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This brief revisits the problem of the semiglobal stabilization of discrete-time linear periodic systems with bounded controls. First, a new state feedback solution based on a parametric periodic Riccati equation is established to solve the problem under the condition that the open-loop characteristic multipliers are strictly inside or on the unit circle. Different from our early solution based on the parametric periodic Lyapunov equation, this new solution does not require any state transformation on the open-loop system and is theoretically more appealing. Second, an output feedback solution is presented in this brief, which complements the two state feedback solutions. Third, as a periodic system model, the multirate sampled-data system is considered, and the problem of semiglobal stabilization for such a system subject to input saturation is solved. In doing so, the connections between the semiglobal stabilizability of the original continuous-time system and of the multirate sampled-data system are established. A numerical example is worked out to illustrate the effectiveness of the proposed design.