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We propose a novel technique to implement microwave arbitrary waveform generation (AWG) in the optical domain based on simultaneous optical spectral shaping and wavelength-to-time mapping in a single linearly chirped fiber Bragg grating (LCFBG). In the proposed approach, the spectrum of an ultrashort optical pulse generated by a mode-locked fiber laser is spectrally shaped and at the same time wavelength-to-time mapped by the LCFBG to generate a microwave pulse with a shape identical to that of the shaped optical spectrum. By designing the LCFBG to have an arbitrary reflection profile, a microwave arbitrary waveform is generated. The key component in the proposed system is the LCFBG. A simple and effective technique to synthesize the LCFBG with an arbitrary reflection response is proposed. An approximate model describing the microwave AWG is derived, which is verified by numerical simulations and a proof-of-concept experiment.