This article addresses the implementation of Finite Impulse Response filters as digital hardware circuits. It formalizes, as a mathematical model, the problem of finding the optimal circuit for a given frequency specification and given input/output fixed-point formats. This model captures at the bit level a wide class of implementations (transposed-form circuits based on truncated shift-and-add adder graphs). It also captures formally the constraints due to the frequency specification, as well as those due to rounding to the output format. This model can be expressed as an Integer Linear Programming (ILP) problem, such that the optimal circuit (in terms of bit-level adders and registers) can be found by standard ILP solvers. This approach allows for a completely automatic tool from a frequency specification to a circuit with user-specified input and output formats. This tool is evaluated (with cost functions modeling FPGAs) on several benchmarks.