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Chirped microwave pulse compression using a photonic microwave filter with nonlinear phase response to implement matched filtering is proposed and investigated. The photonic microwave filter with the required phase response is realized based on optical phase to microwave phase conversion through single-sideband modulation and heterodyne detection. A detailed theoretical analysis on the photonic microwave filter design and the linearly chirped microwave pulse compression is developed. A photonic microwave filter having a quadratic phase response with a bandwidth of 3 GHz is implemented. An application of the photonic microwave filter for linearly chirped microwave pulse compression is investigated.