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An unbalanced temporal pulse-shaping (TPS) system for chirped microwave waveform generation is proposed and demonstrated. The proposed system consists of an ultrashort pulsed source, a Mach-Zehnder modulator and two dispersive elements. The dispersions of the two dispersive elements are opposite in sign, but not identical in magnitude. The entire system is equivalent to a conventional balanced TPS system with two complementary dispersive elements for real-time Fourier transformation and a third dispersive element to achieve a second real-time Fourier transformation. The key contribution of this work is that the third-order dispersion of the dispersive elements is considered, which leads to the generation of a frequency-chirped microwave waveform. A theoretical analysis is performed in which a mathematical model that relates the second- and third-order dispersion of the dispersive elements and the chirp rate of the generated microwave waveform is developed. The theoretical model is then verified by numerical simulations and an experiment. A chirped microwave waveform with different chirp rates of -0.0535 and 0.715 GHz/ns by tuning the third-order dispersion using a tunable chirped fiber Bragg grating is experimentally demonstrated.