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A tunable single-bandpass microwave photonic filter (MPF) with a narrow and flat-top shape operating in high frequency range is proposed and experimentally demonstrated, using a specially designed multi-phase-shift fiber Bragg grating (FBG) cascaded with a programmable optical filter (POF). The key device in the proposed MPF is the multi-phase-shift FBG, providing a flat-top passband with a narrow 3-dB bandwidth of 813 MHz in transmission. Consequently, in the MPF, a single passband with a 3- and 20-dB bandwidths of 840 MHz and 2.43 GHz is achieved, respectively. Moreover, an operating frequency range from 27.1 to 38.1 GHz is realized by tuning the wavelength of the optical carrier and adjusting the transfer function of the POF. To the best of our knowledge, this proposed filter provides the highest tunable frequency coverage in Ka-band ever reported for MPFs with a single narrow and flat-top passband.