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We propose and experimentally demonstrate an approach to generating large time-bandwidth product (TBWP) microwave arbitrary waveforms based on optical pulse shaping using a single spatially discrete chirped fiber Bragg grating (SD-CFBG). The SD-CFBG functions to perform simultaneously spectral slicing, frequency-to-time mapping, and temporal shifting of the input optical pulse, which leads to the generation of an optical pulse burst with the individual pulses in the burst temporally spaced by the time delays determined by the SD-CFBG. With the help of a bandwidth-limited photodetector (PD), a smooth microwave waveform is obtained. The SD-CFBG is fabricated using a linearly chirped phase mask by axially shifting the photosensitive fiber to introduce a spatial spacing between two adjacent sub-gratings during the fabrication process. By properly designing the fiber shifting function, a large TBWP microwave arbitrary waveform with the desired frequency chirping or phase coding can be generated. An equation that relates the fiber shifting function and the microwave waveform frequency chirping is derived. The photonic generation of large TBWP microwave waveforms with a linear, nonlinear and stepped frequency chirping is experimentally demonstrated.