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We present tunable optical filters based on a modified Gires-Tournois interferometer. The back reflection plane of the interferometer is replaced with a one-dimensional micromirror array for phase modulation. Using Gaussian beam optics, we show that the transfer function of the device has the form of the transversal filter in digital signal processing. The design and analysis techniques of conventional digital filters can, therefore, be adapted to tunable optical filters. Both the amplitude and phase of the transfer function can be controlled by reconfiguring the micromirrors. Application examples in dispersion compensation and variable bandwidth bandpass filtering are discussed and experimentally demonstrated.