This paper considers pole assignment and robust pole assignment problems for discrete-time linear periodic systems by using linear periodic state feedback. The monodromy matrix of the closed-loop system is represented in a special form. By combining this special form with our recent result on solutions to a class of generalized Sylvester matrix equations, a complete parametric approach for pole assignment via periodic state feedback is proposed. The free parameters existing in the solutions to pole assignment are further used to achieve robustness performances. The robust pole assignment problem is converted into a nonconvex optimization problem. Numerical examples illustrate the effectiveness of the proposed approaches.