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A novel photonic microwave notch filter based on all-optical mixing using a single narrow-linewidth optical source is proposed. In the proposed filter, the optical carrier emitted from a laser source is externally intensity modulated by a subcarrier signal and then applied to a second intensity modulator, which is driven by a local radio-frequency (RF) source with a dc bias to suppress the even-order sidebands. Since the higher odd-order sidebands have a much lower power than the first-order sidebands, only two first-order sidebands are considered with the subcarrier signal transferred to the two sidebands via optical mixing. By using a linearly chirped fiber Bragg grating as a dispersive device to introduce time delays, a two-tap photonic microwave notch filter is realized. The tunability of the proposed filter is achieved by adjusting the frequency of the local RF signal. Based on the proposed architecture, a two-tap microwave notch filter with a free spectral range tunable from 2.1 to 4.2 GHz is experimentally demonstrated.