  We analyze two families of three-state quantum walks which show the
localization effect. We focus on the role of the initial coin state and its
coherence in controlling the properties of the quantum walk. In particular, we
show that the description of the walk simplifies considerably when the initial
coin state is decomposed in the basis formed by the eigenvectors of the coin
operator. This allows us to express the limit distributions in a much more
convenient form. Consequently, striking features which are hidden in the
standard basis description are easily identified. Moreover, the dependence of
moments of the position distribution on the initial coin state can be analyzed
in full detail. In particular, we find that in the eigenvector basis the even
moments and the localization probability at the origin depend only on
incoherent combination of probabilities. In contrast, odd moments and
localization outside the origin are affected by the coherence of the initial
coin state.
