  We introduce and study a class of discrete-time quantum walks on a
one-dimensional lattice. In contrast to the standard homogeneous quantum walks,
coin operators are inhomogeneous and depend on their positions in this class of
models. The models are shown to be self-dual with respect to the Fourier
transform, which is analogous to the Aubry-Andr\'e model describing the
one-dimensional tight-binding model with a quasi-periodic potential. When the
period of coin operators is incommensurate to the lattice spacing, we
rigorously show that the limit distribution of the quantum walk is localized at
the origin. We also numerically study the eigenvalues of the one-step time
evolution operator and find the Hofstadter butterfly spectrum which indicates
the fractal nature of this class of quantum walks.
