In this paper we consider discrete-time piecewise affine systems with boolean inputs, outputs and states and show that they can be represented in a logic canonical form where the logic variables influence the switching between different submodels but not the continuous-valued dynamics. We exploit this representation for studying Lagrange stability and developing performance analysis procedures based on linear matrix inequalities. Moreover, by using arguments from dissipativity theory for non-linear systems, we generalize our approach to solve the H-infinity analysis problem.