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A novel approach to generating a chirped microwave waveform with continuously tunable chirp rate based on a temporal interferometer incorporating an optically pumped linearly chirped fiber Bragg grating (LCFBG) is proposed and demonstrated. The temporal interferometer is realized using a Mach-Zehnder interferometer (MZI) that incorporates an LCFBG and a dispersion compensating fiber to generate a temporal interference pattern with an instantaneous frequency that is linearly proportional to time. A linearly chirped microwave waveform with its shape that is identical to the temporal interference pattern is generated at the output of a photodetector. The tuning of the chirp rate of the generated waveform is realized by optically pumping the LCFBG that is written in an erbium-ytterbium co-doped fiber with different pumping powers. The key advantage of using optical pumping over external thermal tuning or mechanical tuning to tune the dispersion of the LCFBG is that the dispersion can be tuned at a high speed and controlled remotely. Moreover, the undesirable birefringence effects existing in the mechanical tuning technique can also be avoided. A theoretical analysis is performed that is verified by numerical simulations and an experiment. A linearly chirped microwave waveform with a tunable chirp rate from 79 to 64 GHz/ns by changing the injection current to the pumping laser diode from 0 to 100 mA is generated. The experimental results also show that the central frequency of the generated chirped microwave waveform can be changed by tuning the longitudinal offset of the MZI.