The theory and performance of the adaptive discrete cosine transform filter is examined. The discrete cosine transform filter is a realization of an FIR filter as the cascade of an all-zero FIR filter with a bank of IIR digital resonators. Each bank has a single magnitude and a single phase. The result of such a realization is that each coefficient can be directly identified with an amplitude or a phase of the transfer function at a particular frequency when run as an adaptive filter. The update method examined is the LMS algorithm, with the coefficients being adapted as the magnitudes and phases at a full range of frequencies.