A continuously tunable microwave photonic notch filter with complex coefficient based on phase modulation is proposed and demonstrated. The complex coefficient is generated using a Fourier-domain optical processor (FD-OP) to control the amplitude and phase of the optical carrier and radio-frequency (RF) phase modulation sidebands. By controlling the FD-OP, the frequency response of the filter can be tuned in the full free spectral range (FSR) without changing the shape and the FSR of the frequency response. The results show that the center frequency of the notch filter can be continuously tuned from 17.582 GHz to 29.311 GHz with FSR of 11.729 GHz. The shape of the frequency response keeps unchanged when the phase is tuned.