  While the methodology of band structure unfolding has appeared in several
publications, the original derivations of the unfolding formulas can be
considerably simplified by using the $k$-projection method. In this work, more
transparent derivations of unfolded spectral weights are given by using the
projection operator approach. A range of illustrative examples are also
presented which include finite and random one-dimensional chains, and
Kekul\'e-textured honeycomb lattice.
