Determining fixed-point formats for a digital filter implementation using the worst-case peak gain measure

In this article, we focus on the Fixed-Point implementation of Linear Time Invariant (LTI) filters in state- space representation. For that purpose, we give an algorithm to determine the Fixed-Point Formats of all the involved variables (states and outputs). For the sake of generality, the algorithm works in the case of Multiple Inputs Multiple Outputs filters. The computational errors in the intermediate steps of the filter evaluation as well as their accumulation over time are fully taken into account. We handle several rounding modes (round to nearest and truncation) for two's-complement-based Fixed-Point Arithmetic. Our approach is fully rigorous in the way that the output Fixed-Point formats are shown to be free of overflows and we do not use any (non-exhaustive) filter simulation steps but proofs.