilted antenna element at a magical phase value?" - Why would it? \$\endgroup\$ Commented Feb 7, 2022 at 19:06
  • \$\begingroup\$ Less optimal is not a thing. Something is either optimal with respect to some criteria it is not. \$\endgroup\$
    – vicatcu
    Commented Feb 8, 2022 at 1:08

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When discussing a linear array of elements, assuming the elements are identical (uniform linear array), the pattern can be written as: $$G(\theta) = G_A(\theta)G_E(\theta)$$ where \$G\$ is the total pattern, \$G_A\$ is the array factor and \$G_E\$ is the element factor. This is know as the pattern multiplication principle.

The array factor gain, dependent only on the location of the elements, will typically be independent of scan angle. The element factor is the antenna pattern of a single element and typically has a scan angle dependent gain.

If one (or more) of the elements are rotated, then the elements are not identical and the array is no longer uniform. The rotated element would now have a different antenna pattern than the rest. Having elements with varying gain leads to "similarity loss" in an array antenna (see slide 8 of Phased Array Radar (PAR) Performance Considerations- Impact of Errors).

Having a non-uniform array could provide better performance in a specific direction but would result in degraded performance in other directions.

(for reference, this is the article from the original question: Phase Array Antenna Patterns- Part 1: Linear Array Beam Characteristics and Array Factor)

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