We propose a novel structure of complex-tap microwave photonic filter (MPF) employing an incoherent broadband optical source (BOS) and a programmable optical spectrum processor. By tailoring the optical spectral amplitude and phase, arbitrary complex continuous-time impulse responses of the MPF can be constructed. Frequency responses with a single flat-top, highly chirped, or arbitrary-shape passband are demonstrated, respectively. The passband center can also be tuned in a wide range only limited by the opto-electrical devices. To the best of our knowledge, it is the first demonstration of an incoherent-BOS-based MPF which is single-bandpass, widely tunable, and highly reconfigurable with complex taps.