  We focus on index theory for chirally symmetric discrete-time quantum walks
on the one-dimensional integer lattice. Such a discrete-time quantum walk model
can be characterised as a pair of a unitary self-adjoint operator $\varGamma$
and a unitary time-evolution operator $U,$ satisfying the chiral symmetry
condition $U^* = \varGamma U \varGamma.$ The significance of this index theory
lies in the fact that the index we assign to the pair $(\varGamma,U)$ gives a
lower bound for the number of symmetry protected edge-states associated with
the time-evolution $U.$ The symmetry protection of edge-states is one of the
important features of the bulk-edge correspondence. The purpose of the present
paper is to revisit the well-known bulk-edge correspondence for the split-step
quantum walk on the one-dimensional integer lattice. The existing mathematics
literature makes use of a fundamental assumption, known as the $2$-phase
condition, but we completely replace it by the so-called asymptotically
periodic assumption in this article. This generalisation heavily relies on
analysis of some topological invariants associated with Toeplitz operators.
