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A novel approach to implementing a narrow-passband and frequency-tunable microwave photonic filter (MPF) based on phase-modulation to intensity-modulation conversion in a phase-shifted fiber Bragg grating (PS-FBG) is proposed and experimentally demonstrated. In the proposed MPF, a phase-modulated signal is sent to a PS-FBG. If one of the sidebands falls in the notch of the PS-FBG, the phase-modulated signal is converted to an intensity-modulated signal. Due to the ultra-narrow notch of the PS-FBG, a microwave filter with an ultra-narrow passband is realized. The tunability of the microwave filter is achieved by tuning the wavelength of the optical carrier. A theoretical analysis is performed in which the value of the phase shift and the location of the phase shift in the PS-FBG on the frequency response of the MPF are studied. Two PS-FBGs with different reflection bandwidths and different phase-shift values introduced at the center of the gratings are fabricated and incorporated into the proposed MPF. For the two PS-FBGs, the 3-dB bandwidths are 120 and 60 MHz and the tunable ranges are 5.5 and 15 GHz.