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A continuously tunable microwave photonic filter with a complex coefficient is proposed and demonstrated. The complex coefficient is implemented by a polarization-modulator-based photonic microwave phase shifter, which enables a tunable phase shift from -180° to 180° in 10-40 GHz. An experiment is carried out. By simply controlling the photonic microwave phase shifter, the frequency response of the microwave photonic filter is tuned over the full free spectral range (FSR) without changing the shape and the FSR of the frequency response. The proposed filter shows a rich set of features, including simple operation, large bandwidth, and high tuning speed.