We investigate the structure of $N$-length discrete signals h satisfying h*h=h that vanish on a given set of indices. We motivate this problem from examples in sampling, Fuglede's conjecture, and orthogonal interpolation of bandlimited signals. When N=p M is a prime power, we characterize all such h with a prescribed zero set in terms of base-p expansions of nonzero indices in F-1h. © 1991-2012 IEEE.