  In this paper, we consider periodicity for space-inhomogeneous quantum walks
on the cycle. For isospectral coin cases, we propose a spectral analysis. Based
on the analysis, we extend the result for periodicity for Hadamard walk to some
isospectral coin cases. For non-isospectral coin cases, we consider the the
system that uses only one general coin at the origin and the identity coin at
the other sites. In this case, we show that the periodicity of the general coin
at the origin determines the periodicity for the whole system.
