  In this paper, we generalize and develop results of Queffelec allowing us to
characterize the spectrum of an aperiodic substitution in $\mathbb{Z}^d$ by
describing the Fourier coefficients of mutually singular measures of pure type
giving rise to the maximal spectral type of its Koopman representation. We note
that this is done without the assumptions of primitivity or trivial height, and
provides a simple algorithm for determining singularity to Lebesgue spectrum
for such substitutions. This is used to show that the spectrum of any aperiodic
bijective and commutative $\mathbb{Z}^d$ substitution on a finite alphabet is
purely singular. Additionally, we use the algorithm to show singularity of the
spectrum for Queffelec's noncommutative bijective substitution, as well as the
Table tiling, answering an open question of Solomyak.
