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A method for tuning the operation frequency of a dielectric resonator antenna (DRA) in the design stage is proposed. The method employs a parasitic annular slot, which can be applied to both the linearly-polarized (LP) and circularly-polarized (CP) DRAs. To demonstrate the former, an annular-slot-excited hemispherical DRA is used. By changing the radius of the parasitic slot, the operating frequency of the antenna can be adjusted while maintaining good impedance match and stable radiation patterns. For the CP case, a CP DRA excited by a shorted annular slot is used for the demonstration of this idea. It is found that good impedance matches and axial ratios can be obtained across the tunable frequency range. The radiation patterns are, again, stable across the frequency range. The problems are formulated using the Green's function approach, with the unknown slot currents solved using the method of moments. Measurements were carried out to verify the theory, and reasonable agreement between them is obtained.