Shaped line-source antenna beams can be synthesised by introducing com
plex roots in a Taylor line source distribution to achieve null fillin
g in the shaped region. Optimisation of these roots and others allows
individual control of the amplitude of ripples and the heights of unsh
aped sidelobes up to any specified order. The corresponding continuous
aperture distribution can be made pure real by appropriate root-pairi
ng, which makes the results applicable, via sampling, to resonantly sp
aced linear arrays.