Array Pattern is DFT of Weighting

array pattern:

DFT of w*

k = 0 .. (N-1)

thus we have:

note DFT is periodic:

for any integer m

k = 0 .. (N-1)

angle index

thus:

use appropriate value of m to keep sinq in [-1, 1] as k goes from 0 to (N-1)