Digital filters are small iterative algorithms, used as basic bricks in signal processing (filters) and control theory (controllers). They receive as input a stream of values, and output another stream of values, computed from their internal state and from the previous inputs. These systems can be found in communication, aeronautics, automotive, robotics, etc. As the application domain may be critical, we aim at providing a formal guarantee of the good behavior of these algorithms in time-domain. In particular, we formally proved in Coq some error analysis theorems about digital filters, namely the Worst-Case Peak Gain theorem and the existence of a filter characterizing the difference between the exact filter and the implemented one. Moreover, the digital signal processing literature provides us with many equivalent algorithms, called realizations. We formally defined and proved the equivalence of several realizations (Direct Forms and State-Space).