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An optical approach to generating continuously tunable chirped microwave waveforms using a tilted fiber Bragg grating (TFBG) written in an erbium/ytterbium (Er/Yb) codoped fiber is proposed. By pumping the TFBG, the magnitude and group delay responses of the cladding mode resonances are changed, which can be used to implement a photonic microwave delay-line filter with increasing or decreasing tap spacing. If an ultranarrow pulse is sent to the photonic microwave delay-line filter, a pulse burst with increasing or decreasing pulse spacing is generated. The photodetection of the pulse burst would lead to the generation of a chirped microwave waveform. The proposed technique is demonstrated by an experiment in which a chirped microwave waveform with a tunable chirp rate from 1.8 to 7 GHz/ns is generated.