  One-dimensional discrete-time quantum walks show a rich spectrum of
topological phases that have so far been exclusively analysed in momentum
space. In this work we introduce an alternative approach to topology which is
based on the scattering matrix of a quantum walk, adapting concepts from
time-independent systems. For gapped quantum walks, topological invariants at
quasienergies 0 and {\pi} probe directly the existence of protected boundary
states, while quantum walks with a non-trivial quasienergy winding have a
discrete number of perfectly transmistting unidirectional modes. Our
classification provides a unified framework that includes all known types of
topology in one dimensional discrete-time quantum walks and is very well suited
for the analysis of finite size and disorder effects. We provide a simple
scheme to directly measure the topological invariants in an optical quantum
walk experiment.
