  We study a discrete-time quantum walk (QW) on the line with a single phase at
the origin which was introduced and studied by Wojcik et al.[1]. We call the
model "Wojcik model" here. Konno et al.[2] investigated other types of QWs with
one defect at the origin. They presented a method which gives the stationary
measure corresponding to localization for the QWs by use of the generating
functions splitted in positive and negative parts respectively. In this paper,
we call the method "the splitted generating function method (the SGF method)".
To clarify in detail which QW is appropriate for the method in the case of
study is one of the important challenges to investigate localization properties
for various QWs. As for the Wojcik model, we solve the eigenvalue problem by
the SGF method and our results agree with Ref.[1]. From the solution of the
problem, we derive a stationary measure with an exponential decay for the
position. The explicit expression for the stationary measure is symmetric for
the origin and ensures localization depending on the initial coin state.
