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A fiber-based approach for linear and continuous chirp control in photonics microwave pulse generation is proposed and demonstrated. The technique is based on spectral pulse shaping combined with nonlinear frequency-to-time mapping (FTM). Particularly, the proposed fiber-optics method provides an unprecedented arbitrary chirp-rate control by combining spectral shaping induced by dispersion unbalance in a fiber interferometer, resulting in a nonlinear-chirp frequency interference pattern, with nonlinear FTM. Full linear chirp reconfigurability, including positive, negative and zero chirp rates, is achieved from a single fiber-optics platform by simply varying the relative delay between the interferometer arms. Moreover, a simple balanced photodetection strategy is implemented to achieve dc-free microwave pulses with significantly improved noise figures. High-quality dc-free gigahertz-frequency microwave sinusoids were successfully generated with central frequencies ranging from 100 MHz to 25 GHz and chirp rates ranging from -160 MHz/ns to +160 MHz/ns.