  We consider a one-dimensional space-inhomogeneous discrete time quantum walk.
This model is the Hadamard walk with one defect at the origin which is
different from the model introduced by Wojcik et al. [14]. We obtain a
stationary measure of the model by solving the eigenvalue problem and an
asymptotic behaviour of the return probability by the path counting approach.
Moreover, we get the time-averaged limit measure using the space-time
generating function method. The measure is symmetric for the origin and
independent of the initial coin state at the starting point. So localization
depends only on the parameter which determines the model.
